Reading Meeting 20: Structuralism in Mathematics : Friday, February 16 (16:00-18:00 CET)

Location: Science Park 904 (A1.12)

Structuralism: Modal Set-Theoretic Structuralism

For our final meeting on structuralism, we cover the modal set-theoretic structuralism from Parsons and Linnebø. This rounds out the book we have been following and also our time with structuralism (for now). 

Reading Meeting 19: Structuralism in Mathematics : Sui Generis and Modalities - Friday, January 26 (15:00-17:00 CET)

Location: B0.209 (SP 904)

Structuralism Part III: Sui Generis and Modal Structuralism

For our next meeting on structuralism, we will be covering the two views championed by the authors of the textboock: Shapiro's ante rem, or Sui Generis, structuralism and Hellman's Modal Structuralism.

Main Readings 

Chapter 5 and 6 in Hellman, G., & Shapiro, S. (2018). Mathematical Structuralism (Elements in the Philosophy of Mathematics). Cambridge: Cambridge University Press. doi:10.1017/9781108582933

Reading Meeting 18: Structuralism in Mathematics: Categories - Friday, December 15th (14:00CET)

Location: G3.02 (SP 904)

For our second meeting on Structuralism in mathematics, we will be focusing on the category-theoretical side of the discussion. Stemming from work in algebraic topology, category theory has since the 1950's become an indisposable tool for mainstream mathematics and is seen by some to encode the structuralist philosophy into mathematics itself. We will be reading and discussing some thoughts around the plausibility of using category-theoretic foundations for mathematics and whether this really follows the structuralist philosophy.

Reading Meeting 17: Structuralism in Mathematics - Friday, December 1st (15:00 CET)

Location: Science Park B1.19M

We are now covering the theory of structuralism in mathematics. This first meeting will be an overview of the general position and its various parts together with specific focus on set-theoretic structuralism. Over the course of the coming meetings, we will spend some time on various conceptions of structuralism and their proponents.

Reading Meeting 16: Responses to Higher-Order Logic - Friday, November 10th (16:00 CET)

Location: Science Park A1.06 

We are continuing our discussion about Higher-Order Logic (HOL), this time focusing specifically on arguments against adopting Higher-Order logics for various purposes. These arguments range from (HOL) being set theory in disguise, the (lack of) applicability of (HOL) for its intended purposes, and the serious metalogical issues the logic faces.

Reading Meeting 15: Philosophy of Higher Order Logic - Thursday, October 12th (17:00 CET)

Location: Science Park 904 B1.19D

The reading group in the philosophy of mathematics is back and we hold the first meeting on the 12th of October. The topic will be Higher-Order Logic. We will base the discussion around Stewart Shapiro's chapter "Higher-order logic" in the Oxford Handbook of Philosophy of Mathematics and Logic.

Reading Meeting 13: Basics of Mereology - - Thursday, May 12th (19:00 CET)

Location: Science Park B1.19A

This time we take a recourse to mereology! In his talk, Elias Bronner will introduce the philosophical motivations for the field, Leśniewski's nominalistic position.  Further, we will discuss basic mereology and its relation to Boolean Algebras. Elias will conclude with touching upon a recent interpretation of mereology within set theory by J.D. Hamkins which suggests that mereology is too weak of a system to serve as a foundation for mathematics.

Reading Meeting 12: Mathematical Intuition - Thursday, April 21th (19:00 CET)

Location: Science Park B1.13D

This time we continued Derek So's discussion of phenomenology, its recent developments and relations to linear logic and embodied cognition.

Main Readings 

• Tieszen, R. Mathematical Intuition: Phenomenology and Mathematical Knowledge. Dodrecht-Boston-London: Kluwer Academic Publishers.

Additional Readings

• Zahavi, D. Husserl's Phenomenology, Stanford Univ. Press, pp. 14-26.

• A fragment on categorial intuition by Jan Gronwald.

Reading Meeting 11: Mathematical Intuition - Thursday,  April 7th (19:00 CET)

Location: Science Park B1.19C

In relation to the first chapter of "Mathematical Intuition" by Richard Tiszen, we discussed some notions of Husserl's phenomenology that pertain to the categorial intuition.

Derek So introduced us to some basic concepts of Husserl's phenomenology. We focused on intentionality, synthetic and eidetic intuition. Then briefly discussed the different conceptions of intuition in more contemporary phenomenology.

Main Readings

• Tieszen, R. Mathematical Intuition: Phenomenology and Mathematical Knowledge. Dodrecht-Boston-London: Kluwer Academic Publishers. SUBCHAPTER 2. OF CH. 2, "INTENTIONALITY AND INTUITION", pp. 21-25.

• Zahavi, D. Husserl's Phenomenology, Stanford Univ. Press, pp. 14-26.

• A fragment on categorial intuition by Jan Gronwald.

Reading Meeting 10: Mathematical Intuition - Friday, March 11th (17:00 CET)

Location: Science Park B1.19H

We partially discussed the 1st chapter from "Mathematical Intuition" by Richard Tiszen.

Main Readings 

• Tieszen, R. Mathematical Intuition: Phenomenology and Mathematical Knowledge. Dodrecht-Boston-London: Kluwer Academic Publishers.

Reading Meeting 9: The Philosophical Misconceptions of the Incompleteness Theorem - Friday, February 25th (17:00 CET)

We discussed some popular misinterpretations of Gödel's Theorem.

These were (1) Lucas/Penrose style of arguments against mechanism, (2) GIT as a confirmation of Platonism, (3) The "postmodern" interpretation.

The talk was given by Jan Gronwald.

Main Readings 

• (Benacerraf, 1967) God, the Devil and Gödel. in: The Monist 51(1): pp. 9-32.

• (Copeland and Shagrir, 2013), Turing versus Gödel on Computability and the Mind, in: B. Copeland, C. Posy, O. Shagrir (ed.), „Computability: Turing, Gödel, Church, and Beyond”. Cambridge, Mass.: MIT Press.

• (Krajewski, 2020) On the Anti-Mechanist Arguments Based on Gödel’s Theorem, in: „Studia Semiotyczne”, doi.org/10.26333/sts.xxxiv1.02.

• (Shapiro, 2016) Idealization, Mechanism, and Knowability, in:  L. Horsten, P. Welch (ed.), „Gödel’s Disjunction: The scope and limits of mathematical knowledge”. Oxford Univ. Press

• (Wang, 1974). From Mathematics to Philosophy. Oxon: Routledge

Additional Bibliography 

• (Lucas, 1961) Minds, Machines and Gödel, in: „Philosophy”, Vol. 36, No. 137. Cambridge: Cambridge Univ. Press

• (Turing, 1950) Computing machinery and intelligence. Mind, 433-460

Reading Meeting 8: Wang's Paradox; Dummet's Case against Strict Finitism - Friday, April 16nd (18:00 CET)

Tomasz will introduce us to Michael Dummet's famous argument against strict finitism in the philosophy of mathematics. Dummet observed that every strict finitist is committed to the paradox arising with the use of vague expressions – the Sorites paradox – and concluded that “strict finitism is, therefore, an untenable position”. Or is it?

Slides used in the presentation are available here.

Main reading: Dummett, M. (1975). Wang's paradox. Synthese, 30(3), 301-324.

Additional Bibliography

• Magidor, O. (2012). Strict finitism and the happy sorites. Journal of philosophical logic, 41(2), 471-491.

• Wright, C. (1982). Strict finitism. Synthese, 51(2), 203-282.

• Dietz, R., & Moruzzi, S. (2010). Introduction. [in:] Dietz, R., & Moruzzi, S. (Eds.). (2010). Cuts and clouds: Vaguenesss, its nature and its logic. Oxford University Press 1-19.

Reading Meeting 7: Conceptions of the Continuum, February 19th (18:00 CET)

Next reading will be Solomon Feferman's Conceptions of the Continuum, available here.

Marta will present the paper, followed by a discussion.

Slides used in the presentation are available here.

Reading Meeting 6: Gödel's Incompleteness Theorems, Free Will and Mathematical Thought, February 5th (18:00 CET)

Our next reading meeting will revolve around Solomon Feferman's article Gödel's Incompleteness Theorems, Free Will and Mathematical Thought. 

The paper can be freely accessed from the Mathematics Department site at the University of Stanford.

During the meeting, Ezra will present the paper, followed by a discussion.

Slides used in the presentation are available here.

Reading Meeting 5:  Why Philosophers Should Care about Computational Complexity, January 22nd (18:00 CET)

Next reading meeting will touch on theoretical computer science. We will be seeing how computational complexity can potentially offer new insights into philosophy of mathematics. We will be reading Scott Aaronson's survey paper Why Philosophers Should Care about Computational Complexity.

For the meeting, we expect attendants to read in advance sections 1-5, 8-9 and 12 of the paper, available here. 

The meeting will have a presentation by Andrea and Noel, followed, as usual, by a discussion.

Slides used in the presentation are available here. Noel's paper about proof complexity is available here.

Reading Meeting 4: Towards a Philosophy of Music, January 8th, 2020 (18:00 CET)  

We will be reading Iannis Xenakis' Formalized Music: Thought and Mathematics in Composition, where mathematics and music come together.

During the session, Paul Maurice will present the paper, followed by a discussion.

Reading Meeting 3: Mathematics and Transcendental Phenomenology, December 11th, 2020 (18:00 CET)  

Our third reading will be Richard Tieszen's  Richard's Mathematics and Transcendental Phenomenology.

For this session we expect attendants to have read in advance the full second chapter from the book containing the essay; that is:

Tieszen, R. (2005). Mathematics and Transcendental Phenomenology. In Phenomenology, Logic, and the Philosophy of Mathematics (pp. 46–68),  Chapter 2. Cambridge: Cambridge University Press.

The text can be accessed freely by University of Amsterdam students through this link to the library catalogue. (Recently there have been problems when accessing online collections, so it might be necessary to open the link on an incognito tab).

During the session, Derek will present the paper, followed by a discussion.

Slides used in the presentation are available here.

Reading Meeting 2: What Numbers Could Not Be, November 27th, 2020 (18:00 CET)  

Our second reading will be Paul Benacerraf's What Numbers Could Not Be.

For this session we expect attendants to have read in advance:

Benacerraf, Paul. "What Numbers Could Not Be." The Philosophical Review 74, no. 1 (1965): 47-73.

The article can be accessed freely by University of Amsterdam students through JSTOR, by logging in with the institution.

During the session, Evan will present the paper, followed by a discussion.

Slides used in the presentation will be uploaded soon.

Reading Meeting 1: Towards a Semiotics of Mathematics, November 13th, 2020 (18:00 CET) 

Our opening reading will be Brian Rotman's Towards a Semiotics of Mathematics, where mathematics is presented as an activity essentially done through writing. In order to understand what it means to read and write mathematics and what limitations this imposes on mathematics, Rotman introduces a semiotic model drawing from both Peircean and continental semiotics.

For this session we expect attendants to have read in advance the pages 97-111 from Rotman's seminal paper:

Rotman, B. (2006). Towards a semiotics of mathematics. In 18 Unconventional Essays on the Nature of Mathematics (pp. 97-127). Springer, New York, NY.

The book and this chapter can be be accessed freely by University of Amsterdam students through the link on the UvA library catalogue.

During the session, Noel will present the main points of Rotman's model, both from the paper and some other texts by Rotman, followed by a discussion on the text.

Slides used in the presentation are available here. Noel's overview paper on Rotman's work: God, Infinity and Language: Semiotic Perspectives on the Philosophy of Mathematics (2019).

Debate 2: Is Second-Order Logic Set Theory in Sheep's Clothing? - Friday, April 2nd (18:00 CET)

Tibo will give a short presentation supporting the (in)famous Quine thesis that SOL is in reality ST, followed by a debate. The rules will be uploaded in due time.

Slides used during the presentation are available here.

Suggested Bibliography (They are freely accessible from your UvA student account)

• Quine W.V.O., Philosophy of Logic (2nd Edition), Chapter 5: 'The Scope of Logic', Harvard University Press, 1986. [PDF accessible from the UvA Library]

• Boolos,G.S., On Second-Order Logic, The Journal of Philosophy, Vol.72, No.16, pp. 509-527, 1975, URL=<https://www.jstor.org/stable/2025179 >. [Boolos' paper against Quine's Thesis]

• Resnik M.D., Second-Order Logic Still Wild, The Journal of Philosophy, Vol.85, No.2, pp.75-87, 1988, URL=<https://www.jstor.org/stable/2026993 >. [Resnik's response to Boolos]

• Väänänen, Jouko, "Second-order and Higher-order Logic", The Stanford Encyclopedia of Philosophy (Fall 2020 Edition), Edward N. Zalta (ed.),
  URL=<https://plato.stanford.edu/archives/fall2020/entries/logic-higher-order/>.

• Van Benthem J., Doets K. (1983) Higher-Order Logic. In: Gabbay D., Guenthner F. (eds) Handbook of Philosophical Logic. Synthese Library (Studies in Epistemology, Logic, Methodology, and Philosophy of Science), vol 164. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7066-3_4

Debate 1: Is the Physical World a Mathematical Structure? - Friday, March 12th (18:00 CET)

On March 12th we will try a debate format.  Rover will present the main arguments of M. Tegmark's Mathematical Universe Hypothesis (MUH) followed by the debate.  Rules of the debate will be announced soon. Meanwhile,  Tegmark's paper can be found here.

Slides used during the presentation are available here.

Guest Talk 5: Joel David Hamkins | Book Presentation: 'Lectures on the Philosophy of Mathematics'  - Friday, March 19th (18:00 CET)

Professor Joel David Hamkins will present to Φ-Math his upcoming book Lectures on the Philosophy of Mathematics. The presentation will contain an overview of the book's contents and motivation with a focus on selected philosophical problems tackled in it, followed by a discussion/questions from attendants.

Guest Talk 4: Anna Bellomo | Bolzano, Collections, Sets and Infinity  -  February 26th (18:00 CET)

Anna Bellomo presented her ongoing PhD research on Bolzano's conceptions of infinity, as well as her involvement on the e-Ideas framework.

'In the philosophy of mathematics circles, Bernard Bolzano (1871-1848) is mostly known for two things: his contributions to the so-called rigorisation of analysis, and his proto-Cantorian theory of size for infinite sets. In this talk, I will focus on the latter and summarise some recent findings suggesting that, contrary to what has been so far the default interpretation of Bolzano's treatment of the countable infinite, his focus was not a theory of size for infinite sets, but solving some problems relating to the treatment of (non-convergent) infinite sequences.'

Slides used during the presentation are available here.

Guest Talk 3: Pieter Adriaans | An Information Theoretical Perspective on the Separation of the classes P and NP, by - February 12th (18:00 CET)

The P vs. NP problem, one of the seven Millenium Problems, is one of the most relevant unsolved question in theoretical computer science. The progress in the last decade, however, has been little. Can information theory and philosophy of information provide new insights as to why these classes should be distinct (or the same)? Pieter Adriaans will be offering a talk on the subject, followed by a discussion.

'In this talk, I will present a perspective on the P vs. NP problem in the context of philosophy of information. P is the class of decision problems that can be solved in time polynomial to the length of their input by a deterministic computer. NP is the class of problems that can be solved in polynomial time by a non-deterministic computer. This description gives a direct relation with the philosophical question of determinism versus non-determinism and the problem of the interaction between information and computation. This suggests that a careful analysis of the flow of information through computational processes might help us to understand the P vs. NP problem better. Unfortunately, the existing theories of information are not very adequate for this purpose. Shannon information is based on a statistical notion of entropy that is not directly applicable. Kolmogorov complexity defines a structural concept of entropy that is asymptotic and not computable. Apart from that the classes of problems known to be in NP do not have a structure that easily facilitates the application of information theory. There is, for example, no univocal theory of information measurement for finite sets of numbers. This complicates an information theoretical analysis of the subset sum problem and related problems in NP considerably. I therefore propose to look at a new class of problems that I call Multiple Mutual Key (MMK) decision problems. The case for MMK not being in P is at least prima facie stronger than for other problems in NP since the checking function is a repeated application of one time pad code ciphers, which is a provably safe encryption technique. Apart from that problems in MMK are constraint free: any binary string of an adequate length defines an MMK decision problem. This implies that the average case complexity is high and makes it easy to apply maximum entropy techniques.'

Guest Talk 2: Dean McHugh | Newcomb's Paradox, by - January 29th, 2020 (18:00 CET)

On the last Friday of January, we will have the pleasure to host  Dean McHugh presenting Newcomb's Paradox, a game-theoretical problem in decision theory with many diverse philosophical implications.

Guest Talk 1: Luca Incurvati | Book Presentation: 'Conceptions of Set and the Foundations of Mathematics' - January 15th, 2020 (18:00 CET)  

Luca Incurvati will be presenting his recently published book Conceptions of Set and the Foundations of Mathematics. The book is accessible through the UvA Library.

Book summary:

Sets are central to mathematics and its foundations, but what are they? In this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with which these conceptions are associated. He shows that the conceptual landscape includes not only the naïve and iterative conceptions but also the limitation of size conception, the definite conception, the stratified conception and the graph conception. In addition, he presents a novel, minimalist account of the iterative conception which does not require the existence of a relation of metaphysical dependence between a set and its members. His book will be of interest to researchers and advanced students in logic and the philosophy of mathematics.

Slides used during the presentation are available here.

Φ-Tea 3: Your least and most favourite aspects of PoM - Friday, March 5th (18:00 CET)

Another informal social gathering of our group for tea time! A good opportunity to get to know better the new members. 

The topic: things that impressed and irritated us the most in the PoM during the 3 months the group runs. Of course, people can bring their own most and least favorite things regarding PoM outside of the group's activity.    

Φ-MAS: Gödel's Ontological Argument, December 21th, 2020 (19:00 CET)  

For this special Christmas meeting, Rodrigo will be introducing Kurt Gödel's argument about the existence of God, a classic medieval argument formalized in modal logic. 

No reading preparation is required.

During the session, Rodrigo will present the argument, followed by a discussion.

Slides used during the presentation are available here. See also a handout with the proof.

Φ-Tea 2: The Linguistics of Numerals, December 4th, 2020 (19:00 CET)  

What are numeral words, and how do they work in different languages? Can we extract meaningful insights on the nature of numbers by inspecting the linguistics of the words that denote them?

During the session, Bobby will present an overview of the topic and the different research approaches to date, followed by a discussion.

Slides used in the presentation are available here.

Φ-Tea 1: Constructive vs. Non-Constructive Proofs, November 20th, 2020 (18:00 CET)

We will join for a Zoom tea where we will discuss constructive and non-constructive proofs.

We ask participants, although not necessary, to have a constructive and a non-constructive proof of a theorem they find interesting or illustrative, along with a relevant philosophical input.