Structure, structure and more structure

I was expecting to write about a paper I found recently by Oran Magal, a post doc at McGill University, On the mathematical nature of logic. I was attracted to the paper because the title was followed by the phrase Featuring P. Bernays and K. Gödel

I’m often intrigued by disputes over whether mathematics can be reduced to logic or whether logic is, in fact, mathematics, because these disputes often remind me of questions addressed by cognitive science, questions related to how the mind uses abstraction to build meaning. This particular paper acknowledges, in the end, that its purpose is two-fold. It makes the philosophical argument that an examination of the interrelationship between mathematics and logic shows that “a central characteristic of each has an essential role within the other” But the paper is also a historical reconstruction and analysis of the ideas presented by Bernays, Hilbert and Gödel (the detail of which is not particularly relevant to my concerns). It was Bernays’ perspective that I was most interested in pursuing.

Magal begins with the observation that

the relationship between logic and mathematics is especially close, closer than between logic and any other discipline, since the very language of logic is arguably designed to capture the conceptual structure of what we express and prove in mathematics.

While some have seen logic as more general than mathematics, there has also been the view that mathematics is more general than logic. It is here that Magal introduces Bernays’ idea that logic and mathematics are equally abstract but in different directions. And so they cannot be derived one from the other but must be developed side-by-side. When logic is stripped of content it becomes the study of inference, of things like negation and implication. But while logical abstraction leaves the logical terms constant, according to Bernays, mathematical abstraction leaves structural properties constant. These structural properties do seem to be the content of mathematics, and what makes mathematics so powerful.

Magal describes how Bernays understands Hilbert’s axiomatic treatment of geometry. Here, the purely mathematical part of knowledge is separated from geometry (where geometry is thought of as the science of spatial figures) and is then investigated directly.

The spatial relationships are, as it were, mapped into the sphere of the abstract mathematical in which the structure of their interconnections appears as an object of pure mathematical thought. This structure is subjected to a mode of investigation that concentrates only on the logical relations and is indifferent to the question of the factual truth, that is, the question whether the geometrical connections determined by the axioms are found in reality (or even in our spatial intuition). (Bernays, 1922a, p. 192) (emphasis added)

Magal then uses abstract algebra to illustrate the point:

To understand Bernays’ point, that this is a structural direction of abstraction, and the sense in which this is a mathematical treatment of logic, it is useful to compare this to abstract algebra. The algebra familiar to everyone from our school days abstracts away from particular calculations, and discusses the rules that hold generally (the invariants, in mathematical terminology) while the variable letters are allowed to stand for any numbers whatsoever. Abstract algebra goes further, and ‘forgets’ not just which number the variables stand for, but also what the basic operations standardly mean. The sign ‘+’ need not necessarily stand for addition. Rather, the sign ‘+’ stands for anything which obeys a few rules; for example, the rule that a+ b= b+ a, that a+ 0= a, and so on. Remember that the symbol ‘a’ need not stand for a number, and the numeral ‘0’ need not stand for the number zero, merely for something that plays the same role with respect to the symbol ‘+ ’ that zero plays with respect to addition. By following this sort of reasoning, one arrives at an abstract algebra; a mathematical study of what happens when the formal rules are held invariant, but the meaning of the signs is deliberately ‘forgotten’. This leads to the study of general structures such as groups, rings, and fields, with immensely broad applicability in mathematics, not restricted to operations on numbers.

Again the key to the discussion is the question of content. When mathematics is viewed as a variant of logic it could easily be judged to have no specific content. The various arguments presented are complex, and not everyone writes with respect to the same logic. But the consistency of Bernays’ argument is most interesting to me. He is very clear on the question of content in mathematics. And reading this sent me back to another of his essays, where he is responding to Wittgenstein’s thoughts on the foundations of mathematics is 1959. Here he challenges Wittgenstein’s view with the nothingness of color.

Where, however, does the initial conviction of Wittgenstein’s arise that in the region of mathematics there is no proper knowledge about objects, but that everything here can only be techniques, standards and customary attitudes, He certainly reasons: `There is nothing here at all to which knowing could refer.’  That is bound up, as already mentioned, with the circumstance that he does not recognize any kind of phenomenology. What probably induces his opposition here are such phrases as the one which refers to the `essence’ of a colour; here the word `essence’ evokes the idea of hidden properties of the color, whereas colors as such are nothing other than what is evident in their manifest properties and relations. But this does not prevent such properties and relations from being the content of objective statements; colors are not just a nothing….That in the region of colors and sounds the phenomenological investigation is still in its beginnings, is certainly bound up with the fact that it has no great importance for theoretical physics, since in physics we are induced, at an early stage, to eliminate colors and sounds as qualities. Mathematics, however, can be regarded as the theoretical phenomenology of structures. In fact, what contrasts phenomenologically with the qualitative is not the quantitative, as is taught by traditional philosophy, but the structural, i.e. the forms of being aside and after, and of being composite, etc., with all the concepts and laws that relate to them. (emphasis added)

Near the end of the essay he makes a reference to the Leibnizian conception of the characteristica universalis which, Bernays says was intended “to establish a concept-world which would make possible an understanding of all connections existing in reality. This dream of Leibniz’s (which it seems Gödel thought feasible) is probably the subject of another blog. But in closing I would make the following remarks:

Cognitive scientists have found that abstraction is fundamental to how the body builds meaning or brings structure to its world. This is true in visual processes where we find cells in the visual system that respond only to things like verticality, and it is seen in studies that show that a child’s maturing awareness seems to begin with simple abstractions. Mathematics is the powerful enigma that it is because it cuts right into the heart of how we see and how we find meaning.