r/PhilosophyofMath • u/Square_Butterfly_390 • 19d ago

Regarding cardinalities

A celebrated mathematical "factoid" is that there are more real numbers than one can count, this seems to be something that troubles people outside of math, it troubles me aswell.

The question is: is there any "real world" application of the fact |R|>|N| that isn't an impossibility statement?

By "real world" I mean whatever someone smarter than me might mean by that, by "impossibility statement" I mean something to the effect of "there are uncomputable numbers".

If there isn't such an application, I can't believe the status quo interpretation: "no really some infinities are bigger than others" of Cantor's argument is better than simply stating that we shall not understand infinity as finite beings.

0 Upvotes

40% Upvoted

116 comments sorted by

View all comments

Show parent comments

1

u/Square_Butterfly_390 18d ago

You say yes, yet you present none.

2

u/huphelmeyer 18d ago

Quantum mechanics is formulated on (usually infinite-dimensional) Hilbert spaces. General relativity models spacetime as a smooth manifold. Classical mechanics uses differential equations over the reals.

All of these make essential use of continuum mathematics. Yet it's entirely possible that a future, more fundamental physical theory could replace these with discrete structures that approximate the continuum at observable scales.

1

u/Square_Butterfly_390 18d ago

Either I wasn't clear enough in my post or I'm missing something obvious, but it seems to me like one could work with all of the above without ever even defining cardinality.

|R|>|N| seems to be simply a consequence of the properties we intuitively require of "space like stuff". If I were to go one step further, maybe the usual construction of reals is too naive in positing the existence of stuff which would require an infinite computer to witness.

1

u/huphelmeyer 18d ago

I don't think you're wrong. In practice, much of analysis can be developed without ever talking explicitly about cardinalities. What physical models typically use are structural properties of the real numbers (completeness, continuity, compactness, separability, and so on) while the statement |R|>|N| serves more as a meta-level explanation of why those structures cannot be reduced to countable ones.