Philosophy of set theory and mathematics
Osnova sekce
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Organization of classes.
We will proceed by reading relevant articles and discussing them from various angles. The class will be organized as follows: I will introduce a certain philosophical topic and possibly add some necessary mathematical content. Then there will be a 60 minute part for reading papers (you can bring your own notebooks or tablets or use the computers in the adjacent rooms). In the final part we will be discussing the results.
Additionally, each student will pick an article and present it to the other students during the semester.
How do we identify papers we will read?
We will start with some well-established articles and add new ones according to the interests of the students.
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Please see articles and resources in this sections. They will be the primary focus of this class, but we will consider other articles as well.
The collection "Philosophy of mathematics" (Eds. Benacerraf, Putnam) contains "old" influential papers, some of them are certainly worth reading. We will read some of them.
The collection "The Oxford Handbook of Philosophy of Mathematics and Logic" (Ed. Shapiro) contains more recent papers, some of them again worth reading. We will read some of them.
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Studenti musíOznačit jako hotovo
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Studenti musíOznačit jako hotovo
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Studenti musíOznačit jako hotovo
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We have discussed Hilbert article on the infinite and Moore's historical context. We have illustrated the use of infinitary methods in analysis of arithmetics using Loeb measure on omega_1-saturated models. See Cutland's article below for more details on non-standard analysis and measures.
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In the next topic, we will extend our focus to type theory. Originally developed by Russell and Whitehead in response to set-theoretic paradoxes, it started to be influential due to connections with functional programming and constructive mathematics. Let us start by reading a paper by Wadler, "Propositions as types" and Zach "The significance of Curry-Howard isomorphism" to gain some initial understanding of the concepts.
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With the advent of forcing, concepts of infinity extended from "size", i.e., how hight a given infinity is in terms cardinals, to "width", i.e., to combinatorial properties making the universe wide in the low parts of the cumulative hierarchy. See the enclosed link for a review of "logical" and "mathematical" compactness principles (corresponding to infinity concepts).
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Please read the following two papers and think about summarizing them both. Indentify key ideas (according to you), in both of them.
1] Benacerraf, "What number could not be".
2] Hilbert, "On the infinite".
Both to be found in Philosophy of Mathematics (Benacerraf, Putnam, Eds).
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Please read Sections 10.1 and 10.2 from Barton and Friedman, "Chapter 10: Set theory and structures", page 223 in "Reflections on the foundations of mathematics".
Note: If you want to read more, you can read the whole paper. We will see next class how it goes.
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We will discuss some of the differences between category theory and set theory as they are viewed by Mac Lane (category theory) and Mathias (set theory). You can read Mac Lane's paper "Categorical algebra and set-theoretic foundations" as a concise statement of Mac Lane's position. But more interesting is the exchange between Mac Lane and Mathias given it the collection Set theory of the continuum (Judah, Just, Woodin Eds) in the sections "What is Mac Lane missing?" on page 113 and "Is Mathias an Ontologist?" on page 119. All materials can be downloaded in the section "Recommended reading" above.
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Read the paper by Macintyre "Model theory: Geometrical and set-theoretic aspects and prospects". Think how it connects with our discussions of the role of set theory vs. category theory. For understanding Macintyre's position, read first abou this life and mathematical interests, for instance here:
https://en.wikipedia.org/wiki/Angus_Macintyre
Compare with the paper of Baldwin "The Dividing Line Methodology: Model Theory Motivating Set Theory".
Focus on the fact how both see Shelah's contributions as essential, but interpret them differently. -
Read Putnam's paper "Mathematics without foundations" and think about the following:
1] Do you agree that one of the two ways of looking at mathematics -- which Putnam states --, i.e. ``Mathematics as Modal Logic'', can be understood with little loss of meaning as the simple provability in the first order-logic?
2] What is the other way in Putnam's paper?
3] Do you think (argue your point) that every mathematical statement has a truth value?
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If interested you can also read Putnam's paper "What is mathematical truth"?
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Read Balaguer's and Benacerraf's papers. Balaguer argues it is possible to formulate a convincing argument for platonism in mathematics which includes an explanation of how mathematicians obtain knowledge of mathematical objects (epistemology). It is an answer to an earlier Benacerraf's paper which claims that platonism is incompatible with naturalistic epistemology.
Think about the following points:
- What is naturalistic epistemology, and why it should be incompatible with platonism?
- What is FBP (full-blooded platonism)? Why does Balaguer think that it solves the problem of Benacerraf?
- Compare FBP with Hilbert's (implicit?) position that consistency implies existence.
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Studenti musíOznačit jako hotovo
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The class on Monday March 28 is cancelled. Use the time to read up on Hilbert's programme which we will discuss on April 4.
- Zach's paper provides a summary and discusses current development.
- Smorynski's paper is well-known and provide detailed historical context.
- If interested, read the link on epsilon-calculus.
- For more mathematical details regarding the transfinite induction in PA, with more context for espsilon_0, read the paper Transfinite induction in Peano arithmetic, Richard Sommer, APAL (1995).
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Category theory is sometimes proposed as an alternative foundational framework for mathematics. Read the following (you can download them below):
1] Read the introductory section from the book Category Theory by Steve Awoday to get some understanding of the basics of category theory.
2] Read the paper Categorical foundations of mathematics by J. P. Marquis in Review of Symbolic Logic (2013). This is a response to Feferman's paper (which is also included here for reference).
3] If you are interested, you can also read another Marquis's paper Category theory and the foundations of mathematics, Synthese (1995).
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Category theory may have "type-theoretical flavour", in the sense that category theory deals with objects of different categories and analyses functions (functors) between categories. However, category theory is not a foundational framework and by itself does not aspire to play the role of an alternative foundation for mathematics.
Type theory may aspire to be an alternative foundation. While it seemed to be (almost) forgotten after its introduction by Russell and Whitehead, its connection to computers brought it back to life. The idea of "constructive computation" may be suitably formulated in type-theory (as a logical framework), with the intuitionistic position - and since a computer program is by definition "constructive", this framework has had its proponents.
Recently, there has been some effort to connect and merge type theory as a logical framework with mathematical ideas of category theory, especially homotopy theory. This synthesis is called Homotopy Type Theory, and has beem sometimes proposed as an alternative (constructivist) framework for whole mathematics.
Read the links below (preferably in the order given) to get some idea regarding these notions.
In addition, you can read a paper by Philip Wadler, "Propositions as Types", which explains the connection between type theory and logical syntax.
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Studenti musíOznačit jako hotovo
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Woodin has been an influential mathematician, and his opinions regarding CH have been widely discussed. Read below his popular account of his position. Unlike AC, the status of CH is complex, and therefore Woodin's argument is complex as well.
It is not necessary to understand all the technical details, just try to get some idea as to what a solution to CH should be like. Part I is easier to read, Part II is more technical. The paper also introduces the Axiom of Determinacy, which has been proposed as an alternative to AC.
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Studenti musíOznačit jako hotovo
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Studenti musíOznačit jako hotovo
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We will read a more philosophical considerations regarding infinity, this time from our colleague from FFUK: Vojtech Kolman.
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Based on your preferences, please browse through the papers and a book listed below.
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Studenti musíOznačit jako hotovo
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Studenti musíOznačit jako hotovo
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Studenti musíOznačit jako hotovo
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Studenti musíOznačit jako hotovo
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