by Gene Callahan
The point I wish to make here has been made before, notably, by Lewis Carroll in his essay “What the Tortoise Said to Achilles“, as well as by Wittgenstein, in his work on what it means to “follow a rule,” and by Gödel in his famous paper on undecideability. But, as I recently encountered a very, very bright young philosopher who seemed unaware of the import of such arguments, it is, perhaps, a point worth making once again.
The contention at hand is that, contrary to those who hold that mathematical knowledge offers us an example of objective truths that are of a non-physical nature, mathematical truth is “simply” a matter of positing some arbitrary set of definitions and rules for drawing conclusions from them. The budding philosopher in question contended that the truth of the proposition “2 + 2 = 4” says nothing about the existence of non-material yet objective truths, since its truth is a mere consequence of the accepted definitions of the symbols ‘2’, ‘+’, ‘=’, and ‘4’. No metaphysical truth is revealed in the proposition, anymore than would be revealed by a rule that declared, if you hear “waaaa,” then you ought to respond “xeeee.”
As I see it, and as Carroll, Wittgenstein, and Gödel so ably indicated, the problem with this contention is that nothing at all follows from accepting the definition of each of those terms concerning the truth of “2 + 2 = 4,” in the absence of embracing a notion of ‘truth’ that lies beyond any formal system itself. Why?
Let’s say you are conversing with a chap who doubts that this is all a matter of definition. You take out your copy of Principia Mathematica and explain to him how, in that system, ‘2’, ‘+’, ‘=’, and ‘4’ are defined. He declares that he understands all of those definitions, and is willing to accept them during your discussion. “Therefore,” you declare, “you must acknowledge that 2 + 2 = 4!”
“No,” he says, “I believe 2 + 2 = 5.”
“How can you say that?” you respond. “You accepted all of my definitions — don’t you see my result follows from them?”
“I’m sorry,” he replies, “but I don’t recall you offering me a definition of ‘follows from,’ and, since you are contending that all of this is simply a matter of accepting certain, purely conventional, definitions, I can’t understand why you think what ‘follows from’ these definitions must be the same for me as it is for you.”
As Carroll demonstrated, any attempt to supply more and more formal definitions and rules to patch up this difficulty simply leads to an infinite regress. Without some non-formal sense of what is ‘implied’ by any formal operation or system, i.e., some inkling that there are non-physical facts of the matter that are objectively true, there is simply no way to determine whether or not someone has drawn the ‘correct’ conclusions from a formal system.