Formalism

Formalism is a philosophy of mathematics, which holds that math is just a game of formal symbol manipulation, and that mathematical objects per se don't exist at all.

Formalism has the same flavor as subjectivism.

Many formalists are only formalists with respect to some parts of math. This includes most working mathematicians, insofar as they hold formalist premises. For example, Hilbert held that finite math is meaningful, but infinite math is just a game of formal symbol manipulation.

Logicism

Logicism is a version of formalism, which believes that the rules of the game are reducible to the rules of logic. Logic here refers to logic as understood by Frege, rather than logic as understood by e.g. Aristotle.

Examples

Here I will put examples of working mathematicians saying things based on a formalist premise.

I should mostly put definitions

A huge chunk of written mathematics is formalist. But it is the hardest to identify.

Whenever you have a proof or a definition that is super formal, that uses the technicalities of the definitions rather than explanations, there's probably a formalist definition lurking around somewhere.