Is mathematics a science?

From Merriam-Webster: Under definition 2 (especially 2a) mathematics can be described as a science, definitely.However, when I think of "science" I am thinking of definition 3; in this case, I would say that mathematics is not a science.

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Many of the most important mathematical research papers consist mainly of informal prose with relatively little use of symbols.

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This is true, because mathematicians think in terms of the concepts (and, like anyone else, would find reading a paper full of nothing but abstract symbols rather dry). However, in principle, any paper in pure mathematics can be translated into pure symbolic logic, with symbols representing the various objects and relations, and each step being due to a specific rule of logic.At the beginning of the last century, Russell and Whitehead began a program to translate all of mathematics into pure symbolic logic; it turned out to be much more involved than they had anticipated and couldn't come close to translating all of mathematics. But they did show that ultimately, all mathematics is subfield of logic.-

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There is a lot more to mathematics than symbol manipulation.

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The prose that you see in a paper in pure mathematics is describing what is actually symbol manipulation. Nonetheless, you are correct; it does take a certain amount of creativity and intuition to figure out what is an important enough theorem to prove, and to figure out the path (the exact manipulations of symbols, or the prose description of those manipulations) the proof is going to take.

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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.)

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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.

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).

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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

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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.

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

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Cool! My life seemed to revolve around topological groups for several years (though usually Lie groups in my case).

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My graduate studies was done at an institution that was predominantly algebraists, with a small geometry/toplogy contingent. Yet, somehow, I managed to leave without learning any more algebra than was necessary to pass the qualifying exams. Weird place.

What I meant by "weird" was the way people didn't seem very interested in discussing their research -- which is why I, not being an algebraist, could avoid learning any algebra. It seems very different than when I was studying physics -- where everyone was always talking about their research -- but maybe this is more typical of departments of mathematics.

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It's those of us in theo physics that tend to get converted to at very least a weak Platonism because we keep finding all of the maths supposedly reserved for "logic puzzles and symbol games" firmly embedded in the universe.

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Yeah, I can see how the illusion that your work has something profound to say about reality is even stronger for the theoretical physicists.

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Hmmm... that's Star Trek isn't it? It's so hard drawing a distinction sometimes

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I understand. Everytime I read the news, I feel like I'm in a film noir.

I would think that the selection process would be the person writing the proof.Not being a Platonist, I would say that theorems don't prove themselves.

Sorry that it's taken me so long to get back to you, Brad.

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I really do think that whatever that "math" is it is pure in itself no matter how it sorts difference of natural and artifical selection in the statistical normal distribution approximation.

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While my contention is that modern mathematics is pure symbol manipulation, I certainly do not dispute that it is the mental concepts given to the symbols by the human mathematicians that make the practice of mathematics possible. I should also admit that it is, in the end, the correspondence between certain concepts in mathematics and concepts in the sciences (especially the physical sciences) that largely determine which fields of mathematics and the directions of research are "useful". As it should be, since the main motivation for mathematics (at least for non-mathematicians) is its utility.