## Weekly outline

• ### Non-ecludian geometries

The discovery (or formulation) of non-eucludian geometries can be seen as a major turn from "truth" to "consistency". The focus started to shift from "the true description of reality" to formal systems which are useful in predicting and computing physical phenomena, while their truth started to be considered as an unrelated question.  For computations, the consistency of the formal systems gained prominence.

We will read Chapter III: The discovery of non-ecludian geometry and Chapter VIII: The consistency of the non-euclidian geometries from the book Introduction to non-euclidian geometry by Harold E. Wolfe. See below.

• ### Peano and arithmetics

Giuseppe Peano published in 1889 a small book where he set out to put arithemetics (and by extension real analysis and more) onto formal ground by providing both a convenient formalism and axioms. His work became the source of modern axiomatization and also of notation, but it had some defects: most notably, he lacked the rules for the underlying logic for derivation of proofs. The axiomatization of logic (first-order logic, second-order logic, etc) had to wait for 30 years.

We will read the original small book of Peano. The text is preceded by a careful commentary and explanations by Jean van Heijenoort, and is a part of a well prepared book which collects similar articles and papers (I recommend it for reading):

Jean van Heijenoort, From Frege to Godel: A source book in mathematical logic, 1879-1931, Harvard University Press, 1967.

• ### Dedekind and the continuity of the real line

At the end of the 19th century, Dedekind published his solution to the "problem of the continuity of the real line". It finds an elegant way of rigorously defining the reals out of rational line and infinite subsets of the rational line (cuts).

Read and answer the following in preparation for the class on November 1.

• Read the chapter Continuity and irrational numbers from Essays on the theory of numbers (1901) by R. Dedekind (you can download it below).
• Make sure you understand Dedekind's reasoning and write it down for yourself in current terminology.
• Does the construction require the existence of inifnite sets of rationals? If so, how does Dedekind argue for infinity?

Infinite sets were viewed as a doubtful concept by many mathematicans. Dedekind published earlier (1888) an essay on natural numbers, in an attempt to axiomatize the notion of an number.

Read and answer the following in preparation for the class on November 1.

• Read the chapter The nature and meaning of numbers from Essays on the theory of numbers (1901) by R. Dedekind (you can download it below). This is a translation of the original paper in German (1888) Was sind und was sollen die Zahlen?
• Focus on the definition of natural numbers and on the framework in which this is done: does it rely on logic (which one) or on set theory (in which formulation), or both?
• The essays sparked a well-known controversy in which an Oberlehrer Keferstein attacted Dedekind's essays. A letter exchange followed in which Dedekind attempted to clarify his position. See below Letter to Keferstein. Read it and summarize what it says.
• ### A step back: an excluded middle?

The discussions above due to Peano and Dedekind freely used the classical logic. Further development of set theory in works of Cantor, Zermelo, Koenig, von Neumann and others (see in further sections) also assumed that the laws of logic - though not properly formalized as yet (because the need to formalize logic followed only after the attempts to formalize mathematics) - are clear and obvious. Let us stop at an important dissenting voice: we will read a translation of an address by L. E. J. Brouwer from 1923 in which he attacks the law of the excluded middle. At the beginning of the 20th century, Brouwer stood for intuitionism while Hilbert (which we'll met later) stood for formalism. The first-order predicate logic which is predominant today is based on Hilbert's work.

• Read the address On the significance of the principle of excluded middle in mathematics, especially in function theory (1923).
• Can give an example of a mathematical statement whose validity Brouwer question?
Compare with the current view and intepretation of intuitionism as described in Standford encyclopedia (see the link below).
• ### Formalization of logic: from high hopes to limitations

We will read some of the early communications of Russell, Frege and Hilbert from the first few years of 1900's which are concerned with formalizations of arithmetics and by extension of the whole mathematics (in particular of set theory which started to emerge as an axiomatic system around that time). The short letters between Russell and Frege show the awarness of the problems which formalization might encounter, but also a mild optimism with regard to the possibility of resolving them. Hilbert's address is perhaps more optimistic, in particular when he mentions the consistency of whole mathematics, which he took to be a desirable (and achievable) goal.

These optimistic communications seem too optimistic in light of Goedel's result from 1931 on incompleteness which we also include here.

Remark. While formalization of logic was successfully carried out by Hilbert and Ackermann in the end of 1920's (see also next Remark), it failed to provide the desired outcome of ensuring the consistency and completeness of mathematics, carried out in the formal system (this outcome was blocked by Goedel's results). The hopes that mathematics could be reduced to simple manipulation of symbols (in a formal system) was of a theoretical concern at the time, but would be a more practical one today with powerful computers around. However, Goedel's results put an early stop to this "easy" approach to mathematics.

Remark. An earlier result of Goedel in 1930 (also included here) showed that the system of Hilbert and Ackermann correctly captures the connection between truth and the proof (completeness theorem). This is still seen as a very important and desirable feature of first-order logic.

• ### First order or second order?

Natural axiomatizion of both set theory and arithmetics is "second-order' in the sense that both the REPLACEMENT in set theory and INDUCTION in arithmetics quantifies over collections of objects of the universe.

However, second-order axiomatization has problems related to completentess, and therefore does not deliver "what it promises".

Usually, mathematicians favour first-order logic, while philosophers may argue for second-order.

G. S. Boolos: On second-order logic
J. Vaananen: Second-order logic and foundations of mathematics

Formulate your own position on this matter for a discussion.

• ### Non-classical logics (many-valued, modal, substructural...)

In the next few topics we will deal with non-classical logics from different perspective. Their motivation ranges from reactions to the (perceived) problems of classical logics (as in Brouwer), desire to add a new expressive power (model logics), to use logical machinary to analyse computations (substructural logics, etc.). And there is always some motivation to touch on philosophical issues.

Before we start, we need to get familiar with the notion of algebraic sementics which plays an important role in many non-classical logics.

in particular Section 5.

For many purposes, it suffices to start with some knowledge of lattices

https://en.wikipedia.org/wiki/Lattice_(order)#Bounded_lattice

Boolean algebras

https://en.wikipedia.org/wiki/Boolean_algebra_(structure)

and Heyting algebras

https://en.wikipedia.org/wiki/Heyting_algebra

Boolean algebras are algebras for the classical logic, while Heyting algebras are algebras for intuitionistic logic (in the sense of algebraic semantics introduced above).

For more details, see the lecture notes for the Boolean algebras.
• ### Many-valued logics

See the article in Stanford Encyclopedia of Philosophy for an overview: https://plato.stanford.edu/entries/logic-manyvalued/.

• ### Modal logics

Let us first read the Stanford's entry on modal logics to get a perspective on things:

• ### Substructural logics

https://plato.stanford.edu/entries/logic-substructural/
• ### Cateogory theory and Homotopy type theory

First read Section 1, Categories, pages 1-28, from the book Category Theory by Steve Awodey. This will give a brief background and introduction to category theory from the mathematical point of view.

Then read the paper Category theory as the foundations of mathematics: Philosophical Excavations by J.-P. Marquis, Synthese.

Finally formulate your own position regarding the status of category theory as an alternative  framework for mathematics.

Finally, read Voevodsky personal account of his motivations for creating the so called Homotopy type theory (look to Wiki for more information).

• ### Model theory and logical dreams

Model theory is a large subject. We will first discuss Lowenheim-Skolem theorems and their generalizations which often lead to large cardinals (set-theoretical model theory).

With regard to model theory, set theory and logic, read a popular paper by Shelah.

• ### Independence results for set theory: forcing

We have discussed the method of forcing which was devised by P. Cohen in 1962. It is a general method which is used to show that some interesting set-theoretical and mathematical statements are independent over the standard set-theoretical axioms. For a quick introduction with not some many details, see my preprint A quick guide to independence results in set theory. For more details about forcing, see Kunen's book: Set theory. I also include personal recollections of Cohen, regarding his discovery of forcing.