2/26-27 "Philosophy of Mathematics and Logic" Meeting

"Philosophy of Mathematics and Logic" Meeting

Organized by the Research Center for Thinking and Behavioral Judgment, Keio University

in cooperation with the Global Research Center for Logic and Sensitivity, Keio University


  • 2014年2月26日(水)-27日(木)
  • February 26th - 27th, 2014
  • 慶應義塾大学三田キャンパス 東館6階G-Sec Lab (最寄駅:JR 田町、地下鉄三田又は赤羽橋) (キャンパスマップ)
  • G-Sec Lab, 6th Floor of East Research Building, Mita campus of Keio University. (Campus Map)

数学と論理の哲学に関する研究集会を開催します。1日目はHusserlian Philosophy of Mathematics 及び Hilbert Schools, Gödel などとの関連についての議論をし、2日目はこれに加えてPhilosophy of Logic and Foundations of Mathematics を議論する予定です。

- 2/26 (Feb 26) Husserl's Philosophy of Mathematics and related issues (including Hilbert)
- 2/27 (Feb 27) Current State of Philosophy of Mathematics and Logic


講演者:

Speakers:

  • Mirja Hartimo (University of Helsinki, Finland)
  • Jariro da Silva (Sãn Paulo University, Brazil)
  • Vincent Gerard (University of Poitier, France)
  • Stefania Centrone (University of Oldenburg, Germany)
  • Toshihiro Suzuki (Sophia University, Japan)
  • Chen Bo (Peking University, China)
  • Helmut Schwichtenberg (University of Munich, Germany)
  • Walter Dean (University of Warwick, UK)
  • Salvatore Florio (Kansas Sate University, USA)
  • Hidenori Kurokawa (Kobe University, Japan)
  • Richard Tieszen (UC San Jose, USA) (video presentation)
  • その他ゲスト講演者およびプログラムはこちらにアップデートされます
  • Mitsuhiro Okada (Keio University, Japan)

ゲストコメンテーター:

Guest Commentators:

  • Ryota Akiyoshi (Kyoto University, Japan)
  • Sam Sanders (Gehnt University, Belgium)

参加方法:

参加自由・参加費無料。ただし会場準備のため氏名、所属を明記して「Philosophy of Mathematics Meeting参加希望」と件名を記したメールを事務局宛にお送りください。

This meeting is open to everyone.

*発表言語は英語
ポスター(PDF)はこちら


Program:

Feb.26
9:40
Jariro da Silva (Sãn Paulo University, Brazil)
“Husserl on Pure and Applied Mathematics”
10:40
Mirja Hartimo (University of Helsinki)
“Deeply Respectful Friendship: Husserl and Hilbert on the foundations of mathematics”
11:50
Mitsuhiro Okada (Keio University)
“Husserl and Hilbert on Completeness and Husserl’s Term Rewrite Based-Theory of Manifolds”
12:30 Lunch Break
14:00
Stefania Centrone (University of Oldenburg)
“On Husserl and Hilbert in Göttingen”
15:00
Vincent Gérard (Université de Poitiers)
“Bolzano's criticism of the Port Royal's Law and the commentaries of the School of Brentano: Kreibig, Franklówna, Husserl”
16:30 Tea Break
17:00
Richard Tieszen (UC San Jose)
“Monads and Mathematics: Gödel, Leibniz and Husserl” (Video talk)
18:00 Discussion
18:30 Reception Party
Feb.27
9:40
Salvatore Florio (Kansas Sate University)
“On the Innocence and Determinacy of Plural Quantification” (with Øystein Linnebo)
10:40
Toshihiro Suzuki (Sophia University)
“Phenomenology of mathematical perception”
11:20 Discussion on Philosophy of Mathematics with the guest speakers of the first day
12:00 Lunch Break
13:20
Walter Dean (University of Warwick)
“Cuts, consistency, and commitment”
14:20
Hidenori Kurokawa (Kobe University)
“Kreisel's theory of constructions and the second clause”
15:00 Break
15:20
Helmut Schwichtenberg (University of Munich)
“An approach to constructive mathematics”
16:20
Chen Bo (Beijing University)
“Troubles with Quine’s Philosophy of Logic”
17:20 Discussion on Foundations of logic and mathematics

Abstracts:

Speaker:
Richard Tieszen (UC San Jose)
Title:
"Monads and Mathematics: Gödel, Leibniz and Husserl"
Abstract:

Kurt Gödel began to study Edmund Husserl's phenomenology in 1959. On the basis of his discussions with Gödel, Hao Wang (A Logical Journey: From Gödel to Philosophy, p. 166) tells us that "Gödel 's own main aim in philosophy was to develop metaphysics -- specifically, something like the monadology of Leibniz transformed into exact theory -- with the help of phenomenology." In 1928 Husserl ("Phenomenology", Encyclopedia Britannica draft) wrote that "The ideal of the future is essentially that of phenomenologically based ("philosophical") sciences, in unitary relation to an absolute theory of monads." In the Cartesian Meditations and other works Husserl identifies 'monads' (in his sense) with 'transcendental egos in their full concreteness'. Phenomenology is represented in these works as transcendental eidetic monadology. In the first part of my talk I explore some prospects for a Gödelian monadology that result from this identification, with reference to texts of Gödel, Wang’s reports, and aspects of Leibniz's original monadology. The second half of the talk will link these reflections to philosophical consequences of the incompleteness theorems for Hilbert’s program, Carnap’s early view of mathematics as syntax, and the claim that minds are (Turing) machines. (For background see my recent book, After Gödel: Platonism and Rationalism in Mathematics and Logic, Oxford University Press.)

Speaker:
Mirja Hartimo (University of Helsinki)
Title:
"Husserl and Hilbert on the foundations of mathematics"
Abstract:

Edmund Husserl cherished the friendship of David Hilbert who was his closest colleague and defender during his years in Göttingen. The respectful relationship between the two extended to their work and continued for the rest of their lives. Like Hilbert, Husserl adopted an axiomatic approach to mathematics by the turn of the century. Husserl maintained and further developed his axiomatic model theoretical view of formal mathematics in the 1920s. In his view the axiomatic theories of formal mathematics are fulfilled by the evidence of Deutlichkeit. The source of this kind of evidence is in the “harmonious unity of possible experience,” that is understood to be a model for the theory in question. At the same time Husserl appreciated the “phenomenological spirit” of Hilbert’s program to analyze the presuppositions and basic concepts of mathematics. But Husserl was critical of basing the foundations of mathematics on the intuition of extra-mathematical signs instead of objects. Husserl’s “analytic reduction” of formal mathematics to logic of truth and hence to intuition of extra-mathematical objects in the Formal and Transcendental Logic (1929) was Husserl’s suggestion for how such an analysis should proceed. According to Husserl such an analysis is not of interest for formal mathematics and thus he appears to have thought that for mathematics model theoretical consistency is entirely sufficient. Such an analysis however is needed for philosophical reasons to show how formal mathematics relates to truth and how to avoid equivocations that may lead to paradoxes.

Speaker:
Jairo José da Silva (Sãn Paulo University)
Title:
"Husserl on Pure and Applied Mathematics"
Abstract:

I want, in my talk, to present an overview of Husserl’s philosophy of mathematics as I interpret it, but without indulging in excessive exegetics.
For my purposes, mathematics can be categorized into either pure or applied; pure mathematics being either contentual or formal. Theories of contentual mathematics have materially determined domains, by which I mean domains that are extensions of concepts that predate their theories. Before arithmetic there was the concept of number (and before that the life-world practice of numbering and counting from which the arithmetical concept of number originated). The concept of physical space, to give another example, predates physical geometry. Formal mathematical theories, on the other hand, create their own (formal) concepts; for example, the theory of quaternions or abstract Riemannian geometry. Applied mathematics is the application of formal or formally abstracted contentual mathematics, conveniently reinterpreted, in the empirical sciences and the practical life.
In all Husserl’s major works we can find instigating discussions on mathematics in all its forms and guises. Philosophy of Arithmetic of 1891 contains a logical-epistemological justification of symbolic (i.e., algorithmic, non-intuitive) arithmetic, an issue that was conveniently generalized and extensively discussed in minor works written during the following decade, particularly the Göttingen Double Lectures of 1901. In Logical Investigations (1900-1901) and Ideas I (1913) one can find Husserl’s mature views on pure mathematics, contentual and formal, and in The Crisis of European Sciences and Transcendental Phenomenology (first published in 1954, but containing material that dates back to 1935-6) an enlightening discussion and assessment of the application of mathematics in empirical science, in particular how this is even possible and a delimitation of its scope Husserl deemed adequate.
So, there is plenty from where to draw to sketch my proposed overview. Husserl’s philosophy of mathematics, however, cannot be adequately understood independently of the correct understanding of some logical and phenomenological issues, such as his concept of definiteness and the idea he so emphatically defended (for example, in Formal and Transcendental Logic, 1969 (1929), and Crisis) that science, mathematical or empirical, depends on presuppositions of a transcendental nature. I plan to show that Husserl’s views on logic and mathematics form a coherent and imposing whole demanding serious consideration from philosophers and scientists.

Speaker:
Helmut Schwichtenberg (University of Munich)
Title:
"An approach to constructive mathematics"
Abstract:

We start from minimal logic with only implication and universal quantification as connectives. Then (natural) deductions in logic can be viewed as lambda terms (Curry-Howard correspondence). Next we extend logic to mathematics. We take as base type data free algebras (booleans, natural numbers) and (Scott-Ersov) continuous functionals for higher types. Computable functionals are introduced by their defining equations. We allow inductively defined predicates, given by their defining clauses and the least-fixed-point scheme. Examples are (Leibniz) equality and the missing connectives disjunction and universal quantification. Then falsity can be defined (the boolean constants true and false are Leibniz equal), and ex-falso-quodlibet can be proved. By the Goedel-Gentzen negative translation classical logic is a fragment. In the setting an internal realizability interpretation is possible. This gives a method to extract from a proof its computational content, and automatically verify (by a formalized soundness theorem) the correctness of the resulting program.

Speaker:
Stefania Centrone (University of Oldenburg)
Title:
"On Husserl and Hilbert in Göttingen"
Abstract:

Husserl’s Double Lecture before the Mathematical Society of Göttingen
The issue of imaginary numbers, and, more precisely, of the “logical meaning of the calculatory passage through the imaginary,” is the specific topic of Husserl´s famous Double-Lecture presented to the Matematical Society of Göttingen in winter 1901. In his many references to the Double-Lecture Husserl observes that some important ideas which he presented on that occasion, in particular, his notion of definiteness, were subsequently taken over, without acknowledgement, in the logical investigations of Hilbert’s school. My talk focuses on Husserl's Double-Lecture, in particular, on the two notions of definitness Husserl presented in that Lecture.

Speaker:
Hidenori Kurokawa (Kobe University)
Title:
"Kreisel's theory of constructions and the second clause"
Abstract:

In this talk, we explore some issues concerning the theory of constructions developed by Kreisel and Goodman. The theory of constructions was an attempt to formalize the notion of constructive proof which occurs in the traditional BHK interpretation of the intuitionistic logical constants. There are two well-known issues about the theory. One is that a strong version is inconsistent -- the so-called "Kreisel-Goodman paradox". The other is the so-called "second clause" which occurs in Kreisel's amended version of the clauses for implication, negation, and the universal quantifier. The second clause was added by Kreisel in order to handle a complication pertaining to the decidability of the proof predicate and was commonly included in formulations of the BHK interpretation until the late 1970s. It has since been dismissed by most constructive logicians, sometimes on the basis of the claim that it is the cause of the Kreisel-Goodman paradox. We examine this claim and argue that the it rests on a mistake. We also discuss some reasons why the second clause is not only harmless but also potentially useful. Finally, we highlight what we consider to be the real cause of the paradox by using the framework of an quantified extension of Artemov's logic of proofs.

Speaker:
Mitsuhiro Okada (Keio University)
Title:
"Husserl and Hilbert on Completeness and Husserl’s Term Rewrite Based-Theory of Manifolds"
Abstract:

Hilbert and Husserl presented axiomatic arithmetic theories in different ways and proposed two different notions of “completeness” for arithmetic, at the turning of the 20th Century (1900-1901). The former led to the completion axiom, the latter completion of rewriting. We look into the latter in comparison with the former. The key notion to understand the latter is the notion of definite multiplicity or manifold (Mannigfaltigkeit). We show that his notion of multiplicity is understood by means of term rewrite theory in a very coherent manner, and that his notion of “definite” is understood as the relational web (or tissue) structure, the core part of which is a “convergent” term rewrite proof structure. We examine how Husserl introduced his term rewrite theory in 1901 in the context of a controversy with Hilbert on the notion of completeness, and in the context of solving the problem of the use of imaginaries in mathematics, which was an important issue in the foundations of mathematics in the period.

Speaker:
Salvatore Florio (Kansas State University)
Title:
"On the Innocence and Determinacy of Plural Quantification" (with Øystein Linnebo)
Abstract:

Plural logic has recently become an important component of the philosopher’s toolkit. Interest in this logic is motivated in large part by two alleged virtues that it is thought to possess, namely ontological innocence and expressive power. As appealing as this common picture of plural logic may be, we believe that it is far too optimistic. Our aim is to develop an alternative picture, one in which both alleged virtues of plural logic—ontological innocence and expressive power—are much less significant than they are made out to be. We first develop a set-free Henkin semantics for plural logic and argue for its philosophical legitimacy. Then we reconsider the alleged virtues of plural logic in light of the new semantics. As a result, the role of plural logic as a philosophical tool appears substantially diminished.

Speaker:
Walter Dean (University of Warwick)
Title:
"Cuts, consistency, and commitment"
Abstract:

A reflection principle is a statement or schema which seeks to express the soundness of a mathematical theory T within its own language. For instance, the so-called local reflection principle for Peano arithmetic can be understood to assert that any sentence provable in PA is true in the standard model.
In recent philosophical debates about the role of the concept of truth in mathematical reasoning, it has been repeatedly claimed that acceptance of a theory T implicitly entails commitment to some form of reflection principle for T. In opposition to this view, I will suggest (on the basis of classical results of Kreisel, Levy, and Schmerl) that the justification of reflection principles is in fact closely related to that of mathematical and transfinite induction. Onthis basis, I will then argue that consideration of an argument arising in the philosophical debate prompts us to reconsider the potential significance of another set of classical results (due to Kreisel, Takeuti, and Pudlak) about the provability of consistency statements for cut-restricted provability predicates.

Speaker:
Toshihiro Suzuki (Sophia University)
Title:
"Phenomenology of mathematical perception"
Abstract:

Mathematicians often describe their intuitive understanding of the mathematical objects with the words “mathematical perception”, and claim that it is similar to visual perception. Their claims are normally regarded as of little philosophical significance. In my talk, I will show how we can interpret their claims within the phenomenological framework of Husserl. I do not focus on the question about philosophical justification of their claims. By analyzing the similarity between mathematical intuition and visual perception phenomenologically, I would like to show how we can use phenomenology as a tool in order to understand and imagine what it is like to have mathematical intuition.

Speaker:
Chen Bo (Peking University)
Title:
"Troubles with Quine’s Philosophy of Logic"
Abstract:

This paper will discuss four ‘paradoxes’ in Quine’s philosophy of logic, including Katz’s revisability paradox, the paradox of revisability and bad translation, the paradox of revisability and deviance, and the paradox of revising logic by using logic. The author argues that two paradoxes are real, the other two are apparent and pseudo, being able to be explained away.


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公開日時: February 03, 2014 11:32


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