Re: Computational Math and Terence Tao's three stages of mathematics

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 <[email protected]> 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 <[email protected]> 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
>>
>>
>>
>>
>>
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>>
>>
>

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