That's merely an assertion that you can do that for all numbers I could give you. I have a mapping that does that - I start a counter from 1, and for every subsequent different number I'm given, I increment it.
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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timeasmymeasure
no subject
Date: 2009-02-20 11:00 pm (UTC)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.