REB quoted Bailey & Borwein: > "Note that the above activities are, for the most part, quite similar to the > role of laboratory experimentation in the physical and biological sciences.”
Yes, this makes sense. I think my initial confusion arose from a conflation of “experimental math” with “theoretical math” (the common thread being “tentative conclusions”, but of course I should have noticed the difference in terminology, which indicated it was the differences which are important, not the similarities). I really like the idea of “math labs”, and I wish they’d been part of my primary school curriculum. We had computer labs, which were awesome, but math was taught on a blackboard, as a set of revelations from dead gods (the ancient Greeks and their geometers, the incomparable Newton and his calculus of variations, and so on). The subject was still very engaging (because unlike, e.g. Spanish, it did not depend rote memorization [which is not a strength of mine], and if you ever got stuck on something, you always had the option to re-derive it from first principles), but I think it would have been even better if we’d been given an opportunity to play around and see some of the observations which had led the giants of history to *derive* these conclusions, rather than being handed them as opaque axioms [1] (though, to be fair, we did get more of this in later years, like HS and of course college). > "To be precise, by experimental mathematics, we mean the methodology of doing > mathematics that includes the use of computations for: > 1. Gaining insight and intuition. > 2. Discovering new patterns and relationships. > 3. Using graphical displays to suggest underlying mathematical principles. > 4. Testing and especially falsifying conjectures. > 5. Exploring a possible result to see if it is worth formal proof > 6. Suggesting approaches for formal proof > 7. Replacing lengthy hand derivations with computer-based derivations. > 8. Confirming analytically derived results." > > Well, at least the first 5 items described the role J played for me in the > past couple of years. Ah yes, the light breaks! This is what I was looking for, and now that you point it out, I can’t help but agree. J is helpful for at least the first 5 bullets, and depending on how you use it, maybe all but #7. That said, my personal take, probably colored by my own background and predilections is that J is more suitable for exploring a different kingdom, one which borders Math, and sometimes competes for its citizens: computer science. Which, as they say, is about as much about computers as astronomy is about telescopes. In particular, I think the kind of “science” J is suited to is closer to Wolfram’s NKS, which everyone else calls automata. This is because proofs and formulae are perfect, complete, and static, like jewels. Programs, by contrast, are active, and have some surprising and very often unpredictable emergent properties [2]. It is not far fetched to consider a bee or an ant or as a relatively simple program. And yet hives of them are intelligent, in some meaningful sense of that word (and of course humans are just big masses of tiny independent cells). One of the most memorable and enjoyable examples (for me, anyway) of using J for this kind of work was your (REB’s) exploration of Grey Codes a few years back. I distinctly remember writing a Grey Code function I thought must be close to the limit in performance, because it used bitwise functions, and was about as close as you could come to writing a C or assembler Grey Code program without actually leaving J proper. But then you went and beat me anyway. By a not insignificant margin. > In that process I stumbled upon Hilbert curves and it ended up by > discovering, with the help of J, a new type of Hilbert curves, > hyper-orthogonal ones, which now is a main subject of the PhD-thesis I am > working on. I’m now going to use this as a testimonial when I am talking to a non-convert about J. -Dan [1] If I understand correctly, this one of, if not the primary, motivation behind recent education reform bills here in the US, which, after implementation and a few years of practical experience, are now being denounced by both ends of the political spectrum as “impractical” (which I find disappointing). [2] This is one reason I tend to call BS on people in the industry who claim to have “solved" horizontal or elastic scalability: unlimited scalability is a myth, and we’re likely to locate the actual holy grail before we can just write an arbitrary naive program that someone else will figure out how to scale up for us. ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
