Indeed, some people misrepresent Hilbert's formalism [1] to say that mathematical statements have no intrinsic meaning but are rather just a "game of symbols".

Indeed, some people misrepresent Hilbert's formalism [1] to say that mathematical statements have no intrinsic meaning but are rather just a "game of symbols". What he really said was that we could split mathematical statements into "real" ones, dealing with the finitary, and "ideal" statements, with infinitary terms. He said that the manipulation of ideal statements could be considered a possibly meaningless "game of symbols", but such a game can be used to help prove finitary, "real" statements that are meaningful. As long as you cannot arrive at a contradiction involving finitary statements, infinitary ones are useful. In that view, infinitary statements could be considered meaningless (in the sense that they don't themselves describe anything real) yet not nonsensical. [1]: https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathe...