Fiona T Doherty

Ramsey’s Universals has not been served well by its critics and, as a result, the real and important contentions of Ramsey’s essay are often obscured in discussion of his work. This thesis is intended to form the beginning of an attempt to rectify this by offering an exposition and critique of Ramsey’s essay that is particularly sensitive to the background context and purpose of the essay as a whole and to the subtle structure of the argumentation within it. The construction of the arguments in Universals is so intricate that to assess any of its arguments without placing them in the context of the overall strategy of Ramsey’s essay is to grossly underestimate them. For this reason, most of the labour in this thesis will be directed towards articulating Ramsey’s concerns in his essay and the way in which Ramsey’s arguments are supposed to supplement each other in order to establish his main contentions. These tasks take up the first two chapters. Only then will the third chapter consider one particular argument- the incomprehensible trinity argument- and assess whether it is successful by first identifying the role that the argument is intended to play in the overall structure of Ramsey’s argument and then asking whether or not the argument can be said to fulfil such a role. This final chapter is a mere beginning towards a proper critique of Ramsey’s difficult and subtle argumentation in Universals. Nonetheless it serves as an example of how offering a critique of Ramsey which places his arguments within the context and concerns of the essay as a whole, while it does not immediately vindicate them of all criticism, shows them to be far more subtle and robust than they have been estimated to be.

Structuralism about mathematical objects and structuralist accounts of physical computation both face indeterminacy objections. For the former, the problem arises for cases such as the complex roots i and \, for which a automorphism can be defined, thus establishing the structural identity of these importantly distinct mathematical objects. In the case of the latter, the problem arises for logical duals such as AND and OR, which have invertible structural profiles :369–400, 2001). This makes their physical implementations indeterminate, in the sense that their structural profiles alone cannot establish whether a given physical component is an AND-gate or an OR-gate. Doherty has recently shown both problems to be analogous, and has argued that computational structuralism is threatened with the absurd conclusion that computational digits might be indiscernible, such that, if structural properties are all that we have to go on, the binary digit 0 must be treated as identical to the binary digit 1. However, we think that a solution to the indiscernibility problem for mathematical structuralists, drawing on the work of David Hilbert, can be adapted for the analogous problem in the computational case, thereby rescuing the structuralist approach to physical computation.

I show that the indeterminacy problem for computational structuralists is in fact far more problematic than even the harshest critic of structuralism has realised; it is not a bullet which can be bitten by structuralists as previously thought. Roughly, this is because the structural indeterminacy of logic-gates such as AND/OR is caused by the structural identity of the binary computational digits 0/1 themselves. I provide a proof that pure computational structuralism is untenable because structural indeterminacy entails absurd consequences - namely, that there is only one binary computational digit. I conclude that accounting for individuation is a more important desiderata for a theory of computation than even that of triviality.

ABSTRACT This paper reveals David Hilbert’s position in the philosophy of mathematics, circa 1900, to be a form of non-eliminative structuralism, predating his formalism. I argue that Hilbert withstands the pressing objections put to him by Frege in the course of the Frege-Hilbert controversy in virtue of this early structuralist approach. To demonstrate that this historical position deserves contemporary attention I show that Hilbertian structuralism avoids a recent wave of objections against non-eliminative structuralists to the effect that they cannot distinguish between structurally identical but importantly distinct mathematical objects, such as the complex roots of $-1$.