Chris and Dennis, (01)
Just some additional comments on some of the points. (02)
DT> As indicated by the previous comments and articles, imagery
> and its copartner imagination, trump mathematics. (03)
CM> Imagery and imagination play critical roles in solving problems
> in physics, mechanical engineering, and ontology alike. It is
> mathematics that then provides the means to turn images and
> imaginings  our *ideas*  into theory and thence into concrete
> solutions to real problems. (04)
I agree with Chris (and so would David Hestenes and the people he
quoted in his article). For anyone who missed the previous note,
see http://modeling.asu.edu/R&E/SecretsGenius.pdf (05)
Following are my elaborations of some points that Hestenes made: (06)
1. Poincaré and Lorentz discovered most of the mathematics that
Einstein required for the theory of special relativity. (07)
2. But they did not have Einstein's physical insight (the imagery
that Einstein used in his Gedanken experiments) to provide an
adequate interpretation. (08)
3. Einstein was not as talented a mathematician, but he was able
to discover a brilliant interpretation of the mathematics
that Poincaré and Lorentz developed. (09)
4. Einstein's interpretation also led him to apply some rather
simple calculus to the formulas that Poincaré and Lorentz had
previously derived. That process led him to his most famous
equation, E = mc^2. (010)
5. Without the insight, Poincaré never took the step that derived
the famous equation, but without Poincaré's math, Einstein
wouldn't have had the starting formulas from which to do the
derivation. (011)
6. Even after Einstein did the math and published his famous paper,
Poincaré still did not believe Einstein's interpretation. He
was not convinced that matter could be converted to energy,
as the equation predicted. (012)
None of this implies that imagery "trumps" mathematics or makes it
irrelevant or inappropriate. Without the math, Einstein might have
had an interesting idea, but he wouldn't have been able to state it
in a form that anybody else could use. But given the math, Einstein
had the physical insight to interpret it more effectively. (013)
Another example that Hestenes cited was Faraday's insights into
the lines of force of electric and magnetic fields. Faraday was
an experimental physicist with little training in math. But he had
a fantastic ability to visualize force fields and to use those images
to devise remarkable experiments that gave deep insight into the
interactions of electric and magnetic fields. (014)
Maxwell had a very strong training in both physics and mathematics,
and he had lengthy discussions with Faraday to learn his way of
thinking. As a result, Maxwell was able to formulate his four
famous equations that characterized the electromagnetic phenomena.
Those four equations have stood the test of time, and they have
been used for all the calculations of radio, TV, and light waves
and many other phenomena over the past century and a half. (015)
There are some individuals who have both a highly developed
mathematical talent and a highly developed mental imagery. One
of them was Nikola Tesla, who had a fantastic ability to visualize
the electromagnetic fields in alternatingcurrent motors, *and*
he also had a very strong mathematical talent to apply Maxwell's
equations to design the detailed specifications for the motors.
See http://www.teslasociety.com/teslamotors.htm (016)
Moral of the story: Both mathematics and imagery are essential for
science, and there is no conflict whatever between them. They fit
together beautifully. But different people have different talents,
and collaboration is valuable. (017)
John (018)
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