- #1

"Don't panic!"

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*equality between two mathematical objects*actually means. This maybe a very stupid question, so apologies in advance, but if I'm honest I struggle to come up with an answer that doesn't involve using the notion of

*sameness*which seems very unsatisfactory as

*sameness*is not a well-defined concept.

So far what I've come up with is that if we have two mathematical quantities ##A## and ##B## that each possesses particular mathematical properties, then the expression ##A=B## (assuming that ##A## and ##B## are

*constant*mathematical quantities, i.e. they don't change their form) is the statement that the two quantities do not describe distinct mathematical objects are in fact different representations of a single mathematical object - they are identical in all respects (the fact that ##A=B## is essentially saying that ##A## and ##B## share an identical set of mathematical properties and neither has additional properties not shared by the other (note that I'm trying to avoid using the word

*same*)).

If ##A## and ##B## are

*not constant*quantities however, then the expression ##A=B## is actually a proposition that is only true for specific values of ##A## and ##B##. In this case, we may say that if ##A=B## then ##A## and ##B## simultaneously denote a single mathematical property.

Finally, we can set ##A## and ##B## to a particular numerical values, such that, if ##A=B## then they are just different labels for a unique numerical value.

I hope this is somewhat clear. Any insight into the situation would be much appreciated.