This document discusses the limits of artificial intelligence as related to Gödel's incompleteness theorem. It contains the following key points: 1. Gödel's incompleteness theorem showed that within any given formal system, there are statements that cannot be proven or disproven within that system. 2. This suggests that the human mind is more than just a machine, as it can sometimes solve problems that machines cannot. 3. Roger Penrose argued that phenomena like non-periodic tilings and mathematical intuition demonstrate capabilities of the human mind that go beyond computation. 4. While Gödel did not intend to make an anti-mechanist argument, his theorem implies that the mind cannot be a purely

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

2. 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

3. src: www.adamwalanus.pl/2016/chaitin/160519-1804-19.jpg
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

4. src: www.adamwalanus.pl/2016/chaitin/160519-1804-19.jpg
“Either mathematics is too big for the
human mind or the human mind is
more than a machine.”
Kurt Gödel

8. src: www.adamwalanus.pl/2016/chaitin/160519-1804-19.jpg
“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

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

10. Gödel's Incompleteness Theorem - Numberphile
src: www.adamwalanus.pl/2016/chaitin/160519-1804-19.jpg
https://youtu.be/O4ndIDcDSGc

11. Gödel's Incompleteness extra 1 - human above machines?
src: www.adamwalanus.pl/2016/chaitin/160519-1804-19.jpg
https://youtu.be/mccoBBf0VDM up to 6:40

12. 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,
https://commons.wikimedia.org/w/index.php?curid=9732247
By Geometry guy at English Wikipedia, CC BY-SA 3.0,
https://commons.wikimedia.org/w/index.php?curid=30611873
Kite and dart protitiles

13. By PrzemekMajewski - Own work, CC BY-SA 4.0,
https://commons.wikimedia.org/w/index.php?curid=40160867

14. Roger Penrose: Strong A.I. vs. The Euclidean Plane
src: www.adamwalanus.pl/2016/chaitin/160519-1804-19.jpg
https://youtu.be/wFBzMEE5eaE

15. Strong A.I. vs. The Calculation Problem. Commutative property.
src: www.adamwalanus.pl/2016/chaitin/160519-1804-19.jpg
https://youtu.be/WcmcB1KUBYc

16. src: www.adamwalanus.pl/2016/chaitin/160519-1804-19.jpg
“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. ”

17. 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"

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

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

20. 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