Suppose, like stock-market traders trying to guess today's opening price and price movements, we aim to predict the current and future values of an un

We can predict the future - by Joel David Hamkins

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2024-09-01 14:30:04

Suppose, like stock-market traders trying to guess today's opening price and price movements, we aim to predict the current and future values of an unknown function based on the observed history of previous values. The true function f:ℝ→ℝ will be revealed only gradually, and we seek to predict the current value f( t) at the current time t in light of the historical data, as well as to predict how the function will continue beyond t into the immediate future.

Let us begin with the seemingly easier task of predicting the present. We want to predict the current value f( t) of the function f on the basis of its historical values before time t. Of course, if we knew somehow that the function were continuous, then we could easily predict the current value simply as the limit of the previous values.

The continuous case is thus a trivial kind of case for predicting the present—the limit value will be correct in every continuous instance.