Friday, July 1, 2016
Contributed by Mikhail Filippov, Varun Prasad and Semir Zeki
Ever since the description of the neural correlates of the experience of mathematical beauty, we have been wondering to what extent mathematical beauty falls into the category of biological beauty.
This has led us to enquire further into the extent to which mathematics itself constitutes a study of the brain’s logical architecture; in other words, the extent to which the study of mathematics also belongs in a branch of biology and more specifically neurobiology.
We start by enquiring into the processes which led one of the most interesting mathematicians of the last century, namely Srinivasa Ramanujan, to his conclusions.
They are commonly referred to as INTUITION.
But what is intuition?
The term is commonly used to signify that a significant insight or conclusion has been reached without thinking and without reasoning. This is true in all languages to which we (the writers of this post) have access. One would no doubt find definitions which are more sophisticated but, as the quotations below show, the absence of logical process in having an intuition is the most common definition and reflects, in fact, what the lay person usually means by it.
The term has been much written about recently, especially since the publication of a book about Ramanujan by Robert Kanigel entitled The Man who Knew Infinity, recently made into a film.
Consider the following definitions of intuition, which exclude any reasoning process:
“Direct perception of truth, fact, etc., independent of any reasoning process, immediate apprehension”
“An immediate cognition of an object not inferred or determined by previous cognition of the same object” (from Dictionary.com)
“The immediate apprehension of an object by the mind without the intervention of any reasoning process; a particular act of such apprehension.” (from Oxford English Dictionary)
The French definitions given in Larousse are even more explicit in this regard:
“Connaissance directe, immédiate de la vérité, sans recours au raisonnement, à l'expérience.”
“Sentiment irraisonné, non vérifiable qu'un événement va se produire, que quelque chose existe”
One Italian dictionary defines intuition as follows:
“Intuizione: Conoscenza diretta e immediata di una verità, che si manifesta allo spirito senza bisogno di ricorrere al ragionamento, considerata talora come forma privilegiata di conoscenza che consente, superando gli schemi dell’intelletto, una più vera e profonda comprensione”
while its Spanish counterpart states:
“Percepción clara e inmediata de una idea o situación, sin necesidad de razonamiento lógico”
Nor are such definitions restricted to Western European languages. Much the same definition appears in Japanese.
"直感" = to capture things by feeling rather than reasoning or discussion.
"直観" is to directly understand the essence of things without relying on reasoning.
We give these definitions in different languages only to show that much the same applies to all. Central to most (but not all) definitions is the absence of reasoning or logical thinking during the intuitive process or its result. Hence, the dictionary definitions given do indeed reflect the way in which the term is commonly understood.
Other definitions come closer to the arguments we give below; they make no reference to the absence of reasoning logic, but only to the absence of proof or evidence.
For example, the Merriam-Webster and the Free Dictionary define intuition as follows, respectively:
“A natural ability or power that makes it possible to know something without any proof or evidence: a feeling that guides a person to act a certain way without fully understanding why”
“Something that is known or understood without proof or evidence” (from Merriam Webster)
“The faculty of knowing or understanding something without reasoning or proof”
(from Free Dictionary )
which is not dis-similar to the definition given in the Great Soviet Encyclopaedia
“ИНТУИЦИЯ(позднелат. intuitio, от лат. intueor - пристально смотрю), способность постижения истины путём прямого её усмотрения без обоснования с помощью доказательства.”
(Intuition is an ability to comprehend the truth through direct discovery without its justification with the proof)
We propose below an alternative definition that may be obvious to some but is not to many and therefore worth giving:
An intuition is an unconscious logical brain process with an outcome or conclusion in the form of a statement or proposition. But whether the outcome of the intuitive process is “right” or “wrong”, or “correct” or “incorrect”, can only be determined by a conscious logical process.
The closest dictionary definition to this that we know of is to be found in the Russian Dictionary of Psychology, which of course targets a more specialized audience:
“Интуиция (от лат. intueri – пристально, внимательно смотреть) - мыслительный процесс, состоящий в практически моментальном нахождении решения задачи при недостаточной осознанности логических связей.”
(Intuition - thought process allowing almost instant finding of the solutions to the problem with the lack of awareness of logical connections.)
Mathematics is a subject in which intuition is often invoked.
But the end result of the unconscious logical process that is intuition can only be “right” or “wrong” (correct or incorrect) when consciously scrutinized.
As examples of “right” and “wrong” mathematical intuitions consider the following:
A right (correct) intuition: Pick a point at random on the Earth (assume that the Earth is a sphere). The probability that the point picked lies in the northern hemisphere is 50%.
Most students of mathematics (i.e. those who have enough knowledge to understand the above statements, but who do not know if they are right or wrong) would intuitively guess this statement to be true and it is.
A wrong (incorrect) intuition: Pick a real number randomly. The probability that the real number picked is rational is zero.
Most mathematics students would intuitively guess this statement to be false (you can obviously pick a rational number!). However, it is correct.
But the conclusion that they are correct or incorrect can only be reached through a conscious logical process.
Ramanujan was reluctant to submit his intuitions to the conscious process of deductive logic, until Hardy brought him to England and forced him to do so – i.e., to provide proofs for his intuitions – a conscious process.
The absence of any logical process or reasoning in the intuitive process is not the only weakness of the definitions of intuition; some also exclude the role of experience in reaching conclusions through intuition, as in the Larousse Dictionary or the Dictionary.com definitions given above.
We believe, however, that to have an intuition in any area, one must have experience of that area or knowledge of it, to provide a conclusion or statement, whether correct or incorrect.
Since we suspect that there is only a limited set of deductive logical processes in the brain, it follows that the same logical processes must be used to derive intuitions in different domains; what distinguishes intuitions in different domains, and the logical processes that lead to them, is past experience and knowledge in the relevant domain.
The result of this unconscious logical process (the intuition) depends on initial conditions or inputs, which are based on previous (conscious) knowledge, consciously or unconsciously obtained.
Our proposed definition raises interesting and important issues and leads to the suggestion that
a. There are many (but a limited number of) logical processes, which operate in the unconscious state.
b. These processes are undisciplined and unruly but still obey some sort of brain logical process.
c. They become disciplined and eliminated by revisiting them, and the conclusions to which they lead, in the conscious state.
d. The latter eliminates many of the undisciplined and vagrant unconscious logical processing possibilities, thus stabilizing the logical processing systems of the brain.
The study of intuition in mathematics thus belongs as well to neurobiology. Or, put another way, mathematicians are also covert neurobiologists.
© Mikhail Filippov, Varun Prasad and Semir Zeki