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University of Toronto St. George
Dan Dolderman

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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