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In the introduction to Bourbaki vol. 1, we read:

"It goes without saying that the description of the formalized language is made in ordinary language, just as the rules of chess are. We do not propose to enter into a discussion of the psychological and metaphysical problems which underlie the use of ordinary language in such circumstances (for example, the possibility of recognizing that a letter of the alphabet is "the same" in two different places on the page, etc.)."

I would really like to read more about such "psychological and metaphysical" aspects mentioned here. What is meant by "letter of alphabet being the same in two places"?

I'd appreciate it if someone could introduce some literature/books that discuss these issues.

Ali
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    Try to write "Oklahoma" twice on one page. There will be small differences between the two in terms of penstrokes, etc.. So are they $\mathit{really}$ the same letters? Do they $\mathit{really}$ say the same word? Do they $\mathit{really}$ refer to the US state, or an idea of Oklahoma? Is the answer the same if you live there as if you've never been there? If you like these questions, don't bother searching for literature, just befriend a wastrel sophomore marijuana addict: you'll tire of the topic soon enough. – realdonaldtrump Jun 10 '18 at 03:44
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    See Types and Tokens. – Mauro ALLEGRANZA Jun 10 '18 at 08:13
  • For a little bit more detailed discussion, see ; George Tourlakis, Lectures in Logic and Set Theory. Volume 1: Mathematical Logic Cambridge UP (2003), page 5-on. – Mauro ALLEGRANZA Jun 10 '18 at 08:17
  • My guess is that what Bourbaki meant to write is that the description is written in metalanguage but were uncomfortable relying on distinctions like language/metalanguage because they may go counter to their (flawed IMHO) philosophy of mathematics. – Mikhail Katz Jun 10 '18 at 12:48
  • @MauroALLEGRANZA The Tourlakis book is awesome! Thanks. – Ali Jun 10 '18 at 18:55
  • You are welcome :-) Tourlakis' point of view may be defined as formalist: thus, it agrees with Bourbaki's approach. – Mauro ALLEGRANZA Jun 10 '18 at 20:07

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The point is that you need a meta-system MS in which to do your reasoning about the formalized language you are setting up, which could for example be FOL. You also need to work within MS to describe the formal system you wish to study, such as ZFC. In some modern logic textbooks, they will state that MS is ZFC or something like that, but that is a little misleading because there is still no way to describe ZFC without at the bottom using natural language NL. This circularity at the bottom cannot be escaped. It is not even just the problem that you use physical symbols that are slightly different with each instance of the same alphabet (as Mauro pointed out), but also there is the problem that you cannot describe the rules of any system without at some point relying on your pre-existing understanding of the concepts of "if" and "repeat"!

Psychologically, you have to use your own human mind to interpret any mathematical statement, and in doing so you have to somehow form a mental construct that you believe the author intends to convey (through whatever physical media). Who knows whether you succeeded? How can you ever know? Nevertheless, you subjectively do know whether practical mathematics seems to work empirically in enabling you to accurately predict things about the real world, so there is reason to believe that you must be somehow learning correctly what your mathematics teachers taught you.

Philosophically, one has to realize that there can be no purely mathematical reasoning about the real world, because there is simply no way to refer to any real world object without using some 'real-world hook', such as some natural language phrase! Symbols remain completely meaningless until you imbue it with meaning via some interpretation. Even classical logic itself was designed precisely to capture reasoning about our one reality. Why did we even have the concept of "true" and "false"? Obviously, because (vaguely speaking) every sufficiently precise statement about reality either holds or does not hold (and never both). A careful analysis along these lines also totally dispels the common paradoxes.

user21820
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