[FOM] more on Hilbert's 6th problem

Harvey Friedman friedman at math.ohio-state.edu
Sun Jan 25 00:08:07 EST 2004

On 1/24/04 2:48 PM, "Martin Davis" <martin at eipye.com> wrote:

> 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.

This is certainly correct. I'm not sure that after foundational thinking
gets fully developed and spreads universally, that it will always be the
case. It should become much easier to inject more sense in the early stages
than it is now. But you say will be the case for a while yet.

>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? 

Euler's. Also: this is long long before f.o.m., before people had any idea
how to make sense of things the way they do now.

>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."

A less clear cut example, partly because Italian algebraic geometry is less
important and powerful and central than early calculus. But again, this is
early stuff, before people knew how to make sense of things.

The only disagreement about this that I sense between us, is our respective
attitudes if:

the Italian algebraic geometers discouraged work leading towards making
sense of it, on the grounds, say, that it already makes sense, or that it is
silly to worry about making sense of it.
> 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.

This is the main point of my posting:

Why do you think that Hilbert would ask for a coherent treatment of
probability theory but NOT physics? Not even classical physics?

I repeat what I wrote before:

>>1) major clarifying foundations for even elementary quantum mechanics is
>>probably hopeless until we get a far better grip on the foundations of less
>>mysterious physics;

>>2) in fact, we don't even have a good enough idea as to what foundations of
>>physical science should or could look like, in the first place.

>>I submit that Hilbert would substantially agree with this.

> 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
> level.

It appears that you think that I am opposed to doing


which is in between sense and nonsense. On the contrary, the only thing I am
opposed to is

*doing halfsense and pretending that it makes sense, and behaving as if
there is little value in making sense of the halfsense*

In fact, what do you think I am doing when I do metametamathematics, or even
metametametamathematics right here on the FOM?

Of course some of what I say is, in some sense, halfsense, because I can't
yet turn it into systematic knowledge. I am trying to lay the basis for it
later becoming systematic knowledge.

But I don't pretend that it is systematic knowledge, yet. I do, however,
feel that it is valuable to pay attention to it, and that I am right about
what I say, and people who disagree with me are wrong.

But that is far short of high quality knowledge. What it is is halfsense.
> 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,

just a few

> 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.)

Nice story about Millikan. I guess you think that Hilbert agreed with
Millikan, and FOR THAT REASON, wrote his 6th problem that way?
> 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.

What I don't yet see is how that makes you think that Hilbert wouldn't have
written problem #6 the way he did now, just adding some other mysteries -
e.g., elementary quantum mechanics and so forth.

The same impulses that led Hilbert to work so hard in f.o.m. would have led
Hilbert to problem #6. And as I said above, I also think that Hilbert would
have embraced the idea that foundations of physics should not skip over
elementary physics, and go right into state of the art physics.

Similarly, it would be wholly counterproductive for f.o.m. in the early days
to have skipped over foundational issues surrounding basic mathematics, and
try to move right away to issues such as the necessary use of impredicative
definitions or the necessary use of large cardinals.

Harvey Friedman

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