Online research in philosophy

Traditionally, in the philosophy of mathematics realists claim that mathematical objects exist independently of the human mind, whereas idealists regard them as mental constructions dependent upon human thought.It is tempting for realists to support their view by appeal to our widespread agreement on mathematical results. Roughly speaking, our agreement is explained by the fact that these results are about the same mathematical objects. It is alleged that the idealist’s appeal to mental constructions precludes any such explanation. I argue that realism and idealism, as above characterized, are equally effective (or problematic) in accounting for our widespread mathematical agreement.Both accounts are descriptivist for they take mathematical statements to be true if and only if they correctly describe mathematical objects. By contrast, non-descriptivist accounts take mathematical statements to be rule-like and mathematical symbols to be non-referential. I suggest that non-descriptivism provides a simpler and more natural explanation for our widespread agreement on mathematical results than any descriptivist account.

This paper is a discussion of Gödel's arguments for a Platonistic conception of mathematical objects. I review the arguments that Gödel offers in different papers, and compare them to unpublished material (from Gödel's Nachlass). My claim is that Gödel's later arguments simply intend to establish that mathematical knowledge cannot be accounted for by a reflexive analysis of our mental acts. In other words, there is at the basis of mathematics some data whose constitution cannot be explained by introspective analysis. This does not mean that mathematics is independent of the human mind, but only that it is independent of our ?conscious acts and decisions?, to use Gödel's own words. Mathematical objects may then have been created by the human mind, but if so, the process of creation cannot be completely analysed and re-enacted. Such a thesis is weaker than some of the statements that Gödel made about his conceptual realism. However, there is evidence that Gödel seriously considered this weak thesis, or a position depending only on this weak thesis. He also criticized Husserl's Phenomenology from this point of view.

This article examines one aspect of Thomas Aquinas' understanding of abstraction. It shows in which way, according to Aquinas, universal material objects and individual material objects are the starting point for mathematical objects. It comes to the conclusion that for Aquinas there are not only universal mathematical objects (circle, line), but also individual mathematical objects (this circle, that line). Universal mathematical objects are properties of universal material objects and individual mathematical objects are properties of individual material objects. One type of abstractio formae leads from individual material objects to universal mathematical objects, a second type from universal material objects to universal mathematical objects, and a third type from individual material objects to individual mathematical objects. Therefore, the concept of abstractio formae is ambiguous.

The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their dispensability cannot be demonstrated and, hence, there is no good reason for believing in the existence of mathematical objects which are genuinely platonic. Therefore, indispensability, whether true or false, does not support platonism. Mathematical platonists claim that at least some of the objects which are the subject matter of pure mathematics (e.g. numbers, sets, groups) actually exist. Furthermore, they claim that these objects differ radically from the concrete objects (trees, cats, stars, molecules) which inhabit the material world. We take the standard platonistic position to include the claim that platonic objects lack spatio-temporal location and causal powers. Many (perhaps most) mathematical platonists subscribe to this view.1 But some who call themselves (or might be called) mathematical platonists..

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.

I offer an explanation of how subjects are able to perceive the intrinsic spatial properties of objects, given that subjects always perceive from a particular location. The argument proceeds in two steps. First, I argue that a conception of space is necessary to perceive the intrinsic spatial properties of objects. This conception of space is spelled out by showing that perceiving intrinsic properties requires perceiving objects as the kind of things that are perceivable from other locations. Second, I show that having such a conception of space presupposes that a subject represent her location in relation to perceived objects. More precisely the thesis is that a subject represents her location as the location from which she both perceives objects and would act in relation to objects were she to act. So I argue that perception depends on the capacity to know what it would be to act in relation to objects.

In this paper I propose a position in the ontology of mathematics which is inspired mainly by a case study in the mathematical discipline if-theory. The main theses of this position are that mathematical objects are introduced by mathematicians and that after mathematical objects have been introduced, they exist as objectively accessible abstract objects.

Do mathematical objects exist in some realm inaccessible to our senses? It may be tempting to deny this. For how we could come to know mathematical truths, if such knowledge must arise from causal interaction with non-empirical objects? Among current positions, literalists argue that mathematical objects simply exist in the empirical world, and fictionalists hold that, strictly speaking, mathematical objects do not exist but are rather harmless fictions. Both positions have been ascribed to Aristotle. The ascription of literalism to Aristotle, however, commits Aristotle to the unattractive view that mathematics studies but a small fragment of the physical world; and there is evidence that Aristotle would deny the literalist position that mathematical objects are perceivable. The ascription of fictionalism also faces a difficult challenge: there is evidence that Aristotle would deny the fictionalist position that mathematics is false. I will argue that, in Aristotle’s view, the fiction of mathematics is not to treat what does not exist as if existing but to treat mathematical objects with an ontological status they lack. This form of fictionalism is consistent with holding that mathematics is true.

What is perception doing in mathematical reasoning? To address this question, I discuss the role of perception in geometric reasoning. Perception of the shape properties of concrete diagrams provides, I argue, a surrogate consciousness of the shape properties of the abstract geometric objects depicted in the diagrams. Some of what perception is not doing in mathematical reasoning is also discussed. I take issue with both Parsons and Maddy. Parsons claims that we perceive a certain type of abstract object. Maddy claims (at least at one time claimed) that perception provides the basis for intuition of mathematical sets. 1 Mathematical reasoning with diagrams 2 Do we perceive abstract objects? 3 Do we perceive mathematical sets?

In Mathematical Thought and Its Objects, Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a “nature” than that confers on them.