Mathematical Torment

[Krispy]

There's a lot of questions in contemporary times about how profitable formalizing everything is, but it's still the dominant practice. And it separates the men from the kids.

[frederec]

Originally Posted by Krispy

I was a philosopher. I never took any philosophy of mathematics, but if you follow the logic courses down the rabbit hole long enough you get a good gist of it. All of modern philosophy is backboned on Frege's philosophy of language questions -- and by extension modern predicate logics -- so you learn it or fail. Understanding metalogic is a must otherwise you won't be able to go toe to toe with the big boys when they throw down set theory and lambda calculus to prove their philosophical repose.

Fascinating. I've never taken a philosophy course either, so this is totally foreign territory for me. Interesting, though.

I've come to be mildly fascinated by the way every field pretty much has its own language associated with it, and they get more complex and demanding the further you go. And some take it to impressive extremes.

[Krispy]

An egotistical logician might want to say that it all reduces to logic. Logicism, it's called. But Godel famously destroyed that possibility. But when it comes down to it, the languages of math and logic aren't so different. I'm honestly not sure why a logical calculus can't be translated to a mathematical calculus and vice versa. I need to revisit my metalogic.

[frederec]

Originally Posted by Krispy

There's a lot of questions in contemporary times about how profitable formalizing everything is, but it's still the dominant practice. And it separates the men from the kids.

In mathematics this sort of approach seems to have different levels of popularity depending on the field, the decade, and sometimes the country.

For instance, the secret mathematical society which published under the name Bourbaki was all about formalizing everything and doing everything in the purest, most elegant way possible. But they did this to the exclusion of any sort of exercises which most people argue is the primary way of learning mathematics.

It's a silly point of pride that the only book I have that's all in French is a Bourbaki book that I think is their only one that's never been translated to English. But it has a cool theorem that I've never found in an English book.

[Krispy]

So I don't know a terrible lot about theoretical mathematics, but I was under the impression that you couldn't do informal math. Like, part of what made math math was that it was formal. Same with logic. Part of bucking formaliztion is bucking the idea that the only acceptable proofs for truthiness are those of the mathematical variety.

[frederec]

Originally Posted by Krispy

So I don't know a terrible lot about theoretical mathematics, but I was under the impression that you couldn't do informal math. Like, part of what made math math was that it was formal. Same with logic. Part of bucking formaliztion is bucking the idea that the only acceptable proofs for truthiness are those of the mathematical variety.

In theoretical math, you're pretty much correct. There is plenty of "math by doodling," but that's essentially brainstorming. If you want to publish something, it does need to be pretty formal. But there can be levels of minutiae in presenting proofs, and a lot of that is a matter of style. There are jokes out there about "methods of proof" which are various intimidation and obfuscation tactics used in proofs (often by teachers). I need to try to see if I can find that list, it's amusing.

I myself have ridden the line between pure mathematics and applied mathematics. And in applied mathematics, the need for formality is much less. A lot of time the formality could be there if you wanted to spend the time to go there (and I was the type of person that would approach applied mathematics from a pure perspective), but most people in the field don't care that much.

[frederec]

Here we go. The most common used by students is "proof by example." The most common used by professors is "proof by intimidation." You know, "I've got a PhD from Awesome University and you're a stupid undergrad. Don't argue with me, you idiot."

[Krispy]

Funny thing is I knew what webpage you were going to link without looking.

There is only one way to get a proof done on time and that's proof by fallacy.