The question about mathematics which has baffled most commentators for a long time is the question of the meaning of mathematical statements.  Wittgenstein introduced the idea that the meaning of a statement was to be seen in its use and usage, but pure mathematics appears to have no obvious use —except perhaps to glorify the concept and activity of mathematics.  We have to remember, though, that before the discovery of the computer, pure mathematical insights which led to simplifications were the only way to compute modelling results. They were thin on the ground, but as a consequence, they were literally priceless. So pure mathematics had a cast iron justification: it was in the business of better understanding, and finding computability short-cuts for working mathematics. Until the 17^th century research in pure mathematics was almost entirely an amateur affair anyway.  It was funded by scholars themselves or their friends, and didn’t need to field a public justification.

The principal reason why a few people pursued questions in pure mathematics, was probably curiosity, and especially curiosity about some uncanny patterns which recur again and again. Why?  The subject as a whole had an immense background source of meaning stemming from its successful modelling uses.  So valid explanations of puzzling results in maths itself naturally provoked interest. In a word, mathematics was like a free-standing game, with the added good name of a mass of known, successful mathematical modelling giving it credibility.

So pure mathematics is a meaning bubble —quite separate from ordinary language— not unlike chess, but with the added attraction of a good track record —having been used extensively in science and public projects and to great effect since Ancient Greece.

Though the meaning of mathematical statements is confined to the bubble, it operates on much the same basis as ordinary meaning in ordinary conversation.

Ordinary language is used in various modes, questions, imperatives, orders, etc. but the conversational mode is all about sharing our recently acquired awareness of corners of our world with others. A conversational statement often has the form

1. a.A reference X is made.
2. b.An ascription that X has the quality Y is made.

It tells our listener —if they believe us— that going to X will lead them to finding Y.

The potential references we use are like a subset (our personal subset) of a huge total of possible references. This archive of potential references, and the language used to assemble it, is like a public map of the world. 

An indicative statement we make tells our listener what to expect if she or he goes to the place the reference indicates. It is a tiny bit of potentially predictive know-how.

Similar features are present in mathematical statements. The statement <<The squareroot of 169 is 13>> tell the reader that if they go to look for the number, which when squared equals 169 (the reference) and they apply a squareroot process correctly to find it, the result will be 13. Again it is a tiny bit of predictive know-how.

It was John Lucas, many years ago (in the journal Philosophy 1969), who sorted out what it means to say that a mathematical statement is true. It is to say that, as a source of predictive know-how within the bubble of pure maths, it can be trusted.

Lucas says  (p. 175) 

<<A true friend is one I can trust. And trust, too, is what I can repose in true propositions. In telling you that a friend, a die, a line, or a proposition is true, I am telling you that that you can trust him [or her] or it, that he [or she] or it is trustworthy, that he [or she] or it is worthy of your trust and will not let you down.>>.  [Squarebrackets added by today’s author.]

Trustworthiness is an intrinsically personal quality, and the amount of it a speaker has is a consequence of their perceived credibility.  The meaning conveyed by a statement depends essentially, as in ordinary conversation, on the writer’s trustworthiness. (The meaning “of” the statement is roughly the meaning it would convey if uttered under normal conditions by a fully trustworthy speaker.) In the case of ordinary conversation, if you think the author is trustworthy, you internalise the extra tiny bit of predictive know-how they are offering you. If you don’t, you don’t. 

So there isn’t really a problem about the meaning of individual statements in pure mathematics.  They work on the same basis as ordinary language.  ‘The Problem’ of meaning in mathematics is really about the meaning of the whole activity. This has been heavily mystified and misunderstood for a very long time. Many pure mathematicians are sceptical about the idea that mathematical “applications” are a source of genuine meaning.  This probably arises from their experiences at school and university. During their early years they again and again used textbooks written by authors who had no interest in, or understanding of, “applications”. But these authors nevertheless invented nominal, supposed artificial, “applications” as a way of sugaring the pill —for the reader— of practising using the routines.  This sub-standard practice has given the very idea of ‘applications’ a bad name.

Incidentally such authors use the term ‘application’ to describe a bundle of mathematics pressed into service to deliver answers in the real world.  This is already a question-begging idiom, because the main point of mathematics is to provide models (analogies) of not-yet-realised situations to forecast what their implications will be.  We don’t say that we “apply” our cars to the road: travelling on roads is what cars do. Similarly modelling a theory or a project is what mathematics does.  

NOTE: it would be wrong to call this interpretation of mathematics the “Peircean view”. It is a “projective interpretation of mathematics” because it is about the interesting possibilities arising from new theories and new projects. Peirce himself showed little interest in modelling.   

CHRISTOPHER ORMELL January 1^st 2022.