


History and Philosophy of Mathematics
Org: Duncan Melville (St. Lawrence) [PDF]
 FRANCINE ABELES, Kean University, NJ
Lewis Carroll's Visual and Formal Logics
[PDF] 
Charles L. Dodgson (Lewis Carroll) first published his visual method
in The Game of Logic, a book published in 1886, extending it
ten years later in Symbolic Logic, Part I. Originally designed
to teach the theory of inference in Aristotelian logic and to improve
on the earlier diagrammatic methods of Leonhard Euler (1772) and John
Venn (1880), Carroll's method has not been considered seriously as a
visual logic system.
In the two parts of Symbolic Logic, the second part first
published in 1977, Dodgson developed a formal logic in which he set
down valid rules for making inferences.
In this paper, I will describe the methods he invented to mechanize
reasoning in his formal logic, and demonstrate the superiority of his
diagrammatic method over Euler's and Venn's methods.
 AMY ACKERBERGHASTINGS, Maryland U. College
John Playfair in letters
[PDF] 
This talk is a progress report on an effort to locate, transcribe, and
annotate all of the surviving correspondence written by John Playfair
(17481819), Professor of Mathematics and then of Natural Philosophy
at Edinburgh University. The letters reflect Playfair's wideranging
interests in philosophy, geology, anthropology, chemistry,
architecture, literature, and drama. They contain lists of his
prominent friends: John Leslie, Dugald Stewart, William Robertson,
John Rennie, Mary and Agnes Berry, Lord and Lady Minto, Archibald
Constable, and so on. They share opinions on how the British
government handled the American Revolution and Napoleonic Wars. And,
woven throughout the correspondence, are references to Playfair's
lifelong concern for mathematics and its history, including his
analysis of observations made at Schehallien, papers prepared for the
Transactions of the Royal Society of Edinburgh, and his
supplement to the Encyclopaedia Britannica.
 TOM ARCHIBALD, Simon Fraser University, Burnaby, BC V5A 1S6
Mathematics and the First World War
[PDF] 
Following the declaration of war in 1914, scientific communications
were interrupted. This occurred formally, so that (for example)
German journals were no longer sent to libraries or individuals in
enemy states. There was also an informal side, with many individuals
deciding to cease correspondence with colleagues on the other side.
This talk will give some examples of consequences internationally both
during and after the war, as well as discussing activities by a few
people who chose to resist this path.
 DAVID BELLHOUSE, Department of Statistical and Actuarial Sciences, University
of Western Ontario, London, ON N6A 5B7
A War of Words in Pictures: the dispute between Montmort and
De Moivre over the probability calculus
[PDF] 
In 1708 Pierre Rémond de Montmort published his book Essay
d'analyse sur les jeux de hazard, an analysis of games of chance of
the time using probability theory. Three years later Abraham de
Moivre published his treatise De Mensura Sortis solving various
problems in games of chance, again using probability theory. Montmort
felt that De Moivre had plagiarized his work. In 1718 De Moivre
published an expanded version of his original Latin treatise under the
name The Doctrine of Chances. The dispute is described from
surviving publications and letters. Both the Essay d'analyse
and The Doctrine of Chances contain engravings that describe in
pictures the nature and importance of their work. These pictures are
analyzed in the context of the dispute.
 ROBERT BRADLEY, Adelphi University, Garden City, NY
The Genoese Lottery and the Partition Function
[PDF] 
In 1749, King Frederic the Great sought Euler's mathematical counsel
concerning the establishment of a state lottery. The combinatorial
issues involved in the analysis of this game of chance, known as the
Genoese Lottery, piqued Euler's curiosity. As a consequence, he wrote
four memoirs over the course of his career examining questions arising
from this lottery. We will survey these works, paying particular
attention to the use of the partition function in the second one.
 ED COHEN, Ottawa, Ontario
The Iranian Calendars
[PDF] 
We study the modern Iranian (Persian) calendar after first considering
older Iranian calendars. In examining the modern Iranian (solar)
calendar, we also discuss arithmetical formulas necessary to convert
it reciprocally into the Gregorian calendar.
There will moreover be a short investigation on the Islamic (or Muslim
lunar) calendar vis à vis the modern Iranian calendar. A few
centuries after the modern Iranian calendar was started, Omar
Khayyám (10481131) studied the exactitude of the leap year and
made important contributions. Finally, we examine the political
situation of the 20th century, which changed the various calendars
back and forth, eventually settling again on the main calendar.
 ANTONELLA CUPILLARI, Penn State Erie, Station Road, Erie, PA 16563
The sixtyfourth article of the Instituzioni Analitiche
[PDF] 
In 1748, after ten years of hard work, Maria Gaetana Agnesi
(17181799) published the first Calculus book designed for teaching
and written in Italian: Instituzioni Analitiche ad uso della
Gioventu' Italiana (Analytic Institutions for the use of the
Italian Youth). In the introduction to her work, Agnesi wrote:
... when considering the Integral Calculus, the Reader will find
a completely new method for Polynomials, which has not appeared
anywhere else; it belongs to the famous and never sufficiently
praised Count Jacopo Riccati, Nobleman very proficient in all
sciences, and well known in the literary world. He wanted to do me
the favor of letting me know about it [the method], favor that I did
not deserve, and I want to give him, and the Public, the appropriate
justice, as it should properly be done. What was this new method,
presented in the sixtyfourth article of the book? Was it really
about polynomials? Is it as useful as Agnesi seemed to think?
 DAVID DeVIDI, Department of Philosophy, University of Waterloo, Waterloo,
ON N2L 3G1
Logical pluralism and the municipal bylaws of thought
[PDF] 
Is there any sense in which it is both interesting and
true that there is a plurality of logics? There are, of
course, a multiplicity of systems traveling under the name `logic':
various modal, deontic, combinatorial, constructive, paraconsistent,
relevant, higher order, free, and other `logics', not to mention
impoverished ancestors like Aristotelean syllogism, that differ from
the standard firstorder predicate logic favored by mathematicians and
philosophers. But for all that, there might be a plurality of logics
in only a trivial or uninteresting sense.
In this paper the prospects for logical pluralism are investigated. In
particular, currently popular defenses of pluralism, such as the one
due to J. C. Beall and Greg Restall, are investigated and found to yield
just such an uninteresting logical plurality. An alternative version
of pluralism is sketched, beginning with the observation that a
variety of traditional accounts of what distinguishes logical from
nonlogical principles, usually regarded as equivalent, actually draw
the logicnonlogic line in different places.
 THOMAS DRUCKER, University of WisconsinWhitewater, Whitewater, WI 53190
Serendipity in Mathematics
[PDF] 
Robert K. Merton's `On the Shoulders of Giants' has had a good deal to
offer the community of historians of mathematics as well as historians
of science in general. His posthumous book on serendipity has come in
for rather harsher treatment by both groups. This talk is designed to
point out some of the theoretical features of the book which apply to
mathematics and to illustrate how this fits into Merton's general view
of the rationality of the scientific and mathematical enterprises.
 CRAIG FRASER, Inst. Hist. Phil. Sci. Tech., Victoria College, University
of Toronto, Toronto, ON M5S 1K7
Theoretical Cosmology and Observational Astronomy Circa 1930
[PDF] 
That the invention of geometric cosmological models based on general
relativity occurred at the same time that Vesto Slipher and Milton
Humanson were documenting systematic large nebular redshifts seems to
have been a coincidence. Edwin Hubble in 1936 explicitly associated
the expandinguniverse interpretation of his redshift law with
relativistic cosmology; for Hubble, universal expansion was a
theoretical notion rooted in relativity theory. The paper explores
various historical questions concerning the relationship between
theory and observation in cosmology around 1930.
 ROGER GODARD, Department of Mathematics, Royal Military College of Canada
Convexity
[PDF] 
By the middle of the 19th century, it was recognized that Euler's
gamma function had some special properties. One of them will be
convexity. A curve is convex if the following is true: take two
points on the curve and join them by a straight line; then the portion
of the curve between the points lies below the line. A convex
function cannot look like a camel's back! It corresponds to a
fundamental geometric concept of a function. In this work, we present
some concepts developed by Jensen in 1906, in Sur les fonctions
convexes et les inégalités entre les valeurs moyennes, and
mainly Minkowski's work. We discuss some important applications of
convexity in variational calculus, linear programming and nonlinear
programming.
 HARDY GRANT, York University, Toronto
Greek Mathematics and Greek Science
[PDF] 
The application of mathematics to the physical world was rather more
problematic for the Greeks than it is for us. I shall here be
concerned not with the achievements of the Greeks in this line, which
are generally well known, but with the underlying conditions, the
philosophical and cultural assumptions which tended to encourage or to
inhibit the endeavour.
Relevant issues include the compartmentalization of knowledge, debate
over the desirability and possibility of abstraction, and the
widespread doubt that we can have true knowledge of the changeable.
 ALEXANDER JONES, University of Toronto, Toronto, ON M5S 2E8
Enigmas of the Keskinto Astronomical Inscription
[PDF] 
Just over a hundred years ago, a Greek stone inscription was
discovered on the island of Rhodes, at a site near Lindos called
Keskinto. The inscription, of which only the last fifteen lines are
partially preserved, dates from about 100 B.C. and contains a set of
periodicities for the motions of the planets. The relations between
some of the numbers were explained by Paul Tannery shortly after the
text of the inscription was published in 1895, but little progress has
been made since in making sense of the astronomical and mathematical
principles underlying the inscription. The present talk will discuss
the problems and present some new tentative solutions.
 MIRIAM LIPSHUTZYEVICK, Rutgers the State University (Retired); 22 Pelham Street,
Princeton, NJ 08540
Paul Lévy and the Dichotomy between the Normal and other
Stable
[PDF] 
The ubiquity of describing the statistics of characteristics in large
populations by a normal distribution is commonly accepted. Thus for
instance Hernstein and Murray in their widely disseminated book
The Bell Curve describe the Normal Distribution as:
"A common way in which natural phenomena arrange themselves
approximately."
In fact their model for the distribution of IQ's is mathematically way
off the mark in satisfying the criteria for a normal distribution.
A more precise, yet still off the mark definition is in Jim Holt's
article in the January 4, 2005, New Yorker Magazine:
"As a matter of mathematics the Bell Curve is guaranteed to arise
whenever some variable is determined by lots of little causes (like
human height, health, diet) operating more or less independently."
The great French mathematician Paul Lévy in writing his classic 1924
Calcul des Probabilitésin spite of the fact that the
eminent mathematicians Borel and Deltheil felt it unnecessary to make
the notion of probability more mathematically precise rather than to
rely on common sense reasoningintended to systematically develop
and use the method of characteristic functions in order to simplify
proofs about limit laws.
A sufficient condition for the sums of a large number of "individually
small", independent random variables to approach the normal
distributioni.e., for the Central Limit Theorem to
holdwas first given by Liapounoff in 1901 and a more general one by
Lindeberg in 1922. Paul Lévy used this simpler method to derive
Lindeberg's condition. In so doing he put his finger on the essential
necessary condition and its meaning for the Central Limit Theorem to
hold. This condition states that the components of the sum be not
only "individually negligible" (small) with respect to their total
sum, but that they be "uniformly negligible" (we might use the term
"collectively negligible"), i.e., the probability that
even the largest component random variable be of the order of the
magnitude of the sum, must be negligible.
Lévy showed that in case this condition is not satisfied there exist
families of limiting distributions for sums of independent random
variables, among which the socalled Stable Laws of Index a,
where 0 < a < 2. Here the approach to the limiting
distribution is determined by the contribution of the few largest
components in the sum. Consequently the probability of values of the
sum which deviate from the mean (or the median in case a < 1)
by a large amount is considerable. The "tail" of the probability
distribution of the largest component in the case where the sum
converges to a stable distribution of index a, as well as the
"tail" of the limiting stable distribution decrease as the function
x^{a}. The limiting distribution of the sum reflects that of
its largest component terms. This dichotomy defines the "domains of
attraction" of the normal vs. those of the other stable
distributions.
The statistics of social phenomena in which stable distributions
prevail such as wealth, power, batting averages, intellectual
accomplishments, physical beauty, etc., are hardly ever
discussed in the popular culture where they most emphatically deserve
more attention.
 AMIROUCHE MOKTEFI, IRIST (Strasbourg)/LHPS (Nancy), France
How did Lewis Carroll become a logician?
[PDF] 
It's well known that Lewis Carroll, the famous author of the Alice
books, was a mathematician. His works include essentially manuals of
Euclidean geometry, a treatise on determinants, popular textbooks on
logic and collections of problems and puzzles. The majority of these
works were signed with his real name: Charles L. Dodgson. The logical
works are an exception. In effect, Carroll signed his two textbooks
The Game of Logic (1886) and Symbolic Logic (1896) and his
two articles in Mind: "A logical Paradox" (1894) and "What the
Tortoise said to Achilles" (1895) with his "literary" pseudonym.
This state of affairs led to a number of prejudices and
misunderstandings which influenced the reception of the work. People
thought that Carroll's work was intended for children, that he
considered logic a game and that his logical work completed and
concluded his literary work. Even when his works revealed accurate
ideas and discoveries, commentators claimed Carroll was not fully
"conscious" of the depth of his works.
In this paper, I will essentially try to contextualise Lewis Carroll's
logical work according to three themes: first, historically according
to the development of the new logic in the nineteenth century; then
geographically, by focusing on the academic British background; and
finally personally by situating Carroll's logical works in relation to
the rest of his work. From this, we can correct certain received
ideas which harm the understanding of the work. Also, we will be
able to suggest new ways to explain the use which Carroll made of his
pseudonym, the growing interest which he had in logic, and finally the
status he gave it.
 DAVID ORENSTEIN, Toronto
`Greenwich? Ca n'importe. Où est PARIS?' A local
postConquest eclipsebased longitude calculation at Quebec
[PDF] 
One of the teaching fathers at the Séminaire de Québec observed
the October 27, 1780, solar eclipse. In an unsigned manuscript
attributed to ThomasLaurent Bédard, superior of the Séminaire, he
followed the methods of Lalande's ASTRONOMIE, using the copy from the
seminary's library, still on site in the archive. From this detailed
observation he calculated the longitude of Quebec relative to Paris,
not Greenwich, England, despite the British Conquest.
 ED SANDIFER, Western Connecticut State University, Danbury, CT 06810
Euler's Calculus Texts
[PDF] 
Euler's fourvolume 2500 page calculus text is often described as the
origins or foundations of the modern calculus curriculum. This idea
should not be accepted without some reflections. For example, few
modern mathematics curricula include properties of elliptic integrals,
as does Euler's Integral Calculus. We describe the content and intent
of Euler's calculus course, and make some comparisons with the modern
curriculum.
 JONATHAN SELDIN, University of Lethbridge, Lethbridge, AB T1K 3M4
Curry's Formalism as Structuralism
[PDF] 
H. B. Curry is known for a philosophy of mathematics which he called
"formalism". However, most people who know anything of his
philosophy identify it with an early version which dates to 1939,
relatively early in his career. In this early version, Curry proposed
that mathematics be defined as the "science of formal systems",
where he had in mind a definition of formal system somewhat different
from the usual notion. Among the criticisms of this proposed
definition are the suggestion that under this definition there could
have been no mathematics before there were formal systems, a little
more than a century ago.
In this paper, I will quote from Curry's later work to show that this
criticism does not apply to his mature philosophy, and that his mature
version of formalism is a form of structuralism.
 JOEL SILVERBERG, Roger Williams University, Bristol, RI 02809
The Mathematics of Navigation as Taught in Private Venture
Schools, Academies, and Colleges in the New England Colonies,
17251850
[PDF] 
Curricula and textbooks used in colleges, academies, and private
schools in the newly formed United States between 1776 and 1826 show a
gradual evolution from a vague exposure to whatever mathematical
works, primarily of British origin, were in the possession of tutors,
teachers, and faculty, to a more structured plan in which students
progressed through arithmetic, algebra, geometry, and trigonometry,
culminating in an application of these areas of mathematics to
navigation and surveying, supported by texts specifically written for
that purpose.
While these navigational topics can be found in documents dating from
the earliest years of the 18th century, reflecting the study of
navigation from private tutors and almanac makers, by the second
decade of the 19th century the teaching of navigation had divided into
at least two paths: one for professional mariners typified by the work
of Nathaniel Bowditch, and a second as a capstone liberal arts
experience (intended to teach scientific and mathematical principals)
at colleges and universities, exemplified by texts written by Jeremiah
Day, Professor of Mathematics and eventually President at Yale
University.
This talk will present the details of these curricula and the ways in
which navigation was used to illuminate the principles of Geometry and
Trigonometry for the students of early America.
 JIM TATTERSALL, Providence College, Providence, RI 02918
Arthur Buchheim and an Interpolation Formula
[PDF] 
Arthur Buchheim was a shortlived mathematician of great promise. He
attended the City of London School when Edwin A. Abbott (
Flatland: A Romance in Many Dimensions) was headmaster. Buchheim
received his undergraduate degree from Oxford. He left England for a
time to study under Felix Klein in Leipzig. Upon his return, he
accepted a position as mathematical master at the Manchester Grammar
School. In the short span of seven years, and in deteriorating
health, he published twentyfour papers on a wide variety of
mathematical topics. Sylvester claimed that "had his life had been
spared, I think we may safely say of him what Newton said of Cotes,
that if he had lived, we should have known something." We focus our
attention on Buchheim's work and accomplishments.
 ROBERT THOMAS, University of Manitoba, Winnipeg, MB R3T 2N2
Mathematics as a science
[PDF] 
While mathematics can be regarded as both a science and as an art and
benefits from the tension between those two motivations for practising
it, philosophies of mathematics often do not take either of these
often defended views with much seriousness. While I am unable to
offer philosophical elaboration of the art view (though not wishing to
disparage it), I present how it is that I see my science view of
mathematics as connected to similar views of other sciences.
 CAMERON ZWARICH, University of Waterloo, Waterloo, ON N2L 3G1
Philosophical Implications of Recent Work in Set Theory
[PDF] 
Cohen's method of forcing allows one to show that many concrete
mathematical statements are undecidable from the axioms of ZFC
(ZermeloFraenkel set theory with the Axiom of Choice). A natural
question arises: Can one repair the weaknesses in ZFC exposed by
forcing, and if so, to what extent? The investigation of this
question involves the study of higher axioms of infinity (the
socalled "large cardinal axioms"), their canonical models, and the
determinacy of infinite games. Recently, Woodin has developed
Wlogic, a strong extension of firstorder logic that is the
natural logic given by the method of forcing. He has also developed a
transfinite proof system for Wlogic, and isolated the W
Conjecture, which is essentially completeness for this logic and its
corresponding proof system. A proof of the W Conjecture would
quantify the limits of forcing, provide a possible solution to the
Continuum Hypothesis, show that those large cardinal axioms which
admit an inner model theory of the kind that we know today are
"cofinal" amongst all large cardinal axioms, and challenge the
popular conception that there is no need for additional axioms of set
theory.

