The paper aims to establish if Grassmann’s notion of an extensive form involved an epistemological change in the understanding of geometry and of mathematical knowledge. Firstly, it will examine if an ontological shift in geometry is determined by the vectorial representation of extended magnitudes. Giving up homogeneity, and considering geometry as an application of extension theory, Grassmann developed a different notion of a geometrical object, based on abstract constraints concerning the construction of forms rather than on the homogeneity conditions required (...) by the modern version of the theory of proportions. Secondly, Grassmann’s conception of mathematical knowledge will be investigated. Parting from the traditional definition of mathematics as a science of magnitudes, Grassmann considered mathematical forms as particulars rather than universals: the classification of the branches of mathematics was thus based on different operational rules, rather than on empirical criteria of abstraction or on the distinction of different species belonging to a common genus. It will be argued that a different notion of generalization is thus involved, and that the knowledge of mathematical forms relies on the understanding of the rules of generation of the forms themselves. Finally, the paper will analyse if Grassmann’s approach in the first edition of the Ausdehnungslehre should be explained in terms of the notion of purity of method, and if it clashes with Grassmann’s later conventionalism. Although in the second edition the features of the operations are chosen by convention, as it is the case for the anti-commutative property of the multiplication, the choice is oriented by our understanding of the resulting forms: a simplification in the algebraic calculus need not correspond to a simplification in the ‘dimensional’ interpretation of the result of the multiplicative operation. (shrink)

This paper raises and then discusses a puzzle concerning the ontological commitments of mathematical principles. The main focus here is Hume's Principle—a statement that, embedded in second-order logic, allows for a deduction of the second-order Peano axioms. The puzzle aims to put pressure on so-called epistemic rejectionism, a position that rejects the analytic status of Hume's Principle. The upshot will be to elicit a new and very basic disagreement between epistemic rejectionism and the neo-Fregeans, defenders of the analytic status of (...) Hume's Principle, which will provide a new angle from which properly to assess and re-evaluate the current debate. (shrink)

In a recent book C.S. Jenkins proposes a theory of arithmetical knowledge which reconciles realism about arithmetic with the a priori character of our knowledge of it. Her basic idea is that arithmetical concepts are grounded in experience and it is through experience that they are connected to reality. I argue that the account fails because Jenkins’s central concept, the concept for grounding, is inadequate. Grounding as she defines it does not suffice for realism, and by revising the definition we (...) would abandon the idea that grounding is experiential. Her account falls prey to a problem of which Locke, whom she regards as a source of inspiration was aware and which he avoided by choosing anti-realism about mathematics. (shrink)

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. (shrink)

In this article, I discuss Hawthorne’s contextualist solution to Benacerraf’s dilemma. He wants to find a satisfactory epistemology to go with realist ontology, namely with causally inaccessible mathematical and modal entities. I claim that he is unsuccessful. The contextualist theories of knowledge attributions were primarily developed as a response to the skeptical argument based on the deductive closure principle. Hawthorne uses the same strategy in his attempt to solve the epistemologist puzzle facing the proponents of mathematical and modal realism, but (...) this problem is of a different nature than the skeptical one. The contextualist theory of knowledge attributions cannot help us with the question about the nature of mathematical and modal reality and how they can be known. I further argue that Hawthorne’s account does not say anything about a priori status of mathematical and modal knowledge. Later, Hawthorne adds to his account an implausible claim that in some contexts a gettierized belief counts as knowledge. (shrink)

That the history and the philosophy of science have been united in a form of disciplinary marriage is a fact. There are pressing questions about the state of this union. Discourse on a New Method: Reinvigorating the Marriage of History and Philosophy of Science is a state of the union address, but also an articulation of compelling and well-defended positions on strategies for making progress in the history and philosophy of science.

Kant’s account of space as an infinite given magnitude in the Critique of Pure Reason is paradoxical, since infinite magnitudes go beyond the limits of possible experience. Michael Friedman’s and Charles Parsons’s accounts make sense of geometrical construction, but I argue that they do not resolve the paradox. I argue that metaphysical space is based on the ability of the subject to generate distinctly oriented spatial magnitudes of invariant scalar quantity through translation or rotation. The set of determinately oriented, constructed (...) geometrical spaces is a proper subset of metaphysical space, thus, metaphysical space is infinite. Kant’s paradoxical doctrine of metaphysical space is necessary to reconcile his empiricism with his transcendental idealism. (shrink)

How might one explain the reliability of one's a priori beliefs? What if anything is implied about the ontology of a certain realm of knowledge by the possibility of explaining one's reliability about that realm? Very little, or so it is argued here.

Summary According to Kant, arithmetic judgements are not analytic since they are about our practice of operating with figures and things in a certain way. Hence the empiricist thesis that any meaningful assertion is either analytic or synthetic a posteriori seems to be refuted (§§ 1, 2). Using syntax and semantics of truth-conditional logic Frege nevertheless shows that arithmetic can be understood as a system of quasi-analytic sentences speaking about numbers as abstract entities (§§ 3, 4). Axiomatic set theory, however, (...) conceals the connection between (internal) truth-functional arithmetic and our (external) practice of counting and computing (§§ 5–7). — In spite of the insights truth-conditional semantics provides for a non-psychological understanding of mathematical thinking, it is neither a general theory of meaning and analyticity nor a foundation of a general sense-criterion (§§ 8, 9). (shrink)

In "Dynamics of Reason" (2001), Michael Friedman advocates a neo-Kantian perspective for philosophy of science that addresses the problem of scientific change and opposes both Quine's naturalism and Kuhn's relativism. This critical notice of Friedman's book focuses on the "relativized a priori" principles articulated by Friedman. Friedman's arguments against Quine and Kuhn are subsequently evaluated. It is concluded that Friedman succeeds in illustrating deficiencies of Quine's naturalism, however, he fails to sufficiently establish a "rational" basis for theory-choice and, hence, his (...) argument against Kuhn's "arational" view is unsuccessful. (shrink)

Alberto Casullo ("Necessity, Certainty, and the A Priori", Canadian Journal of Philosophy 18, 1988) argues that arithmetical propositions could be disconfirmed by appeal to an invented scenario, wherein our standard counting procedures indicate that 2 + 2 != 4. Our best response to such a scenario would be, Casullo suggests, to accept the results of the counting procedures, and give up standard arithmetic. While Casullo's scenario avoids arguments against previous "disconfirming" scenarios, it founders on the assumption, common to scenario and (...) response, that arithmetic might be independent of standard counting procedures. Here I show, by attention to tallying as the simplest form of counting, that this assumption is incoherent: given standard counting procedures, then (on pain of irrationality) arithmetical theory follows. (shrink)

The computer played an essential role in the proof given by Kenneth Appel and Kenneth Henken of the Four-Color Theorem (4CT).1 First proposed in 1852 by Francis Guthrie, the four color problem is to determine whether four colors are sufficient to color any map on a plane so that no adjacent regions have the same color. Appel and Heken’s proof involves a lemma that a certain ‘avoidable’ set U of configurations is reducible. The proof of this critical lemma requires certain (...) combinatorial checks which are too long to do by hand. The job was done by an IBM 370/168, using over 1200 hours of computer time. In 1977, Appel and Heken, assisted by John Koch, published the proof, and the 4CT has since been considered an established result. No one has seen the entire proof of the reducibility lemma. It was too long to print out; even if it had been, no one would be able to run through it step by step. (shrink)