Discernment and Discovery: Intuition in Mathematics and Sciences.

If one does research on the sayings of almost all great minds they have indirectly pointed out that all ideas came to them through intuition. Then in this post, let us see what a mathematician or a scientist has to say about intuition in the sense how he/she interprets the word.

Mathematicians have been concerned about induction, inference, and intuition in their own field, not only because they bring in invention, innovation, and discovery just as all other groups of scholars do, but because certain aspects of the field of mathematics are very closely related to pure logic and the link with philosophy is extremely close.

Blaise Pascal famous for the Pascal's law and the unit of pressure in physics which is termed after him had to say the following which clearly shows that he is emphasising the fact that intuition lies beyond reason.

"The heart has its reasons which reason knows nothing of... We know the truth not only by the reason, but by the heart." 

Polya (1954) finds mathematics particularly suited for the processes of inference and induction for the following reasons :

1. Entities are well defined and unambiguous.

2. Clear difference between induction and proof.

3. Proofs when attained are undisputed, beyond controversy and final.

In spite of this high degree of certitude he points out that on the contrary, discoveries in mathematics are the product of plausible reasoning, inference and analogy which are nothing but processes subsumed under intuition according to Bruner(1961).

Polya says "When you have satisfied yourself (intuitively) that the theorem is true, you start proving it (intellectually or rigorously)."

For Poincare intuition is a fact of life among mathematicians and differentiates from the logic by describing two of his contemporaries:

"M. Meray wants to prove that a binomial equation always has a root, or, that an angle may always be subdivided. If there is any truth that we think we know by direct intuition, it is this. Who could doubt it but in eyes of Meray it is not at all evident and to prove it he needs several pages.

On the other look at Professor Klein: To determine whether on a given Riemann surface there always exists a function admitting of given singularities, he connects two of its points with two poles of a battery. The current, says he, must pass and the distribution of this current on the surface will define a function whose singularities will be precisley those called for.

Doubtless Professor Klein well knows he has given here only a sketch and finds in it, if not a rigorous demonstration, at least a kind of moral certainty. A logician would have rejected with horror such a conception, because in his mind it would never have originated.(Poincare 1913, p.211)"

Poincare speaks of intuitive mathematicians as geometers because they frequently project visually and take visual evidence as direct intuitive proof.

Some of his most pertinent quotes on the faculty of intuitive insights are as follows: 

"It is through science that we prove, but through intuition that we discover. To invent is to discern, to choose."

"Discovery in mathematics, does not consist in making new combinations with mathematical entities that are already known. That can be done by any one and the combinations that could be formed would be infinite in number, and greater part of them would be absolutely devoid of interest. Discovery consists precisely in not constructing useless combinations, but in constructing those that are useful, which are an infinitely small minority. Discovery is discernment, selection." (Poincare 1913, p. 386)

Poincare further asserts strongly that the special aesthetic sensibility is what makes a mathematical discoverer, and he who does not have the sensibility will never make discoveries.(1913, p.392)

Another aspect of intuition described by Poincare is his analysis of a sudden apprehension of discovery, which is accompanied by certainty and is the product of great concentrated effort which is unsuccessful, followed by a waiting period and finally by an illumination-the intuitively selected combination which is then subjected to further strenuous analysis. It never happens that unconscious work supplies ready-made the result of a lengthy calculation in which we have only to apply fixed rules. (p.394)

Karl Popper, a philosopher refers to intuition as a starting point in the scientific method: There is no such thing as a logical method of having new ideas or a logical reconstruction of this process. Every discovery contains an irrational element or a creative intuition. (Goldberg, p. 21)

Alfred Sheldrake, a new age scientist has conjectured that all natural systems are regulated by invisible organizing, morphogenetic fields that provide a blueprint of all like object form and behaviour. This invisible pattern may be apprehended in the structures of nature through intuitive insight into spiritual unity with nature as well as in experiments done in quantum physics, biology, and Chrystal chemistry.

Einstein on intuition and the discovery of natural laws, had to say - "There are no logical paths to these laws, only intuition resting on sympathic understanding of experience can reach them. The really valuable thing is intuition. The intuitive mind is a sacred gift and the rational mind is a faithful servant. We have created a society that honors the servant and has forgotten the gift." (Goldberg, p.21)

Isaac Newton hinted at intuitive insights as, "A man may imagine things that are false, but he can only understand things that are true. Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things. No great discovery was ever made without a bold guess."

While John Maynard Keynes on Isaac Newton said, "It was his intuition which was preeminently extraordinary. [He was] So happy in his conjectures that he seemed to know more than he could have possibly any hope of proving. The proofs were . . . addressed afterwards; they were not the instrument of discovery." (Goldberg, p. 21)

What then is mathematics if it is not a unique rigorous, logical structure? It is a series of great intuitions carefully sifted, refined, and organized by the logic men are willing and able to apply at any time. (Morris Kline, a mathematics professor 1980)

Helmholtz had his happy thoughts which "often enough crept quietly into my thinking without my suspecting their importance ... in other cases they arrived suddenly, without any effort on my part.... They like especially to make their appearance while I was taking an easy walk over wooded hills in sunny weather !"

And Gauss, referring to an arithmetical theorem which he had unsuccessfully tried to prove for years, wrote how "like a sudden flash of lightening, the riddle happened to be solved. I myself cannot say what was the connecting thread which connected what I previously knew with what made my success possible."(Jaynes, p.43)

Jeremy Hayward a physicist proposes that intuitive insight is the bridge between nihilism and eternalism. The intrinsic perception of the primordial intelligence that perceives the phenomenal world is indeed the ongoing intuitive insight that binds all to the unified whole. It is not an unnatural power over the natural world but a sense of the fullness of value in each aspect of this existence. Hayward attempts to correlate natural phenomena and the natural intuitive insight, as the innate and primordial wisdom in the world as it is. Intuition is to be fully what is being done, thought, or felt with no self-consciousness, no split mind; an openness and sense of inquisitiveness into the environment (Hayward, p.261).

Finally Alexander Grothendieck, considered to be the greatest mathematician of the last century, whose emphasis on the role of universal properties across varied mathematical structures brought Category theory into focus as an alternative to set theory left around 20000 pages of scribbles stored in 5 boxes now at University of Montpellier. That he thought intuitively in geometric figures is seen in some of sample scribbles attached below:

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In conclusion it can be ascertained that scientists or mathematicians were always fascinated with intuitive insights and emphasised intuition above everything in their endeavours.