MathematicsFinal Thoughts

Considering the strengths & limits of math

Guiding/Knowledge Questions• How can we analyze the certainty of claims in

mathematics?• If math is an arbitrary system of formal logic, why

are there so many applications in the real world? Do we create math to mirror the world or is the world intrinsically mathematical?

• Is math discovered or invented?• What roles do the ways of knowing play in

mathematics?• How does it compare/contrast with the other AOKs?

What does math offer?

• See if you can generate a list of what values math teaches, what kinds or types of knowledge it offers

Values• A respect for universal truths

• Authority is in subject, not individual

• Power of abstractions such as patterns

• A respect for consistency

• A clear focus on problem-solving

• Beauty and elegance in reasoning

• Other ideas?

GÖDEL’S Incompleteness Theorem: The Limits of Logic?

• What do the following have in common:It is impossible for written language to

communicate anything.“Everything I tell you is a lie.”

• It is impossible to prove that a formal mathematical system is free from contradiction.

• We cannot have both completeness & certainty

Choose 1 and Reflect• “The enormous usefulness of mathematics

in the natural sciences is something bordering on the miraculous.” Wigner

• “The more mathematics is abut reality, the less certain it is. The less it has to do with reality, the more certain it becomes.” Albert Einstein

• Math began as discovery, description of the real world and the need to see patterns, but now it is beyond reality. Polanyi