system
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human
Objections to Mathematical Formalism?

I would consider myself a formalist in my understand of mathematics since to me it seems like the only way to hold modern mathematics together. What are the major objections to it?
gpt
Godel's incompleteness theorems is often the reason why Hibert's formalism project came crashing down. If formalism is true then it would be the case that we could construct a computer that could iterate through strings of symbols to produce proof after proof, which seems too good to be true in light of the incompleteness theorems.

A result of his incompleteness theorem is that mathematics cannot be formalized in any way, such that the formalization is both consistent (will produce only true theorems) and complete (will produce all true theorems.)

The completeness theorem states that mathematics cannot prove its own consistency.

So why is this a bad thing?  
1) How would you know if a formalization such as PA or ZF is consistent? If you use some metatheory, you would have to prove the consistency of the metatheory as well. You can appeal to intuition or some infinite metameta...metatheory, but the first is not formalism and the second doesn't show how maths is analytical.  
2) Inability to show something is a not a theorem. How do you know that 0=1 is not a theorem in mathematics? Maybe after a million derivations you would get that result. And if that result does come out, would you accept it or would you need to reevaluate your mathematics? Should you ascribe a 50% chance in believing non-theorems since mathematics would have a 50% chance of being consistent? Seems rather dubious!