Mathematics, Intuitivity, and Beauty

Jacques Maritain, "No Knowledge Without Intuitivity," in Untrammeled Approaches, trans. Bernard Doering (Notre Dame, IN: University of Notre Dame Press, 2017), 345-348.

"Finally, with regard to the second degree of abstraction, the one that is proper to the mathematical sciences, I have a prefatory remark to make first.  Up to the present, when I spoke of the intuitivity of the mind, I always added: the intuitivity of the mind turned toward the real because what I was first and foremost concerned with was philosophical knowledge.  But in the case of mathematical sciences, it is not toward the real but toward beings of reason that the mind and its intuitivity are turned.  Separated from from corporal substance [346] and from the world of sensible experience by an abstraction which, this time, isolates its object from the real and recasts it in an idealist manner, extension and quantity actually constitute for thought a multiform noetic universe set apart, which infinitely overflows the conditions proper t the world of matter such as it exists in reality and such as our senses and our intellect lay hold on it at the first degree of abstraction.

This multiform and teeming universe, not of existing things, but of pure objects of thought, on the one hand is constructed by the mind with regards the manner in which it takes form beginning with the abstraction which draws out the first principles of things, and at the same time is independent of the mind with regard to its properties (some basic quiddity or basic definition having been first freely posited, constitutive of some branch or other of the mathematical sciences) because this is a universe of pure rational necessities.  The ideal entities which compose them have their own consistency and their own structures that are just as firm, and their own laws which are just as inflexible as those of the real world; so that if it is true to say that the mind freely causes them to appear before it, it is equally true to say that they unveil themselves to the mind that explores them.  It is as if from the ocean of the imagination, the intellect, with a wave of a magic wand, causes islands to emerge or archipelagos or mountains covered with eternal snows, which have no existence except as ideals or as pure objects of thought, which the mind sets itself to scrutinize.  There are no sciences freer than the mathematical sciences, which in a sense create their own objects (as is the case with art), but they are not forms of art, they remain sciences, whose truth consists in conformity, if not to existing things, at least to pure objects of thought which, once their ideal existence has been posited, depend solely on that ideal existence for their entire logical development and all their properties.  This is a type of knowledge that is completely foreign to physics (since it is concerned with a purely ideal object), but which is a marvelous instrument of physics...

But let us leave all that aside.  What I consider essential to notice here is that, at the second degree of abstraction, the intuitivity of the mind—turning [347] its attention this time not to reality (even if by means of the imagination) but to this or that system of ideal entities which the mathematical imagination gives rise to—here too, plays a fundamental, indispensable role in knowledge.

And since there is a question here of ideal systems which, however consistent they may be in themselves, are created by the mathematical imagination, is not our concern here above all with an intuitivity of the artistic or aesthetic order?  Intregetas, consonantia, claritas: integrity, consonance or harmony, splendor or clarity, is it not because he is in search of these three elements typical of beauty, to which his soul is interiorly attuned, that the mathematician suddenly sees a principle of intelligible fecundity and intelligible organizing power open up to his particular science new avenues of approach to a higher synthesis?

Here, I find perfectly sufficient the testimony of an eminent scholar in a mathematical area for which, despite my incompetence, I have a particular reverence: topology, which Leibniz, who conceived the idea, called analysis situs (much later Riemann won for it its scientific status), and which studies the relative positions of geometrical entities, not by means of nubmer and measurement, but solely in their qualitative properties or their "ordered relationships."

'The first essential bond between mathematics and the arts,' wrote Marson Morse, 'is found in the fact that discovery in mathematics is not a matter of logic.  It is rather the result of mysterious powers which no one understands., and in which the unconscious recognition of beauty must play an important part.  Out of an infinity of designs, a mathematician chooses one pattern for beauty's sake, and pulls it down to earth, no one knows how.  Afterwards, the logic of words and of form sets the pattern right.  Only then can one tell someone else.  The first pattern remains in the shadows of the mind.'

[He goes on citing examples of Poincaré and Gauss.]

[348, Morse continues... 'The creative scientist lives in the "wilderness of logic" where reason is the handmaiden and not the master.  I shun all monuments that are coldly legible.  I prefer the world where images turn their face in every direction, like the masks of Picasso.  It is the hour before the break of day when science turns in the womb, and, waiting, I am sorry that there is between us no sign and no language except by the mirrors of necessity.  I am grateful for the poets who suspect the twilight zone.'

You will notice that mathematical intuitivity is not poetic intuitivity, for in the latter, it is the subject and the object, the self of the poet and the world which are revealed together to the mind [to understand aright, see his Art and Scholasticism, Creative Intuition in Art and Poetry, and The Situation of Poetry], whereas the intuitivity of the mathematician is of a wholly objective order, as is the case with the intuitivity of the mind in any knowledge properly so called..."