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Comment by IAmBroom | Hacker News Reader

> You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities).

Citation needed.

Especially since there are well-established math proofs of irrational numbers.

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The argument is essentially that you can only measure things to finite precision. And for any measurement you've made at this finite precision, there exist both infinitely rational and irrational numbers. So it's impossible to rule out that the actual value you measured is one of those infinitely many rational numbers.

This argument feels like it's assuming the conclusion. If in principle it is only possible to measure quantities to finite precision, then it follows logically that we couldn't tell the difference between a rational and real universe. The question is, is the premise true here?

AFAIK it would take an infinite amount of time to measure something to infinite precision, at least by the usual ways we’d think to do so…. I suppose one could assume a universe where that somehow isn’t the case, but (to my knowledge) that’s firmly in science-fiction territory.

I don't think time and measurement precision are necessarily related in that way. You can measure weight with increased precision by using a more precise scale, without increasing the time it takes to do the measurement.

There are limits to precision there too. The amount of available matter to build something out of and the size you can build down to before quantum effects interfere.

The example was only to illustrate that measurement precision is independent of the time it takes to perform the measurement.

The real point is that it takes infinite energy to get infinite precision.

Let me add that we have no clue how to do a measurement that doesn't involve a photon somewhere, which means that it's pure science fiction to think of infinite precision for anything small enough to be disturbed by a low-energy photon.

I'm not making the case that it is possible to make measurements with infinite precision. I'm making the case that the argument "It is not possible to make measurements with infinite precision, therefore we cannot tell if we live in a rational or a real world." is begging the question. The conclusion follows logically from the premise. Unless the argument is just "we can't currently distinguish between a rational and a real world", but that seems trivial.

If I'm carrying a single apple, I can measure the number of apples I'm carrying to infinite precision. I'm carrying 1.000... apples.

You're implicitly assuming your conclusion by calling it a "single" apple, which means exactly one. "Apple" is an imprecise concept, but they're often sufficiently similar that we can neglect the differences between them and count them as if they're identical objects, but this is a simplification we impose for practical purposes.

Even for elementary particles, we can't be sure that all electrons, say, are exactly alike. They appear to be, and so we have no reason yet to treat them differently, but because of the imprecision of our measurements it could be that they have minutely different masses or charges. I'm not saying that's plausible, only that we don't know with certainty

> Especially since there are well-established math proofs of irrational numbers.

The logic is circular, simply because mathematicians are the ones who invented irrationals. Of course they have proofs on them. They also have proofs on lots of things that don't exist in this universe.

And as I pointed out elsewhere, many analysis textbooks define a real number to be "a (converging) sequence of rationals". The notion of convergence is defined before reals even enter into the picture, and a real number is merely the identifier for a given converging sequence of rationals.

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