Re: [Phys-l] What comes first, the equation or the explanation?

[John Denker <jsd@av8n.com>]

On 12/21/2011 05:44 PM, chuck britton wrote:

> Feynman's famous Diagrams certainly fit in here somewhere.
>
> The diagrams provide very simple visualizations of some (so I've 
> heard) complex calculations.

Yes, that's a good example.

Continuing that line of thought, a more modest example of the
same thing occurs in electronic circuit analysis.  
 -- There are graphical symbols for resistors, capacitors,
  inductors, et cetera.
 -- There are also equations that give the current versus 
  voltage relationships for such devices.

This gives us another way of answering the original question:
which comes to mind first, the circuit diagram or the equation?

One answer is, it depends on the context.  It depends on the
Gestalt.
 -- If I'm in the midst of drawing a circuit diagram, and
  somebody tells me to add a low-pass filter, I will draw
  the graphical resistor and capacitor symbols.
 -- OTOH if I'm in the midst of writing down the transfer
  function algebraically, and somebody tells me to add a
  low pass filter, I'm going to write 1/(1+sRC).
 ++ On the third hand, this is consistent with my previous
  answer, which is that neither of these representations
  comes "first" because they are so tightly linked that I
  cannot think of one without thinking of the other.
 ++ Perhaps even more importantly, one should never confuse
  a symbol with the thing symbolized.  Neither the algebraic
  "R" symbol nor the diagrammatic "squiggly line" symbol
  are real resistors;  they're just symbols.  My mental
  model of what a resistor "should" includes all sorts of 
  symbols and models, algebraic and otherwise, but all of 
  that is only an approximation to what a real physical 
  resistor actually does.

Let's be clear:  
 a) Far and away the most important thing is the real, 
  physical resistor.  
 b) We get to build a mental model of the resistor.  This 
  model starts out simple and gets more sophisticated
  over time.
 c) At a much lower priority, we can -- maybe -- label
  some aspects of the model as "conceptual" or "mathematical"
  or whatever.
 d) At an even lower priority, we talk about ordering these
  aspects ... but this is generally not worth the trouble,
  and often quite impossible.

Also I stand by my assertion that there is an /iterative/
process.  Suppose I diagram a circuit, and analyze it
algebraically, and then build it, and then measure its
actual operation.  If the measurements don't agree with
predictions, I go back and draw a more-refined diagram,
including parasitics and other nonidealities.  And so
forth, iteratively.

=============

Regarding the principle that learning proceeds from the
known to the unknown:

> This implies linearity;

I wouldn't have said that.  Learning is sequential insofar as 
learning depends on communication, which is sequential.  There
are lots of things that are not sequential.  Juggling involves
lots of stuff going on at the same time.  /Learning/ to juggle
is even more complicated.  Ditto for skiing, and learning to
ski.  Ditto for riding a bike, and learning to ride. 

Ditto for chess.  It is famously unhelpful to ask a chess
master to explain his move.  He knows what the right move is,
but cannot explain the reasons for it.  Of course he /has/
reasons;  the problem is not the lack of reasons but rather
an overabundance of reasons.

Occasionally I get a student who refuses to think about more
than one thing at once, in which case I assign homework:
Learn to juggle.  Learn what your brain feels like when there
are N things going on at once.

> The sequence of topics is essentially the same in all textbooks.

I definitely wouldn't have said that.  Given ten things that
are known and ten things that we would like to learn, there
are something like ten factorial different sequences.  In
particular, some physics courses do F=ma before talking about
momentum and energy, but others do energy and then momentum
and then F=ma.  It works just fine either way, and lots of
other ways besides.  De gustibus non disputandum.

Actually, the only thing that makes sense to me is the spiral
approach, where we introduce a new idea, connect it to one or
two old ideas, and keep coming back to it over time, gradually
building more and more connections.....

The connections exist in some very high dimensional space,
maybe 100 dimensions or 1000 dimensions, maybe more.  In
any case, there is a theorem (Brouwer, 1911) that says you
cannot map something from one dimension to another in a way
that is one-to-one and continuous.  I call this the "flower
pressing theorem".  It means you cannot serialize any nontrivial 
topic in any natural way;  you *will* have to skip some things 
and come back to them later.

I am reminded of what William James had to say about this circa
1898.  What I call "connections" he called "associates".  Talking
about ideas and how they are recalled:

> Each of the associates is a hook to which it hangs, a means to fish
> it up when sunk below the surface. Together they form a network of
> attachments by which it is woven into the entire tissue of our
> thought. The 'secret of a good memory' is thus the secret of forming
> diverse and multiple associations with every fact we care to retain.

A student who has made only a few connections might be tempted
to arrange them "in order" in one dimension ... whereas a
professional has made so many connections that it is quite 
impossible to fish up one idea all by itself, or to place the 
ideas in any natural order.  I can choose some arbitrary order, 
but others may choose differently, and I myself might choose 
differently tomorrow.

=============

I also think it is a mistake to classify students too strictly
as "pictorial" thinkers or "mathematical" thinkers.  They should
be taught to approach every problem from multiple angles.  We should 
not typecast them, nor encourage them to typecast themselves.  A
memorable example of what I'm talking about is George Herman Ruth, 
who was for several years classified as a pitcher.
  http://www.baberuth.com/biography/

Perhaps an even better example is Richard Feynman, who is famous
for his diagrams and for his ability to visualize and conceptualize
the physics ... but was also prodigiously good with numbers and 
equations.

Also, remember Galileo:  Was he a theorist or an experimentalist?
Answer: he was prodigiously good at both.  His greatest contribution
was the *unification* of theory and experiment.