Lately, as one does, I have been reading up on the philosophy, and especially epistemology of mathematics. In doing so, I chanced upon the great, late philosopher of mathematics, Thomas Tymoczko. In his paper “Computers, Proofs and Mathematicians: A Philosophical Investigation of the Four-Color Proof” (1980) Tymoczko gives five very sensible criteria for a realistic epistemology of mathematics. I will quote all five in full with some comments interspersed.
- Drop the image of an isolated mathematician and admit that mathematical knowledge is public knowledge, the knowledge of a community of mathematicians. Recognizing this community will help us recognize the communication between mathematicians that constitutes the doing of mathematics and grounds mathematical knowledge. (Tymoczko 1980, 136.)
Indeed. All knowledge worth the moniker is public. Thus, all sensible epistemology will be social epistemology, and this will also hold for the epistemology of mathematics.
- Drop the image as mathematicians as infallible. Recognizing the fallible nature of mathematical knowledge will help us to recognize the mechanisms by which mathematicians individually and in the community work to minimize error. (Id., 137)
Here, we have shades of Wittgenstein, especially his On Certainty where Wittgenstein points out that a “mathematical proposition has been obtained by a series of actions that are in no way different from the actions of the rest of our lives, and are in the same degree liable to forgetfulness, oversight and illusion” (§651). We are often led astray by the fact that inductive reasoning is fallible through-and-through while deductive reasoning is only fallible insofar as we are fallible. But fallible we are.
- Once mathematics is recognized as fallible, there will be less difficulty in admitting tools as ingredients of mathematical knowledge supplementing proofs. Mathematical tools might provide an entry into the epistemology for the more general physical and cultural setting of the mathematical community. (Tymoczko 1980, 137)
We are limited beings. We need hammers to drive nails efficiently. Similarly we need calculators and computers, compasses and rulers, pens and paper, to do mathematics efficiently. All these things are cultural creations. Our epistemology should reflect this.
- On the other side of the epistemological coin, we can broaden our horizons beyond statements. In the first place a recognition of the richness of mathematical discourse – assertions, questions, conjectures, hypotheses, lemmas, criticisms, etc., – is needed to describe mathematical communication. In the second place the foci of mathematical knowledge are often not individual statements but theories or even branches of mathematics. Reductionist epistemology holds that there must be one basic mathematical theory (be it logic, set theory, formal systems, etc.). But in fact since Godel’s work we’ve known that no single reduction is satisfactory. Mathematics is not one, but many. There is always more than one mathematical theory in use at a given time. So we must consider more seriously the various relationships between mathematicians and the various overlapping theories being developed. (Id., 137)
This opens up a lot of new avenues. Tymoczko explicitly calls for a sociological and historical outlook in the epistemology of mathematics. I wholeheartedly agree. It is very hard indeed to understand what people like Frege and Hilbert were doing in the foundations of mathematics, if one does not understand the mathematics that had come before (cf. Grattan-Guinness 2000). The same goes for the mathematics of today. To see what mathematicians are doing and why, one needs to understand the current – and past – state of play, including the mathematical community.
- Finally, epistemology should abandon the idea of explaining how mathematicians go from knowing nothing to knowing modern mathematics. It is much more realistic to concentrate on mathematical development and try to explain how mathematicians who already know x are able to learn y. We should thereby shift our sights from the one big unmanageable question to a lot of smaller, more tractable ones. (Id., 137)
Here, again, the role of the history of mathematics comes to play. We need not explain how we got from pre-Euclidean geometry to differential geometry. However, we do need to explain how we got from Cantor’s set theory to the state of play, say, in the 1930s. And that story is a lot messier than a lot of people might like. For more on this, I heartily recommend Ivor Grattan-Guinness’ (2000) wonderful history of logic and set theory.
These five criteria were offered as a way of providing “satisfying answers about checkability, rigor and computer supplements in proofs” (Tymoczko 1980, 137) But should we not take heed of these criteria already because we should strive to understand – and do justice to – this peculiar human phenomenon called “mathematics”? And how else could we even accomplish that?
Grattan-Guinness, Ivor (2000): The Search for Mathematical Roots, 1870-1940. Princeton NJ: Princeton University Press.
Tymoczko, Thomas (1980): Computers, Proofs and Mathematicians: A Philosophical Investigation of the Four-Color Proof. Mathematics Magazine, Vol. 53, No. 3, 131–138.
Wittgenstein, Ludwig (1969): On Certainty, Oxford: Basil Blackwell.