Online research in philosophy

This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti?Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the ?no miracles? argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific ?realists? should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism.

This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti-Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the 'no miracles' argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific 'realists' should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism.

A response is given here to Benacerraf's 1973 argument that mathematical platonism is incompatible with a naturalistic epistemology. Unlike almost all previous platonist responses to Benacerraf, the response given here is positive rather than negative; that is, rather than trying to find a problem with Benacerraf's argument, I accept his challenge and meet it head on by constructing an epistemology of abstract (i.e., aspatial and atemporal) mathematical objects. Thus, I show that spatio-temporal creatures like ourselves can attain knowledge about mathematical objects by simply explaininghow they can do this. My argument is based upon the adoption of a particular version of platonism — full-blooded platonism — which asserts that any mathematical object which possiblycould exist actuallydoes exist.

According to standard mathematical platonism, mathematical entities (numbers, sets, etc.) are abstract entities. As such, they lack causal powers and spatio-temporal location. Platonists owe us an account of how we acquire knowledge of this inaccessible mathematical realm. Some recent versions of mathematical platonism postulate a plenitude of mathematical entities, and Mark Balaguer has argued that, given the existence of such a plenitude, the attainment of mathematical knowledge is rendered non-problematic. I assess his epistemology for such a profligate platonism and find it unsatisfactory because it lacks an adequate semantics, in particular, an adequate account of reference.

A common objection raised against naturalism is that anaturalized epistemology cannot account for the essential normative character of epistemology. Following an analysis of different ways in which this charge could be understood, it will be argued that either epistemology is not normative in the relevant sense, or if it is, then in a way which a naturalized epistemology can account for with an instrumental and hypothetical model of normativity. Naturalism is here captured by the two doctrines of empiricism and gradualism. Epistemology is a descriptive discipline about what knowledge is and under what conditions a knowledge-claim is justified. However, we can choose to adopt a standard of justification and by doing so be evaluated by it. In this sense our epistemic practices have a normative character, but this is a form of normativity a naturalized epistemology can make room for. The normativity objection thus fails. However, in the course of this discussion, as yet another attempt to clarify the normativity objection, such a naturalistic model will be contrasted with Donald Davidson's theory of interpretation. Even though this comparison will not improve upon the negative verdict upon the original objection, it will be argued that naturalism cannot accept Davidson's theory since it contains at least one constitutive principle – the principle of charity – whose epistemic status is incompatible with the naturalistic doctrine of gradualism. So, if this principle has this role, then epistemology cannot be naturalized.

A common objection raised against naturalism is that a naturalized epistemology cannot account for the essential normative character of epistemology. Following an analysis of different ways in which this charge could be understood, it will be argued that either epistemology is not normative in the relevant sense, or if it is, then in a way which a naturalized epistemology can account for with an instrumental and hypothetical model of normativity. Naturalism is here captured by the two doctrines of empiricism and gradualism. Epistemology is a descriptive discipline about what knowledge is and under what conditions a knowledge-claim is justified. However, we can choose to adopt a standard of justification and by doing so be evaluated by it. In this sense our epistemic practices have a normative character, but this is a form of normativity a naturalized epistemology can make room for. The normativity objection thus fails. However, in the course of this discussion, as yet another attempt to clarify the normativity objection, such a naturalistic model will be contrasted with Donald Davidson's theory of interpretation. Even though this comparison will not improve upon the negative verdict upon the original objection, it will be argued that naturalism cannot accept Davidson's theory since it contains at least one constitutive principle -- the principle of charity -- whose epistemic status is incompatible with the naturalistic doctrine of gradualism. So, if this principle has this role, then epistemology cannot be naturalized.

A common objection raised against naturalism is that a naturalized epistemology cannot account for the essential normative character of epistemology. Following an analysis of different ways in which this charge could be understood, it will be argued that either epistemology is not normative in the relevant sense, or if it is, then in a way which a naturalized epistemology can account for with an instrumental and hypothetical model of normativity. Naturalism is here captured by the two doctrines of empiricism and gradualism. Epistemology is a descriptive discipline about what knowledge is and under what conditions a knowledge-claim is justified. However, we can choose to adopt a standard of justification and by doing so be evaluated by it. In this sense our epistemic practices have a normative character, but this is a form of normativity a naturalized epistemology can make room for. The normativity objection thus fails. However, in the course of this discussion, as yet another attempt to clarify the normativity objection, such a naturalistic model will be contrasted with Donald Davidson’s theory of interpretation. Even though this comparison will not improve upon the negative verdict upon the original objection, it will be argued that naturalism cannot accept Davidson’s theory since it contains at least one constitutive principle – the principle of charity – whose epistemic status is incompatible with the naturalistic doctrine of gradualism. So, if this principle has this role, then epistemology cannot be naturalized.

Mathematical explanation -- What is naturalism? -- Perception, practice, and ideal agents: Kitcher's naturalism -- Just metaphor?: Lakoff's language -- Seeing with the mind's eye: the Platonist alternative -- Semi-naturalists and reluctant realists -- A life of its own?: Maddy and mathematical autonomy.

It is argued here that mathematical objects cannot be simultaneously abstract and perceptible. Thus, naturalized versions of mathematical platonism, such as the one advocated by Penelope Maddy, are unintelligble. Thus, platonists cannot respond to Benacerrafian epistemological arguments against their view vias Maddy-style naturalization. Finally, it is also argued that naturalized platonists cannot respond to this situation by abandoning abstractness (that is, platonism); they must abandon perceptibility (that is, naturalism).

A discussion of views first presented by this author and Edward Zalta in 1995 in the paper “Naturalized Platonism vs. Platonized Naturalism”. That paper presents an application of Zalta’s “object theory” to the ontology of mathematics, and claims that there is a plenitude of abstract objects, all the creatures of distinct mathematical theories. After a summary of the position, two questions concerning the view are singled out for discussion: just how many mathematical objects there are by our account, and the nature of the properties we use to characterize abstract objects. The difference between the authors in more recent developments of the view are also discussed.