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Do integers exist? Do sequences of integers exist? If they do, so do the real numbers. This is because any positive real number has a decimal expansion which determines it. For example pi = 3.1415926535... or 1/3 = 0.3333333...
Real numbers are abstractions. And abstractions are embedded in Physics. So if you believe that Physics is real then Real numbers are also real .
If there is no need for them to exist, Ockham's Razor says we should not assume them.
Suppose they don't exist, then what are we talking about? We exist, and we do stuff (such as writing blog posts). So, numbers -- all numbers -- are stuff we do. And write about.
They don't need to be more than that. We don't need them to be more than that. If we did, we would be out of luck, anyway.
Earlier quoted context omitted.
Abstractions are emergent properties of the laws of physics.
The “laws of physics” are mathematical models (abstractions) created by humans. And those abstractions are embedded in the physical world (our brains, written text, magnetic disks etc.)
If I understand correctly, the problem is not so much real numbers, but real number that are not otherwise defined(integers, rational numbers, floating point numbers, etc) call them pure real numbers. Because while it is easy to imagine there are numbers that are not defined, the minute you try define what they are... they are now defined, thus the problem/paradox, can something exist that is not defined?
For example, axioms of real numbers state that increasing bounded sequences have a limit, and it is a real number. It could be that this leads to some contradiction somewhere which we don’t know about.
For example, if you erroneously thought that rational numbers in fact themselves satisfy these axioms, you could construct an increasing sequence whose squares converge to 2, and yet the sequence itself doesn’t converge to a rational (because sqrt(2) is not rational). Perhaps some such contradiction exist for real numbers, and in that respect they could “not exist”. That would mean the axioms are somehow inconsistent.
As the article states, we can construct real numbers with Dedekind cuts that only rely on rationals; I didn’t realise there is any nuance to that definition.
You don’t seem to have a clear definition of what exists is
> ...accepting the existence of the reals means that there are numbers that exist but can never be described ...we can never interact with these numbers even conceptually
is also full of vagueness: what is "be described"/"interact"?
That includes all rationals along with irrational that we have a way to approximate by algorithms - like sqrt(2) and PI. It does not include the incalculable numbers like Omega.
Being linked to algorithms, such numbers are only countable many, so they cannot contain "most" of irrational and therefore they lack most of "Real numbers".
There are mathematicians that take this view seriously, like "Constructivists" and "finitists". For example: https://www.youtube.com/watch?v=REeaT2mWj6Y
Earlier quoted context omitted.
The subset of numbers for which you can compute successively better approximations, such as pi and e, is countable. It’s called the set of computable numbers and contains everything that we could even in theory ever describe or “use” somehow. The actually uncountable subset of reals is by definition forever out of our reach, which is what makes uncountable sets… rather ontologically suspicious.
This is a matter of opinion. I do not find anything ontologically suspicious about uncountable sets. What I would really found ontologically suspicious would be the existence of convergent sequences of numbers without a limit or the inexistence of continuity, e.g. the impossibility of knowing whether moving from one side of a curve to the other side of the curve will intersect the curve or not. Even if most programme…
The real numbers are a very convenient abstraction, but they are also a leaky abstraction (edit: when paired with axiom of choice), which is anathema to mathematics.
We can still study the real numbers, they are still valid in some abstract sense, and they are convenient, but an element of the set of reals minus computables can never be constructed, so you'll never actually need to deal with them.