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Re: [Axiom-developer] Computational Math and Terence Tao's three stages

From: Tim Daly
Subject: Re: [Axiom-developer] Computational Math and Terence Tao's three stages of mathematics
Date: Thu, 8 Dec 2016 04:56:40 -0500

See Feynman's Knowing versus Understanding:
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 <address@hidden> 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 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?


On Fri, Dec 2, 2016 at 6:09 PM, Tim Daly <address@hidden> wrote:

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.


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