What can we gain by losing infinity?

Contrarian thinking can be great because it taps into the intuition that the masses are mostly followers who can be led anywhere, not critical thinkers who've deeply examined what they believe. Being contrarian, then, is akin to staking out a new leadership position.

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.

I don’t understand, and I hope it’s just bad writing.

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.

Take the approximate number of subatomic particles in the universe, call it Ω. Define the largest number as Ω² and the smallest number as -Ω², and define the number of decimal numbers between each integer number as Ω², evenly spaced. That should be more than enough numbers. Redefine Ω with each new discovery in physics.

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.

> To Zeilberger, believing in infinity is like believing in God. It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena. But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is.

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.

finite moments. cherish them.

In school I developed a strong hunch that continuity and infinity are "convenient delusions" we have that allow us to process the otherwise horrific complexity of the world. Experiencing time, sound, or visual motion as continuous, rather than discrete signal inputs is so much simpler. Similarly, the mathematical tricks and shortcuts we can use on well behaved continuous functions are both "unreasonably effective" and... probably not grounded in actual reality[1]? But damn are they convenient.

[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!

Normally amps only go up to ten… but this one goes to eleven. …it’s one louder ain’t it!?!

It's not a new idea, and it's a challenging one to investigate. Without real numbers (that are infinitely long) most of the calculus stops working. And everything that depends on it.

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 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.

The first thing that came to mind reading the article is that you need only 60ish digits of pi to calculate the circumference of the universe with a resolution of a Planck length, or something like that. You can have all the digits you want, but at some point you are beyond what is possible in reality, and giving back wrong answers for what you are trying to achieve.

And no discussion of Zeno? Pish.

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?