Philosophers often use 'numerically' in contrast with 'qualitatively' when speaking of identity or sameness. If I tell you that I drive the same car as Jane, that is ambiguous: it could mean that Jane and I drive one and the same car, or it could mean that Jane and I drive the same make and model of car, but not one and the same car. To take a second example, six bottles of beer in a typical six-pack are numerically distinct but qualitatively identical. Suppose you want a beer from the six-pack. It won't matter which bottle of the six I hand you since they are all qualitatively the same (qualitatively identical) in respect of both bottle and contents, at least with regard to the properties that you would find relevant such as quantity, taste, inebriatory potential, etc. If I hand you a beer and you say you want a different beer from the same six-pack, you mean a numerically different one. If I reply by saying that they are all the same, I mean they are all qualitatively the same.
If A and B are numerically identical, it follows that they are one and the same. A and B are one, not two. If A and B are qualitatively identical, it does not follow that they are one and the same. But they might be. For if A and B are numerically identical, then they share all properties, in which case they are qualitatively identical. Furthermore, if A and B are qualitatively identical, it does not follow that they share every property: it suffices that they share some properties.
To see this, suppose that you and I both order the 'monster chimichanga' at the local Mexican eatery. We have ordered the same item, qualitatively speaking. But it turns out that the one served to you is slightly more 'monstrous' (a wee bit bigger) than mine. That doesn't change the fact that they are qualitatively the same or qualitatively identical as I use these phrases. The chimis are two, not one, hence numerically different. They are the same in that they share most properties.
I suppose we could nuance this by distinguishing strict from loose qualitative identity. Strict implies indiscernibility; loose does not.
Can One Step Twice into the Same River?
Stephen Law (HT: Sed Contra) thinks one can make short work of a Heraclitean puzzle if one observes the numerical-qualitative distinction:
If you jump into a river and then jump in again, the river will have changed in the interim. So it won't be the same. But if it's not the same river, then the number of rivers that you jump into is two, not one. It seems we're forced to accept the paradoxical - indeed, absurd - conclusion that you can't jump into one and the same river twice. Being forced into such a paradox by a seemingly cogent argument is a common philosophical predicament.
This particular puzzle is fairly easily solved: the paradoxical conclusion that the number of rivers jumped into is two not one is generated by a faulty inference. Philosophers distinguish at least two kinds of identity or sameness. Numerical identity holds where the number of objects is one, not two (as when we discover that Hesperus, the evening star, is identical with Phosphorus, the morning star). Qualitative identity holds where two objects share the same qualities (e.g. two billiard balls that are molecule-for molecule duplicates of each other, for example). We use the expression 'the same' to refer to both sorts of identity. Having made this conceptual clarification, we can now see that the argument that generates our paradox trades on an ambiguity. It involves a slide from the true premise that the river jumped in the second time isn't qualitatively 'the same' to the conclusion that it is not numerically 'the same'. We fail to spot the flaw in the reasoning because the words 'the same' are used in each case. But now the paradox is resolved: we don't have to accept that absurd conclusion. Here's an example of how, by unpacking and clarifying concepts, it is possible to solve a classical philosophical puzzle. Perhaps not all philosophical puzzles can be solved by such means, but at least one can.
Not so fast. Although superficially plausible, the above solution/dissolution of the puzzle begs the question against the doctrine of Heraclitean flux. Law goes at Heraclitus with the numerical-qualitative identity distinction. But this distinction presupposes a distinction between individuals and qualities. Given this distinction one can say that one and the same individual has different qualities at different times. Thus one and the same river is stepped into at different times. But on a doctrine of Heraclitean flux, there are no individuals that remain self-same over time. There is no substrate of change. Change cuts so deep that it cannot be confined to the properties of a thing leaving the thing, as the substrate of change, relatively unchanged. For Heracliteans as for Buddhists, it's flux all the way down.
Law taxes Heraclitus with an illicit inferential slide from
The river jumped into the second time is not qualitatively the same
The river jumped into the second time is not numerically the same.
But there is no equivocation on 'same' unless we can sustain a distinction between the thing and its properties. Is this distinction unproblematic? Of course not. It reeks with problems. Just what is a thing in distinction from its properties? A Bergmannian bare particular? An Armstrongian thin particular? An Aristotelian primary substance? There are problems galore with these conceptions. Has anyone ever really clarified the notion of prote ousia in Aristotle? Nope. Is a thing a bundle of its properties? More problems. And what is a property? An abstract object? In what sense of 'abstract'? A universal? A trope? Will you say that there are no properties at all, only predicates? And what about the thing's HAVING of properties? What is that? Instantiation? Is instantiation a relation? If yes, does it sire Bradley's Regress? Are properties/concepts perhaps unsaturated in Frege's sense? Can sense be made of that? Is HAVING some sort of containment relation? Are the properties of a thing ontological constituents of it? And what could that mean? And so it goes.
We are presented with a puzzle and a seeming absurdity: There is no stepping twice into the same river. The Moorean rebuttal comes quickly: Of course, there is! Common sense, convinced that it is right, attempts to dissolve the puzzle by making a simple distinction between numerical and qualitative identity. The dissolution seems to work — but only if we remain on the surface of the troubled waters. Think a little more and you realize that the distinction presupposes a deeper distinction between thing and properties. But now we are launched into a labyrinth of ontological problems for which there is no accepted solution. The unclarity of the individual-property distinction percolates back upwards to disturb the numerical-qualitative distinction.
Law has not definitively solved the Heraclitean puzzle.
The Numerical-Qualitative Distinction is Valid at the Level of Ordinary Language
We need to make the distinction, of course: it is fallout from, and exegesis of, ordinary usage. 'Same' is indeed ambiguous in ordinary English. The distinction does useful work at the level of ordinary language. The Heraclitean, however, need not be taken as contesting, at that level, the truth that one can step twice into the same river. He is making a metaphysical claim: there is in reality, below the level of conventional talk and understanding, radical flux. If so, there is nothing that remains self-same over time, such as a river, into which one can step twice.
To think clearly and avoid confusion one must observe the distinction between numerical and qualitative identity. But this distinction, which is serviceable enough for ordinary purposes, rests on a distinction, that of individual and property, which is metaphysically murky. Therefore, the common sense distinction cannot be used to dispatch the Heracliteans' metaphysical claim.
The deep metaphilosophical issue here concerns the role and status of Moorean rebuttals to seemingly crazy metaphysical claims. The illustrious Peter van Inwagen famously denies the existence of artifacts. But he is not crazy, and you won't be able to blow him out of the water with some simple-minded distinction.