Ontology of Mathematics

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An ontological theorist generally begins his discussion with a preconceived notion of what

kind of thing an object will turn out to be. Instead, we will here begin with a Thomassonian

approach to the ontology of mathematics. First, let us consider what happens when we

rst come to determine a mathematical proposition (which I will use synonymously with

'mathematical entitty'). A mathematician does not feel as though he creates mathematical

theories. Pythagoras can hardly be thought to have created the claim that a2 + b2 = c2. It

becomes clear that a mathematical proposition is a discovered one; that is, we would hardly

nd ourselves contending that Pythagoras created his famous theorem. Regardless of who

discovers it, the same mathematical proposition would be discovered. However, what exactly

is Pythagoras discovering when he puts a2 + b2 = c2 to paper (or papyrus)? Pythagoras

is certainly not noting the existence of the formula, but, rather, he is noticing the relation

between a hypoteneuse and its sides. This relationship comes to be expressed in his formula.

So we already see that while a genuine relationship exists between a hypoteneuse and its

sides, a genuine theorem is contingent on language; the language in this case is that of


We are met, then, with two questions. The rst is whether we should consider the terms

of mathematics, such as wo" or four," to abstract or concrete. The second is whether

we should consider the relationships expressed by mathematics, such as one plus one" or

wo sets of two," to be abstract or concrete. A ctional approach, as that suggested by

Harty Field [3], would say that mathematical terms are literally false. So, when we say


'a2 + b2 = c2 is true' wha...

... middle of paper ... them [1]. This question aside, it seems

to me that the most plausible candidate for an ontology of mathematics is that of abstract

artifacts. As with all theories, there will likely be problems which will arise, but I believe I

have given ample evidence in the case against constructivism and ctionalism and signi cant

evidence for an abstract artifact mathematical theory.


[1] Benacerraf, Paul. Mathematical Truth" Journal of Philosophy 70 (1973), pp. 661-80.

[2] Brown, J.R. Constructivism." Philosophy of Mathematics Handout.

[3] Field, Harty. Introduction: ctionalism, epistemology and modality." Realism, Mathe-

matics, and Modality Handout.

[4] McEvoy, Mark. Miscellaneous class notes.

[5] Thomasson, Amie. Fictional Characters as Abstract Artifacts". Fiction and Metaphysics

(Cambridge: Cambridge University Press, 1999), pp. 5-23, 155, 156.
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