Numbers can be anything
This thesis is aimed at an interdisciplinary audience, so it is appropriate for me to declare my own training and approach. My undergraduate training was in philosophy and pure mathematics, since graduating I've dabbled in mathematical logic and set theory, but my academic development has been concentrated on philosophy. My philosophical style is dyed in the wool analytic, and this thesis is primarily a work of analytic philosophy, which means that the arguments of this thesis are meant to stand up on their own. They are meant to be agreed (or disagreed) with as opposed to being believed (or disbelieved). This can be contrasted with work found in the empirical sciences, in which we need to take on trust the results that a scientist reports. If a scientist lacked integrity and was to get a reputation for falsifying data then we would have reason to call her work into question. I, however, make no claim about my own integrity (or lack of it): However, the only way that I could falsify a philosophical argument is by knowingly introducing flaws so that it arrives at the conclusions I want, and the faulty reasoning required would be left out in the open for all to see. I advise the reader to treat with suspicion any demand that this thesis might make on your trust, and I will endeavour to keep the resulting doubts down to a minimum.
The thesis was initially inspired by two pieces of work: Raymond Wilder's Mathematics as a Cultural System [Wilder, 1981] in which Wilder presents the treatment of mathematics as a subculture of our general culture as a way of looking at mathematics, and Penelope Maddy's "Indispensability and practice" [Maddy, 1992]. Maddy sets out what she thinks the aims of philosophy of mathematics should be; she writes
These are, in my view, proper goals for the philosophy of mathematics. We, as philosophers of mathematics, should provide an account of mathematics as practiced, and we should make a contribution to unraveling the conceptual confusions of contemporary mathematics. [Maddy, 1992, p. 276]
She then goes on to point out two major problems with her own philosophy of mathematics which prevent it from reaching her goal. This thesis grew out of an attempt to address Maddy's difficulties in a way which led to the use of anthropological concepts taken from Wilder's work.
Maddy is a mathematical realist. She argues that because mathematics is indispensable to our best theories of the world we have as much reason to believe in the objects posited by mathematics as we do to believe in the physical objects, such as electrons, posited by our best theories [Maddy, 1990]. This is an argument I found convincing at the time. However, as she later pointed out [Maddy, 1992, pp. 286-289], this argument is not in accord with the practices of mathematicians. Her "mathematical practice objection" is that if her indispensability argument is correct then the solution to at least one mathematical problem (whether or not there is a definite answer to the question "Are all sets Lebesgue measurable?") depends on the state of our best physical theories, so mathematicians would be watching the development of physics in an attempt to solve this problem, which is not the case.
One possible solution to Maddy's problem is to say that although there is good reason for believing in physically existent mathematics (possibly as properties of physical entities), it does not automatically follow that mathematicians are concerned with the study of this physical mathematics. A rough and ready scenario can be set up using the concept of culture, which Raymond Wilder describes as the "totality of the individual world views... united by bonds of communication" [Wilder, 1981, p.8]. Mathematicians tend to talk about mathematics to other mathematicians, and there is a language barrier which makes it difficult to talk (in detail at least) about the subject to those who are not trained to interpret the terms they use in the technical sense intended, and a similar situation holds for physicists. Mathematicians and physicists each form their own cultures, united by the bonds of the technical communication which also keeps out those not initiated in the respective subject. So we have the subculture of mathematicians and the subculture of physicists. Within the subculture of physicists "doing mathematics" is the type of practice described by Maddy, whilst within the subculture of mathematicians "doing mathematics" is a practice similar to Hartry Field's fictionalism (as described in Realism, Mathematics and Modality [Field, 1989]), in which mathematics is a form of fiction and statements such as "2+2=4" are true in much the same way as "Oliver Twist lived in London" is true. It is the mathematical practices of physicists that is influenced by the considerations of the mathematical practice objection. Mathematicians are concerned with what is effectively a work of fiction and so need not take into account considerations of what might be the case in the real world, and it is mostly only mathematicians who care about the Lebesgue measurability of sets.
Of course the above scenario is incredibly over-simplified. Mathematicians and physicists cannot be split so easily into two camps. I just want to use it to demonstrate that there is a philosophically interesting possibility of there being more than one mathematics. Indeed I would go so far as to suggest that it is possible that every individual has a unique mathematics due to her personality, experience and training. If it is the case that there are multiple mathematics, then the traditional approach to the problems of the philosophy of mathematics (attempting to describe the epistemology and the ontology of mathematics) is misguided. The difficulty is that both epistemology and ontology are applicable only to the individual; the epistemology of mathematics is concerned with what reasons individuals have for believing in mathematics, and the ontology of mathematics is concerned with explaining what individuals are referring to when they talk about mathematics. If we could assume that everybody is doing the same thing when they do mathematics then an account of the mathematics of one individual would be sufficient to be true of every one; if this is not the case then our account must either be sufficiently general to allow for the variation in mathematics or we must state for whom our account is true.
© Patrick Killeen 1995