Vasco Grilo🔸 comments on Evidence, cluelessness, and the long term—Hilary Greaves

[Vasco Grilo🔸]

This seems to be arguing against standard mathematics.

Yes and no:

The answer depends on what mathematics is about. Insofar as mathematics is simply an intellectual game, then the logical breakdown of infinitudes does not amount to much; mathematicians can continue to play with infinities in classical mathematics and infinite sets in transfinite mathematics just as gamers can continue role-playing in fictional worlds. On the other hand, insofar as mathematical operations are used in attempts to reveal the nature of the Universe, then the logical incoherence of infinitudes means that they reveal nothing of nature since logical contradictions cannot refer to real circumstances.

Kip argues:

The answer to revising both transfinite mathematics and classical mathematics is to replace the traditional use of infinitude with an alternative concept—indefinitude.

(...)

If a value is “indefinite,” then that means either (A) it would be found to be the terminating value in a set of values if the set could be measured or counted, but is either too minute or too vast to be measured or counted in actual practice or (B) it is finite in the sense of being currently undefined as the next value, or sequence of values, beyond the highest or lowest value that can be actually computed or invented for a series.

(...)

What will be called “the indefinite” shall refer to any set that has so many members that the highest value(s) in the set is indefinite according to A.

However:

Functionally speaking then [reference to example], nothing is lost by replacing infinity with indefiniteness—values that were traditionally thought to go on “infinitely” could just as well be thought of as going on “indefinitely.”

I think the crux of the disagreement is described here (emphasis added by me):

Many mathematicians have assumed that the infinite really is a property of nature [and one of the axioms of set theory], existing in a Platonic world on its own apart from minds attempting to conceive it, and that Cantor’s transfinite mathematics teaches us how this property of nature operates. But that is a big assumption, and one which we are not intellectually compelled to make. It could equally be assumed that infinite sets are mathematical inventions, in which case the rules for manipulating infinite sets do not indicate “how the infinite works” as if those rules are merely descriptions of an infinite set’s behavior that mathematicians discover like the principles of atomic motion in a condensed gas are discovered by physicists. That is, it could just as well be supposed that the infinite is an invented idea that refers to no existing natural property at all. If that position is correct, the rules for calculating infinite sets are only “discovered” in the same way that a new strategy in chess is discovered. This kind of “discovery” is actually a form of invention; it is the invention of new rules of inference for manipulating concepts in a pattern coherent with the rules previously established for those concepts.

If this is correct, then the rules of transfinite mathematics are not really the discovery of some phenomenon independent of human activity, but simply the invention of a system of inference. Further, because the traditional view of infinite sets contains self-contradictions, that system of inference has no coherent application to understanding reality in terms of measurement. Infinite sets and transfinite mathematics are better interpreted as elements of a mathematical game rather than a means for understanding the quantitative aspects of nature. Transfinite mathematics is therefore actually misleading about the nature of real sets of things. That is, the “infinite sets” of transfinite mathematics not only do not refer to real sets in nature, but actually lead us astray in understanding the quantifiable aspects of reality. Consequently, Cantor’s math ought to be rejected as a tool for investigating reality even if it is saved as a kind of academic game.

In other words:

Claiming that infinite sets of objects can exist because the rules for calculating them remain consistent as long as you don’t allow subtraction and division is like saying square circles can exist because we could create a formula that allows them to be used, provided some qualifications are put in place that don’t allow us to expose the contradictions resulting from attempting to calculate a round square or a circle with corners.

(...) The result would be that square circles are not any less self-contradictory; all that we would end up proving in the construction of such a formula is that a coherent game of square circle calculation can be made as long as the rules are limited in an ad hoc fashion so that the illogic of square circles is not allowed to be exposed by taking the concept to its, well, logical end. Transfinite mathematics is in the same boat—the game is coherent only because we won’t allow logic to proceed down its natural path so that the self-contradictory nature of its subject would be exposed. This allows the idea of computable infinities in the real world to retain the illusion of being logically coherent.

Regarding:

Maybe I’ve misunderstood and they don’t mean infinity is logically impossible even in mathematics, just only physically.

Kip rejects the existence of infinities in both physics and math. The real world does not allow for contradiction, so infinities have to be rejected in physics. In math, it can exist, but Kip argues that it is better to revise it to the extent math is supposed to decribe the real world (see quotations above).