# Midwest PhilMath Workshop 17 (MWPMW 17)

The seventeenth annual Midwest PhilMath Workshop (MWPMW 17) will take place at the U of Notre Dame on Saturday, November 12th and Sunday, November 13th.

If you would like to give a talk, please email a draft or summary to Tim Bays, Paddy Blanchette, Mic Detlefsen and Curtis Franks by September 18th.

Also, if you plan to attend, please register here: MWPMW 17 Registration

We're pleased to have Pr. Michael Rathjen (Pure Mathematics, U of Leeds) and Pr. Lydia Patton (Philosophy, Virginia Tech) join us this year for featured talks.

Pr. Rathjen will give a talk titled: "On Feferman's Second Conjecture".

Pr. Patton will give a talk titled: "Fishbones, Wheels, Eyes, and Butterflies: Is There a Unified Account of Mathematical and Physical Modeling?".

As with last year, please book your own hotel room. You may do so here: MWPMW 17 Room Reservation

Please continue to monitor this page for further information. We expect to publish the full program around the first of October.

All sessions of the workshop will be held in room 129 of Debartolo Hall. Since our workshop meets on a weekend, only select doors of this hall will be available for entry. These are the doors located on the west side of Debartolo Hall towards the north end of the building.

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**Schedule**

Session I:

#### Saturday, November 12, 2016

#### Room 129 of Debartolo Hall

**1. 9:00am-10:00am**

**• Silvia De Toffoli (Stanford University), "Thinking with Diagrams: The Case of Mathematics"**

Summary:

Visual representations of various kinds are ubiquitous in pure and applied mathematics, in the natural and social sciences, as well as in many other human activities. By investigating these visualizations, many questions arise: What are they? How do they function? What are the conditions of their correct use? Why are they, at times, such effective aids to cognition? What are their epistemological roles? In this talk, I will focus on diagrams in mathematics, not exclusively in geometry where diagrams are common, but in different mathematical domains, such as knot theory and homological algebra. Despite the extreme variety of representations in mathematics, I will try to distill fundamental properties of the nature and use of mathematical diagrams. I will argue that diagrams are not static illustrations simply recording information, but dynamic displays for advancing thought. An effective diagram, or a sequence of diagrams, sets the relevant reasoning into material, visual form. By manipulating these concrete external representations in prescribed ways, information about abstract mathematical structures can be obtained without going through a process of formal calculation. In this way cognitive abilities that no doubt evolved in order to manipulate concrete objects can be re-deployed in the abstract realm of mathematics.

**2. 10:05am-11:05am**

**• Neil Tennant (The Ohio State University), "Core Logic and the Gödel Phenomena"**

Summary:

Classical Core Logic is paraconsistent. It does not have the rule *Ex Falso Quodlibet*, and it affords no proof of the Lewis Paradoxes A,¬A:B and A,¬A:¬B. Nevertheless, every classically valid sequent has a subsequent provable in Classical Core Logic.

A similar result holds for Core Logic: every intuitionistically valid sequent has a Core-provable subsequent.

In both systems there are distinct, non-trivial inconsistent theories. Inconsistency of a set Δ of sentences is still defined as the deducibility of absurdity (⊥) from Δ.

Consider the suggestion that we should adopt Classical Core Logic for the formalization of deductive reasoning in classical mathematics; and that we should adopt Core Logic to do the same for constructive mathematics.

One foundationalist worry in response to this suggestion is that we might thereby deprive ourselves of certain famous results in foundations. For, the worry goes, Gödel’s Incompleteness Theorems crucially involve the very notion of consistency. And the proofs of those theorems might very well rely on arbitrary propositions being derivable from ⊥, in exactly the way that both Intuitionistic and Classical Logic assume.

This study aims to allay such a worry completely. It shows that Gödel’s First Incompleteness Theorem for *classical* arithmetic is *Core*-provable.

**3. 11:10am-12:10pm**

**• Michael Rathjen (University of Leeds), "On Feferman's second conjecture"**

Summary:

In addition to his conjecture about the indeterminacy of **CH** relative to semi-intuitionistic set theory, Solomon Feferman stated another conjecture concerning the relationship between two types of predicates in such set theories, namely that the collection of Δ_{1 }predicates and the collection of predicates for which the law of excluded middle holds should coincide. The talk will address this conjecture.

**Lunch 12:30pm-1:55pm, Private Dining Room, **Morris Inn

### Session II:

#### Saturday, November 12, 2016

#### Room 129 of Debartolo Hall

**4. 2:00pm-3:00pm**

**• Walter Dean (University of Warwick), “Incompleteness via paradox (and completeness)”**

Summary:

This talk will explore a method for uniformly transforming the paradoxes of naive set theory and semantics into formal incompleteness results originally due to Georg Kreisel and Hao Wang. I will first trace the origins of this method in relation to Gödel’s proof of the completeness theorem for first-order logic and its subsequent arithmetization by Hilbert and Bernays in their* Grundlagen der Mathematik*. I will then describe how the method can be applied to construct arithmetical statements formally independent of systems of set theory and second-order arithmetic via formalizations of Russell’s paradox and the Liar (and time permitting also the Skolem and Richard paradoxes). Finally, I will consider the significance of the Kreisel-Wang method with respect to both the Hilbert program and subsequent developments.

**5. 3:05pm-4:05pm**

**• Mate Szabo & Patrick Walsh (Carnegie Mellon University), "Gödel's and Post's proofs of the incompleteness theorem"**

Summary:

Emil Post worked on questions of incompleteness and undecidability already in the 1920s. To some extent he anticipated Gödel's results, but his work only saw publication much later, in 1965. Instead of trying to claim priority to Gödel, Post emphasized that:

"with the Principia Mathematica as a common starting point, the roads followed towards our common conclusions are so different that much may be gained from a comparison of these parallel evolutions."

We take up this comparison. After we survey Post's approach and Gödel's proof, we distill and emphasize two key dissimilarities based on their different methodologies.

**6. 4:15pm-5:15pm**

**• Roy Cook (University of Minnesota-Twin Cities), “Dangerous Distinctions”**

Summary:

The drawing of distinctions is one of the analytic philosopher's most fundamental tools. Of course, unbridled distinction-making has been criticized from within other philosophical perspectives (e.g. 'postmodern' philosophy and feminist philosophy). In this talk, however, I will show that distinctions are problematic by the analytic philosopher's own lights. In particular, I will show that the following claims are inconsistent:

(1) For every condition C, there is a distinction between those objects that have C and those that do not.

(2) Distinctions are objects (since, ,e.g. we can count them, or make distinctions between different types of distinctions).

(3) There is at least one object that is not a distinction.

I conclude by demonstrating that, although there is some formal similarity, this puzzle is distinct from the Russell paradox, and I quickly sketch two consistent formal systems for distinctions that parallel, roughly, ZFC and neo-logicist accounts of set theory respectively.

**7. 5:20pm-6:20pm**

**• Lydia Patton (Virginia Tech University), “Fishbones, Wheels, Eyes, and Butterflies: Is There a Unified Account of Mathematical and Physical Modeling?”**

Summary:

David Hilbert saw the connection between physics and mathematics as the “nerve” that invigorates mathematical reasoning, and Quine and Putnam argued that mathematics is indispensable to physical reasoning. But investigation of contemporary techniques of mathematical modeling compels the question: How do we construct meaningful, testable physical models using “non-natural”, abstract quantities and transformations, such as conformal maps and dimensionless constants? The paper concludes with an analysis of the dual role of geometrical and topographical reasoning in applied mathematics: providing understanding or explanation, on the one hand, and theory testing and explorative reasoning and experiment, on the other. In mathematics, just as in physics, the success or failure of techniques and experiments in proof suggests ways to alter or to strengthen the structure of existing theory.

####
**Reception & Dinner 6:30pm-8:30pm, Private Dining Room,** Morris Inn

### Session III

#### Sunday, November 13, 2016

#### Room 129 of Debartolo Hall

**8. 9:00am-10:00am**

**• John Baldwin (University of Illinois-Chicago), “Philosophical implications of the paradigm shift in model theory”**

Summary:

The paradigm shift that swept model theory in the 1970’s really occurred in two stages. During the first stage in the 1950’s and 1960’s the focus switched from the study of properties of logics to properties of theories. In the second stage, Shelah’s decisive step was to move from merely identifying some fruitful properties (e.g. complete, model complete, ℵ_{1}-categorical) that might hold of a theory to a systematic classification of complete first order theories.

Model theorists now undertake a systematic search for a finite set of syntactic conditions which divide first order theories into disjoint classes such that models of different theories in the same class have similar mathematical properties. With this framework one can compare different areas of mathematics by checking where theories formalizing them lie in the classification. From the standpoint of the philosophy of mathematical practice the focus is changed from justifying the reliability of mathematical results to the clear understanding and organization of mathematical concepts.

This shift sheds new light on a number of philosophical issues. For example, how does formal logic play a significant role in mathematics beyond the old metaphor of ‘the analysis of methods of reasoning’? What constitutes a paradigm shift in mathematics?

**9. 10:05am-11:05am**

**• Arezoo Islami (Stanford University), “The applicability of mathematics: a historical approach”**

Summary:

The applicability problem is the problem of explaining why mathematics is effective in the natural sciences. Different schools (in philosophy of mathematics) have given seemingly conflicting answers to the applicability problem. The conflict is solved if we understand mathematics against a historical background, which “progresses” between broadly speaking algebraic and geometrical structures, As a result of seeing mathematics in its historical context, I hope to show that the applicability problem is ill-posed once it is asked as a general, ahistorical question. Our answer to the question of why integral and differential calculus, for instance, is useful in classical mechanics is different from our response to why algebra was useful to the Renaissance merchant.

**10. 11:10am-12:10pm**

**• Richard Samuels, Stewart Shapiro and Eric Snyder (The Ohio State University), “Scottish Neologicism, Frege's Constraint, and the Frege-Heck Condition”**

Summary:

The Neologicists Crispin Wright and Bob Hale both argue that our actual conception of the natural numbers is correctly characterized by Hume's Principle, not the familiar Dedekind-Peano Axioms, precisely because the former but not the latter meets what has come to be known as Frege's Constraint. Most recently, Hale argues that transitive counting is essential to possessing those concepts, and so ought to be built directly into their defining characterization. However, we argue that if transitive counting is the application relevant to satisfying Frege's Constraint, as Hale explicitly and Wright implicitly assumes, then neither the Dedekind-Peano Axioms nor Hume's Principle will satisfy it. On the other hand, if derivation is allowed in the explanation of transitive counting, then Hume's Principle will in fact satisfy Frege's Constraint, but so will the Dedekind-Peano Axioms. In either case, satisfying Frege's Constraint will not adjudicate in favor of one of these as correctly characterizing our actual natural number concepts.

### 12:15pm Adjournment

#### Emergency Information

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.