Davis and Hersh muse (1981: 321) that “the typical working mathematician is a Platonist on weekdays and a formalist on Sundays.” Platonism, is in a sense, the ultimate abstraction. It is very seductive and convenient default mode of thinking for many mathematicians. Indeed, for Davis and Hersh (1981: 349), Platonism resembles “an underground religion” which “is observed in private and rarely mentioned in public.”
Davis and Hersh inform us (1981: 318) that Platonist view mathematical objects as real. They exist “outside the space and time of physical existence” and are objective facts “quite independent of our knowledge of them.” Thus, for the Platonist:
a mathematician is an empirical scientist like a geologist; he cannot invent anything, because it is all there already. All he can do is discover.
Davis and Hersh characterize the formalist approach to mathematics as “the science of rigorous proof” based on undefined terms called “axioms.” Davis and Hersh contrast (1981: 340) the extreme formalist perspective with Platonism:
One cannot assert that a theorem is true, any more than one can assert that the axioms are true… Thus the statements of mathematical theorems have no content at all; they are not about anything. On the other hand, according to the formalist, they are free of any possible doubt or error, because the process of rigorous proof and deduction leaves no gaps or loopholes.
Strictly speaking, pure mathematics is only about itself. When it is applied to a physical situation as in physics or engineering it becomes a metaphor or simplification. Jacob Bronowski (1978: 70) is careful to differentiate between pure mathematics, “as an abstract system,” and mathematics as “a formal language for extracting something from the universe.” According to Davis and Hersh (1981: 319) Formalists hold that:
When a formula is given a physical interpretation, it acquires meaning, and may be true or false… as a purely mathematical formula, it has no meaning and no truth value.
Bronowski, Jacob (1978) The Origins of Knowledge and Imagination. Yale University Press, Princeton.
Davis, Philip J. and Hersh, Reuben (1981) The Mathematical Experience. Houghton Mifflin, Boston and New York.