Table Of ContentMathematid, Philosopbical, RtIigious and Spontancous
Students' Expldons of the Psmadox of Achilles and the
Tortoise
Matheman'cs and Statistics
Resmted in partid Fulfïhent of the Requirerncnts
for the Degree of Master in The Teaching of Mathematics
Concordia University
Montre& Quebec, Canada
August 1997
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MathematicaC, Philosophical, Religious and âpontaneous Studenb'
Explanations of the Paradox of Achilles and the Tortoise
c.
Certain areas in mathematics seem to possess deep secrets. Such are the areas of
mathematics that deal with the concepts of infinity. The concepts of infinity have ahys
stirred great emotions and produœd seemingly unsovable paradoxes. One such
paradox is Zeno's paradox of Achiller.
We begin our research by examining thu times in whkh Zeno Iived, the
intelledual arguments of that time, and the reasons why Zeno formulated his
paradoxes- We will also examine what effects this paradox M do n the development of
mathematics. This analysis will indude Anstotie's formulation of the paradox and the
general problems of actual infinity. The two viaws on the structure of matter and how
the paradox is dealt &th according to each view will also be covered. Ne* we will
examine various ways the paradox could be explained: From a mathematid point of
view we will examine the paradox in ternis of limitt, transfinite numbers and thmugh
geometric proofs. Then we will examine some of the philosophical explanations, and
how the lwo views on the structure of matter explain the paradox Finally we wil
examine how the concept of infinity is dealt with in Jewish philosophy and what bearing
this rnay have on the explanation of the paadon We will condude by Iistening to two
pairs of students' spontaneous explanation of the paradox and examining if students
background may have any affect on the way they explain and understand the paradox
It is with great pleasure and appreclreclatiothna t I acknowledge the
help of my thesis adn'sor. Dr. Anna Sierpinska, a tnie educator in
every sense of the word Wthout her guidance. knowledge,
encouragement and patience this thesis would never have been
completed.
I also wish to express rny appreciation to the fawlty and staff of the
MTM department for the financial assistance and the teaching
opportunities they have granted me.
I owe a great deal to my wife Risha and daughter Devora Leah for
being understanding and encouraging me. I thank you both.
The Basic Rmciple of dl basic principlcs d the pillar o f ds cimces
is to rralue thet that is a Fitst king who brought cvay cxisting dimg
hto being.
Moses Moimonides (Misneh Torah,B ook I Chqpter I)
Table of Contents
The Philorophicd Questions Undcnglng Zcno9rP aradoxes.
......,
What bdong to mithemitics and what bclongs to physia? oooo21
Chapter 2:Tbc Mathemitid Explinations of the Parados ofA chiües.
Espluution in Ternu of Tninsfiiite Numùer~.-......~..-~..."I19
Chapter 3: Phiiosophid Expl.nritiom oft he Purdox of Achilla
Chapter 4: Reügïour Expbtionr of Zeno's Puidox
Appendices:
Most people view mathematics as a science with no ambiguities. It is
believed that in mathematics everything can be proved and solved in a dear cut
way. True, most amas of mathematics could be viewed in thb light, but them are
many paradoxes that have aept their way into the development of mathematical
theories. Some have been solved while other linger on süII stickhg around like
an annoying virus that won't go away. Many mathematicians believe that there is
no real point wasting time dealing with these paradoxes if they have no pradital
effect on mathematical theory and application.
One such paradox is the paradox of Achilles and the Tortoise. This
paradox was posed in the fffh century BC by Zeno of Elea. The present thesis
examines the paradox in its historical context and attempts to understand what
was its objective. It will look at the infiuenœ that this paradox has had on
mathematics and how it is resolved from both a mathematical and philosophicel
point of view,
To this analysis, based mainly on the study of the existing literature, I
have added two original elements: a religious explanation of the paradox and
accounts of two pairs' of students spontaneous explanation of it: one pair with a
strong Jewish religious background and another with a calculus background.
In conclusion, I will examine if there is anything to be gained by
introducing the paradox of Achilles and the Tortoise into the teaching of
. mathematics. At what age does it becorne appropriate, if at all. to present this
and other paradoxes of infinity to the students? How would introduction of these
paradoxes Med students' intuitive understanding of infinitii,
Many students Say that they love mathematics (If mis is really loving
mathematics, is a separate question). They argue You a h y sk now if you got
the corred answer. In mathematics you aiways know exadly what is being asked
of you, it's al1 so wt and drÿ. For the most part, mis is bue but fortunafely there
are some areas in mathematics that are not at al1 so ciear and not always is
there unanimity amongst leading mathematidam as to what the final answer or
best approach is.
Wth the development of Calwlus many such problems have occurred.
Most Calculus teachers, when introducing the notion of limit. division by zero or
infinity, choose to handle these topics with a very qui& and general overview of
the phiIosophical angle and proceed to instfuct their skidents to dmply accept
the concept of limit as being that number that a fundion approaches and Yor al1
intents and purposes' is that number.
Is this really the best approach to introduœ students to the exciting and
wonderful world of mathematics. After all. as math teachefs, we are given the
opportunity to excite our students and give them something to think about
Should we allow the opportunity to arouse the passions of out students to pass
us by? By introducing students to paradoxes are we giving the student a better
understanding of the subjed? Are we forcing them to think, really deep and hard
