Wikipedia:Reference desk/Archives/Mathematics/2010 July 21

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July 21

Planck's constant divided by wavelength

What does planck's constant divided by wavelength signify? $p={\frac {h}{\lambda }}$ WHAT IS p? —Preceding unsigned comment added by 115.178.29.142 (talk) 03:44, 21 July 2010 (UTC)

p is momentum. Bo Jacoby (talk) 04:50, 21 July 2010 (UTC).

are there any axioms you can't get rid of?

are there any axioms that you must include in any axiomatic system, you can't get rid of it? (Or base a consistent system with its converse as an axiom)? 84.153.179.98 (talk) 09:57, 21 July 2010 (UTC)

The answer to the first question is no, an axiomatic system with no axioms is just fine. I can't parse the thing in the bracket.—Emil J. 11:33, 21 July 2010 (UTC)

The part in parentheses means "[Is there a statement A such that] there is no consistent system in which one of the axioms is the converse of A", though the OP probably wanted negation rather than converse. Maybe $\forall x(x=x)$ qualifies - a consistent system can't AFAIK have an axiom $\neg \forall x(x=x)$. -- Meni Rosenfeld (talk) 11:52, 21 July 2010 (UTC)

Only if the OP's meaning of an "axiomatic system" is a theory in classical first-order logic. If we are talking arbitrary axiomatic systems, then a single axiom can never be inconsistent by itself (unless the language of the system only consists of one formula).—Emil J. 13:33, 21 July 2010 (UTC)

OP here. I just added the parenthetical clause (yes, I meant "negation") to complete "are there any axioms which you MUST include in direct or negated form as one of your axioms if you are to have a consistent axiomatic system".84.153.179.98 (talk) 16:29, 21 July 2010 (UTC)

What is your definition of an inconsistent axiomatic system? Isn't any axiomatic system with a false axiom automatically inconsistent? Meni's axiom is clearly false, whatever universe you are working with (even the empty universe). --Tango (talk) 16:34, 21 July 2010 (UTC)

@Tango: Axiomatic systems are purely syntactic, truth and falsity have nothing to do with it. Furthermore, what you are saying would only apply to semantics of classical first-order logic anyway, as I already wrote. In general, there is no reason why you should interpret the whole thing in a first-order structure, why you should interpret = as identity, $\forall x$ as universal quantifier, and $\neg$ as negation, and therefore there is no reason why the formula should be a priori "false". Oh, and the definition of an inconsistent axiomatic system is that it proves all formulas.

@84: Adding axioms can never turn an inconsistent system into a consistent one.—Emil J. 16:46, 21 July 2010 (UTC)

Not a Number in IEEE floating point never equals itself. That's an easy way for people to test for problems like 0/0. ;-) Dmcq (talk) 17:11, 21 July 2010 (UTC)

You don't need to be using first-order logic for your notation to have some meaning. If you can't know what the symbol "=" means then you can't use it, since it makes your axioms meaningless. The notation is inconsequential, it's the meaning of the axioms that is important. The meaning of $\neg \forall x(x=x)$ is clear (it means there is some x in the universe that doesn't equal itself) and that meaning is self-evidently false (I would argue that this means NaN cannot be considered an entity, it's just a notational shorthand). Since, as we know, a false premise can be used to prove anything, any axiomatic system including that axiom is inconsistent under your definition. --Tango (talk) 22:30, 22 July 2010 (UTC)

No, if you are working in a system rich enought that it's sound to infer $\forall x(x=x)$ then $\neg \forall x(x=x)$ is clearly inconsisent. However OP did not specify any such system. $\neg \forall x(x=x)$ is consistent in first order logic when working without equalitySorry I had to change my text here, I have always worked without equality so I assumed that inclusion of equality had to be stated., and as such is a counterexample to the statement that it is incosistent by itself. I also have a beef with the statement "Since, as we know, a false premise can be used to prove anything", since there are logical systems that avoid the principle of explosion. However this is irrelevant to the discussion at hand though, since as long as we are working in a system where $\neg \forall x(x=x)$ can be shown to be false then the axiom set including it is inconsistent under that system even if the system is nontrivial. Taemyr (talk) 23:07, 22 July 2010 (UTC)

Can you supply a reference for your claim that there are systems with a concept of equality (the use of the symbol '=' means we are clearly working in a system with equality) but where it is possible for something to not equal itself? --Tango (talk) 23:23, 22 July 2010 (UTC)

In primitive recursive arithmetic (PRA), the language is not powerful enough to form quotient spaces as one does routinely in set theory. Nonetheless, one can define the binary relation called "equality" in such a way as to replace the quotient construction. For instance, Skolem defined non-standard models of N in the 1930s. One can then form the field of fractions, which mimicks the rationals. If one takes those elements that are finite, and defines "=" among them to be the relation of being infinitely close, then one obtains a kind of a model of the real line in PRA. Here "equality" is interpreted as the relation of being infinitely close. My understanding is that one of the main issues in category theory is the complexity of the issues of "equality" and "equivalence" in mathematics. Tkuvho (talk) 08:40, 25 July 2010 (UTC)

Surely a number is infinitely close to itself, so that is not a counterexample to my claim. There are lots of different meanings to "equality" in different fields, but they are all equivalence relations (which includes symmetry), to the best of my knowledge. --Tango (talk) 15:06, 25 July 2010 (UTC)

Did you mean reflexivity? -- Meni Rosenfeld (talk) 08:17, 26 July 2010 (UTC)

Yes, thanks. --Tango (talk) 14:40, 26 July 2010 (UTC)

Sorry, but I don't see any reason to assume that '=' is equality. There are usually axioms present that tells us that this is so, but those axioms are not present in the considered axiom set. Taemyr (talk) 17:11, 26 July 2010 (UTC)

what if there were a proof

what if a proof were discovered, requiring NO axioms, that any system with at least 1 axiom is inconsistent? Would mathematicians everywhere hang up their hats and resign and retire? 84.153.179.98 (talk) 10:01, 21 July 2010 (UTC)

Well, such a proof would be incorrect because there is a simple counterexample. The system with the single axiom "True" is consistent - not a useful system, but definitely consistent. Gandalf61 (talk) 10:50, 21 July 2010 (UTC)

use your imagination. Imagine that there is a flaw to the counterexample, and in fact, even the system with the single axiom "True" is inconsistent... what then? 84.153.179.98 (talk) 15:48, 21 July 2010 (UTC)

You are asking me to imagine a nonsensical situation, in which provably false things can be true. I suppose in such an imaginary world I would hang up my hat and retire. I would also put on my hat and continue working for pay- no sense obeying basic rules of logic in such a world, right? Staecker (talk) 16:22, 21 July 2010 (UTC)

Well, if you start with a false assumption (and the single axiom "true" being inconsistent is self-evidently a false assumption) you can prove anything, so you can chose what would happen then yourself and you'll be right. --Tango (talk) 16:25, 21 July 2010 (UTC)

Before we go further... what is the axiom True? What does it hold? 84.153.179.98 (talk) 16:31, 21 July 2010 (UTC)

It is simply a logical statement that is always true. --Tango (talk) 16:36, 21 July 2010 (UTC)

Are you referring to this logical statement when you write True or are you quoting it in full? If you are referring to it, can you quote it instead? If you are quoting it, can you help me understand why it is a statement? WHen I compare it to a statement like "Two sets are equal (are the same set) if they have the same elements" it seems "True" doesn't hold anything at all. By that metric, is {} a statement, is 42 a statement? I'm not dumb, I just don't know how mathematics view statements... 84.153.179.98 (talk) 16:58, 21 July 2010 (UTC)

This is not a discussion forum. If your question cannot be written in the form: "Help me find a reference on the topic...", then it should not be asked and should not be answered. Your question is currently in the form: "Please discuss this nonsense that I made up." -- kainaw™ 16:34, 21 July 2010 (UTC)

sorry, I am curious as to the ramifications of all but the most most trivially useless mathematics being proved inconsistent a priori. It can be written as you suggest. 84.153.179.98 (talk) 16:58, 21 July 2010 (UTC)

{} and 42 aren't logical statements because they don't evaluate to a true/false value, while a statement like "Two sets are equal (are the same set) if they have the same elements" does. The statement True also evaluates to a true/false value, although in a much more trivial way then "Two sets are equal (are the same set) if they have the same elements".

Also, I think your question is ok. Although this is technically a reference desk, we get a lot of "help me understand this concept" questions that people are usually happy to answer. Rckrone (talk) 19:36, 21 July 2010 (UTC)

thanks for this final sentence, as now I do need help understanding concept since I am now utterly confused. I thought, when crafting axiomatic systems, that it doesn't matter whether the axioms are true! They just need to be consistent in their ramifications (if you want to end up with a consistent axiomatic system, that is). So, by that logic, the axiomatic system "1. False" is no worse than the axiomatic system "1. True", since as long as it's consistent, it doesn't matter if the system's axioms are "true". Am I right on this count? 19:48, 21 July 2010 (UTC) —Preceding unsigned comment added by 84.153.184.144 (talk)

also, is {} itself a self-consistent axiomatic system, just one where you can't prove anything useful? 84.153.184.144 (talk) 19:50, 21 July 2010 (UTC)

An axiomatic system with False as the only axiom (or as any of its axioms for that matter) is inconsistent. If we choose a statement A, then False implies A and False implies not A. Emil J mentioned in response to your first question that an axiomatic system with no axioms is fine. I'm pretty sure it'll get you the same stuff as the system with True as the only axiom. Rckrone (talk) 00:53, 22 July 2010 (UTC)

Why does "False" imply something (e.g. A; also: not A), whereas "True" doesn't imply anything? This doesn't make sense to me. It seems to me neither of them are statements that have "implications", unlike the statement that two sets are equal if they have the same elements... 92.229.13.132 (talk) 10:22, 22 July 2010 (UTC)

Because people here are confusing two different definitions of an axiomatic system, as I already pointed out twice. If an axiomatic system is simply a consequence relation given by a set of rules (and axioms, which are just rules with no premises), then the system axiomatized by False is, up to an inessential change in notation, the same as the one axiomatized by True, and it is consistent. If, on the other hand, an axiomatic system is an extension of first-order classical logic by additional axioms, then the system axiomatized by True is the same as the system with no axioms (as True is already provable in the underlying logic), whereas the system axiomatized by False is inconsistent, because anything can be derived from False by rules of classical logic.—Emil J. 11:07, 22 July 2010 (UTC)

(edit conflict)Because for any propositions P and Q,

$(P\land \lnot P)\rightarrow Q$

is an identity. So if P is both true (because it is an axiom) and false (because its value is "False") then any proposition Q can be proved as follows:

$P$

$\lnot P$

$(P\land \lnot P)\rightarrow Q$

$Q$

Remember that we are using the proof theory model of truth here - a proposition is true in a system if it can be proved in the system, and false if its opposite (formally, its "contradiction") can be proved in the system. Gandalf61 (talk) 11:21, 22 July 2010 (UTC)

All right, this is really good stuff. Now can you tell me what it means? Specifically, could you read the notations above in English sentences, since the notation is new for me. Thanks. 92.229.13.132 (talk) 11:33, 22 July 2010 (UTC)

See our article on propositional calculus. Gandalf61 (talk) 12:12, 22 July 2010 (UTC)

Actually I think the scenario is not so unimaginable as others have suggested, and it's a useful piece of conceptual analysis to explore this sort of thing.

But we need to state it more precisely. In Hilbert-style deduction, we have just one rule of inference, namely modus ponens, but we have a bunch of logical axioms, which are not ordinarily counted as "axioms". So one way of phrasing the question would be, "can we derive a contradiction, Hilbert-style, using the logical axioms alone"?

Alternatively, we could do Gentzen-style deduction (see sequent calculus), which has no logical axioms at all, but about fourteen rules of inference. So we could similarly ask, "can we derive a contradiction, Gentzen-style, using no axioms at all"?

These are both, on their face, combinatorial questions. We believe very strongly that the answer to both is "no". But it is reasonable to ask for a justification of that belief. The justification does, in my view, turn out to have ontological content; it relies on a belief in at least some minimal abstract objects, and probably on a belief that there is no finite limit to the size of such an object.

Now, as to your other question, about whether it matters whether the axioms are true -- yes, it absolutely matters whether they're true. Mathematics is not just a formal game — it attempts to describe something, and if it describes it wrong, then it's wrong, even if the axioms are formally consistent. --Trovatore (talk) 01:12, 22 July 2010 (UTC)

Oh, I forgot: You also asked what we would do in such a scenario. My guess is, not much different in practice. Mathematics works; this is observable fact. In a scenario such as you describe, we'd have to rethink very very hard why it works, but I wouldn't really expect it to just stop working. --Trovatore (talk) 01:15, 22 July 2010 (UTC)

I came across a similar question many years ago wikisource:A Book of Nursery Songs and Rhymes/Nursery Songs/XXXV. IF ALL THE WORLD WERE PAPER. :) I think I'll wait until it happens before worrying about it. Dmcq (talk) 11:25, 22 July 2010 (UTC)

Yeah, it's not about being worried. And it's not about finding a backup plan for an eventuality that, to put it mildly, none of us expects to happen.

It's about exploring, and possibly clarifying, the concepts involved.

Can we imagine a coherent response to such an eventuality, or would we all really just decide to start breathing purple and walking on seventeen? If we can imagine a coherent response, does it maybe impact the question of whether even logic has an empirical component, at least in terms of how we come to know it?

I think we have a couple of possibly relevant articles, with titles like Is logic empirical? and Quasi-empiricism in mathematics. --Trovatore (talk) 20:12, 22 July 2010 (UTC)

codimension?

Codimension is the difference between dimensions of two spaces. If the dimensions are n and m, the codimension is n-m. Is there a word for n/m in this case? (Assume it's an integer if you like.) Staecker (talk) 16:19, 21 July 2010 (UTC)

Not that I'm aware of. What would be the use of that number? Things only get named if somebody uses them. Since dimensions are integers (excluding weird fractal stuff), I wouldn't expect a situation requiring you to divide two dimensions would come up naturally. --Tango (talk) 16:28, 21 July 2010 (UTC)

I'm not just looking for names of random things, and I wouldn't be surprised at all if there isn't a word for what I want. I have proved a bunch of theorems about maps from X to Y which rely on the hypothesis that X is of dimension kn and Y is of dimension n for some integers k and n. I'd like to just describe this as "X and Y have dimension-quotient k" or something like that. I don't just want to say that the dimension of Y divides that of X because I want to mention specifically the value k. If there's no word, I'll just make one up (I just did!), or just say it the long way each time. But I'd like to know if there's already a word. Staecker (talk) 16:45, 21 July 2010 (UTC)

Setting up triangles for the ambiguous case

Given the dimensions of an SSA triangle, does the order in which I place them (A-B-C) matter? I am having trouble setting up my triangle so I can easily calculate the height. for example, given Gamma = 60deg, side b = 20, and side c = 10√3. how would I set this up? I've calculated the height to be 8.7 and 17.3.... but I'm pretty sure I am wrong. How can I do this? Thank you. —Preceding unsigned comment added by 24.205.126.22 (talk) 19:13, 21 July 2010 (UTC)

I think you have two problems: First, a given triangle has 3 different heights, one from each of the sides so if you want to find the height you need to specify which one. The other problem is that SSA usually doesn't describe a unique triangle. There are typically two triangles with a given side b, side c and angle γ opposite side c. Is it supposed to be a right triangle? Rckrone (talk) 19:50, 21 July 2010 (UTC)

The sixty degrees can be either of the angles not between the two given sides. Dbfirs 19:57, 21 July 2010 (UTC)

The usual convention for greek letter angles is α opposite a, β opposite b and γ opposite c. Anyway, I should clarify, in this case the sides and angles do describe a unique triangle because of how things work out assuming γ is opposite c, but in general that's something you need to be careful about regarding SSA. Rckrone (talk) 20:03, 21 July 2010 (UTC)

Sorry, I hadn't read the question properly. The triangle is not a difficult one. I suggest that you calculate the height from A to side BC and think about your result. Dbfirs 20:22, 21 July 2010 (UTC)

My original question was about properly setting up the triangle to get accurate results. I found that when I swapped the positions of B and C and calculates the height I got the answer I was looking for (height = side C). I understand which sides are opposite the angles, but what I need to know is where to place the corners of the triangle (A B C) so I can get the correct answer the first time. Thanks for the help thus far. —Preceding unsigned comment added by 64.134.236.5 (talk) 01:25, 22 July 2010 (UTC)

Ah, I see. The convention is that side a is opposite to vertex A, side b opposite to vertex B and of course side c opposite to vertex C. In the UK, the angles are usually named after the vertices, whereas, in the USA, the first three letters of the Greek alphabet are used for the angles. Dbfirs 06:09, 22 July 2010 (UTC)