Re: photelectricity, history, and PAC Learning

[John Denker <jsd@MONMOUTH.COM>]

There has been some discussion of whether it makes sense to analyze the
photoelectric effect from a "19th-century physics" point of view.  This has
led to discussion of point such as
        -- when was the cathode-ray e/m ratio first measured
        -- when were work functions first measured
        -- et cetera
This will at best lead far afield, and at worst will lead to quite a messy
morass, as discusses further below.

But first, let us make contact with another thread:
At 12:57 PM 11/7/00 -0500, Ludwik Kowalski wrote:
> Nobody wants to teach this [History of Science] course and
> it may be my turn to take the burden. I know it can be great
> course but not everybody is a good teacher for it. I am very
> hesitant.

Points to ponder:

1) Teaching the history of science is remarkably different from teaching
science.

2) Kuhn does a pretty good job of explaining why point (1) must be true.  I
like to compare the actual structure of scientific progress to a perennial
plant that suffers quite a bit of "winter kill" each year:

After first year:

             /
            /
           /
          /
         /
        /


After second year:

        \
         \
          \    .
           \  .
            \.
            /
           /
          /


After third year:

                  /
                 /
                /
          .    /
           .  /
            ./
             \
              \
              /
             /
            /

That is, making progress in science typically requires UNLEARNING some of
the current year's "state of the art" knowledge.  (Of course the real
growth is much more complicated and multi-dimensional than ascii sketches
can indicate, and the timescale is decades not years.)

2b) This means in particular that it is a big mistake to think that there
is a self-consistent theory of "19th-century physics" serving as a
foundation on which 20th-century physics is erected.  This is wrong physics
as well as wrong history.

Looking back and picking _selected_ 19th-century ideas with 20/20
hindsight, you can get any answer you want.  Selecting only the ideas that
have stood the test of 100 years' time throws a false light on
history.  Lenard in 1899 (barely in the 19th century) performed crucial
photoelectric experiments, but he did not conclude that the cathode rays
were particles;  he thought they were waves in the ether!  I don't see any
way Lenard could have come up with an acceptable 19th-century theoretical
explanation of the photoelectric effect.  Prior to the experimental
observations, I think any reasonable 19th-century theorist would have
predicted zero photocurrent;  questions about the wavelength-dependence and
intensity-dependence simply would not have come up.

3) This also means that Ludwik is wise to be "hesitant" about signing up to
teach history of science.

-- When teaching science, we typically follow the shortest path from the
root to the current growing tip.
-- When teaching the history of science, if we have any semblance of
honesty, we must discuss the dead side-branches.  This makes the history of
science a MORE COMPLICATED and MORE CONFUSING subject than plain old
science.  This leads to the ironic situation where lazy students think they
will have an easier time if they sign up for "history of science" rather
than "real science".

Training and experience suitable for teaching science is far from
sufficient for teaching history of science.

This also makes a connection with another thread:
At 01:50 PM 11/7/00 -0500, Robert A Cohen wrote,
regarding the notion that a "law" might be a "proven theory"
> Most of my pre-service teachers do think that

In an earlier message I made it clear I didn't believe that, but I wasn't
clear about what I _did_ believe.... So here goes:

4) Essentially nothing in mainstream physics is "proven" in the usual sense
of the word.
     Here I use the term "mainstream" physics to exclude the specialized
     subfield known as mathematical physics.  That is a perfectly
     respectable subfield, but it is too advanced and too specialized
     to be relevant to the current threads.

Kuhn speaks of "falsifiable" theories.  Any theory worth of the name must
be falsifiable.  If it doesn't make predictions then it's not falsifiable
and it doesn't deserve to be called a _scientific_ theory; it's just a
bunch of hot air.

Note that there is no talk of proving a theory correct;  the best one say
is that the theory could have been proven wrong but has survived so far.

These are crucial ideas, central to the way most modern scientists think
about science.  If these ideas are new or unclear to you, please go read
_The Structure of Scientific Revolutions_;  I don't have time to
communicate the whole book via email.

5) In fact, not only do we have essentially no theories that can be proven
correct, most of the quantitative ideas in physics are provably inexact.
        -- We know F=ma is inexact
        -- We know F=GmM/r^2 is inexact
        -- et cetera
These may be asymptotically exact in certain limits, and they may be
extremely useful without being exact, but they remain inexact in any
real-world situation.

Also as Feynman pointed out (volume III section 2-6), even if we knew the
laws exactly, we don't know the initial conditions exactly, and if we want
to predict farther into the future we need _exponentially_ more accuracy.

6) Let me close with a positive statement about what I consider the state
of the art when it comes to validating theories.  Suppose:
  *) we have a source of experimental data, and
  *) we agree to consider only a particular set of theories (possibly
containing some adjustable parameters).

Then:
  a) We can use the data to choose which of the contending theories is the
best (and to fix the adjustable parameters).
  b) We can make firm statistical predictions as to how well the chosen
theory will predict future data from the same source.  These predictions
take the form of lower AND UPPER bounds on the probability that the chosen
theory will predict future data within some specified accuracy.

These powerful and elegant ideas are discussed in the literature under the
name PAC Learning.  PAC is short for Probably-Almost-Correct.  The "Almost"
refers to the specified accuracy bounds, and the "Probably" refers to the
probability that predictions will fall within those bounds.

So the best statement we can make is that typical physics ideas are
PAC.  We can predict that F=ma within some high accuracy with very high
probability under ordinary conditions.