The article doesn’t really tell us what is gained by rejecting infinity.

And in general, why not also reject zero, negative numbers, irrational numbers, complex numbers, uncomputable numbers, etc.?

Seems like an article about quacks that can’t even agree on what the bounds and rules of their quackery are.

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Animats 2 hours ago | parent | next [-]

> The article doesn’t really tell us what is gained by rejecting infinity.

Decidability. The issues around undecidability all involve the lack of an upper bound. In a finite deterministic space, everything is decidable, although some things may be too costly computationally to decide.

There are several ways to go for decidability. The brute force way is computer arithmetic - there is no number larger than 2^64-1. That's how we get things done on computers, but proofs about numbers with finite upper bounds need lots of special cases. Mathematicians hate that.

I used to work on this sort of thing, using Boyer-Moore theory. That's a lot like the Peano axioms. There is (ZERO), and (ADD1 (ZERO)), and (ADD1 (ADD1 (ZERO))), etc. Everything is constructive and has an unambiguous representation in a LISP-like form. You can have recursive functions. But they must be proven to terminate, by having a nonnegative value which decreases on each recursive call. There is a distinction between "infinite" and "arbitrarily large". You can talk about arbitrarily large numbers, but you cannot get to 1/2 + 1/4 + 1/8 ... = 1. You can have integers and rational numbers of arbitrary size, but not reals.

Set theory was interesting. Rather than axiomatic set theory, I was using lists as sets, with the constraints that no value could be duplicated and the list must be ordered. Equality is strict - two things are equal only if the elements are all equal, compared element by element. It's possible to prove the usual axioms of set theory via this route. The ordered criterion requires proving things about ordered list insertion to get there. It's ugly and needs machine proofs.

I was doing this back in the early 1980s, when machine proofs were frowned upon. Mathematicians were still upset about the four-color theorem proof. It's all case analysis, with thousands of cases. That's more acceptable today.

Looked at in this light, infinity is a labor-saving device to eliminate special cases, at a potential cost in soundness.

	▲	svnt 7 minutes ago \| parent [-]
		> Looked at in this light, infinity is a labor-saving device to eliminate special cases, at a potential cost in soundness. Or it is something that clearly conceptually exists, and makes simplistic reductionist viewpoints impossible to prove, which frustrates those who attempt to extend them into twisted metaphysical conjectures.

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jcgrillo 3 hours ago | parent | prev [-]

All indications seem to be that things are only lost, not gained. But that doesn't mean it doesn't hew closer to how things actually are. But if that's how reality actually is, then developing a rigorous understanding of it can only be a good thing, right?

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bmenrigh 3 hours ago | parent [-]

Rejecting infinity is a purely philosophical stance that doesn’t teach us anything about reality.

There is a big difference between “infinity doesn’t exist” and “infinity doesn’t exist physically”.

I should also add that the resolution of zeno’s paradox in the form of calculus where and infinite set of steps can occur in a finite time (or infinite set of distance can span a finite total distance) is conceptually very simple and useful. Rejecting it as unphysical, or saying it must imply time or space come in discrete chunks, is not contributing to an understanding of reality unless the rejection also comes with a set of testable (in principle) predictions.

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jcgrillo 2 hours ago | parent [-]

> There is a big difference between “infinity doesn’t exist” and “infinity doesn’t exist physically”.

Is there? I think one could make a decent case for "nothing exists which doesn't exist physically[1]".

[1] https://plato.stanford.edu/entries/physicalism/

EDIT: you could even probably claim "nothing exists which isn't physically measureable" which may or may not be a stronger claim depending on your point of view.

EDIT AGAIN: rate limited by this dogshit website :D but I'll respond to this comment here:

> Which is exactly why I mentioned rejection of zero, negative numbers, etc. You can reject them, but doing so just throws away useful tools without gaining anything in return.

Yeah! I fully agree. I can see no obvious benefit to rejecting these powerful tools. However, important discoveries often happen in non-obvious directions, and exploring unexplored territory is generally worthwhile. So the fact that it doesn't seem immediately useful doesn't mean it's not worth trying!

	▲	bmenrigh 2 hours ago \| parent [-]
		Which is exactly why I mentioned rejection of zero, negative numbers, etc. You can reject them, but doing so just throws away useful tools without gaining anything in return.