Hypercomputation and the Human Mind

Selmer BringsjordProfessor 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 8.11.05selmer@rpi.edu http://www.rpi.edu/~brings

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.

The Rensselaer AI & Reasoning Lab(The RAIR Lab)

A while back,RPI StrategicInvestment

Cracking Project; “Superteaching”

Slate (IntelligenceAnalysis)

Item generation(theorem proving-basedgeneration)

synthetic characters/psychological time

Wargaming

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

hypothesis generation; AI in support of IA

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Cognitive Science and artificial intelligence (AI) assume that

cognition is computation.

Newell’s “20 Questions” Paper

• The study of the mind is painfully fragmentary, but at any rate...

• Man is a computer.

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.”

Anderson’s Recent “Newell Test” Paper in BBS

• The study of the mind is less fragmentary now, and certainly...

• Humans are computers.

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.

R1 R2 R3

If R5 is non-empty, take a bead away.,,,

If they’re right, this is what, at bottom, we are!

R4 R5 R6 R7 R8 R9

“Humble” Circle of Computation

Turing machines

Abaci

Register machines(modern high-speeddigital computers)

Recursive functions

“Humble” Circle of Computation

Turing machines

Abaci

Register machines(modern high-speeddigital computers)

Recursive functions

-calculus

“Humble” Circle of Computation

Turing machines

Abaci

Register machines(modern high-speeddigital computers)

Recursive functions

-calculus

programming languages(Lisp, C++, ...)

“Humble” Circle of Computation

Turing machines

Abaci

Register machines(modern high-speeddigital computers)

Recursive functions

-calculusprogramming languages(Lisp, C++, ...)

logic programming(in first-order logic)

But we’re not mere carbon-basedcreatures somewhere in this circle

(of mere animal life)!

The Mathematical Landscape

Space of all information processes

The “Turing Limit”

Turing machines, abaci, programs, …

Hypercomputation, now exploding with activity revolutionizing computer science, mathematics, physics, etc.

Analog Chaotic Neural Networks, Zeus Machines, Weyl Machines, P/Poly Machines, …

} AI & Cog Sci

What are we?

“TuringLimit”

Turing machines, abaci, programs, …

Analog Chaotic Neural Networks, Zeus Machines, Weyl Machines, P/Poly Machines, …

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)

So what are these hypercomputers you’re talking

about?We know what computers are.

What’s a hypercomputer?

Bertrand Russell (1936):“Ambrose says it is logically impossible [for a man] to run throughthe 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.”

A Hypercomputer: Zeus Machine

7th Grade Math Problem, Anticipating Elementary Calculus

€

limn →∞

1

4n= 0

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.

The Mountaineering “Argument”Imagine a race of creatures:ipas (ih-pah-s). Ipas areinformation-processingcreatures. They live on the HighPeaks of Hypercomputation, and are capable of phenomenal feats of information processing (suchas cracking the halting problem).We know this because some humansare able to climb the mountainsand study the ipas. In fact, thesehumans have proved all kindsof things about the capability ofipas, and in general know whatthe essence of ipa cognition is.

And yet it is said that no humansare capable of ipa mentation, but are forever limited to living in theValley of Turing Machines.

Ipas

Humans

Argument from Infinitary Reasoning(from Superminds)

Proof. Assume that all human reasoning is at or beneaththe Turing Limit. Then all chains of human reasoning(e.g., proofs) are identical to some chain of reasoningexpressed in first-order logic. But there are many chainsof human reasoning in infinitary logics, and we know that such chains in infinitary logic cannot possibly beexpressed in first-order logic. Contradiction! So, byindirect proof, the starting assumption (which has led toabsurdity) is false. I.e., it’s not true that all human reasoningis at or beneath the Turing Limit. QED

If we are superminds, then, by the way, the received view

concerning our origins explodes...

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).

(2) Theorem Ã: No algorithmic process can produce a device capable of hypercomputation.

(3) 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)

If we are superminds, how then should we proceed, in concrete

terms?

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

For Further Reading...

For Further ReadingNew Proof (one of the seventeen) that Minds Hypercompute: 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: Recent Proof that Minds Hypercompute: Bringsjord, S. & Arkoudas, K. (2004) “The Modal Bringsjord, S. & Arkoudas, K. (2004) “The Modal Argument for Hypercomputing Minds” Argument for Hypercomputing Minds” Theoretical Computer ScienceTheoretical Computer Science 317: 317: 167-190. Offprint available from top level of Bringsjord’s web site.

Argument #1 Against Pinkerian View that We are Evolved ComputersArgument #1 Against Pinkerian View that We are Evolved Computers: Bringsjord, S. : Bringsjord, S. (2001) (2001) “Are We Evolved Computers? A Critical Review of Steven Pinker’s How the Mind Works” Philosophical Psychology 14.2:227-243.

Paper Exhibiting Human Infinitary ReasoningPaper Exhibiting Human Infinitary Reasoning: Bringsjord, S. & van Heuveln, B. (2003) : Bringsjord, S. & van Heuveln, B. (2003) ““The Mental Eye Defense of an Infinitized Version of Yablo’s Paradox,” Analysis 63.1: 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 Sciences 26: 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.

END