Inconsistency in mathematics and the mathematics of inconsistency

No one will dispute, looking at the history of mathematics, that there are plenty of moments where mathematics is “in trouble”, when paradoxes and inconsistencies crop up and anomalies multiply. This need not lead, however, to the view that mathematics is intrinsically inconsistent, as it is compatible with the view that these are just transient moments. Once the problems are resolved, consistency (in some sense or other) is restored. Even when one accepts this view, what remains is the question what mathematicians do during such a transient moment? This requires some method or other to reason with inconsistencies. But there is more: what if one accepts the view that mathematics is always in a phase of transience? In short, that mathematics is basically inconsistent? Do we then not need a mathematics of inconsistency? This paper wants to explore these issues, using classic examples such as infinitesimals, complex numbers, and infinity.     

Finite mathematics

A system of finite mathematics is proposed that has all of the power of classical mathematics. I believe that finite mathematics is not committed to any form of infinity, actual or potential, either within its theories or in the metalanguage employed to specify them. I show in detail that its commitments to the infinite are no stronger than those of primitive recursive arithmetic. The finite mathematics of sets is comprehensible and usable on its own terms, without appeal to any form of the infinite. That makes it possible to, without circularity, obtain the axioms of full Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC) by extrapolating (in a precisely defined technical sense) from natural principles concerning finite sets, including indefinitely large ones. The existence of such a method of extrapolation makes it possible to give a comparatively direct account of how we obtain knowledge of the mathematical infinite. The starting point for finite mathematics is Mycielski's work on locally finite theories.I would like to thank Jeff Barrett, Akeel Bilgrami, Leigh Cauman, John Collins, William Craig, Gary Feinberg, Haim Gaifman, Yair Guttmann, Hidé Ishiguro, Isaac Levi, James Lewis, Vann McGee, Sidney Morgenbesser, George Shiber, Sarah Stebbins, Mark Steiner, and an anonymous referee for encouragement and various useful suggestions. The research described in this article and the preparation of the article were supported in part by the Columbia University Council for Research in the Humanities.     

Understanding understanding mathematics

In this paper we look at some of the ingredients and processes involved in the understanding of mathematics. We analyze elements of mathematical knowledge, organize them in a coherent way and take note of certain classes of items that share noteworthy roles in understanding. We thus build a conceptual framework in which to talk about mathematical knowledge. We then use this representation to describe the acquisition of understanding. We also report on classroom experience with these ideas.     

Professor Gasking on mathematics

Book Information Matters of the Mind. Matters of the Mind Lyons William Edinburgh Edinburgh University Press 2001 xxix + 288 Paperback By Lyons William. Edinburgh University Press. Edinburgh. Pp. xxix + 288. Paperback:,     

Wittgenstein's philosophies of mathematics

Wittgenstein's philosophy of mathematics has long been notorious. Part of the problem is that it has not been recognized that Wittgenstein, in fact, had two chief post-Tractatus conceptions of mathematics. I have labelled these the calculus conception and the language-game conception. The calculus conception forms a distinct middle period. The goal of my article is to provide a new framework for examining Wittgenstein's philosophies of mathematics and the evolution of his career as a whole. I posit the Hardyian Picture, modelled on the Augustinian Picture, to provide a structure for Wittgenstein's work on the philosophy of mathematics. Wittgenstein's calculus period has not been properly recognized, so I give a detailed account of the tenets of that stage in Wittgenstein's career. Wittgenstein's notorious remarks on contradiction are the test case for my theory of his transition. I show that the bizarreness of those remarks is largely due to the calculus conception, but that Wittgenstein's later language-game account of mathematics keeps the rejection of the Hardyian Picture while correcting the calculus conception's mistakes.The following abbreviations are used in this article to refer to Wittgenstein's works: WWK: Ludwig Wittgenstein and the Vienna Circle: Conversations Recorded by Friedrich Waismann, ed. B. F. McGuinness, trans. J. Schulte and B. F. McGuinness, Oxford: Basil Blackwell, 1979; CAM I: Wittgenstein's Lectures: Cambridge, 1930–32, ed. D. Lee, Chicago: University of Chicago Press, 1982; CAM II: Wittgenstein's Lectures: Cambridge, 1932–35; ed. A. Ambrose, Chicago: University of Chicago Press, 1982; PG: Philosophical Grammar, ed. R. Rhees, trans. A. Kenny, Oxford: Basil Blackwell, 1974; BIB: The Blue and Brown Books, Oxford: Basil Blackwell, 1958; LFM: Wittgenstein's Lectures on the Foundations of Mathematics: Cambridge, 1939, ed. C. Diamond, Ithaca: Cornell University Press, 1976; RFM: Remarks on the Foundations of Mathematics, ed. G. H. von Wright, R. Rhees, G. E. M. Anscombe, trans. G. E. M. Anscombe, revised ed., Cambridge: MIT Press, 1978; PI: Philosophical Investigations, ed. G. E. M. Anscombe, R. Rhees, trans. G. E. M. Anscombe, New York: Macmillan Company, 1953; Z: Zettel, ed. G. E. M. Anscombe, G. H. von Wright, trans. G. E. M. Anscombe, Berkeley and Los Angeles: University of California Press, 1970.References to PI and Z are to remark number; references to RFM are to part number (Roman numerals) and remark number (Arabic numerals); and references to the other works are to page numbers. As the evolutionary nature of Wittgenstein's work is an important theme of this article, following the abbreviation for the book in the text I have put in brackets the date of the book or the part of the book from which the quotation comes.     

Apriority and applied mathematics

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.I am especially indebted to Michael Friedman for his valuable suggestions and criticisms of this paper. I would also like to thank Mark Wilson, Anil Gupta, and Arthur Fine for their comments and encouragement.     

Syntax-directed discovery in mathematics

It is shown how mathematical discoveries such as De Moivre's theorem can result from patterns among the symbols of existing formulae and that significant mathematical analogies are often syntactic rather than semantic, for the good reason that mathematical proofs are always syntactic, in the sense of employing only formal operations on symbols. This radically extends the Lakatos approach to mathematical discovery by allowing proof-directed concepts to generate new theorems from scratch instead of just as evolutionary modifications to some existing theorem. The emphasis upon syntax and proof permits discoveries to go beyond the limits of any prevailing semantics. It also helps explain the shortcomings of inductive AI systems of mathematics learning such as Lenat's AM, in which proof has played no part in the formation of concepts and conjectures.     

Informal versus formal mathematics

We discuss Kunen’s algorithmic implementation of a proof for the Paris–Harrington theorem, and the author’s and da Costa’s proposed “exotic” formulation for the P = NP hypothesis. Out of those two examples we ponder the relation between mathematics within an axiomatic framework, and intuitive or informal mathematics. The author is Visiting Researcher at IEA/USP, Professor of Communications, Emeritus, at the Federal University in Rio de Janeiro, and a full member of the Brazilian Academy of Philosophy.     

Epistemic injustice in mathematics

We investigate how epistemic injustice can manifest itself in mathematical practices. We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively—we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics—and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept to highlight a potential danger of intellectual enculturation.