Midwest PhilMath Workshop (MWPMW 10)

• Abstracts

• Friday, October 9, 2009, 4:00--5:30

  Hayes-Hurley Center, room 125

  Tony Martin (Departments of Mathematics and Philosophy, UCLA), "Evidence in Set Theory"
  Nik Weaver (Dept. of Mathematics, Washington U), "Which ordinals are predicatively provable?"
  Hugh Woodin (Dept. of Mathematics, U of California--Berkeley), "The search for ultimate L"

  Tony Martin, "Evidence in Set Theory"

  The sort of evidence in set theory most often discussed is evidence for adopting axioms. Gödel declared in his papers on the continuum hypothesis that a certain kind of evidence for a new axiom would make us have to assume its truth, just as analogous evidence in physics would make us accept a physical theory. It is sometimes said that Gödel was arguing from an assumption about the relation of the epistemology of mathematics to that of natural science. It has also been said that whether or not he was correct depends on what philosophy of mathematics is correct. Examples like this one this raise a number of questions. What should count as good evidence for adopting new axioms of set theory? Can evidence only be evaluated relative to a philosophical view (realist, naturalist, formalist, etc.) about set theory? In particular, does evaluation have to depend an account of what it is to adopt an axiom? I will discuss these questions via a few examples.

  Nik Weaver, "Which ordinals are predicatively provable?"
  
  We must distinguish between the goal of developing axiomatic
  theories which are proof-theoretically as strong as possible
  and the goal of developing strong theories *which we can be
  confident are arithmetically sound*. I will argue that the problem of determining which ordinals are predicatively provable is essential to the latter project. This problem has an interesting history but has been neglected in recent decades and is still very open.

  Hugh Woodin, "The search for ultimate L"
  
  It is now known that the inner model problem for essentially all large cardinal axioms reduces to the case of exactly one supercompact cardinal. I will discuss the Inner Model Programand its role in modern Set Theory.

  Saturday, October 10

  Session I: Special Symposium on Set Theory & Philosophy

  Tony Martin, "Mathematical Objects & Mathematical Concepts"

  I will sketch a view that mathematics is about mathematical concepts and that mathematical objects play no role in pure mathematics. This view is in important respects like the view of Gödel (and, of course, in important respects different), and I will point out similarities and the differences. The main concepts I will discuss are those of natural numbers and and sets. On my view such concepts differ from ordinary concepts only in that they have turned out to be sharp enough to support mathematical study. Some mathematical concepts concepts are fully sharp and some perhaps are not. Among the reasons for doubting that mathematical objects are the subject matter of mathematics are: (1) Benacerraf-style epistemological reasons; (2) the strange kinds of objects that people trying to explain the role of mathematical objects take such objects to be; (3) the fact that object-based accounts of mathematics could not account for any there being set-theoretical statements that lack truth values. (I am open minded about whether there are such statements, but it seems a fact that we do not know there are not any.) Roughly speaking, I would say that I a statement of, e.g., set theory is true if it would have to be true in any structure instantiating the concept of set. One might think of this " would have to be'' as a logical notion. One might also think of it as a modal notion, but not all the modal laws of Geoffrey Hellman's modal theory are valid for the notion.

  Nik Weaver, "Disbelieving the power set axiom"
  
  I will make a case for adopting an attitude of extreme skepticism toward the power set axiom --- not only its philosophical justifiability, but also its practical value, its aesthetic value, and the soundness of its arithmetical consequences.

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  Hugh Woodin, "Multiverse views and the search for ultimate (mathematical) truth"
  
  Does the Continuum Hypothesis have an answer? If so does this
  require a conception of the universe of sets which answers the other known(formally) unsolvable problems of mathematics? Perhaps the best we can do is a multiverse view which leaves some questions, maybe even that of the Continuum Hypothesis unanswered.

  Session II

  Geoffrey Hellman, "Reflections on Replacement andReflection"

  We begin by contrasting two main standpoints regarding the axiom of Replacement (esp. 2'd-order) of ZFC^2, viz. an "absolutist-platonist" ("Fregean") one vs. a "relativist-structuralist" ("Hilbertian"). We illustrate with Boolos' late skeptical views. Next we take up pursuit of small large cardinals in a structuralist spirit (e.g. Zermelo, Putnam, and my (earlier) self), highlighting (1) axioms as defining conditions, esp. largeness conditions, and (2) extendability principles. Finally, we reexamine reflection principles (bypassed in my earlier work), showing how a modal-structural scheme can be formulated without even implicit reference to
  a fixed universe of sets, and we show how this can recover uses by Tait (2005) of 2d-order reflection to obtain many indescribable cardinals. We close with a "bad company" challenge requiring further attention.   

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  Madeline Muntersbjorn, "Poincaré & Mathematical Discovery"

  InScience and Hypothesis (1902) Poincaré remarks that, "mathematical reasoning has of itself a kind of creative virtue” (3). The discipline grows in scope and power over time despite the fact that, “The very possibility of mathematical science seems an insoluble contradiction” (1). Mathematics exhibits more unity than other sciences even though our mathematical abilities vary widely from person to person. One explanation is that mathematics concerns only necessary relations between abstract objects that hold in all possible worlds. Famously, Poincaré rejectslogicism: If mathematical results are necessary and obvious, why are they so hard for so many people to see? Is all of mathematics really reducible to one great tautology, a=a? Poincaré’s positive view is hard to label precisely. Russell (1914) decried Poincaré’s philosophy of mathematics as old-fashioned and “broadly Kantian.” Recently, Folina (1992) describes him as “neo-Kantian” while Zahar (2001) prefers “quasi-Kantian.” These Kantian labels fit when we restrict our attention to foundational concerns surrounding mathematical certainty. However, Poincaré does not distinguish sharply between the philosophy and psychology of mathematics and his views on mathematics are, for this reason, not limited to questions of rigor. While some contend that models of mathematical discovery are impossible because the process is too personal and idiosyncratic, Poincaré identifies distinct phases all mathematical discoveries exhibit. In general, every science, including mathematics, proceeds from descriptive inquiry to comparative study to explanatory principles. In particular, mathematics proceeds from intuitive insights to informal sketches to formal proofs. By attending to the different representational systems employed during distinct phases of mathematical discovery, including external symbols and internal schemas, we present both a more coherent interpretation of Poincaré as well as a framework for a more compelling account of the unity of mathematics.

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  Norma Goethe & Michele Friend, "Confronting Philosophical Ideals of Proof." The received view concerning mathematical proof is that (1) all ideas used in a proof can be expressed as propositions and (2) that these propositions expressing fundamental principles of reasoning in mathematics should be transmitted by means of an axiomatic proof. We include natural deduction methods of proof amongst the axiomatic systems.Both theses come from a broadly “Fregean-logicist” influence on the philosophy of mathematics. We shall concentrate on (2). The received view is familiar from Hilbert and Gentzen.Hilbert makes two strong claims that are of interest here: (I) Formal proofs “are carried out according to certain definite rules, in whichthe technique of our thinkingis expressed”. (II) Our thinking actually proceeds according to these rules which form “aclosed system that can be discovered and definitely stated”. From (II) it follows that axiomatic proofs can be definitely stated in a finite number of steps. In other words, axiomatic proofs are finite. They “bottom-out” at the axioms. Gentzen’s (1969), claims are stronger than Hilbert’s. For Gentzen, (III) Formal proofs have a "close affinity to actual reasoning", in his natural deduction systems.In particular, such proofs reflect as "accurately as possible theactual logical reasoninginvolved in mathematical proofs".Our aim is to advocate a more naturalist approach to the philosophical investigationof the notion of proof. In the approach, we think of mathematical proofs as a means of (i) convincing other mathematicians, but more importantly (ii) inviting further and deeper mathematical enquiry.

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  Benedikt Löwe, "Idealization in epistemology of mathematics"
  
  The process leading from real-world data via stable phenomena to an idealization is well known in philosophy of science, and beyond that in all areas of philosophy. Many philosophical concepts are idealizations or abstractions. In particular in areas where philosophical concepts are meant to describe human activity, idealizations are not without an effect on practice (in analogy to the 'participant observation' methodology of
  anthropology).
  
  In philosophy of mathematics, one such idealization (with a non-negligible effect on mathematical practice) is the move from the various informal forms of proof towards the formalized notion of a derivation in a formal system. We shall discuss its impact on philosophy of mathematics and various approaches to philosophy of mathematics that seem to ignore the fact that formal proof is an idealization.

  Sunday, October 11

  Session III

  Neil Tennant, "The Schröder-Bernstein Theorem"

  The Schröder-Bernstein theorem is essential to the theory of in?nite cardinalities. It says that if two sets can be mapped one-one into each other, then they can be mapped one-oneontoeach other. First we prove the theorem in a set-theoretical setting, at perhaps a higher level of rigor than is usual. The higher level of rigor is important in allowing one to identify exactly what concepts are involved, and how they feature in the reasoning employed. At the next stage we reformulate the leading idea in the proof by appeal to the essential concepts isolated, in order to cast it in a natural-deduction format.

  This format uses introduction and elimination rules of inference to characterize the logical and mathematical concepts involved. It allows one to prove the celebrated result in an ontologically more economical way, by means of rules that provide logical analyses of the concepts involved.

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  Gregory Lavers, "Frege the conventionalist and Carnap the Fregean"

  In this paper I examine the fundamental views on the nature of logical and mathematical truth of both Frege and Carnap. I argue that their positions are much closer than is standardly assumed. I attempt to establish this point on two fronts. First, I argue that Frege is not the metaphysical realist that he is standardly taken to be. Second, I argue that Carnap, where he does differ from Frege, can be seen to do so because of mathematical results proved in the early twentieth century. The differences in their views are, then, not primarily philosophical differences. Also, it might be thought that Frege was interested in analyzing our ordinary mathematical
  notions, while Carnap was interested in the construction of arbitrary systems. I argue that this is not the case: our ordinary notions play an even more important role in Carnap’s philosophy of mathematics than they do in Frege’s.

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  Kent Schmor, "Against Intolerance: A Defense of Carnap's Philosophy of Mathematics"

  Michael Friedman (1988), as well as Warren Goldfarb and Thomas Ricketts (1992), have argued, in different ways, that Carnap did not fully appreciatethe impact of Gödel's incompleteness results on theprospect for a purely syntactic characterization of the analyticity of logic and mathematics.I defend Carnap against both versions of this objection.I also offer a new interpretation of therelationship Carnap’s Principle of Tolerance and his minimalist conception of philosophy.

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  Tim McCarthy, "On the Stability of Gödel’s Second Incompleteness Theorem"

  The stability problem for Gödel’s second incompleteness theorem is the question of whether the consistency claim for a theory T of the relevant type, while unprovable in T under Gödel’s arithmetization, may yet be provable under another. I suggest in this paper that the arithmetization of syntax for T should be viewed as a case of inter-theoretic reduction: the theory reduced is a given syntactic description of T, the reducing theory an arithmetic fragment of T.From that perspective, a number of structural constraints on arithmetic representations of the provability concept for T can be motivated. These constraints are a basis for Gödel’s second theorem, and it turns out that they generate the Hilbert-Bernays-Löb derivability conditions, and thus the standard abstract form of Gödel’s second, if the relevant provability predicate is Sigma 0-1.