## Tracking the World's Great Unsolved Math Mysteries 221

Posted
by
samzenpus

from the another-piece-of-pi dept.

from the another-piece-of-pi dept.

coondoggie writes

*"Some math problems are as old as the wind, experts say, and many remain truly unsolved. But a new open source-based site from the American Institute of Mathematics looks to help track work done and solve long-standing and difficult math problems. The Institute, along with the National Science Foundation, has opened the AIM Problem Lists site to offer an organized and annotated collection of unsolved problems, and previously unsolved problems, in a specialized area of mathematics research. The problem list provides a snapshot of the current state of research in a particular research area, letting experts track new developments, and newcomers gain a perspective on the subject."*
## Re:Strange point (Score:4, Informative)

## If you really want to run into trouble (Score:5, Informative)

See http://rjlipton.wordpress.com/2009/11/12/more-on-mathematical-diseases/ [wordpress.com] for unsolved problems which are all really simple and also really addicting to think about. For many of these, the best way to stop thinking about one of them is to start thinking about another.

I'm actually a bit puzzled as to why this is a Slashdot article. If I wanted to point to something new in the way people are doing math I'd point to Math Overflow http://mathoverflow.net/ [mathoverflow.net] where many professional mathematicians, grad students and others are active. It is essentially a centralized system for people to post math questions and get math answers from people who know. It is very cool. It also is highly addictive to read.

## Re:Check out the Collatz Conjecture... (Score:3, Informative)

"Very little code"? Bah! Kids these days...

This [pastebin.com] will run on any system where `dc` is installed.

## Re:Check out the Collatz Conjecture... (Score:1, Informative)

CAF is a notorious usenet troll. You can safely ignore anything he writes. (Also note: arxiv is not peer reviewed.)

## Re:Math cannot exist before wind. (Score:3, Informative)

The only way it can be proven that mathematics is wholly artificial is to prove that the set of all mathematical "things" that are fundamental is equal to the empty set. ie: there is nothing - not a single property, not a single result - that is true everywhere, including Goedel's Theorum. If even something as simple as Goedel's Theorum is universal, then there exists at least one part of mathematics that is not invented but is wholly natural.

Since the only real constraint that Goedel's Theorem imposes is that there is not a finite set of axioms that can characterize all "sufficiently interesting" mathematics (i.e., that any finite axiomatization is necessarily incomplete) I don't see where you're going with that. It's a construct all the same and no amount of philosophical bullshit will change that.

Now, here we run into a problem. If Goedel's Theorum is not a universal result, but an artifice, then it is also false because it would have to be possible to create a counter-example and the theory states no counter-example of this kind can exist.

The problem is that you're into the space of self-referential mathematics when you're using Goedel's results (they're a necessary part of it, which is where the "sufficiently interesting" really comes from) so your argument just doesn't work too well. In particular, there most certainly is mathematics possible in systems that do not support the expression of Goedel's Theorem, but it's pretty dull stuff (no integers, for example). You can build on that base in various ways, but once you've got self-referentiality then you can get something equivalent to Goedel, and you're stuck. Or you can plunge on, by adding more rules and axioms which will either let you say more (but not everything) or render the whole edifice bollocks. But not one bit of this says anything about whether it is natural or not.

Surely that seals the argument right there and then. Those who argue mathematics is wholly artificial must be arguing Goedel's Theorum is false. All other cases do not prohibit the theorum from being true. Thus, if there is sound reason for believing the theorum true, there is sound reason for excluding the notion that mathematics is an artifice.

Your argument is full of bollocks. Goedel's Theorem is a consequence of a particular level of complexity of a formal system, and once it holds, you've got to a stage where you know there must be truths about the system that cannot be derived from a finite set of axioms about the system. But such a system can be pure artifice (e.g. by moving symbols around according to clearly stated rules, which is obviously non-real) and if one such system can be non-real, you've not proved that your favorite one ("mathematics") is real on that basis either.

Note that I'm not trying to demonstrate that mathematics is artificial. I'm just pointing out that you've not shown that it is not. (I actually suspect this is a matter for philosophers and not mathematicians since the observable effect is the same in either case.)