such as that held by Rene Thom is that once accepting a set of natural givens, called the axioms, the rest of the knowledge of this reality grows in the form of theorems that relate to the axioms and each other through their logical consistency. Knowledge of reality is moved by the ever-forward mathematical refinement of a priori conditions to do away with the theorems’ exceptions, called counter examples.
The Hebraic Bible’s view of signifiers such as words and symbols Is close to, but not identical with, the Platonic view of mathematical formalism. According to the Torah, God made the word with words. God spoke and the world became real. The Aramaic for “Il create in speaking” is avara k’davara , or as the magician says, as he waves his wand over an apparently empty black high hat, abracadabra. The Hebrew word for word, davar, also signifies thing. This view contrasts with the mathematical formalists, among them Hilbert, who considered the signifiers of abstract mathematics simply symbols used in a game, the rules of which being arbitrary, must include proofs of consistencies among them. Consistency from the point of view of physics was addressed by Hertz, in Die Prinzipien der Mechanik,(1894), where he expressed the formalist theoretical physicist’s work as “...within our own minds we create images or symbols of the external objects, and we construct them in such a way that the logically necessary consequences of the images are again the images of the physically necessary consequences of the objects.”
In another set of related contrasts, the constructionist mathematician will argue that mathematical assertions are only true if they can be demonstrated, found or constructed. In contrast, the classical school of mathematics can develop the case for the truth of mathematical statements if they are consistent with field’s network of theorems and proofs, even if, up to the current time, no specific example of this truth can be demonstrated. The former can be thought of as a builder, the latter as a discoverer. For example, suppose we try to make a proposition about perfect numbers where a perfect number is defined as being equal to half the sum of its divisors. Using the perfect number 6, we find that its non-identity divisors are 1, 2, and 3 and half of their sum = 6. Our proposition: either there exists an odd
perfect number, or else there exists no odd perfect number. An expression of this
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