[FOM] proofs by contradiction in (classical?) Physics

[Vaughan Pratt]

On 9/18/2011 12:41 PM, Fouche wrote:
> Can one use proofs by contradiction in Physics? And if it is possible,
> why they are so rare?

Because proofs are rare in physics?

To find a proof in a physics book I had to go through several in my 
collection before coming to Byron and Fuller's "Mathematics of Classical 
and Quantum Physics."  Flipping pages at random, the first proof I ran 
across was a five-line argument for the Fundamental Theorem of Algebra, 
of all things. As luck would have it, it began "Assume the contrary."

Logical thought could be considered to arise from taking the concept of 
"most significant bit" to the extreme of "only significant bit" in a 
given context.  That's the currency in the courtroom: guilty or not guilty.

It is natural to want more bits (was it murder in the first degree or 
merely manslaughter?), but this need is dealt with in the language 
rather than its semantics: the jury may be asked to deliver binary 
verdicts on more than one charge: murder, and separately manslaughter.

The same goes for other areas.  Instead of refining the semantics one 
refines the language with suitable modalities for that purpose, such as 
"maybe," "usually," "rarely," in natural language discourse, "inside the 
unit circle," "on the punctured plane," in mathematics, and so on.

Intuitionism arose from Brouwer's insight into the impossibility of 100% 
confidence in asserting the first bit to be zero without first ruling 
out the possibility that all the remaining bits are one.  Brouwer wanted 
to embed this uncertainty into the semantics of mathematics, but this 
can just as readily be handled in its language, which most 
mathematicians seem to find more convenient.  When one says "practically 
zero" one does not mean something precise such as "plus or minus .001%" 
since that just relocates the problem, rather one means that there is 
some associated uncertainty, which if desired can be quantified but only 
to within limits that themselves are uncertain.

> Is there some intrinsical problem in proving a
> physical fact assuming its contrary? Is it "formally" correct, in the
> system we use to build physical models? Is the "tertium non datur" true
> in classical or quantum Physics?

See above.  What we can say about physics is as much a question about 
language as it is about physics.  LEM can be falsified in one logical 
framework that embeds naturally in another that validates it.  In 
particular every Heyting algebra extends canonically to a Boolean 
algebra by taking its Stone-Priestley dual, forgetting the Priestley 
order to make it a Stone space, and dualizing back.

>
> My two cents are that everybody studying Physics must face an intrinsic
> fuzzyness given by indeterminacy;

Exactly so.

> but even if we restrict to a classical
> framework things are not so easy: what does "assuming ~P" mean, in a
> framework where P can be a _real_ phenomenon (hence true or false by
> mere perception; maybe in a framework where "P is true" is a necessary
> truth)?
> Take this as a joke, but there's a big number of mathematicians
> convinced that Physics is nothing more than a branch of Geometry
> (classical, differential or algebraic, it's not here the place to
> discuss this);

That could work if geometry could be made a natural setting for the 
study of linear operators and their eigenvectors.  Can it?  Its 
underlying geometry is projective, but this forgets crucial structure.

If not, those reducing physics to geometry may find it hard to imagine a 
God that plays dice.

  can the previous questions be restated into a more
> general one, say "Is geometry intrinsically non-boolean/without tertium
> non datur?"

Again, this can be reduced to a question about language.  However not 
all geometries have that concern.  One appealing feature of 
combinatorial topology is that it's a discrete discipline, like number 
theory, that is not burdened with the precision questions that arise 
with continuum-based geometry.  It has this in common with the rational 
field, which suffices for the objects of affine geometry at all finite 
dimensions, though not their transformations, which introduce 
multilinearity.  The Euclidean field (closure of the rationals under 
square root) suffices for the objects of Euclidean geometry (again at 
all finite dimensions) but I'm not aware of any practical way of 
exploiting its discreteness, unlike the rationals where it's easy.  It's 
therefore easier to fall back on the continuum for Euclidean objects, 
and *a fortiori* for either affine or Euclidean transformations, than to 
try to live by the Euclidean field alone.

Intuitionism is alive and well in computer science, but you asked about 
physics.

Vaughan Pratt