Let’s see if a rewrite of Collingwood’s Aesthetic Theory and Artistic Practice makes sense:
The first object of any inquiry is to find out the truth about something; and the first business of metamathematical theory is to discover the truth about mathematics in general, that is to say, to answer the question ‘what is mathematics?’
There is a tension between those who believe this metamathematics should be normative and those who believe it should be descriptive.
When we have decided what mathematics is, can we then go on to use this knowledge for the improvement of our practices as mathematicians or our tastes as judges of mathematics, or can we not?
The difference really turns on this: a normative science assumes that the subject matter of its study is not realized, that there is a difference between what it ought to be and what it is; whereas a descriptive science assumes that its subject matter is already all it ought to be and what it is, that its proper nature is realized in the facts as they stand.
I will say at once that I do not think either of these conceptions can give us a satifactory answer to our question.
It was obvious to the Greeks that the philosophical sciences are normative. This was challenged by descriptive approaches to activities of the mind, then by realist metaphysical approaches. Mill described how we learn arithmetic from induction through playing with pebbles. Helmholtz, Poincare and others explored our geometric faculties. Most Anglophone philosophy of mathematics post-1930 has addressed metaphysical questions: “What is mathematics about?”, “If abstract entities, how do we know about them?”, “If structures, what is a structure?”, “If merely logical truths, or ‘if-thens’, why adopt a language which appears to refer?”.
Both the psychological view of philosophy, as the study of the thinking or acting mind, and the metaphysical view of philosophy, as the study of the ultimate nature of the real world, are compatible with the doctrine that the philosophical sciences have a purely theoretical interest, and are completely devoid of practical value.
But advocates of the descriptive point to benefits of their work (1) that reflection may increase reverence for the field and (2) the clearing away of misleading false theories. E.g., Wittgenstein answering Hilbert “No one shall drive us out of the paradise which Cantor created for us.” (On the infinite)
I would say, “I wouldn’t dream of trying to drive anyone out of this paradise.” I would try to do something quite different: I would try to show you that it is not a paradise—so that you’ll leave of your own accord. I would say, “You’re welcome to this; just look about you.”.
“Mathematical logic” has completely distorted the thinking of mathematicians and philosophers by declaring a superficial interpretation of the forms of our everyday language to be an analysis of the structures of facts. In this, of course, it has only continued to build on the Aristotelian logic.‘ (p. 156, No. 48)
…the ‘curse of the invasion of mathematics by mathematical logic that any proposition can now be represented in mathematical notation and we thus feel obliged to understand it, although this way of writing is really only the translation of vague, ordinary prose’ (p. 155, No. 46).
If you can show there are numbers bigger than the infinite, your head whirls. This may be the chief reason they were invented.
The misunderstandings we are going to deal with [e.g. platonism] are misunderstandings without which the calculus [e.g. set theory] would never have been invented, being of no other use, where the interest is centred entirely on the words which accompany the piece of mathematics you make. Lectures on the Foundations of Mathematics, 1939, 16-17)
Continuing the transcription of Collingwood:
But if bad ethics may lead to bad conduct, whatever the reason, the relation of ethics to conduct is not simply that of a description to the things described. Bad astronomy does not derange the movement of the stars.
Why should bad theory be able to lead to bad practice, and yet good theory not be able to lead to good practice? Were it so, philosophy should certainly be avoided.
Here we could have a lengthy case study of the roots and demise of the metamathematical thinking of some generation to mimic Collingwood’s study of English naturalism in art. Perhaps we might look at logicism (see Between logic and physics)
Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.
But did research mathematics subscribe to this view? Or
mathematics is the study of mental objects with reproducible properties,
Mathematics is “the study of general abstract systems, each one of which is an edifice built of specified abstract elements and structured by the presence of arbitrary but unambiguously specified relations among them.” [Stone, p. 717]
Having placed such historically-situated views of mathematics in their context, we continue.
We have travelled a long way from the conception of metamathematics as a normative science. But we have at any rate, I think, seen reason to abandon the idea of metamathematics, or any other philosophical science, as an ex posto facto generalized description of a group of facts existing independently of the being described.
The business of metamathematics, then, is to determine the ideal of mathematics, the goal towards which mathematical production is directed.
One can’t do mathematics without realising that one is doing so.
…the knowledge that he is trying to write a mathematical proof/paper/book implies the knowledge of what a mathematical proof/paper/book is; or rather, since the mathematical proof/paper/book is not yet written, of what a mathematical proof/paper/book should be. This knowledge is a philosophy of mathematics, and such a philosophy must therefore preside over the birth of every work of mathematics–unless indeed there are works of mathematics which the mathematician lets fall inadvertently, without suspecting it, as Monsieur Jourdain spoke prose. Setting aside these possible by-blows, every work of mathematics is conceived through the agency of some theory, concerning the nature of mathematics as such. No doubt the theory changes; continued experience in mathematical work is very likely to change it; but because theories change as experience changes, it does not follow that the theory is a mere description of the experience. It would be truer to describe the experience as an attempt to put the theory into practice.
There is thus a reciprocal relation between metamathematical theory and mathematical practice. To suppose that a metamathematics can be worked out in vacuo, apart from all experience of actual mathematical work, and then used as a normative science laying down once for all a code of rules that mathematics must obey if it is to be genuine mathematics, is to suppose an absurdity. If that is all that is meant by the conception of metamathematics as a normative science, the conception is a chimera. But the conception of metamathematics as a descriptive science, following after the facts and merely noting their characteristic features, is no less chimerical. The true relation between metamathematical theory and mathematical practice would seem, rather, to be of such a kind that neither can exist in isolation from the other. Mathematics cannot exist without a theory of mathematics, because unless the mathematician has such a theory in his own mind–unless, for example, he can set before himself the end of [example of aim of mathematics]–he does not know what he is trying to do, and therefore is not trying to do anything. For this reason it is useless, and worse than useless, for professional philosophers to advise professional mathematicians to leave metamathematical theory alone and get ahead with their own business, the writing of papers or what not, leaving philosophy to the philosophers. The advice is useless because the mathematicians can never take it, and it is worse than useless because it shows the philosophers to be bad philosophers.
The theory of mathematics cannot exist without mathematics, not because it merely describes mathematics, as entomology cannot exist without insects, but because it is an organic element within the process by which works of mathematics come into being, and it cannot exist except as an element in that process. No doubt, a particular theory of mathematics may be extracted from its place in such a process and isolated for expert scrutiny. It is right and necessary that this should be done; and here the mathematician is wrong if he tells the philosopher to keep his hands off metamathematical theory, because that is his business, not the philosopher’s, and the theory that serves him in the creation of his own works of mathematics is thereby justified. The mathematician is wrong because a theory is a theory, and must stand or fall by its merits as such; and to say ‘my theory is justified by its fruits’ is to expose oneself to the retort ‘if your theory was a better theory, your papers might be better papers’.
But to separate the mathematician from the philosopher in this way, and set them at loggerheads, is to create trouble. Much of our difficulty over the whole problem of normative sciences is due to the fact that philosophy is supposed to be incarnate in certain eccentric, ridiculous, and vaguely disquieting persons called philosophers. It is these persons who, since they possess philosophy, possess the normative rights which belong to the philosophical sciences; and people who are interested in science, or morals, or art, naturally resent being ordered about and told how to conduct their own affairs by these shadowy figures. And it is equally natural that professional philosophers should hasten to reassure them by insisting that they personally make no such claim and, on the contrary, regard philosophy not as normative but merely as descriptive. Both parties are frighted by false fire. Mathematics and philosophy may be professions, but they are more than that; they are universal human interests, and this is indeed the only reason why the professional mathematician or professional philosopher has an audience. He speaks, not to his fellow-professionals, but to the mathematician or the philosopher in all men.
The philosophy of mathematics, then, may be a department of study to the professional philosopher, but to the mathematician it is a matter of life and death. The philosopher may neglect metamathematics, but the mathematician cannot; he must decide what mathematics is, or he cannot pursue it. If he decides wrong, he will pursue it wrong. This is the purpose or function of metamathematical theory in its relation to mathematical practice. The mathematician, as a rational being, must know what he is doing, or he cannot do it. In so far as he is an mathematician, his knowledge of what he is doing is his philosophy of mathematics.
It seems to me therefore that the philosophy of mathematics is not a system of thoughts which philosophers think about mathematicians; it is a system of thoughts which mathematicians think about themselves in so far as they are able to philosophize, and philosophers think about themselves in so far as they are interested in mathematics. In either case, its purpose is the same. The mind, in its intellectual function, is trying to understand itself in its mathematical function. The mathematics about which we philosophize is not a ready-made fact, it is something which we are trying to do, and by understanding what we are trying to do we come to be able to do it better. The philosophy of mathematics is therefore not a description of what metamathematical facts are, nor yet an attempt to force them into being what they are not: it is the attempt of mathematics itself to understand itself and, through understanding itself, to become itself. That is why mathematicians, even if they care little about philosophy in general, cannot help caring about the philosophy of mathematics. For there is no escape from the dilemma: either a mathematician does not know what he is doing, or else he has a philosophy of mathematics, a metamathematical theory expressing the principles by which he tries to guide his mathematical practice.
If we follow Collingwood, we will try to understand the philosophies of the leading practitioners through the ages. This isn’t merely to report them and their alterations, but also to be allowed to judge them: Were they adequate to the practice of the day? Were the tensions present within them recognised and resolved, if only partially, by successor philosophies? It means the philosophy of mathematics attending very closely to mathematical practice.
We can focus on more specific issues, such as changing conceptions of space. Dieudonne writes
In this volume, an attempt is made to show how geometers have been led progressively to broaden their ideas. The reader will discover here several fundamental ideas that are revived time after time in different guises, marking the incessant effort toward better understanding of geometric phenomena: transformations and correspondences, invariants, “infinitely near” points, extensions of geometric objects by the “adjunction” of new “points”. (History of Algebraic Geometry, VII)
Imagine a book to imitate Collingwood’s The Idea of Nature, called The (Mathematical) Idea of Space.