Kobe Philosophy of Mathematics Seminar
Kobe Philosophy of Mathematics Seminar is a bi-weekly seminar that focuses on the broad issues in the philosophy of mathematics. The format of this seminar is hybrid (in person or by Zoom). This Seminar receives support from the Japan Society for the Promotion of Science (JSPS).
| Organizers: | Professor Makoto Kikuchi |
| Dr. Zhao Fan | |
| Contact: | fanzhao@people.kobe-u.ac.jp |
Forthcoming talks
- July 16, 2024 (14:00-15:00)
This talk will take place in Room 421, 3rd Building for Research of Science and Technology, Kobe University.
Speaker Walter Dean (Warwick University) Title Incompleteness via Paradox and Completeness Abstract This paper explores the relationship borne by the
traditional paradoxes of set theory and semantics to formal
incompleteness phenomena. A central tool is the application of the
Arithmetized Completeness Theorem to systems of second-order
arithmetic and set theory in which various "paradoxical notions'' for
first-order languages can be formalized. I will first discuss the
setting in which this result was originally presented by Hilbert &
Bernays (1939) and also how it was later adapted by Kreisel (1950) and
Wang (1955) in order to obtain formal undecidability results. A
generalization of this method will then be presented whereby Russell's
paradox, a variant of Mirimanoff's paradox, the Liar, and the
Grelling–Nelson paradox may be uniformly transformed into
incompleteness theorems. Some additional observations are then framed
relating these results to the unification of the set theoretic and
semantic paradoxes, the intensionality of arithmetization (in the
sense of Feferman, 1960), and axiomatic theories of truth.
- July 16, 2024 (15:30-16:30)
This talk will take place in Room 421, 3rd Building for Research of Science and Technology, Kobe University.
Speaker Walter Dean (Warwick University) Title On Kripke on Gödel on Church's Thesis and the Entscheidungsproblem (joint work with Juliette Kennedy, Helsinki) Abstract The first goal of this paper is to reconstruct and evaluate
an argument for Church's Thesis which was first explicitly formulated
by Saul Kripke (2013) but is closely related to the frame of Gödel's
original papers on the Completeness and Incompleteness theorems
(1929/30, 1931). Kripke's argument relies on three premises: 1)
"computation is a form of deduction"; 2) Hilbert's Thesis -- i.e. the
claim that the content of a mathematical proposition can always be
faithfully expressed in a first-order language; 3) the Completeness
Theorem itself. We will highlight how subsequent work on decision
problems in computer science strengthens Kripke's argument for 1)
while Kreisel's (1967) squeezing argument for logical validity
strengthens his argument for 2). We will next present a
formalization of Kripke's additional claim that premises 1) and 2) can
be used to argue directly for the undecidability of the
Entscheidungsproblem when combine with Theorems in IX and X in Gödel's
1931 paper (whose role is otherwise somewhat mysterious). This
brings into further focus not only Gödel's early thinking about the
significance of undecidable problems but also the role of details like
finite axiomatizability and the possibility of formalizing metatheory
(including the Completeness Theorem) to the resolution of the
Entscheidungsproblem.
- July 18, 2024 (15:00-16:00)
This talk will take place in Room 421, 3rd Building for Research of Science and Technology, Kobe University.
Speaker Walter Dean (Warwick University) Title From Real Analysis to the Sorites Paradox via Reverse Mathematics" (joint work with Sam Sanders, Bochum) Abstract This paper represents an application of "reverse
philosophy" as applied to the sorites paradox -- i.e. the method of 1)
using the methods of Reverse Mathematics to assess the strength of the
principles required to sustain a given philosophical argument and 2)
building on this analysis to propose novel objections and replies.
After highlighting the role of measurement theory in theorizing about
vagueness, we illustrate the dependence of the traditional discrete
form of the sorites on Hölder's representation theorem for ordered
Archimedean groups. While it is known that Hölder’s Theorem is
derivable in RCA0, we also consider two forms of the sorites which
rest on non-constructive principles: the continuous sorites of Weber &
Colyvan (2010) and a variant we refer to as the covering sorites.
After considering antecedents from the history of real analysis, we
show in the setting of second-order arithmetic that the former depends
on the existence of suprema and thus to Arithmetical Comprehension
(ACA0) while the latter depends on the Heine-Borel Theorem and thus to
Weak König's Lemma (WKL0). I will then illustrate how recursive
counterexamples to these principles (Specker sequences and singular
covers) can be understood as providing resolutions to the
corresponding paradoxes while also indicating how such constructions
can be viewed as incompleteness theorems in the spirit of Talk 1.
Past talks
- December 15, 2023 (10:30-12:00)
This talk will be presented online.
Speaker Elkind Landon (Western Kentucky University) Title A theorem of infinity for Z-Principia Mathematica Abstract I here develop a new, foundationless simple-type grammar to replace Principia Mathematica's well-founded simple-type grammar. Rewriting the axiom schemata of Principia in foundationless simple-types, or Z-types, gives us a new system, ZPM. Adding to ZPM a plausible new axiom schema, Z*107, allows us prove Infinity in every type. Z*107 is a plausible new axiom schema because, as I will argue, it is a logical truth of ZPM. Further, using Z*107 to prove Infinity is not circular: the new axiom alone does not secure a proof of Infinity, but crucially relies on heterogeneous relations. So using Z*107 to prove Infinity is not question-begging. In this talk I also relate this result to previous discussions of Infinity and its status in the Principia-Logicist project.
- July 28, 2023 (10:40-12:10)
This talk will take place in Room 421, 3rd Building for Research of Science and Technology, Kobe University.
Speaker Sebastian Sunday Grève (Peking University) Title Gödel vs Wittgenstein: Mathematical Proof, Truth, and Language-games Abstract pdf
- April 21, 2023 (10:40-12:10)
This talk was presented online.
Speaker Zach Weber (University of Otago) Title Paraconsistency: past, present, future Abstract There are many approaches to paraconsistency, ranging from the very moderate to the more radical. In this talk I will explore and extend the more radical end of the spectrum, where there are truth-value gluts. The aim is to evaluate the philosophical and technical tenability of such an approach. I will show that there are very significant technical challenges to face on this sort of radical approach, but that there is good philosophical support for facing these challenges.
- Feburary 24, 2023 (10:30-12:00)
This talk was presented online.
Speaker Stewart Shapiro (Ohio State University) Title Semantics and Logic: the Meaning of Logical Terms Abstract pdf
This talk will take place in Room 421, 3rd Building for Research of Science and Technology, Kobe University.
| Speaker | Walter Dean (Warwick University) | |
|---|---|---|
| Title | Incompleteness via Paradox and Completeness | |
| Abstract | This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various "paradoxical notions'' for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) in order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell's paradox, a variant of Mirimanoff's paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth. |
This talk will take place in Room 421, 3rd Building for Research of Science and Technology, Kobe University.
| Speaker | Walter Dean (Warwick University) | |
|---|---|---|
| Title | On Kripke on Gödel on Church's Thesis and the Entscheidungsproblem (joint work with Juliette Kennedy, Helsinki) | |
| Abstract | The first goal of this paper is to reconstruct and evaluate an argument for Church's Thesis which was first explicitly formulated by Saul Kripke (2013) but is closely related to the frame of Gödel's original papers on the Completeness and Incompleteness theorems (1929/30, 1931). Kripke's argument relies on three premises: 1) "computation is a form of deduction"; 2) Hilbert's Thesis -- i.e. the claim that the content of a mathematical proposition can always be faithfully expressed in a first-order language; 3) the Completeness Theorem itself. We will highlight how subsequent work on decision problems in computer science strengthens Kripke's argument for 1) while Kreisel's (1967) squeezing argument for logical validity strengthens his argument for 2). We will next present a formalization of Kripke's additional claim that premises 1) and 2) can be used to argue directly for the undecidability of the Entscheidungsproblem when combine with Theorems in IX and X in Gödel's 1931 paper (whose role is otherwise somewhat mysterious). This brings into further focus not only Gödel's early thinking about the significance of undecidable problems but also the role of details like finite axiomatizability and the possibility of formalizing metatheory (including the Completeness Theorem) to the resolution of the Entscheidungsproblem. |
This talk will take place in Room 421, 3rd Building for Research of Science and Technology, Kobe University.
| Speaker | Walter Dean (Warwick University) | |
|---|---|---|
| Title | From Real Analysis to the Sorites Paradox via Reverse Mathematics" (joint work with Sam Sanders, Bochum) | |
| Abstract | This paper represents an application of "reverse philosophy" as applied to the sorites paradox -- i.e. the method of 1) using the methods of Reverse Mathematics to assess the strength of the principles required to sustain a given philosophical argument and 2) building on this analysis to propose novel objections and replies. After highlighting the role of measurement theory in theorizing about vagueness, we illustrate the dependence of the traditional discrete form of the sorites on Hölder's representation theorem for ordered Archimedean groups. While it is known that Hölder’s Theorem is derivable in RCA0, we also consider two forms of the sorites which rest on non-constructive principles: the continuous sorites of Weber & Colyvan (2010) and a variant we refer to as the covering sorites. After considering antecedents from the history of real analysis, we show in the setting of second-order arithmetic that the former depends on the existence of suprema and thus to Arithmetical Comprehension (ACA0) while the latter depends on the Heine-Borel Theorem and thus to Weak König's Lemma (WKL0). I will then illustrate how recursive counterexamples to these principles (Specker sequences and singular covers) can be understood as providing resolutions to the corresponding paradoxes while also indicating how such constructions can be viewed as incompleteness theorems in the spirit of Talk 1. |
- December 15, 2023 (10:30-12:00)
This talk will be presented online.
Speaker Elkind Landon (Western Kentucky University) Title A theorem of infinity for Z-Principia Mathematica Abstract I here develop a new, foundationless simple-type grammar to replace Principia Mathematica's well-founded simple-type grammar. Rewriting the axiom schemata of Principia in foundationless simple-types, or Z-types, gives us a new system, ZPM. Adding to ZPM a plausible new axiom schema, Z*107, allows us prove Infinity in every type. Z*107 is a plausible new axiom schema because, as I will argue, it is a logical truth of ZPM. Further, using Z*107 to prove Infinity is not circular: the new axiom alone does not secure a proof of Infinity, but crucially relies on heterogeneous relations. So using Z*107 to prove Infinity is not question-begging. In this talk I also relate this result to previous discussions of Infinity and its status in the Principia-Logicist project. - July 28, 2023 (10:40-12:10)
This talk will take place in Room 421, 3rd Building for Research of Science and Technology, Kobe University.
Speaker Sebastian Sunday Grève (Peking University) Title Gödel vs Wittgenstein: Mathematical Proof, Truth, and Language-games Abstract pdf - April 21, 2023 (10:40-12:10)
This talk was presented online.
Speaker Zach Weber (University of Otago) Title Paraconsistency: past, present, future Abstract There are many approaches to paraconsistency, ranging from the very moderate to the more radical. In this talk I will explore and extend the more radical end of the spectrum, where there are truth-value gluts. The aim is to evaluate the philosophical and technical tenability of such an approach. I will show that there are very significant technical challenges to face on this sort of radical approach, but that there is good philosophical support for facing these challenges. - Feburary 24, 2023 (10:30-12:00)
This talk was presented online.
Speaker Stewart Shapiro (Ohio State University) Title Semantics and Logic: the Meaning of Logical Terms Abstract pdf