Formalism and Realism - a comparison

This blog post continuous a series of posts on the philosophy of mathematics and is will therefore not go into detail on the meaning of the various technical terms used.

As we already discussed formalism and realism in class, I will try to summarise what makes the difference between the two and especially what parts of realism are rejected by the formalist position:

Formalism, in my opinion, is the most anti-realist position I came across so far since it rejects realism in the most basic philosophical questions: in ontology, epistemology and truth value.

The differences found in ontology is probably captured best when reviewing the two positions concerning this central question of mathematical philosophy: While the realist believes in mathematical objects existing independently of us, the formalist not only rejects this idea, but even avoids talking about objects, since he doesn’t need them for his ”playing with formal sentences”. Therefore, the formalist takes even a step further away from realism than does constructivism in this regard since a constructivist at least sticks with the notion of objects. As a result, the realist when proving a theorem speaks of discovering mathematics and believes that he found a piece of (mathematical) truth while the formalist would argue he was just playing in his system of formal sentences and found a statement that he can deduce using the systems axioms.

In epistemology the formalist position introduced a new method of proof: the formal proof which to a strict formalist is the only one acceptable, meaning the formalist rejects to all the other proof methods used by realism. This in combination with the formalist ontology yields to the formalist, if at all, only believing in manipulation-meaning when asked about what one could give meaning to in mathematics and therefore, rejecting all the various ways a realist would give meaning to his mathematical objects and findings.

The way realists consider truth-value, namely as independent on us, is rejected by formalism as well since they see truth-value just as another property of formal sentences one could introduce inside his theory if one wishes to and which, in particular, has no greater meaning outside this theory.

Suming this up, all these differences especially lead to a very different approach towards proofs: The formalist strongly rejects the whole process of a realist proving a theorem using quite various methods to find out the a-priori truth behind a mathematical object/sentence which would have been there independently of us and suggests the mathematician chooses the system he wants to work in right now by choosing formal sentences as his starting points (axioms) and uses formal proofs to find out more about it, but can always just come up with another formal system and ”play” inside this other one since there is no right system to do mathematics in.

But which parts of realism are compatible with formalism then?

First of all, they agree on trivial systems being bad/uninteresting systems.

Secondly, the mathematical practice is dominated by a so-called formalist-realist schizophrenia, pointing at the fact that most mathematicians see themselves as formalist, but would rather go for a realist approach when needing to come up with the idea for a proof. Furthermore, due to the effort needed to formulate a formal proof, many mathematicians don’t actually prove formally, but, as suggested by Hilbert, formulate an heuristic proof which they believe could be turned into a formal one. This approach, honestly, also appeals to me.

Thirdly, due to the some open questions in formalism the formalist needs to decide between being constructivistic or realistic on some meta-level and since many (formalist) mathematicians agree on choosing the realist position at this point, there seems to be at least some agreement between the realist and formalist position, even though they seem to differ heavily in the first place.