Limits of AI. The Gödelian argument. Complexity Explorers Krakow #2

Limits of AI
Gödelian Argument
Complexity Explorers Kraków
Marcin Stępień @marcinstepien Kraków 2018-11-14

src: www.adamwalanus.pl/2016/chaitin/160519-1804-19.jpg
“Kurt Gödel's achievement in modern logic is
singular and monumental – indeed it is more than a
monument, it is a landmark which will remain visible
far in space and time. ... The subject of logic has
certainly completely changed its nature and
possibilities with Gödel's achievement.”
John von Neumann

Photo by Cmichel67 - Own work, CC BY-SA 4.0,
https://commons.wikimedia.org/w/index.php?curid=47606311
Gödel’s theorem
“Tells us what we don’t know and can’t know. It sets a
fundamental and inescapable limit on knowledge of
what is. It pinpoints the boundaries of ignorance -
not just human ignorance, but that of any sentient
being.”
Paul Davies
Foreword to “Thinking about Gödel and Turing” by Gregory J Chaitin, 2007

“Either mathematics is too big for the
human mind or the human mind is
more than a machine.”
Kurt Gödel

“There can be no machine which will
distinguish provable formulae of the system
from unprovable ...
On the other hand if a mathematician is
confronted with such a problem he would
search around and find new methods of proof.”
Alan Turing
Lecture to the London Mathematical Society 20 II 1947

Inspired by Epimenides
paradox
All Cretans are liars
- Epimenides from Crete

Gödel's Incompleteness Theorem - Numberphile
https://youtu.be/O4ndIDcDSGc

Gödel's Incompleteness extra 1 - human above machines?
https://youtu.be/mccoBBf0VDM up to 6:40

Penrose Tiling
❏ Non-periodic tiling generated by
an aperiodic set of prototiles.
❏ The aperiodicity of prototiles
implies that a shifted copy of a
tiling will never match the original.
Photo ty Solarflare100 - Own work, CC BY 3.0,
By Geometry guy at English Wikipedia, CC BY-SA 3.0,
Kite and dart protitiles

By PrzemekMajewski - Own work, CC BY-SA 4.0,

Roger Penrose: Strong A.I. vs. The Euclidean Plane
https://youtu.be/wFBzMEE5eaE

Strong A.I. vs. The Calculation Problem. Commutative property.
https://youtu.be/WcmcB1KUBYc

“My incompleteness theorem makes it likely that
mind is not mechanical, or else mind cannot
understand its own mechanism. If my result is taken
together with the rationalistic attitude which
Hilbert had and which was not refuted by my
results, then [we can infer] the sharp result that
mind is not mechanical. This is so, because, if the
mind were a machine, there would, contrary to this
rationalistic attitude, exist number-theoretic
questions undecidable for the human mind. ”

It is not about
anti-mechanist view
● Gödel was a convinced dualist *
* "The Implications of Gödel's Theorem" J.R. Lucas 1998
● Penrose calls himself "a very
materialistic and physicalist kind of
person"

What can be
answered
What cannot be
answered
Encoded rules,
axioms
System
(Turing complete
computing device
+ data)

System
(Turing complete
computing device
+ data)
Human’s insight
Encoded rules,
axioms
Created /
learned
input
What can be
answered
What cannot be
answered

Views classification by Scott Aaronson, further critique
1. Consciousness is reducible to computation (the view of strong-AI
proponents)
2. Sure, consciousness can be simulated by a computer, but the simulation
couldn't produce "real understanding" (John Searle's view)
3. Consciousness can't even be simulated by computer, but nevertheless has a
scientific explanation (Penrose's own view, according to Shadows [Of The
Mind])
4. Consciousness doesn't have a scientific explanation at all (the view of 99% of
everyone who ever lived)
Scott Aaronson “Quantum Computing since Democritus” https://www.scottaaronson.com/democritus/lec10.5.html

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