andrewducker | Interesting Links for 05-06-2022

Happily, since there's no such thing as a fish, all these statements about fish are in fact statements about elements of the empty set, and hence vacuously true, no matter how confusing :-)

I didn't realise that any statements about an empty set, presumably including contradictory ones, were true!

It's one of the more subtle ways in which mathematical jargon differs from colloquial usage.

In maths, a statement such as "All Xs have property Y" is taken to be exactly identical to "There is no X which does not have property Y". And therefore, if there are no X at all, the latter statement is true, and the former statement (being equivalent) is also regarded as true.

In colloquial usage this seems counterintuitive because there's generally expected to be some secondary implication that there are at least some X in the first place, and perhaps even a lot of them – enough to be worth using the word "all". (If there were only one or two Xs, you'd be more likely to say "The only X [has some property]" or "Both the Xs [have it]" than "All Xs do"; if there weren't any X at all, you likely wouldn't mention it in the first place.)

But in maths, we like to keep the semantics as simple as possible, and avoid secondary connotations of that kind. So "all X are Y" just means "there is no X that is not Y", with no implication at all of how many X there are, including whether there are any at all.

Hence, any statement about a class of objects that there aren't any of at all is taken to be true. For instance, every exponent of a counterexample to Fermat's last theorem is even – and at the same time, every such exponent is also odd. And the fact that those statements sound contradictory does not make them actually contradictory, precisely because there is no actual element of the set available to have both properties at once.

(Indeed, a common technique for proof by contradiction is to show that some set is empty by demonstrating that any element of it would have to have two conflicting properties. Which reflects another important reason why the mathematicians' use of language is kept free of those secondary implications: sometimes, you don't know yet whether there exist any objects in the class you're talking about, and you have to be able to talk about it regardless without any accidental implication that you might have the least idea!)

Thank you, that makes perfect sense!

Interesting Links for 05-06-2022

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