From: JulioHuato
Date: Thu, 29 Jan 1998

With regards to Kol'man & Yanovskaya's piece. They say they agree with Hegel's view of math's object. Quoting Hegel, they say that the "space forms and quantity relations" of the world constitute the object of math. Now, since in math, there's a correspondence between "space forms" and "quantity relations" and they can be expressed in terms of each other (although, say, "space forms" beyond R^3 are not "space" in an intuitive sense), then they are "quantity relations" in general. Then K&Y go on to criticize Hegel for considering math, "as do today's formalists, only from the aspect of its inner logical consistency, and not of its objective truth, i.e. only as a calculation, but not as a science which has its own subject of research."

Here's my question: Is math another CONCRETE science, in the sense of being concerned with some aspects of REALITY (space forms and quantity relations) to which it constantly has to refer for verification? Or, is it, as most mathematicians would have it, a FORMAL science, in the sense of being EXCLUSIVELY concerned with its own internal logical consistency once a few universal definitions and axioms are laid out and accepted as "valid" (not necessarily as "objective")? What's the essence of mathematics, the objectivity of its definitions, axioms, and theorems or its mode of reasoning whatever the relation of such definitions, axioms, and theorems to the objective world? Why?

IMO, what K&Y imply is that the main results of mathematics have to be tested like you test physical or biological theories and hypotheses. Even though math is not concerned with the specific nature of processes, it is still concerned with the "quantity relations" aspect of REALITY, which means that mathematical reasoning per se is incomplete. But how do you test a mathematical theorem directly against empirical reality? Moreover, how do you test axioms and definitions? What kind of practice validates a mathematical definition, axiom, or theorem?

J.