Is mathematics a science?

Symbolic logic is mathematics. Mathematics, as I'm sure you must know, is more than just number theory. Boolean algebra could be considered basic mathematics restricted to the "values" 0 and 1.Even the name "boolean" comes from George Boole, the mathematician who invented what he called "the calculus of logic." Logic (formal logic, at least) and math are the same thing, a connection that has been make starkly clear by the work of mathematicians like Russel and Godel, and others. Which is pretty much what I said. Mathematics is a language with a strict grammar - so strict that it ensures that properly-formed utterances (defined in math as "derived, via valid transformations, from accepted axioms") will be meaningful, even though they can be formed via the grammar without reference to their meaning.

I disagree with that. The more usual statement is that mathematical proofs can be translated into formal proofs. However mathematics papers can contain a lot more than proofs. It is a widely held view that Goedel's incompleteness theorem showed that this goal was (and is) unachievable. An online review of Goedel's theorem can be found here. Quine argued against that view in his highly respected paper "Truth by Convention".

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However mathematics papers can contain a lot more than proofs.

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Well, I'm not sure what area of mathematics you have studied, but having read many papers in pure mathematics I can say that a paper in pure mathematics is nothing but proofs. Maybe this is a peculiarity of functional analysis, but I doubt it.-

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It is a widely held view that Goedel's incompleteness theorem showed that this goal was (and is) unachievable.

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What Godel showed was (1) no system of logic complex enough to support Peano arithmetic can prove its own self-consistency, and (2) any system of logic complex enough to support Peano arithmetic will have well-defined statements that can neither be proven to be true nor proven to be false. I'm not sure if either of these are relevant to the exact point I was making, but maybe I wasn't clear as to my point.-

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Quine argued against that....

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Your link seems to be talking about a different subject altogether, but I've only glanced at it quickly, and so it is possible that I missed the point in the essay you feel is important.-My point was very specific.Starting with a decent system of logic, say the Russel-Saunders system, and the Zermelo-Frankel axioms of set theory, one can, through relatively simple, yet very tedious, applications of logical operations arrive at any theorem that has been correctly proved in any current journal or text book in pure mathematics. The prose expositions in these papers and texts are mostly a description of the logical operations that would be done had these proofs been done in logic.(There may be an introduction explaining the recent history of the subject, and maybe a paragraph or two in the body of the paper explaining the motivation for the proof; maybe this is what you meant when you said that a mathematical paper can contain "a lot more" than proofs, although "a lot more" is still overstating the case.)

There are other "grammers" that do not exist and yet are purely cognizable FROM Boole, x=x^3 and higher powers GIVEN THE PROSE in his{rules of thought} work. It remains to be seen if these "lingos" will be called "math" of today or "organons" anon. Claiming the plurification of Boole categorically will never occurr is more dangerous than asserting the small probability that God exists not existing.This message has been edited by Brad McFall, 09-01-2005 03:31 PM

My main area happens to be functional analysis. That's correct. But many people see it as showing that there are mathematical statements which cannot be proved from the axioms, but which we can see as being true (and prove true by means of a meta-theory).I'm not quite sure I agree with that conclusion, but it is what mathematical platonists tend to conclude. Sorry about that. It was the best link I managed to find, but it wasn't a very good one. Quine's paper is, quite explicitely, about mathematical truth. The link tries to describe it in a broader context, and that tends to confuse the issue. Incidently, the paper is probably not worth reading unless you are into the philosophy of mathematics. But why should we start with Zermelo-Frankel? Surely those are not encoded in our genes.From my perspective, the development of axiom systems such as ZF is an important part of mathematics. That is where much of the creativity and inventiveness is needed. By looking at only proofs derived from axioms, and ignoring the origin of the axiom systems, you omit some of what I consider to be the most important parts of mathematics. You cannot get from ZFC to, say, the Hahn-Banach theorem (of functional analysis) merely by tedious applications of logical operations. You first have to introduce new concepts, new definitions, new axioms (such as the defining axioms of a Banach space). And introducing such new concepts is, in my opinion, an important part of mathematics.

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My main area happens to be functional analysis.

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What area? Mine was operator algebras -- C*-algebras to be exact.

Topological groups and function spaces over such groups. However, I've been mainly doing computer science for quite a while now.

Do you have a Ph.D.? If so, what was your dissertation topic? Or did you do a dissertation for a Master's? Same question: what was the topic?Sorry for being nosy -- just curious about what other people work on (or have worked on).

Ph.D. was related to fixed point properies on topological groups.

But this is, in part, what considering different topoi is all about. And it's not exactly an inactive area of research... What makes you think we can't build Banach spaces from ZF? Not that I've ever tried but there's nothing especially exotic about a Banach space. Cool! My life seemed to revolve around topological groups for several years (though usually Lie groups in my case).

Sure, I agree with you. But you cannot do it "through relatively simple, yet very tedious, applications of logical operations" as was suggested in Message 18. For example, you would first need to introduce the notions from point set topology. That doesn't look too hard when you have at your disposal a rich natural language, but how are you going to do that when all you have is first order predicate calculus and the axioms of ZFC?

With much patience and even greater resolve How long did it take R&S to get to 1 + 1 = 2 ?But surely this is the same as asking how long it would take to come up with micro-biology armed with just a knowledge of QCD and electroweak? It's a long job but in principle it can be done.

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That doesn't look too hard when you have at your disposal a rich natural language, but how are you going to do that when all you have is first order predicate calculus and the axioms of ZFC?

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Are you saying that microevolution cannot lead to macroevolution given enough time? Yow!Edited to add:Looking over your post again, it seems to me that you may be saying that rewriting all of current mathematics using only predicate logic would be very difficult, very tedious, very time consuming, and, in the end, not very interesting or edifying. If that is what you are saying, then I completely agree -- I believe that was the main conclusion of Russell and Whitehead's work! All I am saying is that it is possible in principle to rewrite all of contemporary mathematics (that is, the axioms, definitions, and theorems and their proofs) using just predicate logic and the ZF set theory axioms.You may still not believe this, but since I am not actually all that interested in doing it myself (I am fascinated by symbolic logic, but mostly as a spectator) I'm not all that interested in proving it. Besides, Russell and Whitehead spend decades and only managed to produce a couple of volumes, and I suspect they didn't get much beyond elementary analysis.This message has been edited by Chiroptera, 02-Sep-2005 01:56 PM

No, they're not. Isn't that sort of the fatal flaw in mathematical platonism? That the axiomatic conditions that mathematical reasoning depends on are ultimately arbitrary? Not to be uncharitable to math platonists but their position seems to be based more on a need for their work not to simply be logic puzzles and symbol games rather than an actual "math" that exists somewhere in the universe.

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Not to be uncharitable to math platonists but their position seems to be based more on a need for their work not to simply be logic puzzles and symbol games rather than an actual "math" that exists somewhere in the universe.

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Heh. There was a faculty member at my previous institution; we were having a discussion about a paper that I was reading for a class he was leading.I made the statemnet that the theory we were discussing was a powerful tool that was invented to solve the problems we were discussing, or something to that effect.He replied (in a Russian accent), "Math is not invented. It is discovered."I replied, "Oh, yes, that is an important controversey, whether mathematics is invented or discovered.""Is no controversey. Mathematics is discovered."I've always maintained that discretion is the better part of valor, and so I quickly agreed with him and went on about the problems in the paper.