Thanks, that was interesting for a bit but then went nowhere.
To save others the time, as I understand it, their disagreement is about the meaning of the equals sign. I suppose they're right, because we do use different definitions of equality for different needs, both within math and in other fields, and of course without a good definition all sides agree to, people can find themselves talking past one another.
For what it's worth, the definition of equality I like best is the one based on symbolic manipulation - two symbols a and b are equal iff substituting a by b in ANY statement does not modify the statement's truth value.
> To save others the time, as I understand it, their disagreement is about the meaning of the equals sign.
I think it's more general than that. The meaning of symbols is not fixed, but a matter of intersubjective agreement and context. "2+2=4" if we intend that "2" represents the natural number subsequent to one and preceding three; and we intend that "+" refers to infix addition; and we intend that "=" refers to quantitative equality; and we intend that "4" represents the natural number subsequent to three and preceding five. This is actually not an "iff" statement because there are an infinite amount of other meanings we could reassign to the symbols contained within the proposition in order to result in a true statement.
But intent is a property of the speaker, not the phoneme. Intent does not inhere in the message. Intent must be inferred from context and previous usage of the terms.
> without a good definition all sides agree to, people can find themselves talking past one another.
I think this happens a lot and its why people think other people deny things like "2+2=4." Then there is a whole class of people who take advantage of ambiguity to instrumentally employ fallacies of equivocation.
