I am a finitist and constructionist at heart.

Sure, mathematical abstractions and infinite structures are fun to play around with..

But go ahead and actually provide me the list of all naturals. You can not. Ever.

jojomodding 6 days ago | parent | next [-]

But for any list of naturals you give me I can give you one with more natural numbers on it.

	▲	moi2388 6 days ago \| parent [-]
		Exactly.

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lo_zamoyski 7 days ago | parent | prev | next [-]

This might be conflating two concerns, though, which is the reality of numbers and their actual existence.

Compare Plato with Aristotle. Plato held that the all forms exist in some third realm, so numbers would be counted among them. Aristotle, however, held that forms exist in particular instantiations or in minds that abstracted them from reality. (Aquinas could be said to synthesize both views in the sense that forms exist in particulars and in minds, but also have their origin in God, thus making God a sort of third realm, in a way. Neo-Platonists would view the "mind of God" similarly.)

Now, in the Aristotelian view, numbers are quantities abstracted from concrete reality (indeed, quantity is one of the categories), but they are not substantial forms, as you will not see instances of numbers as substances in the world. They're abstractions of accidental forms. Furthermore, a form needn't be instantiated actually, but can exist potentially. This is how he resolves Zeno's paradoxes. You can divide a length an infinite number of times - or in a CS context, you can apply the successor function indefinitely - but only potentially; as a matter of actuality, you have not divided a length an infinite number of times.

So, for Aristotle, you have a finite plurality of things that are potentially infinitely divisible, or a finite series of actions that can be potentially infinitely repeated or whatever.

For a contemporary realist, Aristotelian treatment of math, James Franklin is worth checking out [0].

[0] https://web.maths.unsw.edu.au/~jim/structmath.html

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moi2388 7 days ago | parent [-]

If something can exist theoretically but not practically, your theory is wrong.

But go ahead and divide a length an infinite number of times then. And actually infinite, not as it tends to infinity.

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lo_zamoyski 6 days ago | parent [-]

Take the time to understand the subject matter, because your first sentence doesn't makes sense.

	▲	moi2388 6 days ago \| parent [-]
		It does.

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griffzhowl 6 days ago | parent | prev [-]

> But go ahead and actually provide me the list of all naturals. You can not. Ever.

But how did you come to this conclusion unless by assuming that there are infinitely many natural numbers?

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moi2388 5 days ago | parent [-]

Proof by contradiction. Heard mathematicians like that.

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griffzhowl 5 days ago | parent [-]

Ironically, finitists and constructivists don't like proof by contradiction...

I agree though, that you have come up with a contradiction. Specifically, because you seem to believe these two statements:

There are finitely many natural numbers.

Given any finite list of natural numbers, we can always produce another natural number not on that list.

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moi2388 5 days ago | parent [-]

Leading to yet another finite list. You will never actually end up with an infinity. You’ll run out of space, you’ll run out of time.

It “exists” just like magic in Harry Potter exists. Not really.

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griffzhowl 4 days ago | parent [-]

No one thinks you can actually write an infinite list.

What it means for there to be infinitely many natural numbers is that for any finite list, there are natural numbers not on that list (something you appear to agree with). If there were finitely many natural numbers this wouldn't be true

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moi2388 4 days ago | parent [-]

Sure they do. Cantors argument, infinite sums, mathematicians keep using infinite lists as if they do exist.

Let’s not even begin about the axiom of choice. Or transcendental numbers.

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griffzhowl 3 days ago | parent [-]

You're also treating the natural numbers as infinite if you think there can't be a finite list containing all of them

	▲	moi2388 2 days ago \| parent [-]
		Depends. Not the way I look at them. But this discussion is getting off topic (my own fault), so perhaps we should continue it somewhere else.