Posts Tagged ‘Henri Poincaré’
Quote of the Day
The principal aim of mathematical teaching is to develop certain faculties of the mind, and among them intuition is not the least precious. It is through it that the mathematical world remains in contact with the real world, and if pure mathematics could do without it, it would always be necessary to have recourse to it to fill up the chasm which separates the symbol from reality. The practician will always have need for it, and for one pure geometer there should be a hundred practicians.
Quote of the Day
The accident of a rhyme calls forth a system from the shadow.
—Henri Poincaré, quoted by Jean Cocteau in Le Rappel à l’ordre
The hierarchy of facts
What I have attempted to explain…is how the scientist is to set about making a selection of the innumerable facts that are offered to his curiosity, since he is compelled to make a selection, if only by the natural infirmity of his mind, though a selection is always a sacrifice…
There is a hierarchy of facts. Some are without any positive bearing, and teach us nothing but themselves. The scientist who ascertains them learns nothing but facts, and becomes no better able to foresee new facts. Such facts, it seems, occur but once, and are not destined to be repeated.
There are, on the other hand, facts that give a large return, each of which teaches us a new law. And since he is obliged to make a selection, it is to these latter facts that the scientist must devote himself.
No doubt this classification is relative, and arises from the frailty of our mind…No doubt a vaster and a keener mind than ours would judge otherwise. But that matters little; it is not this superior mind that we have to use, but our own.
Quote of the Day
Among chosen combinations [of ideas] the most fertile will often be those formed of elements drawn from domains which are far apart. Most combinations so formed would be entirely sterile, but certain among them, very rare, are the most fruitful of all.
The act of combination
In the empty room you’re trying to connect the dots, linking A to B to C to maybe come up with H. Scratching is a means to identifying A, and if you can get to A, you’ve got a grip on a slippery rock wall. You’ve got purchase. You can move on to B, which is mandatory. You cannot stop with one idea. You don’t really have a workable idea until you combine two ideas.
I have coined the term “bisociation” in order to make a distinction between the routine skills of thinking in a single “plane,” as it were, and the creative act, which, as I shall try to show, always operates on more than one plane.
If there is any novelty in the suggestion I am about to make—and I must confess I fear there is—it lies only in the juxtaposition of ideas.
Every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations, and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination.
Scientists who have made important original contributions have often had wide interests or have taken up the study of a subject different from the one in which they were originally trained. Originality often consists in finding connections or analogies between two or more objects or ideas not previously shown to have any bearing on each other.
It is obvious that invention or discovery, be it in mathematics or anywhere else, takes place by combining ideas.
The philosopher must form a new combination of ideas concerning the combination of ideas.
The essential possibility of [metaphor] lies in the broad ontological fact that new qualities and new meanings can emerge, simply come into being, out of some hitherto ungrouped combination of elements.
Instead of thoughts of concrete things patiently following one another in a beaten track of habitual suggestion, we have the most abrupt cross-cuts and transitions from one idea to another, the most rarefied abstractions and discriminations, the most unheard of combination of elements, the subtlest associations of analogy; in a word, we seem suddenly introduced into a seething cauldron of ideas, where everything is fizzling and bobbling about in a state of bewildering activity, where partnerships can be joined or loosened in an instant, treadmill routine is unknown, and the unexpected seems only law.
What is intuition?
One can have memory of the future as well as of the past. Memory of the future is usually called instinct in animals, intuition in human beings.
Pure analysis puts at our disposal a multitude of procedures whose infallibility it guarantees; it opens to us a thousand different ways on which we can embark in all confidence; we are assured of meeting there no obstacles; but of all these ways, which will lead us most promptly to our goal? Who shall tell us which to choose? We need a faculty which makes us see the end from afar, and intuition is this faculty. It is necessary to the explorer for choosing his route; it is not less so to the one following his trail who wants to know why he chose it…Logic, which alone can give certainty, is the instrument of demonstration; intuition is the instrument of invention.
Intuition is a mode of gathering.
Mathematical intuition need not be conceived of as a facility giving an immediate knowledge of the objects concerned. Rather it seems that, as in the case of physical experience, we form our ideas also of those objects on the basis of something else which is immediately given.
[Intuition] grasps a succession which is not juxtaposition, a growth from within, the uninterrupted prolongation of the past into a present which is already blending into the future. It is the direct vision of the mind by the mind.
Intuition is the ability not to construct solutions to problems in a rational manner, but rather to produce them spontaneously (holistically) according to situational demands…Intuition is, we could perhaps say, a fire that lights itself.
My intuition made me work…Intuition makes us look at unrelated facts and then think about them until they can all be brought under one law.
By intuition I do not mean the fluctuating testimony of the senses or the deceptive judgment of the imagination as it botches things together, but the conception of a clear and attentive mind, which is so easy and distinct that there can be no room for doubt about what we are understanding…Intuition is the indubitable conception of a clear and attentive mind which proceeds solely from the light of reason.
Intuition [in Nobel laureates] is closely associated with a sense of direction; it is more often about finding a path than arriving at an answer or reaching a goal. The ascent of intuition is rooted in extended, varied experience of the object of research: although it may feel as if it comes out of the blue, it does not come out of the blue.
Quote of the Day
It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better.
Alec Nevala-Lee 