Obviously, equations are an important feature of mathematics, but there is a problem:
"identity, an objector may urge, cannot be anything at all: two terms plainly are not identical, and one term cannot be, for what is it identical with?" as Bertrand Russell wrote.
This quote is from a blog post of John Baez, which I found fascinating because it exactly reflects my own worries about this question when I was a kid. Of course, then nobody was willing to seriously discuss this with me and it was the first time that I had to conclude that adults are less smart than I initially thought.
A similar problem was caused by my observation that one cannot make valid statements about non-existent
things, e.g. the sentence "pink unicorns do not exist" has no meaning, because by definition there is nothing it could be about.
I was convinced for a while that the two problems are related but I am not so sure about this anymore. (*)
I really do regret that the internet did not exist yet when I was a kid, because it would have helped me to refine my worries tremendously (x) and e.g. think about the 'other worlds strategy' earlier (btw I discussed one problem with the idea of other worlds here). On the other hand, I
am glad, for obvious reasons, that it did not exist then.
Of course, writing about a non-existent internet is problematic, so instead let me end this blog post with a link to a rather short proof of the famous equation 1+1=2, a proposition which is occasionally useful.
(*) We leave the question whether I really am (or was) the same person as I was (or am?) as a child for another time(!).
However, then I was convinced that if n is a non-existent thing, e.g. the integer n which solves the equation n*n=7, it is not necessarily true that n=n and therefore the concept of mathematical identities is problematic because of non-existent things. But I cannot be sure that I remember this argument from my childhood correctly.
(x) E.g. with a link to Black's symmetric universe.
added later: If you followed this link, then perhaps you already saw also Rene Descartes' argument about human identity: I cannot doubt that I am, however I can doubt that my body (including my brain) exists; it follows that I am not (just) my body.
This conclusion, approximately 350 years before The Matrix, about the mind-body problem was also the first important contribution to the Copenhagen interpretation imho.
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