UPDATE: Albert Lautman – “Mathematics and Reality”
UPDATE (12/17/2012): There is a much better translation available in Albert Lautman’s Mathematics, Ideas and the Physical Real. Go buy that.
(Here is a translation of Albert Lautman’s “Mathematiques et Réalité”. Its a bit rough but I will try to clean it up over the next couple of days.)
“Mathematics and Reality”
The logicians of the Vienna School maintain that the formal study of scientific language must be the only object of the philosophy of science. It is a difficult thesis to accept for those philosophies that consider it their sole task to establish a coherent theory of the relationship between logic and the real. There is physical reality and there is the miracle of explanation, it is only that it needs more developed mathematical theories in order to interpret it. There is in the same way mathematical reality and it is similar object of admiration to see certain domains resist their exploration in a way, so that one must approach them with new methods. It is just as when analysis is introduced into arithmetic or topology into the theory of functions. A philosophy of science that would be unable to carry the entirety of the study into this solidarity between domains of reality and the methods of investigation would be singularly without interest. Philosophy is not the effect of mathematical nature. If logico-mathematical rigor can seduce it, it is certainly not because it permits the establishment of a system of tautological propositions, but because it shines a light on the tie between rules and their domain. It, at the same time, produces a curious event that is for the logicians an obstacle to eliminate, becoming for the philosopher the highest object of interest. It is a question of all the “material” or “realist” implications that the logicist is obliged to accept: this is the well-known axioms of Russell, the axiom of infinity and the axiom of reducibility. It is above all, following Wittgenstein, the affirmation that all true propositions correspond to an event of the world, which involves the entire procession of restrictions and precautions for logic. In particular, any proposition relating to the set of propositons, every logical syntax, in the sense of Carnap, is impossible since it would need for itself the power to consider, correlatively, the world as a totality, which is impossible.
The logicistian of the Vienna School continuously affirm their complete accord with the school of Hilbert. Yet nothing could be more questionable. In the logicist school, in the style of Russell, one strives to find the atomic constituents of every mathematical proposition. The arithmetic operations are defined starting from the primitive notions of elements and classes and the concepts of analysis are defined by extension starting from arithmetic. The concept of number plays here a central role and this role is again augmented by the arithmetization of logic in the style of the work of Gödel and Carnap. The primacy of the concept of number, nevertheless, seems not to be confirmed by the development of modern mathematics. Poincaré had already indicated in relation to the theory of dimension that the arithmetization of mathematics does not always correspond to the true nature of things. Herman Weyl has, in the introduction to his work Gruppentheorie und Quantenmechanik, established a distinction that appears fundamental to us and will have to be accounted for in all future philosophies of mathematcs. It distinguishes two currents in mathematics. The first, issuing from Indian and Arab thinkers, puts the concept of number into relief and succeeds in the theory of functions of complex variables. The second is the Greek point of view in which every domain carries with it a system of characteristic numbers. It is primacy of the geometric idea of the domain over that of whole numbers.
The axiomatics of Hilbert and his students, far from wanting to restore the mathematics of sets as only a promotion of arithmetic, tend, on the contrary, to free for every domain of study a system of axioms as such, the union of conditions implied by the axioms emerging at the same time as the domain and the valid operations within this domain. It is also that one is axiomatically constituted in the modern algebraic theory of groups, of ideas, of the system of hypercomplex numbers, etc.
The consideration of a purely formal mathematics must therefore give order to a dualism of a topological structure and the functional properties in relation to this structure. In a similar way, the formalist presentation of theories: axiomatization is only a question of greater rigor. The object studied is not the set of propositions derived from the axioms, but organized being, structures, totality, having an anatomy and physiology proper. Citing as an example Hilbert’s space defined by the axioms that confer on it a structure appropriate to the solution of integral equations. The significant point of view here is the synthesis of the necessary conditions and not this analysis of first notions.
This same synthesis of the domain and the operation finds itself in physics under a slightly different point of view. Carnap, sometimes, seems to consider the relationship between mathematics and physics as that between the form and the matter. Mathematics would provide the system of coordinates in their inscription of physical givens. This conception hardly appears defendable seeing that modern physics, far from maintaining the distinction of a geometric form and a physical matter, combines, on the contrary, spation-temporal and material givens in the common framework of a mode of synthetic representation of phenomena; that this were by the tensorial representation of the theory of relativity or by the Hamiltonian equation in mechanics. Thus, one witnesses for every system with a simultaneous and reciprocal determination of container and contents. It is again a determination proper to every domain, the interior of which no longer subsists for the distinction between matter and form. Carnap seems, it is true, to have another theory of the relationship between mathematics and physics, conforming much more closely to his logistic tendancy. It considers physics, not as the science of real facts, but as a language in which one expresses experimentally verifiable statements. This language is submissive to the rules of syntax, of mathematical nature when it is uniformly valid in all their defined domains, of physical nature when their determinations vary with experience. There is the new affirmation of the universality of mathematical laws by opposition to the variation of physical givens. It seem to us that this conception does not reflect that the variation of physical givens only make sense in relation to the presumed variety of varying sizes, and this variety is physics. Carnap’s example from page 131 of his book The Logsche Syntax der Sprache is characteristic. When, we say, the components of the fundamental metric tensor are constants, it is a mathematical law; when they change they observe laws of physics. The real philosophical problem would rather be of the knowledge of how a differential geometry could become the theory of gravitation. This relation between geometry and physics is the proof of an intelligibility of the universe. It results in the development of the mind, in a way, to structure the universe in profound harmony with the nature of this universe. One conceives that this penetration of the real by human intelligence has no meaning for those excessively formalistic. These, in effect, would rather see in these pretensions of the mind to know nature a relevant approach to the studies of Lévy-Bruhl. Understanding would be for them a mystical belief analogous to the participation of the subject to the object in the primitive soul. The term of participation has in philosophy another origin more noble and Brunschvieg has justly denounced the confusion of the two senses. The participation of the sensible in the intelligible in Plato can be identified. If the first contacts with the sensible are not sensations and emotions, the constitution of mathematical physics gives us access to the real via the knowledge of the structure it has been given. Similarly, it is impossible to speak of the independent reality of the modes of thought according to which it leaves it to dread and disparage the merely mathematical language indifferent to the reality that it decries, philosophy engages itself through an attitude of meditation where it should achieve the secrets of nature. There is, therefore, no reason to maintain the distinction, made by the Vienna School, between rational knowledge and intuitive experience, between Erkennen and Erleben. In wanting to remove the relationship between thought and the real, as in refusing to give science the value of a spiritual experience, one risks only having a shadow of science, rejecting the spirit in the search for the real towards the violent attitudes where raison plays no part. This is a resignation that the philosophy of science must not accept.