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Discrete mathematics

The discrete mathematics paragraph of this article was misleading. All the areas mentioned include both discrete and continuous mathematics.

Information theory isn't only discrete mathematics, information theory also applies to analog signals. See Differential entropy.

Analog signal processing is also a form of computation.

Theoretical computer science considers both discrete and continuous computational processes, and both discrete and continuous input/output:

• Continuous computability theory:
  • Computable analysis
  • Algorithmic complexity theory

• Continuous complexity theory:
  • Complexity theory of continuous time computation using dynamical systems or other continuous models of continuous computation.
  • Complexity theory of numerical analysis (various approaches including Information-based complexity, algebraic complexity theory)

Including the question of P!=NP over Real numbers

• Analog coding theory

• "analog encryption" OR "analogue encryption"

• analog vlsi

• "analog Automata" OR "continuous Automata"

• Graph theory
  • continuous graphs

For now, I've added the above information to the article, but the whole section on discrete mathematics should be deleted, and its content redistributed to elsewhere in the article. Bethnim (talk) 13:37, 23 March 2010 (UTC)

I've moved combinatorics to the structure section, and that just leaves Theoretical computer science so the discrete section is renamed to Theoretical computer science. Bethnim (talk) 13:54, 23 March 2010 (UTC)

Maths not part of logic?

User Keifer Wolfowitz seems to be claiming completely without any references of [or? Kiefer.Wolfowitz (talk) 19:57, 23 March 2010 (UTC) ]

My edit noted that logic includes the study of fallacies, etc. and cited Peirce, so editor Wolfkeeper's statement is false. C.S. Peirce observed that logic relied on mathematics, which is the science of necessary reasoning according to his father B. Peirce, who is quoted in this article. Kiefer.Wolfowitz (talk) 19:58, 23 March 2010 (UTC)

proof that Maths is not part of logic. Note that logic includes the study of inconsistent systems and many other things, and it does not seem that there is any part of mathematics that is not part of logic in the broad sense. But there are clearly parts of logic which are not usually considered mathematical.- Wolfkeeper 19:43, 23 March 2010 (UTC)

He also seems to be maintaining a claim that maths is actually not logical in the article; with an apparent oblique reference to Gödel's incompleteness; this is frankly a bizarre non sequitor and completely unreferenced.- Wolfkeeper 19:43, 23 March 2010 (UTC)

My understanding is that logic is a branch of mathematics. Stephen B Streater (talk) 19:47, 23 March 2010 (UTC)

Bertrand Russell and Alfred North Whitehead proved in Principia Mathematica that all of mathematics can be broken down into a simple and small set of logical propositions ("axioms"). Godel's Incompleteness was an extension of that work. It shows that as a logical system the system of mathematics we use contains statements that cannot be proven to be true or false. and if you add axioms to try to fix this gap, it will contradict itself.

So if you are representing his views correctly, then he is misunderstanding godel's proof and it's relation to bertrand and whiteheads: namely, that which could be deduced from the simple and elementary fact that in math there are no contradictions: and that is that they don't contradict each other. That's how it is that we have computers; as alan turing's essay "computable numbers (see computable number) was in turn an extension of godel's work, and led to the digital computer - a machine that theoretically can do any and every mathematical operation. Kevin Baas^talk 19:52, 23 March 2010 (UTC)

(ec)No, I think you've got a few misunderstandings in there. First of all the system of Principia Mathematica is no longer used in any serious way; its importance is historical. Its notation is bizarre and the system as a whole is Baroque, to the point that almost no one really studies it in detail anymore. Certainly I have not, which makes it hard for me to say exactly what part of mathematics can or cannot be formalized in the system of PM, but it is definitely not "all mathematics".

More seriously, you're missing the fact that Goedel was at the opposite end of the philosophical spectrum from Russell and Whitehead, and his work is in large part a demonstration that what Russell and Whitehead sought was unachievable. Russell and Whitehead were part of the logicist school, that sought to reduce all of mathematics to logic (that is, to tautology), to show that mathematical statements were analytic propositions. Goedel on the other hand was a realist/Platonist, at least in his later years.

Goedel's theorems do not strictly speaking refute the claim that mathematical propositions are analytic; that claim is not mathematical enough to be subject to mathematical refutation. However they definitely put severe roadblocks in the way of the most expansive logicist dreams.

There are other errors in your comments but this is not the place to discuss them. The point relevant to editing the article is that we must not make claims such as "mathematics is part of logic" or "logic is part of mathematics", because there is no consensus among workers in the field that either of these statements is true. We can, however, attribute such claims to various thinkers, giving due weight according to their recognition and importance. --Trovatore (talk) 20:15, 23 March 2010 (UTC)

"First of all the system of Principia Mathematica is no longer used in any serious way..." you seem to be misunderstanding the entire point of principia mathematica. it is not supposed to be pedagogical in any way, it was meant as one big proof and it's generally accepted among mathemticians to be a formally rigorous and successful one. All the stuff about it being barouqe, use a bizarre notation, etc. is all non-sequitor. and please don't bring up "russel's paradox", we both know that has nothing to do with this.

"More seriously, you're missing the fact that Goedel was at the opposite end of the philosophical spectrum" - actually, far from being "serious", that is utterly irrelevant.

RE: "Goedel's theorems do not strictly speaking refute...'": Who are you talking to? I'm not sure that's really pertinent at all to the discussion. and i have no idea where you're going with that or why it matters.

RE: "There are other errors in your comments..." all your "errors" have so far been your errors in interpretation and/or simply non-sequitors and straw men. so far it is yet to be demonstrated that there are any errors in what i actually said, and your tangents have left me completely unpersuaded. Kevin Baas^talk 21:41, 23 March 2010 (UTC)

You can have contradictions in maths of course; proof by reaching a contradiction is considered part of mathematics (albeit somewhat controversial sometimes). Contradictions are certainly generated and studied.- Wolfkeeper 20:07, 23 March 2010 (UTC)

I mean that the formal system we call mathematics does not itself contain contradictions. I am saying something different here. I am saying that the formal system we call "mathematics" is consistent. Kevin Baas^talk 21:41, 23 March 2010 (UTC)

::: Please provide a reference to a reliable source (unlike Principia Mathematica) that logic covers mathematics. Until then, I won't condescend to discuss your original research. Then strive for consensus before ignoring the warning hidden in the opening sentence. Thanks. Kiefer.Wolfowitz (talk) 20:05, 23 March 2010 (UTC)

It's usually understood that the foundations of maths are in philosophy and logic. And the claim of the article right now is that maths isn't logic at all!!!- Wolfkeeper 20:07, 23 March 2010 (UTC)

RE "At all" Not only the scholarship but the italicized clichés of Dan Brown.Kiefer.Wolfowitz (talk) 20:10, 23 March 2010 (UTC)

Funny. Try the thing that 's staring you right in the face. it's all 1's and 0's and digital (i.e. logical) circuitry. everybody knows that. and i just explained it in the above paragraph. look up turing machine. in addition to the references, both internal and external, that i gave you in the above paragraph. if you can't understand what i wrote or what the references say, that's your problem, not mine. And those papers and articles i referenced are not my research. But that's utterly obvious. As regards "I won't condescend to..." clearly you don't know what "condescend" means. Let me enlighten you: it's exactly what you were doing in the sentence that you started out with "I won't condescend...". Oh the irony! I'll try to recuse myself from this discussion as, judging from these things, [1] seems like something i'm going to have a particular difficult time avoiding, myself. Kevin Baas^talk 20:15, 23 March 2010 (UTC)

There's no need for personal attacks or displays of emotion in our discussion about logic - it's illogical, Captain ;-) What we should do is, as was pointed out above, report what good sources have claimed, after reaching consensus here. These ideas were not born fully formed, and it is not surprising that there is some inconsistency in their use. But the subject is big, and it is important to keep it concisely worded. Stephen B Streater (talk) 20:37, 23 March 2010 (UTC)

someone brought up the dichotomy: "mathematics is part of logic" or "logic is part of mathematics". it's a false dichotomy. logic is a branch of mathematics AND mathematics is a formal system. simple. Kevin Baas^talk 21:23, 23 March 2010 (UTC)

Tutorial websites

http://www.mathscentre.ac.uk/students.php http://www.mathtutor.ac.uk/ I would add them to the article but it is locked. Please could someone add it when it is unlocked, as I may not return here for years. Thanks 89.240.44.159 (talk) 12:42, 19 April 2010 (UTC)

It's locked for a reason. If not already on Wikipedia, useful information about the subject should ideally be added into the relevant articles here. But Wikipedia is not a tutorial, or a directory of tutorials - see WP:NOT. Stephen B Streater (talk) 20:42, 16 May 2010 (UTC)

Definition of mathematics

In my humble opinion, more attention should be paid to the definition of mathematics. The first line of the article - as far as I know - is not a recognised definition, more a sketchy impression what mathematics is roughly like.

In a true "mathematical" spirit, the question must be answered first whether the concept "mathematics" can be defined at all. A mathematician told me that it is fundamentally impossible to give such a definition, at least to mathematicians (but I did not fully understand his reasoning).

Is mathematics perhaps just a term culturally attached to a rather arbitrary choice of some forms of logical reasoning? Perhaps a definition can only be given if a purpose is decided first (e.g. there are "ontological" and "teleological" definitions: the former try to grasp the essence of a concept, the latter are a choice that can be practical or unpractical, but is never correct or incorrect).

On reason to be strict in the definition of mathematics is the continuing debate about the patentability of mathematical algorithms. Some argue that all algorithms are inherently mathematical, I am inclined to believe that none of them is: only the proof of an algorithm is - perhaps - mathematics. Is all mathematics inherently so fundamental knowledge that it should never be withdrawn from the public domain? But many mechanical inventions basically are geometric, and geometry is a branch of mathematics.

Is there a fundamental difference between philosophy and mathematics? Or is it just cultural: mathematicans prove, philosophers cite. Both are disciplines that do not depend on observations (which is afaik a reason not to consider them "science" at all, in some perceptions - which must be a deception for a PhD in maths!). Maths often deal with quantities, but e.g. boolean algebra doesn't. Math's use shorthands - formulas - but the Pythagoras theorem in plain language is still mathematics.

Who responds to the challenge? Rbakels (talk) 19:54, 12 August 2010 (UTC)

No one, I hope.

Before you step in this one, please look back in the archives of this page. --Trovatore (talk) 19:56, 12 August 2010 (UTC)

Please explain. I don't want to consult an archive of a discussion page to find the very definition of the topic of a lemma! Rbakels (talk) 20:05, 12 August 2010 (UTC)

The current definition is the result of a very long discussion over several years, with many editors taking a lot of people points of view into account. Every word has been discussed and a near consensus arrived at. Most alternatives have been discussed somewhere and you will find a reluctance to repeat arguments which have already been had. As you yourself point out there is not one universally agreed definition so it would not be right for us to give one, the sketchy impression is perhaps the best which can be achieved.--Salix (talk): 22:12, 12 August 2010 (UTC)

Ah, I see. But that perhaps confirms that mathematics can not be defined at all - is it just a loose designation of some forms of logic? Or is it characterized by the methodology? Mathematicians let logic speak for themselves - logicians from a philosophical tradition cite authorities, in as many footnotes as possible. I think there is a need for a (non-)existence proof of the concept "mathematics". This may not be too interesting to mathematicians themselves, but I guess the prime purpose of an encyclopedia is to define the topics of its lemmas, and, as I said, it has practical relevance, like in the political and legal field. Perhaps a summary of the (failed) debate you refer to does the job. If only to prevent other nasty people to ask the same nasty questions I asked again ;) Rbakels (talk) 04:35, 13 August 2010 (UTC)

Have you seen the page Definitions of mathematics that might be a more appropriate venue for this discussion. It seems that there are a number of different schools of thought about the definition with Realists, Intuitionist, and Formalist. As such the question of the definition becomes a philosophical rather than a mathematical problem.--Salix (talk): 06:37, 13 August 2010 (UTC)

Thanks, very helpful. Would it perhaps make sense to link the "mathematics" article to the "definitions of mathematics" article. Other people like me may expect a Wikipedia lemma on topic "X" to contain the definition of "X" itself????? Rbakels (talk) 09:31, 13 August 2010 (UTC)

There is a link - Definitions of mathematics is top of the "See also" list. Gandalf61 (talk) 12:54, 13 August 2010 (UTC)

It does have a definition. Not a completely precise definition, not a definition in the mathematical sense, but a definition along the lines of one you might expect at articles like dog or apple. That's what's appropriate to this sort of article. In articles about a mathematical object, say an ultrafilter, you expect a precise definition. But mathematics is not a mathematical object.

(Just by the way, I recognize your use of the word lemma, but not everyone will; it's not a terribly common usage in English, outside of specialized works on linguistics.) --Trovatore (talk) 20:51, 14 August 2010 (UTC)

In Our Time

The BBC programme In Our Time presented by Melvyn Bragg has an episode which may be about this subject (if not moving this note to the appropriate talk page earns cookies). You can add it to "External links" by pasting * {{In Our Time|Mathematics|p00545hk}}. Rich Farmbrough, 03:17, 16 September 2010 (UTC).

Infinity picture

An image displaying the infinity symbol eight times seems unnecessary to me, and I think that removing it would greatly improve the article's aesthetic quality. —Preceding unsigned comment added by 131.104.240.36 (talk) 10:28, 20 October 2010 (UTC)

Proof of Pythagoras' theorem

It doesn't do much for me either. How about replacing it with this featured picture (right) of a proof of Pythagoras' theorem? --Avenue (talk) 13:18, 20 October 2010 (UTC)

The infinity does match the text better, which is on notation, and there's already a diagram demonstrating Pythagoras's theorem further down. I can't immediately think of something to replace the current image though.--JohnBlackburne^words_deeds 13:29, 20 October 2010 (UTC)

At least on the small screen I'm using, the infinity image appears opposite a paragraph about proof, and is at least a paragraph away from our text about notation. I don't think it's a great illustration of notation either - why does it really matter that the infinity symbol can be drawn slightly differently in various typefaces? I do take your point about Pythagoras' theorem appearing further down, though. --Avenue (talk) 15:17, 20 October 2010 (UTC)

The parallel postulate is the fifth axiom underlying Euclidean geometry, and the most controversial.

How about this image depicting the parallel postulate? This would tie in with the discussion of axiomatic systems in that section's final paragraph. --Avenue (talk) 15:42, 20 October 2010 (UTC)

That doesn't really work either, as it doesn't illustrate the parallel postulate, or at least not without far more explanation (and it's not controversial in modern mathematics). But it's on history, not mathematical theory, so detailed explanation is just distracting. What about something from commons:Category:History_of_mathematics ? A few things stand out as historic and interesting to me, such as file:Yanghui_triangle.PNG, or one of these commons:Category:Geometria by Augustin Hirschvogel.--JohnBlackburne^words_deeds 22:48, 20 October 2010 (UTC)

Most fundamental concepts already have an illustration of some sort, and the section aready has a picture of Euler. Perhaps the image should just be removed? Jeffrey Daniel (talk) 18:05, 21 October 2010 (UTC)

Main image

Though I have nothing but respect for Euclid, I'm not sure that Raphael's painting of him is the best lead image for this article. It seems to me that something more lively or visually interesting might be preferable, similar to the approach used by the biology and chemistry articles.

I would suggest a pair of images, similar to the layout on the chemistry article. My pick would be the two images on the right. The first image illustrates geometry (specifically Desargues' theorem), and was obtained from Portal:Mathematics/Featured picture archive. The second is a picture of the Mandelbrot set, and is a mathematics-related featured picture on Wikipedia (see Wikipedia:Featured pictures/Sciences/Mathematics). Jim.belk (talk) 00:16, 16 November 2010 (UTC)

External Links Section

I propose that we add www.onlinemathcircle.com to the external links section. A good explanation of what it is would be: "A Community Dedicated to Making Mathematics More Open" —Preceding unsigned comment added by Shrig94 (talk • contribs) 02:14, 21 November 2010 (UTC)

No, as there's nothing there: it's a new site with no content, and even if it were a large, active site it's not clear it should be included or which if any articles it's relevant to (maths competitions or particular topics for example).--JohnBlackburne^words_deeds 12:18, 21 November 2010 (UTC)

Awards section

It seems inappropriate to me to advertise prizes in this article. The article is about mathematics, not mathematicians. There are no such sections in the articles about politics or economics, for instance. The insertion appears to have been placed arbitrarily into a section in which it does not belong, in any case. 131.111.184.95 (talk) 08:37, 15 December 2010 (UTC)

The point of the brief mention of the Fields Medal is in the context of the question of whether or not mathematics is a science. If mathematics is a science, why are mathematicians awarded a Fields Medal rather than a Nobel Prize? Your reason for not mentioning the Fields Medal, that the article is about mathematics, not mathematicians, seems odd. Every article on an academic subject mentions major contributers to that subject. Would you mention calculus but not mention Newton? Rick Norwood (talk) 13:32, 15 December 2010 (UTC)

The existence or non-existence of the Fields Medal (or any other award) has no relevance to the question of whether mathematics is a science. Your reasoning about the Nobel Prize does not seem to make any sense. Is peace or literature a science because they have Nobel Prizes? Does computer science or geology fail to be a science because they lack Nobel Prizes? Again, the existence of an award has no relevance to the question of whether a subject is considered a science. Hence, the mention of awards certainly does not belong in that section of the article. As for the article more generally, it may be the case that articles on academic subjects mention major contributors to the subject. This, however, is not relevant to the issue being discussed. The issue is whether a particular prize or prizes should be mentioned; this has nothing to do with mentioning contributors. Euler is an example of a contributor; the Fields Medal (or any other award) is not. Newton was awarded a slew of honors for his work; should the article on calculus list all of them? 131.111.184.95 (talk) 19:28, 16 December 2010 (UTC)

Let's agree to disagree. Rick Norwood (talk) 14:19, 17 December 2010 (UTC)

I agree with 131.111.184.95's argument that whether there is a Nobel Prize has nothing to do with whether mathematics is a science. And the material on Hilbert's 23 problems seems irrelevant. On the other hand, I agree with Rick Norwood that it's reasonable to have a brief discussion of the mathematical profession(s) in this article --- especially since the public seems to have many misconceptions about what mathematicians do --- with a link to the full article at Mathematician. So I vote that we refactor this material. Mgnbar (talk) 17:50, 18 December 2010 (UTC)

I've made a new section and dumped the material there, for lack of a better place. It clearly needs some work to integrate into the article. LJosil (talk) 22:54, 30 December 2010 (UTC)

Goedel and logicism

In reference to the disputed claim that

However, in the 1930s important work in mathematical logic showed that mathematics cannot be reduced to logic

I have to say that I agree that this assertion should not appear without qualification, because not everyone agrees (for example Torkel Franzen did not agree). However it is a serious and widely held point of view, likely the majority view, and ought to be represented. I suspect that Wolfkeeper and Kevin Bass don't understand the argument for how Goedel's theorems might be said to refute logicism, and this is a bit subtle and I'm not going to go into it right now. It is strictly speaking beside the point anyway for purposes of editing the encyclopedia — the challenge is to source the statement, and to correctly attribute it, not to a single thinker because it's a widely held view, but to a current of mathematical thought. --Trovatore (talk) 21:24, 23 March 2010 (UTC)

If that line is refering to godel's work it is totally nothing like anything godel showed. Kevin Baas^talk 22:05, 23 March 2010 (UTC)

as to it being a "serious and widely held point of view", quite the opposite is true, as Axiomatic_set_theory#Applications spells out in no uncertain terms:

Nearly all mathematical concepts are now defined formally in terms of sets and set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, and vector spaces are all defined as sets having various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of relations is entirely grounded in set theory.

Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most or even all mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second order logic. For example, properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set.

Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept that (in principle) theorems in these areas can be derived from the relevant definitions and the axioms of set theory. Few full derivations of complex mathematical theorems from set theory have been formally verified, however, because such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath, includes derivations of more than 10,000 theorems starting from the ZFC axioms and using first order logic.

Kevin Baas^talk 22:08, 23 March 2010 (UTC)

You seem to be taking the position that set theory is a part of logic, which is a controversial claim and not generally accepted.

(Here by logic I'm talking about logic in the strict sense, not "mathematical logic" — of course everyone agrees that set theory is part of mathematical logic. But mathematical logic is not logic; it's mathematical logic.)

The logical character of first-order logic is not (much) in dispute. The logical character of the ZFC axioms, on the other hand, is very much in dispute. I think it is fair to say that most workers in the field consider the ZFC axioms to be, depending on their philosophical school, either formal assertions or substantive claims, but in either case not mere logical necessities. --Trovatore (talk) 22:20, 23 March 2010 (UTC)

You're not making any sense to me. for instance, "think it is fair to say that most workers in the field consider the ZFC axioms to be, depending on their philosophical school, either formal assertions or substantive claims, but in either case not mere logical necessities." - your speaking of axioms as "substantive claims" and questioning whether they are "logical neccessities". that doesn't make any sense to me. a "susbstantive claim" might be a premise perse, but then axioms are things that act on premises. a logical neccesity is something that follows from the use of axioms, not something that precedes (or constitutes!) them. if there are really philosophers that are conflating these things, as you claim, then they are seriously confused. you also seem to have contradicted yourself regarding logic and set theory. that doesn't make any sense to me. we seem to be talking past each other. In any case I quoted directly from an article there so it's not me talking past you there but wikipedia. and then i add in contradiction of the dichotormy "math part of logic or logic part of math: "logic is a branch of mathematics and mathematics is a formal system". and you can read the intor to the formal syatem article to see what relation i am saying math has to logic in that sense. so there it is in either case wikipedia speaking, and i stand by wikipedia's interpretation and if you're ever confused about what i mean i mean what wikipedia means. Kevin Baas^talk 12:57, 24 March 2010 (UTC)

And regarding logic and sets, -- as you should already know from reading the material i cited from Axiomatic set theory#Applications -- see first-order logic and second-order logic. Kevin Baas^talk

I do think we're talking past each other. Let me explain about "reducing mathematics to logic", in the sense that the logicists wanted to do it, what that would mean if it could be done.

The idea was that the truths of mathematics should be purely logical truths; that is, understood correctly, they should not require any assumptions at all, but should simply be expressions of making valid inferences. So if you need axioms, then you have not reduced mathematics to logic — at best, you've reduced it to logic plus those axioms. Unless of course the axioms themselves are logically necessary.

To take an example, consider the equation 0+0=0. Arguably I can rephrase this as saying if I have two buckets, and there's nothing in either bucket, and there's nothing that's in both buckets, then there's nothing in that's in either bucket. Replacing sets by predicates, I could express this as follows:

$\forall x\lnot P(x)\land \forall x\lnot Q(x)\land \forall x\lnot (P(x)\land Q(x))$

$\Rightarrow$

$\forall x\lnot (P(x)\lor Q(x))$

Now, the above statement is a logical truth, in the sense that it doesn't matter how you interpret the predicate symbols P and Q. You can prove it (say, using Gentzen-style sequent calculus) without using any axioms whatsoever. And arguably it captures the meaning of the equation 0+0=0.

The logicists wanted to do something like that for all of mathematics. Whether the Goedel theorems proved that this is impossible is a matter of debate, and depends considerably on what you consider to be "logic". --Trovatore (talk) 18:24, 24 March 2010 (UTC)

Ok, so what i'm getting is that the "purely logical" is like "non-tautological tautology" or "premises based solely on axioms that have no premises" - and in any case a self-contradicting mental fixation for the loosely grounded. now while axiomatic systems can be well grounded an their interpretations pretty smooth, and one can argue that they are natural expressions of nature insofar as they come from us and we are part of nature, and clearly my computer here can do just fine with them so there must be something physically innate about them. but arguably that's nonlinear dynamics and emergence that makes the analog parts of a computer operate digitally. so you're (i mean said philosophers, not you) trying to ground the physical application of logic in a neccessary discrete process, but what actually gives rise to it is emergent nonlinear analog processes. and you're bending the term "logic" to refer to those analog processes when you know deep down inside that there's something innately different about them, and then conflating the bent definition w/the original one.

so ya, that "logic" (pun intended) is transparently dubious. and maybe it was part of some of the more "religious" mathemeticians of old (e.g. Pythagoros), and might make for a good historical note and an interesting insight into insanity. but we definitely shouldn't write it in any way that makes it sound like the idea that math is not a formal system or any absurd proposition like that is at all credible, which is what it sounded like to me. Kevin Baas^talk 19:39, 24 March 2010 (UTC)

also be it said that the idea "So if you need axioms, then you have not reduced mathematics to logic — at best, you've reduced it to logic plus those axioms. Unless of course the axioms themselves are logically necessary." - the axioms are somewhat arbitrary, actually, the real restriction is their topological (for lack of a better word) relation to each other. i.e. a computer's "operations" or "instructions" can be called "axioms", thus in the original turing machine you can have a set of axioms. but turing also showed that there are quite an innumerable set of turing machines w/different "axioms" which are all equivalent to the universal turing machine. (and to cite some real world practical examples: IBM compatible, apple, cray, DEC Alpha.) that is, they can all emulate each other. so the only thing really "unique" or non-arbitrary about them is that they can emulate each other. i.e., roughly, their place in the Chomsky_hierarchy (namely, Turing_completeness), quite irrespective of the grammar and axioms they use. I might be a little off-topic on this, but that's what it reminds me of, in any case. Kevin Baas^talk 19:50, 24 March 2010 (UTC)

okay, now reading logicism i see more what you mean: that the traditional set of "logic" rules apparently need to be augmented with a few more rules (whcih can be expressed in terms of the original set) in order to be, as it were, turing complete. and then this is a weaker sense of "logical" in that the set of rules needs to be augmented so. i never meant to imply that they didn't and i actually think it rather insignificant that they do. whose to say that the original set of "logic" rules isn't any more arbitrary than the "augmentations"? us?! talk about arbitrary! Kevin Baas^talk 20:48, 24 March 2010 (UTC)

These are terribly complicated issues and there is no general agreement on them. The talk page for the mathematics article is not the right forum to talk about them. If you are interested I strongly recommend the work of Harvard's Peter Koellner. He has a brilliant article in the current issue of the Bulletin of Symbolic Logic. You might also be interested in his On the question of absolute undecidability, which you can find online. --Trovatore (talk) 20:54, 24 March 2010 (UTC)

(unindent)they seem rather straightforward to me, but yes, yes, i shall digress. the issue is the phrase "...important work in mathematical logic showed that mathematics cannot be reduced to logic." , with which i still take issue with, if for slightly altered reasons. by "reduced to logic" i read something like "proofs of theorems be expressible in a reduced formal grammar", and I maintain that they can, e.g. a turing machine or a ZFS+AOC system. and i contend that the subtler point that you need a few axioms beyond the basic and,or, in, not operations, which are noentheless expressible with those operations, and that some egregious purists might balk at that is a bit trivial and a bit to fine to be stated so boldly, apart from the phrasing as worded being -- as we have just witnessed -- misleading. Kevin Baas^talk 21:02, 24 March 2010 (UTC)

as it's worded now "logic alone" it's a little better but i still thinks it's ambiguous on a point this is, ultimatley, rather subtle. Kevin Baas^talk 21:10, 24 March 2010 (UTC)

I changed mathematicians to non-mathematicians, as the sources only include beliefs of non-mathematicians. Roger (talk) 02:48, 8 January 2011 (UTC)

There's a lot of nonsense said above (on both sides of the argument, if there is one) that I will not comment about. However there is a simple issue that needs no technical arguments. The current version, which Trovatore keeps reverting to, starts: "Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper." I think all mathematicians would agree that mathematics is not experimentally falsifiable; it is hard to imagine how an experiment could falsify mathematics. It is quite conceivable that mathematics, or some part of it, will one day be found to be inconsistent (remember Russell's paradox?), but if it happens, it will have nothing to do with experiment. The universe has been found to not be a Euclidean space, but that does not affect the work of Euclid (which is not entirely rigorous anyway, but that is another issue) any more than it affects hyperbolic geometry (which does not model the universe either). I'm unsure what Popper actually though about mathematics (his WP article does not mention mathematics as subject at all), but I think falsifiability can only be taken to characterize empirical science, which mathematics simply isn't. As an aside, it would be more interesting to know if many philosophers believe that philosophical theories are experimentally falsifiable. But I digress.

Next sentence: "However, in the 1930s Gödel's incompleteness theorems convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that 'most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently.'". OK, so (if I understand this correctly) Popper believes (around 1995, presumably a bit before his death in 1994) that mathematical hypotheses are conjectures that could be experimentally falsified, which just shows he doesn't understand what a hypothesis is in mathematics. But in any case there is no relation with Gödel's findings of the 1930's; I think this sentence makes a completely unjustified link between them and Popper's quote (in which "even recently" is unlikely to mean "before 1930"). And I would like to know what kind of non-logical element "many mathematicians" would like to invoke to resolve statements that according to the incompleteness theorem cannot be decided by logic alone.

So to get to the point I really wanted to make, there are two issues mixed up in this sentence which in fact are totally unrelated: (1) the question of whether mathematical axioms are assumptions about reality that could be experimentally falsified, and (2) the question whether in principle all mathematical statements can be proved or disproved in an appropriate formal logical system. Gödel's incompleteness theorem shows that (under mild assumptions on what "mathematical statement" and "formal system" mean) the answer to (2) has to be "no". So there will be in every theory some statements that cannot be decided from the axioms of the theory by pure logic. But that is miles away from anything involved in question (1). First of all such a statement is not an axiom of the theory, or a hypothesis of any particular theorem, which are simply assumed to be true in order for the theory/theorem to be applicable; it is a statement whose truth or falsehood one might think to be deducible from the (assumed) truth of the axioms, but in fact is not. And second, axioms have long since ended being considered to be "self-evident truths", they are just starting points of a theory that implicitly determine what the theory is about; they are meaningless in reality, or in any other theory. Take the axioms of your favorite theory: groups, probability, topology, mathematical analysis, (and yes) geometry, set theory, even mathematical logic itself. Which means that the answer to (1) is also "no", but with no relation whatsoever to Gödel's work.

And logicism in all this? It certainly never assumed that mathematics needs no axioms. It is inconceivable to base say geometry on pure logic without some axioms telling what geometric notions like "point" and "line" mean. It also does not affirm that all mathematical statements can be decided by pure logic (from a given set of axioms). What it probably does affirm (I'm in no way an expert on this) is that apart from the rules of logic and the axioms, mathematics needs no vague kind or reasoning based on things that are "obvious" without being able to be formalized. It definitely rejects the idea that there are mathematical statements whose truth is open to experimental verification or falsification, but that does not distinguish it from other schools of thought (Popper notwithstanding). Marc van Leeuwen (talk) 14:03, 10 January 2011 (UTC)

I agree that Trovatore keeps reverting to statements that are not true, and not commonly believed by mathematicians. At best, the statements are confusing and misleading. If some mathematicians do believe these statements, then there should be a citation to a source, so the interested reader can learn just what the mathematician is really saying. As it is, the whole section is confusing. Roger (talk) 21:14, 10 January 2011 (UTC)

Roger, to be honest, I don't think you understand these issues well enough to comment on them. Evidence for this would be your change to the claim that Goedel showed that "some mathematical statements are undecidable". That's completely wrong. Goedel did not show that any individual statement was "undecidable". He did show that certain foundational theories as a whole were undecidable in the sense of a decision problem, but that's different. He also showed that, for any given formal theory satisfying certain conditions, there are statements that are undecidable in a different sense in that theory (that is, that they are independent of it), but not that they are undecidable full stop.

Marc van Leeuwen's claims are more subtle and I will need time to evaluate and respond. --Trovatore (talk) 21:54, 10 January 2011 (UTC)

Trovatore, I said that Goedel showed that some mathematical statements are undecidable. Your gripe seems to be that I did not refer to a particular formal system and add the qualification "in that theory" to that sentence. There is of course a whole article explaining the matter in detail. Please address the substance of what I said, and skip the ad hominem attacks. Roger (talk) 22:57, 10 January 2011 (UTC)

It's a total misrepresentation to say he showed certain statements were undecidable. That's the kind of nonsense you read in the popularizations. The Goedel sentence of PA is undecidable in PA, for example, but it is not "undecidable"; it's true. --Trovatore (talk) 23:03, 10 January 2011 (UTC)

Yes, of course "undecidable" means undecidable in a formal theory. See Decidability (logic). I see that you used the term "question of absolute undecidability" above to refer to whether a statement could be undecidable in any reasonable theory. I did not say anything about absolute undecidability. Roger (talk) 23:28, 10 January 2011 (UTC)

OK, let me just respond to a couple of things in Marc's comments. I may well not respond to everything (it's kind of long and I'm sitting here in an airport). First, absolutely some mathematicians take the view that mathematical statements can be experimentally falsifiable. The experiment in question is the discovery of a proof.

Take an example: Consider the claim that there is an absolute powerset of the set of all natural numbers. That is, that there exists, in whatever probably non-physical but nevertheless real sense, a set that has as elements all subsets of the naturals, so that none are left out.

This is a claim about the world. It says that the world has such an object in it.

Can it be falsified, in principle? Absolutely yes. The existence of such a set implies, for example, that the first-order theory called second-order arithmetic (slightly confusing maybe — it's a theory of two-sorted first-order logic, with one sort for naturals and another sort for sets of naturals; the variables for sets of naturals is what makes us call it "second-order arithmetic") cannot prove the sentence 0=1. Attempts to prove 0=1 in second-order arithmetic are experiments, that attempt, among other things, to falsify the existence of P(N). If any such experiment ever succeeded, we would know that P(N) does not in fact exist.

Now, this may not be (probably is not) the majority view, but it is certainly not true that no mathematicians hold it.

On the other hand, the view that Goedel refuted logicism is, I think, the majority view, and therefore does not need to be attributed to individual mathematicians. I am on less sure ground in discussing what logicism actually claimed (past tense — it's not really a viable current school, though there are "neo-logicists"), as I have not closely read Russell or Whitehead or Frege. However I think Marc is incorrect; I think it really did assert that you could do mathematics without assuming anything. You would still need definitions, but not axioms in the sense of actual synthetic assertions. --Trovatore (talk) 22:33, 10 January 2011 (UTC)

Trovatore, Russell, Whitehead, and Frege did not claim to be able to do mathematics without axioms. Please read up on the subject before claiming to represent the opinions of most mathematicians. If you are right about what most mathematicians believe, then you ought to be able to give a source that explains what those mathematicians believe. Roger (talk) 22:57, 10 January 2011 (UTC)

As I say, I'm not an expert on what the logicists thought in detail, but it is certainly now generally agreed that whatever it was, it was wrong. The turning point seems to have been Goedel, whether or not his theorems actually refuted logicism (Franzen thought not, which I do need to go back and figure out why he thought that). --Trovatore (talk) 07:41, 11 January 2011 (UTC)

Not to anyone particular, but to the attitude at the heart of this whole fight: please do not use philosophical waffle without being clear about the mathematics first, or you will be part of the grand scheme to reduce philosophy to random bits of pretentious babble about other subjects as spouted by those who have never actually studied the other subjects for their own sake.

The infinity sign

Dudes. Why is there a bunch of infinity signs in this article? They serve no purpose. I suggest we remove it, —Preceding unsigned comment added by 134.173.58.98 (talk) 04:07, 27 January 2011 (UTC)

I agree. What is the point of having the infinity sign in eight typefaces? If you really want something illustrating notation, an equation would do. Otherwise, this part of the article has enough illustrations; this one could simply be deleted. --seberle (talk) 18:05, 8 February 2011 (UTC)

I agree. Paul August ☎ 18:38, 8 February 2011 (UTC)

Mathematics by language

Some races, talk mathematics structure within their plain language, and have no written symbols to prove it. One example might be that according to a person who lives where there is no winter, he/she might only know snow and ice, but to a person who lives where it is winter for half the year, there are numerous kinds of snow definitions. There is granular, powdered, drifting, hard packed, just right to make an igloo, too hard even to leave tracks and I can go on and on, and another person will understand just what I am describing in exact detail. —Preceding unsigned comment added by 65.181.32.135 (talk) 16:58, 9 February 2011 (UTC)

If I understand correctly, your main point is that speakers of different languages speak mathematics differently. This is interesting but requires clarification and strong citations. Perhaps you also intended to express the idea that mathematics is a "language", separate from colloquial, natural languages. This is problematic. As an English speaker from the USA, I speak mathematics within my plain language. For example, I might say "Three plus eight is eleven." or "The antiderivative of x squared with respect to x is one third x cubed, plus C.". In writing I usually (but not always) abbreviate "three" as "3", "plus" as "+", etc. Mgnbar (talk) 17:15, 9 February 2011 (UTC)

Criticism of Definition of Mathematics, and General Remarks About the Article

This is a terrible article starting with a shit definition of maths justified by poor sources, and suitable only for children. How is this definition any different for an equivalent definition of physics by replacing the word mathematics for the word physics? Quantity, change and space, are physical concepts, mathematics deals with concepts that are patently not physical. That leaves structure. To say that algebraic topology, finite geometry, graph theory etc deals with structure is uninformative and misleading

The following are examples of awful sentences:

Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered.[8]

Mathematics arises from many different kinds of problems.

Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area.

Number theory also holds two problems widely considered to be unsolved: the twin prime conjecture and Goldbach's conjecture.

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set.

The study of space originates with geometry – in particular, Euclidean geometry.

Who the fuck writes this shit? Do they read what they wrote?

The only half-decent math entries on wikipedia, tend to be those that are too technical for idiots to fake, and even then they tend to be verbose, repetitious and awkwardly phrased. —Preceding unsigned comment added by 86.27.195.112 (talk) 13:42, 20 February 2011 (UTC)

In what way is this sentence (for example) awful? Are you upset by the content, or the writing, or what?

Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered.[8]

It is difficult to write an article on a topic this big, with which so many people have experience at varying levels. Much of the article is the result of long argument to achieve near-consensus --- i.e. writing by committee --- which inevitably produces compromises. The definition of mathematics is a particularly sticky point. If you would sincerely like to improve Wikipedia, then propose your own text. Also, please keep in mind Wikipedia's policies on civility. Mgnbar (talk) 14:29, 20 February 2011 (UTC)

A bit presumptuous in the opening, and why can't I edit it?

"....establish truth by rigorous deduction from appropriately chosen axioms and definitions" This is not uncontroversial. It seems like what mathematicians wnat to believe about themselves, more than something factual. Also, why can't I edit this article? Sincerely, Mythirdself. —Preceding unsigned comment added by Mythirdself (talk • contribs) 19:10, 28 April 2011 (UTC)

The reason you can't edit it is that is was vandalised so much that we were forced to prevent unregistered and new users from editing it. You should be able to edit it now though (you need ten edits and that was your tenth). Hut 8.5 19:15, 28 April 2011 (UTC)

Before you edit anything in the opening, you should probably run it by the Talk Page first. The opening has had a very long history of edits and discussion (most of it in archives). Some of the opening is agreed compromise after long debate. --seberle (talk) 02:13, 29 April 2011 (UTC)

The metacomment there actually refers to the first sentence only. Tkuvho (talk) 04:28, 29 April 2011 (UTC)

The reference for the sentence I criticized isn't even comprehensible. It's (I assume) an author: "^ Jourdain". What's the work, page number, etc.? —Preceding unsigned comment added by Mythirdself (talk • contribs) 23:31, 29 April 2011 (UTC)

I agree the "Jourdain" reference is bizarre and that both the reference and the sentence need correcting. However, I also question the new statement "Since David Hilbert's time, it has become customary to view mathematical activity as establishing truth by rigorous deduction from appropriately chosen axioms and definitions." I'm a bit outside my area of expertise, but this is not my understanding of Hilbert's formalism. I thought Hilbert actually claimed the precise opposite -- that formal mathematics was arbitrary and in no way connected with real world "truth". Euclid, on the other hand, assumed his system to establish mathematical "truth", right? Is there a reference for this new statement? --seberle (talk) 00:53, 30 April 2011 (UTC)

Although I won't defend the sentence completely, I think I can put some nuance to the "precise opposite". As I see it, mathematical reasoning had been based for ages on an intuitive notion of what must be true in some Platonic mathematical universe, but Hilbert considered this too vague and open for philosophical debate (and no doubt things like non-Euclidean geometry had eroded the status of a Platonic universe). So he wanted to save the notion of truth by equating it to provability in a formal system; a notion sufficiently precise that no debate would be possible, and which itself is open to to mathematical study. In this point of view axioms are beyond discussion: playing the game means accepting the axioms (but one can choose to play different games). But I think that Hilbert still believed that the properties of an ideal mathematical universe in (say) number theory, analysis or algebra could be completely captured in a set of axioms; those that accept the truth of those axioms then cannot doubt the truth of theorems derived from them. So Hilbert is still out to capture "truth" in some world, not "real" but ideal, in a rigorous way. In some aspects Hilbert's project has been extremely successful, but 20th century developments in set theory and logic have led to a divorce between "truth" and "provability", which cannot be equated; the former necessarily remains outside our grasp, so we will have to settle for the latter. Marc van Leeuwen (talk) 06:22, 30 April 2011 (UTC)

The idea that Hilbert thought mathematics was an arbitrary system is one of the pernicious duds in the literature that does not stand up to scholarly scrutiny. Fortunately, there are quotes to disprove this. One of them is reproduced in the recent article

• Karin Usadi Katz and Mikhail G. Katz (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. Foundations of Science. DOI: 10.1007/s10699-011-9223-1 [2] See arxiv Tkuvho (talk) 20:31, 30 April 2011 (UTC)

The way Hilbert thought mathematics was arbitrary may be an oversimplification, but I agree with Marc van Leeuwen that Hilbert divorced provability from truth and this continues to this day. Therefore the new statement in the opening that "Since David Hilbert's time, it has become customary to view mathematical activity as establishing truth" is still wrong, or at least misleading. Am I missing something? At the very least, we need support for such a statement. At best the sentence should be removed or modified for clarity. --seberle (talk) 03:26, 1 May 2011 (UTC)

I read the opening, out of curiosity, to see where I might assign presumption. The "Since David Hilbert's time ..." conjecture is where I find the most improvable arena. I hope Mythirdself, having started this thread, will return favor and enunciate their (the three of them) concerns as well as edit content where improvement can follow such collaboration. I recommend being bold in such endeavors, and for page watchers to exercise good faith in anticipation. Imagine how stifling it may be, asking a new editor to run their ideas through the talk page before editing. Seriously, that is contrary to many core principles. And good luck creating a better lead, a thing that seems quite doable. My76Strat (talk) 03:47, 1 May 2011 (UTC)

Hilbert as well as Peano introduced axiomatisations of geometry that created a new paradigm for doing mathematics, the one that dominates our thinking about mathematics today. A few decades later, Hilbert was involved in the project together with Bernays of providing a metamathematical finitistic basis for mathematics. As far as truth versus provability is concerned, there is certainly a large (though not universal) consensus today that they are different, so it is hard to see what to Blaim Hilbert for here. Tkuvho (talk) 03:58, 1 May 2011 (UTC)

The more I read this lead, the more I think the subject is Modern mathematics opposed to Mathematics. From the broad title, it is reasonably debatable that Hilbert is not a necessary element of the lead at all. I think much of the presumption derives from this possibility. This is not an indictment against Hilbert, but a critique of the lead summary for this article. My76Strat (talk) 05:11, 1 May 2011 (UTC)

I just wanted to add a quick comment concerning the problem of "truth" versus "provability". The simplest solution would be to sidestep this dichotomy altogether by replacing the mention of "truth" by "correctness". Also, while truth and provability aren't the same, certainly probability implies truth, so one could even leave "truth" in place. I don't see anything wrong with saying that mathematicians try to convince each other of the "truth" of their theorems. What I do see as problematic, and which was the original problem pointed out by editor Mythirdself, is the unqualified claim that mathematics can be defined as deriving theorems from axioms. Such a claim is simplistic and presumptuous, as originally pointed out by Mythirdself, and probably agreed to by 95% of wiki readers. Tkuvho (talk) 06:41, 1 May 2011 (UTC)

Just a few comments. @seberle, I did not say that Hilbert divorced provability from truth, quite to the contrary, I said he wanted to equate them. The divorce I would attribute more to such things as Gödel's incompleteness theorems. @Tkuvho, indeed when I said one needs to settle for provability if truth is beyond grasp, I meant "the whole truth" is beyond grasp, but provability still implies truth, so we are left with "nothing but the truth". That is, assuming our basic axiomatization (say ZF) is consistent, which we all believe, but which unfortunately is (if I understand correctly, not being a logician) also out of grasp for the kind of formal proof one can give within the axiomatization. In my opinion "Since David Hilbert's time, it has become customary to view mathematical activity as establishing truth" is acceptable, in particular since the rest of the sentence makes clear that truth is in fact obtained through proof; saying "establishing provability by [providing proof]" does not make for a nice statement, even if it is nominally more correct. Marc van Leeuwen (talk) 11:42, 1 May 2011 (UTC)

Sorry for my misrepresentation of what you said, Marc van Leeuwen. I sometimes write these comments too quickly! The latest edit of the opening is a slight improvement. Please keep in mind, everyone, that mathematicians' ideas about "truth" may not accord with popular ideas or philosophical ideas of truth. For example, Marc van Leeuwen wrote: "... so we are left with "nothing but the truth". That is, assuming our basic axiomatization (say ZF) ..." In other words, mathematical "truth" depends upon the axiomatic system chosen. Is the Axiom of Choice "true"? It all depends, and in the end it depends on the mathematician's (arbitrary) choice. This kind of truth has no bearing on what most people think of as "real world truth". I emphasize this only because we are writing an opening paragraph for the general public and the average person is likely to understand the opening now as saying that mathematicians use mathematics to find real-world truth. This contains some truth -- mathematicians do believe mathematics can be used to model the real world -- but it is also somewhat misleading because mathematical "truth" (e.g. Banach–Tarski or transfinite numbers) does not have quite the same meaning as real world "truth". That's my two cents worth. --seberle (talk) 16:52, 1 May 2011 (UTC)

Science

I removed this quote from the lede, because of undue weight:

• "Some mathematicians, such as Vladimir Arnold, define Mathematics as an empirical science."

Arnold wrote that mathematics is a branch of physics, among other entertaining absurdities.  Kiefer.Wolfowitz 08:51, 2 May 2011 (UTC)

I removed the following section, which seems to be far below the rest of this article in quality, and which also seems to give undue weight to speculations:

Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper. However, in the 1930s Gödel's incompleteness theoremsconvinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently." Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself.

An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the scientific method.

The opinions of mathematicians on this matter are varied. Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics.

I have reverted this removal. I think this material is important and disagree that it's "lower quality". --Trovatore (talk) 08:58, 2 May 2011 (UTC)

A section littered with "citation needed" and "who?" tags has lower quality, obviously, than sections that don't. Sentences like "In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels" need deletion, but others can do the excision. Your other edits don't seem to enjoy much support.  Kiefer.Wolfowitz 09:02, 2 May 2011 (UTC)

They are not "my edits" per se, not my original edits anyway. I learned quite a bit from those sections.  ::::The fundamental sociological problem here is that most mathematicians are exposed to formalist ideas when they begin to study mathematics seriously. Formalism is very attractive on a superficial level. If they never go into math logic (and most don't) they may never really think about it much, and just kind of drift along accepting a formalist narrative by default. That probably describes a large fraction of the editorship here, grosso modo, and leads to a constant danger of this article tilting towards a formalist POV without a realist or empiricist counterweight. I try to provide that counterweight. --Trovatore (talk) 09:08, 2 May 2011 (UTC)

What does your latest opinion have to do with improving the article?  Kiefer.Wolfowitz 09:12, 2 May 2011 (UTC)

My view is that, in removing the material you removed, you removed an important exploration of alternatives to formalism. Restoring it, therefore, improved the article. --Trovatore (talk) 09:15, 2 May 2011 (UTC)

Kiefer.Wolfowitz, when you describe mathematics from Euclid onwards as formal derivation of consequences from axioms, you are not expressing anyone's opinions but yours. The most productive mathematicians of the 18th and 19th centuries respectively were Euler and Cauchy. I challenge anyone to produce a single derivation of consequences from axioms in Euler and Cauchy. You are talking dead wood, you are not talking mathematics. Take it from a research mathematician. Tkuvho (talk) 09:17, 2 May 2011 (UTC)

I reduced the axiomatic-fixation of the lede---just check the history. I also noted something about mathematical models of reality.  Kiefer.Wolfowitz 09:22, 2 May 2011 (UTC)

When talking about research mathematicians of the caliber of Arnold, I would recommend a higher dose of reverence. He may have exaggerated when he called mathematics a branch of physics, but his sentiment that mathematics is also a kind of experiental science is shared by many research mathematicians, if not by zealous wikipedians. Tkuvho (talk) 09:19, 2 May 2011 (UTC)

This is an OR claim, that is a digression from improving the article. There are plenty of notable writings by Jon Borwein or Richard Varga about experimental mathematics that do not engage in absurdities, like claiming that mathematics is a branch of physics, or claiming that "mathematics is defined as an experimental science".  Kiefer.Wolfowitz 09:59, 2 May 2011 (UTC)

Editing this article is not cowboys and indians. Why are you smearing yourself with the war-paint of "research mathematician"?  Kiefer.Wolfowitz 09:24, 2 May 2011 (UTC)

Don't worry, I sided with the indians on that one ;) But you should realize that your view of mathematics as Euclid-style formal derivation is a very reductionist view. The Elements are a finished product of a long process of which we have no record. We don't even know if it was really written by Euclid. The process is what constitutes mathematics, not the finished product. At any rate, that's what everyone thought until the 20th century. Tkuvho (talk) 09:32, 2 May 2011 (UTC)

Let me repeat. I have toned down the logicist and foundationalist and ontological -crap aspects of this article---the ontological discussion is worse than the others. The weight on "philosophical" issues is undue. I added the sentence about mathematics and reality, not about the reality status of mathematics, in the lead paragraph. Please stop accusing me of having a Euclidean/formalist view of mathematics, when I have only indicated that axiomatic reasoning is thousands of years older than your heroes.

Secondly, you restored the statement that mathematics establishes truth by logical reasoning. On the contrary, I removed that statement because mathematicians rarely talk about truth. Mathematicians discuss properties of mathematical objects and consequences of hypotheses. Further, since the time of Peirce (with Fregean backsliding) competent logicians have known that there are many logics, with working mathematicians playing loosy-goosy with whatever logic suffices to resolve problems.  Kiefer.Wolfowitz 09:47, 2 May 2011 (UTC)

Not true that mathematicians rarely talk about truth. Really we mostly talk about truth. Proof is a means of establishing truth, but the main emphasis is generally on what the state of affairs is, not on how it is established.

I note that the "science" section seems to have been moved to the bottom of the article, where it has no connection with anything around it. It belongs together with discussion of foundations. --Trovatore (talk) 16:39, 2 May 2011 (UTC)

Lede

I made some edits.

1. "formulate newconjectures, and develop arguments called proofs to convince their peers of their theorems. Since David Hilbert's time, it has become customary to view mathematical activity as establishing truth by rigorousdeduction from appropriately chosen axioms and definitions." was replaced with "Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are formal arguments sufficient to convince other mathematicians of their validity. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. However, mathematical proofs are less formal and painstaking than proofs in mathematical logic. Since the late 1800s, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions."
2. "The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions". Albert Einstein, on the other hand, stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."" was replaced by "Regardless of the ontology of mathematical objects, there is agreement that mathematics is a formal science rather than an empirical science." Then I removed that distraction, which doesn't belong in the lead.
3. I corrected many misstatements about natural science, and I moved a long and poor-quality section on science below, following the description of mathematical fields.

IMHO, the article needs a discussion of the unity of mathematics, how the same objects appear in apparently disparate fields of inquiry, such as the role of groups in complex analysis (homotopy or homology), geometry, and polynomial equations. (Peirce referred to this as surprising to find the same _____ in an African jungle and the Alaskan Klondike!)

 Kiefer.Wolfowitz 12:26, 1 May 2011 (UTC)

Kudos to your efforts which have manifest a substantial improvement in the lead summary for this subject; IMO. My76Strat (talk) 02:23, 2 May 2011 (UTC)

Thanks!  Kiefer.Wolfowitz 08:37, 2 May 2011 (UTC)

The stuff in the lead is not bad but overemphasizes the axiomatic approach at the expense of the quasi-empirical or realistic one. I'll think about how that might be improved. The science section needs to be moved back together with foundations, where it belongs. --Trovatore (talk) 20:11, 26 May 2011 (UTC)

Mathematics and (Baconian) natural science

I moved that section to the bottom, because imho it still reads like an essay, rather than an encyclopedia article, and its weight may be undue. Editor Trovatore disagrees, so this is worth a discussion.  Kiefer.Wolfowitz 20:24, 26 May 2011 (UTC)

From the start, I think the link to "space" in the first line:

Mathematics is the study of quantity, structure, space, and change.

Is probably meant to link to the http://en.wikipedia.org/wiki/Space_(mathematics) page instead, though neither of them come anywhere close to being explanations of "Space" that belongs in the definition of "Mathematics"...

Neither space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction.

Nor http://en.wikipedia.org/wiki/Space_(mathematics) In mathematics, a space is a set with some added structure.

Seem to be relevant descriptions for the more abstract concept "space" of which Mathematics studies.

The intro / declaration from the Mathematics portal seems to be a more fundamental description: Mathematics, from the Greek: μαθηματικά or mathēmatiká, is the study of patterns. Such patterns include quantities (numbers) and their operations, interrelations, combinations and abstractions; and of space configurations and their structure, measurement, transformations, and generalizations. Mathematics evolved through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the systematic study of positions, shapes and motions of abstract objects. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.

---

There's a typo in the Applied Mathematics section. Looks like someone only half pasted a quote: 'formulation and study of mathematical models. — Preceding unsigned comment added by 94.195.50.242 (talk) 10:30, 30 May 2011 (UTC)

Lede

Let's try to work on a version of the lede here instead of getting into an edit war. As I mentioned, I object to tracing the axiomatic method back to Euclid in describing modern mathematics, because this was simply not the case before Peano, Hilbert, and Co, and we don't really know what Euclid and his contemporaries did. Describing mathematics flat out as formal derivation of theorems from axioms is philosophically naive. Tkuvho (talk) 10:42, 2 May 2011 (UTC)

I agree with your statements, but I wish that you would acknowledge your previous misattribution of naiveties to me---not that my skin is thin but my respect for you is substantial.

There should be no message of Peano, Hilbert, and Co. in the lead, because they are ignored in the body, per MOS. On the other hand, we should agree that for 2000 years, there was essentially one book of deductive mathematics, The elements of Euclid (and nobody cares about the historical author, at least for this article). We should also mention Euler's Analysis of the infinite as the calculus textbook for c. 200 years.

The article's discussion of pure mathematics and putting foundations ahead of mathematics is distorted. Only in the 20th century did there appear professional mathematicians ignorant of applications and not trying to solve the urgent scientific problems of the day. It is undue weight to describe pure mathematics with so little concern with applications.  Kiefer.Wolfowitz 11:02, 2 May 2011 (UTC)

I did not say you were naive, I said that thinking of modern mathematics as formal derivation from axioms is naive. Peano and Hilbert should certainly not be ignored in the body since they are responsible for the modern axiomatic philosophy. I can't see why Benjamin Peirce should be mentioned but not Hilbert. How many mathematicians will agree with you on this choice? Certainly Euclid can be mentioned, but as I pointed out already it would be inaccurate to portray 17-19 century mathematics as dominated by anything at all resembling an axiomatic approach, except in some isolated areas. Tkuvho (talk) 11:42, 2 May 2011 (UTC)

Good: We avoid discussing axiomatics, then. (I tried to reduce the axiomatics's undue weight in the lead.)

There are several Pierces. Benjamin Peirce's definition is notable (and I provided an improved reference to the edition cited, which was edited by CSP, his son). I do not say that Renaissance & Enlightenment mathematics was axiomatic, only that Euclid was upheld as the example of axiomatic reasoning during that time (until c. 1850).

I suppose that you admit that the current definition of mathematics in this article, follows popular authors, but cannot be taken as serious. (The present article does not seem to meet the requirements of reliability, imho.) B/CS Peirce's definition can be taken seriously, on the other hand. Andrew Gleason wrote a similar definition of mathematics, if my memory is correct. Next week, I should be able to provide references to further discussion.  Kiefer.Wolfowitz 12:38, 2 May 2011 (UTC)

I agree that Einstein's definition is a bit of a joke. Hilbert's definition is more serious, and he is certainly more of a heavyweight than Benjamine Peirce. Tkuvho (talk) 16:15, 2 May 2011 (UTC)

B. Peirce's definition appears in the edition edited by his son, who is taken rather seriously by philosophers and logicians. CSP's New Elements of Mathematics has a long discussion of alternative definitions of mathematics, which is worth consideration.  Kiefer.Wolfowitz

OK, I did not know that. Thanks for pointing it out. We should make a note of that. CSP is certainly more of a heavyweight, perhaps even competing with Hilbert. At any rate, I find Hilbert's remark more informative. Tkuvho (talk) 16:43, 2 May 2011 (UTC)

I should have applauded your addition of Hilbert's quote. CS Peirce has similar remarks, claiming that every mathematician is a Platonist wanting to find the truth, and nobody can explain the advance of mathematics (and more generally science) by just pretending that it's a formal game (or establishment of consensus): This was one motivation for my changing the wording of the account of proofs---which do more than convince others.  Kiefer.Wolfowitz 17:00, 2 May 2011 (UTC)

Thanks, great. My next project is an AfD for formal science, which is a fabrication. Tkuvho (talk) 17:05, 2 May 2011 (UTC)

• Groan* I know that it is a lost cause to plead to a logician for restraint and allowances for convention and the fallible misguided masses. ;) Can you suggest another alternative phrase for mathematical logic, mathematical statistics, mathematics, and theoretical computer science, and perhaps information theory?  Kiefer.Wolfowitz 17:28, 2 May 2011 (UTC)

I have the highest respect for all these fields. How about "living sciences"? Tkuvho (talk) 17:33, 2 May 2011 (UTC)

I still have philosophical problems with this: "Since the pioneering work of Giuseppe Peano, David Hilbert, and others on axiomatic systems in the late 1800s, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions." The first sentence seems more applicable to logic than math. The second is a tautology: is the definition of "a good model" that which "provides insight and predictions?" Isn't insight subjective? — Preceding unsigned comment added by Mythirdself (talk • contribs) 18:25, 26 May 2011 (UTC)

The first sentence has undue weight on foundations, perhaps reflecting the interests of active writers. "Axiom-chasing" & foundations are a big concern for a small number of research mathematicians, roughly say the same number as pursuing applications in quantum mechanics.

A good model makes a cognitive difference, and is subjective in the sense that good mathematical models don't matter to mollusks or mud. You can find a discussion of modeling in the SIAM reprinted book on Mathematics Applied to Deterministic Problems in the Natural Sciences or von Neumann's articleor C.S. Pierce's, etc.  Kiefer.Wolfowitz 19:44, 26 May 2011 (UTC)

I agree. We can work to reduce the foundational bias of the lede. If you look up the history of the page, you will see that it evolved in the right direction: the page used to start out with a categorical announcement to the effect that mathematics is a formal science, period. I assume that meant it was deriving consequences from postulates, no? Of course, most working mathematicians don't work that way. Tkuvho (talk) 19:59, 26 May 2011 (UTC)

Quine's discussion of mathematics as the most formalized part of our language (roughly) is another useful statement by an authority. I made some comments at Applied Mathematics's talk page, urging this article to discuss applications more. (Laplace and Fourier leading to Harmonic analysis, and quantum mechanics and Hilbert space, etc.)  Kiefer.Wolfowitz 20:09, 26 May 2011 (UTC)

I'm not sure quoting a famous and idiosyncratic figure like Quinne is a good idea. He's brilliant, but he has his own very Quinnian views. Something blander (hey, it's an encyclopedia not a revolution) might be more appropriate. Mythirdself (talk) 18:37, 2 June 2011 (UTC)

Unity despite specialization

Mathematics undergoes specialization, yet continuity within mathematics is maintained by the discovery and elaboration of structures/theories whose powerful abstractions provide fresh insight. This quote form Charles Sanders Peirce is probably too long for inclusion:

The host of men who achieve the bulk of each year's new discoveries are

mostly confined to narrow ranges. For that reason you would expect the arbitrary hypotheses of the different mathematicians to shoot out in every direction into the boundless void of arbitrariness. But you do not find any such thing. On the contrary, what you find is that men working in fields as remote from one another as the African diamond fields are from the Klondike reproduce the same forms of novel hypothesis. Riemann had apparently never heard of his contemporary Listing. The latter was a naturalistic geometer, occupied with the shapes of leaves and birds' nests, while the former was working upon analytical functions. And yet that which seems the most arbitrary in the ideas created by the two men are one and the same form. This phenomenon is not an isolated one; it characterizes the mathematics of our times, as is, indeed, well known. All this crowd of creators of forms for which the real world affords no parallel, each man arbitrarily following his own sweet will, are, as we now begin to discern, gradually uncovering one great cosmos of forms, a world of potential being.

Charles Sanders Peirce (``CP 1.646)

About this [| Edits]

Hi, Perhaps the user has not understood the comments made to 'change the syntax'. The [| reference book]'s [| introduction], [| Development of Philosophy], etc. are indeed exercises in glorifying Greek Mathematics. How is this a reliable source is just one question, the point here is 'The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.' while at other places like India [| you may find the same], another source. The book therefore can only be interpreted as a reference to beginning of the systematic study of mathematics in its own right in the Ancient Greece. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 17:00, 6 June 2011 (UTC)

The edit in question involved changing "began with the Ancient Greeks" into "began in the Ancient Greeks". I don't see how that change has anything to do with reliability of whatever sources; it says the same, just with bad syntax, as far as I can tell. Marc van Leeuwen (talk) 08:59, 7 June 2011 (UTC)

Thanks for clarifying, though I later changed it to in Ancient Greece which again was removed. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 13:50, 7 June 2011 (UTC)

About this [| Edit]

Hi, I would like to know why are the changes reverted after deleting 4 sources in the guise of statement that one source is unreliable without giving any proofs. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 07:10, 6 June 2011 (UTC)

Doing a spot check a second of the 4 sources says nothing about mathematics in India, it is about early dentistry. Does you have a page ref on the French work? Thenub314 (talk) 17:38, 6 June 2011 (UTC)

Your edits were reverted because most standard sources on the history of Mathematics (e.g. Boyer and Merzbach A History of Mathematics, Wiley, 1991) place Egyptian and Babylonian mathematics far earlier than Indian mathematics. The sources you are using are also unreliable/irrelevant. Subhash Kak is a well-known Hindutva nationalist whose "work" in this area has drawn widespread criticism. He is moreover not a historian of science or mathematics. Another of your sources is a paper on archeo-dentistry. What does that have to do with mathematics? As for your other two sources, would you be so kind as to quote what they actually say on the subject? Let's face it, the Indus Valley civilization didn't leave behind any mathematical or even written records of any kind, so to claim that mathematics began with it is nonsense. Athenean (talk) 17:36, 6 June 2011 (UTC)

[[:Image:Neolithic mehrgarh.jpg|thumb|right|x108px| text|Early farming village in Mehrgarh, c. 7000 BCE, with houses built with mud bricks. (Musée Guimet, Paris).]]

Excavated ruins of Mohenjo-daro, with the Great Bath in the front.

Again, let us see sources claiming 'Subhash Kak is a well-known Hindutva nationalist whose "work" in this area has drawn widespread criticism. He is moreover not a historian of science or mathematics', which is repeated again without proof. About the rest of the two sources, I will see where I can get details. That "Indus Valley civilization didn't leave behind any mathematical or even written records of any kind" is again another assertion without proof.

Here is the actual source [| with images!!] next to it. It does look like solid geometry, maths, simple algebra, weight and economics to me, but I may be wrong here.

Lets see if I can show a visual proof here - the images you see on the right from IVC and Mehrgarh - present structures that only a deluded would feel could be done with no sense of Geometry.

[| Early civilizations of India, an article by Bernard Sergent, CNRS researcher President of the Society of French mythology] clearly mentions "Le système de mesure indusien a été explicité. The measurement system has been explained indusien. On possède même trois règles graduées. It even has three rulers. L'une, du site de Lothal, a pour plus petite unité une longueur de 1,7 mm. One, the site of Lothal, a smaller unit for a length of 1.7 mm. Les systèmes de mesure varient extrêmement d'une civilisation à l'autre. Measurement systems vary extremely from one civilization to another. Or, le premier système de l'Inde, défini dans l' Arthasâstra de Kautilya, sans doute au IVe siècle avant notre ère, donne une unité minima de longueur de 17, 86 mm, le décuple presque exact de la mesure indusienne. However, the first system of India, as defined in the Arthasastra of Kautilya, probably in the fourth century BC, gives a unit minimum length of 17, 86 mm, almost exactly ten times the extent indusienne."

Another quote "As one scholar of Finland, Asko Parpola, comes to assume. The geometrical arrangement and oriented along the cardinal points of the cities of the Indus involves celestial observations. after the Rg-Veda, who knows at this point that myths, which he makes vague references - uses a tracking system that has absolutely heavenly no parallel elsewhere in the field Indo-European. This is the system naksatra: a set of twenty-seven points celestial defined by bright stars or constellations, which define the circuit of the moon. The only thing that looks like this is a Chinese counterpart, and also in ancient Chinese culture that are naksatra in Indian culture. Parpola, which I summarize very, reasons thus: the establishment of a system as rigorous for Astronomy in India Brahmin surprise, he cons easily explained by the Indus civilization, which had developed measurement system and was interested in heaven..." ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 19:19, 6 June 2011 (UTC)

Another source - The Shatapatha Brahmana (शतपथ ब्राह्मण śatapatha brāhmaṇa, "Brahmana of one-hundred paths", abbreviated ŚB) is one of the prose texts describing the Vedic ritual, associated with the Shukla Yajurveda. It survives in two recensions, Madhyandina (ŚBM, of the vājasaneyi madhyandina śākhā) and Kanva (ŚBK, of the kāṇva śākhā), with the former having the eponymous 100 adhyayas,7624 kandikas in 14 books, and the latter 104 adhyayas,6806 kandikas in 17 books. Linguistically, it belongs to the latest part of the Brahmana period of Vedic Sanskrit (i.e. roughly the 8th to 6th centuries BCE, Iron Age India). REF:- Keith, Aitareya Aranyaka, p. 38 (Introduction): "by common consent, the Satapatha is one of the youngest of the great Brahmanas"; footnotes: "Cf. Macdonell, Sanskrit Literature, pp. 203, 217. The Jaiminiya may be younger, cf. its use of aadi, Whitney, P.A.O.S, May 1883, p.xii."

Another source - [| engineering skill] of Harappans. A few pages about this page (backword and forward) are also interesting. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 20:14, 6 June 2011 (UTC)

Yet another source, [| grammar] and more.

Grammar? Engineering? This is insane. If you don't know that Subhash Kak is not a reliable source, you might want to start by reading the article about him. The earliest mathematical documents from India are from the first millennium BC, whereas those from Egypt and Mesopotamia are from a millennium earlier, hence much much older. Your sources prove nothing. Bricks and rulers are not "mathematics". The IVC did not leave behind a single mathematical document. Images of mud-brick huts also prove nothing. Athenean (talk) 08:00, 7 June 2011 (UTC)

Is this how Wikipedia works? Calling facts presented with secondary sources as insane and then giving random statements without and proofs? Wikipedia is becoming digital cesspool where secondary sources are not asked from supposedly high-borne who can talk anything and get away with it. Where are the secondary sources to support all the wild assertions that begin the arguments and are maintained throughout, effectively ignoring the whole discussion? ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 08:08, 7 June 2011 (UTC)

Start by reading an actual source on the history of mathematics, like Boyer & Merzbach which I have already mentioned above. And yes, showing pictures of mud-brick villages and random irrelevant sources about grammar and engineering is quite weird. What does this have to do with mathematics? Can you show me a single mathematical document from the IVC? I didn't think so. Athenean (talk) 08:14, 7 June 2011 (UTC)

Please throw some light on 'actual source' terminology. Besides it is indeed weird to know from Wikipedia that Engineering has nothing to do with Mathematics and I am alarmed by such a random statements on this page, especially after a secondary source -this- is presented exactly to clarify the same, with numbers and all. Another reference if needed [here]. About 'mathematical documents' - it would be again primary source! Please refrain to demand anything other than secondary sources on talk pages, it is against standards. By the way, where are primary sources from Greece that are understood as 'mathematical documents' and why are these considered on this pages?

About the source mentioned A History of Mathematics, By Carl B. Boyer, Uta C. Merzbach, here is a link for uninformed [| from the same source], about Early Indian Mathematics. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 11:17, 7 June 2011 (UTC)

Please read your own edit carefully. You claim that mathematics more advanced than that employed in trade and measurement appeared in 3000 BC also in India, but your source refers only to exactly this kind of less complex mathematics, trade and measurements as "weights and measures" were used fire and foremost in trade and bartering. And, besides, to support your major claim you'd need anyway a fuller discussion than this one sentence tidbit. Gun Powder Ma (talk) 12:05, 7 June 2011 (UTC)

Well, I don't consider this as 'personal shortcut to glory' when it is the people of the Indus Valley who should be credited for the whole arrangements, not I am presuming 'more advanced versus less advanced mentality' here from either side, for I am not a bigot.

A better treatment of the 'weights and measures':-

THERE IS INDEED A SCALE, symbolic(and actual usage) of calibrations I presume and 3000BC or still remote dating of IVC (another source - yes, 3000 BC, yet another source).

More about the THE SCALE- the ruler - a few more details, so also some common sense in words presented for 'better treatment'- 'indicative of numerate culture with well-established, centralized system of weights and measures' some common sense explained in words (yes, numerate culture is the word) and 'decimal system of linear measurements' suggested by THE SCALE. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 13:18, 7 June 2011 (UTC)

The claim that the "Indians" developed mathematics, alongside Egypt and Mesopotamia, as early as 3000 BC is hyperbolical for two reasons:

1. In 3000 BC, there was still no Indian civilization, but rather forerunner copper and bronze cultures. The only high culture potentially capable of producing advanced mathematics was the Indus Valley Civilization, but this had not even entered its mature period and its designation as "Indian" is problematic to say the least (and consequently avoided in the WP article, so WP consensus).
2. The only known possible script on the entire Indian subcontinent was the Indus 'script' which appeared by the mid-3rd millennium BC. This 'script', however, is still undeciphered so it is impossible to make any inferences as to its contents, including its mathematical level and use of numbers. Gun Powder Ma (talk) 09:31, 7 June 2011 (UTC)

I agree. I've just read the article by Bernard Sergent referred to above (in the French original), and it makes clear most of all that whatever written traces there are from the Indus civilization cannot be deciphered (for lack of any "Rosetta stone" for it); this would certainly exclude having any trace of a "study of mathematics in its own right". It does not exclude having any trace of mathematical activity (one could imagine finding tables of some sort that suggest such activity) but the cited article does not in fact mention any mathematics at all. It does refer to traces of Indus culture which indicate measuring and astronomical observation (and of which it is discussed in how far any heritage of it can be found in more recent cultures). This does not seem to justify in any way including "Indians" (whatever that may designate) in the sentence "More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy". With all due respect for the Indus culture, there seems to be no ground for mentioning it in this article. Marc van Leeuwen (talk) 12:43, 7 June 2011 (UTC)

The source for the above line "More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy" - just says "Kline 1990, Chapter 1."! The thinly detailed reference material stays though anything else is refuted with one reason or another.

Is this how Wikipedia works? Another reference is A History of Greek Mathematics:

From Thales to Euclid, Volume 1 - here that goes endlessly about glory of Greek Culture right from the beginning which is just fine, but that does not mean no one else thought of anything else. The nature of references here is indeed disturbing. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 13:32, 7 June 2011 (UTC)

I think you are confusing mathematics with the various rules-of-thumb used to build structures. Standard reference works agree that Greek mathematics began at an earlier date than Indian mathematics. That in no way minimizes the many important contributions of Indian mathematics. Rick Norwood (talk) 14:19, 7 June 2011 (UTC)

Where are the "Rosetta stone" proofs of these Greek Mathematics? Is there no SCALE?

So the current understanding I assume as per 'Standard reference works'(making it possible to disregard any other secondary sources), in spite of elaborate town planning (along with stone built houses, elaborate drainage system, adequate water supply etc.), great uniformity in the Harappan products throughout vast territories benefiting both producer and consumer, uniform trade mechanisms introduced across cities (including weights), and understanding on Numbers, Space and Time can hardly be passed off as rule of thumb 'just to build structures' when the line in consideration itself talks of maths 'for taxation and other financial calculations, for building and construction, and for astronomy'! ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 15:19, 7 June 2011 (UTC)

It sounds like a review of the English Wikipedia policies on reliable sources, original research and synthesis are in order. --John (User:Jwy/talk) 16:13, 7 June 2011 (UTC)

Indeed, considering the random introduction of terms like 'standard sources on the history of Mathematics', 'mathematical document', 'actual source', "'script', however, is still undeciphered so it is impossible to make any inferences as to its contents, including its mathematical level and use of numbers", 'rules-of-thumb used to build structures', 'Standard reference works' etc. are thrown around as excuses to ignore secondary sources. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 06:56, 8 June 2011 (UTC)

I thought discussions on Wikipedia would be at a higher level than the usual Internet fare. Especially for the article on mathematics. Does being educated and sharing a good cause reduce the incidence of flame-wars? — Preceding unsigned comment added by Mythirdself (talk • contribs) 01:45, 9 June 2011 (UTC)

I am sorry to inform you that education only serves to make flame wars worse. When your educated your more likely to believe your are right. Usenet was started by academics you know. Thenub314 (talk) 02:44, 9 June 2011 (UTC)

Just found a page on Ruler on Wikipedia for information. I am sure it applies to mathematics to an extent. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 19:52, 16 July 2011 (UTC)

Thisthat2011, Wikipedia talk page guidelines explicitly say, Never address another user in a headline. You seem to do this over and over again. Please desist. Please change the headlines of the section and the one below to more neutral ones. Fowler&fowler«Talk» 01:00, 19 July 2011 (UTC)

Done! ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 06:20, 19 July 2011 (UTC)

Etymology

I think the Etymology from Semitic Arabic is the important thing. Because it tells the history of Mathematic just from the etymology and begin crediting some specific mathematic such as Al-Jabar, Arithmatic, Logarithma (read Loharitema), and Algorithm. It opens mind that the Greek doesn't invent this specific mathematic since The Arab invent the specific system number which lead to many specific mathematic knowledge such as Al-Jabar, Arithmatic, Logarithma, and Algorithm. See all the Al- infront of words, explain they come from Arab with Arabic names and Arabic system number. So be honest and let's tell the truth to the world this mathematic come from Arab. The Arab own the system number and put God name in Arab on mathematic knowledge. For the editor Please be honest because this is an important encyclopedy. I hope the etymolog from Semitic Arab soon move to the main page. And Please Complete the article about Al-jabar, Arithmatic, Log, Logarithma and Algorithma it self. — Preceding unsigned comment added by 202.152.202.248 (talk) 04:06, 28 July 2011 (UTC)

The New Oxford American Dictionary says that mathematics and arithmetic come from Greek. Every other source that I've ever seen agrees. I admit that this Wikipedia article is not well-referenced on the etymology. But you propose no references at all for your claims. Frankly, they sound like original research. So please back them up with reliable sources.

You also request that the Wikipedia articles Logarithm, Algorithm, etc. be improved. That is quite reasonable. However, this talk page exists to discuss the article Mathematics only, not all of the mathematics articles on Wikipedia. For that discussion, please go to Wikipedia talk:Wikiproject Mathematics. Mgnbar (talk) 19:54, 28 July 2011 (UTC)

Fields of mathematics

Now that the lead sentence has (weak) citations, which is great, there is still the issue that "quantity, structure, space, change" are listed as "fields of mathematics". Mathematicians simply do not talk like this. Maybe philosophers of math talk like this? Even if so, it needs citation, and equal weight should be given to how mathematicians talk about math. Most fields of math are about specific problems (or classes of problems, or sets of axioms) that draw on more than one of these four "ingredients".

Further, if the graphics are supposed to suggest that group theory is a subdiscipline of "structure" (let's accept that it is, for now), then they also suggest that "complex numbers" is a subdiscipline of "quantity". Is "complex numbers" supposed to be complex analysis? Or is that a subdiscipline of "change"? I know that this task is difficult, but the current solution is not good. We could do better by simply citing from the American Mathematical Society's (or another comparable organization's) classifications and copying text from the appropriate Wikipedia articles. We should also describe historical views on the organization of mathematics (e.g. as indistinguishable from physics or natural philosophy?). The current solution seems neither contemporary nor historical. Mgnbar (talk) 15:59, 22 July 2011 (UTC)

This shopping-cart definition of mathematics comes from De Morgan and Hamilton in the 19th century, where it apparently comes from Kant (assuming GF from CSPeirce). It provides popularly accessible overviews of the main fields of pure mathematics, namely number theory/algebra, topology/geometry, and analysis. However, it is obviously incomplete, ignoring mathematical logic and many of the traditional applied fields.  Kiefer.Wolfowitz 17:47, 1 August 2011 (UTC)

Here is how the American Mathematical Society currently demarcates the fields of mathematics when reporting statistics on new Ph.D.s in the USA (see [3]). The numbers give some idea of the relative sizes of these fields. Of course, the numbers vary from year to year, the mix might be different in other countries, and one may argue whether statistics, education, etc. are fields of math proper. Mgnbar (talk) 15:55, 24 August 2011 (UTC)

Field	2010	2009
Algebra, Number Theory	230	223
Real, Complex, Functional, and Harmonic Analysis	102	98
Geometry, Topology	149	139
Discrete Math, Combinatorics, Logic, Computer Science	116	141
Probability	73	82
Statistics, Biostatistics	495	483
Applied Math	229	169
Numerical Analysis, Approximations	88	93
Nonlinear Optimization, Control	27	24
Differential, Integral, and Difference Equations	102	117
Math Education	15	14
Other	6	22
Total	1632	1605

Edit: Added column for 2009 results (see [4]). Mgnbar (talk) 18:19, 24 August 2011 (UTC)

And here is a summary of the AMS's 2010 Mathematical Subject Classification (see [5]). I have not changed the order of the names, but I have introduced break points, to improve readability and to try to reconcile somewhat with the coarser classification given above.

• General; History; Mathematical Logic and Foundations; Combinatorics; Order, Lattices, and Ordered Algebraic Structures
• General Algebraic Systems; Number Theory; Field Theory and Polynomials; Commutative Algebra; Algebraic Geometry; Linear and Multilinear Algebra, Matrix Theory; Associative Rings and Algebras; Nonassociative Rings and Algebras; Category Theory, Homological Algebra; K-Theory; Group Theory and Generalizations; Topological Groups, Lie Groups
• Real Functions; Measure and Integration; Functions of a Complex Variable; Potential Theory; Several Complex Variables and Analytic Spaces; Special Functions; Ordinary Differential Equations; Partial Differential Equations; Dynamical Systems and Ergodic Theory; Difference and Functional Equations; Sequences, Series, Summability; Approximations and Expansions; Harmonic Analysis on Euclidean Spaces; Abstract Harmonic Analysis; Integral Transforms, Operational Calculus; Integral Equations; Functional Analysis; Operator Theory; Calculus of Variations and Optimal Control, Optimization
• Geometry; Convex and Discrete Geometry; Differential Geometry; General Topology; Algebraic Topology; Manifolds and Cell Complexes; Global Analysis, Analysis on Manifolds
• Probability Theory and Stochastic Processes; Statistics
• Numerical Analysis; Computer Science; Mechanics of Particles and Systems; Mechanics of Deformable Solids; Fluid Mechanics; Optics, Electromagnetic Theory; Classical Thermodynamics, Heat Transfer; Quantum Theory; Statistical Mechanics, Structure of Matter; Relativity and Gravitational Theory; Astronomy and Astrophysics; Geophysics; Operations Research, Mathematical Programming; Game Theory, Economics, Social and Behavioral Sciences; Biology and Other Natural Sciences; Systems Theory, Control; Information and Communication, Circuits
• Mathematics Education

Mgnbar (talk) 16:42, 24 August 2011 (UTC)

One thing about the table which is striking is the number in probability/statistics. I've always found it odd how these are grouped under Applied mathematics in the article whereas I think they deserve a second level header. In most UK departments I've been at have been split along pure/applied/stats lines. --Salix (talk): 16:50, 24 August 2011 (UTC)

Most of these statistics degrees are trade-school theses of little theoretical interest even among statisticians. (They would be at the level a top 70 program in mathematics in the USA, say.) Few have any interest to mathematical statistics or theoretical statistics and fewer still would interest many mathematicians of either pure or applied bent, unless these mathematicians would have intellectual curiosity or scientific interests .... ;) Sensible persons generally agree that probability is a field of mathematics: Applied probability often is closer to operations research and industrial engineering than to statistics, which is the mathematical theory of scientific practice. and less of an engineering discipline.  Kiefer.Wolfowitz 12:39, 25 August 2011 (UTC)

For what it's worth, in my experience in the USA, small colleges tend to have one department (pure/applied/stats), while large universities tend to have two (pure/applied and stats) or three (pure, applied, and stats). And most of the probability theory I've seen in my life (not much, admittedly) has seemed to be quite pure rather than applied. Mgnbar (talk) 12:16, 25 August 2011 (UTC)

Kiefer Wolfowitz: In case this was not clear, I am not talking about changing the intro. I am talking about changing the section "Fields of mathematics", to use terminology that a mathematician would recognize. This can be done in a reader-friendly way. For example, the treatment of the field "Analysis" could begin "Analysis is the branch of mathematics that has emerged from the study of change." Or something about calculus. You get the idea. My point is that we can use grown-up words from the 21st Century. Mgnbar (talk) 12:16, 25 August 2011 (UTC)

Mathematicians might define analysis as the study of interesting questions, descended from infinitesimal calculus, involving topological linear spaces. But that definition would not be suitable for this article.  Kiefer.Wolfowitz 12:44, 25 August 2011 (UTC)

If that was intended as a response to what I wrote, then it misses. There is an enormous gulf between what I proposed ("change") and what you replied with ("topological linear spaces"). Mgnbar (talk) 13:27, 25 August 2011 (UTC)

Edit request from 180.242.11.235, 27 July 2011

Etymology

The word "mathematics" (jovo or java in modern language) comes from the semitic hebrew Arabic (Arabic: محمد‎ Muḥammad, pronounced [mʊˈħæmmæd], which means The Praised Man or The Honourable Man, The Man was the Mesenger of the religion of Islam(born 570, Mecca, Arabia [now in Saudi Arabia]—died June 8, 632, Medina) , and is considered by Muslims to be a messenger and prophet of God, the last law-bearer in a series of Islamic prophets, and, by most Muslims,the last prophet of God as taught by the Quran. This Original word is Mohammadika which means Your Praised Man or Your Honourable Man. Muhammad in Al-Qur'an, had been told by God to read (in Al-Alaq first verse) and to seek refuge to Lord (keeper) of Calculation (Robbilfalaq) (it means Allah) in Al-Falaq first verse.

The word mathematicis irrelevant with máthēma comes from μανθάνω (manthano) in ancient Greek and from μαθαίνω (mathaino) in modern Greek, both of which mean to learn Since The Arabian originally invent the zero system number and the Decimal system number (read Al-Qur'an see the system number for Al-Qur'an Juz) which mean the Greek (Pythagoras and Archimedes)and The Roman didn't know the Arabic system number until Muhammad spread Islam. The word Mathematic become Jovo or Java in modern time it came from the original word in Arabic محمد‎. From Al-Qur'an the Arabian muslim originally know all human Calculation is relative than God calculation so in the mathematic knowledge they use Al-Asmaul-Husna(99 names of the only one God of Islam) as the name of this knowledge such Ar-Rahim (Arithmatic, see rythm from "rhyme" and rhyme from rahim), Allah (log), Allah Ar-Rahim (logarithm), Al-Gofur Ar-Rahim (AlGorithm, some say from Al-Khawarijm) and Al-Jabar (Al-Gebra). Since this knowledge come from Arab soon people call it "mathematics" refers to Muhammad.

110.137.147.244 (talk) 14:18, 27 July 2011 (UTC)

This requires citation. The citations you offer are just numbers. What are the cited works? Mgnbar (talk) 15:24, 27 July 2011 (UTC)

All of the citations are there, you just can't see them because there's no "reflist" on this page. Also, if you just look at the info while editing, its there. However, there's no way this is going in the article, because just looking at the citations, they sure don't seem reliable (except for Britannica, but that only cites the info about Muhammad, not the info about mathematics). This is religious historical revisionism--yes, Islamic scholars invented a lot of math. That doesn't mean that somehow the term "mathematics" derives from a non-Greco origin. Qwyrxian (talk) 10:07, 29 July 2011 (UTC)

Oops; thanks for pointing out how I should have been able to see the refs. None of the three web links offered has ANY mention of math. Mgnbar (talk) 14:11, 30 July 2011 (UTC)

there are refrences/citations for this. It will added one by one, but if some editor has the refrences/citations just added in this etymology article (to complete the etymology article).

The newly added reference [6] describes the accomplishments of Muslim mathematicians, but still does not support the claim that the word mathematics comes from Arabic, as far as I can see. Mgnbar (talk) 06:33, 1 August 2011 (UTC)

One-line definition

I have been musing on the definition of mathenatics as "the study of quantity, structure, space, and change". It strikes me that I could take my camera, get out there and contrast a single human being with a crowd, compare the structure of a leaf with a network of roads, picture the clever use of a tiny volume in a yacht and the humbling vastness of the mountains, and document the changes of seasons. This would provide me material for a show "the study of quantity, structure, space, and change", and we would not recognise any of it as mathematics.

Rather than characterise mathematics by WHAT it studies (even though the list is very compact and comprehensive), I would attempt to characterise, in one sentence, HOW it does it. So my two cents: "Mathematics is the art of rigorous abstract thinking"

• art: as in artificial and artisan: human made, and a skill
• rigorous: the greek started it, then came the study of axioms, the widespread use of manipulation of symbols etc...
• abstract: the moment someone realised that sharing 5 roots or 5 fruits between 3 people is the same challenge, (taste does not matter) we were started...
• thinking: as in "thinking before acting", mathematics as a tool to conquer the world. And seen like that, mathematical concept do not preexist their invention, even though some of them are so uncommonly useful - contentious obviously!

Does it make sense?

Obviously the proposed definition touches on many elements already well discussed in the article.

Philippe Maincon (talk) 17:24, 15 July 2011 (UTC)

This has been discussed heavily. You raise good points, but your points seem like original research to me. Unfortunately, so does the current one-line definition in the article. It lacks citation. At the risk of beating this to death yet again, let me give two definitions of mathematics with citations:

• "Mathematics is the science of quantity and space" (p. 6).
• "The study of mental objects with reproducible properties is called mathematics" (p. 399).

The source is The Mathematical Experience, winner of the American Book Award and a classic of popular mathematics. Mgnbar (talk) 17:38, 15 July 2011 (UTC)

The battle over the lead occupied a great deal of time and effort, and many sources were cited, which appear in the article. Generally, the lead reflects the article, and the footnotes appear in the body of the article. The "quantity, structure, space, and change" formulation was a compromise, and is reflected in several other articles, and in the subsections of this article.

For what it is worth, I was on the opposing side of the battle, favoring, "Mathematics is the body of knowledge discovered by pure reason, as contrasted with science, the body of knowledge discovered by experiment." This formulation lost, because most major reference works define mathematics by the subjects it studies, rather than by its method.

It may be that someday the lead may be changed, but anyone attempting such a change needs to prepare for strong resistance. Rick Norwood (talk) 12:53, 19 July 2011 (UTC)

I'm a fan of both "Mathematics is the science of quantity and space" and of well-sourced defns., but all things considered the current version is a pretty stable compromise. The proposed version is much too broad to define math. Philosophers, for example, would surely claim they do the same thing, and no mathematician would describe what he did for a living as "Think rigorously abstractly." JJL (talk) 17:00, 19 July 2011 (UTC)

Rick Norwood: I know that this has been battled; I said so twice in my post. Your comment that the lead reflects the article is spot-on. I have always viewed the overarching article structure of "quantity, structure, space, and change" as a gross violation of NOR. (How many mathematicians would recognize "change" as a "field of mathematics"?!) In the past, I have proposed that the article instead categorize fields of math based on the American Mathematical Society's (or other similar organization's) categories for new Ph.D.s: logic, algebra/combinatorics/number theory, geometry/topology, analysis, applied, etc. This idea gained no traction. Mgnbar (talk) 17:55, 19 July 2011 (UTC)

There are many competing definitions of mathematics. Some, like most definitions of topics, define mathematics by the things it studies; others by how it studies them. It's not Wikipedia's place to settle the matter. —Ben Kovitz (talk) 03:38, 23 August 2011 (UTC)

Peirce: "The science that draws necessary conclusions" (1870)

Charles Sanders Peirce's New Elements of Mathematics has an thorough and stimulating discussion of previous definitions of mathematics.  Kiefer.Wolfowitz 10:49, 29 July 2011 (UTC)

What's your point? The last time I checked, Peirce's view was mentioned. What else do you want? Peirce is a fascinating figure, but his outlook is very far from universally shared among experts. --Trovatore (talk) 21:00, 29 July 2011 (UTC)

It seems that his point was to directly respond to the issue of defining mathematics, by pointing to a discussion of definitions of mathematics. See page 25 of the linked material. Kiefer Wolfowitz's comment is exactly relevant. Mgnbar (talk) 14:02, 30 July 2011 (UTC)

It's a book by Peirce. Peirce is not a neutral observer; he's a proponent of a particular POV. That POV should be mentioned, and indeed, it is mentioned. --Trovatore (talk) 21:15, 30 July 2011 (UTC)

Peirce's discussion of definitions, on page 25 and even earlier beginning on page 3, discusses the history of older definitions better than this article, and therefore it is a resource for somebody wishing to improve this article to 1900 standards.  Kiefer.Wolfowitz 22:46, 30 July 2011 (UTC) Peirce criticizes the Kantian definition of mathematics, which devolved via De Morgan & Hamilton and others to this articles's shopping list and political compromise.

C.S. Peirce's definition is "mathematics is ... the study of hypotheses, or mental creations, with a view towards drawing necessary conclusions" (p.4) He demolishes the shopping-cart definition (parroted still on Wikipedia) on pages 5-7, as (in conclusion) "probably ... the very worst". K.W. 03:20, 31 July 2011 (UTC)

Anything other than a "shopping-cart definition" is necessarily going to be POV. The ideal solution would be simply to dispense with "defining" mathematics in the lead, and leave the various definitions people have proposed, all of which carry philosophical baggage, to later in the article. Everyone knows, roughly speaking, what mathematics is, and nothing we can put in the lead is going to be better than roughly speaking in any case. So it's a useless exercise to try to define it at all in that location.

Unfortunately, purely pro forma, we have to put something there; it's expected. I think the current definition does about as little damage as anything that has been proposed. --Trovatore (talk) 04:08, 31 July 2011 (UTC)

Peirce's definition has never had widespread acceptance. Here is Florian Cajori mentioning dissatisfaction with it in 1919. Any attempt to characterize the essence of mathematics takes a side in a long-running philosophic controversy; see Definitions of mathematics for a small sample. I think we are wise to follow the other standard reference works, which have also opted for a "shopping-cart definition". Whatever its flaws, our definition lays out the topics covered in the body of the article, which is the most you can realistically hope for. —Ben Kovitz (talk) 03:26, 23 August 2011 (UTC)

Definition as description vs definition as demarcation

Looking over the above exchanges, I find that I have left some things unexplained that I now believe I know how to say more clearly, and on a related note, that I also do not like the first paragraph as it stands. (Well, I never did really, I just thought it was "least bad", but I now find that it has evolved in a way that I think is suboptimal, and I have a candidate point in the past where I think the text was better, that we should consider a starting point.)

What is a definition? In mathematics itself, our definitions, say of a "ring" just for example, provide precise demarcations. They divide things into rings and non-rings, with nothing in between (ignoring quibbles about whether you require a unit). It's not unnatural that mathematicians would like to be able to provide that sort of a definition for mathematics itself.

Perhaps such a project is possible; I am skeptical, but for the sake of argument suppose that it is possible to define "mathematics" in such a way that it precisely demarcates that which is mathematics from that which is not mathematics. We are left with the problem that no such definition is agreed among mathematicians or philosophers of mathematics. It is not Wikipedia's function to pick one from among them. We simply may not do that; it is a blatant violation of WP:NPOV.

To preserve neutrality, we could futz around with competing definitions and say who uses them. But I hope everyone agrees that the lead is not the place for that. The article is about mathematics, not about how to define the term mathematics, and more than two or three sentences is too much to spend on the definition in the lead section.

But luckily, we don't have to. All we have to do is recognize that the sort of "definition" required in the lead paragraph is not a demarcation at all. The first sentence of dog does not give you the information required to divide all objects into dog and non-dog, and it can't be expected to. Rather, it gives you enough information to identify what the article is talking about, and possibly tells you something you might not have known about what is included. (Are dingoes dogs? Does mathematics deal with things other than numbers?)

That's why an NPOV definition in the lead will necessarily have a "shopping-cart" aspect to it. It can't demarcate what is mathematics from what is not, but it can say some things that we all agree are mathematics, including some that some readers may not have realized are mathematics. --Trovatore (talk) 07:50, 1 August 2011 (UTC)

The better historical lead

Here are the first three sentences as they stand: (I'll leave the refs but won't put a reflist):

Mathematics (from Greek μάθημα (máthēma) — knowledge, study, learning) is the study of quantity, structure, space, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity.

Here is the version I like better, from 1 February 2009:

Mathematics is the academic discipline, and its supporting body of knowledge, that involves the study of such concepts as quantity, structure, space and change. Some practitioners of mathematics maintain that mathematics is the science of pattern, and that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions".

Here's the main reason I like it better: The current version pretends to be a demarcation. It lists four (vague) things; those are mathematics, nothing else is. As I argue above, that cannot possibly meet NPOV. The Feb 2009 version does not; rather, it lists some of the things that mathematics studies, without claiming to exhaust the subject.

That's not to say it can't be improved. I would change "the academic discipline" to "an academic discipline", making the non-demarcative nature more explicit. Also I would probably incorporate some of the language of the current version as well; I have no detailed proposal at this time for how to do that. But I hope my main point is clear. --Trovatore (talk) 07:50, 1 August 2011 (UTC)

I'm only going to address one of your points: "academic discipline" was discussed, and it was pointed out that while many mathematicians work in academia, many others work in industry, and historically there were many "amateur" mathematicians, and therefore "study" is better than the more limited "academic discipline". Rick Norwood (talk) 12:36, 1 August 2011 (UTC)

Overall I agree with Trovatore and I prefer the Feb 2009 version. Precise definitions about the practice of math are not necessary or even possible. (Topology as a math concept is precisely defined, but topology as a discipline of math is only vaguely defined.) The Feb 2009 version offers several views, as concisely as possible, which to me is ideal. I agree with Rick Norwood that the "academic" is undesirable. Mgnbar (talk) 13:13, 1 August 2011 (UTC)

Yes I prefer the 2009 version, which seems to capture most views on what mathematics is.--Salix (talk): 13:48, 1 August 2011 (UTC)

I applaud Trovatore's clear and stimulating discussion today. However, the 2009 definition is problematic because of several issues

1. the academization error, as noted by Mgnbar and by Rick Norwood (who referenced archived discussions here).
2. "Rigorous deduction": It is clear that mathematical proofs aim at convincing other mathematicians and at meeting the contemporary standards of rigor, but non-logical mathematical proofs fall far short of the rigor displayed in proofs in philosophical logic, especially in mathematical logic (or logic programming with relevant intuitionist logic or other paradises ...).
3. "Some practitioners of mathematics": should be "some research mathematicians and philosophers of mathematics". ("mathematical practitioners" include "junior high-school teachers of mathematics and mathematical educationalists", which are unreliable sources here).
4. The intentional definition from Benjamin Peirce should be upgraded to the definition by C.S. Peirce, noted above.
5. There should be the mention of mathematics as "the science of quantity" (sic.), which was popular in 19th century German and has been embalmed in many "social sciences". This is the definition of mathematics most familiar to the general public (who remember their time with arithmetic and plane geometry (and perhaps $\mathbb {R} [x,y]$ or univariate calculus). This definition's inadequacy became apparent with the rise of projective geometry and topology.
6. It should mention that mathematics is dynamic: It develops by adding new fields, like mathematical statistics (Laplace, Gauss, and Karl Pearson, and for the needs of physics, biology, and insurance), or functional analysis and operator theory (for the needs of physics and classical mathematics, e.g. differential & integral equations). The fields of set theory and general topology were developed to unify and better understand the foundations of other mathematical disciplines, illustrating the importance of abstraction/abduction/retroduction/hypothesis formation. It also drops fields like projective geometry and summability theory, which were central to mathematics c. 1870, but now are studied as examples of more general theories; invariant theory has made a comeback.
7. The definition focuses on pure mathematics, which has been practiced by a rather narrow group of mathematicians beginning in the 20th century. The best mathematicians, like Kolmogorov and von Neumann, contributed to the development of science and modern technology. I believe that the NSF noted c. 1996 that most mathematics covered by mathematical reviews were in non-mathematical academic departments. Some discussion of vital applied-mathematics, e.g. by Kolmogorov or Arno'ld or Atiyah or Lax etc., should appear.

Those caveats aside, the 2009 definition is better because it clearly notes the traditional major fields of mathematics (analysis, algebra/number theory, geometry & topology) in popular versions "change, structure & quantity, space", as conventional.

Despite my criticisms of the 2009 definition, it has several virtues:

• It signals the incompleteness of this shopping-cart definition of mathematics with the phrase "such as". That incompleteness then motivates perhaps the most prominent, accessible definition of mathematics, Benjamin Peirce's (which should be replaced by C.S. Peirce's).

Thanks,  Kiefer.Wolfowitz 17:41, 1 August 2011 (UTC)

P.S. popularizers of mathematics like Steen and Devlin (who notoriously discussed "Bayesian mathematics" (sic.) on Science Friday would seem to be not the highest quality sources. What about definitions from Saunders MacLane or von Neumann or Kolmogorov (e.g. in his leadership of the 3 volume survey of mathematics) or Peter Lax or Garrett Birkhoff or Michael Atiyah, etc.? (The younger Fields Medalists Tim Gowers or Terrence Tao are smart and have written popular treatments of mathematics, but they are not yet as broad or as profound as the fellows I mention, who have led the development of whole fields of mathematics.)  Kiefer.Wolfowitz 17:35, 1 August 2011 (UTC)

I agree with your criticisms of the Feb 2009 lead (except #4). Here are a couple more. (1) Quoting Peirce's definition in the lead makes it sound more widely accepted than it is. (2) "Some practitioners" introduces a philosophical argument even before we've said much of anything about math. I do like that the Feb 2009 definition is worded to suggest that the "shopping cart" does not exhaust the subject. Maybe pursuing that idea further could lead to really good first para, simultaneously (loosely) demarcating the subject and introducing it. —Ben Kovitz (talk) 04:02, 23 August 2011 (UTC)

I'm really against demarcating it whatsoever in the opening paragraph. --Trovatore (talk) 04:22, 23 August 2011 (UTC)

Oops, I thought we were in agreement. Maybe I confused matters by misusing the word "demarcating". I'll try explaining what I think is both our concern again, a little differently. Normally an article leads with a definition of the topic, telling some kind of essential characteristic that sets it apart. With mathematics, though, all attempts to define it are controversial. But, there is near-universal agreement about what subjects mathematics includes. So, we can serve the same purpose as an ordinary definition by listing some subjects within mathematics. But, no list could cover all of mathematics, so it would be wrong to present it as if it were exhaustive. Is that your concern regarding demarcation? I'm proposing to provide a list but word it so it's clearly just some examples of subtopics to give the flavor, with no pretense at exhaustion. Maybe something in the spirit of, "Mathematics is the study of stuff like numbers, shapes, spatial relationships, correspondences between sets, probability, symmetry, paths through networks, approximation, infinity, ways to shuffle groups of things, and, well, you get the idea, eh?" OK, not so long and not so informal, but you get the idea, eh? —Ben Kovitz (talk) 15:17, 23 August 2011 (UTC)

As unsatisfactory as the "quantity, structure, space, and change" list is, it is now embedded not only in this article but in several other Wikipedia articles. If we want to avoid giving different "definitions" in different articles, we need to either change it everywhere, or keep what we've got. I don't think "qss&c" is so bad. How about just adding "such as" in front of the list? Rick Norwood (talk) 14:03, 24 August 2011 (UTC)

Instead of saying "which are arguments sufficient to convince other mathematicians of their validity", it should something along the lines of "which are arguments sufficient to logically imply some mathematical statement. Of course, interpretation of these arguments are subject to human error, as in any other field." The former quote almost implies that mathematicians only convince, and don't prove. Consider this page http://en.wikipedia.org/wiki/Physician. It has the line "...which is concerned with promoting, maintaining or restoring human health through the study, diagnosis, and treatment of disease, injury and other physical and mental impairments" however, it doesn't add "Any method of restoring health is not certain, but only has convinced other doctors that it will restore health." — Preceding unsigned comment added by CBribiescas (talk • contribs) 16:06, 27 September 2011 (UTC)

Edit request from Knwlgc, 13 September 2011

{{edit semi-protected}} After " Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered.[10]" add "At times it is difficult to know where pure mathematics ends and applied mathematics begins." Knwlgc (talk) 05:12, 13 September 2011 (UTC)

Well, it's a true statement; I doubt anyone here really wants to argue that the line between pure and applied mathematics is sharp. But what's your rationale for adding it at that point in the text? What problem are we trying to solve? --Trovatore (talk) 23:28, 14 September 2011 (UTC)

The preceding sentences in the article introduce the concepts of 'pure' and 'applied' mathematics. Suppose you are someone unfamiliar with either the concept of 'pure' or 'applied' mathematics, then having read a brief statement about both you might be curious as to their relation with one another. In some way mathematics as a whole is characterized by the relation between its parts, and, having come to the article to read about mathematics as a whole, it would be reasonable to assume you might want to know a bit about the relation between the specific parts 'pure' and 'applied' mathematics. Since it would seem relevant to make a brief comment on the nature of their relation with one another, I suggested a statement which is seemingly unobtrusive, undoubtable, vague, but which points towards the complex relation between pure and applied mathematics. My goal is this: paint a picture of mathematics that is enticing and suggestive enough that someone reading a wikipedia article might want to know more about mathematics as a whole or about the relation between its parts. Knwlgc (talk) 17:54, 16 September 2011 (UTC)

I think that it's reasonable to add. A non-mathematician may not know that the line between pure and applied is fuzzy, because they may not realize that disciplines of math are fuzzy at all. I'd like a citation though. Mgnbar (talk) 19:53, 16 September 2011 (UTC)

In the preface to Keener's Principles of Applied mathematics he states "Much of applied mathematical analysis can be summarized by the observation that we continually attempt to reduce our problems to ones that we already know how to solve." While this is not the same as saying "At times it is difficult to know where pure mathematics ends and applied mathematics begins," it has some significant baring on this statement. Many of the problems which "we already know how to solve" are both 'pure' and 'applied' and it may not be clear where pure mathematics ends and applied mathematics begins in any such reduction. I should also say that I am not an expert in either 'pure' or 'applied' mathematics and thus do not have access to the knowledge needed to prove the statement "The line between pure and applied mathematics is vague." either true or false (or if it is anything more than nonsense).Knwlgc (talk) 23:35, 16 September 2011 (UTC)

I now believe that my statement may fall into the category of "original research" as I can not find a direct source which states clearly or precisely "The boundary between pure mathematics and applied mathematics is vague". Since I do not have experience identifying what is or is not O.R. I am willing to discard my edit request.Knwlgc (talk) 23:35, 16 September 2011 (UTC)

Procedural note: I'm removing this 'edit semi-prot request' for now, pending consensus  Chzz  ►  01:37, 17 September 2011 (UTC)

My request still exists, it has not gone away. It was assumed when I sent in my edit request that there was not consensus (otherwise this article would not need semi-protection), how has the state of my request changed since its conception? I should add that an admission of willingness to discard my edit request, is not a request to discard my edit request (had it been then I would have discarded it).Knwlgc (talk) 03:08, 17 September 2011 (UTC)

Since everyone seems to agree that your statement is true, I've shortened it slightly and added it as a clause in the final sentence. Rick Norwood (talk) 11:52, 17 September 2011 (UTC)

The sentence "Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, but there is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered." is a run on sentence. Perhaps one could change it to "Additionally, mathematicians engage in pure mathematics without having any application in mind. Though, there is no clear distinction between pure and applied mathematics as pure mathematics sometimes has practical applications."Knwlgc (talk) 16:52, 17 September 2011 (UTC)

I disagree that it is a run-on sentence; I have no trouble parsing it. But the sentence is quite long. So I have broken it into two pieces in the article. Mgnbar (talk) 22:11, 17 September 2011 (UTC)

I am not a grammarian: my claim that the sentence was a run-on has no baring on whether the sentence was or was not a run-on. I apologize for any confusion my comment may have caused.Knwlgc (talk) 06:05, 18 September 2011 (UTC)

There is no clear line separating run-on sentences from non-run-on sentences, and sentences that began as short sentences often turn into run-on sentences by subsequent edits, but occasionally editors disagree as to at which point this actually happened. Marc van Leeuwen (talk) 09:46, 21 September 2011 (UTC)

Marc, I think we need to add that to the page on run-on sentences. I hope it's not semi-protected. Knwlgc (talk) 00:22, 27 September 2011 (UTC)

Unintended article misinformation? This perspective from field leadership.

The work in the Centre for Experimental and Constructive Mathematics is more than twenty years ahead of Mathematics as an international discipline. This has been inside information which might now be public.

It seems as though there could have been some confusion about what Applied Mathematics was. Generally, Applied Mathematics has been a variety of subsets of ad hoc amalgamations of Theoretical Physics, Statistics, Computing Science, Mathematics, Engineering Science, and almost anything else. The common denominator clarifying what Applied Mathematics has been was tricky to find. If/when ad hoc disciplines (an oxymoron) try existing primarily as theoretical constructions built for the purpose of trying to get money any old way, the result is internal organizational inefficiency. It would be unfair and dreamy of Mathematics, as the international discipline this is, to ask other organizations to have internal cohesion due to our recent update of the definition of if, without first demonstrating what we mean by internal cohesion.

(Instantiations and examples differ slightly: instantiations are generalizable and examples may have generalizable properties and features. However, part of this work includes teaching mathematics to seven billion people, thus for now I preferentially use the word example.)

An example of confusion arising from lack of internal organizational cohesion due to presence of ad hoc discipline, is the 50% vote of support Jonathan and Peter Borwein received from participating voters (abstention rate unknown) for establishing the Centre for Experimental and Constructive Mathematics at Simon Fraser University, a tie which was broken in preference of establishing Experimental and Constructive Mathematics by someone in senior administration circa 1992, and the upshot of which includes the Organic Mathematics Project which singularly redefined Mathematics online education and collaboration; correction of Aristotle; redefinition of the word if; free demonstrative and instructional tutoring services for the world's central banks with respect to the additive and multiplicative identities; open questions including where Mathematics proofs come from, ownership of intellectual property in collaborative processes, how Mathematics and Mathematically informed disciplines develop communities; and the intellectual property ownership question: who owns Mayer Amschel Rothschild's intellectual property conceived circa 1794, still in circulation, and which I previously cited in my work as osmosis.

Generally and obviously, a discipline is not yet qualified to offer its services to customers until after the discipline has demonstrated the same expertise internally. Having organizations' internal and external services match works. The Centre for Experimental and Constructive Mathematics and our network is perfect for driving optimization by osmosis and naturally occurring, real selection processes. Therefore handling the question << what is Mathematics >> is part of this constructive instruction. This exemplifies what Applied Mathematics really is, both in this self-referencing demonstration and explanatory definition update, and in real world ubiquitous application across all disciplines everywhere; therefore we acknowledge Mathematics was previously domesticated partly under Philosophy and partly under Science, and might be correctly understood as a profession under the Institute for Electric and Electronic Engineers, who as an organization has the highest standards in ethics and professional conduct. This Applied Mathematics includes Information Theory and Computer Architecture. Having the discipline Mathematics perfectly located under the IEEE solves all problems related to franchising Mathematics other than my unique personal problem if The Rothschild Family prefers to take me to court for accidental intellectual property theft.

References:

http://www.cecm.sfu.ca/

http://www.cecm.sfu.ca/organics/project/

http://www.ieee.org/index.html

Founder by Amos Elon, ISBN 0 670 86857 4

JenniferProkhorov (talk) 19:23, 19 October 2011 (UTC)

I don't understand. In any event, this talk page is specifically for discussing changes to the Wikipedia Mathematics article. It is not a general discussion forum about mathematics or its philosophy. So please describe what concrete changes you wish to make to this article. Mgnbar (talk) 22:04, 19 October 2011 (UTC)

mathematical science

Does mathematics really belong to the mathematical sciences? Mathematical sciences says

• "Mathematical sciences is a broad term that refers to those academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper"
• "Computer science, computational science, operations research, cryptology, econometrics, theoretical physics, and actuarial science are other fields that may be considered mathematical sciences."Brad7777 (talk) 15:30, 6 November 2011 (UTC)

The question is, what is mathematics? I think mathematics is that body of knowledge arrived at by deduction from axioms. But the dictionary disagrees with me, and says mathematics is the study of numbers and geometry. Wikipedia says mathematics is the study of quantity, space, structure, and change. The Wikipedia definition was a compromise, not perfect, but nobody wants to open that particular can of worms again.

One moderately authoritative list of what mathematics is is the AMS subject classification, which lists 00A06 Mathematics for nonmathematicians (engineering, social sciences,etc.); 03A10 Logic in the philosophy of science; 03B70 Logic in computer science; 08A70 Applications of universal algebra in computer science; 35Q68 PDEs in connection with computer science; 35Q91 PDEs in connection with game theory, economics, social and behavioral sciences; 35Q92 PDEs in connection with biology and other natural sciences; 46B85 Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science; 46N60 Applications in biology and other sciences; 47N50 Applications in the physical sciences; 47N60 Applications in chemistry and life sciences; 62P05 Applications to actuarial sciences and financial mathematics; 62P10 Applications to biology and medical sciences; 62P25 Applications to social sciences; and whole sections on 68-XX COMPUTER SCIENCE; 90Bxx Operations research and management science; 91-XX GAME THEORY, ECONOMICS, SOCIAL AND BEHAVIORAL SCIENCES; and 92-XX BIOLOGY AND OTHER NATURAL SCIENCES. And that's not even getting into the final section on math ed. What does that tell us?

It seems to say that we should differentiate between mathematics, and applications of mathematics. It also seems to suggest that computer science, game theory, economics, social and behavioral science, and biology and other natural science are close enough to mathematics to get sections of their own. And where the hell do you put statistics?

Once you reject my definition, and try to define mathematics by the subjects it investigates, I think the task is hopeless. Try to tell me a subject that is not investigated using mathematics.

Rick Norwood (talk) 18:43, 6 November 2011 (UTC)

Mathematics isn't primarily mathematical in nature. It is COMPLETELY mathematical in nature.

Mathematical Symbols: Clickability,Hyperlinking, Own Pages

I don't know of a better place to discuss this than the wiki page for Mathematics, although the question pertains to any page under the section of Mathematics.

Can we implement mathematical symbols that are ALSO hyperlinks to wiki pages for each symbol?

I do not think all mathematical symbols have wiki pages, but I don't see why not.

At least they could link to a relevant page in which the symbol is heavily used.

I believe this would make it significantly easier to learn mathematics from wikipedia. — Preceding unsigned comment added by 140.247.59.253 (talk) 23:45, 27 July 2011 (UTC)

List of mathematical symbols gives a extensive list of symbols.--Salix (talk): 08:03, 28 July 2011 (UTC)

The idea is not just to have a list of symbols, but to use hyperlink versions of the symbols on any or all wiki pages within Mathematics. Like many special terms, the first usage of a symbol on any wiki page could be a hyperlink version of the symbol. Much in the same way that unique terms can be clicked on to bring the wiki-reader to the definition of that term, so too could she more quickly learn about the mathematical symbols that crop up in whatever section of mathematics she is currently browsing. — Preceding unsigned comment added by 140.247.59.84 (talk) 15:29, 9 August 2011 (UTC)

Sounds great: do you have an implementation available?Knwlgc (talk) 06:12, 18 September 2011 (UTC)

I don't know HTML really, but if you look at List of mathematical symbols many are hyperlinks, i.e. =. We could change the first usage of any symbol on any wikipage to such a hyperlink version to the symbol's page, and make a new page for it if it does not already exist. Are there wikipedia guidelines regarding such a change? Personally I don't see any reason not to. — Preceding unsigned comment added by 140.247.59.29 (talk • contribs)

Seems like a good way to build the web. No need for HTML— wiki markup is pretty easy to use. For example, [[π]] makes a link to π. Also, on talk pages, four tildes ~~~~ "signs" your comment, adding a date/time stamp.

@others, is there a better place to promote this idea? Seems like the kind of thing that would go well if multiple people start picking at it. __ Just plain Bill (talk) 01:21, 30 November 2011 (UTC) see below Just plain Bill (talk) 02:51, 30 November 2011 (UTC)

I'm sure we've had this discussion before as I remember thinking it a very bad idea at the time, for various reasons.

• symbols are often very small, so links are difficult to see.
• it doesn't work with PNG formulae generated by LaTeX rendering, which is used very often as either editor preference or because the formulae are too difficult to render in HTML
• Some symbols like ≤, ≥, ±, ⊆, ⊻ are already 'underlined', while adding links to others will change their meaning, to make them look like those or other underlined symbols (links always underlined is a user option).

If a symbol really needs explaining then add a sentence, e.g. from Euler's formula:

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x,

$e^{ix}=\cos x+i\sin x\$

where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians.

This is far clearer than linking any symbol.--JohnBlackburne^words_deeds 02:34, 30 November 2011 (UTC)

Even better. Was unaware of previous discussion, and those points seem valid. __ Just plain Bill (talk) 02:51, 30 November 2011 (UTC)

Quotations paragraph

Why do we have long paragraph of quotations about mathematics in the lead? Shouldn't that go in Wikiquote? Kaldari (talk) 22:18, 4 December 2011 (UTC)

Mathematosis?

How about adding "Mathematosis" to the "See Also" section? 164.107.189.191 (talk) 14:37, 6 December 2011 (UTC)

Because there is no such article. There once was but it was deleted: Wikipedia:Articles for deletion/Mathematosis.--JohnBlackburne^words_deeds 15:23, 6 December 2011 (UTC)

Galileo's Death Year

It's nice to see this right in the first paragraph of a significant article: "Galileo Galilei (1564-1942) said" I never knew the man lived to be almost 400. Good job, Wikipedia. And the article is locked so I can't even fix this boneheaded error. Ugh. — Preceding unsigned comment added by 131.193.127.17 (talk) 16:03, 6 December 2011 (UTC)

Fixed, thanks !--JohnBlackburne^words_deeds 16:10, 6 December 2011 (UTC)

That's the downside of "protecting pages". Protecting pages can "protect" vandalism, yet it can also "protect" from perfectly harmless and beneficial contributions. Creating an account is one way to circumvent this issue, but if you're going to edit one freakin' little typo rather than being a lifelong editor, there is no incentive to register, much less contribute. 164.107.189.191 (talk) 16:58, 6 December 2011 (UTC)

"However, mathematical proofs are less formal and painstaking than proofs in mathematical logic."

hi.

I would replace

"However, mathematical proofs are less formal and painstaking than proofs in mathematical logic"

with

"Mathematical proofs are written in a formal language provided/analysed by mathematical logic".

The main reason for this exchange is that mathematical logic is itself a part of math! Therefore, the above statement means something like "trains are faster than TGVs". It is just nonsense. Another reason is that proofs in e.g. algebra are just as formal and painstaking as proofs in mathematical logic...

best regards a phd-student in math — Preceding unsigned comment added by 138.246.2.177 (talk) 17:48, 20 December 2011 (UTC)

agreed. as a computer programmer i would also like to note that the statement is nonsensical in that they both use a completely strict and formal grammar, and it is the grammar and rules thereof that determines the formalness, not what is said with it, so to say one is more or less formal than the other is absurd. one may have more letters or steps in one proof vs another, but in either case each step of the proof is no less a faithful application of the grammar rules than any other. anyways, if the change hasn't been made yet, i'm going to make it. Kevin Baas^talk 17:59, 20 December 2011 (UTC)

Hold on a minute here. The text as given is certainly bad, but the proposed "correction" is if anything even worse. Proofs are not to be identified with formal proofs. Proofs are arguments directed at human mathematicians (including oneself); they are not ordinarily in any formally specified language.

The reason the existing text is bad is that mathematical logic is a branch of mathematics, and proofs in mathematical logic need not be any more or less formal than in any other branch. In that sense the IP contributor is right. But the proposed correction is wrong. --Trovatore (talk) 18:12, 20 December 2011 (UTC)

alright, suggestions? Kevin Baas^talk 18:33, 20 December 2011 (UTC)

I'd just remove the sentence outright. I think the idea it was trying to convey is precisely the opposite of what it says now, namely that mathematical proofs are not ordinarily completely formal. But it did a bad job of that, and we don't need that idea at that point in the text, because there's nothing there it's contrasting with. --Trovatore (talk) 18:59, 20 December 2011 (UTC)

Mathematical proofs really are less formal and painstaking than proofs in mathematical logic. Open any math book to a proof. I'll pick one at random off the shelf behind me, and open it to a random page. "Proof: Recall that a subspace Y of L is said to be convex if for every pair of points a, b of Y with a < b, the entire interval [a, b] of points of L lies in Y." I think this is fairly typical of how a mathematical proof is written. Now, compare with a proof in mathematical logic:

• (1) (~B->(~A->~B)) (L1)
• (2) (~A->~B)->(B->A) (L3)
• (3) (~B->(B->A)) (1),(2) HS.

More formal. More painstaking.

Mathematicians usually assume that the kinds of proofs we do in our work could, if necessary, be reduced to mathematical logic, but we never, in practice, do that.

Rick Norwood (talk) 19:13, 20 December 2011 (UTC)

The problem is the phrase "proof in mathematical logic". The term mathematical logic doesn't normally mean things like your lines (1), (2), (3); it normally means set theory, model theory, recursion theory, and proof theory. Those are branches of mathematics, and proofs in those branches of math are not really different in character from proofs in, say, differential topology. --Trovatore (talk) 19:19, 20 December 2011 (UTC)

Mathematical logic usually means mathematical logic, as in Hamilton's Logic for Mathematicians or Manin's A Course in Mathematical Logic. The other topics you mention are in Foundations, rather than in Mathematical Logic. I agree that the proofs in essentially all areas of mathematics except formal mathematical logic are in the metalanguage rather than in the object language. Rick Norwood (talk) 19:53, 20 December 2011 (UTC)

I tend to agree with trovatore that the difference between logic and other fields of mathematics is not the level of painstaking care and/or formality (the idea of a logician as some kind of an OCD is an erroneous one), but rather in the field of investigation. Tkuvho (talk) 20:02, 20 December 2011 (UTC)

The main point here is that both the grad student and the computer programmer above are confused about the relationship between formal mathematical logic and ordinary mathematical proofs. Mathematical proofs are almost never written in formal language, rather they are written in the metalanguage. And the are almost never analyzed using mathematical logic, they are analyzed by a mathematician trusting his ability to think logically. Rick Norwood (talk) 20:06, 20 December 2011 (UTC)

The term mathematical logic will not work here. I'm sorry, but it means what I said and does not mean what you said. Other than that I basically agree with you. --Trovatore (talk) 20:23, 20 December 2011 (UTC)

Sorry, I'll amend that. It can mean what you said. It just doesn't usually. The main use of the term is as in The Handbook of Mathematical Logic. --Trovatore (talk) 20:27, 20 December 2011 (UTC)

there is no confusion here. i know what is a mathematical proof and what is not. and i understand perfectly the relationship between formal mathematical logic and ordinary mathematical proofs. and i couldn't care less what a few nondescript sloppy pompous self-described "mathematicians" have to say about there oxymoronicly non-rigourous "mathematical proofs". Kevin Baas^talk 21:33, 20 December 2011 (UTC)

when ever i hear "proof" in mathematics i think "formal proof". after all, the whole point of mathematics is to formally prove or disprove propositions. (well, there's applications outside of that, obviously, but you get the point.) if you want to talk about some loose logic in an informal grammar and call it a "proof", well that's your perogative, but from the first time i learned about mathematical proofs (in middle school, mind you - i was an advanced student), we never did it that way. (always w/formal grammar. we wrote it in informal grammar in the left margin, yes, but the formal grammar, or at least exactly stating the official name of the rule applied, was mandatory.) and i will never trust a mathematician whose "proofs" cannot be directly translated into a formal grammar and shown to be valid and consistent. and if i were a math teacher, they'd get an F.

having said that, i agree with what someone said earlier that it might be best to just remove the sentence altogether. Kevin Baas^talk 21:19, 20 December 2011 (UTC)

03-XX   Mathematical logic and foundations 
 03-00   General reference works (handbooks, dictionaries, bibliographies, etc.) 
 03-01   Instructional exposition (textbooks, tutorial papers, etc.) 
 03-02   Research exposition (monographs, survey articles) 
 03-03   Historical (must also be assigned at least one classification number from Section 01) 
 03-04   Explicit machine computation and programs (not the theory of computation or programming) 
 03-06   Proceedings, conferences, collections, etc. 
03Axx  Philosophical aspects of logic and foundations 
03Bxx  General logic 
03Cxx  Model theory 
03Dxx  Computability and recursion theory 
03Exx  Set theory 
03Fxx  Proof theory and constructive mathematics 
03Gxx  Algebraic logic 
03Hxx  Nonstandard models [See also 03C62]

previous unsigned comment by User:Rick Norwood 21:25, 20 December 2011 (UTC)

thanks for that, rick. see? "Proof theory and constructive mathematics" one section. notice the absence of a separate, independant section on "Proof theory and constructive mathematics for mathematical logic, specifically, which is for some reason different". Kevin Baas^talk 21:27, 20 December 2011 (UTC)

Kevin_Baas: while Travatore and I disagree on some things, we both understand the difference between a "formal proof", which can be checked by a computer program, and most mathematical proofs, which cannot be checked by a computer program. Yes, most writers of mathematics use formal grammer, in the sense that we use the grammar of authors, not of twitter. But in mathematics "formal proof" has a more technical meaning. It doesn't mean the same thing as "rigorous". It has to do with the distinction between an object language, in which the formal proof is written, and a metalanguage, in which this paragraph is written. The point on which we disagree is over which of the topics on the AMS list fall under "mathematical logic" and which under "foundations". Rick Norwood (talk) 21:29, 20 December 2011 (UTC)

I am not saying that a mathematical proof must be written in a formal grammar. i am saying that it must be capable of being faithfully and unambiguously represented in one. Kevin Baas^talk 21:35, 20 December 2011 (UTC)

I reverted your change because, though you say one thing above, the change you made says another. We need to either omit this entirely or find a way of saying it that is both intelligible to the layperson and mathematically accurate. Rick Norwood (talk) 14:14, 21 December 2011 (UTC)

So, Kevin Baas, you agree that a typical mathematical proof is not "written in a formal language provided/analysed by mathematical logic", but merely could be written in such a language? So the proposed change, as written, is incorrect? Mgnbar (talk) 14:42, 21 December 2011 (UTC)

Ah, and therein lies the danger of insufficient rigor. Where I to take off my proverbial mathematics hat for a while, I would say to the first sentence, "close enough". In the second sentence, however, you put forth a causal proposition so i have to put it back on, and say, which proposal, and how does that follow? Also i think the sentence should just be removed altogether. It doesn't do anything there anyways, besides add confusion, that is. Kevin Baas^talk 15:41, 21 December 2011 (UTC)

I see, the one offered by anon. Well, you got a lot of subjective words there. For instance, if he would have said "formal grammar" it would'be been stricter. "formal language" however, could just be normal everyday english, but where care is taken in one's communication to unambiguously present spatial relations. In that case, i'd say no definition of "mathematical proof", however broad, would include a situation where the proof is less rigourous and sound than any other. And that is where I take issue with the original sentence. TYhis is math here. It's either 100% proven, or 0%. There is no middle ground. Kevin Baas^talk 15:49, 21 December 2011 (UTC)

And you see now the dangers in using informal language (<-langugage here, not grammar) when describing formal concepts. Case in point of something ambigiuous which by my criteria would ipso facto not qualify as part of a valid mathematical proof. Kevin Baas^talk 15:56, 21 December 2011 (UTC)

In the mathematical sense of the phrase "formal language", it is not possible to be "more formal" or "less formal". A "formal language" is one where the proofs depend only on the form the symbols take, and not on the meaning of the symbols. If you prefer "formal grammar", that is also used in the same sense. I'm primarily following Hamilton's Logic for Mathematicians.

But, back to the point of this discussion. I agree the disputed sentence should either be improved or, if nobody can come up with a good way to improve it, removed.

Rick Norwood (talk) 16:29, 21 December 2011 (UTC)

A new suggestion:

"For convenience, most proofs are written in a metalanguage and, therefore, have to deal with the insufficiencies of each metalanguage. Nonetheless, mathematical proofs should (!) be written in such a way that a mathematician could translate them into a more formal language with an unambiguous grammar. This more formal proof could then even be checked by an computer. Usually, one of these more formal languages is taught at the beginning of mathematical logic"

btw.: less attacking and more suggestions and we could have closed this secition yesterday....!!! — Preceding unsigned comment added by 138.246.2.177 (talk) 17:14, 21 December 2011 (UTC)

Galileo in lead third paragraph, quote

I actuallyLOVE the lead, and appreciate the approach of quoting a few important folks as they described mathematics. However, I found the context lacking, especially for Galileo's quote. It is too often that the poor metaphors of natural law and language are used to describe mathematics. Such conceptions are invaluable to its history and this article, but there is a responsibility to more properly contextualize this paragraph I'm question. I believe the final sentence, a quote by Einsteiniis intended to achieve this effect, but j would rather see a punchline less punched if it meant clarity that could prevent further propagation of naive interpretations, despite also being valuable for other reasons. — Preceding unsigned comment added by 67.161.64.224 (talk) 07:43, 6 January 2012 (UTC)

See also section

The choice of entries in the See also section looks rather arbitrary to me and is, in my opinion, quite uneven. Iatromathematicians is an extreme obscure topic. Why is Self-similarity on this list? Any suggestions for a criterion to decide what should be on this list?  --Lambiam 12:42, 12 January 2012 (UTC)

I agree and have been bold and reduced it to what I think makes sense. There's certainly no need to mention individual topics: the article mentions many, including all major ones; they are all available in the template below. The same goes for particular aspects of mathematics education/ability, particular lists and particular groups. That's left the definition article and two relating mathematics to other topics. I would suggest only similar articles are added, i.e. not too narrow/specialist and not covered by one of the existing links (so no topics, no lists).--JohnBlackburne^words_deeds 13:04, 12 January 2012 (UTC)

Edit request on 20 January 2012

Please transpose the words space and quantity. It was recently found that the "all pages lead to philosophy" loop was broken by an edit to the mathematics page and the problem would be easily fixed by transposing these two nouns. menaing of the page would not be changed and many meme-ers would be made very happy. Thank you.

66.99.120.222 (talk) 15:24, 20 January 2012 (UTC)

Sorry, to edit articles to cause them to lead to philosophy violates the idea that all articles naturally lead to philosophy. Also, the current word order is used throughout the article and in several other articles. Rick Norwood (talk) 16:02, 20 January 2012 (UTC)

.................................................................................................. — Preceding unsigned comment added by 205.125.65.84 (talk) 15:46, 26 January 2012 (UTC)

Definition

MATH is defined as the expression of logic through the use of quantity.

It is often believed to be a determining science - which means it has causal properties. This is not true. A symbol cannot have any bearing on physical or metaphysical phenomena. At its best, it expresses logic or the scale and change of entities. At its worst...well let's not talk about it...:))))

LOGIC can be expressed in different forms - Math, Words, Pictures — Preceding unsigned comment added by 117.207.152.172 (talk) 13:15, 17 March 2012 (UTC)

People have been trying to define mathematics in terms of logic, quantity, abstraction, you name it, for hundreds of years (thousands if you count Aristotle). Please see Definitions of mathematics for a sampling. —Ben Kovitz (talk) 10:07, 11 June 2012 (UTC)

Verification failed

The first sentence doesn't agree with the citation. The reference says that mathematics is primarily about abstraction, not simply that it is about space, quantity etc. Removing the part about abstraction significantly changes the expression and the meaning, it does not encompass the same range.Teapeat (talk) 03:47, 19 March 2012 (UTC) For example boolean logic is not really, in any normal sense, to do with space, quantity, structure or change, but is normally considered to be part of Mathematics. It is however an abstraction of logic.Teapeat (talk) 03:47, 19 March 2012 (UTC)

Given this, I am considering the verification of the reference given to have failed, and the removal of that part to be original research.Teapeat (talk) 03:47, 19 March 2012 (UTC)

This discussion went on for years. Years! Roughly from 2008 to 2010, but don't quote me on that. I was on your side, but I lost, and the current version is a compromise that took a very long time to reach. If you want to reopen this discussion, just be sure you are prepared to spend a lot of hours over the next few years trying to resolve the issue. Rick Norwood (talk) 17:40, 8 April 2012 (UTC)

"Idealization" seems better. Tkuvho (talk) 08:37, 22 May 2012 (UTC)

"Idealization" is precisely a great way of describing the creation of mathematical concepts, and not merely scientific theories. Consider, for example, ideal points in projective geometry; ideals in number theory; finally the real line itself which is indeed an idealization. This term is more descriptive and informative than "abstraction". Furthermore, abstraction currently leads to the article abstraction (mathematics) which is rather silly. Tkuvho (talk) 09:14, 22 May 2012 (UTC)

The process of generalisation and decontextualisation in mathematics is most commonly called "abstraction", not "idealization" (quick test - compare google hits for "abstraction mathematics" and "idealization mathematics" - the ratio is about 7:1). A good exposition of this process is Chapter 2 Numbers and abstraction in Mathematics: A Very Short Introduction by Timothy Gowers. But I agree that abstraction (mathematics) could do with sourcing and improving. Gandalf61 (talk) 10:36, 22 May 2012 (UTC)

The cited source is not reliable. It's a web page about a university math/comp-sci department. It's probably just copying us, and it appears to have misattributed the famous Galileo quotation to Newton. The virtues of our definition are that it accords well with the kind of definition offered in other reference works, and it reflects the body of the article that we actually have. Please see Definitions of mathematics. I would favor removing the citation and leaving our definition unsourced, but wording it in such a way that it does not come across as the final word on the subject. How about we simply mention the lack of any consensus on the definition of mathematics, and link to Definitions of mathematics? —Ben Kovitz (talk) 10:02, 11 June 2012 (UTC)

I just made some changes: a new section on definitions, and a rewording of our opening sentence so it addresses the lack of consensus on a definition of mathematics, but is still informative and reflects the body of the article. I removed the old source that was probably quoting us, and added some new sources that I think are pretty solid, with the exception of the two I found for calculus (as the study of change). It would be better to have a single good source for that. I found a pretty amusing source on the inadequacy of any definition to cover all of mathematics: a published critique of this very article in a general survey of mathematics.

There are still problems with the lead. The lead is now five paragraphs, which is too long. Much of that bulk says rather doubtful things that are not covered in the article, no doubt the residue of long-past soapboxing. I think that by merely summarizing the body of the article, the lead could easily be brought back into shape. But, I'm done for the day. —Ben Kovitz (talk) 23:22, 16 June 2012 (UTC)

Edit request on 29 March 2012

MATH: Mental Abuse To Humans — Preceding unsigned comment added by Ilovenickelback17 (talk • contribs) 00:53, 29 March 2012 (UTC)

Can you provide a reliable source for this? Bzweebl (talk • contribs) 22:42, 5 July 2012 (UTC)

Request For Comments on recent "awkward" changes

Please post your comments regarding recent "awkward editings" performed by me on the article page regarding:

1) Adding <br>s in order to make citations more readable;

2) Hyperlinking David Hilbert;

3) Citations made in cursive;

that have been reverted motivating that with "please to to avoid html breaks" (whatever the "double to" means).

Thanks.

Maurice Carbonaro (talk) 08:00, 20 July 2012 (UTC)

I hadn't noticed this, and it looks like you have put a similar remark on your talk page. So I will partly copy from there:

1) The breaks and indenting: I found the suddenly interrupted flow of the article with a non-bulleted indented list slightly disturbing.

2) The hyperlink from "David Hilbert": Hilbert was already wikilinked (not hyperlinked) in the article lead. This is sufficient - see wp:REPEATLINK.

3) The cursive citations: I don't really mind these. AFAIC, feel free to re-italicise them.

- DVdm (talk) 10:56, 20 July 2012 (UTC)

Fine. I have re-italicised the cursive citations. Please feel free to use talk pages more in the future. Thanks. Maurice Carbonaro (talk) 07:11, 2 August 2012 (UTC)

What is the ultimate goal of mathematics?

Math seems to be going on in multiple different directions. But if mathematicians could define a goal that all of their research amounts to (like why we're trying to solve the Riemann hypothesis and other unsolved problems, and/or what they have in common), that would be great to include here. (Hint: It's definitely not 42 or NaN.) 68.173.113.106 (talk) 21:48, 6 March 2012 (UTC)

What I mean is, some people want to investigate topology, others want to solve P versus NP, still others want to do advanced complex analysis. Personally, I'm looking into mathematical finance (even though I'm just a kid). So what do all of these approaches have in common? If at all, why is it important? (Added to 68.173.113.106 (talk)'s previous comment on 21:53, 6 March 2012 (UTC))

Read Charles Sanders Peirce's Reasoning and the Logic of Things, which has a nice forward by Hilary Putnam.  Kiefer.Wolfowitz 22:45, 6 March 2012 (UTC)

The goal is to first identify and then determine the relative size or amount of something.WFPM (talk) 22:05, 18 March 2012 (UTC) And Chemistry tells you what something is and Physics tells you what something does. And Science gives you a hierarchic list of things to think about.WFPM (talk) 22:19, 18 March 2012 (UTC)

This talk page is not a forum to discuss the ultimate goal of mathematics.

Now, it is potentially a place to discuss whether the article should talk about the ultimate goal of mathematics. My feeling, at the moment, is no, it should not. There is no agreement about whether there is an ultimate goal nor, if so, what it is. We already have enough of a nightmare regarding the "definition" of mathematics, and there we kind of have to talk about it at least a little, because pro forma we have to have something resembling a definition in the lead.

That said, if there were a recognized debate about the ultimate goal of mathematics, I suppose that debate could plausibly be covered in that article. But I really have not noticed that this topic is much discussed, even by philosophers of mathematics (and much less by mathematicians), and yes, I do know some philosophers of mathematics and have followed their discussions to some extent. --Trovatore (talk) 22:28, 18 March 2012 (UTC)

I guess I just thought that mathematics started out with counting (enumeration) and went on from there. Is that illogical?WFPM (talk) 23:09, 18 March 2012 (UTC)

See my first point: We're not here to discuss whether that's "illogical" or not. If you want to discuss the question on its merits, as opposed to what to do with the article, please look for some other forum.

Now certainly, questions on the merits could come up in the context of what to do with the article, but I do not see any serious proposal on the table, just a vague hope from the OP that something relating to the question might be included here. I have given my first impression on that question (I don't think at the moment that we should say anything, at least not in terms so grand as an "ultimate goal"); my points are certainly subject to refutation, but I don't see how your comment goes in that direction. --Trovatore (talk) 23:20, 18 March 2012 (UTC)

I don't think there is a "goal" of mathematics. It's a "tool" used in science to organize and make understandable our ideas. And the article does a good job of talking about the history of the subject matter, but not as its use as a tool in science or technology. It leaves out about the slide rule for instance. And about the evolution of mathematical ideas like the square root sample size and the unknown number approximation routine, which are all unimportant details I know, but important to the importance of an understanding of the subject matter.WFPM (talk) 00:06, 19 March 2012 (UTC) But it sure has gotten more complicated, and in some places even more complicated than the science problems we're trying to understand with it.WFPM (talk) 00:58, 19 March 2012 (UTC)

There is no goal of mathematics, neither is mathematics a "tool" for science and technology. Mathematics is what happened when people sat down and applied their intuition to abstract ideas. It just "is", and if it is anything at all it is a natural outgrowth of the human intuition. It's even surprising that it's useful for understanding the universe/real world whatsoever (and maybe it isn't). 85.229.130.18 (talk) 10:47, 4 May 2012 (UTC)

I think there has been some debate, or at least assertion of opinions by authorities, about something sort of like the goal of mathematics. Hardy and Halmos both weighed in on the value or superiority of pure mathematics over applied mathematics. I can't believe there aren't some people who pooh-pooh pure mathematics. (Well, I know there are, because they've pooh-poohed to me personally, just not in writing with recognized importance.) I think many people have pointed out that pure mathematics often "pays off" in unexpected practical ways, such as (ironically enough) Hardy's number-theory work bearing practical fruit in public-key cryptography. The article already does cover these topics to some extent, as in Mathematics#Inspiration, pure and applied mathematics, and aesthetics. It could probably be done a little more deliberately. —Ben Kovitz (talk) 10:27, 11 June 2012 (UTC)

Do we have to find an ultimate goal? Perhaps a better posed question is: Can we find traits common to all branches of mathematics? Quantity, logic, and intuition are all partially correct but not generic enough (e.g., algebra intentionally leaves quantity unspecified). How about "Mathematics seeks to construct abstract entities and discover their properties through logic based on a limited set of axioms." Of course, in trying to be generic I have introduced some terminology, and others are welcome to improve upon this. — Preceding unsigned comment added by 71.236.24.128 (talk) 22:01, 18 August 2012 (UTC)

Math as a hobby

I think math as a hobby could become a useful addition to the article if someone worked hard enough on writing it.Marvin Ray Burns (talk) 02:16, 19 May 2012 (UTC)

Can you recommend a good source or two to summarize? —Ben Kovitz (talk) 10:26, 12 June 2012 (UTC)

The Book of Numbers by Bently shows that a although a lot of early numeric discoveries were done by mathematical giants such as Pythagoras, Euclid, Newton, etc., the giants were at one time mere hobbyists.Marvin Ray Burns (talk) 01:20, 10 September 2012 (UTC)

There are good examples of math as a hobby, but none of the three names you mention seems to fit well under that heading. Rick Norwood (talk) 11:54, 10 September 2012 (UTC)

Benjamin Peirce a logicist?

A C.S. Peirce scholar brought this line in the article to my attention:

Two examples of logicist definitions are "Mathematics is the science that draws necessary conclusions" (Benjamin Peirce)[24] and "All Mathematics is Symbolic Logic" (Bertrand Russell).[25]

Russell's quote is quite logicist, but Benjamin Peirce's quote doesn't seem so, since logicism usually means the idea that much or all of mathematics is reducible to logic, not merely the idea that mathematical conclusions are logical or deductive. Have anti-logicists generally held that mathematical conclusions are not generally deductive?
Now, the B. Peirce quote is from "Linear Associative Algebra" in which he goes on to say, beginning lower on the same page (the article's first page),

Mathematics, under this definition, belongs to every enquiry, moral as well as physical. Even the rules of logic, by which it is rigidly bound, could not be deduced without its aid. The laws of argument admit of simple statement, but they must be curiously transposed before they can be applied to the living speech and verified by observation. In its pure and simple form, the syllogism cannot be directly compared with all experience, or it would not have required an Aristotle to discover it. It must be transmuted into the all the possible shapes in which reason loves to clothe itself. The transmmutation is the mathematical process in the establishment of the law. [....]

That really doesn't sound like logicism at all. It's more to say the same as his son C.S. Peirce said, that the mathematician aids the logician, not vice versa. (I'm not arguing about whether they were right, just that that was their view.) Anyway, I suggest that Benjamin Peirce's definition of maths not be characterized as "logicist." The Tetrast (talk) 01:00, 7 September 2012 (UTC).

Nobody has commented, so I'm not sure whether anybody cares. I propose to revise the line in question to

An example of a logicist definition is "All Mathematics is Symbolic Logic" (Bertrand Russell).[25].

If nobody comments during the coming five days or so, I'll go ahead and make the change. The Tetrast (talk) 02:56, 8 September 2012 (UTC)

I care. I agree that we should not call Benjamin Peirce a logicist. I'm no expert, but I believe the literature says that he led the way toward logicism but predated it. Strictly speaking, the quotation doesn't say that he was a logicist; the definition is meant to give the spirit of logicism. Still, the juxtaposition certainly suggests that Pierce was a logicist. Can you think of a graceful way to word that paragraph so that Peirce's definition is still included? (A nuanced but still brief exposition of Peirce's and Russell's definitions is here. Could you double-check that, too?) Many people come up with the "necessary conclusions" definition on their own, and they think it's the only reasonable definition. It would be nice if the main Mathematics article mentioned that definition and its originator, alongside the main competing definitions. Also, I don't think Russell's definition is clear to someone who doesn't know much about the field; the Peirce definition is much clearer. —Ben Kovitz (talk) 21:37, 10 September 2012 (UTC)

I just had an idea: distinguish between logicism and defining mathematics in terms of logic—since they're not the same, anyway. This provided a nice way to put Peirce's definition first, just by going in chronological order. It's in this version. Please take a look, see what you think, and improve. —Ben Kovitz (talk) 22:59, 10 September 2012 (UTC)

I would not even agree that Benjamin Peirce offers a "definition of mathematics in terms of logic". "The science that draws necessary conclusions" could only be said to be drafted "in terms of logic" if logic is the only science where necessary conclusions are found. But one might equally argue that mathematics and logic both deal with necessary conclusions but not in the same way. Just like philosophy and psychology both deal with the mind but not in the same way. I would rather amend the whole paragraph to read something like this:

"In Principia Mathematica, Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proven entirely in terms of symbolic logic. A logicist definition of mathematics is Russell's "All Mathematics is Symbolic Logic."[25] Related but more subtle views were put forward by Benjamin Peirce, who wrote that mathematics is "the science that draws necessary conclusions" [24], and his son Charles, who wrote that logic is "the science of drawing necessary conclusions".

I can reference the latter quote if this change is liked. (Greatcathy (talk) 09:31, 14 September 2012 (UTC))

Thanks, Greatcathy. I understand Benjamin Peirce's definition to be in terms of logic in that "drawing conclusions" and "necessary conclusions" are entirely concepts from logic. That is, it distinguishes mathematics from other topics by the character of its propositions and reasoning rather than by the content of its propositions. Maybe you can find a way to improve the wording in the article so this point is made clearly.

Here are a couple quotations that might be helpful for figuring out what to say in this paragraph:

1. From Boyer & Merzbach's History of Mathematics: "[Benjamin Peirce's] son was in wholehearted agreement with this view, as a result of Boole's influence, but he stressed that mathematics and logic are not the same. 'Mathematics is purely hypothetical: it produces nothing but conditional propositions. Logic, on the contrary, is categorical in its assertions.' This distinction was to be argued further throughout the mathematical world in the first half of the twentieth century." I take the "but" to mean that Benjamin Peirce's definition is so clearly related to logic, it suggests that the two are easily confused, and C.S. Peirce found it necessary to explain the distinction.

2. Here's an excerpt from C.S. Peirce's writing: "…[T]he mathematician does not conceive it to be any part of his duty to verify the facts stated. … All features that have no bearing on the relation of the premises to the conclusion are effaced and obliterated. … [T]he mathematician frames a pure hypothesis stripped of all features which do not concern the drawing of consequences from it, and this he does without inquiring or caring whether it agrees with the actual facts or not; and secondly, he proceeds to draw necessary consequences from that hypothesis."

It would be best, of course, if we can refer to more-recent commentaries that reflect scholarly consensus on the meaning and significance of the B. Peirce definition.

I'd prefer not to include both "[mathematics is] the science that draws necessary conclusions" and "[logic is] the science of drawing necessary conclusions". I think it's inviting confusion. If there is a strong reason to include them both, let's make that explicit in the text. If they're both important but require more exposition, then maybe the two juxtaposed definitions should go into Definitions of mathematics. BTW, I invite you to have a look at that article, too; it also needs many improvements, and you might have a lot to contribute. —Ben Kovitz (talk) 01:53, 19 September 2012 (UTC)

Why mathematics is more important than physics and chemistry?

We have mathematics subject since we are in elementary school, but not physics or chemistry until high school180.194.246.163 (talk) 10:14, 12 November 2012 (UTC)

Which is more important is a matter of opinion. You need math to understand physics and chemistry. That is one reason math comes first. Also, the public schools are paid for by taxpayers. Physics and chemistry labs are expensive. Mathematics only requires pencil and paper. Rick Norwood (talk) 12:58, 12 November 2012 (UTC)

Group theory picture

As a graduate student in group theory, I do find it a bit alarming that this picture of a rubix cube has become the cornerstone of Wikipedia's imagery on groups. Yes, the rubix cube does form a group under composition of turns, but is it a key, or even interesting example of hows a group can function? I motion that it is changed to something more relevant to the field itself (e.g. an illustration of a dihedron group might be nice as it is the lodestone of much of most introductory texts). If I am alone on this issue, I will simply retract my argument, but I do find it a tad annoying. — Preceding unsigned comment added by 76.126.169.150 (talk) 03:46, 13 January 2013 (UTC)

system of equations

Hello, i have a problem with my homework. and i was wondering if you can help me with it. — Preceding unsigned comment added by Mickenson45 (talk • contribs) 00:41, 6 March 2013 (UTC)

No, sorry. This page is for discussing proposed improvements to the mathematics article. However you can ask at the mathematics reference desk. The operative word there is "help", not "do it for you" — first, write out what you've done so far, explain where you're stuck, and ask for help getting past that point. --Trovatore (talk) 01:18, 6 March 2013 (UTC)

Lead paragraph (sorry)

"Mathematics as profession" does not at all treat the right topic. "Renowned Math prizes" would be a more appropriate title. — Preceding unsigned comment added by 85.224.152.237 (talk) 20:20, 6 April 2013 (UTC)

I agree with the fact that the content of "Mathematics as profession" has nothing to do with it's title. "Math Prizes" or "Math Awards" would a more appropriate title for that content. Anyway, would be a good thing to have a section titled "Mathematics as profession" or something similar to deal with the professional jobs where mathematicians apply their knowledge. MickMurillo (talk) 21:21, 3 May 2013 (UTC)

The philosophy of mathematics, the branch of philosophy that studies mathematical assumptions, foundations, and implications, is one of the biggest branches of philosophy in the world. This branch never stops growing, from Thales, a great mathematician from Ancient Greece, to V. N. Bhat, a small mathematician from India, they have all added what little they could. Every idea and theory has helped this art grow. Every year mathematicians discover something new that helps us understand mathematics a little better. The mathematicians don't MAKE or ADD to the subject, they DISCOVER something that is already there. This is the crux of the matter, you cannot make something in math, everything is just there. There can, however, be another way to interpret the art. The Arabic numerals, which were actually made in India but were carried to England by Arabic Traders, are known as the language of math. However, if you lived in Babylon in prehistoric times, you would have a whole different way to express mathematics. The Arabic numerals aren't math, they are just a way to interpret the art. However that means that mathematics cannot be defined because there is no definite way to express it. From the plastic ladybugs used by teachers in 2nd grade to help you add and subtract, to the Arabic numerals used around the world, they are just different languages used to express math, just like all the different languages in the world that are used to express people. Math is a never ending problem, that can be used to solve problems. Just like you can never count to infinity using a language of math, you can never be finished with math. There will always be a new theory to try or a new method that explains another one of life's great questions. -- Professor Captiosus — Preceding unsigned comment added by Professor Captiosus (talk • contribs) 17:29, 20 January 2013 (UTC)

Given the warning in the article pseudo-code about changing the opening, I decided to bring my proposal here, since the way it currently reads is awkward, in my opinion. I propose:

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of abstract objects and the logical relationships among such objects. Mathematics encompasses topics including quantity, structure, space,and change, although it has no generally accepted definition.

...for the following reasons: 1) Mathematical entities, as represented by symbolic notation, are abstract. They are well defined for the purposes of an axiomatic system, and the rules and operations between mathematical objects are logical in character. I'm sure there is no objection here. 2) The topics listed are not well defined, and while math is used to study things like structure and change, it is NOT the study of naturally occurring structure or change, but the study of abstract representations of such. That is to say, the quantum zeno effect negates the direct correspondence with reality of infinitesimal calculus. Topology deals with abstract surfaces ect. 3) The "and more" is amateurish and doesn't do anything to inform a reader about what mathematics actually is. 4) I don't even know what is meant by "the abstract study of subjects" - the verb study is surely only undertaken by a physical human or physical computer, an abstract object cannot "study" anything as far as I know. The word "subjects" is too ambiguous and probably incorrect. In colloquial parlance "subjects" can mean topics of learning in school or whatnot, but "fields" or "disciplines" works better if I understand the connotation correctly. In any event, Math is the the discipline that makes use of well-defined abstract objects and manipulates them by logical rules and operations. One might even include "rigorous" before "study" in my proposed intro, but it's not particularly important. What I see as important is to do away with "abstract study" because nobody even knows what abstract study is. Either everything that could conceivably studied is abstract or nothing is. My point is that I could see dog feces on my shoe and look at it carefully and in my brain associate dog feces with my previous understanding of dog feces, and the structure of the smear on my shoe could be "structure", and by the current lead paragraph I would be doing math. To me, a definition that excludes nothing is a poor definition. -Fcb981(talk:contribs) 00:56, 29 August 2012 (UTC)

This is, as you might guess, a discussion that has been held many times before. The conflict is between what non-mathematical sources say mathematics is and what mathematicians say mathematics is. While I agree with you, non-mathematical sources such as dictionaries and encyclopedias say mathematics is the study of numbers and shapes. Wikipedia reflects sources, not truth. Sigh! Rick Norwood (talk) 17:21, 29 August 2012 (UTC)

Fcb981, I sympathize with your complaints about the opening quasi-definition: "abstract study" is weird, the mathematical study of dynamics is awfully static and certainly different from how change is studied in the natural sciences, etc. I'd like to see a better definition. However, here are the three big reasons favoring the current definition:

(1) The breakdown of math into major topics corresponds to the structure of the article, at least the subheadings under "Pure mathematics". A good lead should summarize the body of the article; see WP:LEAD.

(2) The definition of mathematics is highly controversial. Please see Definitions of mathematics for a sampling of leading definitions and the unresolved conflict between them. There is not even a consensus among mathematicians on how to define mathematics; see Mura, Robert (1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences". Educational Studies in Mathematics. 25 (4): 375–385. {{cite journal}}: Unknown parameter |month= ignored (help). For this reason, it's not appropriate for us to give a proper definition that draws a clear boundary between what is and is not mathematics; we settle for a rough distinction that leaves the boundary indeterminate.

(3) As Rick Norwood said, Wikipedia is a summary of sources, not truth for which there is no consensus in previously published, reliable sources. Lacking a consensus among mathematicians, we've opted for a definition similar to that of other general reference works, but tuned a bit to the body of the article. The current definition has its flaws, but it's certainly well-sourced (except for "change").

These reasons don't mean that the current opening can't or shouldn't be greatly improved. They're obstacles that need to be overcome in order to reach that improvement. Here are three things that you can do. First, edit the body of the article to the point where it calls for rewriting the definition; see WP:LEADFOLLOWSBODY. The body of the article is begging for all kinds of improvement, and those would likely do as much or more to get across the nature of mathematics than rewriting the definition. Second, find a way to word the new definition so it doesn't take one side in the conflict over definitions. Your current proposal takes one side, or at least opposes some of the leading definitions. And third, find good sources to support the new definition. The reasons you gave above refer to the nature of mathematics, not to sources.

The reasons you gave regarding clarity are well taken, though. I just changed "subjects" to "topics". Of course, this is only a small improvement. It might also be possible to improve the "abstract" part without first overcoming the obstacles described above. —Ben Kovitz (talk) 21:03, 10 September 2012 (UTC)

Well...does seem to be written by mathematicians and not writers.  :-) In addition to starting with a single awkward sentence (instead of a well formed paragraph), it then goes on to provide a laundry list of quotes about math from famous people. I feel that the field is so broad and there's no "Joe Math" who started it, so a good lead on the topic should probably avoid naming any specific mathematicians as best possible. I'm also about the "holy trinity of paragraphs" so I'd say paragraph one would go approximately like this (solve for X)

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the abstract study of topics including quantity,[2] structure,[3] space,[2] and change.[4][5] While the term has been around since X[?], there is no generally accepted definition of everything it encompasses [7][8], and practicing Mathematicians analyze many other properties [6]. Though often pursued for its own sake, many of today's scientists rely heavily on applied mathematics for their work—and practical uses for what began as pure mathematics are frequently discovered.[17]

For what would be the third paragraph, start with a variant of Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.[11] and finish with a couple more sentences about applications. The second paragraph is whatever's left when the quotes are cut—basically expanding on the definitions of the field and its sub-branches. Well, there's my "vision statement". Metaeducation (talk) 08:09, 6 January 2013 (UTC)

I agree that the laundry list of quotes should be deleted wholesale and replaced with a summary of the body of the article, per MOS:LEAD. It looks like several hours' work. Want to give it a go? I haven't felt much enthusiasm for doing this, because the body is such a poor overview of mathematics. But it would still be a fine step forward. —Ben Kovitz (talk) 13:38, 20 May 2013 (UTC)

Wrong link in lead paragraph

Wrong link in lead paragraph - the space link should go to Space (mathematics) not Space. — Preceding unsigned comment added by 173.79.197.180 (talk) 02:02, 27 February 2013 (UTC)

Please sign your talk page messages with four tildes (~~~~). Thanks.

I'm not sure about that. The lead says that mathematics is the "study of topics encompassing quantity, structure, space, change, and other properties." So i.m.o. in this context space is meant to be taken in the more general sense. - DVdm (talk) 09:12, 27 February 2013 (UTC)

Yes. Geometry has traditionally been defined as "the science of space", not "the science of mathematical spaces" (which are not the same as space in the ordinary sense, though partly inspired by it). —Ben Kovitz (talk) 13:21, 20 May 2013 (UTC)

Pure mathematics on top section

It is just a minor change but i cant touch it.

On top section last paragraph,it said, "Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind."

It is ambiguous. A mathematician can engage in "pure mathematics" without mathematics and any application in mind. The sentence isn't exactly false but ambiguous. If i can edit i would just delete ", or mathematics for its own sake, ". If you think we must mention something like "pure mathematics often has mathematics in mind", then try split the sentence in better shape.

Also,this wiki article is trivial and important for all, extra care on wordings/semantics must be given, so it doesn't spread any misleading information.14.198.221.131 (talk) 16:34, 23 December 2012 (UTC)

I'm having some difficulty understanding you. Maybe this will help: the phrase "or mathematics for its own sake" is there to clarify the meaning of "pure mathematics" for people not familiar with it. —Ben Kovitz (talk) 13:18, 21 May 2013 (UTC)