Thinking About Thought

Thought must consider thought itself.

Our consideration does not require an a priori definition of thought. In fact it is better if we do not have a definition to begin with. Useful definitions are usually arrived at as a result of well-managed consideration, and not in advance of the consideration. This is such a basic of good thinking that I am amazed so few take it into account. In particular, many recent discussions regarding consciousness, artificial intelligence, and the nature of reality appear to me to begin by presuming the fundamental character of the phenomenon they would ostensibly describe, and to proceed by means of argument, pro and con. This is not thinking, but the short-circuit of thinking.

The method of prior definition, argument, and counter-argument is the inevitable method of politics, and perhaps that is why it has become so prevalent in those fields that are supposed to value clear thinking: Perhaps it is because so much of our thinking has become politicized.

This method may show occasional partial successes, when the prior definition is a lucky hit, but it is an especially weak method for discovering the character of an uncharted phenomenon and one that is most probably polyvalent. Even simple logic, that limited hand-child of thought, teaches us that the disproof of one contention is not the proof of its opposite.

So we see that we may think freely about thought without first considering its ultimate character, insisting on its sources, or creating a pre-emptive inventory of its uses and productions.

When we speak of thought, we know on a fundamental level what activity we are speaking of. All we are seeking to do in thinking about thinking is deliver that fundamental awareness into the realm of thought itself. We do not have to raise linguistic questions, or questions as to whether computers are “thinking” in the same sense we are, or questions as to whether thought itself is merely an illusory by-product of the brain’s “software.”

All we really have to do is observe thought, and see what it does, and see how it moves, and see what it spends itself on. If we do that well enough, and long enough, I am sure that we will understand thought more deeply.

And here I see that I do entertain at least one prior assumption about thought. I assume that the purpose of thought is understanding. Understanding, not the display of expertise. Understanding, not statistics. Understanding, not the catalogue of a ramifying hierarchy of facts.

Why do humans seek to understand? Some might suppose it is to gain mastery. I suppose it is to create harmony. At the deepest level, perhaps harmony and mastery are the same. But if so, it is at a level that only harmony can reach, that mastery alone cannot touch. But what sort of harmony?

Many people appear to feel that the function of thought is to lift us out of the physical realm into the reaches of “higher” existence—the ideal, the abstract, the spiritual. I am beginning to feel that its function is actually to apprehend those “higher” modes of being, and return them to the body.

Why do we seek harmony? Well, what else is there to do?

You may well disagree with my assumption that the purpose of thought is understanding. I mention it here in an attempt to be as clear and honest as possible. I don’t intend to argue the assumption, because I do not wish it to function as one of those prior definitions. I wish this essay to be not a persuasion, but an invitation to observe. So you may safely trot along beside me for a while and see if you think any genuine observation is occurring.


The Mathematical Model of Thought

It is currently impossible to speak of thought itself without reference to mathematics. I love mathematics, and have had some degree of training in the subject. Mathematics is a valuable tool of thought. But I do not feel that it is identical to thought, or that it is the essence of thought.

Thought is prior to mathematics, and is the more fundamental behavior. Thought created mathematics. Mathematics did not create thought.

This would seem to be a perfectly evident statement. Yet in almost all contemporary fields of thought or research, the mathematical model predominates, even in those fields for which it is ill-suited.

We have the spectacle of physicists and mathematicians writing books presuming to explain, in more or less mathematical terms, such matters as the origin of the universe or the nature of consciousness, when we have no assurance, in either case, that these matters are mathematically treatable.

(A proviso: I am aware that when cosmologists speak of the “beginning” of the universe, in a first Planck interval and Planck distance, they are speaking of a description that seems to fit most of the facts and that is mathematically consistent. In fact, I accept their model (provisionally). Nevertheless, we have no assurance that the beginning they model is the “actual” beginning. How could we know, unless we were there to compare it to our model? Please note further that I do not maintain these questions will never be mathematically treatable—that would be as great a presumption. I am simply saying that we have as yet little indication.)

We have the further spectacle of thinkers in fields which are resolutely a-mathematical imitating the mathematical model—as when a French literary philosopher misconstrues the meaning of i, the square root of negative one, as some sort of universal phallic quality; as when the painter speaks of the uncertainty principle; as when the poet writes “experimental” work.

By “mathematics” I mean to include, for the time being, all of those deeply mathematized branches of science which give us so much conceptual trouble. Has no one ever observed that it is precisely in the mathematized branches of science that we do begin to have conceptual troubles?

Developments in biology in recent years have been as amazing and far-ranging in their effects as those in physics, and actually now promise (or threaten) to bring greater transformations into our lives. And yet, although developments in biology have certainly upset many of our earlier and long-held notions, they have never posed us the paradoxes regarding the fundamental nature of reality that developments in physics have posed.

I am sure that many would assume this is because physics is the more “fundamental” science, the science that pushes closer to the limits of the knowable. We typically create a neat little hierarchy running from physics at the extreme through chemistry then into biology, treating each successive discipline as “fuzzier” and therefore less fundamental. (The categories themselves get fuzzier after biology—which science comes next? Anthropology? Ecology? Where does psychology fit in, how far down the line?)

But is this hierarchy of fundamentalism in fact an accurate description? If it is, how do we know it is? Mightn’t we just as easily assume that the arrangement describes no hierarchy at all? That it describes nothing more than the relative ease with which mathematics may be applied to the given disciplines? That it is, in short, a mirror of our own assumptions?

Why do we assume that the paradoxes of physics are the paradoxes of nature? Is it not possible that they are the paradoxes of mathematics? Is it not possible that when we come up against Heisenberg’s uncertainty principle or against the four-slit photon experiment, we are simply meeting Goedel’s Unprovability Theorem in yet another guise?

Please observe that I am very carefully not saying two things: (1) I am not saying that the universe is ultimately knowable, and (2) I am not saying that only our stupid mathematics is getting in the way of knowing it.

Mathematics is a powerful tool for understanding, and we would be fools to abandon it. On the other hand, I at least would be deeply surprised if the universe (or multiverse) were fully knowable by any of its denizens.

What I am saying is that mathematics is only a method of thought, and has limits which thought itself does not have. (Thought has its own different and I think somehow larger limits—but I will discuss that in another place.)

In our age, because of its manifold successes, we have forgotten the limits of mathematics. That is, although we are all aware of Goedel’s Theorem, we treat it as a special result, of interest only to mathematicians, and almost universally fail to draw the implication that, in our study of reality, mathematics can never be a completely adequate tool.

There are many subjects worthy of genuine and rigorous thought which however may not be treated effectively by mathematics. In our culture these subjects have become marginalized, and assigned to the weaker thinkers. They are frequently treated as having no real existence whatsoever.

An example is the question of the existence of divinity. The fact that we cannot run an experiment on God or gods does not mean that we cannot think clearly and powerfully and usefully on the subject. It merely means that we cannot mathematize the subject. But because most strong thinkers today can only think mathematically, the question is usually dismissed out of hand.

Or worse, the question is assigned to theologians to chew over, so that it may become hopelessly confused, buried under a debris of ill-begotten concepts, and eventually completely useless to most humans.

Wait a minute, some might say: We have already had powerful thinkers going over the question of divinity. It happened in the late European middle ages, just prior to (and probably instrumental in) the various renaissances. And look at all the nonsense and folly that came out of that: angels dancing on the head of a pin, the perfection proof of God’s existence.

Not so, I would say. You’re looking at it the wrong way round. Your example in fact supports the point I am making.

What happened in the middle ages was that scholars discovered classical logic, and promptly went around applying it to every theological question they could come up with. Of course the result was absurdity, because those questions were not amenable to the application of logic in the first place.

Mathematics, as has been adequately demonstrated for many years now, may be understood as logic. The operations of mathematics are the operations of logic. They are the operations of logic on a selected and carefully defined set of concepts, but they are the operations of logic.

It is easy to see in retrospect that what the medieval theologians were doing was foolish and wrongheaded. Perhaps they were simply intoxicated with the beauty and potency of this new, to them, method of thought. It was such a powerful method, perhaps they thought it was powerful enough to do anything.

What is harder to see is that when we today dismiss such questions from all rigorous thought because they are not amenable to mathematics, we are committing a very similar, if reciprocal, error.

The question of the existence of divinity is a fundamental question, a very important question. It matters intrinsically to the life of every creature. We may never be able to resolve the question. But we are surrendering far too much territory to thoughtlessness if we do not consider it.

I believe it is possible to think productively and powerfully on such questions, even if we may not think entirely mathematically. I believe we may think with clarity, honesty, and precision, corrected but not ruled by logic. We may at least get rid of nonsense. And while we may not be able ever to settle whether there is a divinity, we could probably come to some reasonable concurrence about what the nature of that divinity might be if it did in fact exist.

We could create, to put it another way, a theoretical model of heaven.

“Not as a god,” Wallace Stevens says, “but as a god might be.”

The question of the existence of divinity is only one of many questions which are important but about which we no longer allow ourselves to do any serious thinking. We do not allow ourselves to do so because it is patent that we cannot yet think about these questions mathematically.

We have, then, a situation in which we either address extremely important questions in a spuriously mathematical manner, or do not address them at all. Surely this is not acceptable. Surely we can find a potency and efficacy in thought that does not depend entirely on mathematics.


The Limits of Mathematics

In order to discuss a sort of thinking which does not depend entirely on the mathematical model, it will be useful to first describe the limits of mathematical thought. Mathematicians and logicians who are familiar with Goedel’s Theorem, the Liar’s Paradox, or Russell’s set-of-all-sets paradox may already have a very good understanding of these limits.

I would like to present a more generally accessible description.

No system can fully comprehend the system of which it is a part—in chaos theory, this means sensitive dependence on initial conditions.

Descriptions of reality are systems. They may be thought of as emergent phenomena, and therefore by any approach they are parts of larger systems which themselves are parts of reality.

In human terms, no mind can understand itself fully. In cosmological terms, no model we create can approximate more than a portion of reality. In logical terms, no system can be both consistent and complete: paradox.

Quantum zero.

All logic and all mathematics, to put it another way, are discontinuous at the singularity of paradox. If we have no taste for paradox, we shall be unhappy creatures, for our existence is not only beyond mathematics, but beyond imagination.


Analysis and Synthesis

To think most powerfully and fruitfully, you must think more fundamentally. The most powerful tool of thought is not analysis, which after all is always only logic, but the ability to make connections.

But not all connections are equal. Some are indeed rich and deep, at once evocative and explanatory. Others are merely amusing, although there is frequently genius and passion in the merely amusing. Others are frivolous, a waste of time, like the artificial categories of information on Jeopardy. Still others are perverse, persisted in for the wrong reasons and against all harmony.

If connection is the most powerful tool of thought, how do we select, as individuals and then as tribes, and—one hopes—eventually as humans, those connections which are the most valuable?

We assume an identity between valuable connections and fundamental connections. Anyone may argue this identity does not exist, but we are free to posit it, use it as a working hypothesis, and see whether our results encourage us to continue with the hypothesis or abandon it.

The question then becomes: How do we identify the more fundamental connections? The answer is very simple: root and branch. How many roots (how many antecedents, and antecedents of antecedents) feed into the connection? And how many precedents and precedents of precedents radiate from it? And one further question: How fundamental are the rootlets and the branchlets themselves?

This last question of course brings the task of evaluating a connection into a sort of unprovable circularity. But we are not after proof, remember. We are after better and more useful thought. And we find ourselves in a world in which our attempts to estimate value must always be somewhat circular and self-referential. Only the self needs value. It needs it because it conceives of itself as distinct from everything else. Rankings arise with identities.

There is an interesting question regarding the way in which values arise in any thinking system. On the internet, there is something called a “driver” which powers search engines. The way this driver works, apparently, is to keep count of the most frequently used sites or clues and to strengthen those connections in proportion to the frequency with which they occur.

Such a system is probably a good model for how the human nervous system (or any other intelligence) has evolved. Such a system is also notably self-referential or fed back, and feedback, we have learned, is the hallmark of chaotic systems—in economics, for example, the theory of increasing returns.


Thought and Wellbeing

The essence of strong thought is not in tricks of memory, or in the ability to manipulate increasingly high-order abstractions. These in fact are what computers are so far capable of doing, and all they are capable of doing, and they are not the primary strengths of the human mind, though there are certainly sports out there with fantastic memories, as well as English professors who can deconstruct mathematics and mathematicians who can prove poetry doesn’t exist.

In the same way, though we admire LeBron James or Clayton Kershaw very greatly for their gifts, and for the perfection they have brought to those gifts, we do not mistake for overall good health the ability to jump vertically four feet in the air from a standing stop, or the ability to fling a baseball at a hundred miles an hour at a dime-sized target 60 feet 6 inches away.

Good health is the vigor of a body, any body, with reference to the whole of that body and not its exceptional abilities. So: a strong mind is not so much a mind with unusual abilities in certain arenas as a healthy mind, a mind that uses its own potential to the maximum. Any non-damaged human mind is capable of remarkable feats of thought, given its maximum health.

We may train our bodies to health, barring accidents and illness, and we may likewise train our minds to think powerfully. It’s a rigorous training, unless we happen to be one of those born with the ability. There are such physical sports, born with a constitution of such innate well-being that ordinary living can hardly damage it. There are such mental sports.

I consider health of thought to be the first requirement of thinking. It is a discipline little understood as a discipline—that is, as a path—and as a result, there are few masters to teach us, though there are all sorts of partial masters who may train us to produce prodigal results in a narrow arena.

And what is health of thought?

It is characterized by flexibility, connectivity, clarity, and the willingness to recognize preconceptions and surrender them. Thought is a gift of the being, as nimbleness, coordination, efficiency, and freshness are the gifts of the body.

Objectivity is an ideal, not a fact. Honesty is the only science.


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