Between Disorder and Order

If incubation, whether for a large matter or a small one, is an unconscious process, then it is worth asking what kind of logic it follows and how it occurs. We often assume that because thought is verbal, unconscious reasoning is somehow inappropriate. Einstein might have disagreed, but he too said that fully conscious thought is only one extreme of a spectrum that can never be reached in pure form. There is always an unconscious part mixed into the conscious part of thought.

I am no specialist in this field, but I want to leave a few personal observations about conscious and unconscious thinking. We feel that we think with words and build sentences. That is true not only when speaking with others but also when quietly studying. If someone asked us to think about a problem without using words, we would probably become completely helpless. We cannot solve a problem in our minds without formalizing reasoning in language. The language itself does not matter, but words must be involved.

Yet our way of thinking is not based only on words. Before we begin to think or speak a sentence, we already need to know where it is going. We have grammatical rules to follow. Usually we do not begin a sentence with a negating word, and if we do not know what we are trying to say, we stop speaking. But the moment the word of negation appears in the mind, we already know the next verb and the overall sentence. In such moments the entire sentence must already exist in the mind in a nonverbal form before it is expressed in words.

Formalizing thought through language is extremely important. Words have force, and words pull on other words. In that sense, they function almost like a mathematical algorithm. Just as an algorithm can carry mathematical reasoning nearly on its own, words have a life that draws out further words and allows us to reason and use formal logic. Formalizing thought consciously in language is probably useful for memory as well. If we do not do that, what we think may be much harder to retain. Still, nonverbal thought must come before verbal thought. Thought is historically far older than language, and there is nothing strange about that. Human language has existed for only tens of thousands of years, and it is hard to believe that humans did not think before language emerged. Looking at animals, or at children before they learn to speak, it is difficult to believe that they do not think in any form at all.

Unfortunately, it is not easy to know what kind of logic nonverbal thought follows. One reason is that logic itself is based on language, so using language to study nonverbal thought is nearly impossible. But unconscious thought is crucial for generating new ideas. It is used during the long incubation period mentioned by Poincare and Hadamard, and it also forms the basis of the more fundamental phenomenon of mathematical intuition.

Ordinarily, the proof of a theorem is made of many consecutive steps, and only after deduction following deduction does one arrive at the answer. Yet except in very rare cases, that is not how a theorem is first proved. Usually the proposition is first formed as a whole. With some sense of where it starts and where it ends, the intermediate steps are then set up and linked through necessary proofs. It is rather like building a bridge. First you decide where it begins and where it ends, then you place the middle supports, and only at the end do you lay down the roadway. Just as a sentence must first arise in an overall form before it is formalized in words, a proof too must first exist roughly in the mathematician's mind before it passes into deductive stages.

This helps explain why there are many valid theorems whose first proofs turned out to be wrong. A mathematician may correctly formalize a theorem and find a path forward, yet still make mistakes in the middle steps. If the intuition behind the theorem is sound, there is often another correct method for carrying out the difficult part, or a slightly different way to reach the same conclusion. Mathematicians often speak of the "meaning" of a theorem, expressed in nonformal language and grounded in analogy, similarity, metaphor, and intuition. That meaning justifies the original intuition, but because it cannot be fully formalized, it feels imprecise and unfit for strict literature, something that can only really be shared among friends.