The Historical Achievement
Whitehead begins Science and the Modern World with a meditation on the seventeenth century's discovery that nature is mathematical. Galileo, Descartes, and Newton showed that motion, force, and gravity could be precisely described by equations — and that these equations predicted new phenomena with astonishing accuracy. This was not obvious in advance; it was a discovery that the relational structure of nature corresponds to the relational structures that mathematicians had developed for purely internal reasons.
The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. So long as you are dealing with pure mathematics, you are in the realm of complete and absolute abstraction.
Form and Actuality
On Whitehead's account, mathematics works because it describes eternal objects — the abstract forms that actual occasions realise in their concrescence. The laws of nature are not external constraints imposed on matter but descriptions of the patterns in which occasions typically select from available forms. A mathematical law captures a deep habit of nature: a regularity in how the creative advance has proceeded. This is why the laws hold — not because they are logically necessary but because the relevant occasions have consistently chosen the relevant forms.
The seventeenth century had finally produced a scheme of scientific thought framed by mathematicians, for the use of mathematicians. The great characteristic of the mathematical mind is its capacity for dealing with abstractions; and for eliciting from them clear-cut demonstrative trains of reasoning, entirely satisfactory so long as it is those abstractions which you want to think about.
The Limits of Formalisation
Whitehead is also alert to what mathematics cannot capture. A mathematical description of a storm is not the storm; a differential equation of neural activity is not the experience of pain. The formal structures that mathematics excels at describing are abstractions from the full richness of actual occasions, and that fullness — including the qualitative, valuative, experiential dimensions — exceeds any formalisation. Science needs mathematics, but a philosophy of science that identifies nature with its mathematical description has already committed the fallacy of misplaced concreteness.
The first two chapters of Science and the Modern World are among Whitehead's most historically informed, tracing the development of mathematical physics from the Greeks through Newton. Whitehead himself was co-author (with Bertrand Russell) of Principia Mathematica (1910–13), so his reflections on the limits and nature of formalism come from a position of genuine expertise.
Related Concepts
