NYU Philosophy of Mathematics Conference


Friday, April 3rd:

(Talks on this day will be located in 5 Washington Place, room 101.  Map)

   2:30pm - 3:00pm - Check-In

      Pick up folders, name tags, etc.

    3:00pm - 4:30pm - Joel David Hamkins

      The Set-Theoretical Multiverse

Set theorists commonly take their subject as constituting an ontological foundation for the rest of mathematics, in the sense that other abstract mathematical objects can be construed fundamentally as sets, and in this way, they regard the set-theoretical universe as the universe of all mathematics. The common view among set theorists---perhaps the orthodox view---exhibits a two-fold realist or Platonic nature, asserting first that mathematical objects exist as sets and secondly, that these sets enjoy a real mathematical existence, accumulating to form the universe of all sets. A principal task of set theory, on this view, is to discover the fundamental truths of this cumulative set-theoretical universe. I emphasize that this orthodox view holds the set-theoretical universe to be unique---it is the universe of all sets---and on this view, interesting set-theoretical questions, such as the Continuum Hypothesis, will have definitive final answers in this universe.  Proponents of the view point to the increasingly stable body of regularity features flowing from the large cardinal hierarchy as indicating in broad strokes that we are on the right track towards the final answers to these set theoretical questions. A paradox for the orthodox view, however, is the fact that the most powerful tools in set theory are most naturally viewed as methods for constructing alternative set-theoretical universes. With forcing and other methods, we seem to glimpse into alternative mathematical worlds. In this talk, I shall defend a contrary view, the Multiverse view, which takes these other worlds at face value and holds that there are many set-theoretical universes. This is a realist position, granting these universes a full mathematical existence, and does not reduce via proof and completeness to a version of formalism. The multiverse view remains Platonist, but it is second-order Platonism, that is, Platonism about universes. I shall argue that set theory is now mature enough to fruitfully adopt and analyze this view. I propose a number of multiverse axioms, provide a multiverse consistency proof, and describe some recent results in set theory that illustrate the multiverse perspective, while engaging pleasantly with various philosophical views on the nature of mathematical existence.

    4:45pm - 6:15pm - Haim Gaifman

      Structuralism, Formal Proofs, and Realism

Structuralism appeals to a general primitive notion of structure, which is not defined in set theory (or any other mathematical theory); for set theory itself appeals to some such notion ─ the “intended interpretation” that justifies its axioms. Structuralism makes possible a robust notion of mathematical truth, while avoiding confusions that arise out of bad questions about “mathematical objects”. There are many simple examples of self-evident arguments grounded in structural understanding, which establish far from trivial results, requiring no formal proofs. Some constitute “rock bottom” mathematical truths. I shall discuss the role of formal proofs and the interplay of formalism and structural understanding in mathematical investigation of the individual, the community, and in a historical perspective. Gödel’s results imply an unbridgeable gap between truth, which relies on structures, and provability. They make possible a substantial realistic (or ‘Platonist”) position in the foundations of mathematics. Often, independence results have been taken to imply a weakening of mathematical realism (or “Platonism”). This is an oversimplified misleading picture. I shall go on to discuss the situation in set theory and the similarities and differences between it and the historical example of non-Euclidean geometries.

Saturday, April 4th:

(Talks on this day will be located in 100 Washington Square East (the Silver Center) room 714.  Map)

    10:30am - 12:00pm - Neil Tennant

       Core Logic

When we evaluate sentences in a model, we make inferential steps of various kinds. These can be codified in an inferentialist theory of evaluation-proofs and evaluation-disproofs. The rules involved are austere and highly intuitive. They allow one to determine truth-values of sentences modulo basic facts about the model in question. True sentences are deduced; false ones are refuted (that is, absurdity is deduced from them). These evaluation rules generalize naturally to deductive rules allowing complex sentences both in place of the 'basic facts', and in place of absurdity. One thereby attains core logic. It is constructive and relevant. Its introduction and elimination rules are in epistemically gainful harmony. Its proofs are in normal form. Core logic is a unified system for both constructing truth-makers and transmitting truth. It helps one to do away with any hard and fast distinction between syntax and semantics---or at least to replace it with a picture of 'purely syntactic' deductions lying at one end of a spectrum, and 'heavily semantic' evaluations at the other. The key to this unifying account is a generalized form of 'cut-elimination' (note the scare quotes).

    12:00pm - 1:30pm - Lunch

    1:30pm - 3:00pm - Stephen Simpson

     Toward Objectivity in Mathematics

We present some ideas in furtherance of objectivity in mathematics.  We call for closer integration of mathematics with the rest of human knowledge.  We point out some insights that can be drawn from  some current research programs in foundations of mathematics.

    3:15pm - 4:45pm - John Burgess

     Putting Structuralism in its Place

One textbook may introduce the real numbers in Cantor's way, and another in Dedekind's, and the mathematical community as a whole will be completely indifferent to the choice between the two. This sort of phenomenon was famously called to the attention of philosophers by Paul Benacerraf. It will be argued that structuralism in philosophy of mathematics is a mistake, a generalization of Benacerraf's observation in the wrong direction, resulting from philosophers' preoccupation with ontology.

Sunday, April 5th:

(Talks on this day will be located in 100 Washington Square East (the Silver Center) room 714.  Map)

    10:00am - 11:30am - Stewart Shapiro

     An “i” for an i:  Singular Terms, Uniqueness, and Reference

There is an interesting logical/semantic issue with some mathematical languages and theories. In the language of (pure) complex analysis, the two square roots of -1 are indiscernible: anything true of one of them is true of the other. So how does the singular term ‘i’ manage to pick out a unique object? This is perhaps the most prominent example of the phenomenon, but there are some others. The issue is related to matters concerning the use of definite descriptions and singular pronouns, such as donkey anaphora and the problem of indistinguishable participants.  Taking a cue from some work in linguistics and the philosophy of language, I suggest that i functions like a parameter in natural deduction systems. This may require some rethinking of the role of singular terms, at least in mathematical languages.

    11:45am - 1:15pm - Peter Koellner

    Truth in Mathematics:  The Question of Pluralism

The discovery of non-Euclidean geometries (in the 19th century) undermined the claim that Euclidean geometry is the one true geometry and instead led to a plurality of geometries no one of which could be said (without qualifi cation) to be "truer" than the others. In a similar spirit many have claimed that the discovery of independence results for arithmetic and set theory (in the 20th century) has undermined the claim that there is one true arithmetic or set theory and that instead we are left with a plurality of systems no one of which can be said to be "truer" than the others. In this talk I will examine such pluralist conceptions of arithmetic and set theory. I will begin with an examination of what I take to be the most sophisticated and developed version of the pluralist view to date -- namely, that of Carnap in ``The Logical Syntax of Language'' -- and I will argue that this approach is problematic and that the pluralism involved is too radical. In the remainder of the talk I will investigate the question of what it would take to establish a more reasonable pluralism. This will involve mapping out a mathematical scenario (using a recent result proved jointly with Hugh Woodin) in which the pluralist could arguably maintain that pluralism has been secured.

    1:15pm - 2:45pm - Lunch

    2:45pm - 4:15pm - 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.

I shall survey the current state of affairs as I see and argue that we are at a critical crossroads.

    4:30pm - 6:00pm - Kai Hauser

     Intuition and Mathematical Objects

The view that mathematics deals with ideal objects to which we have epistemic access by a kind of perception (‘intuition’) has troubled many thinkers.  Using ideas from Husserl’s phenomenology, I will take a fresh look at these matters.  The upshot of this approach is that there are non-material objects and that they can be recognized in a process very closely related to ordinary perception.  In fact, the perception of physical objects may be regarded as a special case of this more universal way of recognizing objects of any kind.