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I have a good example why practical mathematics needs philosophy. In mathematical analysis and computer science, natural numbers and infinities have different c
by js8 4y ago
I have a good example why practical mathematics needs philosophy. In mathematical analysis and computer science, natural numbers and infinities have different character.
In most of mathematics (like number theory or analysis), it's the countable infinite objects and their properties that are being studied, like natural numbers or infinite sequences. The finite sets themselves are considered somewhat trivial (natural) objects, and uncountable sets are mostly a universe where things happen. As a consequence, for mathematicians, all natural numbers are the same, no matter how large.
In computer science, I would argue, the view is quite different. The interest of computer science is in medium-sized but finite sets (roughly between 2^6 and 2^(2^6)). The small sets of size less than 2^6 are trivial (you can brute force them without a computer), while with sets that are larger than 2^64 we almost don't need to compute with (they are like limits what can a DB store), and numbers above 2^(2^64) are pretty meaningless. So you can see, different natural numbers matter differently, some are important and some are less, they are not treated as identical objects.
Since computer science is kinda part of mathematics, you can see the philosophical focus is different.
- bibanez 4y agoStill, what you prove for all naturals in Math still works for a finite number of them so the limitations in CS are just arbitrary, I don't think they matter much philosophically
- js8 4y agoThey matter in what problems each field focuses on. And the idea that the natural numbers are all the same, but infinities have different character can be seen as arbitrary as well (cf. set theory).
- ogennadi 4y agoWe need INFINITY because no matter how much storage we have, we'll always have problems that need even more storage to solve?
- srcreigh 4y agoThis view of CS seems to exclude two of the most important topics - undecidability and NP hardness. the main techniques in both of these areas require infinite sets. For undecidability, it’s the infinite set of proposed halting problem TMs, and for NP hardness, it’s the infinite set of solutions to a given problem. Runtime analysis also requires numbers larger than 2^64 — any algorithm which is not O(n) is practically faced with these numbers. Sorting, matrix multiplication, etc.
- js8 4y agoYes you can have mathematical techniques in CS, I am talking about focus or emphasis of the discipline. Obviously, there are gonna be subjects on the fringes (set theory is another example). Similar to what you say, mathematical analysis uses uncountable sets (especially the ones above reals). But they're just a theater, not the actors. The actors are the countable sets - sequences, continuous functions.. And I think similar situation is in CS, where infinity, while useful, is just a theater to uncover properties of medium-sized finite objects. For many people, philosophy is kind of "obvious babbling". But as you can see, that's what we engage here. It's interesting to think about this in the context of Skolem's paradox - I think the two views (the mathematical and the CS emphasis on nature of numbers) are actually equally valid.
- srcreigh 4y agoI disagree my friend. CS is math; they aren’t separate. The topics are at the core, not on the fringe. And if we need infinity to understand the small objects, then infinity is more important than the small objects themselves.