Computers Can't Do Math

What is a concept? It is an abstracted universal in the intellect. Consider the example of triangularity. We can predicate this concept of any concrete, physical triangle, but none of these concrete instances are triangularity itself, and triangularity itself is not a concrete instance. Indeed, any concrete instance is necessarily determined to be this triangle, but not that triangle, while a concept as abstracted form is true of all triangles. An analysis of the concept allows us to discover properties of triangles, e.g., the property that the inner angles must sum to 180 degrees. Such properties hold for all concrete instances. These properties are themselves abstracta, not concrete things in the world. Triangularity is a matter of semantics, because it is the essential meaning of a triangle, what makes it the kind of thing it is.

The same can be said of quantities. Just as you will never encounter naked triangularity, as such, in the world, you will never meet the number 3. However, you will encounter collections of three things, things measured to be 3, things with three sides, and so on.

Now take a physical machine that is simulating a Turing machine. Can the machine add? Well, strictly speaking, no! You can, however, simulate addition, to a point, by representing numbers using a system of symbols and using syntactic rules to manipulate strings of these symbols such that they result in new strings that correspond to the numbers that addition would produce. But these strings aren't quantities, they aren't numbers per se, but representations. Your Turing machine's tape can be relabeled, and the interpretation of the strings on that tape are inherently ambiguous. They can be read, for most practical purposes, according to an interpretation that assigns to them a consistent numerical meaning, but there is no numerical meaning in them per se, any more than the ink in a book arranged in the shape of "cat" is the concept of Cat. You could assign to them a completely different interpretation, and there would be no inherent reason to prefer one over the other. Only the interpreter's purposes fix the interpretation. But the meaning assigned, the concepts themselves, are just the meanings themselves. They are the intelligible content that the interpreter assigns to those symbols, and this intelligible content does not reside in the physical computer.

So, if there is any magical thinking, I claim that is rests on the side of those who claim, rather flippantly, that machines can think.