Holes in the Heart of Reason

Philosophical Reflections IV

The world is objectively real, blithely uninterested in our beliefs and wishes. This enables us to discover its nature by testing our ideas against what actually happens, as reality reacts to our tests according to its nature and not according to our whims.

The application of this principle, the development of science and technology, has led to astounding progress in the understanding of nature and in the quality of life of human beings. Yet despite this, many still yearn for the spurious comforts of subjectivism. Rationality and science, the greatest liberators of mankind in our history, are decried and despised as inferior to mystical insights.

Indeed, mysticism and subjectivism do their best to find comfort in science itself. Anything which seems to allow a role for mysticism, or which might limit the power of reason, is eagerly seized upon. Quantum mechanics in physics, and Gödel’s theorem in mathematics, are prime examples of this attack from within.

The Attack on Science and Reason

Michael Crichton’s science-fiction book Jurassic Park (a good story ruined by appalling philosophical commentary) contains a speech which sums up this attitude:

“The great intellectual justification of science has vanished… Science has claimed the power to eventually control everything, through its understanding of natural laws. But in the twentieth century, that claim has been shattered beyond repair. First, Heisenberg’s uncertainty principle set limits on what we could know about the subatomic world. Oh well, we say. None of us lives in a subatomic world … Then Gödel’s theorem set similar limits to mathematics, the formal language of science. Mathematicians used to think that their language had some special inherent trueness that derived from the laws of logic. Now we know that what we call ‘reason’ is just an arbitrary game.”

Strong claims indeed. So let us examine these issues in detail.

Quantum Physics and Uncertainty

The Uncertainty Principle of quantum physics (the physics of subatomic particles) states that certain pairs of quantum characteristics, such as position and momentum, cannot both simultaneously be measured precisely. This has given rise to two basic mystical interpretations. One is Crichton’s: if the Uncertainty Principle implies that there is knowledge inherently beyond the reach of science, some superior means of knowledge may exist which can reach it.

That claim is empty. By the best interpretation of the Uncertainty Principle, quantum states don’t actually have fixed values until the interaction which “collapses” or “observes” them occurs. For example, light travels as an electromagnetic wave, but its energy is absorbed in discrete packets or photons. A wave does not have a fixed position: it is “spread out” through space. Thus, one can’t find the exact position of a “travelling photon”, not because it is information science can’t reach, but because there is no such thing as an exact position for it. A photon only has a “position” when it is absorbed, which “collapses” it to the point where it is absorbed. But then, it is no longer travelling, and its “wave” characteristics can’t be measured since they no longer exist. This is why we can’t simultaneously measure both wave-states and particle-states: the two states can never exist simultaneously.

I propose that the Uncertainty Principle, in this form, is a logical necessity from the nature of quanta, and thus a remarkable validation of logic itself. At the core of logic lie the Law of Identity (a thing is itself) and its corollary, the Law of Non-Contradiction (a thing cannot simultaneously be both “X” and “not X”). Quantum nature is so foreign to our normal perceptions that we cannot visualise it. We can only visualise and verbalise its qualities in the mutually contradictory terms of waves and particles. Yet the laws of logic demand that, to the extent that the wave-nature and particle-nature of quanta are true descriptions of their real nature, both cannot exist simultaneously: this would violate the Law of Non-Contradiction. If both contradictory parts couldexist at once, then one might have a case against the universality of logic. That they can’t, even in realms far outside direct human experience, shows how deeply the principles of logic are woven into the very fabric of reality.

Quantum Physics and Subjectivism

From the above springs the second mystical interpretation of quantum physics: if quantum states have no fixed quantity until they are observed, this implies the primacy of consciousness, the rule of mind over reality (e.g. see TableAus, August 1991, pp 10-11).

I have discussed this subjectivist claim extensively in an earlier article (Subjectivism, Reality and Quantum Physics [TableAus, October 1991]). The truth of the matter is that observation is irrelevant: quanta have no fixed values until they interact appropriately with other quanta, a purely physical phenomenon that has nothing to do with consciousness. These interactions, and consequent “fixing” of values, occur all the time, whether someone is watching or not. The error arises simply because we observe via those quantum interactions, leading to a confusion between means and causes.

I believe that a deeper theory is to be found behind quantum physics, one which more completely explains the details, one which more exactly describes the true nature of quanta, one which can explain how and why quantum states “collapse” in any instance. But even if not, the present state of quantum physics provides no comfort for mysticism. Indeed, it is a tribute to the power of reason and science that a world so bizarre, so impossible to visualise, can be known at all, let alone understood so precisely.

Gödel’s Theorem: Beginning

Gödel’s Theorem, translated from Mathematese into English, grandly states: “A formal system cannot be both complete and consistent.” It was originally developed with regard to “typographical number theory” (TNT), a mathematical system which aimed to deduce everything about numbers.

Basically, Gödel showed that by a system of “Gödel numbering”, anything can be expressed as numbers. Therefore, TNT itself can be reduced to a numerical system. Thus to be complete (able to describe everything about numerical systems), TNT must completely describe itself. Now, looking at it from outside, we can make this true statement about TNT: “TNT cannot consistently assert this statement.” As this is a true statement about TNT, then to be complete and consistent, TNT must be able to consistently assert that it cannot consistently assert it! So if TNT is complete, it can’t be consistent; it can only be consistent if it is incomplete (unable to generate such paradoxes)! A detailed exposition can be found in Douglas Hofstadter’s book Gödel, Escher, Bach: An Eternal Golden Braid.

This rather loopy theorem has excited the interest of mystics and subjectivists, who see in it the hope of demolishing the foundations of logic, mathematics and science.

The first steps along this path were attacks on the possibility of artificial intelligence, in an attempt to place living consciousness on a special, non-material plane. The argument went: computer intelligence must be based on a formal mathematical system, which cannot be both complete and consistent, therefore true artificial intelligence is not possible. Hofstadter convincingly demolished such arguments. It is all very well for us to look at machine intelligence from the outside, point the finger, & exclaim: “Ha! You can’t consistently say ‘machines cannot consistently assert this sentence’, whereas I can consistently say it”, but we ourselves are capable of the same self-referential paradoxes. The machine can point at us and say the same sort of things.

Gödel’s Theorem: Mysticism

Since we ourselves are as vulnerable to Gödel’s theorem as a machine, the guns then are turned on reason itself. Jurassic Park has already been quoted. Such ideas are developed in more detail in The Mind of God by Paul Davies, a book much praised and much touted by mystics the world over. A scientist admitting not only the need for mysticism, but its superiority over reason!

“Gödel’s theorem warns us that the axiomatic method of making logical deductions from given assumptions cannot in general provide a system that is both provably complete and consistent. There will always be truth that lies beyond, that cannot be reached from a finite collection of axioms. The search for a closed logical scheme that provides a complete and self-consistent explanation for everything is thus doomed to failure.”

So what can we do? Why, it’s obvious:

“We are barred from ultimate knowledge, from ultimate explanation, by the very rules of reasoning that prompt us to seek such an explanation in the first place. If we wish to progress beyond, we have to embrace a different concept of ‘understanding’ from that of rational explanation. Possibly the mystical path is a way to such understanding. Maybe [mystical insights] provide the only route beyond the limits to which science and philosophy can take us, the only possible path to the Ultimate.”

Gödel’s Theorem: Truth

So, starting from the grandiose statement that a formal system cannot be both complete and consistent, we’ve inexorably progressed to the failure of reason and science, the necessity for mystical insight to plug the gaping hole at the heart of logic itself.

This is a classic case of an idea being given a sweeping description which hides its derivation, then being completely cut from its roots and allowed to float unfettered to the heights of fantasy.

Gödel’s theorem proves no more than its derivation proves: that any system powerful enough to refer to itself, can make self-referential paradoxes. I am a human being: conscious of the world, conscious of my own existence, conscious of truth and falsehood. Therefore, I can say “I cannot consistently make this statement”, or more simply, “this sentence is false”, or more simply still, “I am lying”. So my ability to be “complete” makes me “inconsistent”: if I’m telling the truth I’m lying, and if I’m lying I’m telling the truth.

Make no mistake: Gödel’s theorem imposes no other limit. I can know every fact about the universe, every physical law, all chemistry, all biology, all psychology: it denies me none of it. By its nature, by its own phraseology, Gödel’s theorem allows a rational being or even a formal system to have complete understanding of absolutely everything: provided it pays the “price” of being able to articulate self-referential paradoxes. That price, I for one am happy to pay.

So this is the great truth which lies forever beyond the grasp of logic, mathematics and science. This is the liberating insight, the holistic grasp of reality, which only mysticism can reach. The ability to say, “I am lying”, and mean it.

© 1992, 1996 Robin Craig: first published in TableAus.