Certainly you can build a branch of mathematics without an axiom of infinity, and that’s fine, it’s math over finite sets.

However, an axiom of infinity is independent, it doesn’t contradict anything in standard formalizations, and so it doesn’t make sense to say “infinity is wrong”.

He may think the axiom of infinity isn’t satisfied by our real physical world, but that’s not a math question! There’s nothing logically inconsistent about infinite sets nor their axiomatizations.

When the author says we cannot truly observe infinity, what does that mean? Infinity is a mathematical symbol we can observe. We can't observe infinitely many objects, but even if we could, it wouldn't be the same as observing infinity. You can't observe the number one by observing one stone.

I think there is some confusion in this article between symbols and what they can stand for, and I can't help but wonder if that same confusion is at the root of ideas like ultrafinitism.

For my personal opinion, strict finitism provides a richer field of study than potential infinitism or actual infinitism. Compare this to Errett Bishop's constructive analysis that requires the calculation of bounds to real numbers, instead of classical analysis only requiring that a real number exists. Much more difficult, though more precise.

I found "On Feasible Numbers" by Vladimir Sazonov to have application for computers. In a feasible mathematics, a large number fails to exist (say, 2^512), but a proof of contradiction must exceed such a large size (perhaps larger than the universe). Likewise, we have unix time that tries to count forever, so we should pick a storage size so large that counting exceeds the heat death of the universe. 10^100 years worth of Planck seconds fits in 501 bits, so round that to 512 bits. 512 bits of time ought to be enough for anybody :)

https://jeanpaulvanbendegem.be/home/papers/strict-finitism/

Friends don't let friends do Platonism.

For real, if you're a formalist you can ask these foundational questions without fear of this kind of dread; they become methodological rather than some kind of metaphysical mess.

I'm hoping this is just bad writing from Quanta rather than something "ultrafinitists" truly believe.

I really don't think it's that complicated. Even pre-schoolers, competing to see who can say the highest number, quickly learn the concept of infinity. Or elementary school students trying to write 1/3 as a decimal.

Of course you need to be careful mapping infinity onto the physical world. But as a mathematical concept, there is absolutely nothing wrong with it.

> Mathematicians can construct a form of calculus without infinity, for instance, cutting infinitesimal limits out of the picture entirely.

This seems like a useful concept that also doesn't require denying the very obvious concept of infinity.

> computers handle math just fine

strong disagree tbh

Finite unbounded admits all the natural numbers. Allowing infinity to represent no-finite-limit, as apposed to trying to shoehorn actualized infinities (general real numbers inclusive of “unconstructible” “non-uniquely specifiable” numbers, and “concrete” higher-order Cantor cardinalities, whatever that could mean) into local structures.

Anyways, I enjoyed reading the perspective of a mathematician on this.

If this seems too conservative to you, like if for some reason you want to talk about the volume of the universe in terms of the width of an up-quark or whatever, feel free to tack on some modifier to my proposed number system.

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.

[1] EDIT: the reasoning is simple, if naive: the largest quantities we can measure are not, in fact, infinitely large, and the smallest ones we can measure are not, in fact, infinitesimally small. So until you show me an infinitesimal or an infinity, you're just making them up!

BTW, the article is really badly written.

People ask whats the point? For me the study of the infinitesimal vs finite has really helped me better understand issues of precision and approximation in computers. I feel like I know exactly why 1/3 plus 1/5 is not exactly 8/15 in my Calculator app. Or why points in my 3d object face are not coplanar after rotation. Or why games have weird glitches when your character is too far from origin point. Or why a spreadsheet shows rounding issues

> computers handle math just fine with a finite allowance of digits.

Go try and write yourself a robust algorithm to do booleans on polygons or calculate a voronoi diagram. The finite nature of floating point is the mother of all leaky abstractions and bites you in the arse any time you think you are smart enough to roll your own algorithms.

The notion of "believing in" axioms is absurd ... as absurd as believing in the rules of chess and disbelieving the rules of checkers. Each set of rules or axioms forms a system (possibly degenerate if the axioms are inconsistent). The rules, axioms, and systems aren't "true" or "false" -- that's a category mistake. Studying the systems resulting from the Peano axioms or ZFC is a worthwhile endeavor. Studying the systems resulting from finitist axioms may well be too, but the nonexistence of infinities in the latter doesn't mean that they don't exist in the former--that's crackpottery. Mathematics has room for both sorts of systems.

Which axiomatic systems best model the world is a different matter. Now we're in an empirical realm, where there are observations, evidence, facts. And observational reports are necessarily finite, so even if there are "real" infinities they can't be demonstrated. But "all models are wrong", so both infinite and finitist axiom systems might serve as good approximations.

Likewise with computer systems--all actual computer systems are finite state machines, but it's convenient and useful to model them as Turing Machines that allow for both infinite non-halting systems and finite halting systems.

And since both this medium and I are finite, I will stop there.

>“Infinity may or may not exist; God may or may not exist,” he said. “But in mathematics, there should not be any place, neither for infinity nor God.”

>much as, Zeilberger might say, science brought doubt to God’s doorstep.

>But one day, he added, mathematicians will look back and see that this crackpot, like those of yore who questioned gods and superstitions, was right. “Luckily, heretics are no longer burned at the stake.”

LOL. What is this guy's problem?

The space of contrarian ideas is vast, and most of them are probably bad, but, nevertheless, the willingness to hold unconventional, internally consistent views should be celebrated, because it increases diversity of thought. Our collective hive mind grows stronger through heresy.

However, I like my heresy with a splash of axiomatic precision, which is sadly lacking in this article.

Perhaps we can recover some of it by treating the infinitely variable values as approximations of the more discrete values and then somehow proving that the errors from them stay bounded, for at least some interesting problems.

The idea that nothing is demonstrative of infinity is clearly incorrect.

Take the screen you're reading this on. One pixel is composed of a bunch of different atoms, and once you get down to one of them, that atom subdivides into a bunch of subatomic particles, some of which even have mass. Let's take one of those for argument's sake. Split that, and you get some quarks.

Now let's imagine that's the smallest you can go. We can still talk about half of a down quark, or half of that, etc. Say, uh, infinitely so. There you go, everything is infinite. That wasn't so hard was it?