AMS/IMU Special session on history and philosophy of mathematics, Bar Ilan University, 16 and 17 june '14

Mon. 11:15-13:15, Mon. 16:00-19:10, Tue. 11:30-13:30

List of abstracts by author name in alphabetical order (Bascelli, Błaszczyk, Ehrlich, Herzberg, Kanovei, Knobloch, Marquis, Reeder, Sherry, Steiner)
Tiziana Bascelli, Italy

Geometric structures at the foundations of dynamics: Galileo's and Torricelli's science of motion

Abstract. Galileo presented his pioneering theory about local motion in the book Two New Sciences (1638), in the form of a geometric treatise. At the beginning of the book he presents the principle and definition from which, later on, he deduces properties of motion. In order to prove that a body falling from rest moves downward with a uniformly accelerated motion he has to introduce a new concept of speed and describe time, space, and speed by means of geometry. The application of geometry to objects that belong to natural philosophy implies that the main properties of both, geometric objects and those of the physical world are homogeneous. I will aim to show to which extent geometric objects match categories of natural philosophy. Galileo's principle of uniformly accelerated motion was not evident to his contemporaries, who questioned whether it was valid. Evangelista Torricelli supported Galileo's view but only after inverting the roles of the principle of accelerated motion and main theorems. He also developed further Galileo's scientific achievements applying principles of mechanics. I will aim to show Torricelli's proof of existence of a special acceleration for every solid body, the one we now call 'gravity'.

Piotr Błaszczyk, Institute of Mathematics, Pedagogical University of Cracow, Poland

Continuum versus continuity. Zeno and his modern rivals revisited

Abstract. The thesis of the Achilles paradox is clear and precise: "in a race the quickest runner can never overtake the slowest" (Aristotle, Physics), "it is impossible for him [Achilles] to overtake the tortoise when pursuing it" (Simplicius, On Aristotle's Physics 6). However, in modern interpretations the question whether Achilles can overtake the tortoise is turned into the question whether he can catch up with it: "Achilles never catches the tortoise" (Black 1951), "Achilles will never catch up with [the tortoise]" (Vlastos 1967), "[Achilles] never catches the tortoise" (Huggett 2010). Moreover, the standard solution to the paradox consists in determining the "rendezvous point", where Achilles "comes abreast with the tortoise" (Grünbaum 1967). We show that there is a hidden assumption both in the very statement of the paradox and its standard solution that to overtake the tortoise Achilles has to catch up with it. We provide a mathematical model for the race such that Achilles overtakes the tortoise and there is no point at which he catches up with it. In our model the space is Euclidean, i.e. it represents the ancient Greek continuum, the race (movement) is continuous, and we can also represent in our model the standard solution to the Achilles paradox based on the modern notion of limit.

Philip Ehrlich, Department of Philosophy, Ohio University Athens, OH 45701

Cantorian and non-Cantorian Theories of Finite, Infinite and Infinitesimal Numbers and the Unification Thereof

Abstract. In addition to Cantor's well-known systems of infinite cardinals and ordinals, there were a variety of other systems of actual infinite numbers that emerged in the decades bracketing the turn of the twentieth-century. Two grew out of the work of Paul du Bois-Reymond [1870-71, 1875, 1877, 1882], Otto Stolz [1883], Felix Hausdorff [1909] and G. H. Hardy [1910] on the rates of growth of real functions, and the others emerged from the pioneering investigations of non-Archimedean ordered algebraic and geometric systems by Giuseppe Veronese [1892], Tullio Levi-Civita [1892, 1898], David Hilbert [1899] and Hans Hahn [1907]. Unlike Cantor's systems, which solely embrace finite numbers alongside his familiar infinite numbers, the other just-said non-Cantorian number systems, like the more recent hyperreal number systems associated with Abraham Robinson's nonstandard approach to analysis, embody finite, infinite and infinitesimal numbers.
In [Ehrlich 2012], we show how the above-mentioned Cantorian and non-Cantorian number systems admit a striking unification in the author's [Ehrlich 2001] algebraico-tree-theoretic approach to J. H. Conway's system of surreal numbers. Building on the above, in this paper we will provide introductions to the aforementioned non-Cantorian theories of the finite, infinite and infinitesimal that emerged in the decades bracketing the turn of the twentieth-century, explain the motivation for their introduction, outline the roles these and related theories play in contemporary mathematics and discuss the relations between these theories and the better-known theories of Cantor and Robinson that emerge from the just-said unification. It is the author's hope that by drawing attention to the spectrum of theories of the infinite and the infinitesimal that have emerged from non-Archimedean mathematics since the latter decades of the 19th century, it will become clear that the standard 20th-century histories and philosophies of the actual infinite and the infinitesimal that are motivated largely by Cantor's theory of the infinite and by non-standard analysis are not only limited in scope but are inspired by an account of late 19th- and early 20th-century mathematics that is as mathematically myopic as it is historically flawed.
Philip Ehrlich, Number Systems with Simplicity Hierarchies: A Generalization of Conway's Theory of Surreal Numbers, The Journal of Symbolic Logic 66 (2001), pp. 1231-1258.
Philip Ehrlich, The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small, The Bulletin of Symbolic Logic 18 (2012), pp. 1-45

Frederik Herzberg, Institute for Mathematical Economics, Universitat Bielefeld, Bielefeld, Germany

Epistemic justification, formally: Alternatives to foundationalism and an exercise in mathematical philosophy

Abstract. An important problem in epistemology concerns the question of finding necessary and/or sufficient conditions for beliefs to be justified (so-called "doxastic justification"). It has been commonly assumed, based on an influential interpretation of Aristotle's Organon, that non-skeptical positions with respect to doxastic justification fall within one of three categories: foundationalism, coherentism, and infinitism. Foundationalism claims that regresses of justification always have to end in some -- foundational -- belief; coherentism denies that the structure of doxastic justification is linear; infinitism allows for infinite regresses of justification. For a long time, infinitism has not been considered a serious contender. In recent years, Peter Klein (1998ff) and Jeanne Peijnenburg (2007ff) have argued the case for infinitism from a traditional analytic and Bayesian epistemological perspective, respectively. Coherentism, however, has been in a defensive mood for more than two decades since impossibility theorems for coherence measures appeared and one of its chief proponents, Laurence BonJour, abandoned that position.
We shall show that a rigorous mathematical analysis entails: (i) that there is common ground between weak versions of foundationalism, coherentism, infinitism, (ii) that common arguments against infinitism fail on a large scale; (iii) that a graded, BonJourian notion of coherence admits a thoroughly probabilistic formalisation. For the refutation of a common argument against infinitism (part of point (ii)), we shall prove a general consistency theorem for regresses of probabilistic doxastic justification, using the Loeb measure construction known from nonstandard probability theory.
These findings possess plausibility for the epistemology of mathematics: Foundationalism is an account of full (as opposed to partial) justification of belief in mathematical propositions given an axiomatic foundation; coherentism accounts for partial justification (based, e.g., on competent authority); and infinite regress of justification seems an appropriate description of arguments regarding competing axiomatic foundations for mathematics.

Vladimir Kanovei (Moscow, IPPI)

On Euler's summation of the alternating factorial series

Abstract. Euler's method of summation of the series 1 - 2 + 6 - 24 + ..., intuitively appealing but hardly rigorous according to commonly accepted mathematical standards, reappeared in the middle of the 20th Century from a very different standpoint. In honor of Euler, his priority has to be reestablished.

Eberhard Knobloch, Institut fur Philosophie, Wissenschaftstheorie, Wissenschafts- und Technikgeschichte, TU Berlin, Germany

From Archimedes to Kepler: Analogies and the dignity of mathematics

Abstract. Archimedes used analogies in the context of discovery. Two examples will illustrate this method: the way in which he found the quantitative value of the surface of a sphere and the use of indivisibles in the strict sense of the word in order to calculate areas and volumes. Kepler admired Archimedes. He wrote his "New stereometry" explicitly referring to the ancient geometer. He liked analogies as his guide as he emphasized, especially when he calculated the volumes of solids. Without knowing Archimedes's "Approach" (falsely called "Method") he used a similar method applying not well defined infinitely small quantities. The most interesting and most difficult problem is the calculation of the volume of an ideal mathematical apple.

Jean-Pierre Marquis, Department of philosophy, University of Montreal, Montreal, Quebec, Canada.

The Evolution of the Idea of Pure Mathematics

Abstract. In this presentation, I will focus on the evolution of the notion of pure mathematics, especially in the 19th and 20th century. I will try to exhibit what I take to be various shifts in the notion itself, from the opposition between pure/mixed mathematics and pure/applied mathematics, to a general notion of pure mathematics based on the axiomatic and abstract method, which characterized prominently the 20th century. The latter notion lead to a view that the applications of mathematics to the real world was no less than a miracle. This latter claim could not have been made and was not even conceivable even at the beginning of the 20th century. I will conclude with remarks on the present situation suggesting that the notion is shifting again along two main axes. The first one has to do with endo-applications of mathematics to itself, via various mathematical bridges constructed in the last fifty years or so, thus bringing a difierent unity to mathematics and a difierent way of thinking of applications itself. The second axis is the development of categorical frameworks that bring at the same time a higher level of abstraction and a different way of thinking about applications.

Patrick Reeder, Kenyon College

Title: Locating success and failure in applying Nonstandard Analysis to Euler's Introductio

Abstract: In Leonhard Euler's seminal work Introductio in Analysin Infinitorum (1748), he readily used infinite numbers and infinitesimals in many of his proofs. We aim to reformulate a group of proofs from the Introductio using concepts and techniques from Abraham Robinson's celebrated Nonstandard Analysis (NSA). We will specifically examine Euler's proofs of the Euler formula, the divergence of the harmonic series and---time permitting---the Wallis product. We argue that NSA possesses the tools to provide appropriate proxies of some--but not all--of the inferential moves found in the Introductio. Each of these proofs relies upon familiar Taylor-MacLaurin series representations of functions like sine, cosine and the natural logarithm. We will show that once we assume the Taylor-MacLaurin series representation, the proofs glide straight through NSA. Now, Euler's own proofs for the Taylor-MacLaurin series also appear in the Introductio. What is striking is that every single one breaks down within NSA. I hope to examine these breakdowns and then provide some (at least preliminary) diagnostic discussion as to why NSA trips over Euler's proofs of these historically significant infinite series.

David Sherry, Northern Arizona University

Leibniz and the Syncategorematic Infinite

Abstract. Leibniz generally conceived of infinitesimals as fictions. I dispute the currently fashionable opinion that Leibniz's infinitesimals are best understood as logical fictions, fictions eliminable by paraphrase. This so-called syncategorematic conception of infinitesimals is present in Leibniz's texts, but there is an alternative, formalist account of infinitesimals there too. I argue that the formalist account makes better sense of Leibniz's frequent analogy between infinitesimals and imaginary roots. Moreover, the formalist interpretation fits better with Leibniz's deepest philosophical convictions.

Mark Jay Steiner, Hebrew Univesity

Lecture 1: "The Silent Revolution in the Philosophy of Ludwig Wittgenstein, 1937"

Abstract. It is customary to distinguish the early Wittgenstein of the Tractatus from the later Wittgenstein of the "Philosophical Investigations." But I believe a revolution of a similar magnitude occurred in the middle of the "later" period of Wittgenstein, August 1937, which included a new approach to the philosophy of mathematics. This approach is a change in the way Wittgenstein perceives the relationship between mathematics and its canonical applications. Instead of seeing mathematical theorems as rules of "grammar" governing mathematical language, he suddenly began to see them as rules governing human activities like counting and measuring. Mathematical theorems became "supervenient" upon these activities, which eliminated the "problem" of the applicability of mathematics.
Lecture 2: "Getting More out of Mathematics than we Put In"
Abstract. Mathematical concepts often have the purpose of facilitating calculation in physics. (Examples: imaginary exponentials instead of trigonometric functions; potentials instead of fields.) Often the increased facility of calculation comes at a price: the mathematical formalism brings in superfluous information (e.g. imaginary solutions). This superfluous information often turns out not to be superfluous at all--on the contrary, it ends up describing phenomena not even known at the time. The lecture will follow the attempts of mathematicians to describe rotations in 3-dimensional space, and the amazing surplus benefits these attempts led to.

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