Paul Crowley (![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
ciphergoth) wrote2009-02-18 11:03 pm
ciphergoth) wrote2009-02-18 11:03 pm
Mathematics poll
Inspired by a similar poll in ![[livejournal.com profile]](https://www.dreamwidth.org/img/external/lj-userinfo.gif)
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]
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]
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For many irrational numbers (an uncountable infinity of the things, actually), the only way for me to express them would be to recite an infinite string of numbers at you. I mean, sure, there are a few that can handily be expressed as 'Pi' or 'root 2', but if your mapping can only cope with those, then it doesn't work for all numbers. And if your 'proof' of your mapping is to have me tell you a number and you give me the integer for it, then you've restricted yourself to proving the mapping for any numbers small enough for me to be able to give them to you. Which isn't very many, and doesn't prove a thing.
You did say it worked for all numbers. Not just 'all numbers you'd be able to give me'.
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That's not what I'm thinking of; my mapping is independent ogf the order in which the numbers are presented to it. (The output depends only on the input; my function has no internal state).
For many irrational numbers (an uncountable infinity of the things, actually), the only way for me to express them would be to recite an infinite string of numbers at you. I mean, sure, there are a few that can handily be expressed as 'Pi' or 'root 2', but if your mapping can only cope with those, then it doesn't work for all numbers.
You catch on a good deal quicker than the last people I presented this to! Yes, indeed, my mapping only works with those numbers for which there is a finite symbolic expansion (I use the ascii values of the characters, raised to powers of 256 to depending on the place-order of each character.)
I submit that any entities that cannot be represented in this way are not numbers; indeed they are not anything at all, and certainly not anything mathematical.
The whole point of maths is that it involves arbitrary manipulations of arbitrary symbols on pieces of paper (and lately on computers). Except they're not arbitrary, because they're constructed in such a way that you can make valid inferences about the real world from them. (And that's the whole point of maths: it's intensely useful such that modern society couldn't exist without it).
If maths is about manipulating arbitrary groups of symbols, then anything that cannot -- not even in principle -- be represented as an arbitrary groups of symbols, isn't part of maths.
I realise that the way I think about maths isn't the way most most mathematicians do; although I suspect my way is equally valid.
You did say it worked for all numbers. Not just 'all numbers you'd be able to give me'.
If you can't give me something, not even in principle, and nor can anyone else, it doesn't exist.
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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.
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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.
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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.
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