Dan, good to read from you.
Sorry my answer took a while, but I attended a lecture on "Infinite bookkeeping", from which I learned that 1-1+1-1+1-1+1-1+1-1+.....= 1r2 (the dots meaning 'ad infinitum') and 1-2+3-4+5-6+.... = _1r4 , unfortunately this cannot be experienced with J. From "Mathematics by Experiment: Plausible Reasoning in the 21st Century", Jonathan Borwein & David Bailey you can learn "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. For the last three items you have to use the more well-known tools like Maple or Mathematica, I've heard. But let me give you a concrete example I experienced. From a certain characteristic of the subcurves of the Hilbert curve, I was calculating the average over all subcurves and got the impression, with the help of J, that the average was below 2, in all dimensions (I calculated, at least). In proving that I divided the problem in subclasses of subcurves, for which I could prove the average was <: 2 in each subclass. But for one class I could not and J told me why, the average there was >2, sometimes. Since I am still convinced of the statement, I try to calculate with J to see whether I can add two classes to level the surplus of the one with what is missing in the other. Still under construction. But far more important, however unusable as a selling point, is the fact that J changes your way of thinking, at least, it did with me. You learn to consider problems more holistically, I would rather say. Don't know how to express it better (in English ...). And this happened for me a long time ago already, in solving a question of Ewart Shaw, more than 11 years ago, http://www.jsoftware.com/pipermail/general/2004-September/018185.html 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. So yes, it is experimental as Bailey & Borwein write: "Note that the above activities are, for the most part, quite similar to the role of laboratory experimentation in the physical and biological sciences. In particular, they are very much in the spirit of what is often termed "computational experimentation" in physical science and engineering, which is why we feel the qualifier "experimental" is particularly appropriate in the term experimental mathematics. We should note that one of the more valuable benefits of the computer- based experimental approach in modern mathematics is its value in rejecting false conjectures (Item 4): A single computational example can save countless hours of human effort that would otherwise be spent attempting to prove false notions." But as I have tried to clarify, J can be much more than just that. R.E. Boss > -----Original Message----- > From: Chat [mailto:[email protected]] On Behalf Of Dan > Bron > Sent: maandag 11 april 2016 14:58 > To: J Chat <[email protected]> > Subject: Re: [Jchat] Computationally Assisted Mathematical Discovery and > Experimental Mathematics > > REB wrote: > > to show J to be one of the best tools for experimental mathematics. > > Is it? I think it might be a great tool for learning mathematics up to the > secondary school (high school) level, but experimental math? Wouldn’t that > be largely symbolic? > > Or, even if numerical, wouldn’t it be better explored using a environment > with more pre-packaged tools, like Matlab or R or something? > > I’m not knocking J here (you all know it’s my intellectual addiction), but > I’d be > interested in hearing your thoughts on what makes J powerful for > experimental math. > > Unless you meant experimental in the sense of a chem lab in high school? A > hands-on, instructor-led training, used as a pedagogical method? > > -Dan > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
