See Feynman's Knowing versus Understanding: https://www.youtube.com/watch?v=NM-zWTU7X-k Calculation and "the right answer" are never enough.
See also: Wholeness and the Implicate Order by Bohm for a discussion of the pilot wave vs quantum theories. On Sat, Dec 3, 2016 at 3:57 PM, Lawrence Bottorff <borg...@gmail.com> wrote: > In my opinion, exactly what S&A are doing in SICP is what math should be > about. A great example is Tree recusion 1.2.2 > <https://xuanji.appspot.com/isicp/1-2-procedures.html> in this nicer > version of SICP (scroll down to 1.2.2). It talks about the Fibonacci series > and how to do it with a Scheme tree recursion -- but then how wasteful that > is due to the duplication. Then they talk about the phi equation. Then they > compare, talking (around) big-O. This is exactly how math -- starting > somewhere in middle or high school should be taught! SICP obviously > emphasized the programming, but that could be flipped. So yeah, pummeling > kids with weak, hand-waving Stage 1 math is a real loser. It's not real > theory, it's not real-world computational. And, as Sal Kahn says (after I > said it for years), American K-12 math is not mastery-oriented, rather, > just give them a letter grade (whatever that is supposed to mean/achieve) > and herd them to the next level . . . deficiencies accumulating, math > phobia building. > > In the real world math is done with electronic digital machines, i.e., > computers. Not even calculators anymore! So when I see second-rate versions > of Stage 1 math being taught sans computer but those ubiquitous > WAY-overpriced "graphing" (sic) calculators in hand, I see red. > > By the way, what do you think of Sussman/Wisdom's *Structure and > Interpretation of Classical Mechanics (2nd ed). *It's a true literate, > tangled code tome. I can imagine NASA hiring a bright young physicist who > mastered Goldstein's *Classical Mechanics *-- but then this kid has no > idea how any of it is actually done in the real world, i.e., how to do it > on a computer. So often the computational side is just an afterthought in > schools. But then another question about SICM. It's using Scheme with a > library Sussman created. It seems to beg the question, When do you write > code versus when do you use CAS systems like Axiom? > > LB > > On Fri, Dec 2, 2016 at 6:09 PM, Tim Daly <axiom...@gmail.com> wrote: > >> https://terrytao.wordpress.com/career-advice/there%E2%80%99s >> -more-to-mathematics-than-rigour-and-proofs/ >> >> One can roughly divide mathematical education into three stages: >> >> 1. The “pre-rigorous” stage, in which mathematics is taught in an >> informal, intuitive manner, based on examples, fuzzy notions, and >> hand-waving. (For instance, calculus is usually first introduced in terms >> of slopes, areas, rates of change, and so forth.) The emphasis is more on >> computation than on theory. This stage generally lasts until the early >> undergraduate years. >> 2. The “rigorous” stage, in which one is now taught that in order to >> do maths “properly”, one needs to work and think in a much more precise >> and >> formal manner (e.g. re-doing calculus by using epsilons and deltas all >> over >> the place). The emphasis is now primarily on theory; and one is expected >> to >> be able to comfortably manipulate abstract mathematical objects without >> focusing too much on what such objects actually “mean”. This stage usually >> occupies the later undergraduate and early graduate years. >> 3. The “post-rigorous” stage, in which one has grown comfortable with >> all the rigorous foundations of one’s chosen field, and is now ready to >> revisit and refine one’s pre-rigorous intuition on the subject, but this >> time with the intuition solidly buttressed by rigorous theory. (For >> instance, in this stage one would be able to quickly and accurately >> perform >> computations in vector calculus by using analogies with scalar calculus, >> or >> informal and semi-rigorous use of infinitesimals, big-O notation, and so >> forth, and be able to convert all such calculations into a rigorous >> argument whenever required.) The emphasis is now on applications, >> intuition, and the “big picture”. This stage usually occupies the late >> graduate years and beyond. >> >> >> I'm of the opinion that computational mathematics is at the first stage. >> We write >> >> code that "sort-of works" and lacks any attempt at formality, even >> failing to >> >> provide literature references. Moving to stage 2 will be a long and >> tedious >> >> task. Axiom has the connections to the proof machinery and is being >> >> decorated to provide some early attempts at proofs using ACL2 and COQ. >> >> This effort is an interesting combination of mathematical proof and >> >> computational proof since both fields underlie the implementation. >> >> Moving computational mathematics up Tao's tower is going to be a long, >> slow, >> >> painul effort but, if memory serves me correctly, so was graduate school. >> >> Tim >> >> >> >> >> >> _______________________________________________ >> Axiom-developer mailing list >> Axiom-developer@nongnu.org >> https://lists.nongnu.org/mailman/listinfo/axiom-developer >> >> >
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