Talk:Mathematics: Difference between revisions

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This is the talk page for discussing improvements to the Mathematics article. This is not a forum for general discussion of the article's subject.

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This should include the William Lowell Putnam Mathematical Competition for college undergraduates in the US and Canada. —Preceding unsigned comment added by 166.82.218.97 (talk • contribs)

Under "Notation, language, rigor", there's a statement "modern mathematical notation has a strict syntax". I'd like to see a citation to back this up, or indeed a link to a wp page describing "the" strict syntax, especially since there's a link for musical notation. In my experience there are a variety of different notations used in math, often varying within fields, between authors in the same field, and sometimes even on the same page of exposition. Gwideman (talk) 14:42, 11 December 2009 (UTC)[reply]

I think the idea being expressed here is not that there is one and only one strict syntax for all of mathematics but rather that the syntax, whatever it may be, is strict, and you cannot, for example, write x+1^n when you mean (x+1)^n. It could probably be expressed more clearly. Rick Norwood (talk) 20:09, 11 December 2009 (UTC)[reply]

Perhaps "precise" would be a better word than "strict"?Paul August ☎ 20:15, 11 December 2009 (UTC)[reply]

I agree that the objective is to convey an idea precisely from author to reader. And agreed that within a subset of notation, the writer has to write it in accordance with that idea, and the reader has to read it the same way. However there are all sorts of ambiguities: Should one write d/dx, or use prime, or a dot? If one reads x', does that mean derivative of x, or a second variable that may or may not be related to an x somewhere else? What does a caret ("hat") mean? Are square brackets an array subscript or an operator of some kind? Does the author's choice of beginning- and end-of-alphabet letters (a, b, c vs x, y, z) distinguish coefficients from variables, or not? What about i, j, k: Variables? Subscript? Sqrt(-1)? Much relies on context and upon the reader being familiar with the author's mathematical "culture". Which is to say, the notation not strict or precise according to some consistent standard, but relies on the reader divining which convention the author is using, or sometimes even what convention the author has invented on the spot.

Anyhow, I'm certainly not qualified to characterize this -- I was hoping someone would point to a reference or two on the subject! Gwideman (talk) 02:22, 12 December 2009 (UTC)[reply]

The following sentence is pasted directly from the lead: "Although incorrectly considered part of mathematics by many, calculations and measurement are features of accountancy and arithmetic."

I am not an authority on mathematics, so perhaps I fall under the category of the many who make incorrect assumptions according to that statement, but the arithmetic lead on this same wikipedia mentions that subject specifically as being "the oldest and most elementary branch of mathematics".

There is definitely a contradiction here.

Measurement too might be argued to be a branch of mathematics (i.e. geometry ("earth-measuring")); Euclid's Elements, which is a treatise on geometry, is specifically mentioned in the lead as an example of mathematics. Zalmoxe (talk) 16:12, 24 December 2009 (UTC)[reply]

There have been several changes for the worse in the lede, including the sentence you mention and another sentence claiming that numerology is mathematics. I've tried to restore things to the stable lede that has existed for a long time. Rick Norwood (talk) 13:47, 6 January 2010 (UTC)[reply]

Thank you Rick. This new old lede [interesting ortographic revival there btw, I wasn't aware of it until now] seems more consistent. Zalmoxe (talk) 05:11, 18 January 2010 (UTC)[reply]

To say that mathematics studies physical objects is misleading. Of course, mathematics is applied to the physics of motion, but the mathematics is first developed with reference to abstract shapes, such as triangles, before considering questions such as the irregularities, imperfections, and discontinuities of any physical triangle. Mathematics first considers ideal motion, usually of a point mass not subject to friction, air-resistance, or uncertainty, before considering all of the messy reality of the physics of actual motion in the real world.

Which should the lede state, abstract objects or physical objects?

Rick Norwood (talk) 14:45, 6 January 2010 (UTC)[reply]

I agree that saying that mathematics "studies studies physical objects is misleading". But the article does not say that. What it says is that "mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects" — that's quite a different thing. The idea in this sentence is to give a sense of where mathematics came from, not what it is. Paul August ☎ 17:05, 7 January 2010 (UTC)[reply]

Good point! Rick Norwood (talk) 12:55, 8 January 2010 (UTC)[reply]

I am not a registered user, but can somebody find a citation for maths vs. math. I'm an American having an argument with a British friend over whether mathematics is plural or singular. I say it is singular, therefore mathematics should be shortened to math. But he insists that mathematics refers to a diversity of strands and is therefore plurally maths. —Preceding unsigned comment added by 137.205.222.238 (talk) 13:08, 13 January 2010 (UTC)[reply]

As well argue over the spelling of "color"/"colour". They do it one way in the US and a different way in England, and there is no right or wrong. The main reason to use "math" is that it gave Tom Lehrer a rhyme. "The guy who taught us math, who never took a bath, acquired a certain measure of renown..." Rick Norwood (talk) 15:06, 13 January 2010 (UTC)[reply]

I've read a topic on this, and the whole math/maths thing can be summed up to a) the Germans and/or the French as American math developed from German mathematik and French mathematiques (s is silent) b) the fact that even in britain "Maths IS my favorite subject" Math is while a plural noun an uncountable noun and treated as singular, because you'd never say "Maths ARE my favorite subject" as subject would have to plural as well and Mathematics is a subject and not subjects.96.3.141.210 (talk) 19:08, 17 January 2010 (UTC)[reply]

And what about "physics", which is grammatically similar but is "la physique" in French (also "mécanique" is considered a distinct subject here, but I doubt whether that explains the singular). Too bad nobody abbreviates it to "phys", and even then, this wouldn't help... Marc van Leeuwen (talk) 10:44, 25 January 2010 (UTC)[reply]

Sorry if this subject has already been discussed (I tried to check) but I find the second paragraph of the lede utter nonsense:

There is debate over whether mathematical objects such as numbers and points exist naturally or are human creations. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions".[5] 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."[6]

Maybe my problem is that I misinterpret "exist naturally"; I can't think of any other meaning than "exist in nature", and in that case I find it hard to imagine any serious debate about the question: numbers and other mathematical abstractions do not exist in nature. I will admit the existence of a black hole at the other end of the galaxy, but not that of a complex number (or natural number for that matter, say 0:-) in my back yard. This is not to say that mathematical abstractions are (uniquely) human inventions, I would expect any extraterrestrial civilisation to come up with the same, or very similar, abstractions.

The two citations do not express opposing views either. Mathematics draws necessary conclusions, but those conclusions apply to the object of mathematics, that is to abstractions. Such conclusions only apply to reality insofar as reality is willing to abide by the laws of mathematical abstractions; since this is not certain, neither is the application of conclusions to reality, which is what Einstein appears to say. Marc van Leeuwen (talk) 10:23, 25 January 2010 (UTC)[reply]

• You need to inform yourself that although one can follow and believe some philosophical line there are others. Take a look at Philosophy of mathematics, in particular to the section Mathematical realism.  franklin  11:31, 25 January 2010 (UTC)[reply]
• Also what one can understand by "exists", or "nature", or "naturally", or even "exists naturally" have many many different possible answers. That paragraph is pointing to the problem of the ontological existence of mathematical objects.  franklin  11:58, 25 January 2010 (UTC)[reply]

I'm not taking any particular philosophical position, I'm just saying this paragraph is not making any clear sense. If "existing naturally" is a reference to one or various ontological positions, this should be made clear. I would have less difficulty with "There is debate over the kind of existence, if any, that can be attributed to mathematical abstractions such as numbers", as it more clearly indicates that the discussion is about "existence", rather than about numbers themselves. Also I do not believe that the two citations belong to the ontological debate you refer to. Peirce does not refer to existence or reality at all, and Einstein most probably (as a physicist) is talking about physical reality, not some kind of Platonic reality. So I'm just saying this paragraph is lousy. Marc van Leeuwen (talk) 12:56, 25 January 2010 (UTC)[reply]

• Well, you starting saying that numbers do not exist in nature and that you expect extraterrestrials to come out with the same abstractions (which is a valid but still very particular philosophical position) as your point for labeling the paragraph as utter nonsense but let's forget that you did. About other ways of writing this paragraph, which is one in the lede of a very general article, there are probably hundreds of possibilities. If you bring some alternative it can be seen if it is a better choice. I don't find the current version a bad one as holes can always be found in three lines about a topic that takes volumes to explain. The two quotations, although there are probably several alternatives, refer to problems linked to the ontology of mathematical abstractions. Peirce's key word is "necessary" and the level in which it is necessary, and Einsteins' to to a possibly antagonistic position again depending what we take as the meaning of "reality". This topics we very hot at Einsteins time and environment. He and Godel probably talked about these things a lot. I wouldn't dare to narrow what he really meant with "reality" there. Again, several options for quotations are available. If you bring some options it can be seen whether they are more suitable than the current ones. One thing that can help is a link inside that paragraph to a more specific article explaining all this. An option is philosophy of mathematics.  franklin  13:59, 25 January 2010 (UTC)[reply]

At worst, numbers exist naturally as singularly probable singularities in nonlinear dynamical systems that exist naturally. thus, to say that they don't exist naturally would be like saying that the non-trivial zeros of the Riemann zeta-function don't exist. Kevin Baas^talk 16:21, 26 January 2010 (UTC)[reply]

• No better explanation exists.  franklin  17:30, 26 January 2010 (UTC)[reply]

Not so, I came up with a clearer one: Consider the vibration of a string. Because of certain mathematical relationships, the harmonics will always be at 2x,3x,4x,5x, etc. of the base frequency. Any thing not a natural number multiple would only dampen or eliminate the vibration. Since "1" is defined simply as "There is a number one.", to say that "natural numbers exist naturally" one only needs to show that there's some kind of physical law that strongly prefers natural number multiples. And that is precisely what we just showed. As they say in the industry: Q.E.D., b@%&@! Kevin Baas^talk 16:10, 29 January 2010 (UTC)[reply]

• I was being ironic, sorry. You need to read a little more about this topic. I guess you can start reading philosophy of mathematics and then whatever related book you can find.  franklin  17:51, 29 January 2010 (UTC)[reply]

Actually for any physical string (and any finite amplitude of vibration) the harmonics will not be exactly as 2x, 3x etc, because of parameters like stiffness of the string that are ignored in the mathematical model of the string. Does that show that natural numbers are not naturally exact integers? No, it just shows that this particular physical problem does not have the exact properties of the mathematical model.

I.e. regardless of the correctness or incorrectness of the diff eq. describing the model, there are more concurrent diff. processes at work. That doesn't diminish my point.

My point was that, so long as there is positive and negative feedback, periodic relationships will inexorably emerge from natural phenomena. And what can "numbers exist naturally" mean if not precisely that? But I digress --- just wanted to make sure my meaning was clear. Kevin Baas^talk

But enough of this; in spite of my initial somewhat provocative language, I did want to pose a serious question, not evoke a philosophical debate. The first sentence of the paragraph is not clearly formulated; at best it indicates a somewhat esoteric philosophical debate that I think does not deserve to be mentioned in the lede of an article on mathematics. The citation by Peirce is not about ontological questions, but an introduction to broadening the sense of the term "mathematics" to more than purely quantitative questions (notably he mentions quaternions as not being covered by that); while understandable in the late 19-th century context, such broadening is no longer relevant since it has been completely integrated into mathematics already. Einstein's quote may be pertinent, but is more about the role of mathematics in the sciences than about the philosophy of mathematics itself. Altogether, the paragraph seems less than helpful to readers who want to learn about mathematics. Marc van Leeuwen (talk) 16:51, 29 January 2010 (UTC)[reply]

• Well, about the form of the paragraph I agree about the possibility of improving it. About the pertinence of having such an information in the lede i think it is more than indispensable. This is the main article about a science or a discipline ( or whatever mathematics is). The first thing you do when you start to study a science is to define its subject of study. In the case of the other sciences, although it is equally difficult to define the subject for most of the people is clear (at least intuitively) what they are. Take for example Biology, it studies living beings. Although it is very hard to define (appropriately) what is a living being it is mush more clear (at least intuitively) what they are. In the case of mathematics, even from a materialist perspective the nature of the subject matter of the science (if it is a science) is less clear. I guess I don't have to point out what is the importance of defining the subject matter of a science but let me just say that that defines its purpose and applicability which then determines its evolution (for example if it develops more as a game like chess, or to things with direct application or both). For example compare the volumes of publications in Physics and Mathematics that are about situations that do not correspond to reality. This example is interesting since these two sciences are very similar in many of the techniques they use. In Physics you always have the oracle of "reality" that is the ultimate verification. As well as in Mathematics it is valid the question about what is the nature of this "real world", and whether it even exists. Now in mathematics the objects have a different nature. There is also no such "oracle" validating the results or even the axioms. that is why the question becomes even more important in this case.  franklin  17:51, 29 January 2010 (UTC)[reply]

The use of Newton is needlessly anglophilic. —Preceding unsigned comment added by 129.120.193.30 (talk) 23:08, 16 February 2010 (UTC)[reply]

Isaac Newton is almost universally acknowledged as one of the greatest mathematicians of all time. To omit him would be anglophobic. Rick Norwood (talk) 13:43, 23 March 2010 (UTC)[reply]

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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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

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)[reply]

(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)[reply]

"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)[reply]

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)[reply]

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)[reply]

::: 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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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:

${\displaystyle \forall x\lnot P(x)\land \forall x\lnot Q(x)\land \forall x\lnot (P(x)\land Q(x))}$

${\displaystyle \Rightarrow }$

${\displaystyle \forall x\lnot (P(x)\lor Q(x))}$

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

(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)[reply]

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)[reply]

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)[reply]