# The History of Mathematical Logic

## Event Information

### Description

Each year since 2015 we have run a one-day conference on mathematics and its history at Birkbeck (University of London) organised by the British Society for the History of Mathematics (BSHM), and supported by the Department of Economics, Mathematics and Statistics at Birkbeck.

This year's event will be on the history of mathematical logic. We chose this theme to honour the eminent mathematical historian Ivor Grattan-Guinness, former President of the BSHM, who died at the end of 2014. So this will be a particularly special conference. We are privileged to have an excellent line-up of speakers: **Adrian Rice, Apostolos Doxiadis, ****Volker Peckhaus, John Dawson, Amirouche Moktefi, Susanne Prediger **and** Michel Serfati**.

Registration will be £5 for students, £10 for BSHM members and Birkbeck staff, and £20 for non-members. This will include tea/coffee and biscuits at break times, but not lunch, as we wanted to keep the registration fee to a minimum. There are numerous places to eat within easy walking distance of Birkbeck, including a coffee shop that sells sandwiches and snacks within Birkbeck itself.

## Programme

*(Abstracts are below the programme)*

9:30 Registration and Coffee

10.00 Introduction and appreciation of Ivor Grattan-Guinness

10.20 Apostolos Doxiadis: *First there is no proof, then there is.*

11.00 Coffee

11.20 Susanne Prediger: *Learning the logical structures of deductive reasoning – insights into mathematics education research*

12.00 John Dawson: *How relevant has logic been to mathematical practice?*

12.40 Lunch

2.00 Volker Peckhaus:* Convolutions of 19th Century Logic*

2.40 Adrian Rice: *“Everybody makes errors”: The intersection of De Morgan’s logic and probability, 1837-1847.*

3.20 Tea

3.40 Michel Serfati: *The search for Laws of Thought. Some mathematical and psychological aspects in the work of Boole*

4.20 Amirouche Moktefi: *Why make things simple when you can make them complicated? An appreciation of Lewis Carroll’s symbolic logic*

5.00 Close

**Abstracts**

Apostolos Doxiadis

*First there is no proof, then there is.*

Historian of mathematics Reviel Netz has written that mathematical proof was invented somewhere in Greece, at “430 BCE plus or minus twenty years”. But the problem with origins is, precisely, that they are origins: what was there before something was there? What could proof have come from, other than from the head of a bearded Greek shouting “eureka”? In an exploration of the cultural milieu of the archaic and classical Greek world, both within but, mostly, outside mathematics, I try to create a cohesive narrative of the genesis of deductive proof, that makes it seem as something less than a miracle.

Susanne Prediger

*Learning the logical structures of deductive reasoning – insights into mathematics education research*

Deductive reasoning is one of the major mathematical practices. It is crucial to assure the validity of conjectures and a deductive theory generation. However, empirical studies have shown that many students in secondary and tertiary mathematics have serious difficulties with deductive reasoning. A reason was identified in students’ challenges to understand the logical structures. The talk offers insights into mathematics education research, which does not only document difficulties but also develop approaches for fostering students’ competencies in deductive reasoning. The approach makes explicit the logical structures and offers scaffolding for composing and expressing formal argumentations. The investigation of students’ process can enhance our understanding of mathematical and linguistic characteristics of deductive reasoning.

John Dawson

*How relevant has logic been to mathematical practice?*

For centuries mathematics has been regarded as the most secure realm of knowledge, the exemplar of rigorous deductive reasoning. Yet for much of its history it has not been such; the logicist view that mathematics is subsumed within logic is now no longer accepted; and few universities today require their mathematics graduates to have had a course in logic. What accounts for this seeming incongruity, and to what extent have results in formal logic actually affected progress in mathematics?

Volker Peckhaus

*Convolutions of 19th Century Logic*

Among Ivor Grattan-Guinness’s everlasting merits we find that he has opened our eyes for the complexity of the development of modern logic, starting from different mathematical needs. His distinction of the two traditions, the tradition of the Algebra of Logic represented by the British mathematician George Boole and the tradition of Mathematical Logic with the German logician Gottlob Frege as the champion, became very influential. In lectures and presentations Ivor Grattan-Guinness always stressed that there was a wall between these traditions precluding interrelations between them. In the paper presented it is shown that the wall had holes. It will focus on German-British interactions in 19th century logic.

Adrian Rice

*“Everybody makes errors”: The intersection of De Morgan’s logic and probability, 1837-1847.*

The work of Augustus De Morgan on symbolic logic in the mid-nineteenth century is familiar to historians of logic and mathematics alike. What is less well known is his work on probability and, more specifically, the use of probabilistic ideas and methods in his logic. The majority of De Morgan’s work on probability was undertaken around 1837-1838, with his earliest publications on logic appearing from 1839, a period which culminated with the publication of his Formal Logic in 1847. This talk examines the overlap between his work on probability theory and logic during the earliest period of his interest in both.

Michel Serfati

*The search for Laws of Thought. Some mathematical and psychological aspects in the work of Boole*

In the middle of the nineteenth century (1854), Georges Boole published an outstanding work of mathematical logic, entitled An investigation of The Laws of Thought (...). His objective was to describe the organization of human thought by means of a new algebra of which he was the inventor. Accordingly, he writes in the opening lines of the book :“The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolic language of a Calculus, and upon this foundation to establish the science of Logic and construct its method“. Such a project was unprecedented, even with the possible exception of Leibniz. Boole’s work and logical construction were the object of very numerous studies, in particular by Ivor Grattan Guinness and myself. My presentation at this Conference will describe the contexts, both mathematical and psychological, in which Boole had worked.

Amirouche Moktefi

*Why make things simple when you can make them complicated? An appreciation of Lewis Carroll’s symbolic logic*

Contrary to his contemporary colleagues, British logician Lewis Carroll (1832–1898) notoriously developed a symbolic logic without dropping the existential import of universal affirmative propositions. This constraint complicated his investigations and the workability of his logic. The object of this talk is to determine the motivations that led Carroll in this direction, the difficulties he faced due to this choice and the benefits he gained from it. It will be argued that Carroll’s decision reflected his belief in the social utility of symbolic logic and allowed him to tackle more efficiently some logical problems that resisted to early symbolic logicians.