Questions Considered l What is computation in the abstract sense? l What can computers do? l What can computers not do? (play basketball, reproduce, hold.

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Presentation transcript:

Questions Considered l What is computation in the abstract sense? l What can computers do? l What can computers not do? (play basketball, reproduce, hold a conversation, …) l What is a Turing machine and why is it important?

l Why is a Turing machine a universal computing device? l Why is the halting problem unsolvable by a computer? l Why are other problems unsolvable by a computer? l How can one classify non-halting Turing computations? l Can a computing device be more powerful than a Turing machine?

l Could quantum mechanics lead to such a device? l Could faster than light transmission lead to such a device? l How are formal logics inherently limited by Goedel’s theorem? l What are the consequences of this limitation? l How many true, unprovable statements are there?

l Why aren’t true, unprovable statements more of a problem for mathematics? l Is the human mind more powerful than a Turing machine? l Is Goedel’s theorem related to this question? l Is human consciousness related to this question? l Do humans have mathematical intuition that cannot be expressed in formal logic?

l Why can’t formal logic fully capture the concepts of l finiteness l integers l infinities beyond the integers l Can a computer be conscious? l Can a computer understand?