There’s more to mathematics than rigour and proofs (2007) • Hacker News Board

Re: There’s more to mathematics than rigour and proofs (2007)

#91

Post by l33t2328 » Thu, Apr 21, 2022, 3:02 PM UTC

ravi-delia wrote: post #87

l33t2328 wrote: post #61

Earlier quoted context omitted.

I’d argue that logic is a subfield of computability theory, and so by extension we (in CS) absorb math.

Like I said, you can indeed construct logic through CS! But you can also obviously construct CS from logic. You can also construct logic through the natural numbers, so number theory absorbs math and by extension CS? Fact is anything reasonably complex can replicate everything else. Exactly one of these fields, however, is specifically carved out as the study of any formal system, and it isn't CS.

I’m arguing that logic is a subfield of CS, so “constructing” CS “from” logic isn’t a contradiction.

Re: There’s more to mathematics than rigour and proofs (2007)

#92

Post by Koshkin » Thu, Apr 21, 2022, 9:35 PM UTC

l33t2328 wrote: post #90

Koshkin wrote: post #85

Earlier quoted context omitted.

How so? Even a real interval of a finite length cannot be covered by any set of intervals of a smaller total length. (Unless the person you are trying to explain this to starts raising questions about the meaning of 'interval' or 'length', in which case the meaning of the original question becomes just as uncertain in the first place.)

> Even a real interval of a finite length cannot be covered by any set of intervals of a smaller total length. You’ve only restated the problem without saying _why_ covering the reals is different.

Because the 'intervals' in question are already 'real'?

Re: There’s more to mathematics than rigour and proofs (2007)

#93

Post by Koshkin » Thu, Apr 21, 2022, 9:43 PM UTC

l33t2328 wrote: post #89

Koshkin wrote: post #86

Earlier quoted context omitted.

It does not look like CS studies the real numbers.

Each real number can be represented as a subset of the natural numbers. CS surely studies these.

But not the representations themselves (which is what's important in this case).

Re: There’s more to mathematics than rigour and proofs (2007)

#94

Post by l33t2328 » Fri, Apr 22, 2022, 4:19 PM UTC

auggierose wrote: post #88

l33t2328 wrote: post #77

Earlier quoted context omitted.

I mean that for any epsilon > 0, you can have a set of intervals of the form (a_i, b_i) where every rational number is in some interval and the sum over all i of b_i - a_i That is, you can cover the rationals with intervals of arbitrarily small total length.

I see. But is that not just a simple conclusion of the fact that the rationals are countable, and that there exists a converging series of positive numbers?

Right but explaining why the reals are different is the tricky part

Re: There’s more to mathematics than rigour and proofs (2007)

#95

Post by qiskit » Sat, Apr 23, 2022, 3:18 PM UTC

l33t2328 wrote: post #43

qiskit wrote: post #13

Earlier quoted context omitted.

CS is math...

I’d argue math is actually (a proper subset of) CS.

CS only involves computable numbers ( Turing ) which are a subset of real numbers. By definition CS is a subset of math, not vice versa.