Mathematics poll

[ciphergoth]

Inspired by a similar poll in palmer1984's journal.

About the "proof" question below: examples of the kind of proof I mean would be a proof that there are infinitely many primes, or that the square root of two is irrational, or of Pythagoras's Theorem. A proof in computer science counts too. By "know a proof off by heart" I mean that you'd be able to convince someone of it at a party, if they had the background to follow the proof.

If you know lots of proofs, feel free to choose one you particularly like in the last question...

[Poll #1351621]

Comments

[fizzyboot.livejournal.com]

You're going to find a lot of math awfully hard to do if you can't think about any uncountably large collections.

The phrase "uncountably large collections" isn't uncountably large. Your brain is a finite object or finite size and there is a limit to the complexity of the things it can think about. The concept of "uncountably large" is not sich a complex thing.

I don't really do maths, but I do do programming. For example, I am designing/implemententing a programming language ATM. If I were to try to enumerate all the programs that could be written in my language, it would take up more than all the matter in the universe. However that does not stop me from reasoning about it, nor does it dtop my computer from running a compiler and interpreter for that language.

If my language could not be espressed in a finite set of symbols, it couldn't exist in any real sense.

[ciphergoth.livejournal.com]

So, numbers that can't be expressed don't "exist" but you're prepared to treat them as if they do "exist" if, for example, you're talking about measure theory. I think you think you're doing something terribly important and clever, but from here it looks more like theology than mathematics.

[fizzyboot.livejournal.com]

So, numbers that can't be expressed don't "exist",

Putative symbol-lists (whether you want to think of them as numbers or otherwise) that cannot in principle be expressed, are not expressed anywhere. In that sense they clearly do not exist.

A movie on my hard disk takes up about 1 GB, i.e. it's an 8,000,000,000 digit long binary number. Do all such numbers "exist"? I submit that the only ones that exist, are those that actually have physical representations somewhere; to say otherwise is to say that a movie exists before filming has started on it, which IMO is ridiculous.

I think you think you're doing something terribly important and clever

No, I think it's bloody obvious.

[ciphergoth.livejournal.com]

Naah, it's pretty much a mathematician's green ink letter...

[elsmi.livejournal.com]

> The phrase "uncountably large collections" isn't uncountably large. Your brain is a finite object or finite size and there is a limit to the complexity of the things it can think about. The concept of "uncountably large" is not sich a complex thing.

Quite so. And one of the inferences we can make in doing that non-infinite reasoning is that, using the relevant senses of the words, there is no mapping with the property you propose. You can't just retreat behind puns when someone calls you on that...

But, you know, room for all kinds. I hope you and Brouwer have a nice beer together.