How real are real numbers? (2004)

Btw that's not true. Cantor's diagonal argument isn't a proof by contradiction, it's a purely constructive argument. The way he made it in his original paper is a bit more technical than this but this is the "modern" version that's a bit easier to put into layman's terms.

Say you say you can construct a (countably finite) list containing all the real numbers. I say I don't care how you made your list I can give you a number that's not on it, and in fact cut your list down to just numbers between 0 and 1. If you have all the real numbers you must have all the numbers between 0 and 1, but even if you make a countably infinite list of numbers between 0 and 1 I'll give you a procedure that will construct a number that's not on your list no matter how you made it.

1) Read the first number on your list. If it has 1 in the first decimal place, make the first decimal place of my number a two otherwise make it a 1. 2) Read the second number on your list. If it has 1 in the second decimal place, make the second decimal place of my number a two otherwise make it a 1. ....

Proceed in that manner.

At the n-th step I read the n-th number on your list. If it has 1 in the n-th decimal place make the n-th decimal place of my number a 2 otherwise make it a 1.

Now: My number is clearly nowhere on your list as it differs at in least one decimal place from every number on your list.

Therefore it is not possible to construct a countably infinite list of real numbers.

As in, not even computable in theory. It is isn't about normal "computable" concerns. It is "proven to never be characterizable".

Which is a class of numbers whose "existence", if that can term can even be applied coherently for undefinable things, is contested, in theory. In practice they certainly do not exist.

1/3 has infinite decimal digits, but is definable with a finite number of symbols, so it is a computable real. Even if we had no algorithm yet to compute those digits.

Try and define a specific number, that requires infinite symbols to define. As far as I am aware of, no part of calculus involves specific values that have no finite definition, except when the need for uncomputable/undefineable reals are asserted on a circular basis (i.e. they are needed to resolve problems with assumptions that already assume them.)

(Note that a number defined by interpreting the infinite digits of pi as mathematical relations, would still be considered a definable number, assuming some form of convergence could be proven. Because pi is finitely defined.)

Waves are pervasive and described by relationships involving those numbers. They show up in other relationships. With important properties such as conservation of energy that any partial precision wouldn't be able to achieve.

Numbers are not just evident by single value measurement, but even more powerfully when they govern a system, where any problem with the definition would result in an easily recognizable failure of an entire theory.

I think "precision" is the wrong way to look at what can mean something or not.

I think the boundary between numbers that "make sense", relative those that don't is better found by looking at the progression of numbers.

From naturals, to integers, to rationals, to algebraic (both non-rational roots, and roots of negatives), all the way to limits and series. (Note that the infinite computation associated with expanding digits is not a definition problem. Even 1/3 requires infinite digits in decimal, but the relationship between 1 and 3 is clear.)

What is true about all these numbers is not precision, but that they emerge from a finite number of relationships.

They can be written exactly, defined perfectly, with finite numbers of symbols. (Meaning, abstracting away notation, with a finite number of relationships.)

And all those types of numbers do show up exactly (for all appearances), in waves, and other relationships. The relationships themselves make predictions more powerful than the practical precisions we might have in measuring single values.

So pi really exists. All kinds of physics would fail if it didn't. That doesn't mean we can make a perfect pi circle with plan length, since that would be an arbitrary test, and if the medium is discrete units, one chosen to a priori fail.

Contrast with: The uncomputable, undefinable numbers, which we can't define, can't measure, etc., and are introduced via shaky (relative to the general body of mathematics) means. They require infinite information to define exactly. Not just measure, but even to define. Which is a remarkable postulation, and is not needed to solve any problems they don't themselves introduce.

If your measurement of energy is 150 +/- 2, you only need a handful of digits to do calculations involving that value that preserve it just fine. Insisting that that billionth digit and more still match is no longer working with the real world.

> The relationships themselves make predictions more powerful than the practical precisions we might have in measuring single values.

You can make a prediction to infinite precision but it's not falsifiable. The infinite precision isn't real any more than aether theory is real.

Any powerful results that impact the real world don't need all that precision.

> if the medium is discrete units, one chosen to a priori fail

I choose discrete units because that's what the universe is, as far as we can measure.

If there's something more subtle than Planck, we can't measure it.

It's possible the universe does round at some point. We can't tell.

Can you describe any theoretical experiment that could tell the difference between perfect pi and thousand digit pi?

> Contrast with: The uncomputable, undefinable numbers,

I agree that there's a stark difference there. But I don't think computability is the specific point where it detaches from reality, it's just where the disconnect gets the most obvious.

Nicolas Gisin: Time, Superdeterminism, & Quantum Gravity

https://www.youtube.com/watch?v=jcHzgy0I6gk