Philosophy of Mathematics by Oystein Linnebo
My rating: 3 of 5 stars

Better Than the Super Bowl

Numbers are words. Numbers exist in the same way that other words exist, namely when they are used in speech or writing. That is to say, they are neither objective nor subjective but inter-subjective. Numbers are names which refer to themselves, and there are specified ways in which numbers may be used with each other. This is their syntax. Like all words, numbers are defined by other words. This is called their semantics. Words are the only things we really know about. We know most about numbers because they are more precisely defined than other words. When we use numbers according to the specified rules, we are able to discover relationships among them, many of which are unexpected. These discoveries constitute the pragmatics of numbers.

This is a summary of my philosophy of mathematics. It is simple, easy to comprehend, comprehensive, and probably wrong. I say this because people far more intelligent than I have agonised over the existence, meaning and practical effect of mathematics without coming to a consensus. I cannot understand why this is so. Linnebo chronicles the last two hundred years or so of philosophical agony about numbers largely through a focus on one man, the German Gottlob Frege. Frege devoted much of his life to proving that arithmetic was entirely rational, that is, that it could be derived from indisputable first principles. He failed.

He failed for technical reasons involving necessary self-contradictions. But some forethought might have suggested that his failure was inevitable for other reasons as well, namely that there is no, and cannot be any, privileged language no matter how well-defined that language is. Language is its own representation. There is nothing beyond or outside of language that can be used to demonstrate its integrity. The logic of any language is contained in its syntax and cannot be gainsaid by reference to any logic applied from outside of itself.* The logics among languages may differ but they cannot be compared in order to determine which is superior. This is the ultimate conclusion of the mathematician Kurt Gödel, who finally killed Frege’s project as a matter of principle.

One implication of my philosophy of mathematics is that by using numbers we submit to them, their syntax, and their semantics in order to experience their pragmatics. No further justification for engaging is mathematics, it seems to me, is required. Nor is there any deep mystery, except perhaps psychological, about why we do this. Submission to mathematics, like submission to any language, is the giving over of one’s life to a society, namely that group which jointly participates in it. To use numbers is not to associate with the gods as the ancient Greeks thought. It is to associate with each other, to accept the judgments of each other, and to be recognised for furthering the implicit social project of the language of our association.

It strikes me as more than odd that Linnebo as well as the mathematicians he analyses simply ignore their own social dedication in their language of choice. The primary pragmatic consequence of mathematics is membership. Membership is achieved even in the failure to achieve any other result in mathematics. So, for example, Frege, and Bertrand Russell and even Gödel continued to ‘speak’ mathematics knowing that it is as ‘flawed’ as any other language. It may not be the key to understanding the universe; but it is nevertheless beautiful, intriguing, seductive, and it is most of all a way to spend a harmless afternoon with friends. Certainly better than watching the Super Bowl for those who prefer words to physical violence.

* This applies to the ‘dialects’ within mathematics as well. As Linnebo points out, the logic of Frege was very different from the logic of Kant for example.

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