Once upon a time I had a Linear Algebra class, in that class we had to, on occasion, do proofs. I, having taking discrete math, already knew of several ways of approaching proofs (e.g. weak/strong inductive, contradiction, direct), as far as I was taught, while theoretically any proof could be converted into a direct proof (albeit often with significant effort), all of them were great tools that were applicable to different situations.
After the first proof we had to do he addressed the class and said “several of you used non-direct proofs. I’ll accept it with full credit this time, but I’ll start marking off for them from now on. In the professional mathematical field non-direct proofs are considered a joke and you’ll be mocked and probably rejected for publication if you ever use anything other than a direct proof. This is for your own good.”
Is this real, or was he just crazy? He was only a grad student (and kind of a douche*, but I digress), but I accept that there may be differences between computer science and math when it comes to the acceptability of proofs. Or maybe the journals or professors he worked with were just strangely elitist and convinced him of that (I’ve certainly met some professors in various fields who did things the One True Way and any deviation is tantamount to idiocy).
I (and the others who took discrete math with me) once used a trick we learned for matrix multiplication in discrete math. Even though we got the right answer, and clearly marked what we were doing, he marked us off because “I see what you’re doing, but in real math we use equations, not this retarded rotation shit.”
He sounds really irritating. There’s not really any strong cultural bias against “non-direct” proof (e.g., proving a statement by reasoning from its negation to a contradiction). One certainly would not be mocked and rejected from publication and all that simply for noting an indirect proof; people use them all the time.
However, I will say that my own aesthetic sense tsk-tsks at presenting in indirect formulation a proof whose gist could just as well be written directly… I find people (not just students, but also professional mathematicians whose aesthetic senses happen to differ from mine) often do this, but to my eyes, the direct formulation is typically much more pleasant and illuminating, when available.
Of course, this may have much to do with the fact that I’ve spent a lot of time thinking about intuitionistic mathematics, where double negation elimination is unavailable and thus reasoning from the negation of a statement to a contradiction produces a weaker result than reasoning to the statement directly.
I’m not calling myself a professional mathematician, but I’ve taken enough classes from people who were, without ever hearing any of them say anything like this, to vote for Crazy.
Direct constructions of examples are one thing, but a rigorous proof is a rigorous proof. Some mathematicians are (or used to be) very picky about the axiom of choice and regard any proof which uses the axiom as nonconstructive. I’m not a logician, but the very foundations of set theory involve axioms of the form “there exists a set…” so proofs which avoid choice aren’t necessarily constructive either.
As to the quote “several of you used non-direct proofs. I’ll accept it with full credit this time, but I’ll start marking off for them from now on. In the professional mathematical field non-direct proofs are considered a joke and you’ll be mocked and probably rejected for publication if you ever use anything other than a direct proof. This is for your own good.” I can only say that the percentage of students in a linear algebra class who will eventually submit an article is around 0. I suspect that since many (if not all) of the arguments he was looking for can be made directly he was imposing his own bias on what is a good argument.
Or maybe, since you were students, he wanted you to do things the “hard way” just for the practice of it, or as exercises in the mental discipline for that kind of stuff.
Of course, it always helps if you can just, you know, see the whole proof instantly in your head. About 35 years ago, a mathy friend asked me if I could prove that 2[sup]n[/sup] - 1 can be prime only if n is prime. I knew it was true, but had never before considered trying to prove it.
I had a proof in my head in about 30 seconds.
Hint:Consider what 2[sup]n[/sup] - 1 looks like when written in binary. And consider what you can you with that if n is not prime. Jragon, did you also use the spaghetti method to find the determinant of a 3x3 matrix?
His actual research/thesis field was, IIRC, game theory. I’m not very familiar with game theory, sadly, but maybe that had something to do with it. Or maybe he was just overly judgmental. Who knows?
Nah, I asked him about it directly once. He was very, very clear that he thought that using any method other than a direct proof (no induction, no contrapositive, no contradiction, and so on) was a ridiculous notion not to be taken seriously. No matter the context. He said he spent days converting proofs into direct ones for his papers, and occasionally would throw out weeks of research because he couldn’t come up with a direct formulation.
Not sure what the “spaghetti” method is. So probably not. I always just memorized 2x2 and used cofactor expansion for everything else.
It’s usually pointless to argue with somebody who has arbitrary authority. They can just declare the rules are whatever they say they are.
But if you want to make the attempt, I’d suggest finding a non-direct proof made by some indisputable expert - some mathematician who won the Fields Medal or the Abel Prize using a non-direct proof. And then ask him if he considers that work to be valid.
I did a master’s degree in algebra, and I saw lots of proofs by contradiction. Sometimes that’s the most elegant or intuitive way of stating the proof, IMO.
Sounds like a “constructivist”/“intuitionist” Constructivism (philosophy of mathematics) - Wikipedia ; there have been several great mathematicians with that philosophy, but as far as I know it’s a minority view in modern mathematics.
One of the most famous proofs ever, Euclid’s proof of the infinite number of prime numbers, is a proof by contradiction. I can’t imagine any mathematician saying that proof was “a joke”.
It’s very often presented that way, but I don’t think it really counts as a proof by contradiction in the form Euclid originally gave it: “Prime numbers are more than any assigned multitude of prime numbers.” In other words, for any finite set of prime numbers, there will always be a prime number not included in the set.
A grad student with crazy ideas, inside the math department!? I knew at least three in my school alone.
A high school math teacher inisisted that the word ‘minus’ applied to written ‘numerals’ only. For the insubstantial, spoken ‘numbers’ one had to use the word ‘subtract’. The details are fuzzy now but I do remember you could fail for getting it wrong.
Constructive proofs are better, in that proving that X exists by coming up with a specific method to construct it is often more useful than proving that X can’t not exist, but at the same time the existence of the latter kind of proof doesn’t mean that a proof of the former kind exists.
Is this a thing now?