Our brains operate through networks of cells. Undefined objects—neurons—are interconnected by inextricable webs of synapses. Each neuron communicates with ten thousand others, and each network of neurons and synapses corresponds to a concept. One consequence of this arrangement is that our ways of thinking are systematically linear. So too are the models generated by our brains. Mathematics is therefore full of linear representations—chiefly corresponding to "degree 1" in algebra and the concept of a "line" in geometry—whether in the form of systems of equations or linear algebra itself, not to mention statistical fitting models, which are primarily and predominantly linear. But it would be naive to imagine that linearity alone could model the complexity of the situations we wish to describe. Mathematicians are determined to discover new ways of thinking with which to develop the mathematics of tomorrow. The notion of a derivative is a first step beyond linearity… while remaining linear locally. A derivative can be viewed as a rate of change in the neighborhood of a point. Thus, if we take increments Δx that are "small enough" for the ratio under consideration not to "differ too much" from the value of the derivative, then for a function f differentiable at x0, we can write:
for x = x0 + Δ x, an expression of the form a + b Δ x, where a and b are constants. A function differentiable at a point is therefore locally linear. This concept is widely used in nonstandard analysis: the increments Δx become infinitesimals, which inhabit the halo of numbers infinitely close to 0. Intuitively, a differentiable function viewed under a microscope is a straight line. The microscope's magnifying power corresponds to the more traditional concept of a neighborhood.
A stroll decided by a coin toss
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What if we study phenomena that evolve over time? If we take t0 to be the present instant, the sign of Δt distinguishes the past (Δt < 0) from the future (Δt > 0). Assuming that a phenomenon is differentiable means assuming that the derivative of its function exists for every sequence of time increments Δt considered. But only the past can be observed! We can therefore measure f (t) only for negative values of Δt. Yet under the differentiability assumption, this is enough to estimate the value of the derivative of f at t0. Knowledge of the past is enough to determine f ’(t0 ), while the present allows us to measure f (t0 ). The past and present therefore suffice to give a local description of the phenomenon's future evolution:
