[FOM] proofs by contradiction in (classical?) Physics
drago at unina.it
Tue Sep 20 08:09:53 EDT 2011
Sunday, September 18, 2011 9:41 PM
Fouche fom at cs.nyu.edu wrote.
Can one use proofs by contradiction in Physics?
Surely. Many results - also in mechanics - have been obtained (since the time of Stevin) by arguments relying on the impossibility of the perpetual motion, a principle from which no direct proof can be obtained.
The most celebrated proof in theoretical physics is Sadi Carnot's proof, reported by most textbooks on thermodynamics.
F.: And if it is possible, why they are so rare?
Because since Descartes and Newton's time theoretical physicists argued by starting from necessary statements (if not metaphysical notions as absolute space and absolute time and divine gravitational force). Only a minority of theoretical physicists argued without the idealistic notions (included in the necessary principles), i.e. by relying on only empirical data. Einstein and Poincaré stressed that there exist two ways of organising a physical theory from experimental data. A century before them, Lazare Carnot illustrated this point by means of the following words concluding his formulation of mechanics, which is at all different from Newton's mechanics (see my paper “A new appraisal of old formulations of mechanics”, Am. J. Phys., 72 (3) 2004, 407-9):
“Among the philosophers who are occupied with research into the laws of motion, some make of mechanics an experimental science, the others, a purely rational science; that is to say, the former in comparing the phenomena of nature break them up […] in order to know what they have in common, and thereby reduce them to a small number of principal facts […]; the others begin with hypotheses, then reasoning in consequence upon their suppositions, come to discover the laws which bodies follow in their motions, if these hypotheses conform to nature, then comparing the results with the phenomena, and finding them to be in accord, conclude from this that their hypothesis is exact.” (L. Carnot, Essai sur les Machines en général, Defay, Dijon, 1783, p. 102).
Before him, D'Alembert stressed the same poiint in the issue "Elemens" of Encyclopédie Française.
Another physical theory which argued in only this way was chemistry. See my paper “Atomism and the reasoning by non-classical logic”, HYLE, 5 (1999) 43-55 (with R. Oliva)
F.: 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?
The only intrinsic problem is given by the way of dealing with the conclusion of an indirect argument starting form a negative statement. The indirect argument reaches the negation of the negative thesis; to obtain the corresponding affrimative statement one has to apply the principle of sufficient reason: nothing is without reason; from which one obtains: there exists a reason for everything; and hence the affirmative thesis. This Leibnizian principle is useful only when direct evidence is lacking, just when the theory does not start from necessary principles, but it is aimed to solve a problem by making use of either common knowledge or experimental data only ; in the case of Sadi Carnot: the problem of which bound exists in the efficiency of the transformations of heat in work; in the case of chemistry, the problem of which are the elements constituting matter.
Is the "tertium non datur" true in classical or quantum Physics?
The physical theories including indirect proofs argue through double negated statements (chemistry: It is not possible that matter is divisible at infinity) which are not equivalent to the corresponding affirmative statements, owing to the lack of evidence of the latter ones (in previous case one has to exhibit the dimension of the finite elements; that was impossible before early 1900, i.e. along one century of development of chemistry.
F.: My two cents are that everybody studying Physics must face an intrinsic fuzzyness given by indeterminacy; 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); can the previous questions be restated into a more general one, say "Is geometry intrinsically non-boolean/without tertium non datur?"
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.
All the best
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