[FOM] more on Hilbert's 6th problem

Martin Davis martin at eipye.com
Sat Jan 24 14:48:05 EST 2004

I wrote:

<<Harvey quotes Hilbert on this. In reading what Hilbert said, it is 
important to keep in mind that this was in 1900 before relativity and 
quantum theory, when it might appear that physics as a field for 
fundamental investigation was coming to an end, and what was left was to 
tidy up the loose ends.>>

I thought of this as a mild uncontroversial reminder that in studying the 
history of ideas, it is important not to forget the historical context. 
However, Harvey has taken issue with this, writing:

<<Is there something that Hilbert wrote that you disagree with?

I doubt if there was ever a time where many reasonable people would agree
that "physics as a field for fundamental investigation was coming to an
end". >>

It is often the case that an intellectual endeavor develops, to begin 
with,  in a rough-and-ready way on the basis of intuition and analogy 
(especially the engine of analogy that the extensibility of a mathematical 
formalism can provide). At such a point insistence on absolute rigor and 
axiomatic foundations can be stifling. Which was more significant for the 
development of mathematics: Berkeley's penetrating foundational critique of 
calculus or Euler's glorious follow-the-formalism-as-far-as-it-takes-me 
efforts? More recently one can point to Italian algebraic geometry, cleaned 
up only in the middle of the 20th century. I think it was E.T. Bell who 
said: "Sufficient is the day to the rigor thereof."

Not that it's important, but I do think that if Hilbert had an inkling of 
what lay ahead for physics, he would not have included his 6th problem on 
his list, at least not in that form. Perhaps he would have contented 
himself with a call for an axiomatic treatment of probability theory.

A good case in point of what was to come is Bohr's theory of the hydrogen 
atom. This was a beautiful simple theory that was in excellent agreement 
with experiment. It predicted correctly the position of the lines 
corresponding to hydrogen in a spectroscopic analysis of light (terrestrial 
or stellar). There was just one problem: the assumptions were inconsistent. 
Bohr assumed that the electron revolved around the nucleus (a proton) like 
a planet around a star, but held by Coulomb force rather than gravity. In 
addition, he assumed that the atom could emit radiation (producing those 
spectral lines). Using classical mechanics this would lead to a 
degeneration in which the electron spiralled into the nucleus. But instead 
Bohr introduced the ad hoc assumption that there is a discrete set of 
energy levels, and that radiation is accompanied by a simple "quantum leap" 
(the energy being an integer multiple of Planck's h) into a lower energy 

As to the feeling at the end of the 19th century that physics might be 
coming to an end, I can say this: when I was a boy many decades ago, I read 
vociferously about developments in contemporary physics, and in that 
literature, the statement was a commonplace. The recent post by Laura Elena 
makes the same point. I believe it was the American Nobel prize-winning 
physicist Millikan who is supposed to have said that future generations of 
physicists would be limited to more accurate measurements of the basic 
physical constants. (If I remember correctly, Millikan was awarded the 
prize for his "oil drop experiment" measuring the charge on an electron, 
and that it later was found out that the value he had come up with was 
seriously off.)

In a later post, Harvey reports on his reading an account of physics at the 
end of the 19th century:

<<I read there about how the physicists felt that there are purely mechanical
explanations of all sorts of phenomena, and they were desperately trying to
give such explanations, but coming up against brick walls.>>

This desperation and "brick walls" are apparent only in hindsight (which is 
notoriously 20-20). At the time, many investigators thought that their 
difficulties would soon be overcome. The development of statistical 
mechanics had shown (although without the rigor that Hilbert was rightly 
asking for in that connection) how the laws of thermodynamics could be 
reduced to the mechanical motion of molecules. There was every expectation 
that light would be brought into the fold.


                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
                          (Add 1 and get 0)

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