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?