1. Computability Thank you for staying close to me!! Learning and thinking More algorithms... computability

2. Goals for Education... ● Memorizing and reproducing facts ● What else?

3. Why does it matter? ●...

4. (Some) Goals for education beyond memorizing and reproducing facts ● Questioning ● Viewing the same problem from different perspectives ● Emphasis on thinking things through, not just accepting what has been said, but really trying to understand it ● Pattern discovery – connecting ideas and facts from one area with those in another ● Problem-solving ● Thinking outside the box

5. The “Age of Reason” ● Around 1870 to 1930 ● Studies of the foundations of mathematics ● Goal was to unify all of mathematics using a small collection of basic principles ● Hope was to prove that all of mathematics could be defined ● Why did people want to do this?

6. Well... People believed that if you could fully unify mathematics in this way, one day (by today) it would be possible to absolutely correctly prove things including “Who should be President?” and “Should we build a mass transit system?”

7. Where it started... ● Simple set theory says: if e equals 2 and set S ={1, 2, 3} then e is an element of set S. ● Cantor believed that one could use set theory, starting with to define the set of all integers (positive and negative whole numbers and 0), then all rational numbers (fractions), and, hence, all real numbers (all decimal numbers)... and hence move on to defining all of mathematics.

8. Russell's Paradox ● The most famous of the logical or set-theoretical paradoxes. ● Simple (naive) set theory ignores the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.

9. Russell's Paradox ● A set can contain sets. For instance, in a set of lines, each line is a set of points. ● And a set can contain itself. ● What if S is the set of all sets that are not members of themselves. If S is a member of itself, then by definition it must not be a member of itself. And if S is not a member of itself, then by definition it must be a member of itself.

10. Not a Popular Notion ● Mathematician David Hilbert: “We will not be expelled from the paradise into which Cantor has led us.”

11. Hilbert's Solution... ● An ambitious program to prove mathematics was consistent, complete, and computable. – Consistent: without contradictions – Complete: all mathematical statements can be proven true or false – Computable: a mechanical device can theoretically be made to automatically determine the truth of any mathematical statement. ● Required all problems and their proofs to be stated completely formally

12. Required... ● All problem statements and proofs had to be state-able using strictly formal methods ● Belief was that formal methods could be used to substitute for things like human judgment, tastes, insights

13. If so... ● Once again, a computer can be created and program to conclusively generate correct answers for all problems.

14. Kurt Gödel ● Logician... in fact, viewed (sometimes with Russell and sometimes alone) as the greatest logician of the 20 th century and, with Aristotle and Frege, one of the three greatest of all time ● In 1931 proposed the Incompleteness Theorem which stated that mathematics based on formal methods could not be both complete and consistent.

15. Incompleteness Theorem ● Any self-consistent axiomatic system powerful enough to describe integer arithmetic will allow for propositions about integers that can be neither proven nor disproven within the axioms

16. Formally ● In any consistent formalization of mathematics that is sufficiently strong to define the concept of natural numbers, one can construct a statement that can be neither proved nor disproved within that system. ● No consistent system can be used to prove its own consistency.

17. Today's alternative to set theory ● von Neumann developed a formal theory based on classes where a class is a restricted form of a set. ● This avoids the paradoxes of set theory but is much more limited.

18. Ross Beveridge on what you can compute and what you can't ● It is critical to learn to evaluate what is decidable/computable and what is not. ● It is simply... wrong... to resolve by opinion that which is computable ● It is impossible to decide by computation what is not computable.

19. Homework for 10/25 ● See if you can come up with two simple examples, one of which is computable and one of which isn't. How do you know? What are the differences? ● -or- Come up with one example of each where you have seen this mis-used

20. The statement on the other side of this page is false.