Michael D. Resnik

Logic: Normative or Descriptive? The Ethics of Belief or a Branch of Psychology?

By a logical theory I mean a formal system together with its semantics, meta-theory, and rules for translating ordinary language into its notation. Logical theories can be used descriptively (for example, to represent particular arguments or to depict the logical form of certain sentences). Here the logician uses the usual methods of empirical science to assess the correctness of his descriptions. However, the most important applications of logical theories are normative, and here, I argue, the epistemology is that of wide reflective equilibrium. The result is that logic not only assesses our inferential practice but also changes it. I tie my discussion to Thagards views concerning the relationship between psychology and logic, arguing against him that psychology has and should have only a peripheral role in normative (and most descriptive) applications of logic.

Mathematical Knowledge and Pattern Cognition

This paper is concerned with the genesis of mathematical knowledge. While some philosophers might argue that mathematics has no real subject matter and thus is not a body of knowledge, I will not try to dissuade them directly. (One might do so by developing a theory of meaning and truth, which together with observations from the sociology of mathematics would imply that mathematical knowledge exists. Mathematicians do seem to make knowledge claims, so all one needs is a theory which shows that here at least appearances are real.) I shall not attempt such a refutation because it seems clear to me that mathematicians do know such things as the Mean Value Theorem, The Fundamental Theorem of Arithmetic, Godels Theorems, etc. Moreover, this is much more evident to me than any philosophical view of mathematics I know of including my own. So I am going to take mathematics as my starting point. Granted the existence of mathematical knowledge, the major problem it poses is that it is a case of the acquisition of objective knowledge and belief with no apparent interaction with an external subject matter. This phenomenon or apparent phenomenon sets mathematics apart from the rest of science and grants a great deal of initial plausibility to the view that mathematics is a priori. On the other hand, the same data make it seem implausible that the epistemology of mathematics is a special case of the general epistemology of science. There are a variety of ways to react to this situation. One can, as Frege and Godel have, posit a special faculty which enables us to perceive external but abstract mathematical objects.1 Or like Brouwer and

On the philosophical significance of consistency proofs

ConclusionWe have seen that despite Fefermans results Gödels second theorem vitiates the use of Hilbert-type epistemological programs and consistency proofs as a response to mathematical skepticism. Thus consistency proofs fail to have the philosophical significance often attributed to them.This does not mean that consistency proofs are of no interest to philosophers. We know that a ‘non-pathological’ consistency proof for a system S will use methods which are not available in S. When S is as strong a system as we are willing to entertain seriously then a consistency proof for it will yield no epistemological gain. But in other cases philosophers might argue that the proof uses methods which are merely different rather than stronger than those available in the system in question. This claim has been made, for example, in the case of the constructive consistency proofs for elementary number theory. Similar philosophical investigations can be made on relative consistency proofs, since these differ from each other in the principles they employ. For example, most relative consistency proofs can be carried out within elementary number theory, but without using the theory of real numbers, no one has been able to prove the consistency of Quines ML relative to that of his NF.What about the consistency of all mathematics or of some strong system for set theory? How do we answer the skeptic? Since here a convincing proof is not possible, we have established that the skeptic demands too much. We cannot be certain that our axioms are free from contradiction and must treat them as hypotheses which may be abandoned or modified in the face of further mathematical experience. This attitude is taken by many foundational workers who also go on to voice opinions about thelikelihood that various systems are consistent. Since these opinions are variously supported by appeals to the clarity of the mathematical concept formalized, the existence or non-existence of ‘weird’ models for the system and actual empirical experience with the system, this is surely a fruitful area for philosophical research.

Holistic Realism: A Response to Katz On Holism and Intuition

In his book, Realistic Rationalism (henceforth RR), and his article, “Mathematics and Metaphilosophy”(henceforth MM), Jerrold Katz develops and defends a philosophy of mathematics that is realist in ontology and rationalist in epistemology. On his view, mathematics contains a body of necessary truths about certain abstract entities that are known a priori though the exercise of reason. Katz believes that his epistemology answers the familiar challenge, so well articulated by Paul Benacerraf, to mathematical realists that they explain how beings like us can acquire knowledge about causally inert, nonspatial, nontemporal, mathematical objects. Katz thinks the key to responding successfully to this challenge is to develop a “no contact epistemology,” that is, one that posits no physical process connecting us to mathematical objects. By basing his account of mathematical knowledge upon reason and intuition, Katz believes that he has avoided mysticism, such as that associated with Plato’s epistemology of recollection, and has given a deeper account than, say, Frege’s view that we grasp or apprehend abstract objects or Godel’s brief comparison of mathematical intuition with ordinary sense perception. Katz thinks that the only other no contact epistemology worthy of consideration is W. V. Quine’s confirmational holism. But he argues that this falls prey to his own Revisability Paradox and is inconsistent (RR, 72–4; MM, 375).

II. Frege as Idealist and then Realist

Michael Dummett argued that Frege was a realist while Hans Sluga countered that he was an objective idealist in the rationalist tradition of Kant and Lotze. Sluga ties Freges idealism to the context principle which he argues Frege never gave up. It is argued that Sluga has correctly interpreted the pre‐1891 Frege while Dummett is correct concerning the later period. It is also claimed that the context principle was dropped prior to 1891 to be replaced by the doctrine of unsaturated entities.

Against Logical Realism

This paper argues against Logical Realism, in particular against the view that there are facts of matters of logic that obtain independently of us, our linguistic conventions and inferential practices. The paper challenges logical realists to provide a non-intuition based epistemology, one which would be compatible with the empiricist and naturalist convictions motivating much recent anti-realist philosophy of mathematics.

On Skolem's Paradox

1. All natural numbers are in the domain. 2. The only other elements of the domain are sets of natural numbers. 3. The denotations of 0, 1, 2, ... are 0, 1, 2, ... respectively; the denotation of N is the set of all natural numbers, and the denotation of S is the set of all sets of natural numbers in the domain; the denotation of Î is the relation of membership between numbers and sets of numbers. Finally, the denotation of J is the function J extended to give some arbitrary value—say 17—for arguments that are not both numbers (that is, one or both of which are sets).

Between Mathematics and Physics

Nothing has been more central philosophy of mathematics than the distinction between mathematical and physical objects. Yet consideration of quantum particles shows the inadequacy of the popular spacetime and causal characterizations of the distinction. It also raises problems for an assumption used recently by Field, Hellman and Horgan, namely, that the mathematical realm is metaphysically independent of the physical one.

Ontology and logic: remarks on hartry field's anti-platonist philosophy of mathematics

In Science without numbers Hartry Field attempted to formulate a nominalist version of Newtonian physics—one free of ontic commitment to numbers, functions or sets—sufficiently strong to have the standard platonist version as a conservative extension. However, when uses for abstract entities kept popping up like hydra heads, Field enriched his logic to avoid them. This paper reviews some of Fields attempts to deflate his ontology by inflating his logic.

Beliefs About Mathematical Objects

As a mathematical platonist, I hold that mathematical objects are causally inert and exist independently of us and our mental lives. This obliges me to explain how we can refer to such alien creatures and acquire knowledge and beliefs about them.1 This paper is a piece of that larger project.