# MWPMW 15

The fifteenth annual meeting of the Midwest PhilMath Workshop will take place October 18-19, 2014 on the campus of the University of Notre Dame.

__Program__

To help us mark this birthday, we are happy to welcome Warren Goldfarb, Joel David Hamkins and Hugh Woodin as featured speakers.

Professors Woodin and Hamkins will participate in a special Saturday morning symposium on the foundations of set theory.

Professor Goldfarb will speak on Gödel's philosophical ideas in the Sunday morning session.

__Reception__

**Friday, October 17: 7:00pm-9:00pm**

All attending the MWPMW 15 are warmly invited to a reception Friday evening from 7:00pm--9:00pm in the Morris Inn, Salon A.

__Schedule of Talks__

All sessions of the workshop will be held in room 129 of DeBartolo Hall.

**Note: Because of ongoing construction and renovation projects, please enter DeBartolo Hall ****using the doors on the west side of the building.**

**Saturday, October 18, 2014**

__Session I:__ 9:00am--12:15pm

• 9:00am--10:00am

Benjamin Rin (U of California-Irvine, Dept of Logic & Philosophy of Science)

"Transfinite Recursion & Computation in the Iterative Conception of Set"

__Abstract:__

Transfinite recursion is an essential component of set theory. In this talk, I look for intrinsically justified reasons for believing in recursion and the notions of higher computation that surround it. In doing this, I consider several kinds of recursion principles and present results concerning their relation to one another. I then consider philosophical motivations for these formal principles coming from the idea that computational notions lie at the core of our conception of set. This is significant because, while the iterative conception of set has been widely recognized as insufficient by itself to establish Replacement and recursion, its supplementation by algorithmic notions suggests a new and philosophically well-motivated reason to believe in such principles.

__Symposium: Perspectives on the foundations of set theory__

__Symposium: Perspectives on the foundations of set theory__

• 10:05am--11:05am

W. Hugh Woodin (Harvard U, Depts of Mathematics & Philosophy)

"Is the universe of sets an ultimate version of Gödel’s constructible universe?"

__Abstract:__

There is now a single specific conjecture which if true would confirm that there is an ultimate version of Gödel's constructible universe, L. This conjecture has emerged from the program within Set Theory which seeks to define canonical universes in which large cardinal axioms can hold.

If this conjecture is false then that entire program will have to be significantly revised. But what if the conjecture is true? Does this provide a compelling case that V = Ultimate L? If not what is the basis for the rejection of this axiom?

I will discuss some arguments for and against this axiom.

• 11:15am--12:15pm

Joel David Hamkins (CUNY, Depts of Mathematics, Philosophy & Computer Science)

"The pluralist perspective on the axiom of constructability"

__Abstract:__

I shall argue that the commonly held V≠L via maximize position, which rejects the axiom of constructibility V = L on the basis that it is restrictive, implicitly takes a stand in the pluralist debate in the philosophy of set theory by presuming an absolute background concept of ordinal. The argument appears to lose its force, in contrast, on an upwardly extensible, multiverse concept of set, in light of the various facts showing that models of set theory generally have extensions to models of V = L inside larger set-theoretic universes.

**Lunch: 12:30pm--1:55pm**

__Session II:__ 2:00pm--4:05pm

• 2:00pm--3:00pm

John Mumma (California State University, San Bernardino, Dept of Philosophy)

"Towards a notion of geometric consequence"

__Abstract:__

The intuitive character of elementary geometry as a mathematical subject is commonly thought to be found in its axioms. In this talk, I propose an alternative account that ties the intuitiveness of elementary geometry to the concepts whereby the geometric conclusion of a proof is judged as following, in a mathematical sense, from its geometric premises. My starting point is the formal proof system *E *developed as an analysis of the proofs of the early books of the* Elements*. Though it can be, given a conventional semantics in terms of Tarskian models, its formalism naturally suggests an interpretation in terms of a notion of *geometric figure*. I describe my strategy for giving an acceptable characterization for the notion, and flesh out its core intuitive component. I then examine how the resulting standard of *geometric consequence* would compare to the standard of logical consequence operative in modern axiomatizations of elementary geometry.

• 3:05pm--4:05pm

Catherine St Croix (U of Michigan, Dept of Philosophy)

"Truth and Mathematical Depth"

__Abstract:__

Within the sciences, theoretical virtues are often said to track truth and guide progress. Judgments of mathematical depth, I argue, can play a similar role. This talk is in two parts. First, I’ll examine some examples of depth judgments, looking at both circumstances of application and content. This study suggests a tentative classification, which will prove useful in the second part of the talk. In particular, we can carve out a species of depth judgments that is well-suited to the guiding role of theoretical virtues. In the second part of the talk, I’ll use this category to illustrate that, contrary to recent work on the topic, focusing on mathematical depth does not allow us to eschew certain familiar ontological disputes within the philosophy of mathematics.

Break 4:05--4:25

__Session III: __4:30pm--6:35pm

• 4:30pm-5:30pm

John Baldwin (U of Illinois-Chicago, Dept of Mathematics, Statistics & Computer Science)

"Axiomatizing changing conceptions of the geometric continuum"

__Abstract:__

We begin with a general account of the goals of axiomatization, introducing several variants (e.g. modest) on Detlefsen's notion of 'complete descriptive axiomatization'. We examine the distinctions between the Greek and modern view of number, magnitude and proportion and consider how this impacts the intent of Hilbert's axiomatization of geometry. We list propositions from Euclid, Archimedes, and Descartes that a modern axiomatization must account for. We argue, as indeed did Hilbert, that propositions concerning polygons, area, and similar triangle are derivable (in their modern interpretation in terms of number) from Hilbert’s first order axioms. We note that Tarski's extension to the first order complete theory *ε*^{2} of geometries over real closed fields grounds the geometry of Descartes as well as Euclid. Then we break new mathematical ground by considering a proposition strangely absent from explicit treatment in Hilbert's geometry: formulas for the circumference and area of a circle. We provide a (complete) first order theory of geometry in which the formula C = πd computes the circumference of a circle but which has non-Archimedean models. We argue that Hilbert's continuity properties show much more than the data set of Greek mathematics and thus are an immodest complete descriptive axiomatization.

• 5:35pm--6:35pm

C. Anthony Anderson & Harry Deutsch (U of California-Santa Barbara, Dept of Philosophy; Illinois State U, Dept of Philosophy)

"Russellian Propositions in a Class Theoretic Setting"

__Abstract:__

In *Principles of Mathematics, *Appendix B, Russell sketches a paradox of propositions that reflects the "anti-Cantorian" view he held at the time that "There cannot be more ranges [classes] of propositions, than there are propositions," and that turns on his view that propositions are finely individuated. In *Principles, *Russell confesses that he "has not succeeded" in discovering the "complete solution" to this paradox, but subsequently, in *Principia Mathematica*, invokes the ramified theory of types as a solution. The paradox was rediscovered by Myhill in 1958 and has become known as the "Russell-Myhill antinomy" (RM). The literature contains several attempts to resolve the paradox, all of which, however, invoke a type theoretic structure. We suggest an untyped solution based on adding propositions to a strong set theory (Morse Kelly set theory) and we show how certain other paradoxes involving classes of propositions can be resolved in this setting. In the course of our discussion we develop an analysis of the notion of a (term) constituent of a proposition and suggest a modified axiom of regularity to reflect the idea that the constituent relation should be well-founded. This allows alternative resolutions of the paradoxes, based on regularity, thus further confirming our approach.

**Social Hour & Dinner: 6:45pm--9:00pm**

**Sunday, October 19, 2014**

__Session IV:__ 9:00am--1:20pm

• 9:00am--10:00am

Sean Morris (Metropolitan State University of Denver, Dept of Philosophy)

"Cantor, Zermelo, and Russell on the Pragmatic Conception of Set"

__Abstract:__

Since the late1960's, there has been widespread agreement among both philosophers and mathematicians that the only conception of set is that of the iterative conception as embodied in the axioms of Zermelo-Fraenkel (ZF) set theory and its related systems. In this paper I argue that this is largely an anomaly in the history of set theory, that up to this point in its history set theory had been a much more pluralistic endeavor. I claim that such an approach to set theory falls into a distinct tradition of set theory that I call "the pragmatic conception of set," a tradition present already at the theory’s founding and that finds its culmination in the work of W.V. Quine. As a philosophical consequence I urge that we reconsider the philosophical and mathematical benefits that research in non-standard set theories may hold.

• 10:05am--11:05am

Warren Goldfarb (Harvard U, Dept of Philosophy)

"On Gödel’s General Philosophical Viewpoint"

__Abstract:__

In about 1960, Gödel wrote a short and cryptic document he titled "My Philosophical Viewpoint", consisting of fourteen numbered sentences. Using this document as a springboard, I propose an account of Gödel's philosophical orientation in metaphysics and epistemology generally, and connect these general views to the more well-known stances he took in philosophy of mathematics. I then articulate some puzzles about Gödel’s thinking on questions of the applicability of mathematics to the physical world, an issue which receives only sporadic attention in his writings.

Break 11:05am--11:20am

• 11:20am--12:20pm

Roy Cook (U of Minnesota-Twin Cities, Dept of Philosophy)

"Frege's definition of cardinal number"

__Abstract:__

In this paper I show that Frege's final definition of cardinal number is uniformly mis-described in the literature: Cardinal numbers are not, loosely speaking, collections of collections (i.e. 1 is not the value-range of the concept holding of the value-ranges of all concepts with one instance). Instead, cardinal numbers are, again speaking loosely, collections of functions. After exploring the formal ways in which this definition differs from the standard, and mistaken, account of it in the literature, I conclude by making some suggestions regarding the consequences this observation has for our interpretation of Frege's mature philosophy of mathematics.

• 12:25pm--1:25pm

Gary Ebbs & Warren Goldfarb (Indiana U-Bloomington, Dept of Philosophy; Harvard U, Dept of Philosophy)

"First-Order Logical Validity and the Hilbert-Bernays Theorem"

__Abstract:__

What we call the Hilbert-Bernays (HB) Theorem establishes that for any satisfiable first-order quantificational schema S, one can write out linguistic expressions that are guaranteed to yield a true sentence of elementary arithmetic when they are substituted for the predicate letters in S. The theorem implies that for interpreted languages L that are rich enough to express elementary arithmetic, a first-order logical schema S is valid---true for all set-theoretical interpretations---if and only if every sentence of L that can be obtained by substituting predicates for predicate letters in S is true. In this talk we will (first) explain and defend W. V. Quine's claim that the HB theorem licenses us to define the first-order logical validity of a schema S in terms of predicate substitution, and (second) clarify the HB theorem by presenting an accessible and illuminating new proof of it.

**Adjournment 1:30pm**

__Arrangements__

We're fortunate to have Mrs. Harriet Baldwin (baldwin.1@nd.edu) as the workshop manager again this year. Sad to say, it will be her last since she has decided to retire this coming January.

Please contact her to reserve a hotel room. Please do that as soon as you can. There will be a press for rooms, so timely reservation is advised.

Please also let Harriet know if you would like to attend the workshop dinner and also if you have any dietary restrictions. All participants of the workshop are cordially invited to attend the dinner as guests of the university.

For those of you familiar with Harriet's friendly and expert handling of the previous workshops, you might also want to add a "thank you" to her. She has been a wonderful partner and friend for the MWPMW over the years.

__Student Participation__

As usual we will have limited funds to help defray expenses for student participants coming from out of town whose departments do not have funds to sufficiently offset their costs. These funds will generally take the form of subventions for lodging. If you are interested in applying for a subvention, please notify Mrs. Baldwin (with a copy to me) as soon as possible, and have an appropriate faculty member from your department email me to confirm that it cannot sufficiently underwrite your expenses.

Please consult this site for fuller information over the coming days and weeks.

If you have further questions, feel free to contact Mrs. Baldwin or myself at the email addresses given above.

__Registration__

It will aid our planning if you register for the workshop. To do so, please contact Harriet Baldwin.