Commentary on an observation of René Thom

Time, Logic, Number, Knowledge:
Goedel, Russell, Brouwer, Thom

Since December 2008 I've been working on a translation of a collection of interviews of René Thom: "Paraboles et Catastrophes". On page 20 of Chapter 1 he comments on Goedel's Theorems. Given that no other science besides mathematics imagines that it can base its own foundations from within, that is to say on logic and axioms alone, it's not surprising to discover that mathematics is also unable to do so.

What this means is the following: Before stating and proving his undecidability theorems, Goedel first shows that the 1st order propositional calculus is both decidable and consistent. Given any proposition in Aristotelian logic, there is a decision procedure, free from contradiction, that can decide if it is true or false.

This is not true the system of integers constructed from Peano's postulates. It follows that the concept of number cannot be derived from the laws of logic; if it could, Arithmetic could be proven to be consistent, which it can't. Since the number concept arises outside of logic it therefore, in some sense, really belongs to the universe of observables, like electrons and bacteria.

One recalls that Russell and Whitehead wrote 3 massive volumes to try to show that Number can be derived from Logic. Their failure reaffirms an idea which, I believe, was enunciated by Wittgenstein: logic is a science without content. It organizes and underlies the rules of inference of other sciences.

Coming from a Kantian perspective, Luitzen Egbertus Jan Brouwer argues that the integer concept is actually derived from the synthetic apriori of Time. Primary intuitions inherent in the conception of time, "before" and "after", and of a zero point identified as the instantaneous "now" are the raw materials from which the human mind extracts the successor function described in Peano's postulates.

I don't think that this is correct. Parenthetically, there is another interpretation of time that one finds in Special and General Relativity, where it is merely another spatial dimension in a Riemannian geometry. The properties of time are thus made subservient to the properties of length.

Notice that in disagreeing with both interpretations of the conception of time, I am taking a position against converse notions. If time is based on length, and metric length is based on number, then Time would derive some of its properties from Number. Conversely, Brouwer maintained that Number was derived from Time.

To my way of thinking, our understanding of the nature of time really has to do with the epistemological categories of Past, Present and Future. "Knowing in time", in terms of these categories takes the form of The Unknowable, The Known, and The Unknown. The apprehension of time cannot be understood apart from the phenomena of knowledge, which don't exist without sentient or knowing minds.


That which is past is forever lost to us, thus permanently unknowable, passed from Being to Non-Being, though it may be reasonably inferred by (very fallible) memory or deduction from laws and principles, (in particular the belief in the stability of nature.)
That which is immediately present is all that we can say is truly known.
That which is unknown may become known in the future.

Under this interpretation, even as Number cannot be derived from Logic (Goedel's Theorems) so Number cannot be derived from Time; nor can Time be derived from Space.

Logic ,and Time in its 3 epistemological phases, are the preconditions for knowledge. Because of Goedel's discoveries, Number now figures with Space as an "external" observable. Number and Space, however, do appear to have a converse dependency. Even as Descartes, in some sense, reduced Geometry to Algebra and Arithmetic, so through the notion of "ratio" one can perhaps derive the number system from Geometry itself . One must give credit to Euclid for this deep insight.

Matter does not enter into this discussion, though it is clear from General Relativity and the recent discoveries of observational cosmology, that matter and space are more inseparable than anything imagined by Leibniz.

Return to

Science Editorials