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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.
In this session we will deal with a critique to some main ideas of Predicativists. They believe that, in some respects, arithmetic has some advantages that analysis and set-theory do not. Koellner puts this idea to the test.
Main Readings:
After talking about the infinite and Frege Arithmetic, it is time to turn to potentialism about set-theory. In this session we will turn to potentiality about the height of the cumulative hierarchy.
Main Readings:
Our second meeting of the year continues along the lines of potentialism. Now we turn to Frege Arithmetic and a genuinely potentialist account of it.
Welcome to PhiMath 2024! In our first meeting for this academic year, we will discuss a paper by Linnebo and Shapiro on the notion of Potential Infinity.
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
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.
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.
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.
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:
Additional Bibliography