The lecture begins with a discussion of the origins of mathematics and a description of the three problems of antiquity. The third problem is the construction of an angle trisection by classical means: compass and straight-edge alone. (pp. 11-12)

a) The French academy stopped accepting proposed classical angle trisections centuries before they were proved impossible in the 1800's. Why do you think the academy did this and do you think the decision was justified?

b) Take a look at the angle trisections due to Archimedes and Apollonius. Why do these not qualify as true classical constructions?

A paradigm is a "model" or a way of thinking about things and a computer is a tool created by man. Do you think that the tools that we create can affect the way we think about things? Try to think of examples from mathematics and examples from other fields and day to day life.(p.13)

Next, the principle philosophies of mathematics are broken down into three categories (pp. 14-17). The first category is Platonism. Platonism views mathematics as the study of pure forms that exist independently of our imagination. The language of geometry and numbers is the language that we use to talk about these pure forms and a proof is constructed to reveal the properties of these forms. The schools of Logicism and Formalism rest on the principle that mathematical statements are true if they can be deduced from a set of "first principles" or axioms. Mathematics becomes a process of defining sets of axioms and deducing statements from these axioms. The Constructivist and Intuitionist schools are founded on the idea that mathematical objects are products of human thought and therefore must be properly defined. For the Constructivists and Intuitionists, mathematical statements, including proofs, must be based on objects whose existence is not merely asserted, but rather established by a description of how the object could be constructed.

Question: The traditional proof the the square root of 2 is irrational proceeds by logical contradiction: you assume that root 2 is rational and that you have expressed it as a fraction in lowest terms, say a/b. You can then show that 2 must divide both 'a' and 'b' which contradicts your assumption that a/b was in lowest terms. You therefore conclude that your initial assumption was incorrect and root 2 cannot be irrational.

a) Do you think a Platonist would accept this proof?

b) What would Logico-Formalists need to accept this proof?

c) Is this a constructive proof? In particular, this proof is based on the assertion that if a number is not rational, then it must be irrational. Would a Constructivist consider this sufficient to show that a number is irrational? (Click here for a link to a page that discusses the Law of the Excluded Middle.)

Click here for an excerpt from an article that Reuben Hersh wrote for the American Mathematical Monthly on new ideas in the philosophy of mathematics. (American Mathematical Monthly. 102, no.6, 1995, pp. 589-594)

Next, the talk examines the nature of proof and truth. In 1935, the German mathematician Kurt Godel showed that from any complete set of axioms, it is possible to construct true statements that are not provable from those axioms. In other words, mathematical truths exist that cannot be proved from first principles. This challenged both the Platonist and the Logico-Formalist perspectives. How do you think Constructivists and Intuitionists respond to Godel's result? (pp. 18-20)

Dr. Borwein next considers the nature of mathematical experiment. Can you think of a mathematical conjecture that you might make that you could use the computer to help you investigate? (pp. 21-32)

The Goldbach conjecture proposes that every even number is the sum of two primes and remains unproven in the Logico-Formalist sense. Dr. Borwein suggests that in the future mathematical research using the computer might be able to show that the Goldbach conjecture is true with probability greater than 0.99999 and that its complete truth could be established with a budget of 10 billion dollars. Would you accept a result such as the one achieved by this imagined research as a proof? Would you be willing to propose new ideas about even numbers based on this result? (pp. 34-35)