Formalism

Formalism is a philosophy of mathematics, which holds that math is just a game of formal symbol manipulation.

Formalism has the same flavor as subjectivism.

Formalism is one of the three major philosophies which dominate the minds of modern mathematics, the others being Intuitionism and Platonism. Very few working mathematicians adhere consistently to any of those three philosophies. Almost all of them tacitly accept the (mutually inconsistent) premises of all three.

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.

For a good example of explicit formalism, see the following quote from Russel and Whitehead about definitions

A definition is a declaration that a certain newly-introduced symbol or combination of symbols is to mean the same as a certain other combination of symbols of which the meaning is already known. [...] We will give the names of definiendum and definiens respectively to what is defined and to that which is defined as its meaning. [...]

[...]

It is to be observed that a definition is, strictly speaking, no part of the subject in which it occurs. For a definition is concerned wholly with the symbols, not with what they symbolize. Moreover it is not true or false, being the expression of a volition, not of a proposition. [...] Theoretically, it is unnecessary ever to give a definition: we might always use the definiens instead, and thus wholly dispense with the definiendum. Thus [...] definitions are no part of our subject, but are, strictly speaking, mere typographical conveniences. Practically, of course, if we introduced no definitions, our formulae would very soon become so lengthy as to be unmanageable; but theoretically, all definitions are superfluous.

[TODO maybe add something to this page about Wittgenstein? or maybe not cuz so few ppl are wittgensteinians...]

Logicism

Logicism is a specific type of formalism, which believes that the rules and concepts of the game are reducible to those of logic (sets, set membership, modus ponens, etc). Logic here refers to logic as understood by Frege, rather than logic as understood by e.g. Aristotle.

For example, Logicism defines the natural numbers to be set-theoretic objects like the following

• $0:=\emptyset$
• $1:=0\cup \{0\}=\{\emptyset \}$
• $2:=1\cup \{1\}=\{\emptyset ,\{\emptyset \}\}$
• $3:=2\cup \{2\}=\{\emptyset ,\{\emptyset \},\{\emptyset ,\{\emptyset \}\}\}$

etc.

Examples

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

The working mathematician almost always has some formalist premises. To some extent, he treats some pieces of math like a game of formal symbol manipulation. However, it is difficult to identify if, in any specific instance, a mathematician is being a formalist. The reason for this is that what is understood by one man as a conceptual argument, could be understood by another man as mere symbol manipulation.

The following example is from Gamelin and Greene, Introduction to Topology:

3.1 Theorem: Let $X$ and $Y$ be topological spaces. A function $f:X\rightarrow Y$ is continuous if and only if $f$ is continuous at every point of $X$.

Proof: Suppose that $f$ is continuous at every point of $X$ and that $V$ is open in $Y$. Let $x\in f^{-1}(V)$. Since $f$ is continuous at $x$, there is an open set $U_{x}$ containing $x$ such that $U_{x}\subset f^{-1}(V)$, or $f(U_{x})\subset V$. Set $U=\cup \{U_{x}:x\in f^{-1}(V)\}$. Then $U$ is open and $f(U)\subset V$, so that $U\subset f^{-1}(V)$. Since $U$ includes each point of $f^{-1}(V)$, $U$ coincides with $f^{-1}(V)$ and $f^{-1}(V)$ is open. Thus $f$ is continuous.

Conversely, suppose that $f$ is continuous. Let $x\in X$ and let $V$ be an open set containing $f(x)$. Then $U=f^{-1}(V)$ is an open set containing $x$ and $f(U)\subset V$. Consequently $f$ is continuous at $x$ for all $x\in X$. $\square$

The reason why I say that this proof is formalistic is that it could be easily understood by someone who knows the formal definitions of the objects involved in the proof, but has no conceptual understanding of what it means for a map to be continuous. Indeed, that situation is the case for many students who are learning about topology for the first time. Although more advanced mathematicians have at least some level of intuitive (non-formalistic) understanding of why these two definitions of continuity are equivalent, that is neither necessary nor sufficient for proving the above theorem.

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.

Refutations

Mathematicians usually don't actually do deductions.

[TODO explain] Gödel refuted formalism, as it was understood by Hilbert's program.