Sir Roger Penrose, a Knight on Tiles, an Emperor of the mind and the only person on earth after Albert Einstein who has the ability to visualize the 4 dimensional space-time and make necessary modifications to General Relativity at the planck scale. The only scientist who firmly believes that Quantum mechanics is both incomplete and as well as inconsistent defending Albert Einstein's famous quote, "God does not play dice". He was the only mathematician who understood the implications of Godel's undecidable theorems to physics and to consciousness very early in his career. The only cosmologist who went beyond the Big Bang and answered the question what happened before it.
Penrose is currently working on his Twistor theory completely alienated from the scientific community who are all working on an alternative theory called as the string theory. However Penrose is mostly known for his series of attacks against the proponents of Strong AI starting from Emperor's New Mind which won the New York Times best seller and one of the most important discovery of the 20th century and continued his attacks in Shadows of the mind which firmly established the conclusion that,
"The inescapable conclusion seems to be: Mathematicians are not using a knowably sound calculation procedure in order to ascertain mathematical truth. We deduce that mathematical understanding – the means whereby mathematicians arrive at their conclusions with respect to mathematical truth – cannot be reduced to blind calculation!"
—Sir Roger Penrose
What this implies is that a machine can never ever surpass human intelligence for all cases. Of course there can be complex machine learning algorithms which will give machines enough ability to beat humans while playing chess or driving a car or recognizing patterns but the inevitable conclusion seems to be a robot can never ever surpass the intelligence of human mathematicians.
The center piece of the argument comes from the Godel-Turing theorem derived by Roger Penrose in his book Shadows of the Mind. The argument goes like there are perfect mathematical statements where a machine will never halt or will run forever.
1. Find a natural number which is not a sum of powers of 2.
- 2. Find an even number greater than 2 which is not the sum of two primes. (Goldbach's conjecture)
- Let us assume A be the set of all computations known to human beings to deduce whether a given computation on a natural number halts or not. Or in other words if a computation C does not halt then A halts giving us the output that C does not infact halt.
Penrose takes us through a series of arguments using Cantor's diagonal argument where he proves that such an sound algorithmic procedure A encompassing all known computations known to human beings if it exists will fail to stop for one given computation where we actually know that the computation C in fact does not stop. Or in other words A will run forever without halting when in fact C does not halt. This is a contradiction the algorithmic procedure A encompassing all known computations was assumed to be the underlying procedure used by human beings to ascertain whether a computation will halt or not but as we have just now proved there will always exist a computation C where we know that the computation C will not halt but the procedure A fails to halt and there by fails to know whether C halts or not and therefore A cannot be the underlying procedure used by human beings as it leads to a contradiction which is quite obvious for anyone to see. Human beings are not Turing machines or a human mind is not a Turing machine.
“Either mathematics is too big for the human mind or the human mind is more than a machine.”
– Kurt Gödel
“Gödel’s Theorem shows that human thought is more complex and less mechanical than anyone had ever believed”
- Rudy Rucker
The above proof is known as the Penrose's version of the Godel-Turing theorem. Since I am a programmer and not a mathematician I completely understand this Turing's version of Penrose's argument against Strong AI compared to the pure Godel's version of it which was summarized by David Chalmers below.
"(1) suppose that “my reasoning powers are captured by some formal system F,” and, given this assumption, “consider the class of statements I can know to be true.” (2) Since I know that I am sound, F is sound, and so is F’, which is simply F plus the assumption (made in (1)) that I am F (incidentally, a sound formal system is one in which only valid arguments can be proven). But then (3) “I know that G(F’) is true, where this is the Gödel sentence of the system F’” (ibid). However, (4) Gödel’s first incompleteness theorem shows that F’ could not see that the Gödel sentence is true. Further, we can infer that (5) I am F’ (since F’ is merely F plus the assumption made in (1) that I am F), and we can also infer that I can see the truth of the Gödel sentence (and therefore given that we are F’, F’ can see the truth of the Gödel sentence). That is, (6) we have reached a contradiction (F’ can both see the truth of the Gödel sentence and cannot see the truth of the Gödel sentence). Therefore, (7) our initial assumption must be false, that is, F, or any formal system whatsoever, cannot capture my reasoning powers."
Human beings can make contact to the Platonic world of numbers and mathematical truths which is not feasible for machines and it is for this reason that they can never ever surpass the intelligence of human mathematicians. A simple bitter truth which Plato and the other pagan mystery religions discovered long before modern science was born. The Empirical world is just a mere shadow copy of the real Platonic realm which exists independent of us.
Now we know that just if you do big number crunching doesn't mean you are thinking and somehow you've made a machine to think. Thinking is much more than just number crunching. The proponents of Strong AI are no where near to it. Even though modern artifical intelligent machines are able to act on their own without being programmed for it explicitly deep down in its chip there still lies a set of implicit machine learning algorithms which does the trick for them. What it means is that any such alogrithm can be transformed into a formal system and the Godel's Incompletness theorems can be applied to it showing that there will always be one statement where human beings know that the statement is true but the machine can never ever utter it. Hats off to Sir Roger Penrose he surely deserves a nobel prize for this discovery.