Abstract
The thoughts in this paper follow a suggestion by Reuben Hersh that pluralists think of mathematical theories as models of other parts of mathematics. Through this lens, the totality of ‘mathematics’ is then a game of interpretation of one theory by another. As a discipline, mathematics is sui generis.
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Notes
- 1.
The word ‘game’ suggests a formalist conception of mathematics, and this conception is not one I wish to explore in this paper. To distance the formalist view from the view here: formalism is the view that all there is to mathematics is formal play, within certain rules. For Hilbert the rules included the finisitic constraint on the signs and manipulations and the consistency of each theory. In contrast, the view examined here is that meaning and interpretation in mathematics is had partly by one theory being translated into the language of another and then finding out the extent to which the theorems, concepts, or ideas of one theory can be recovered by the second. The emphasis is different, there is no requirement concerning formal representation, and there are no rules imposed from the outside.
- 2.
The suggestion was made on the occasion of a conference on mathematical pluralism in Kolkata, in December 2015. The conference was organized by Mihir Chakraborty. Hersh’s paper had the title: Pluralism as Modelling and as Confusion.
- 3.
For example, in some applications of mathematical theories to physical reality, we literally use an inconsistent mathematical theory but are cautious never to ask the questions that draw out the inconsistency. For example, ‘renormalisation’ is a procedure in quantum electrodynamics where we simply eliminate the infinite quantities that occur during certain sorts of calculation. We do this because in the physical world we are measuring there are supposed to be no infinite quantities, and mathematically, this means that the normal arithmetic operations such as addition or division give results that, again, are not supposed to occur in the physical measured world.
- 4.
In the sense meant here, a trivial theory is one where every formula in the language is true or derivable. In a classical or intuitionist theory, if we generate or discover a contradiction that is derivable or true in the theory, then we can derive any formula and its negation. This is also called ‘explosion’ of the theory. Trivialism is what ensues after explosion.
- 5.
Concurrently, and independently, Bolyai was developing imaginary geometry as well. Both denied the parallel postulate of Euclidean geometry and thus proved the independence of the parallel postulate. The story of the development of non-Euclidean geometry is a very nice illustration of theories of mathematics used to model and contrast other theories of mathematics.
- 6.
By ‘meta-logically’ I mean logically in the sense of informal logic. That is, were I to say that ‘we are not logically banned’, this might be interpreted to mean that ‘according to a particular formal representation of logical reasoning, we are not banned’. Moreover, we might be able to choose any formal representation, including invent our own strange one. No. What I mean here is that we do not suddenly find ourselves in the realm of trivialism if we make a perverse interpretation. It might simply be unnatural, ridiculous, strange, uncomfortable, and difficult to make sense of, but not for all that, ‘impossible’.
Bibliography
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Friend, M. (2017). Mathematical Theories as Models. In: Sriraman, B. (eds) Humanizing Mathematics and its Philosophy. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-61231-7_21
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