Grammars of Creation by George Steiner

Author:George Steiner
Language: eng
Format: epub
Publisher: Open Road Integrated Media

IV

To Plato, the point would have been self-evident. It is inconceivable that one should question the life of the mind without addressing mathematics and the sciences which, in the main, derive from the sovereignty of mathematics. Since Galileo and Descartes, this injunction has become theoretically and pragmatically inescapable. It is in mathematics and the sciences that the concepts of creation and of invention, of intuition and of discovery, exhibit their most immediate, visible force.

The difficulty, however, is twofold. Mathematicians and scientists “get on with the job.” Like the proverbial millipede wary of introspective paralysis, they avoid too close a scrutiny of the epistemological foundations of their disciplines. They do not inquire too probingly into the justifying or subversive presuppositions of their manifestly triumphant progress. Even so elemental a challenge as the debate on the axiomatic and logical foundations of pure mathematics, on the internal consistency of axiomatic systems, at the turn of the century, is left in abeyance. There is simply too much to do. In turn, the application of even abstruse mathematical theories to natural sciences and technology reinforces habits of empirical confidence. The “thing works,” whatever the arcane philosophic vexations at its originating depths.

The second difficulty is one of access. The non-mathematician, the outsider to the sciences, can hardly begin to grasp, let alone gauge, the arguments and controversies such as we have them as to the nature of mathematical scientific creation or invention. Ignorant of the languages of mathematics and of their translation into the exact and applied sciences, the listener can scarcely make out even the rudiments of the debate as to whether or not the objects of mathematical operations are, in essence, intuitions, mental artifacts, of realities in an existential sense. One needs considerable familiarity with mathematical symbolism in order to follow the controversies on whether or not there are in pure mathematics “discoveries” or, instead, an autonomous unfolding of a priori, as it were tautological, systems generated from within the human intellect and its deep-seated instinct for speculative, other-worldly play. Homo ludens. If, as Galileo ruled, nature speaks mathematics, far too many of us remain deaf.

Though almost a century old, Henri Poincaré’s paper on mathematical creation is still considered a classic. Poincaré was attentive to the dynamics of algebraic intuition and resolution within his own eminent research. He brought to bear on them both wit and self-observant finesse of an unusual acumen. His autobiographical record of the moments, of the circumstances of seeming hazard, which triggered his famous memoir on Fuchsian functions—Poincaré was putting his foot on the platform of a provincial omnibus when the seminal hunch and concomitant solution struck him—remains exemplary. The “lightning flash” resulted from a lengthy process of prior but unconscious or subliminal work. It was the outcome of a concentrated process of analytical technique below the threshold of Poincaré’s daylit awareness. Enigmatically, argues Poincaré, the sub-conscious self, somehow impregnated with algebraic impulses, has a tact and operative delicacy “in no way inferior to the conscious self.” It is at

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