As previously noted, I couldn’t hack Stephen Wolfram’s big book but I like his way of thinking. This new post from his blog is fun and fascinating. It’s about a 20-year-old kid who met a challenge Wolfram set out earlier this year — with a $25,000 reward attached. Good (if esoteric) reading.

The general concept of “discovering” solutions vs. engineering them seems fairly profound, yeah?

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In some ways that’s always been the difference between mathematicians and engineers. That’s why engineers, scientists, applied mathematicians, et al. can get away with knowing a little math, reducing practical problems to their sticking points, then looking up the solution to the reduced form in a treatise written two hundred years ago by an abstract mathematician who just loved numbers and never saw the practical side of anything.

An astronomer, a physicist and a mathematician were holidaying in Scotland. Looking out from the train window, they saw a black sheep in the middle of a field.

“How interesting,” observed the astronomer, “all scottish sheep are black!”

To which the physicist responded, “No, no! Some Scottish sheep are black!”

The mathematician gazed heavenward in supplication, and then intoned, “In Scotland there exists at least one field, containing at least one sheep, at least one side of which is black.”

…

Mathematics is fundamentally about skepticism. That’s why we don’t accept your phony-baloney computer “proofs.”

Well played, Carmody… well played.

There’s a good book to be written titled “Mathematician! The Art of the Diabolical Counterexample.”

Eh, scientists are all skeptics, but as Bertrand Russell (wasn’t he into math?) said,

I’m sure we’re on the same page with this, but I think it

isrelevant, for example, to the “controversy” surrounding the computer modeling of the Intergovernmental Panel on Climate Change.Yes, but Bertrand Russell was also a master of the diabolical counterexample. Just ask Frege.

Frege had this really beautiful, totally intuitive set theory, which he used as a foundation for predicate calculus and arithmetic and all of mathematics.

The problem was with Frege’s concept of the extension of a set (the objects that satisfy a property that the set defines), and the formal way that Frege defined this concept. It permitted a certain kind of meta-reasoning that led to Russell’s Paradox, which goes something like this.

Some sets include themselves as members, while some don’t. (Let’s say, an ordered set of all ordered sets.) We can define the set S of all sets that don’t include themselves as members. Does S include itself or not? Whether it does or doesn’t, the set’s extension contradicts itself.

There have been lots of attempts to try to reduce mathematics to a closed formal procedure. They’ve all turned out to be deeply, but hardly ever obviously, flawed. That’s what mathematical skepticism is about.

Okay, I concede that the power of the counterexample is greater in mathematics than in softer science. But does that show that mathematicians are more skeptical, or does it just reflect that the field is more controlled?

I appreciate the philosophy of the Russell quote, which recognizes that in the real world a single counterexample may not be enough to discount a working theory, especially if there is no alternative or better theory. Essentially, skepticism serves all rational people, but in a world that is chaotic enough to be practically stochastic, we cannot ally our skepticism with rigid formalism; it must instead be tempered with courage and practicality.

But sorry for all the navel-gazing. In the context of the Wolfram post, there’s another point: in mathematics, it can sometimes be proven that a single counterexample is the

onlycounterexample in the space of possibilities, which is another useful result we don’t get much in softer science.Slashdot is reporting that the 2/3 Turing Machine isn’t universal.

Take that phony computer proofs!

The reply!

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