The massive problem with regarding string manipulations as the foundation of mathematics

Question

Formalists believe that mathematics is just a game of string manipulation, not much different from other games like Ludo or chess. I think string manipulation is an extremely useful way to think about mathematics when you don't care about the foundational questions. In the discussion of foundations, however, this view fails miserbaly. Amongst the many, many flaws: string manipulation involves decision making like "From this string, you can infer that string" or "If the initial string is not this, then you can't infer that string".

We see that the ideas of logic (like If, then, Not, AND, OR) underlie string manipulation. This underlying notion of logic is not arbitrary, but is absolute in nature. For instance, when we define logic itself as a symbol manipulation game, we are free to come up with arbitrary non-standard rules. We are free to come up with rules that can even derive contradictions. This is because the symbols in the game don't have any meaning. When considered as games, standard logic is on an equal footing with non-standard logic.

But, if we consider the rules of logic that underlie the decision making in this string manipulation game, those rules are absolute and not arbitrary. This means that there exists an underlying fundamental logic that is discovered, not invented, and that it serves as the foundation of string manipulation games.

What is the formalist response to this?

Answer 1

For logics, it goes like this:

You have to (and can) show that the results from your string manipulation (your syntactic manipulations) actually are an appropriate model for the semantics. That is, you want your calculus to be complete (can derive all semantically true statements as true) and sound (can only derive semantically true statements as true).

For example, for first order predicate logic, the Sequent calculus can be shown to be sound and complete, so the purely syntactic manipulations of the sequent calculus are enough to reason about first-order predicate logic.

Answer 2

There are many different varieties of formalism in the philosophy of mathematics. 'Game formalism' or the idea that mathematicians are just inventing and playing games of string manipulation is a pretty extreme variety and does not have many advocates. Other varieties are described in the SEP article.

The basic idea behind formalism is that mathematical sentences don't have meanings in and of themselves, though they may be given an interpretation under which they express something true or false. Formalism sits in contradistinction to platonism, under which mathematical sentences express truths about some realm of abstract objects that exist independently of us, and logicism, under which mathematical sentences are a body of logical truths, and intuitionism, under which mathematical sentences are statements of what human cognition is capable of proving, etc. There are many different approaches to the philosophy of mathematics.

For the formalist, mathematics is fundamentally syntactic in nature. On Curry's account of it, formalism understands mathematics to be the study of formal systems, free of philosophical presuppositions, and open to any choice of axioms and rules. Likewise, for Carnap we are at liberty to build any logic and any language we choose. This does not make the choice arbitrary, but it means that we assess formal systems based on how useful they are. Classical logic and classical arithmetic are very useful. This does not make them somehow 'true' or 'correct' in some unique way, just useful.

I do not agree with you that there is something fundamental or absolute about classical logic, nor that we somehow observe classical logic in nature. Logics are human creations; they help us to give structure to our beliefs. Classical logic is the most commonly used logic, but far from the only one. Arguably there are some fundamental notions that underpin the development of logic, such as this thing is not the same as that thing, or this statement is incompatible with that statement. But such considerations do not get you very far. You might be surprised at how much discussion and disagreement there is on the subject of identity and negation.