Fraser MacBride

The basic relations and functions that mathematicians use to identify mathematical objects fail to settle whether mathematical objects of one kind are identical to or distinct from objects of an apparently different kind, and what, if any, intrinsic properties mathematical objects possess. According to one influential interpretation of mathematical discourse, this is because the objects under study are themselves incomplete; they are positions or akin to positions in patterns or structures. Two versions of this idea are examined. It is argued that the evidence adduced in favor of the incompleteness of mathematical objects underdetermines whether it is the objects themselves or our knowledge of them that is incomplete. Also, holding that mathematical objects are incomplete conflicts with the practice of mathematics. The objection that structuralism is committed to the identity of indiscernibles is evaluated and it is also argued that the identification of objects with positions is metaphysically suspect.

According to neo-Fregeans, an expression that is syntactically singular and figures in a true sentence is guaranteed to have some existing thing in the world to pick out. But this approach is confronted by a dilemma. If reality is crystalline, has a structure fixed independently of language, then the view that reality is guaranteed to contain a sufficient plenitude of objects to supply referents for the relevant expressions is left hostage to cosmological fortune. Whereas if reality is plastic then it becomes dubiously coherent to conceive of our ordinary, scientific, and mathematical claims about a diversity of objects as being genuinely true or false of an independent reality. To avoid this dilemma, the neo-Fregeans must abandon the dichotomy that says either our true sentences must ‘mirror’ the structure of a crystalline reality or ‘impose’ structure upon a plastic one. If we can legitimately deny that there is an intelligible question to be raised of how language hooks onto reality, then prima facie we can grant that a sentence S is true without thereby becoming embroiled in the uncomfortable consequences of having to say what makes S true. So ontological quietism holds out the last, best hope for neo-Fregeanism.

According to philosophical folklore Ramsey maintained three propositions in his famous 1925 paper “Universals”: (i) there is no subject-predicate distinction; (ii) there is no particular-universal distinction; (iii) there is no particular-universal distinction because there is no subject-predicate distinction. The ‘first generation’ of Ramsey commentators dismissed “Universals” because they held that whereas predicates may be negated, names may not and so there is a subject-predicate distinction after all. The ‘second generation’ of commentators dismissed “Universals because they held that the absence of a merely linguistic distinction between subject and predicate does not provide any kind of reason for doubting that a truly ontological (i.e. non-linguistic) distinction obtains between particulars and universals. But both first and second-generation criticisms miss their marks because Ramsey did not maintain the three identified propositions. The failure of commentators to appreciate the point and purpose of the position Ramsey actually advanced in “Universals” results from (a) failing to consider the range of different arguments advanced there, (b) looking at “Universals” in isolation from Ramsey’s other papers and (c) failing to consider Ramsey’s writings in the context of the views that Russell and Wittgenstein held during the early 1920s. Seen from this wider perspective Ramsey arguments in “Universals” take on an altogether different significance. They not only anticipate important contemporary developments⎯the resurgence of Humeanism and the doctrine that the existence of universals can only be established a posteriori⎯but also point beyond them.

According to one creation myth, analytic philosophy emerged in Cambridge when Moore and Russell abandoned idealism in favour of naive realism: every word stood for something; it was only after “the Fall,” Russell's discovery of his theory of descriptions, that they realized some complex phrases (“the present King of France”) didn't stand for anything. It has become a commonplace of recent scholarship to object that even before the Fall, Russell acknowledged that such phrases may fail to denote. But we need to go further: even before the Fall, Russell had taken an altogether more discerning approach to the ontology of logic and relations than is usually recognized

One version of the Julius Caesar problem arises when we demand assurance that expressions drawn from different theories or stretches of discourse refer to different things. The counter‐Caesar problem arises when assurance is demanded that expressions drawn from different theories . refer to the same thing. The Julio César problem generalises from the counter‐Caesar problem. It arises when we seek reassurance that expressions drawn from different languages refer to the same kind of things . If the Julio César problem is not resolved then the Fregean account of numbers as objects is cast into doubt, the notion of number left relative to a language. Wright introduced this problem by asking whether there can be such a thing as ‘International Platonism’. After rejecting Hale's attempt to resolve it I argue that the threat posed by the Julio César problem diminishes – even though it cannot be made to logically disappear – once it is recognised that the radical interpretation of an unfamiliar language is inevitably holistic, the evidence available invariably defeasible and consequently Cartesian certainty about the significance of the utterances of a foreign tongue neither to be sought after nor attained

G. F. Stout is famous as an early twentieth century proselyte for abstract particulars, or tropes as they are now often called. He advanced his version of trope theory to avoid the excesses of nominalism on the one hand and realism on the other. But his arguments for tropes have been widely misconceived as metaphysical, e.g. by Armstrong. In this paper, I argue that Stout’s fundamental arguments for tropes were ideological and epistemological rather than metaphysical. He moulded his scheme to fit what is actually given to us in perception, arguing that our epistemic practices would break down in an environment where only universals were given to us.

Neo-Fregeanism contends that knowledge of arithmetic may be acquired by second-order logical reflection upon Hume's principle. Heck argues that Hume's principle doesn't inform ordinary arithmetical reasoning and so knowledge derived from it cannot be genuinely arithmetical. To suppose otherwise, Heck claims, is to fail to comprehend the magnitude of Cantor's conceptual contribution to mathematics. Heck recommends that finite Hume's principle be employed instead to generate arithmetical knowledge. But a better understanding of Cantor's contribution is achieved if it is supposed that Hume's principle really does inform arithmetical practice. More generally, Heck's arguments misconceive the epistemological character of neo-Fregeanism.

Is there a particular‐universal distinction? Ramsey famously advocated scepticism about this distinction. In “Some Formal Ontological Relations” E.J. Lowe argues against Ramsey that a particular‐universal distinction can be made out after all if only we allow ourselves the resources to distinguish between the elements of a four‐fold ontology. But in defence of Ramsey I argue that the case remains to be made in favour of either the four‐fold ontology Lowe recommends or the articulation of a particular‐universal distinction within it. I also argue that the case remains to be made against a spatio‐temporal conception of the particular‐universal distinction.

In this paper I contribute a further element to the case for admitting non-symmetric relations by dismantling the case against them. Armstrong and Dorr have both argued (1) that asymmetric relations give rise to ‘brute necessities’, whilst Dorr further argues (2) that admitting non-symmetric relations generates spurious possibilities and (3) that exploiting work of Goodman and Hazen, we can do without non-symmetric relations anyway. Against (1) I argue that neither Armstrong nor Dorr succeed in avoiding brute necessities themselves. Against (2) and (3) I argue that admitting non-symmetric relations doesn’t give rise to spurious possibilities and Goodman and Hazen’s work cannot be used to establish that non-symmetric relations are dispensable to science and mathematics.

The early David Lewis was a staunch critic of the Truthmaker Principle. To endorse the principle, he argued, is to accept that states of affairs are truthmakers for contingent predications. But states of affairs violate Hume's prohibition of necessary connections between distinct existences. So Lewis offered to replace the Truthmaker Principle with the weaker principle that ‘truth supervenes upon being’. This chapter argues that even this principle violates Hume's prohibition. Later Lewis came to ‘withdraw’ his doubts about the Truthmaker Principle, invoking counterpart theory to show how it is possible to respect the principle whilst admitting only things that do not violate Hume's prohibition. What this really reveals is that the Truthmaker Principle is no explanatory advance on the supervenience principle. Extending Lewis's use of counterpart theory also allows us to explain away the necessary connections that threatened to undermine his earlier statements of supervenience.