Mathematical platonism is the view that abstract mathematical objects exist. Ontological pluralism is the view that there are many modes of existence. This paper examines the prospects for...

Mathematical platonism is the view that abstract mathematical objects exist. Ontological pluralism is the view that there are many modes of existence. This paper examines the prospects for plural platonism, the view that results from combining mathematical platonism and ontological pluralism. I will argue that some forms of platonism are in harmony with ontological pluralism, while other forms of platonism are in tension with it. This shows that there are some interesting connections between the platonism–antiplatonism dispute and recent debates over (...) ontological pluralism. (shrink)

This article motivates and develops a reductive account of the structure of certain physical quantities in terms of their mereology. That is, I argue that quantitative relations like "longer than" or "3.6-times the volume of" can be analyzed in terms of necessary constraints those quantities put on the mereological structure of their instances. The resulting account, I argue, is able to capture the intuition that these quantitative relations are intrinsic to the physical systems they’re called upon to describe and explain.

Putnam rejects having argued in the terms of the argument known in the literature as “the Quine-Putnam indispensability argument”. He considers that mathematics contribution to physics does not have to be interpreted in platonist terms but in his favorite modal variety. The purpose of this paper is to consider Putnam’s acknowledged argument and philosophical position against contemporary so called in the literature ‘fictionalist’ views about applied mathematics. The conclusion will be that the account of the applicability of mathematics that stems (...) from Putnam‘s acknowledged argument can be assimilated to so-called ‘fictionalist’ views about applied mathematics. (shrink)

The ‘indispensability argument’ for the existence of mathematical objects appeals to the role mathematics plays in science. In a series of publications, Joseph Melia has offered a distinctive reply to the indispensability argument. The purpose of this paper is to clarify Melia’s response to the indispensability argument and to advise Melia and his critics on how best to carry forward the debate. We will begin by presenting Melia’s response and diagnosing some recent misunderstandings of it. Then we will discuss four (...) avenues for replying to Melia. We will argue that the three replies pursued in the literature so far are unpromising. We will then propose one new reply that is much more powerful, and—in the light of this—advise participants in the debate where to focus their energies. (shrink)

Several philosophers have suggested that some scientific explanations work not by virtue of describing aspects of the world’s causal history and relations, but rather by citing mathematical facts. This paper investigates what mathematical facts could be in order for them to figure in such “distinctively mathematical” scientific explanations. For “distinctively mathematical explanations” to be explanations by constraint, mathematical language cannot operate in science as representationalism or platonism describes. It can operate as Aristotelian realism describes. That is because Aristotelian realism enables (...) mathematics to apply to the physical world without a morphism’s mediation. (shrink)

A standard formalization of a scientific theory is a system of axioms for that theory in a first-order language (possibly many-sorted; possibly with the membership primitive $$\in$$ ). Suppes (in: Carvallo M (ed) Nature, cognition and system II. Kluwer, Dordrecht, 1992) expressed skepticism about whether there is a “simple or elegant method” for presenting mathematicized scientific theories in such a standard formalization, because they “assume a great deal of mathematics as part of their substructure”. The major difficulties amount to these. (...) First, as the theories of interest are mathematicized, one must specify the underlying applied mathematics base theory, which the physical axioms live on top of. Second, such theories are typically geometric, concerning quantities or trajectories in space/time: so, one must specify the underlying physical geometry. Third, the differential equations involved generally refer to coordinate representations of these physical quantities with respect to some implicit coordinate chart, not to the original quantities. These issues may be resolved. Once this is done, constructing standard formalizations is not so difficult—at least for the theories where the mathematics has been worked out rigorously. Here we give what may be claimed to be a simple and elegant means of doing that. This is for mathematicized scientific theories comprising differential equations for $$\mathbb{R}$$ -valued quantities Q (that is, scalar fields), defined on n (“spatial” or “temporal”) dimensions, taken to be isomorphic to the usual Euclidean space $$\mathbb{R}^n$$. For illustration, I give standard (in a sense, “text-book”) formalizations: for the simple harmonic oscillator equation in one-dimension and for the Laplace equation in two dimensions. (shrink)

Recent debates about mathematical ontology are guided by the view that Platonism's prospects depend on mathematics' explanatory role in science. If mathematics plays an explanatory role, and in the right kind of way, this carries ontological commitment to mathematical objects. Conversely, the assumption goes, if mathematics merely plays a representational role then our world-oriented uses of mathematics fail to commit us to mathematical objects. I argue that it is a mistake to think that mathematical representation is necessarily ontologically innocent and (...) that there is an argument from mathematics' representational capacity to Platonism. Given that it is common ground between the Platonist and nominalist that mathematics plays a representational role in science, this representationalist argument is to be preferred over the explanatory, or enhanced, indispensability argument. (shrink)

Indispensability arguments are among the strongest arguments in support of mathematical realism. Given the controversial nature of their conclusions, it is not surprising that critics have supplied a number of rejoinders to these arguments. In this paper, I focus on one such rejoinder, Melia’s ‘Weasel Response’. The weasel is someone who accepts that scientific theories imply that there are mathematical objects, but then proceeds to ‘take back’ this commitment. While weaseling seems improper, accounts supplied in the literature have failed to (...) explain why. Drawing on examples of weaseling in more mundane contexts, I develop an account of the presumption against weaseling as grounded in a misalignment between two types of commitments. This is good news to the weasel’s opponents. It reinforces that they were right to question the legitimacy of weaseling. This account is also beneficial to the weasel. Uncovering the source of the presumption against weaseling also serves to draw out the challenge that the weasel must meet to override this presumption—what is required to be an ‘honest weasel’. (shrink)

2020 update of the entry "Properties".

The purpose of this thesis is to provide a metaphysic for moral realism and moral perception. This thesis is in two parts. The first is concerned with basic ontology. I begin in chapter 1 with an analysis of causation, demonstrating that substance theory is superior to Humeanism at accounting for our observations; thus I defend a substance ontology. In chapter 2, I address human agency, demonstrating that reasons internalism does not allow for incompatibilist freedom; hence, I affirm reasons are states (...) of affairs. I move on to qualia in chapter 3, demonstrating that arguments both for and against qualia realism involve question begging. In response, I argue that qualia realism is just as reasonable as anti-realism, and affirm realism. Given this, I address the mind-body problem in chapter 4, demonstrating the failure of physicalists to account for mental causation. I conclude that panpsychism, idealism, and substance dualism are equal reasonable alternatives: each allows for intrinsic properties, which are integral to substance ontology. The second part builds a moral ontology, along with an account of moral perceptual knowledge, compatible with the above basic ontology. In chapter 5, I reject moral properties as monadic since this would bar mapping moral concepts onto mind-independent reality. Instead, I argue that moral properties are relations. More specifically, in chapter 6, I make the case that they are features of relations, relations themselves being the total complex states of affairs between one or more substances. This allows moral properties to have their own nonreducible phenomenal quality. To account for moral normativity, I argue that moral features are actually a class of causal features, and briefly argue that final causes determine the normative causal trajectory of all substances. In chapter 7, I conclude this thesis with addressing possible objections to my theory of perception and perceptual knowledge. (shrink)

One of the important discussions in the philosophy of mathematics, is that centered on Benacerraf’s Dilemma. Benacerraf’s dilemma challenges theorists to provide an epistemology and semantics for mathematics, based on their favourite ontology. This challenge is the point on which all philosophies of mathematics are judged, and clarifying how we might acquire mathematical knowledge is one of the main occupations of philosophers of mathematics. In this thesis I argue that this discussion has overlooked an important part of mathematics, namely mathematics (...) as it is exercised by ordinary people. I do so by looking at the different theories that have been put forward in the recent debate, and showing for each of these that they are unable to account for the mathematical practices of ordinary people. In order to show that these practices do need to be accounted for, I also argue that ordinary people are doing mathematics, i.e. that they engage in properly mathematical practices. Because these practices are properly mathematical, they should be accounted for by any philosophy of mathematics. The conclusion of my thesis, then, is that current theories fail to do something that they should do, while remaining neutral on how well they perform when it comes to accounting for the practices of professional mathematicians. (shrink)