> So if "nonsense" breeds meaning, is it really nonsensical?

How bad would the analogy be to a calculation with imaginary numbers that still yields a real result?

Very bad. Ordered pairs of real numbers are neither more nor less real than real numbers themselves.

Would you say that the rationals (equivalence classes of ordered pairs of integers) are "less real" than the integers?

The point of the analogy was that a calculation might involve mathematical concepts that from some point of views may seem iffy on the face of them, but still arrive at a result that does not involve any of the intermediate perceived iffyness.

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_mathe...

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