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

[Vaughan Pratt]

On 9/19/2011 2:09 PM, Hendrik Boom wrote:
 > It has been argued (does anyone remember where?) that intuitionistic
 > math is adequate for physics, because physics theories are established
 > by failing to be falsified.  The resulting negativity means that any
 > classical arguments can be translated by the double-negation
 > interpretation into an intuitionistic argument without loss of
 > applicability to the physics.

So was Newton's corpuscular theory of light intuitionistically valid 
only until Young's double-slit experiment falsified it?  Or was it 
intuitionistically invalid all along?  Or was it intuitionistically 
invalid only for the century between Young's experiment and Einstein's 
demonstration of the photoelectric effect, whose elucidation by quantum 
mechanics can be construed as validation of Newton's theory?

In any event, failure to falsify is clearly not how physical theories 
are established, or string theory would be an established theory today.

It might be more appropriate to reason about physics nonmonotonically. 
Evidence for and against a theory is adduced, and the prevailing 
theories tend to be those with the weight of evidence in their favor, 
but always with the door left open for new evidence to change things yet 
again.

On 9/20/2011 5:09 AM, Antonino Drago wrote:
> Geometry may be non-boolean. Please, read propositions 17-22 of
> /Geometrical studies on Parallel lines/ by Lobachevsky (as an Appendix
> to R. Bonola: /The Theory of Parallels/, Dover); you will recognise both
> indirect proofs and some conclusions which are double negated
> statements: e.g. the final one: "The second assumption [of two parallel
> lines] can likewise be admitted _without_ leading to any _contradiction_
> in the results..."; this conclusion is not equivalent to to the
> statement “The second assumption is consistent”, because Lobachevsky
> could offer only indirect proofs. However, in the propositions following
> the no. 22 he argues from the affirmative version of this hypothesis;
> this fact shows that he applied the princple of suffcient reason to the
> above double negated statement.

Very nice examples.  One could add that geometry need not even be 
intuitionistic, witness projective geometry, whose logic can be 
construed as the quantum logic of Birkhoff and von Neumann.

Vaughan Pratt