121–130 of 141 posts
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, bu…
My favorite math paper is "Is 10^10^10 a Finite Number?" by David van Dantzig. It lies more on the side of philosophy, so many can understand it easily. I first learned about it many years ago from Van Bendegem's list of strict finitism papers, and I would recommend that list for anyone interested in learning more about strict finitism. For my personal opinion, strict finitism provides a richer field of study than po…
Earlier quoted context omitted.
Can you observe 2.34 x 10^456789 apples?
No. I believe that is more apples than there are atoms in the universe, so not only it is impossible to observe, it is a fundamental contradiction with our universal reality. No one and nothing will ever be able to observe or interact that many apples, and so a reference to that many apples is only an abstract mathematical convenience that has no direct bearing to reality. Like infinity. I'm not sure I actually belie…
But none of this can be observed either, which in my book makes your argument a bit weak.
Your "universal reality" is a construction relying in big parts on the mathematics relying on infinity as a concept.
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, bu…
> But in the late 1800s, Georg Cantor and other mathematicians showed that the infinite really can exist. I think, as I understand it, the objection is this. The proposition that infinity is "real", and there are actually infinite (not just very many) things.
As far as I can tell, numbers aren't real either. "Twelve" isn't a thing that exists in itself in the physical universe, it's an abstraction over some features of reality.
"Infinity" is another abstraction, but it's not the same kind of abstraction as "Twelve". It's a further step.
All mathematics is abstractions, layered on each other. See also "God created the integers, all else is the work of man"
Last year I made the mistake of asking ChatGPT what the world would look like if `∞ === -∞` and it took me seriously (I think) and led me on an hours-long dance where in the end it had me trying to prove, mathematically, that `2 > 1` ... and it was at that point I realised that I'm not cut out to think in numbers and maybe it was for the best that I failed my end-of-school Maths exam
[0]: https://en.wikipedia.org/wiki/Projectively_extended_real_lin...
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, bu…
You can idealise it like many things in mathematics, but implementation details fail compared to the abstract ideals.
My favorite math paper is "Is 10^10^10 a Finite Number?" by David van Dantzig. It lies more on the side of philosophy, so many can understand it easily. I first learned about it many years ago from Van Bendegem's list of strict finitism papers, and I would recommend that list for anyone interested in learning more about strict finitism. For my personal opinion, strict finitism provides a richer field of study than po…
2^512 is the number of distinct values of 512 bits ... it sure seems to succeed in existing. SHA-512 is a cryptographic hash that depends on that.
My favorite math paper is "Is 10^10^10 a Finite Number?" by David van Dantzig. It lies more on the side of philosophy, so many can understand it easily. I first learned about it many years ago from Van Bendegem's list of strict finitism papers, and I would recommend that list for anyone interested in learning more about strict finitism. For my personal opinion, strict finitism provides a richer field of study than po…
You know about busy beavers? These programs do fit in few bits, yet the number of states they can reach does not.
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, bu…
Formalism is a very common approach in the philosophy of mathematics-but it isn’t the only one, and it is not the philosophy which motivates ultrafinitism.
Another viewpoint is that mathematical objects somehow really exist; mathematics is more than just a symbol manipulation game. One variation of this is (mathematical) Platonism, which believes they exist in some timeless realm beyond this physical universe; that view has no issue with infinities either, since adherents of this view generally believe that realm to be infinite and filled with infinities.
Yet another view is conceptualism-mathematical objects really exist, but in the human mind. And this is the viewpoint that motivates ultrafinitism - the human mind is finite, so infinite mathematical objects cannot really exist in it, or at least not in the fullness of the sense that finite objects can; and that turns out to be true, not just for infinities, but also for overly large finitudes.
This idea that some mathematical objects are in a philosophical sense “more real” than others is a big motivator of mathematical constructivism-trying to find axioms which respect that philosophical distinction, and work out what the consequences of those axioms are. Ultrafinitism is just a particularly extreme form of constructivism, which adopted a stricter “criterion of reality” for mathematical objects than most constructivists do