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

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

#81

Post by dang »

postingposts wrote: post #68
hither_shores wrote: post #63

Earlier quoted context omitted.

That is not at all what the incompleteness theorems say. The first incompleteness theorem says that for any consistent formal system T (with a recursively enumerable set of axioms) capable expressing of elementary arithmetic, T can express a statement which it can neither prove nor disprove. The second incompleteness theorem says that T can't prove the statement "T is consistent". (I've still glossed over a number of…

Yes, it is. Sorry if you can’t read deeply into it or something. I’m not posting for grad students. I can sense you just like to correct people. Ahhhh I’m so wrong, you’re right?

We've banned this account for continuing to break the site guidelines after we asked you to stop.

https://news.ycombinator.com/newsguidelines.html

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

#82

Post by l33t2328 »

Koshkin wrote: post #75
l33t2328 wrote: post #54

Earlier quoted context omitted.

Cardinality hasn’t much to do with it since there are uncountable sets which you may cover like that.

Sure, but for countable subsets (such as the rationals) it is easy to show.

Sure. Explaining what’s different about the reals is the hard part

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

#83

Post by nextos »

07121941 wrote: post #60
nextos wrote: post #37

Earlier quoted context omitted.

I totally agree with what you say, but there's a lot of low hanging fruit in mathematizing biology. It's not easy, but it's certainly very impactful.

can you expand on that? sounds really interesting

Drop me an email, contact info in profile :)

Otherwise, I will try to come back to this comment in a few days. I'm a bit busy with an urgent deadline, but I'm glad to expand on this later on!

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

#84

Post by postingposts »

dang wrote: post #81
postingposts wrote: post #68

Earlier quoted context omitted.

Yes, it is. Sorry if you can’t read deeply into it or something. I’m not posting for grad students. I can sense you just like to correct people. Ahhhh I’m so wrong, you’re right?

We've banned this account for continuing to break the site guidelines after we asked you to stop. https://news.ycombinator.com/newsguidelines.html

No post body was provided.

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

#85

Post by Koshkin »

l33t2328 wrote: post #82
Koshkin wrote: post #75

Earlier quoted context omitted.

Sure, but for countable subsets (such as the rationals) it is easy to show.

Sure. Explaining what’s different about the reals is the hard part

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.)

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

#86

Post by Koshkin »

l33t2328 wrote: post #56
Koshkin wrote: post #53

Earlier quoted context omitted.

But the real numbers cannot be faithfully represented in a computer!

So? Neither can the list of possible computer programs, yet CS studies them

It does not look like CS studies the real numbers.

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

#87

Post by ravi-delia »

l33t2328 wrote: post #61
ravi-delia wrote: post #58

Earlier quoted context omitted.

Math is at it's broadest the study of formal systems. Computer science is the study of a particular formal system. While it is a powerful enough system to contain all of math within it, there are many such systems nested within each other. Is the Turing machine formalism more powerful? No. Is it more efficient or intuitive? Also no. It requires axiomatic reasoning to construct, and then reproduces it internally. Math…

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.

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

#88

Post by auggierose »

l33t2328 wrote: post #77
auggierose wrote: post #71

Earlier quoted context omitted.

Well, first you would have to explain what you mean by your statement.

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?

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

#89

Post by l33t2328 »

Koshkin wrote: post #86
l33t2328 wrote: post #56

Earlier quoted context omitted.

So? Neither can the list of possible computer programs, yet CS studies them

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.

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

#90

Post by l33t2328 »

Koshkin wrote: post #85
l33t2328 wrote: post #82

Earlier quoted context omitted.

Sure. Explaining what’s different about the reals is the hard part

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.

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