The Universe of Discourse

The mathematical past is a foreign country

A modern presentation of the Peano axioms looks like this:

1. !!0!! is a natural number
2. If !!n!! is a natural number, then so is the result of appending an !!S!! to the beginning of !!n!!
3. Nothing else is a natural number

This baldly states that zero is a natural number.

I think this is a 20th-century development. In 1889, the natural numbers started at !!1!!, not at !!0!!. Peano's Arithmetices principia, nova methodo exposita (1889) is the source of the Peano axioms and in it Peano starts the natural numbers at !!1!!, not at !!0!!:

There's axiom 1: !!1\in\Bbb N!!. No zero. I think starting at !! 0!! may be a Bourbakism.

In a modern presentation we define addition like this:

$$ \begin{array}{rrl} (i) & a + 0 = & a \\ (ii) & a + Sb = & S(a+b) \end{array} $$

Peano doesn't have zero, so he doesn't need item !!(i)!!. His definition just has !!(ii)!!.

But wait, doesn't his inductive definition need to have a base case? Maybe something like this?

\begin{array}{rrl} (i') & a + 1 = & Sa \\ \end{array}

Nope, Peano has nothing like that. But surely the definition must have a base case? How can Peano get around that?

Well, by modern standards, he cheats!

Peano doesn't have a special notation like !!S!! for successor. Where a modern presentation might write !!Sa!! for the successor of the number !!a!!, Peano writes “!!a + 1!!”.

So his version of !!(ii)!! looks like this:

$$ a + (b + 1) = (a + b) + 1 $$

which is pretty much a symbol-for-symbol translation of !!(ii)!!. But if we try to translate !!(i')!! similarly, it looks like this:

$$ a + 1 = a + 1 $$

That's why Peano didn't include it: to him, it was tautological.

But to modern eyes that last formula is deceptive because it equivocates between the "!!+ 1!!" notation that is being used to represent the successor operation (on the right) and the addition operation that Peano is trying to define (on the left). In a modern presentation, we are careful to distinguish between our formal symbol for a successor, and our definition of the addition operation.

Peano, working pre-Frege and pre-Hilbert, doesn't have the same concept of what this means. To Peano, constructing the successor of a number, and adding a number to the constant !!1!!, are the same operation: the successor operation is just adding !!1!!.

But to us, !!Sa!! and !!a+S0!! are different operations that happen to yield the same value. To us, the successor operation is a purely abstract or formal symbol manipulation (“stick an !!S!! on the front”). The fact that it also has an arithmetic interpretation, related to addition, appears only once we contemplate the theorem $$\forall a. a + S0 = Sa.$$ There is nothing like this in Peano.

It's things like this that make it tricky to read older mathematics books. There are deep philosophical differences about what is being done and why, and they are not usually explicit.

Another example: in the 19th century, the abstract presentation of group theory had not yet been invented. The phrase “group” was understood to be short for “group of permutations”, and the important property was closure, specifically closure under composition of permutations. In a 20th century abstract presentation, the closure property is usually passed over without comment. In a modern view, the notation !!G_1\cup G_2!! is not even meaningful, because groups are not sets and you cannot just mix together two sets of group elements without also specifying how to extend the binary operation, perhaps via a free product or something. In the 19th century, !!G_1\cup G_2!! is perfectly ordinary, because !!G_1!! and !!G_2!! are just sets of permutations. One can then ask whether that set is a group — that is, whether it is closed under composition of permutations — and if not, what is the smallest group that contains it.

It's something like a foreign language of a foreign culture. You can try to translate the words, but the underlying ideas may not be the same.

Addendum 20250326

Simon Tatham reminds me that Peano's equivocation has come up here before. I previously discussed a Math SE post in which OP was confused because Bertrand Russell's presentation of the Peano axioms similarly used the notation “!!+ 1!!” for the successor operation, and did not understand why it was not tautological.

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