On the topic of experimentation – yes it is vital to find truths before we have the theory. What bothers me is the article discovers some interesting things by experimentation and completely fails to ask the question of where these things come from, when there is a beautiful theory here, which pretty much any mathematician with a passing acquaintance with number theory would know. If a company which notionally produces mathematical tools doesn’t do this, then i think that’s a bit sad. They had a great opportunity to snare a passing reader with an interesting question, some experimentation, and pointers to theory discussing deep truths. Again, maybe not the point of the article … however, it is good for passing readers to realise that a bit of experimentation can, and usually does, lead deeper when you ask the right questions.

Finally, it gave me a chance to tell the Kepler counterfactual anecdote which i like. I do think this was a case of “oh, that’s cute” and failing to followup and thus missing something really nice.

]]>Actually I’d say that’s unfair… to base 7. Digits in base 10 contain 18% more information than base 7. To correct for this, I’d divide your measure by log(b) where b is the base. This give 22/7 a still stronger lead, but makes 3.14 better than the other convergents.

I think your broader argument about the reliance on brute force neglects the practical reality that almost all problems we face are simpler (or even trivial) if only we know the correct theory. The trouble is that we don’t necessarily know the correct theory (in fact, it might not be known by anyone yet) and there’s a cost and risk in acquiring the relevant theory. When you write “not brute force, but by thinking” what you really mean is “not brute force, but by study and a little thought”, unless you worked out the theory of convergent fractions yourself (and even then, it would be rather haughty to expect everyone else to do it too).

The original Wolfram post demonstrates how a piece of software can help you discover some truths without knowing the theory, though obviously the theory is better if you happen to know it. The post is also about best utilising human memory, and for good or bad, all our memories are configured with a bias to base 10. Personally, I favour the dozenal system (http://hexnet.org/content/excursion-numbers) if only the switching costs weren’t so high, though the reasons for that are connected to working with rational numbers rather than any particular irrational one.

I also think that the original post doesn’t take itself quite as seriously as you treat it. It’s a fun game playing with a tool rather than a proof of anything serious or practical.

Thanks for posting your thoughts.

Cheers,

Mark

If you check out this link: http://www.ereaderiq.com/ and sign up for e-mail updates, they will send you an e-mail every time a new free book is available. I’ve actually read some pretty good free ones, and if you don’t like it you can discard it and lose nothing!

]]>(Because I am the official representative of all blogs everywhere.)

]]>