Ed, (01)
I strongly disagree with that claim: (02)
> The Master Builders knew How; they just didn't know Why. (03)
I agree that they didn't have a mathematically formulated
theory that would enable them to compute the stresses. But
I seriously doubt that they could design cathedrals that might
take a century to build without having a solid understanding
of what keeps a structure from falling down. (04)
I'll get back to that point after discussing the following: (05)
> I suspect you would have seen it differently at the turn
> of the 20th. Think of Edison and Tesla. (06)
An excellent example. Edison and Tesla were as different as
possible in almost every way imaginable. Edison was a trial
anderror, seatofthepants experimentalist who did dozens
or even hundreds of painstaking trials before he got something
that worked. Tesla was an excellent mathematician who worked
out every detail in advance, and his alternating current motors
ran correctly on the first try. (07)
Edison hired Tesla to work in his laboratories, but their
methods of working were so radically different that there was
no way they could collaborate. Before Tesla, nobody believed
that it was possible to design motors that ran on alternating
current. Tesla's designs were truly brilliant, and they were
so complex that it was impossible to discover them by trial
and error  because they wouldn't work at all if the design
wasn't exactly correct. Even today it is difficult to explain
to students how they work. (08)
Tesla was not only an excellent mathematician, he also had
an outstanding ability to visualize electrical and magnetic
fields. In that respect he was similar to Michael Faraday,
who first imagined and described them informally. Faraday
didn't have any training in mathematics, but he taught Maxwell
how to think about those fields. Then Maxwell restated
Faraday's ideas in his four famous equations. (09)
With Maxwell's mathematics, mere mortals who do not have
Faraday's ability to visualize fields can compute the results.
However, I am convinced that Faraday understood those fields
even without the mathematical ability to do the computation. (010)
Tesla combined both abilities: visualization similar to
Faraday's and mathematics similar to Maxwell's. That
combination put him far ahead of Edison in many respects.
Business acumen, unfortunately, was not one of them. (011)
During history, many other famous individuals also had an
outstanding ability to visualize. Leonardo da Vinci was
one of them. His ability was obvious in his paintings, but
it also enabled him to conceive and draw a wide range of
inventions that were far ahead of his time. (012)
Another famous visualizer was Albert Einstein, whose
mathematical ability was modest. His first wife was
apparently a better mathematician than he was. But
Albert could visualize the physics and interpret the
mathematics better than the best physicists of his day. (013)
Although I grew up as a mathematician, I do not believe
that the ability to write formulas is the heart of
mathematics. The most important part is the ability
to visualize the problem and imagine the solution. (014)
Writing down the solution in symbols is bookkeeping.
It's necessary to convince other mathematicians that your
solution is correct. But bookkeeping in mathematics, as
in business, is far less important than the insight to
visualize the solution. (015)
I believe that the cathedral designers had a very solid
theory of how cathedrals should be built, even though
they didn't have the notation or the inclination to write
it down for posterity to verify. (016)
John (017)
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