Hypercomputation and the Human Mind

1. Hypercomputation and the Human Mind Selmer Bringsjord Professor of Logic, Philosophy, Cognitive Science, and Computer Science Rensselaer AI & Reasoning (RAIR) Laboratory Department of Cognitive Science Department of Computer Science Rensselaer Polytechnic Institute (RPI) Troy NY 12180 USA AFRL-R 8.11.05 selmer@rpi.edu http://www.rpi.edu/~brings

2. Next-Generation “Weak” Logic-based AI • Isolate and dissect human ingenuity. • Mathematize a weak correlate to this ingenuity courtesy of advanced logical systems. • Implement this correlate in working programs. • Augment the correlate with machine-specific power.

3. The Rensselaer AI & Reasoning Lab(The RAIR Lab) Wargaming • Cracking Project; “Superteaching” A while back, RPI Strategic Investment hypothesis generation; AI in support of IA Slate (Intelligence Analysis) Item generation (theorem proving-based generation) synthetic characters/psychological time

4. Cognitive Science and artificial intelligence (AI) assume that cognition is computation.

5. Newell’s “20 Questions” Paper • The study of the mind is painfully fragmentary, but at any rate... • Man is a computer.

6. Pinker’s How the Mind Works • “The mind is a system of organs of computation designed by natural selection to solve problems faced by our evolutionary ancestors in their foraging way of life.”

7. Anderson’s Recent “Newell Test” Paper in BBS • The study of the mind is less fragmentary now, and certainly... • Humans are computers.

8. Are Newell & Simon & Co. correct?Are human persons computers? No. They’re wrong. And they can be wrong... even if we are fundamentally information processors.

9. If they’re right, this is what, at bottom, we are! If R5 is non-empty, take a bead away. , , , R1 R2 R3 R4 R5 R6 R7 R8 R9

10. “Humble” Circle of Computation Abaci Recursive functions Register machines (modern high-speed digital computers) Turing machines

11. “Humble” Circle of Computation Abaci Recursive functions Register machines (modern high-speed digital computers) -calculus Turing machines

12. “Humble” Circle of Computation Abaci Recursive functions programming languages (Lisp, C++, ...) -calculus Register machines (modern high-speed digital computers) Turing machines

13. “Humble” Circle of Computation Abaci Recursive functions programming languages (Lisp, C++, ...) -calculus logic programming (in first-order logic) Register machines (modern high-speed digital computers) Turing machines

14. But we’re not mere carbon-basedcreatures somewhere in this circle(of mere animal life)!

15. The Mathematical Landscape Space of all information processes Hypercomputation, now exploding with activity revolutionizing computer science, mathematics, physics, etc. Analog Chaotic Neural Networks, Zeus Machines, Weyl Machines, P/Poly Machines, … } The “Turing Limit” Turing machines, abaci, programs, … AI & Cog Sci

16. What are we? Superminds! Phenomena not capturable in any third-person scheme. e.g., phenomenal consciousness (e.g., that which it feels like to carve ski turns at 40mph) Analog Chaotic Neural Networks, Zeus Machines, Weyl Machines, P/Poly Machines, … “Turing Limit” Turing machines, abaci, programs, …

17. So what are these hypercomputers you’re talking about?We know what computers are.What’s a hypercomputer?

18. Bertrand Russell (1936): “Ambrose says it is logically impossible [for a man] to run through the whole expansion of . I should have said it was medically impossible. ... Might not a man's skill increase so fast that he performed each operation in half the time required for its predecessor? In that case, the whole infinite series would take only twice as long as the first operation.”

19. A Hypercomputer: Zeus Machine

20. 7th Grade Math Problem, Anticipating Elementary Calculus

21. John Anderson: “But Godel shot down Hilbert’s Program with a proof! You’re just claiming.”Ok, fair enough.I have proofs, or at least formal arguments. Seventeen of them, in fact.Today, one intuitive, one precise (synopsis only, of course). For more, see ‘For Further Reading’ @ the end.

22. The Mountaineering “Argument” Imagine a race of creatures: ipas (ih-pah-s). Ipas are information-processing creatures. They live on the High Peaks of Hypercomputation, and are capable of phenomenal feats of information processing (such as cracking the halting problem). We know this because some humans are able to climb the mountains and study the ipas. In fact, these humans have proved all kinds of things about the capability of ipas, and in general know what the essence of ipa cognition is. And yet it is said that no humans are capable of ipa mentation, but are forever limited to living in the Valley of Turing Machines. Ipas Humans

23. Argument from Infinitary Reasoning(from Superminds) Proof. Assume that all human reasoning is at or beneath the Turing Limit. Then all chains of human reasoning (e.g., proofs) are identical to some chain of reasoning expressed in first-order logic. But there are many chains of human reasoning in infinitary logics, and we know that such chains in infinitary logic cannot possibly be expressed in first-order logic. Contradiction! So, by indirect proof, the starting assumption (which has led to absurdity) is false. I.e., it’s not true that all human reasoning is at or beneath the Turing Limit. QED

24. If we are superminds, then, by the way, the received view concerning our origins explodes...

25. A Proof that People Didn’t Evolve (1) If people are the product of evolution, they were produced by an algorithmic process (= by a process at or below the Turing Limit). Theorem Ã: No algorithmic process can produce a device capable of hypercomputation. People are capable of hypercomputation. (4)  People were not produced by an algorithmic process. (from 2 and 3) (5)  People aren’t the product of evolution. (from 4 and 1)

26. If we are superminds, how then should we proceed, in concrete terms?

27. RAIR Lab’s Push Against the Turing Limithttp://www.cs.rpi.edu/~kelleo/busybeaver/

28. For Further Reading...

30. For Further Reading New Proof (one of the seventeen) that Minds Hypercompute: Bringsjord, S., Shilliday, A., Taylor, J., Bram van Heuveln, B., Baumes, J. & Ross, K. (forthcoming) “A New Gödelian Argument for Hypercomputing Minds Based on the Busy Beaver Problem,” Applied Mathematics and Computation. Preprint available at top level of Bringsjord’s web site. Recent Proof that Minds Hypercompute: Bringsjord, S. & Arkoudas, K. (2004) “The Modal Argument for Hypercomputing Minds” Theoretical Computer Science 317: 167-190. Offprint available from top level of Bringsjord’s web site. Argument #1 Against Pinkerian View that We are Evolved Computers: Bringsjord, S. (2001) “Are We Evolved Computers? A Critical Review of Steven Pinker’s How the Mind Works” Philosophical Psychology14.2:227-243. Paper Exhibiting Human Infinitary Reasoning: Bringsjord, S. & van Heuveln, B. (2003) “The Mental Eye Defense of an Infinitized Version of Yablo’s Paradox,” Analysis63.1: 61-70. Book Explaining and Defending View That We’re “Superminds”: Bringsjord, S. & Zenzen, M. (2003) Superminds (Dordrecht, The Netherlands: Kluwer Academic Publishers). Anderson’s “Newell Test” paper: Anderson, J. & Lebiere, C. (2003) “The Newell Test for a Theory of Cognition,” Behavioral and Brain Sciences26: 587-601. Zeus Machines Introduced and Contextualized, and the Humble Circle Fully Explained: Boolos, G. & Jeffrey, R. (1989) Computability and Logic (Cambridge, UK: Cambridge University Press). Phenomenal Consciousness and Cognitive Science: Block, N. (1995) “On a Confusion About a Function of Consciousness,” Behavioral and Brain Sciences 18: 227-247.

31. END