Talk:Philosophy of mathematics/Archive 1

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Archive 1

Archive 2

Archive 3

Early discussion

I don't think this page should be used for people to argue which philosophical line of thought they endorse, instead this forum should be used to constructively coordinate the fruition of this article as one that is coherent and representative of the discipline of philosophy of mathematics as endorsed by the various experts in the field.

What does Julius Caesar=2 mean? Someone needs to patch that up because it doesn't really make sense.

heh. this is going to be *so* weird to people who think math is real...

It's also a total mess as an encyclopedia article.

This article should describe the various different philosophies of mathematics.

it describes three extreme points of view, those being that it is fundamental physical and unchangeable, that it arises entirely from the proof process followed by mathematicians, and that it arises entirely from the structure of the human mind. A fourth extreme point of view, that it is entirely a colonizing process, I didn't deal with, as it's so postmodern.

Also, I doubt it should be "weird" to anyone. Anyone who works with math pretty soon hits some philosophical issues.

yeah but they breeze past them - of 150 mathematicians in one graduating class I know, only two of them paid any attention to the field proper, and they disagree sharply, falling one each into the "embodied mind" and "part of the physical universe" camps. This is not an easy subject to really explain.

A lot of scientists (particularly if you get into the social sciences) don't consider mathematics neutral.

If anyone needs proof that mathematics is not a social construct, look no further than the computer you're sitting at right now.

It differs from the philosophy of science by not taking mathematics as a neutral point of view - rather, investigating such subjects as the willingness to accept mathematical proofs, the validity of induction or analogy, what combination of metaphors constitutes an isomorphism, mathematicians' social capital and the meaning of the well-known collaboration graph.

A lot of 'hard scientists' don't consider 'social science' to be science -- they point to the lack of falsifiability in many of their theories, and assert that abandoning objective standards of mathematical and logical proof removes the basis for falsifiability, leaving only opinion, fashion, and popularity contests.

There are physicists who do think that. But that's a relatively small minority opinion (something like 10% or so). My experience with "math-fetishists" is that they actually tend to be social scientists. Very few physicists (or even mathematicians) seem to believe that an idea that can't be expressed in mathematics isn't an idea at all, however I've known/read a number of social scientists that seem to think that. The problem is that regarding mathematics in social science *greatly* limits the hypothesis that you can form and the theories that you can test. Also, pretty much anyone who does qualitative research would disagree with non-mathematical means non-falsifible, and I'm pretty sure that most physicists don't think this (even though I know of a few who do).

The problem is that regarding mathematics in social science *greatly* limits the hypothesis that you can form and the theories that you can test.

That's probably the attraction for us math-fetishists. Science must be so easy for the math-challenged: you simply have to think of things, write papers and win the support of your peers, without any real chance of bumping up against nasty cold non-human reality.

The trouble is that just because something is stated in terms of mathematics doesn't automatically make it more objective, precise, testable, falsibility or correct. I get *REALLY* annoyed at people who think it does. (I should point out that I have a Ph.D. in physics.)

all of this is what leads to "reasonable method" ideas. Science itself, hard or soft, is basically in doubt at the moment... neutrality of its process is questioned for a lot of reasons... Falsifiability as a doctrine has little or no applicability to basic mathematical axioms... which are too abstract to falsify....

This article is still a mess. It's confusing, misleading, and in some places, downright wrong....

Wrong? Where? Your strange misreading seems to be the problem here:

I guess I'll start by removing the most egregious errors

asks what makes one theory more acceptable than another despite imperfect empirical validation and limits of the scientific method - and assumes mathematics as a neutral point of view.

I'm not sure what neutral point of view means in this context. If the statement is that mathematics doesn't influence how scientists view the world, then its false.

other way around - are you actually *reading* this or just reacting to it? it is quite true that scientists assume mathematics as a neutral point of view, and that philosophy of science has no basic critique of mathematics itself, except for some of the social sciences as above.

No it's not (and I'm speaking as a former physicist). Scientists do *NOT* generally assume that mathematics has a neutral point of view, and a large part of the "working knowledge" of being a physicist, is to be able to critique when math is being used appropriately and when one is better off dispensing with using math all together.

This is gibberish. It's also inaccurate

and collaborators, actually mapped onto the physicists' particle physics foundation ontology or if they were simply another sacred geometry like that of Plato - a useful but limited model that awaited understanding of some deeper ontology.

Explain then why various commentators refer to "Platonic neorealism" in both physics and in mathematics, both arising seemingly from overbelief in Euler's Identity...

Overbelief in Euler's identity? It has nothing to do with Euler's identity. Such a philosophy existed long before Euler's identity. If you haven't noticed, neo-Platonism arose in ROMAN times (around 1500 years before Euler was even born, if you don't know), and has its foundation in far more holistic views of reality than Euler's identity... Perhaps YOUR view arises from the possibility that at least a large portion of your right hemisphere was excised at some point in your life. Khranus

So did Erdös really do work on foundations? Or the 'fundamentals' of number theory? It's not a field I associate with his name. Anyone? Matthew Woodcraft

Agreed. Erdos' program was to re-prove many basic proofs in number theory without any reliance on complex analysis whatsoever... that in itself changed the fundamentals, as it removed reliance on a whole set of methods and assumptions. The well known graph of Erdos' collaborators drives new work in graph theory and research into collaboration on research. "Paul Erdos, more than anyone else, did the most to make mathematics into a social activity"

I see. Did he do any work on foundations? In Erdoös's lifetime, number theory wasn't at the foundation of mathematics; it was typically built up from set theory, yes? Matthew Woodcraft

this was one of the things that changed as a consequence of Erdos' extensive work - someone (I forget who) said Erdos "more than anyone, turned mathematics into a social activity" - literally moving its assumed basis from set theory to proots collaboration by the sheer volume of this work. Few people even in Erdos' early career believed in set theory as the basis of arithmetic...

Various people who understand nothing of the modern theories are hacking this. No one says that scientists aren't influenced by mathematics, quite the other way around, they believe in it overmuch, and in "falsifiability" overmuch, and that ideology is clearly guiding the commentary in the above.

Scientists don't *believe* mathematics any more than you or I *believe* words. Mathematics is a communications device. It is perfectly capable to expressing ideas which are incorrect or non-sense. There are a number

of postmodern theorists who talk about science, but have the unfortunate habit of ascribing beliefs to scientists that they simply do not have.

some, yes, others no. Lakoff is disciplined about this. Zerzan less so. And they often talk about "normal science" only, i.e. "corporate science" etc.

Falsifiability is ideology. Other than strict followers of Popper, it's hard to find anyone who believes that cognitive bias and infrastructural prior investment don't play huge roles in determining what constitutes a "disproof".

In science yes. In mathematics no. In mathematics, you get to define what truth is. You can argue

that social factors influence how much effort there is in disproving a theorem, but I don't think one can plausibly argue that a proof or disproof of a mathematical theorem itself is affected by cognitive bais. Keep in mind that scientists and mathematics use the term "disproof" in radically different ways.

true - cognitive bias in mathematics seems to be all-or-nothing, i.e. either all mathematicians buy into it, or none do. That's one reason why Erdos' work was so important in foundations... you can actually *trace* this spreading of beliefs...

I think that's going a little too far. Certainly over history, the idea of what constitutes a valid proof, or a valid counterexample to a proof, has changed. Matthew Woodcraft

It may be that you have more knowledge in this field, but it seems to me that constitutes a mathematical proof or mathematical disproof is exactly the same now as it was in the time of Euclid. If you can give some examples of mathematical statements and proofs which would be accepted at one time, but not another then I would be appreciative.

Euclid provides a perfectly good example. For a long time, his proofs were thought to be complete. But a couple of hundred years ago, the expected standard of rigour increased, and extra axioms were added that he'd 'missed out'. Matthew Woodcraft

If you are referring to non-Euclidean geometry, then this isn't what happened (if you are referring to something else let me know). What happened was that for a long time, people believed that Euclid's axioms were the only ones that were possible, but they couldn't prove it. They couldn't prove it because it wasn't true. In the 19th century, it was proven that you could change one of the axioms and end up with a system that was consistent. This didn't change the nature of proof, or their rigor, or the validity of any of the proofs.

Actually, this paragraph above is a common misunderstanding of the evolution of proof from Euclid's time. The controversy over non-Euclidean geometry was NOT because people didn't believe you could logically deduce some statement from others, but because Euclidean geometry was seen as true BECAUSE it accurately reflected "reality". In the end, your proof (back in the early 19th century and before) was "true" if it accurately reflected "reality". Of course, logic played a role, but you were allowed to use any kind of starting assumptions that seemed reasonable.

The nature of proof *has* changed, partially because the notion of "truth" has changed. A mathematical statement nowadays is considered "true" as long as you exhibit a model in which it holds. The idea that consistency is what's important and not whether your theory accurately reflects reality is a 20th century notion. Consider another controversy: complex numbers. It took quite a while for their use to be accepted, mainly because of philosophical difficulties people had over them. Eventually mathematicians realized that as long as the use of complex number didn't lead to a contradiction, it was ok to use them; however, the pre-20th century mathematicians weren't able to quite take the plunge and say it was ok to use *anything* which was consistent with the basic foundations of mathematics. Of course, this might have had something to do with the fact that the foundations of mathematics were not as clearly delineated as today.

Which goes to my point that mathematicians and scientists mean something completely different when they talk about disproof. If something in science is "disproven" then its possible that people will change their minds with more evidence. If something in mathematics is "disproven" then that's the end of the story, it won't ever change. Conversely, in mathematics unlike science you have "prove" something to be true. Going back to Euclid, there are a number of things in mathematics that people believe to be true or believe to be false but haven't been proven one way or the other, and sometimes there are surprises. But once something in math has been proven or disproven, then its the end of the story. Science is different. In math, something is true or false because you've defined it to be so.

Again, this is entirely misleading. In addition to my reasons above, let me comment some more on rigor. Some of the proofs in Euclid are nowadays seen as *wrong*. Euclid assumed certain things that were not postulated, such as the fact that circles intersected in zero or two points. However, a closer examination of Euclid, especially his definitions of point, line, etc., indicates that Euclid probably did not have a notion of "axiomatic method". It is debatable whether he actually thought he was setting up a logical system of deduction, as some assume today.

So I stand by my statement. The nature of proof in mathematics hasn't changed since Euclid. The nature of disproof in science can change from decade to decade, and the whole notion of the scientific method happened centuries after Euclid.

No, I'm not talking about non-Euclidean geometry, I'm talking about changes in the standards of rigour. http://www.marco-learningsystems.com/pages/kline/johnny/johnny-chapt5-6.html is the best reference I can find on the Web (it's mostly about other things; the first few paragraphs are relevant). [Heh - I just found a rather clearer explanation right here, at Euclidean geometry.' Matthew Woodcraft

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Why is "metaphysics" relevant, and "ethics" not? Philosophy of mathematics is often framed in more theological terms of ontology, morals, and cosmology... which is not the ethics/epistemology/metaphysics distinction used in philosophy. In particular Godel neatly moved from one to the other, and Wittgenstein and Russell... most of the major figures. Erdos has some very strange compilations of conventional cosmology, which are called Erdosisms.

That's certainly not the usual referent of the term 'Erdösism'. And I don't believe that either morals or cosmology 'often' come up in discussion of the philosophy of mathematics.'

Erdos had his own strange minimal cosmology where God was a "supreme fascist", men who got married were "captured", divorced were "escaped", who stopped doing mathematics were "dead", and who died had "left", etc... Godel had an ontological proof of God, a quite rigorous one... this is cosmology as conventionally understood. So is the study of sacred geometry. Russell and Wittgenstein's famous "elephant dialogue" is more a debate on the morality of doing something versus doing nothing and claiming uncertainty as an excuse... but all right, I'll mostly agree, and say that this just highlights the central role of ontology... theology might be theology because it *does* debate morals and cosmology a lot, and math might be math because it only concerns itself with ontology (minimal set of operations and constants to describe everything else with).

Perhaps it is more correct to say there is a "philosophy of science" but a "theology of mathematics"...?

ontology of mathematics? as one of several foundation ontologies? i.e. we cna say that the particle physics foundation ontology is basic, or that the euler's identity foundation ontology is basic, or that something else explains them both (like human cognition). As Judea Pearl points out this is all part of the "algebra of seeing", and has nothing to do with an "algebra of doing" which he proposes we need, like the other body philosophers.

Snipped from article - the quote preceding them suffices, and these just add blur:

In terms of which, the reasonable method is the best description we can make of the best human mind, and the foundation ontology is the best description we can make of "reality, something out there to be discovered."

As use of mathematics is not confined to scientists making predictions, but is employed also in law and economics and political science, most researchers find it appropriate that "we" be a somewhat larger (some say less disciplined) group than physical scientists, and that reasonable method be perhaps more inclusive than feasible method or scientific method.

I have this concern that this article is written by someone who doesn't have a working knowledge of how mathematicians and scientists think. While there isn't anything wrong with an outside point of view, I'm getting this impression that a lot of the statements assume a world view by mathematicians and scientists that they simply do not hold.

In particular there seems to this assumption that physicists regard math as objective and the preferred means of expressing ideas. In fact, most of the physicists I've worked with regard math as a necessary evil, and if you use an equation to explain something you can uses words to explain, they will nail you to the wall.

it's common to require people in any science to explain something in words first - exposing the idea to a larger number of people using metaphors - and only afterwards to go to the bother of rigorous prediction with an equation. That is the same in economics and political science.

The trouble with the article is that these stereotypes about what physicists actually think are scattered throughout the article, and there is no where where you can say "but many/most/some scientists don't think that."

physicists vary in my experience - some are very aware of foundations problems, others not. But the question of "who's we" in mathematics stands: math is used in very different ways in economics or finance, logic used very differently in law. A complete article would get into the role of math in each and how various standards of proof rely on it, or not... but perhaps a separate article on standards of proof would actually be the answer here?

http://www.rbjones.com/rbjpub/philos/maths/index.htm looks to be a very good site on this subject. Certainly it gives a broader range of topics than our current page.

More comprehensive, yes, it should be listed as a reference. It gives special attention to foundations, as it should, pays maybe too much to logicism which is an old view of mathematics. It's got its own slant.

However, it totally fails to lay out the extreme points of the foundations debate which the current article (despite other failings) does well. If you can fit the material from rbjones into the general framework of the existing article, adding topic lists, etc., more links to new pages on the specific debates, that's ideal I think...

If you can fit the material from rbjones into the general framework of the existing article, adding topic lists, etc., more links to new pages on the specific debates, that's ideal I think...

I agree!

It gives special attention to foundations, as it should, pays maybe too much to logicism which is an old view of mathematics. It's got its own slant.

That's the whole point -- we've all got our own slant. So let's represent older as well as newer ideas, and try fairly to represent all the slants. Even today's brave new ideas will one day be out of fashion -- and old ones may even get their day in the sun again.

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Speaking of which, I keep wanting to deal with theological and ontological methods more generally than just mentioning the current Pope's opinion. A lot of theorists think that "starting from Euler's Identity, taken on faith, and proving that all the rest of reality conspires to make it seem true" is identical methodologically to "starting from God, taken on faith, and proving that all the rest of reality conspires to make Him seem real", i. theology...

A flame war

Axel's most recent edit is highly suspect. What is the justification for removing the list of significant 20th century figures in the Philosophy of Mathematics? Is there any doubt whatsoever about this list?

Yes. most people on the list were not philosophers of mathematics,

they *redefined* and *reframed* the field. Read Alfred North Whitehead on civilization and algebra sometime. The field as you have laid it out is pre-Principia-Mathematica (1911 or so), plus embodied mind tagged on the end, which is not bad as far as it goes, but it hardly lays out the endpoints, and is incomprehensible to newcomers. The TT quote was important for laying out the traditional debate. But never mind, I can work with your basic framework, if you will add these characters that you say were left out:

and the most influential philosophers of mathematics were left out. So I scrapped the whole list. Especially your insistence on Erdos is ridiculous. He may have had his personal philosophy of mathematics, like many

his personal philosophy of mathematics was that it was "a social activity", and

That's crap. Erdos was a Platonist with a capital P. He believed that there's a "Book" out there which contains the most beautiful proofs of all theorems, and it's our job to find those proofs. Erdos was a uniquely social mathematician, that's all. AxelBoldt

Quite true. But what the man *DID* is not relevant to what he *BELIEVED*. Like Einstein effectively proved that God *did* play dice...

he uniquely succeeded in propagating that view into the community of all working mathematicians. Your insistence on cutting him out is ridiculous, as the school of "research into collaboration of research" foundationists who cultishly extend the Erdos Number (which is hardly "tongue in cheek" now) proves.

This is not a school in the philosophy of mathematics, and Erdos has absolutely nothing to do with it, except that his name is attached to the Erdos number. AxelBoldt

"absolutely nothing, except"... and what is the significance of the Erdos Number to the philosophy of mathematics? "absolutely nothing"? There is not one school that would agree with that.

He did not contribute one iota to *YOUR VIEW OF THAT FIELD* because you have consistently denied that proof process or cosmology can matter at all - hardly the view of the meta:three billionth user. So you have cut out the whole view that mathematics is created by social consensus of certain disciplined people - which is the closest to being proven as true.

Erdos had a personal cosmology,

I'm sure he did. And I don't care what it is, nor would any reader of an article about the philosophy of mathematics. AxelBoldt

and by eliminating complex analysis completely from number theory, he built a new ontology for mathematics that was much simpler than the one before, and could be used as part of any foundation ontology that wanted to omit the use of "i".

That's crap. He did no such thing. He proved the prime number theorem with elementary methods, that's it. He never attempted to eliminate complex numbers, and he certainly never tried to build a new foundation ontology. AxelBoldt

he didn't *TRY* to do it - he just *DID* it - it's called "the well-known collaboration graph" - and it has certainly expanded and grown as a study on its own, and is now driving new research into pure graph theory. Your view of his contribution differs from that of "The Man Who Loved Only Numbers", his biography, although I agree he did not *completely* eliminate the "i".

mathematicians do, but he did not contribute one iota to the field. AxelBoldt

I just noted the contribution. If you would rather credit this to me as original, and not to Erdos or his followers, you are going to make me very very famous, and I should thank you.

Boys, boys! Stop this futile wrangling. You make the Middle East seem peaceful :-) Ed Poor

If you keep using your axe on him, he may bolt from the project. Is that what you want? Please, my esteemed colleague AxelBoldt, make a bit more effort to elevate the level of civility in our learned discourse.

I prefer him revealing his obvious ideological biases and political agenda.

And you, Mr. 24, try to be a little less confrontational.

no, the confrontation is useful. check the meta again. There's a lot more there.

Have you ever considered using phrases like, "It seems to me that..." or "It might be a good idea to keep X in the article"?

no. This is a debate about truth itself. We might as well keep it up to the point just before bodily harm is expected. If every male in the world punched one other in the nose hard enough to almost break it, once a day, there'd be no war. And we'd get much better at setting our priorities.

I expect better of you both, as Bush might say. :-) Ed Poor

and Bush is responsible for green-lighting the present disaster by choosing the absurd War on Terror rhetoric, which is actually a "War on Terra". Ed, the right answer here is for Axel and I to keep hacking at articles alternately until only that bare minimum that we cannot defend excising remains... at which point we will know what other articles must be written. The process works, unless we curtail it by cowardice or some "civility" or "authority" (both highly destructive to "truth")

Axel, thanks for sticking at this. I can see why you subordinate philosophers to schools, as opposed to the laundry list approach I took originally. What's missing from this article is a sense of how the schools relate or what common assumptions they make. For instance, it's pretty obvious that an embodied mind theory must relate to a theory in social constructivism - as cognitive bias must relate to culture bias - these schools are connected in non-obvious ways all of which deny Platonism, thus our argument re Erdos. The schools that have been concerned about notation and notation bias, e.g. what counting in base 10 might do to our conclusions or the ease or difficulty of reaching certain conclusions, or the infamous "||-|=|" example, are also inseparable from the proof process or "proof by computer" debates.

One way to deal with this is to add a list of questions at the end, with one or two at the beginning, that the layman can read and comprehend.

Another is to laundry list philosophers of mathematics, as I did earlier, with one-liners about their views or contributions.

But I like the "single phenomenon seen by many schools" approach, something like:

"||-|=|" demonstrates one of the key issues - notation bias. Depending on the number base or whether the phrase is seen this way or upside down as "|=|-||", and depending again on whether counting is in absolute numbers, it becomes true or false entirely depending on its orientation or interpretation.

I know of no shorter example to demonstrate why a philosophy of mathematics is necessary. Or why decisions like how to count, whether to distinguish directions in counting or only distance, and what social conventions to follow in sharing truth mathematically, need treatment outside mathematics.

It would be worthwhile to explain how each school sees this little gem, and to incidentally remark on where they think Euler's Identity, "+" or "i" came from. Those to me all seem to be central questions that differentiate the schools, and to set apart those who question the external consistency of the proof process itself (which only the social constructivists seem to do, claiming it as no more than metaphorical, as you say, just like a science) 24

I have removed these two paragraphs:

The main problem with Formalism is that it doesn't provide a criterion for which axiom systems to study. Why these axioms and not some others? For the formalist, all games are alike, except that some might be more fun or more useful than others. How is it possible that games played with paper and pencil could have any predictive powers over physical reality?

A major problem for Formalism is given by Gödel's second incompleteness theorem: when investigating a sufficiently interesting axiom system, we can never be sure whether the axiom system is inconsistent, i.e. completely pointless

I don't see how these two problems are any different for formalism than for platonism and logicism.

they aren't. Only embodied mind, social constructivism, state definite opinions about these problems, but previous attempts at this article that tried to lay out their views of consistency and metaphor vs. isomorphism were "unpopular". See the history for the edit that mentions Turing's biology investigations, more on Erdos, and other potential answers to this question. Restore any of that text that you find relevant to this issue, although it may require reworking. 24

In all three, mathematicians study those systems they find interesting, but they can't necessarily give a "criterion" for which axiom systems to study.

most mathematicians would say that such criteria are not their job I think, part of the "professionalization" and segementation of the field that has plagued it since Whitehead. 24

In all three, it's impossible to be sure that a sufficiently powerful system is consistent, which is unfortunate. (Also, that last sentence was wrong: we can never know for sure that such systems are consistent, but we can sometimes know that they are inconsistent).

how? inconsistent with what ? There's an intersection with philosophy of science and another intersection with economics, notably Amartya Sen who asked "is internal consistency of choice bizarre?" in a paper in the 80s. If it is, then there's no point investigating internal consistency... except insofar as if affects tolerances of infrastructural systems, e.g. test equipment, technology in general.24

24, an axiom system is called "inconsistent" if you can derive a contradiction from it, which means that you can derive anything from it. AxelBoldt

yes, of course, but that doesn't answer the question. There is still both internal and external consistency, and axiomatic inconsistency is normally considered "internal" (by the Platonist, Formalist, and Logicist schools, which frankly to me seem to be defying Goedel). What is traditionally meant by an "inconsistent" system is that the axioms, after a derivation process, yield a contradiction - proof of consistency is traditionally showing that an axiom set cannot remain consistent if any of the axioms are contradicted or removed, e.g. arithmetic relies on certain axioms and doesn't work if any of them are omitted. However, different fields of mathematics rely on different sets of axioms, and only through relatively intense procedures can they be reconciled, e.g. elliptic equations - modular forms equivalence. Basically all of mathematics is a bunch of islands relying on different sets of axioms. So, the embodied and social-construction (or proof-construction) schools argue that the thing that was traditionally believed to be internal consistency is in fact external (relying on validation by other human senses, other mathematicians), and that the thing that was traditionally believed to be external (the application of mathematics in science) is in fact internal consistency of choice as per a "paradigm", a self-reinforcing belief system of the validating mathematicians or a validating process. Then you have Sen's perspective, which is that any internal consistency of choice is bizarre and that (economic) isomorphism is just more metaphor on internal grounds. True to form, his economics is about refusing to measure the value of human time, simply maximizing the amount of it freed. This guy hit a chord somehow - his whole economic philosophy is lauded in India and in the anti-globalization movement - and he has the 1998 Swedish Bank Prize and the ear of the World Bank. So, there is some weight to this inside-out view of consistency and how one would discover a system to be "inconsistent" by external-to-proofs means - if a community of people cannot live by a system, as Erdos did, it becomes in some way socially inconsistent, and if cognitive science and neurobiology cannot be reduced to instructions capable of generating the mathematics-proving system (i.e. the mathematician's brain), it becomes in some way instructionally inconsistent. Turing was on this track when he studied limits on the expression of neurons. Anyway, all of this is secondary to the core issue of notation bias...(reinforcing this next line)...24

also, I repeat, if you can't explain "||-|=|", you simply cannot explain.24

I think those two problems apply uniquely to Formalism for the following reasons:

Platonists don't study axiom system, they study mathematical entities. If the axioms happen to be inconsistent, than the axioms are simply wrong and we pick new ones that describe the entities better. The entities come first, the axioms are just a crutch. For the formalist, axiom systems are the central object of study. A platonist can reject an axiom system as pointless, a formalist can never do that. The question which axiom systems to study is trivial for the Platonist: those that describe mathematical reality. Formalists don't have such a criterion. I'm not sure about Logicism though, how they find their axiom systems.

It could be argued that formalists try to study consistent systems, just as Platonists study mathematical entities. When a system is discovered to be inconsistent, the Platonists reject it as pointless, and the formalists reject it as trivially uninteresting. Then, the Platonists try to change it to be consistent, so it can better approximate the truth. The formalists try to change it to be consistent, so that it becomes interesting again.

I maintain that Goedel's results are a much more devastating blow to Formalism than to Platonism: the Platonist will simply say "See, I told you so, axiom systems never capture the whole truth that's out there. They are merely a (defective) presentational device."

It might be useful to make a distinction in the article between modern Formalists and those preceding Goedel. It's true that Goedel's results were a blow to the programs of people like Hilbert. But modern Formalists are no longer pursuing a powerful, complete, consistent theory of all mathematics, because they know that is impossible. A modern Formalist is more likely to describe his work as exploring interesting axiom systems, and would say that a system loses interest once it's found to be inconsistent. For the modern Formalists and Platonists, Goedel's theorems are equally inconvenient.

I agree with this more or less as stated. I think both schools must be rejected, or Goedel must be rejected.

First, that is not what was said, and second, Goedel was a Platonist. AxelBoldt

And, if you look carefully at Goedel, especially his ontological proof of God, it's apparent that he considered the "big Book of theorems" (Erdos-speak) to be something more organic than textual, as seemingly did Turing.24

Goedels ontological proof was intended as a study of the precise assumptions inherent in one of Leibniz' arguments, not as an absolute and convincing proof of God's existence.

I will attempt to phrase a statement about Goedel's impact on Platonists and Formalists. AxelBoldt, Monday, April 15, 2002

Regarding the last comment: it is true that we can sometimes know that certain system are inconsistent, but the sentence only referred to interesting axiom systems, which arguably only includes ones whose inconsistency hasen't been established. AxelBoldt, Sunday, April 14, 2002

If a system is said to be interesting today if its inconsistency hasn't been established yet, then the sentence in question was incorrect, because it says we will never know.

yes, agreed again. Perhaps thought the question is not so much "interesting" as "committed" or "actionable", i.e. do we choose to test the proofs or axioms for errors or inconsistency? If so, how? What motivates this? Is it the reputation of the mathematician(s) involved? The degree of investment in technologies or experimental apparatus that are proposed relying on the proofs? There are presumably many systems that are relied upon today that will be proven inconsistent under certain conditions or assumptions in future - i.e. the "islands" in mathematics will have "bridges" between them but will not necessarily form "land masses". Well, this is interesting, as it now appears to have touched on all the issues from my article which weren't covered in Axel's. However there is a question of how far to go here, since we're coming from three different universes. That's why I suggested "||-|=|" as a way to describe the issues to the layman, and then the theories' varying attitudes to logic, set theory, Euler's Identity, consistency and validation of proofs or reputation of mathematicians, could be sketched for those theories that consider it important. 24

Is it fair to say that the philosophy of mathematics is concerned generally with the question of what completeness, consistency, and proof actually are? Or, deepening that, why mathematicians of different schools think that certain things are "interesting" and others not? How an isomorphism differs from a metaphor, and why we would expect in the "technical" (mathematical and scientific) professions a higher degree of trust in a mathematically-stated theory than in a theory stated in other ways? There's also not enough of a relationship between this article and other philosophy articles, e.g. on metaphysics/ontology, epistemology, ethics - each of which has its own influence or parallel in how we view mathematics as a field. I don't think it's a reach to say that Euler and Galois represented two quite different ethics, and that their work reflects that to no small degree. 24

I removed the following:

Some common concerns variously divide and unify the schools, but there are few broadly-accepted 'universal truths' about mathematics. 20th century efforts tended to focus on the foundations problem in mathematics, and where characterized by such programs as David Hilbert's proposal to re-found mathematics as logic. After Kurt Godel demonstrated the futility of this approach, other foundations approaches were attempted. Debate at the end of the 20th century focused on quasi-empiricism in mathematics, spurred in large part by prominent mathematicians like Putnam who defended a non-Platonic form of mathematical realism using quasi-empirical arguments. Thomas Tymoczko and others noted the acceptance of quasi-empirical proofs and extended Putnam's analysis to mathematical practice (proving theorems, publishing papers). This further led to some attempts to more strictly define mathematics itself as a consequence of human internal cognitive structure, of emergent human psychology, or of social organization. At the end of the 20th century, most mathematicians would be likely to accept the statement 'mathematics is a language' but would argue about its special status or the necessity and universality of that language, e.g. to alien species, which had been a key point in Godel's analysis.

There is significant room for debate about the naming of the schools. Many mathematical realists (including Putnam) reject the name 'Platonism' as it assumes Plato's ontology is valid. Those described as 'social constructivist' would argue that they represent the quasi-empirical schools as a whole, and are 'social' only insofar as they recognize the importance of an ongoing debate and consensus-seeking among them. Finally, those mathematicians who focus on lateral unifying or bridging efforts between different fields of mathematics, e.g. the Langlands program of conjectures of isomorphisms between several different fields, may find themselves unrepresented in this list of 'schools'. The quasi-empirical (social constructivist) and empirical (embodied mind) theorists are more concerned with linkages between schools or subfields of mathematics, and would reject the term 'school' as implying some curriculum with a foundation ontology (e.g. what is taught first to the students of mathematics). Nonetheless, early to mid 20th century debates did tend to neatly divide neatly into 'schools', so for historical accuracy and ease of reference, the various positions will be presented in that form here:

Hilbert was a formalist, not a logicist. His program of proving the consistency of the basic axioms with finitary means was ended by Goedel, nothing else. Formalism is alive and well.
It is unfair to skip in this introduction over all the intervening schools and go right to quasi-empiricism, whatever that is.
Godel's analysis has nothing to do with universality of mathematics to aliens.
The Langlands program is not a part of the philosophy of mathematics.

There is really nothing I can save here. This is clearly written by 24. AxelBoldt 23:24 Dec 19, 2002 (UTC)