In 1930 an unassuming young mathematician by the name Kurt Gödel (1906–1978) detonated a mathematical atomic bomb. It completely annihilated the mathematical world of his time.

The devastation was so extensive that the foundations of mathematics had to be rethought. His handiwork is called Gödel’s incompleteness theorem. Its implications are debated even to this day.

So what does the incompleteness theorem say?

Briefly, the theorem states that, for any sufficiently complex mathematical system, it is possible to construct a statement that is not provable or disprovable, yet is also true. This statement is called the Gödel sentence for the mathematical system. In other words, no matter how sophisticated the mathematical system, there are always mathematical truths that lie outside the reach of the system.

In other words, it is impossible to create a mathematical Theory of Everything.

A controversial conclusion

One of the most controversial conclusions that people draw from Gödel’s theorem, which Gödel himself endorsed, is that the human mind transcends mathematics and has access to an infinity realm of truth that lies outside of the physical world. J. R. Lucas and Roger Penrose are the most well known proponents of this argument; hence it is called the Penrose–Lucas argument. You can read Lucas’ writeup here and Penrose’s version is in his book The Emperor’s New Mind: : Concerning Computers, Minds and The Laws of Physics (1989).

Their argument is that humans seem capable of understanding the truth of the Gödel sentence for any mathematical system. If the human mind were itself reducible to a mathematical system, it would have a Gödel sentence. However, at the same time, the human mind would be able to understand that its Gödel sentence is true. This is a contradiction, since the only way a mathematical system can understand that a statement is true is by proving it true. But the Gödel sentence cannot be proven by its mathematical system.

So, because our assumption that the human mind is reducible to a mathematical system results in a contradiction, the assumption must be false. Thus the human mind is not reducible to a mathematical system.

Penrose’s mathematical defense of his conclusion

It is no surprise, in a materialist age, that Lucas and Penrose’s argument is quite controversial. Penrose’s response to the critics is empirical, based on his experience as a mathematician. So, first, some background on his mathematics.

Penrose is famous for his contribution to the theory of black holes, which demonstrated the mathematical conditions for the singularity that forms them.

Incidentally, he is also known for an interesting geometric design called Penrose tiling, which is a way of laying shaped tiles in such a way that there is no generally repeating pattern — but at the same time there are no gaps:

Both results relate to his argument about the human mind and Gödel’s incompleteness theorem. The crux of it is that it is possible for physical reality to depend on Gödel’s incompleteness theorem. Such a mind-blowing idea! How can rocks and trees and water depend on an extremely abstract theory about the limitations of mathematics?

The undecidability problem

With both black holes and tiling, Penrose demonstrated that physical systems have the property of being undecideable. This is a computer science term, but it is derived from Godel’s incompleteness theorem. The upshot of undecidability is that there are some things that computers cannot calculate.

Since all currently known physical laws are calculable by a computer, this is a surprising result. If there are some physical phenomena that are undecidable, this means they cannot be generated by any known physical law.

So now let’s circle back to Penrose’s argument about the human mind…

The argument becomes even more intriguing given what he has proven about the undecidable nature of physical reality. Contrary to physical laws, which cannot solve undecidable problems, the human mind seems to be an escape hatch.

If the human mind has a transcendent ability to perceive mathematical truth, placing it outside of mathematics, then Penrose has found a mechanism that can resolve the undecidability he found in the foundation of physics. In other words, Penrose has shown mathematically that the physical world requires a mind’s transcendent power in order to operate, at least within the two physical systems he has analyzed. So, the world isn’t matter. It isn’t energy. It isn’t even information. The world is fundamentally controlled by a mind. Talk about mind-blowing!

Objections

As you might expect, Penrose’s view is not without critics.

The common objections are:

1. The mind is not a complete mathematical system. It has some random and inconsistent elements which means it can prove its own Gödel sentence.
2. The mind is too complex and inscrutable for humans to ever identify its Gödel sentence.
3. If humans could understand the truth of the mind’s Gödel sentence, this would be a contradiction. Therefore, it is a logical impossibility for humans to understand the truth of the mind’s Gödel sentence.

Penrose thinks that only the second objection has significant weight. His response brings in his experience, and that of his fellow mathematicians, when conducting their mathematical research.

If the critics were right, and the mind’s algorithms are inscrutable, then each mathematician would have a unique way of see mathematical truth. It is true that each mathematician has an idiosyncratic technique for making discoveries, often just coming to them out of the blue. But all mathematicians seem to share the ability to see the truth once it is discovered.

Indiana Jones goes through all sorts of adventures to find the Lost Ark. Yet, once he has found it, it can be appreciated by friend and foe alike. So it is with mathematics. Once a mathematical truth has been discovered, and made understandable, everyone can see the truth. This means that the algorithm, if it exists, is the same for all. And since the algorithm works for everyone, it is not inscrutable, but available to all.

Therefore, if there were indeed an algorithm for the mind, then it has an understandable Gödel sentence. And since the statement is understandable, everyone can see that the statement is true, which is a contradiction.

This is why Penrose finds the Gödelian argument for the mind’s transcendent nature compelling. Based on the universality of mathematical insight, it is impossible for the mind to have a Gödel sentence without resulting in a contradiction. Since the mind cannot have a Gödel sentence, then the mind cannot be a mathematical system. Therefore, Penrose believes, the mind is beyond mathematics and thus it can solve the undecidable problems at the foundation of reality.

It took us a while to get here, but what a trip it has been! Makes me wonder why people waste time with drugs when math can change our very understanding of reality.