And not just for Barbie. This article says that math problems are getting too big for our brains.

Well, that's one of the thing that transhumanism is for. This part bothers me, though:

Math has been the only sure form of knowledge since the ancient Greeks, 2,500 years ago.

You can't prove the sun will rise tomorrow, but you can prove two plus two equals four, always and everywhere.

This begs the definition of the words "knowledge" and "prove." Two plus two can be proven, I suppose (inductively from one plus one equals two), but only within the confines of the mathematics that you're using. It's not "sure" or "knowledge" in any absolute sense.

What they really mean is that some of the tougher mathematical problems are not amenable to classic deductive analytical proofs, but are more reliant on brute-force computations, possible now because we have machines that can perform them in a useful amount of time.

Exactly. And if you change the number base (as in Abstract Algebra and Coding Theory) 1+1 = 0

Or, you could prove that:
2+2=10
2+2=11

Yeah, what would the author of the linked piece make of this line from the Group Theory I'm studying right now: "For simplicity, we will assume that zero does not equal one"! :-)

I'm still upset about receiving a transformed version of Fermat's last Theory as a freaking homework assignment. It fit right in with all the other problems in apparent complexity, and we already knew the way to solve the others was to haul out a large toolbox of transforms and keep guessing until one made the problem look more tractable. Then apply a second transform...

It _looks_ like you can wrap your head around it. But 23 pages later you realize 'ok, I have to do one of the others'. Grrr.

Excuse me, but this is the same BS I've seen repeated too many times in the last 30 yrs. It's like claiming someone's a computer genius because he knows some passwords or executable command switches. It's not genius.

Using brute force methods with computers is similar. Because the set your working with is difficult to succinctly define it's now 'too big for human brains!' Horse pucky!

Yes, we puny humans have limited capacity, but we've learned to deal with infinities and other difficult issues quite well thank you.

Look at the size of the galaxy. Then the universe. Be humble and quit worrying about our poor brains being able to handle it. We will do what we ALWAYS do. We will generalize. Simplicity by generalization is the mark of genius anyway.

I can't agree that humans have limited capacity...maybe in physical form. Our intellect and the ability to adapt it to whatever problem is there is infinite. There is, of course, always the time taken in such endeavors, but eventually, we figure it out. Then again, maybe Barry Bonds would disagree that we're limited physically.

Mac, it's pretty clear that human minds are finite in several concrete ways. First, according to legend, Gauss was the last fully human mathematician who understood the whole realm of mathematics. Everyone since is a spec-ialist. Also, you mention the time problem, as far as mathematics goes, mathematicians often remain productive into their 80's and 90's.

Second, humans won't always be able to understand computer-generated proofs. The problem is that that a long computer proof may already be as compact as it can possibly be. That length might exceed unmodified human limits by a few orders of magnitude.

Due to the stopping time conjecture (to which theorem proving in a sufficiently complicated and interesting axiom set is equivalent), there's even statements which are true, but which you can't prove so using a deterministic machine program quantum or not (they are equivalent in terms of what programs they can run) with infinite storage, much less using all the available resources of the visible universe (which are finite), much less using human brain power.

I'll read this article. But it's kind of snooty to say mathematicians aren't good philosophers. Mathematicians aren't philosophers. And I fail to see (from the news story blurb) why that observation is relevant.

This begs the definition of the words "knowledge" and "prove." Two plus two can be proven, I suppose (inductively from one plus one equals two), but only within the confines of the mathematics that you're using. It's not "sure" or "knowledge" in any absolute sense.

Good observation Rand. My take is that "proof" shouldn't always be taken in a strictly mathematical sense. There's no reason that we can't generalize proof methods to apply the empirical world.

For example, the use of Bayesian and other statistical methods allows you to come up with a proof scheme where the output is the likelihood (or some other consistently assigned weighting system) based on a list of observations and assumptions that a statement is true.

Karl:
Mac, it's pretty clear that human minds are finite in several concrete ways.

No argument there.

First, according to legend, Gauss was the last fully human mathematician who understood the whole realm of mathematics.

Legend usually denotes some story-telling :)

Everyone since is a spec-ialist. Also, you mention the time problem, as far as mathematics goes, mathematicians often remain productive into their 80's and 90's.

What I was trying to allude to is that it can take quite some time to solve a particular problem, well beyond lifetimes of individuals, but mankind (humans) will eventually understand and manipulate the circumstances of that problem.

Second, humans won't always be able to understand computer-generated proofs.

One of the reseasons we invented computers. We used our knowledge to create a tool to help us with another problem. Infinite creativity and drive.

The problem is that that a long computer proof may already be as compact as it can possibly be. That length might exceed unmodified human limits by a few orders of magnitude.

I agree here, but I wasn't using a measuring stick of unmodified for the human mind. I'm thinking closer to the collective thught of mankind itself. I truly believe that we as humans are infinite in ability of our minds. Maybe its a pipe dream, but that's one pipe I'll smoke until they put me in the ground....'sides, makes great background for novels. (Yeah, I'm a writer too.)