Online research in philosophy

Mathematical axioms have traditionally been thought of as obvious or self-evident truths, but current set theoretic work in the search for new axioms belies this conception. This raises epistemological questions about what other forms of justification are possible, and how they should be judged.

It is commonly suggested that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical hypothesis is absolutely undecidable, then it is indeterminate. I shall argue that no such understanding makes all of (a) -- (c) plausible. However, I will identify one understanding of the phrase “absolutely undecidable” which makes both (a) and (c) plausible. This suggests that a new style of mathematical antirealism deserves attention -- one that does not depend on familiar ontological or epistemological concerns. The key idea behind this view is that typical axioms of mathematics are indeterminate because they are relevantly similar to CH.

Recent years have seen a growing acknowledgement within the mathematical community that mathematics is cognitively/socially constructed. Yet to anyone doing mathematics, it seems totally objective. The sensation in pursuing mathematical research is of discovering prior (eternal) truths about an external (abstract) world. Although the community can and does decide which topics to pursue and which axioms to adopt, neither an individual mathematician nor the entire community can choose whether a particular mathematical statement is true or false, based on the given axioms. Moreover, all the evidence suggests that all practitioners work with the same ontology. (My number 7 is exactly the same as yours.) How can we reconcile the notion that people construct mathematics, with this apparent choice-free, predetermined objectivity? I believe the answer is to be found by examining what mathematical thinking is (as a mental activity) and the way the human brain acquired the capacity for mathematical thinking.

We present a range of mathematical theorems whose proofs require unexpectedly strong logical methods, which in some cases go well beyond the usual axioms for mathematics.

In a recent article in this journal Phil. Math., II, v.4 (1989), n.2, pp.? ?] J. Fang argues that we must not be fooled by A.J. Ayer (God rest his soul!) and his cohorts into believing that mathematical knowledge has an analytic a priori status. Even computers, he reminds us, take some amount of time to perform their calculations. The simplicity of Kant's infamous example of a mathematical proposition (7+5=12) is "partly to blame" for "mislead[ing] scholars in the direction of neglecting the temporal element"; yet a brief instant of time is required to grasp even this simple truth. If Kant were alive today, "and if he had had a little more mathematical savvy", Fang explains, he could have used the latest example of the largest prime number (391,581 x 2 216,193 - 1) as a better example of the "synthetic a priori" character of mathematics. The reason Fang is so intent upon emphasizing the temporal character of mathematics is that he wishes to avoid "the uncritical mixing of ... a theology and a philosophy of mathematics." For "in the light of the Computer Age today: finitism is king!" Although Kant's aim was explicitly "to study the 'human' ... faculty", Fang claims that even he did not adequatley emphasize "the clearly and concretely distinguishable line of demarcation between the human and divine faculties.".

Visual thinking -- visual imagination or perception of diagrams and symbol arrays, and mental operations on them -- is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual thinking in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. Drawing from philosophical work on the nature of concepts and from empirical studies of visual perception, mental imagery, and numerical cognition, Giaquinto explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. He shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. Visual Thinking in Mathematics reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences. Clear and concise throughout, it will appeal to scholars and students of philosophy, mathematics, and psychology, as well as anyone with an interest in mathematical thinking.

I argue that we need not accept Quine's holistic conception of mathematics and empirical science. Specifically, I argue that we should reject Quine's holism for two reasons. One, his argument for this position fails to appreciate that the revision of the mathematics employed in scientific theories is often related to an expansion of the possibilities of describing the empirical world, and that this reveals that mathematics serves as a kind of rational framework for empirical theorizing. Two, this holistic conception does not clearly demarcate pure mathematics from applied mathematics. In arguing against Quine, I present a formal account of applied mathematics in which the mathematics employed in an empirical theory plays a role that is analogous to the epistemological role Kant assigned synthetic a priori propositions. According to this account, it is possible to insulate pure mathematics from empirical falsification, and there is a sense in which applied mathematics can also be labeled as a priori.

The distinction between analytic and synthetic propositions, and with that the distinction between a priori and a posteriori truth, is being abandoned in much of analytic philosophy and the philosophy of most of the sciences. These distinctions should also be abandoned in the philosophy of mathematics. In particular, we must recognize the strong empirical component in our mathematical knowledge. The traditional distinction between logic and mathematics, on the one hand, and the natural sciences, on the other, should be dropped. Abstract mathematical objects, like transcendental numbers or Hilbert spaces, are theoretical entities on a par with electromagnetic fields or quarks. Mathematical theories are not primarily logical deductions from axioms obtained by reflection on concepts but, rather, are constructions chosen to solve some collection of problems while fitting smoothly into the other theoretical commitments of the mathematician who formulates them. In other words, a mathematical theory is a scientific theory like any other, no more certain but also no more devoid of content.

In the article the problem of independence in mathematics is discussed. The status of the continuum hypothesis, large cardinal axioms and the axiom of constructablility is presented in some detail. The problem whether incompleteness is really relevant for ordinary mathematics and for empirical science is investigated. Another aim of the article is to give some arguments for the thesis that the problem of reliability and justification of new axioms is well-posed and worthy of attention. In my opinion, investigations concerning the status of independent sentences give insight into our understanding of mathematical concepts, of mathematical knowledge and of the role of mathematics in empirical science.

There has been a significant shift in the discussion of a priori knowledge. The shift is due largely to the influence of Quine. The traditional debate focused on the epistemic status of mathematics and logic. Kant, for example, maintained that arithmetic and geometry provide clear examples of synthetic a priori knowledge and that principles of logic, such as the principle of contradiction, provide the basis for analytic a priori knowledge. Quine’s rejection of the analytic-synthetic distinction and his holistic empiricist account of mathematic and logical knowledge undercut the traditional defenses of the a priori in two ways. First, one could no longer defend the view that mathematical and logical knowledge is a priori solely by rejecting Mill’s inductive empiricism. Moreover, holistic empiricism proved to be a more challenging position to refute than inductive empiricism. Second, the rejection of the analytic-synthetic distinction blocked an alternative defense of the a priori status of mathematics and logic that appealed to their alleged analyticity.