PHILOSOPHY OF MATHEMATICS OR PHILOSOPHY FOR MATHEMATICS? By Henri Laurie.
Review of Øystein Linnebo’s “Philosophy of Mathematics”, Princeton University Press, 2017. (This one is impressionistic; I hope to present a more conventional summary-of-contents review in due course).
I’ve just read Øystein Linnebo’s superb book on the philosophy of mathematics. It is very, very good. Superbly clear, concise, well organised, it gives not only a very accessible introduction but also takes the reader all the way to the cutting edge of what philosophers are doing in the philosophy of mathematics. Above all, Linnebo writes as a fully engaged philosopher and makes his preferred choice of philosophical position clear. But this is no mere polemic: I felt he clearly and forcefully presents the strengths and weaknesses of all the philosophical positions he discusses.
That said, even an introductory text in philosophy these days is not always easy reading. For this one, you do need some familiarity with foundational subjects in mathematics, particularly set theory and logic — he even uses the notation in a few places. Similarly, I felt that his presentation of Euclidean and non-Euclidean geometries would only work for readers who already knew the difference. But this is not a criticism: a meaty book like this one repays the effort one makes in reading it. Any person truly interested in the philosophy of mathematics should be willing to study mathematics, after all.
Wait a minute. Is that really true? The sad fact is that the philosophy OF mathematics is not the same as philosophy FOR mathematics. The book’s great strength is to make clear that mathematics is a fascinating source of philosophical problems and issues. Here’s a sample:
1. What is the nature of existence? One of the perennial questions—are we, as a friend once tried to persuade me, creatures of spirit who happen to have a bodies, or are we merely bodies, or are we bodies that aren’t just bodies? What is a body, anyway? Do immaterial things like spirit really exist? Well, the existence of mathematical objects pose these questions in a way that is very promising. After all, the point is that things like numbers and circles are simpler than bodies and spirits. The question goes back to Plato, to the very foundation of the Western tradition in philosophy (footnote: Sad to say, Linnebo remains totally confined to this tradition—but I would say that in doing so he is entirely fair to the limitations of the philosophy of mathematics).
According to Plato, the objects of mathematics exist in the first place as perfect ideals, in a spiritual realm. We human beings consist partly of spirit and it is this part of us that is able to perceive (perhaps a bit dimly, but nevertheless directly) these perfect objects. But this isn’t merely knowledge of spirit, separate from the world. Indeed, Plato would say that the better we understand world of spirit, which according to him is the world of eternal and perfect objects, the better we understand the material world, which is determined by the world of spirit and indeed is nothing but an imperfect set of copies of the ideals that exist in the spirit world.
The role of reasoning and proof in mathematics and logic is also made clear in Plato’s position. Our material existence is prone to error and failure. Logic and proof however come from the ideal world, as is seen from the fact that perfect logic is itself without any flaw and therefore one of the few direct realisations of spirit in our material world. For Plato, then, there is a natural link between philosophy, mathematics, piety and truth. And it all comes together in the answer to the question: what are we, really? And Plato gave this beautifully simple answer.
However, for centuries now the conception of the world as spirit in the first place has been problematic. One will not find philosophers these days who defend a purely Platonic vision of reality. As Linnebo makes clear, this is powerfully and perhaps most simply seen in the philosophy of mathematics. Without the simplicity of platonic idealism, it becomes very hard to say precisely how it is that mathematics achieves its apparent universality—for example, we believe that all circles, at all times and all places, have the property that their diameters and circumferences have the same ratio, nowadays symbolised with the constant π. But what kind of thing has that sort of existence? Where does it exist? The very circles that have this property themselves do not exist as real objects, composed of real substance. At best, they could only be like the edges of shadow. But such edges would reproduce the imperfections of the body that casts the shadow, and so would not be neither perfectly smooth, and furthermore according to physics even if the edge were perfect, it would induce diffraction and the so the edge of the actual shadow would have some thickness, contrary to a perfect circle.
Linnebo takes several chapters to outline how a philosopher deals with these difficulties, the ideas are never simple, but they are truly fascinating—as philosophy, but (except for a few of the pages devoted to set theory) not as mathematics. Mathematicians are happy to work with mathematical objects, whatever the philosophical conundrums may be that these objects raise.
2. What is the nature of knowledge? What is the nature of truth? Are there such things as universal truths? Is the sensation of understanding, of knowing something, just an illusion? How do we know what we know? Can we know that we know? Is the certainty of mathematics itself an illusion?
Nowadays many of these questions are addressed in psychology, including the psychology of mathematics (see for example the books by [Butterworth] and by [Dehaene] ), but of course the primary tradition in answering these questions is epistemology, one of the three parts in the great division of philosophy (again, according to the Western tradition). And again, if one were willing to grant Plato his ideal world as the determinant of our imperfect material world, everything would fall into place. We know imperfectly because we are imperfect material beings, but we have the possibility of knowing the ideal, perfect world, because we consist partly of spirit, and our spirit is itself of the same nature as the ideal world. By contemplation, by the study of philosophy, and yes, by the study of mathematics, we have access to knowledge of the ideal world, but we must accept that this knowledge is incomplete and hard-won.
Well, as I said, no philosopher these days would be simply Platonist. Moreover, there are many ways to describe how animals (and for that matter, plants and bacteria) have ways of interacting with the world that amount to knowledge that has been acquired, whether by individual learning events or by evolution across many generations in a population. But these will not do for mathematics, nor perhaps for any knowledge that is acquired by purely symbolic means: diagrams, formulae, logic, principles. The problem is to say how it is that pure abstraction interacts with the real world, indeed how it is that as physical beings we can acquire knowledge of things that do not interact with the real physical world?
I don’t think Linnebo does much to say how philosophers salvage epistemology of purely abstract entities in a world where all knowing is by material bodies. He seems to say that knowing is ok, we need not worry about it, if only we solve the problem of existence. But perhaps he would say that in his view mathematical objects are not purely abstract entities.
3. How should we live? Can we, by using reasoning and philosophy, reach good decisions about what to do? In the Western tradition, this question summarises the third great division of philosophy, namely ethics (footnote: Which includes politics, justice and the relationship of philosophy to theology, and also figures in part in many of the more applied aspects of Western philosophy, such as aesthetics and environmental philosophy). But although these questions are probably the oldest in philosophy, and moreover provide interesting links with Indian and Chinese philosophy at the time of Plato, it seems that they have no interest to philosophers of mathematics. Linnebo does not address such questions at all, not even narrowly within mathematics.
Yet surely these are good questions. How should we choose which mathematical questions to study? How should mathematics be applied? Above all, how should mathematics be taught? Does the teaching of mathematics involve any particular values, and if so, how should those values be justified? Do we have a responsibility to make mathematics as widely available as possible? Or on the contrary, should mathematics and mathematical knowledge be limited to those with the most mathematical talent? Is mathematics a value in itself or is it valuable only to the extent that it is useful? Linnebo does not even begin to address these questions, and leaves the impression that, as far as philosophy is concerned, they are not part of the philosophy of mathematics.