# MWPMW 18

The eighteenth annual Midwest PhlMath Workshop (MWPMW 18) will meet on the University of Notre Dame campus Saturday, October 14th and Sunday, October 15th, 2017.

If you would like to give a talk, please email a pdf of your talk or a substantial summary of it to Paddy (pblanche@nd.edu), Tim (tbays@nd.edu), Curtis (cfranks@nd.edu) and me (mdetlef1@nd.edu).

We would like to have all proposals for talks by September 6th so that we can set the program by mid September.

Talks should be 40-45 minutes in length, with 15--20 minutes left for discussion.

This year, we're pleased to have Dr. Karine Chemla (CNRS, SPHERE, UMR 7291, Université de Paris 7-Diderot), Prof. William Tait (Philosophy, University of Chicago) and Prof. Mark Wilson (Philosophy, University of Pittsburgh) join us as featured speakers.

Please click the link to the right to get to the workshop's registration form. Your completing this in a timely manner will aid organization.

We look forward to having you with us at the workshop in October.

Please check these pages for further updates.

Note: For medical emergencies on campus at any time of day or night, dial 1-5555 or 911 from any campus phone, or 574-631-5555 from a cell phone.

## Program & Schedule

All sessions will be in room 101 of the Jordan Hall of Science. Both lunch and dinner on Saturday will also be in Jordan Hall, in the Reading Room.

**I. 9:00am, Saturday, October 14th**

**Joshua Hunt (University of Michigan): "Symmetries in Crystal Field Theory: Convenience vs. Intellectual Significance"**

**Summary:**

Although symmetry arguments in physics and chemistry frequently rely on group representation theory, such symmetry arguments are often unnecessary: non-symmetry-based approaches or approaches eschewing representation theory often suffice to solve the same problems. This raises a question about the nature of these arguments: in what sense are they merely convenient as opposed to intellectually significant? To address this question, I develop a case study from crystal field theory, a part of quantum chemistry. Crystal field theory characterizes how the energy levels of a spherically symmetric metal ion change when placed within an electrostatic field of a finite group, such as the octahedral group. I compare and contrast two methods for treating this problem: an elementary approach vs. a group (representation) theoretic approach. I show how the group theoretic approach provides three methodological advantages that make it more convenient than the elementary approach: it modularizes the crystal field theory problem into more manageable sub-problems; it unifies the problem on the basis of symmetry; and it makes the problem more tractable by replacing many integrals with simple arithmetic. To distinguish convenience from intellectual significance, I argue that modularization and unification are two intellectually significant functional roles whereas the increase in tractability is merely a matter of convenience. I close by considering how differences in functional roles can underpin a wide range of intellectually significant differences in expressive means.

**II. 10:10am, Saturday, October 14th**

**Ofra Rechter (Tel-Aviv University): "Why Is there a Problem about an Axiomatic Presentation of Arithmetic in Kant?"**

**Summary:**

Kant claims that arithmetic has no axioms, and also argues about specific simple singular identities that they cannot be regarded as axioms. One of Kant’s arguments is that then there would be infinitely many of them, but Kant also claims that arithmetic has postulates: “immediately certain practical propositions” that require no proof. Kant’s claims received harsh criticism (notably Frege’s, Gl. §5), and his philosophically initiated readers are rewarded with an abundance of interpretive puzzles. Perhaps the first of them was Kant’s philosophically sympathetic and mathematically informed expositor, Johann Schultz, whose attribution to Kant of the view that arithmetic does have axioms and postulates like geometry was influenced by Schultz’s own attempt to provide a systematic presentation of arithmetic. Kant may have perceived that the time was not ripe for an axiomatization of arithmetic, but what he said in support of his denial of axioms did not convince Schultz. What motivates Kant’s interest in the axiomatic presentation of arithmetic? What might its significance be from Kant’s perspective?

In this talk (a part of a larger project) I argue that the main issue here turns on the observation that to have an axiomatic presentation of arithmetic like the one accepted for geometry one would have to face the issues of iteration or recursion. Schultz has come close to this in his treatment of multiplication as iterated addition.

**III. 11:20am, Saturday, October 14th**

**Karine Chemla (CNRS, Université de Paris 7-Diderot): "Elements of a history of ideality in mathematics"**

**Summary:**

In Kummer’s first public presentation of his “ideale complexe Zahlen” (first published in the 1846 issue of the *Bericht über die zur Bekanntmachung geeigneten Verhandlungen der Königl. Preufs. Akademie der Wissenschaften zu Berlin*), Kummer draws a parallel between ideal elements in geometry and the “prime factors” that he introduces. In the historiography of Kummer’s work in number theory, the other parallels that Kummer drew have been underlined and discussed: the ones with “algebra and analysis”, and with Gauss’s work in number theory, which occur in this first 1846 presentation, and the one with chemistry, which appeared only later. However, the parallel with projective geometry has remained in the shadow. In this talk, I argue that Kummer’s reflection on Poncelet’s introduction of ideal relations in geometry, and the reconceptualization that Chasles offered for this in his 1837 *Aperçu historique* played a key part in Kummer’s introduction of “ideal complex numbers.” This part is clearly perceptible in the structure of the 1846 publication, and I will explain how we can read the effect of Chasles’ reconceptualization in the definitions that Kummer presents. I also argue the parallel between ideality in projective geometry and in Kummer’s work on numbers helps us understand features of the “ideal complex numbers” that have puzzled historians. This episode is interesting at a higher level, since it suggests that the philosophical reflection on the value of generality that geometers like Poncelet and Chasles developed in the context of the shaping of projective geometry was instrumental *as such *in inspiring key developments in other domains of mathematics, and precisely in this case, the introduction of ideal elements more widely in mathematics.

**12:30pm, Lunch (Jordan Hall of Science, Reading Room)**

**IV. 2:00pm, Saturday, October 14th**

**William Tait (University of Chicago): "What Hilbert and Bernays meant by ‘finitism’"**

**Summary:**

In “Finitism” I argued, on the basis of taking the 'finite’ in 'finitism' seriously, that finitist number theory SHOULD coincide with primitive recursive arithmetic, PRA. I will say very little about that argument. Another, historical, question is: What DID Hilbert and Bernays mean by “finitism”? Goedel took the answer to be PRA and I tend to believe that, too, in the soft sense that I don’t believe that they INTENDED to accept results whose proofs go beyond PRA. So the question is not exactly “What did they accept as finitist?”, since there are cases in which results were accepted without full understanding. I addressed the question in an appendix to a collection of essays *The Provenance of Pure Reason*, but the issue remains alive, especially in the hands of Wilfried Sieg, one of the most distinguished historians of Hilbert’s work in foundations of mathematics. His claim seems to be that, not only did the Hilbert school knowingly accept methods that go beyond finitism, but they even accepted the notion of function as a finitist notion, rejecting totally the ‘finite’ in ‘finitism'. I will mainly examine his arguments for this.

**V. 3:10pm, Saturday, October 14th**

**Vera Flocke (New York University): "Carnap’s Defense of Impredicative Definitions"**

**Summary:**

Carnap distinguished between the kinds of questions that in his 1950 article “Empiricism, Semantics and Ontology” he calls “internal” and “external” already in *The Logical Syntax of Language *(1937 [1934]), where this distinction was part of Carnap’s solution to deep problems concerning the foundations of mathematics. Carnap (1937 [1934], p. 114) wanted to show how one can accept simple type theory and the impredicative definitions that it condones without committing oneself to a Platonist view with respect to impredicatively defined properties. He attributed such a “metaphysical conception” to Ramsey, and said that he had “absolutely nothing to do” with it (p. 114). In order to explain the difference between himself and Ramsey, Carnap effectively argued that he (unlike Ramsey) makes an unproblematic “internal” use of higher-order quantifiers. This early application of the internal/external distinction offers important insights into how it should be understood. For example, it is not at all clear what Carnap rejects when he rejects “metaphysics”. Clarifying the relevant characteristics of Ramsey’s view allows to clarify the features of at least one instance of “metaphysics”. In addition to this interpretative contribution, my talk yields a new, Carnapian defense of impredicative definitions, which unlike (for instance) a Gödelian defense does not rest on ascribing any particular ontological status to impredicatively defined properties.

**VI. 4:20pm, Saturday, October 14th**

**Jeffrey Schatz (University of California-Irvine): "On Alternative Interpretations of ‘Maximize’ and the Case for Forcing Axioms"**

**Summary:**

In recent years, the debate regarding extending the axiom system ZFC+LCs has focused on two alternatives: forcing axioms, especially Martin’s Maximum, and axioms arising from the inner model program, especially those asserting that V=Ultimate L. As Maddy argues in her Naturalism in Mathematics that a fundamental goal of set theory is to provide a maximal universe for diverse mathematical structures and theories to “live”, a key part of any decision between these mutually exclusive axiom candidates is determining whether either can be properly said to maximize over the other. Towards this end, this talk will focus on two proposed characterizations of maximization: Steel’s approach based on interpretive power, and Maddy’s approach based on permitting the existence of distinct isomorphism types. After surveying the status of these axiom candidates under the former approach, this talk then looks to the hitherto relatively unexamined question of how they relate under Maddy’s approach. In so doing, this talk will present both mathematical and philosophical challenges arising from this approach, concluding by pointing towards further refinements in the concept of maximize necessary for a complete justification for either axiom candidate.

**VII. 5:30pm, Saturday, October 14th**

**Mark Wilson (University of Pittsburgh): "Classical Mechanics as a Paradigmatic Philosophical Problem"**

**Summary:**

In 1892, Heinrich Hertz complained: "It is exceedingly difficult to expound to thoughtful hearers an introduction to mechanics without being occasionally embarrassed, without feeling tempted now and again to apologize, without wishing to get as quickly as possible over the rudiments, and on to examples which speak for themselves."

The conceptual issues to which Hertz alludes eventually led to a familiar set of expectations as to how such muddles might be addressed, through rigorous axiomatics and clear conceptual analysis (Hertz’ demand was included on the celebrated 1899 list of problems that Hilbert thought that mathematics should address in the century to follow). Recent work in effective computer modeling has suggested a rather different vein in which concerns like Hertz’ can be addressed. I shall outline how this works using several elementary examples.

**Reception & Dinner, 6:45pm (Jordan Hall of Science, Reading Room)**

**VIII. 9:00am, Sunday, October 15th**

**Jamie Tappenden (University of Michigan): "Styles of Mathematical Explanation: Why do**

**Elliptic Functions have Two Periods?"**

**Summary:**

Recent years have seen sustained attention to the topic of explanation as a phenomenon within mathematical practice. This talk will present a historical case study illustrating that, among other things, mathematical explanations can exhibit the same interest-relativity and context-dependence that are found in explanations of physical events. The example is the explanation of the fact that elliptic functions (defined as they were in the nineteenth century) are doubly periodic. Two ways to address the fact – one using techniques characteristic of Bernard Riemann (develop the Riemann surface then integrate on a torus) and another in the style of Weierstrass (represent via the Weierstrass P-function and its derivative) reveal strikingly different mathematical virtues. The explanations are both “good ones”, but for reasons that may be incommensurable. Furthermore, resolving whether or not the reasons are in fact incommensurable will depend on further mathematical discovery.

**IX. 10:10am, Sunday, October 15th**

**Jonatha Ettel (Stanford University): "Mechanical Curves, Mechanism, and the Question of the Essence of Mathematics"**

**Summary:**

The seventeenth century saw the emergence of thoroughly mathematical natural philosophy, one whose central idea is neatly summed up in Descartes’ claim that physics should be based solely on mathematical principles, since the principles of geometry and pure mathematics ``explain all natural phenomena, and enable us to provide certain demonstrations regarding them’’. This claim immediately raises a number of questions: what sorts of principles are to be regarded as genuinely mathematical? do demonstrations from such principles genuinely produce certain knowledge? do such principles provide adequate resources to represent and explain natural phenomena of interest? Answering these questions was rendered more complicated by the fact that mathematics and physics were evolving rapidly during the period. Descartes’ *La Géométrie* contains some of the most thoroughly worked out answers to these questions, yet subsequent developments in physics and mathematics rendered these answers untenable. One issue was the role of transcendental or ``mechanical’’ curves in physical modeling. We consider one such curve---the cycloid---in the geometrical physics of the late seventeenth century, and examine the way that requirements of geometrical style, clarity of demonstration, and mechanical workability are balanced in some of Christiaan Huygens’ results on the cycloidal pendulum and their generalization in Newton’s *Principia*.

**X. 11:20am, Sunday, October 15th**

**Anna Bellomo (University of California-Irvine): "Two conceptions of domain expansion in mathematics"**

**Summary:**

This talk will compare two different conceptualisations of domain expansion in mathematics, as exemplified by Kenneth Manders and, among others, Richard Dedekind, respectively. After presenting how Manders’ ideas relate to the common model-theoretic notions of existentially closed models and model completeness of theories, we argue that the framework yields problematic assessments of some paradigmatic examples of domain expansion. We then move on to the question of how these examples are handled in Dedekind’s framework, and then use the results of this analysis to assess the merits of Manders’ proposal.

**12:30 Adjournment**