PFMRG: The First Phase

PFMRG: The Reading Resources For The First Phase

In the first few sessions we are going to learn about basic questions of philosophy of mathematics and get ourselves familiar with formal logic and formal proofs. Our discussions follow two tracks.

Track I: Basic Questions Of Philosophy Of Mathematics

I have picked two introductory resources:

• Øystein Linnebo. Philosophy of Mathematics, Princeton University Press, 2017
• David Corfield. Towards a Philosophy of Real Mathematics Cambridge University Press, 2006

Linnebo’s book will be our main source. It is an introductory textbook to the philosophy of mathematics. The advanatge of this book to many other books at the same introductory level is its achievement of explaining philosophical issues in a clear and simple language. Also, it does not require a great deal of background in logic or mathematics, but it does not evade mathematical technicalities either It is not easy to achieve this! I think Linnebo does a good job of merging logic and mathematics with their underpinning philosophies: just enough to serve the purposes of the book; after all, this is not a mathematics book.: Since our reading group comprises of people with hugely varying degree of mathematical maturity, this makes the book a great fit for us. Finally, the book covers all important movements and trends in philosophy of mathematics with the exception of Pragmatism (Peirce, Ramsey, and others).

Second, the book does not hide the great prominence it gives to Frege and his philosophy of mathematics throughout the chapters. I personally appreciate this honesty and I wish I could see it more in other publications.

We will read and discuss almost all of its chapters. To be prepared for our first few sessions I recommend reading chapters 1-4. In below, I have included additional supplementary materials for these chapters.

Corfield’s book is complementary to Linnebo’s book; It gives a pluralistic point of view to the philosophy and foundations of mathematics. It is only supplementary for the first phase. I think Corfield’s book will be a great source of ideas when we move on to the second phase of our reading group.

Supplementary Reading Materials For Track I

• We shall talk about Frege’s “context principle”. Linnebo explains it in the book very well but the discussion of it is rather short. However, there is a very nice talk by him about the context principle: The role of Frege’s Context Principle in his epistemology of mathematics, Øystein Linnebo

• A great complementary source is Awodey and Reck’s Frege’s Lectures on Logic: Carnap’s Student Notes, 1910–1914

• Steve Awodey, A. W. Carus. The Turning Point and the Revolution: Philosophy of Mathematics in Logical Empiricism from Tractatus to Logical Syntax. Pages 6 & 7 are extremely relevanet to what we discuss regarding “mathematical objects and mathematical objectivity” in Frege’s philosophy.

Track II: Introduction To Formal Logic & Formal Proofs

For this track we have the following reading materials:

• Bas van Fraassen. Formal semantics and logic. Macmillan Co. New York 1971, newly digitized version 2016.

van Fraassen’s book is based on his lectures in logic at Yale 1966–1968 and Indiana 1969–1970. These lectures were intended specifically for philosophy students. We will have a selected reading of chapters 1-3.

• Aristotle. Prior Analytics, circa 350 BCE Translated by A. J. Jenkinson
• Other translations:
  • Aristotle, Prior Analytics, translated by Robin Smith, Indianapolis: Hackett, 1989.
  • Aristotle, Prior Analytics Book I, translated by Gisela Striker, Oxford: Clarendon Press 2009

Aristotle’s Prior Analytics is the first time in history when Logic is scientifically investigated. The book is famously about deductive reasoning, known as syllogism. This will be our first introduction to deductive reasoning.

Supplementary Reading Materials For Track II

• Brentano’s doctoral dissertation Brentano saw himself not merely as a historical scholar of Aristotle, but as his intimate disciple. Brentano finds that (in Aristotle’s view) being in the sense of the categories, in particular substantial being, is the most basic; all other modes, potential and actual being, being in the sense of the true, etc., stand to it in a relation of well-foundedanalogy. Many of his mature views are prepared in this work. For example his discussion of being in the sense of being true appears to be the foundation of his later non-propositional theory of judgment.

• C. S. Peirce. Collected Papers of Charles Sanders Peirce, volume 7. Harvard University Press, Cambridge, MA, 1935. Edited by C. Hartshorne, P. Weiss, and A.W. Bucks. Paragraph 218.
• Peirce Existential Graphs
• Jeremy Avigad. Reliability of mathematical inference. Synthese, pages 1–23, 2020
• I. Lakatos. Proofs and Refutations. The Logic of Mathematical Discovery. Cambridge University Press. 2015
• William Thurston. On Proof and Progress in Mathematics. 1994.

Formal Logic Begins With Aristotle

The first paragraph of Prior Analytics From the beautiful translation by Gisela Striker gives a pretty good summary of what is in rest of the book:

First, to say about what and of what this is an investigation: it is about demonstration and of demonstrative science. Then, to define what is a premiss, what is a term, and what a syllogism, and which kind of syllogism is perfect and which imperfect. After that, what it is for this to be or not to be in that as in a whole, and what we mean by ‘to be predicated of all’ or ‘of none’.

Aristotle then continues to define syllogism.

A syllogism is an argument in which, certain things being posited, something other than what was laid down results by necessity because these things are so. By ‘because these things are so’ I mean that it results through these, and by ‘resulting through these’ I mean that no term is required from outside for the necessity to come about.

Corcoran-Aristotle’S Natural Deduction System

It will be seen that Aristotie’s theory of deduction contains a self-sufficient natural deduction system which presupposes no other logic. Perhaps the reason that Aristotle’s theory of deduction has been overlooked is that it differs radically from many of the ‘standard’ modern systems. It has no axioms, it involves no truth-functional combinations and it lacks both the explicit and implicit quantifiers (in the modern sense). … My opinion is this: if the Lukasiewicz view is correct then Aristotle cannot be regarded as the founder of logic. Aristotle would merit this title no more than Euclid, Peano, or Zermelo insofar as these men are regarded as founders, respectively, of axiomatic geometry, axiomatie arithmetic and axiomatic set theory. (AristotIe would be merely the founder of ‘the axiomatic theory of universals’.) Each ofthe former three men set down an axiomatization of a body of information without explicitly developing the underlying logic. That is, each of these men put down axioms and regarded as theorems of the system the sentences obtainable from the axioms by logical deductions but without bothering to say what a logical deduction is.