> And I agree that our formalism of integers is simpler and more elegant than our formalism of real numbers. But that could just be because we've done a worse job formalizing real numbers!

Everything you can express in integers you can express in reals, but there are many things expressable in reals not possible in integers. It would be surprising if the formalism for a thing that completely supersets another thing had an equally simple formalism

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xscott 5 days ago | parent | next [-]

> [...] but there are many things expressible in reals not possible in integers

Are you sure there is anything we can express in the Reals that isn't an integer in disguise?

The first answer might be the sqrt(2) or pi, but we can write a finite program to spit out digits of those forever (assuming a Turing machine with integer positions on a countably infinite length tape). The binary encoding of the program represents the number, and it only needs to be finite, not even an integer at infinity.

Then you might say Chaitin's constant, but that's just a name for one value we don't know and can't figure out. You can approximate it to some number of digits, but that doesn't seem good enough to express it. You can prove a program can't emit all the digits indefinitely. And even if you could, is giving one Real number a name enough? Names are countable, and again arguably finite.

It seems to me we can prove there are more Reals than there are Integer or Computable numbers, but we can't "express" more than a finite number of those which aren't computable. Integers in disguise.

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tshaddox 4 days ago | parent [-]

It seems like you're suggesting that mathematicians replace the reals with the computables. This is a reasonable thing to try, and is likely of particular interest to constructivists. There's even this whole field:

https://en.wikipedia.org/wiki/Computable_analysis

	▲	xscott 4 days ago \| parent [-]
		> It seems like you're suggesting that mathematicians replace the reals with the computables. No, not any more. :-) I liked the idea a year or two ago, but I've come to believe that even the Integers are too bizarre to worry about. For now, I'm content just playing with fixed and floating point, maybe with arbitrary precision. Stuff I can reason about. I just think people are a little too casual thinking they are really using the Reals. It might be like Feynman's quote about saying you understand QM.

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tshaddox 5 days ago | parent | prev | next [-]

I'm not suggesting the two formalisms should be equally simple. But surely it's not controversial to claim that formalizing the reals involves much more advanced mathematics (and runs into much deeper problems) than formalizing the integers. I'd argue that this disparity is slightly surprising, given that both the integers and the reals are ubiquitous in essentially all branches and levels of mathematics.

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orlp 5 days ago | parent | prev [-]

Actually in math it's very common for the more general system to be simpler. Compare for example the prime numbers with the integers, or general groups with finite simple groups and the monster group.