ThinkingAboutMath

Thinking about Mathematics (Stewart Shapiro)

  • Very good. Not actionable, but very interesting.
  • Come down with Structuralism
  • Classical logic and impredicative definitions are thoroughly entrenched in contemporary mathematics. Accepted not b/c sanction philosophically, but b/c smooth practice of math needs them.
  • Lowenheim-Skolem Theorems: Notions of natural number finitude, ect. cannot be captured in 1st order theories. Any 1st order theory, if it has infinite models, has unintended models that get these notions wrong.
  • Virtually all of math can be rendered in the language of set-theory, and all known theorems in math can be proved in ZFC (a version of set theory). However many interesting and important questions cannot be decided by ZFC axioms. eg: Cantors Continuum Hypothesis
  • Incompleteness theorem: Realist says it shows there is no axiomatization whose theorems are all and only the truths of arithmetic. "There is more to truth that provability in the deductive system" Arithmetic truth out runs formal provability
  • The incompleteness theorem shows that the undecidable sentence phi is truth of the natural numbers => we have an informal proof of the truth of the formally undecidable sentence. There is more to arithmetic provability than can be derived in any fixed formal axiomatization. => arithmetic truth and informal arithmetic provability both out run what can be produced by a machine.
  • Kant took philosophical measure of the new scientific 19th C developments. The result was a heroic attempt to explain the necessity of math and the apriori nature of mathematical truths, while explaining the place of math in the empirical sciences.
  • Kant claimed: Structure of mathematical reasoning due to the structure of our apparatus of perception.
  • Widely agreed that Kants philosophy of math failed from later developments in math and science eg: non-euclidean Geometry.
  • Frege committed to view that all propositions about numbers are knowable. (T or F)
  • More Incompleteness: Truth values of undecidable statements can be decided by embedding them in a richer structure. Also common practice of mathematicians, eg: invoke structures far richer than a statement they are trying to prove.
  • Most real numbers do not have names.
  • No consistent theory can prove its own consistency (More incompleteness)
  • How can humans, physical organisms in a physical universe, have knowledge of a casually inert realm of abstract objects?
  • Structuralism: The essence of a natural number is its relation to other numbers. The subject matter of arithmetic is a single abstract structure. Set theorist point out that the set theoretic hierarchy contains as many isomorphissms types as possible. that is the point of the theory.

Backlinks