Comment by meroes
10 months ago
> You seem to be imagining that you need to understand Turing machines before you'll be allowed to make statements about some specific computation.
All computations are finite, like you said. In general it’s a mystery to me how infinites of math are indispensable to the best sciences. In a sense it is simple, sure-we just use it when it helps us like you said. But since the indispensability of modern math is still hotly talked about in philosophy, I don’t know that I would call the finitely bound physical using infinity, simple. Do I think the difficulties of a mathematical and computational system with a biggest number (one we can pluck from cosmology as the largest number of useable states before total heat death) are simpler than one that uses infinite infinities at the lowest level, spawning philosophical challenges? I don’t know. I just can’t confidently call the status quo simple.
Computations are finite, but that limit keeps going up. If we picked a number that seemed like "way more than enough" in 1960, it might be less than our phones can do now. If we picked a number from cosmology, what happens if that science changes? "Heat death" is the current consensus of our distant future, but even that is challenged. Until we know what dark energy is, we can't be sure about "heat death". So if cosmology has some breakthrough that changes our understanding, why make it so we have to go revisit all our CS proofs? Using infinity decouples the proofs from irrelevant externalities.
> the indispensability of modern math is still hotly talked about in philosophy
Meh, philosophers have to talk about something. I'll worry about what philosophers say when I see a single useful insight from the entire field. We've been waiting thousands of years, but I'm sure a useful thought is right around the corner.
And the concept of infinity is hardly modern.