Oops. I didn’t see this post when I referenced intuitionism.
Arguably, an even more important, more fundamental, proof by contradiction was done by Aristotle (I think it was Aristotle) to demonstrate the irrationality of 2[sup]1/2[/sup].
It was done by one of Pythagoras’ students, not Aristotle. According to legend the student’s name was Hippasus.
Is there a proof that \sqrt{2} is irrational that doesn’t use contradiction?
The wikipedia article on square root of 2 Square root of 2 - Wikipedia outlines a constructive proof
That must have gone over well in Pythagoras’s circle. (His triangle?)
Quite well, if by “gone over well”, you mean “they drowned him at sea for it.” (According to legend, at least.)
Among mathematicians there are:
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Classical mathematicians that includes at least 95% of all practicing mathematicians. We (since I included myself) accept proof by contradiction, proofs based on the axiom of choice and both finite (ordinary) mathematical induction, as well as transfinite induction (over all ordinals).
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Constructivists. They do not accept any argument that relies on an existence proof, but insist that existence is demonstrated by actually describing a construction. I believe they normally accept finite induction, but reject transfinite induction. I imagine most of them accept the usual argument for the irrationality of sqrt(2).
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Intuitionists. They reject any argument based on excluded middle (that is, proofs by contradiction) and are usually constructivists as well. This would appear to be where your instructor fits.
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Ultrafinitists. The name Yesenin-Volpin, a Russian mathematician and dissident comes to mind. He (I hesitate to say “they” since he has few, if any, followers) rejects all infinite sets. He claims that numbers like 10^{10^{10}} are meaningless. Presumably there is no sharp boundary between meaningful and meaningless numbers, but they gradually lose meaning as they get larger.
Your instructors statement that papers with indirect proofs will be rejected is factually wrong and, assuming he doesn’t actually believe it, is a lie. If he does believe it, he is deluded.
There are a class of proofs that I call “lazy proofs by contradiction” that gain in simplicity and clarity when turned into direct proofs. Which I often see in one of my collaborators. I change them and he agrees, but he never learns the general principle. But many proofs are inherently non-constructive. Since mathematics builds on itself, nearly all of modern mathematics would disappear if even mild constructivism took hold.
For a fascinating example of how at least some of it could be recovered–albeit with a great deal of effort–see Errett Bishop’s book, Constructive Analysis. He explains clearly what he is doing and why. There are a couple of fascinating details. If you use his definition of continuous function (essentially, uniform with a constructive epsilon on any closed interval) then all constructive real-valued functions on the reals is continuous. However, there is (and apparently cannot be) any constructive proof of this fact. He interprets this to mean it is not a theorem, but also he will never be able to construct a counter-example. On the other hand, he claims there is a constructive counter-example to Brouwer’s fixed point theorem. To understand this, you have to realize that not very many real numbers are constructive in Bishop’s sense. Classically there are only countably many, although they cannot be constructively enumerated, so there are uncountably many in Bishop’s sense.
Tragically, Bishop committed suicide. It has been suggested that he was depressed over the failure of his point of view to take over the mathematical world, although this sounds like an over-simplification to me.
Intuitionists are always constructivists, as it’s a school of constructivism (note ultrafinitists are also constructivists). Constructivism is not well defined. There’s many schools, including national ones (e.g. Russian constructivism). There’s lots of different ideas of what constructivist means: is impredicativity allowed, for instance (if you’re a type theorist you could consider whether your type theory includes an impredicative universe like ECC’s Prop below the predicative hierarchy or whether the universe hierarchy only contains predicative universes like MLTT)? Ironically, I think some schools of modern constructivism deny that Bishop is a constructivist.
Also, about the disappearance of constructivists from mathematics departments: that’s because they all decamped to theoretical computer science departments! As for modern works in constructive mathematics along the lines of Bishop: just check out the standard library of any proof assistant based on the Curry-Howard correspondence, like Coq, Agda and so on. There’s an awful lot of constructive mathematics being written with these tools.
This whole situation can be better understood if you view mathematics as fundamentally a game. Mathematicians play this game not for its practical implications (though it happens to have many), but simply because it’s fun. Like any games, it has rules, but some people disagree about what set of rules makes for the most fun game.
Well, one way of looking at it is that being irrational is a negative statement to begin with (defined primarily as the negation of rationality), on which account there is no more “constructive” sense to the irrationality of \sqrt{2} than that the rationality of \sqrt{2} entails a contradiction.
Intuitionistic mathematics is perfectly happy with the standard proof of the irrationality of \sqrt{2}, on this account of irrationality, because it’s perfectly ok to use contradiction to establish a negation; it’s just that intuitionistic mathematics doesn’t generally allow for double negation elimination, which is unnecessary here.
[If one defines irrationality as something other than merely “not rational”, though, then it may well take more work to prove. This is demonstrated in the Wikipedia article linked to by Andy L above, which distinguishes between “not rational” and “at finitesimal distance from each rational”, taking the latter to be the definition of irrationality]
A proof is a proof. (And, as a professional mathematician, I can assure you that in the “professional mathematical field” non-direct proofs are not considered worthy of mockery or rejection from journals, or even commented upon in comparison with the alternative.) While a non-rigorous proof will certainly be rejected from any nontrivial journal, whether a proof is direct has absolutely nothing to do with that. Mathematicians like elegant proofs, which again has absolutely nothing to do with whether a proof is “direct.”
And even at that, they may like elegant proofs, but they’ll still settle for inelegant ones if need be. It’d be very hard to call Wiley’s proof of Fermat’s Last Theorem “elegant”, but he still won the prize for it. Likewise the proof of the four-color conjecture: There may have been some elegance involved in reducing the problem to a few thousand cases, but using a computer to brute-force through those cases was anything but.
Agree completely. The most inelegant proof ever is probably the classification of finite simple groups. Last I heard, it would involve about 10,000 pages of seriously detailed mathematics and there is not a single person who knows the whole thing. The most complicated proof I ever read in enough detail to understand was the theorem (I think it was Gelfand-Schneider) that implied that e and pi were transcendental. It stated that if a_1,…,a_n were real numbers, linearly independent over the rational number field, then e^{a_1},…,e^{a_n} were algebraically independent.
But let me reiterate: ** There is no bias in mathematical journals against indirect proofs or proofs by contradiction.**