TIMETABLE AND DIRECTIONS
What does it mean to do the philosophy, history and sociology of
mathematics in the 21st century?
July 30th, 2010
Hosted by
Department of Science and Technology Studies
University College London
Organized by
Josipa G. Petrunic
ESRC post-doctoral fellow in the History, Philosophy and Sociology of Mathematics
Directions
Wilkins Building, Garden Room
From Euston Square Underground Station
http://crf.casa.ucl.ac.uk/screenRoute.aspx?s=1178&d=785&w=False
From Euston Underground Station
http://crf.casa.ucl.ac.uk/screenRoute.aspx?s=1309&d=785&w=False
From Goodge Street Underground Station
http://crf.casa.ucl.ac.uk/screenRoute.aspx?s=1176&d=185&w=False
Timetable overview (further details below)
8:30 am - 9:00 am Coffee, tea and registration
9:00 am - 9:10 am Introduction
9:10 am - 11:15 am Session One
11:15 am - 11:30 am Break and refreshments
11:30 am - 1:00 pm Session Two
1:00 pm - 2:00 pm Lunch (provided)
2:00 pm - 3:30 pm Session Three
3:30-3:45 Break and refreshments
3:45 - 5:30 pm Session Four and concluding remarks
1
Timetable Details
9:00 am - 9:10 am
Introduction
Organizer, Josipa G Petrunic
9:10 am - 11:15 am
Session One
Broadening the remit of the philosophy and history of mathematics
GILLIES, D.
Philosophy of mathematics and sociology
MEHRTENS, H.
Mathematics as part of general history
LENG, M.
TBA
11:15 am - 11:30 am
Break and refreshments
11:30 am - 1:00 pm
Session Two
Where disciplines collide:
Defining academic practices in the history, philosophy and sociology of mathematics
GRATTAN-GUINNESS, I.
Too historical for the mathematicians, too mathematical for the historians
GRAY, J.
What does it mean to do the philosophy, history [and sociology] of mathematics in the 21st
century?
1:00 pm - 2:00 pm
Lunch (provided in the Garden Room)
2
2:00 pm - 3:30 pm
Session Three
To know mathematics is to learn mathematics:
Philosophical and historical views on mathematical learning
PETRUNIC, J.G.
Platonism, cognitive science and mathematics: what sociological accounts of training and
learning can tell us about how we ‘know’ numbers?
LAWRENCE, S.
Researching the regional history of mathematics and the implications for mathematics
education
3:30-3:45
Coffee, tea and refreshments
3:45 pm - 5:15 pm
Session Four
The implications of historical case studies in mathematics
DURAND-RICHARD, M.J.
Douglas R. Hartree (1897-1958) at the crossroads between physics and mathematics, science
and industry, analogue and digital calculus.
GIAQUINTO, M.
Euclid’s method (Exact title TBA)
5:15 pm to 5:30 pm
Concluding remarks
3
Mathematics between academics and engineering: predicting tides in the 19th century
Marie-José Durand-Richard
SPHERE, UMR 7219 (REHSEIS CNRS-Université Denis Diderot, Paris)
Abstract
Despite the fact mathematics today is essential to the modeling process, it still portrays an
image of a science in which truth depends on logical derivations from first principles, such
that mathematical legitimacy is generated internally.
I would like to show that this view of mathematical truth relies upon a philosophical
approach that conceives of language as a combination of grammar and dictionary. This
approach, clearly expressed by Descartes, was revivified in mathematics, especially in
algebra, in the works of Babbage and Boole at the beginning of the 19th century. Other
approaches to language were developed in the 20th century, some of which have authorized
historians of mathematics to consider « polysémie » not as a corrupted use of language but
rather as an indication of the richness of language. Polysémie argues that several meanings
coexist based on differing contexts. In this way, the uniqueness of meaning derives from the
use of words in very specific contexts.
I shall illustrate how this view of mathematical language might help illuminate the history of
mathematics by referring to the specific example of the mechanization of calculus from
integral calculus to harmonic analysis as it occurred in France and Great Britain. I shall
compare how, within those two different contexts, the relationship between mathematics
and engineering gave rise to differing approaches for predicting tides.
Philosophy of Mathematics and Sociology Donald Gillies Abstract This paper considers whether philosophy of mathematics could benefit by the introduction of some sociology. It begins by considering Lakatos’ arguments that philosophy of science should be kept free of any sociology. An attempt is made to criticize these arguments, and then a positive argument is given for introducing a sociological dimension into the philosophy of mathematics. This argument is that both mathematics and philosophy of mathematics can give rise to a misleading mysticism, while sociology can have a salutary de‐mystifying effect. This argument is illustrated by considering Brouwer’s account of numbers as mental constructions. His account is compared with a sociological account of how numbers were actually constructed historically. This sociological account makes Brouwer’s position look implausible, and constitutes a strong argument against it. Too historical for the mathematicians, too
mathematical for the historians
Ivor Grattan-Guinness
When I started working in the history of mathematics in the mid 1960s, the field
hardly existed at all, especially for the 19th century onwards.
In the mean time, interest has grown considerably, especially among
mathematicians and mathematics educators and also concerning the more modern
periods.
However, the field still lacks a professional base, and interaction with the historians
of science remains rare. I shall muse upon these topics, including some consideration
of the somewhat similar situation pertaining to the history of logic.
What does it mean to do the philosophy, history [and
sociology] of mathematics in the 21st century?
Jeremy Gray
Open University and the University of Warwick
Abstract
I shall briefly consider some aspects of the present state of philosophy of
mathematics and the last few years of the history of mathematics before turning to
the present relationship of history of mathematics to the history of science, which I
shall consider in institutional terms. Then I shall look at some of the intellectual and
institutional challenges facing historians of mathematics over the next few years.
Researching the regional history of mathematics and
the implications for mathematics education
Snezana Lawrence
Bath Spa University
Abstract
In this talk I will describe the research trail from working on the history of
mathematics in the Balkan countries between the 17th and early 20th century.
The questions I will attempt to pose will relate to the inheritance of mathematical
traditions, the transference of mathematical cultures from Western European
nations to the emerging national schools in the Balkan countries, the influence
French, English and German schools exerted on the emerging Balkan mathematical
cultures, and the reasons and possible implications for mathematics education.
Mathematical Explanation: A Point of Contact for History,
Sociology, and Philosophy of Mathematics
Mary Leng
University of Liverpool
Abstract
A complaint that is often raised about philosophers of mathematics is that they pay too little
attention to mathematical practice itself, ignoring the history of the subject and questioning
the relevance of its sociology. Arguably, though, some philosophical questions about
mathematics do not require a detailed understanding of the history or sociology of the
subject for their answers. In choosing between the opposing ontological accounts provided
by full blooded Platonism and mathematical fictionalism, for example, information about the
practice of pure mathematics is of little use. Both views can account for the phenomena
given in mathematical practice in similar ways, so philosophers must look elsewhere to find
arguments in favour of one over the other view. This paper considers a point of contact
where knowledge of the history and sociology of mathematics is essential for a rich
philosophical account - the question of the nature of mathematical explanation - and argues
that this question should be a central one for philosophy of mathematics in the 21st
century.
What does it mean to do the philosophy, history and sociology of mathematics in the 21st century? Herbert Mehrtens, Braunschweig Mathematics as part of general history Abstract My official field is modern history, specializing on history of science and technology; sociology and philosophy are indispensable auxiliary sciences. My perspective on the history of mathematics is that of a historian of culture, taking “culture” in an ethnographic sense, i.e. not without a sociological touch. Trying to grasp the contemporary cultures of mathematics I attempt to analyze and describe them as part of and in interaction with the general culture of the period and the geographical space in view. Further I am trying to write for a general educated academic public, not for mathematicians. I do not deny the worth of history of mathematics for mathematicians, as long as it is up to the standards of professional historiography. But I believe it to be very important to take mathematics as part of general culture and thus to attempt bridge the abyss between mathematics and the lay public. Platonism, cognitive science and mathematics: what sociological
accounts of training and learning can tell us about how we ‘know’
numbers
Josipa G. Petrunic
University College London
Abstract
The philosophy, history and sociology of mathematics have all struggled with similar
questions : “What is mathematics?” and “How do we come to know mathematical
knowledge ?” A plethora of responses have ensued. One rather popular response
has been to revivify the ghost of Plato to claim that mathematics is based on
relationships and objects that reside in a special, a priori third realm of some sort.
We come to know mathematical knowledge by accessing this special realm through
intuition. Another more recent approach has been to revivify the ghost of Immanuel
Kant to argue that knowledge of mathematics is immediate – gained through a
process of “grasping” truths that are immediately clear. Within this camp, some
philosophers of mathematics have appealed to cognitive science as a potential
means of explaining how this grasping occurs. On this account, we come to know
mathematics because it is part of the hard-wired machinery of our brains. There are
a number of problems with both of these approaches, a crucial one being the fact
that often mathematicians claim to “know” certain pieces of knowledge that they
later deem untrue, false or wrong. Neither the Platonist nor the neo-Kantian,
cognitive scientific approach can respond to this issue.
In the hopes of providing a more robust account—one which does respond to the
fact that often in mathematical history practitioners have used what we today deem
to be false, untrue, incorrect, or wrong mathematical knowledge—I will revivify the
sociological ghost of Ludwig Wittgenstein to argue that mathematics is what we
define it to be and what we choose to categorize it as (there being no need for a
priori categories here). This allows for the inclusion of a heterogeneous set of objects,
relationships and claims in a broad category known as “mathematics”. Furthermore,
we come to know mathematical knowledge through a process of training, learning
and normative socialization, all of which are temporal, changing and context
dependent. To demonstrate the historical basis for these sociologically-minded
claims, I will appeal to the various claims about mathematical knowledge made by
British algebraists in the early 19th century.