Some philosophers see into themselves, and some into their times; still others forge an alliance with the future.

Some philosophers see into themselves, and some into their times; still others forge an alliance with the future, scribbling their secrets late at night and speaking in whispers to the insubstantial and impatient souls that are gathered around their study door dying to be born. Years go by and the dust of time collects. A new world is made. Things whir and pop and sizzle. There is the clatter of dropped dishes and laughter in the dark. Heels clack on the pavement. Taxis wheel and honk, and sanitation trucks bang down city streets. The sky is filled with electromagnetic pulses. The smoldering red sun edges over the horizon. Alarm clocks ring or chime or tinkle. Radios burst into chatter and still some silken thread of memory ties the present to the past so that, pausing for just a moment, we see ourselves reflected in a scholar's eyes, the calm smile in the portrait knowing and at peace.

Within the categorical syllogism, ordinary language represents the ordinary flow of inference. Two premises are given; there is a plash of insight, and one step undertaken. The mind hops right along, not quite knowing where it is going but getting there nonetheless. On the right, a checklist does its work. The logician's clamp retains its force of old, but the inferential steps involve no more than the substitution of symbols for symbols, with the anchor of inference embedded in identities. Inference now proceeds from one identity to the next; no plash of insight is involved, only the solid satisfying ratcheting sound of symbols being substituted for symbols.

Ultimately, Leibniz argued, there are only two absolutely simple concepts, God and Nothingness. From these, all other concepts may be constructed, the world, and everything within it, arising from some primordial argument between the deity and nothing whatsoever. And then, by some inscrutable incandescent insight, Leibniz came to see that what is crucial in what he had written is the alternation between God and Nothingness. And for this, the numbers 0 and 1 suffice.

Validity is the touchstone of inference, and truth of judgment: the fact that vichyssoise is cold ratifies the judgment that vichyssoise is, indeed, cold, and the judgment that vichyssoise is cold expresses the fact that vichyssoise is cold.

In the end, every scheme and every science is justified by itself or it is not justified at all.

Leibniz endeavored to provide an account of inference and judgment involving the mechanical play of symbols and very little else. The checklists that result are the first of humanity's intellectual artifacts. They express, they explain, and so they ratify a power of the mind. And, of course, they are artifacts in the process of becoming algorithms.

For the most part, it is true, ordinary men and women regard mathematics with energetic distaste, counting its concepts as rhapsodic as cauliflower. This is a mistake--there is no other word. Where else can the restless human mind find means to tie the infinite in a finite bow?

The desire to see and the desire to ratify what one has seen are desires at odds with one another, if only because they proceed from separate places in the imagination.

The world of shapes, lines, curves, and solids is as varied as the world of numbers, and it is only our long-satisfied possession of Euclidean geometry that offers us the impression, or the illusion, that it has, that world, already been encompassed in a manageable intellectual structure. The lineaments of that structure are well known: as in the rest of life, something is given and something is gotten; but the logic behind those lineaments is apt to pass unnoticed, and it is the logic that controls the system.