you move from a discussion of naturalness to a discussion of mathematical naturalness. I see no reason to suppose that these two terms are the same. Indeed, doing mathematics itself might well be considered by some to be perverse and unnatural behaviour, even though most enjoyable!

It goes without saying that you are welcome to your opinion. But just to review: you said zero was not a counting number, to which I replied in comment 13 by suggesting that zero is the counting number for the empty set. My comment 18 was in reply to something Alexandre wrote.

But, since you’ve brought it up again, I’ll say a bit more about this. In dealing with sets, it is a completely natural thing to construct sets by applying the principle of (bounded) comprehension: given a set X and a predicate P defined on X, we can form the subset {x: P(x)}. In some cases, this may yield the empty set.

For example, if X is the set of positive natural numbers and P(x) is the predicate “x is an odd perfect number”, then it may very well be that this subset is empty; we don’t know yet. It would be perfectly natural for a mathematician, while conducting research on this question, to contemplate the nature of this set. That’s completely standard practice. If in the fullness of time she manages to prove that the set is empty, does it suddenly become something “artificial”?

In a similar vein, I asked Alexandre whether he thought the notion of falsity were also unnatural (or artificial, to use your word). Well?

If you accept that the empty set is not such an unnatural construct after all, isn’t zero just as natural?

(I’m not absolutely sure of this, but I believe the convention of most mathematicians these days is to include 0 when defining the set of natural numbers.)

I merely said that I thought zero was not a “natural number”. I said nothing about doing mathematics without it, which is a different and completely unrelated issue. We use all manner of artifical constructs in mathematics, as in the rest of science and engineering, and so we should. We can even use the concept of zero without zero thereby becoming a natural number.

” I just assert that the result looks very sloppy. Multiplication gets worse.” You can, of course, just assert it. But still…

In Peano Arithmetic with 0, the definition of addition is
x + 0 = x
x + y’ = (x + y)’
and the definition of multiplication is
x * 0 = 0
x * y’ = x * y + x

In Peano without 0 the definition of addition is
x + 1 = x’
x + y’ = (x + y)’
and the definition of multiplication is
x * 1 = x
x * y’ = x * y + x

So why is the definition of addition sloppy? And by what measure does multiplication get worse??

Overnight I thought of another note in response to Alexandre’s comments about the empty set. I think Todd would be behind me in saying that the definite article is actually the least important thing here.

If you’ve got extensionality, then yes you have actual uniqueness. But Todd and I would both be perfectly happy to throw it away and make do with uniqueness up to isomorphism (which does not require extensionality).

I am really surprised by the dogmatism of this statement, and I really disagree. Is then the notion of falsity also ‘unnatural’? The idea of nothing?

Perhaps I don’t understand what you mean by ‘natural’.

The literal uniqueness of ‘the’ empty set (the set of pink unicorns = the set of humans living on Mars) is somewhat irrelevant in my opinion. What really counts (ha ha) is the presence of an initial set: a great convenience for many, many reasons. And initial objects are unique up to unique isomorphism, which for any mathematical purpose is just as good as (actually, more natural than) literal uniqueness. Similarly, one says ‘the’ disjoint union of two sets, even though such a thing is not literally unique, but only so in this categorical sense.

I’ll say this: the empty set is a great convenience, so that it would (to me) be awkward and unnatural to do mathematics without it. It’s ’second nature’, then. I think John Armstrong is saying much the same thing.