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From Kant To Turing

From Kant To Turing. He Sun Max Planck Institute for Informatics. Question 1: Time vs. Space. Time Hierarchy Theorem (1965) P NP PSPACE EXP We can reuse space many times, while we cannot reuse time. What is the relation to the theory of relativity?.

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From Kant To Turing

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  1. From Kant To Turing He Sun Max Planck Institute for Informatics

  2. Question 1: Time vs. Space • Time Hierarchy Theorem (1965) PNPPSPACEEXP • We can reuse space many times, while we cannot reuse time. • What is the relation to the theory of relativity?

  3. Question 2: Randomness in Computer Science 01010000110101010100111 01010101101010001010110 101000101001010111101 OWF f x f(x) • Why randomness is useful in most settings, and ``should” be useless for poly-time algorithms? • Under reasonable assumptions BPP=P (1997). • Why can hardness generate randomness? • One-Way Functions exist iff Pseudo-random Generators exist (1989). • Randomness Evaluated by Computation Models = Randomness Evaluated by Entropy (1999)

  4. Why Philosophers Should Care About Computational Complexity Scott Aaronson • Computational Complexity and Turing Test • Gödel’s idea on evolabilityandLeslie Valiant’sevolability theory • PAC-Learning and the Problem of Induction • Quantum Computing and the Many-Worlds Interpretation • “Traditional ” Proofs and Zero-Knowledge Proofs • Complexity, Space and Time

  5. John von Neumann, 1903-1957 150 publications, 60 in pure mathematics, 30 in physics, and 60 in applied mathematics The Computer & The Brain, 1958

  6. John George Kemeny, 1926-1992 Scientific American, 1955

  7. Alan Turing, 1912-1954 Mind, 1950

  8. Grandjean Burke’s Questionnaire (1974) Burke: Among the people listed above, who have great influence to your work and interests? Gödel: Only Kant is important.

  9. Aristotle, 384 BC – 322 BC Metaphysics

  10. Metaphysics Metaphysics began with the study of the knowledge of God and the nature of a future world. It was concluded early that good conduct would result in happiness in another world as arranged by God.

  11. Immanuel Kant, 1724-1804

  12. Critique of Pure Reason, 1781 “The science of metaphysics must not attempt to reach beyond the limits of possible experience but must discuss only those limits, thus furthering the understanding of ourselves as thinking beings. The human mind is incapable of going beyond experience so as to obtain a knowledge of ultimate reality, because no direct advance can be made from pure ideas to objective existence.”

  13. Critique of Pure Reason, 1781 What can I know? What should I know? What may I hope? What is a Human Being?

  14. Friedrich Ludwig GottlobFrege, 1848 - 1925 Concept-Script: A Formal Language for Pure Thought Modeled on that of Arithmetic

  15. Ludwig Josef Johann Wittgenstein, 1889 - 1951 There is a world that we need to describe. We need some (mathematical, or other) language to express this world; Question: Is what we say in this language the truth of the world? We need to answer (3), by only using the chosen language. Languages can never express the whole truth. TractatusLogico-philosophicus

  16. Lowenheim-Skolem-Tarski Theorem, 1920 Let be a set of formulas in a language of cardinality , and assume that is satisfiable in some infinite structure. Then for any cardinality , there is a structure of cardinality in which is satisfiable.

  17. Gödel Incompleteness Theorem, 1931 In any sufficiently strong formal system there are true arithmetical statements that cannot be proved (in the system). Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

  18. Church-Turing Thesis, 1935 Church: The notion of general recursive functions captures the informal idea of effective procedure. Turing: Whenever there is an effective method for obtaining the values of a mathematical function, the function can be computed by a Turing machine.

  19. Since 1940s

  20. Since 1960s • JurisHartmanis Notebook Entry on Dec.31.1962: • “This was a good year.” • This was a good 50 years. • Philosophers and Computer Scientists have left more questions to each other. Thank You!

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