Emily Grosholz

The growth of mathematical knowledge

Acknowledgments. Introduction. Notes on Contributors. Part I: The Question of Empiricism. The Role of Scientific Theory and Empirical Fact in the Growth of Mathematical Knowledge. 1. Knowledge of Functions in the Growth of Mathematical Knowledge J. Hintikka. Huygens and the Pendulum: From Device to Mathematical Relation M.S. Mahoney. 2. An Empiricist Philosophy of Mathematics and Its Implications for the History of Mathematics D. Gillies. The Mathematization of Chance in the Middle of the 17th Century I. Schneider. Mathematical Empiricism and the Mathematization of Chance: Comment on Gillies and Schneider M. Liston. 3. The Partial Unification of Domains, Hybrids, and the Growth of Mathematical Knowledge E. Grosholz. Hamilton-Jacobi Methods and Weierstrassian Field Theory in the Calculus of Variations C. Fraser. 4. On Mathematical Explanation P. Mancosu. Mathematics and the Reelaboration of Truths F. de Gandt. 5. Penrose and Platonism M. Steiner. On the Mathematics of Spilt Milk M. Wilson. Part II: The Question of Formalism. The Role of Abstraction, Analysis, and Axiomatization in the Growth of Mathematical Knowledge. 1. The Growth of Mathematical Knowledge: An Open World View C. Cellucci. Controversies about Numbers and Functions D. Laugwitz. Epistemology, Ontology, and the Continuum C. Posy. 2. Tacit Knowledge and Mathematical Progress H. Breger. The Quadrature of Parabolic Segments 1635-1658: A Response to Herbert Breger M.M. Muntersbjorn. Mathematical Progress: Ariadnes Thread M. Liston. Voir-Dire in the Case of Mathematical Progress C. Mclarty. 3. The Nature of Progress in Mathematics: The Significance of Analogy H. Sinaceur. Analogy and the Growth of Mathematical Knowledge E. Knobloch. 4. Evolution of the Modes of Systematization of Mathematical Knowledge A. Barabashev. Geometry, the First Universal Language of Mathematics I. Bashmakova, G.S. Smirnova. Part II: The Question of Progress. Criteria for and Characterizations of Progress in Mathematical Knowledge. 1. Mathematical Progress P. Maddy. Some Remarks on Mathematical Progress from a Structuralists Perspective M.D. Resnik. 2. Scientific Progress and Changes in Hierarchies of Scientific Disciplines V. Peckhaus. On the Progress of Mathematics S. Demidov. Attractors of Mathematical Progress: The Complex Dynamics of Mathematical Research K. Mainzer. On Some Determinants of Mathematical Progress C. Thiel.

The Partial Unification of Domains, Hybrids, and the Growth of Mathematical Knowledge

Relations among distinct areas of mathematical activity are most commonly discussed in terms of the reduction of axiomatized theories, where reduction is defined to be the deductive derivation of the axioms of the reduced theory as theorems of the reducing theory. Such derivation requires that the characteristic vocabulary of the reduced theory be redefined in terms of the vocabulary of the reducing theory; these definitions are called bridge laws. In this century, philosophers of mathematics have discussed relations among arithmetic, geometry, predicate logic, and set theory in these terms, and have claimed variously that geometry may be reduced to arithmetic, arithmetic to predicate logic, and arithmetic and geometry to set theory. Their arguments run parallel to those made by philosophers of science, who claim variously that biology may be reduced to chemistry and chemistry to physics, and that within physics, classical mechanics may be reduced to some combination of relativity theory and quantum mechanics.

Two Episodes in the Unification of Logic and Topology

The aims of this essay are twofold. In the first place I would like to exhibit a twentieth-century instance of an important pattern of mathematical reasoning which I have discussed elsewhere in its seventeenth century manifestations (Grosholz [1980] and [1984]). My claim is that growth in mathematical knowledge often results when two or more distinct but related fields are unified by the hypothesis of a partial structural analogy, which allows for the combination of their resources in the solution and discovery of problems. Second, I want to argue that the particular conjunction of logic and topology reveals much of interest about the status of logic as a mathematical discipline. What is the relation of logic to the rest of mathematics? The answer to this question depends upon how one thinks of logic. Among philosophers who would say that it is the canon of the laws of thought, two positions can be distinguished. The deductivist claims that logic provides the inferential patterns we use, independent of what we are reasoning about. So, for example, the first order theory of a mathematical field will consist of axioms couched in logical and extra-logical vocabulary, and purely logical rules of inference to specify the connections between axioms and theorems. Viewed as a canon of inferential patterns, logic has no subject matter of its own. The reductionist adds to the deductivist position the further claim that extralogical vocabulary can be translated into purely logical vocabulary, and that all mathematical principles become theorems of logic as Quine does in his [1953]. If logic has no subject matter, then on this account neither does mathematics; this is the nominalist tack taken by, for example, Charles Chihara [1973]. However, what emerges here is a second role for logic, as a canon of representations for other mathematical objects and procedures which will exhibit their conceptual complexity. When the deductivist tries to prove meta-theorems about inferential patterns, and when the reductionist wonders why logic takes on this peculiar representative function, logic itself emerges as a special subject matter.

Reference and Analysis: The Representation of Time in Galileo, Newton, and Leibniz

Productive scientific discourse must both search for conditions of intelligibility of the object under investigation, and it must enable successful reference. Representations apt for analysis and those apt for reference are typically not the same, and the resultant discourse may thus be heterogeneous and multivalent, a fact missed by philosophers who, like logicians, equate discursive homogeneity with rationality. I show this to be the case in texts where Galileo, Newton and Leibniz investigate the nature of time, and shed new light on the debate about time and space between Clarke (Newton’s spokesman) and Leibniz.

The Use of Models in Petroleum and Natural Gas Engineering

This essay is an inquiry into the formulation of models in the science of petroleum and natural gas engineering. The underlying questions of this essay how adequate some of the fundamental models of this science really are, and what assumptions have been made as the models were created. Our claim is that a good account of the adequacy of models must be strongly pragmatist, for these questions cannot be answered properly without strict attention to human purposes. These purposes include not only the search for a better understanding of geological formations, their natural history and structure, but also classroom instruction for university students, and economically feasible extraction of petroleum and natural gas. These models include machines as well as natural formations, and so too raise the interesting question of how we (pragmatically) model machines. We claim that many of the distortions and over-simplifications in these models are in fact intentional and useful, when we examine the models in light of their pragmatic aims.

Two Leibnizian manuscripts of 1690 concerning differential equations

Abstract Leibniz was very interested in developing techniques for the solution of differential equations. In 1690 he elaborated two manuscripts in which he employed the technique of separating variables. Thus he had to evaluate the logarithm of negative numbers. The present article consists mainly of a critical edition, English translation, and a commentary on these two interesting manuscripts.

A NEW VIEW OF MATHEMATICAL KNOWLEDGE

Philosophers of mathematics have lately begun to realise that the reformation in philosophy of science sparked by Thomas Kuhns The Structure of Scientific Revolutions has bearing on their own discipline. Kuhn and his confreres argue that scientific rationality is not just a matter of the inferential relations holding among propositions of a formalised scientific theory, but includes as well practices and patterns of reasoning which extend, revise and reshape scientific domains. This shift in orientation invokes the more general reflection that causal, genetic factors are as significant for epistemology as logical structure. But philosophy of mathematics has been slow to assimilate such insights. First of all, mathematics, with the demise of theology, has seemed to be the bastion of a priori knowledge. Secondly, philosophers of mathematics are generally trained in philosophy and logic rather than mathematics. Thus they have been too willing to accept the dictum that logic (and set theory) are the keys to mathematical knowledge, and have spent most of their time arguing about formalisations of various parts of mathematics in the language of predicate logic. Philip Kitcher, by contrast, is widely educated in both mathematics and philosophy, and has done extensive independent research in the history of mathematics. Long an active participant in contemporary debates on epistemology, he saw clearly the need for a seachange in philosophy of mathematics and began to work out a position, first in an impressive assortment of articles, and now in the more durable form of a book. The publication of Kitchers The Nature of Mathematical Knowledge is an event of great importance for philosophy of mathematics. And the book is not only instructive, but a pleasure to read. Kitchers prose style is clean, colourful and lucid; his even-handed use of gender with respect to pronouns is also welcome. He knows how to debate, taking the opposition seriously and revising or suspending his own opinions when necessary. He continually anticipates and parries possible objections, and explains dense or difficult passages by examples and reformulations. His arguments are models of clarity. The first four chapters of the book constitute an attack on mathematical apriorism, the philosophical strategy of most philosophers of mathematics who have attempted to account for the distinctive features of mathematical

Reuben Hersh on the growth of mathematical knowledge: Kant, geometry, and number theory

In his reflective writings about mathematics, Reuben Hersh has consistently championed a philosophy of mathematical practice. He argues that if we pay close attention to what mathematicians really do in their research, as they extend mathematical knowledge at the frontier between the known and the conjectured, we see that their work does not only involve deductive reasoning. It also includes plausible reasoning, “analytic” reasoning upward that seeks the conditions of the solvability of problems and the conditions of the intelligibility of mathematical things. We use, he argues, “our mental models of mathematical entities, which are culturally controlled to be mutually congruent within the research community. These socially controlled mental models provide the much-desired “semantics” of mathematical reasoning” (Hersh 2014b, p. 127). Every active mathematician is familiar with a large swathe of established mathematics, “an intricately interconnected web of mutually supporting concepts, which are connected both by plausible and by deductive reasoning,” that include “concepts, algorithms, theories, axiom systems, examples, conjectures and open problems,” and models and applications. Thus, “the body of established mathematics is not a fixed or static set of statements. The new and recent part is in transition” (Ibid, pp. 131–2).

Analysis and Reference in the Study of Astronomical Systems

This chapter examines the use of reference and analysis in the study of astronomical systems from Newton to the present day. I begin by noting that Bas van Fraassen pays more attention to theoretical models that do the work of analysis, while Nancy Cartwright and Margaret Morrisson pay more attention to models where the relevant relation is not satisfaction (as between a meta-language and an object-language), but representation (as between a discursive entity and a thing that exists independent of discourse). I then track the use of both kinds of models in Newton’s Principia, Books I and III, and find strategies of juxtaposition, superposition, and unification. In the era after Newton, problems of analysis were addressed by Euler, Lagrange, Laplace and Hamilton, while problems of reference came to the fore in the empirical work of Herschel and Rosse. The tension between reference and analysis also appears in the debates between Hubble and Zwicky, and in the work of Vera Rubin.

The Representation of Time in the 17th Century

The foregoing arguments apply not only to mathematics but also to the sciences. Here I review the innovative mathematical representations of time in the work of Galileo, Descartes and Newton, and then turn to the debate over whether time is absolute (to be defined analytically) or relative (to be defined referentially) between Newton and his mouthpiece Clarke, and Leibniz. The detailed examination of Leibniz’s treatment of time is also a re-consideration of the methodology of this book, since it was inspired by Leibniz. How shall we think about the ways in which the two kinds of discourse that record empirical compilation and theoretical analysis may be combined in science? Leibniz calls on the Principle of Sufficient Reason to regulate a science that must be both empirical and rationalist, aiming to correlate precise empirical description with the abstract conception of science more geometrico. He encourages us to work out our sciences through successive stages, moving back and forth between concrete taxonomy and abstract systematization.