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Lecture

# Paradox

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University of Toronto St. George

Philosophy

PHL100Y1

Dan Dolderman

Summer

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PARADOXES OF MATHEMATICS
Interdisciplinary mathematics is a bridge which connects the real and the
intuitive to the formal and to the language. The process of interdisciplinary
mathematics is then a process of language creation, and it is where the informal and
the formal meet; this is the place where human interpretation and meaning meets the
structure and the conventional truth in the formal language. This process often results
in confusion and sometimes even to paradoxes.
This reading demonstrates how our theory of human-math dichotomy can deal
with paradoxes of mathematics. At the same time we discuss the relationship between
Syntax and Semantics, structure and meaning in the languages, and in particular in
formal languages. Finally we demonstrate how a paradox of mathematics lives in the
margins of the language of mathematics. Two famous examples of paradoxes of
mathematics are Liars paradox and Zenos paradox. They are said to have great
impact on the development of mathematics.
A Paradox is an argument that seems like a perfectly logical derivation from
perfectly reasonable assumptions, but it comes to a contradiction with its own original
premises.
1. Liars paradox: is about someone who is believed to always lie. What do you
make of it if at some point (s)he claims that whatever (s)he says is a lie. Would
you believe that?
Lets analyze this statement:
THIS STATEMNET IS FALSE
a. There are two cases to study: either this statement is false or it is true (why?)
b. Case 1: assume that the statement is genuine, believable, and true; therefore,
trusting that the statement is a good judge of itself, we must accept that the statement is false! This conclusion is in contradiction with the assumption (that
the statement is true/right/believable/correct). So,
The statement is true the statement is false!
c. Case 2: assume that the statement is false and should not be trusted; then the
content of this statement is not true, so it is not true that it is false; then it has to
be true (why?) Again this conclusion is in contradiction with the assumption,
and
The statement is false the statement is true!
2. Zenos paradox: given a line segment, the Greek Construction method was able to
write a finite set of instructions that would bisect the line (or to find the midpoint of
the line segment); here it goes: Given a line segment AB, draw a circle of radius AB
centered at each of the two points A and B. Then the two circles intersect at two points
C and D. Now the two points C and D define a line segment CD, and this line segment
intersects our original line segment, AB, in a point E. This point E is the midpoint of
AB.
Zenos used this set of logical instructions to outline a paradox about motion. This
paradox remained alive for thousands of years and thinkers have responded to it in
variety of ways. Here is a variation of Zenos paradox: for a particle of virtually no
dimensions (like a point) to travel (on a ruler) from the position marked 0 to the
position marked 1, the particle must first go through the position marked , and then
through 3/4 and then 7/8 and then 15/16, etc. This process of each time bisecting the
remaining distance NEVER ends, and the particle NEVER reaches the location 1; but
in reality the location marked 1 is obviously reached because in reality motion exists.
On the one hand the facts of Geometry suggest that we can bisect a line segment as
often as we wish, and so our particle will require a journey through an infinite process
of bisections to reach the destination, and on the other hand any particle moving with aconstant velocity surely reaches the finite destination in finite time. Then how is
Geometry a mathematical model of the real? Even worse, for the particle to reach the
point , it must first pass through the point and before that it must first go through
1/8, ., and therefore the point wouldnt even be able to start to move!
Syntax versus Semantics
A formal language, like the language of mathematics and computer programming, is a
system in which each element has a predetermined, mechanical role to play. Rules of
grammar and syntax determine the correct way of organizing these elements to form
an acceptable sentence or expression. In a formal language we are concerned with
syntactical correctness and not with the meaning. In fact meaning is secondary to the
function of the syntax; as a matter of fact it is believed that the syntax will help
defining and restricting meaning. In natural languages, like English, however the
meaning h

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