# Expressing identity in mathematics

I considered asking this on math.SE, but I realized this question wasn't really about mathematics.

Suppose I have 4 pens sitting on my desk. So I have a set S = {pen, pen, pen, pen} = {pen}. That's not quite what I wanted: I can't tell the pens apart.

How do I distinguish the pens? Well, I might consider the color of the pen. So, I have a set S = {black pen, black pen, red pen, blue pen} = {black pen, red pen, blue pen}. That doesn't quite express the idea that the pens have an identity either: I can't tell the black pens apart.

So, I might consider the configuration of molecules in the pens. That gives me a set S = {black pen with configuration A, black pen with configuration B, red pen with configuration C, blue pen with configuration D}.

That works, but I'm curious if there is a "standard" way to make the distinction. In programming, the usual way to do this is to use the memory address of the data structure as its identity. Is there a standard way to express identity in mathematics?

(If this is a stupid question, feel free to tell me.)

If I take your question quite literally, — Is there a standard way to express identity in mathematics — then it actually is a question about mathematics.

There is such a thing called a multiset (or bag), which is to a set X as a histogram h: X → ℕ is to a characteristic function χ: X → {0,1}. You could then write {pen, pen, pen, pen} as a valid multi-set of four elements. Precisely how you define a multiset from foundations is not really important; but you certainly can do it, for instance by considering equivalence classes of sequences under permutations, so that {a,b,b,c,c} = {c,b,a,c,b} denotes the equivalence class of both the sequence (a,b,b,c,c) and the sequence (c,b,a,c,b), etc.

This doesn't say anything at all about what we mean by distinguishing similar objects from one another, but it does allow you to refer to the concept more or less directly in formal mathematics.

A longer answer based on what we mean by distinguishing physical objects

If I take your question more figuratively, as hinted at by your discussion about the physical distinguishability of pens and how we can hope to account for them as individual objects, your question is a very pertinent question of philosophy, as it underpins our notions of physics and number. The distinction that we make between different pens of the same colour cannot be done (to the best of our knowledge) for elementary particles such as electrons. Indeed, there was a numerical corrections in statistical mechanics, i.e. the so-called "Gibb's Paradox", which was unexplained until people realized that "this particular particle" isn't always something that was sensible to talk about.

What you're concerned about is not so much a question of identity, but the complementary thing, which is distinctness — that this is not the same as that. The way that we talk about distinctness is to take note if they are equivalent in every way. This is summed up by Leibniz' Principle of Identity: that two objects that have identical properties are in fact the same.

How do we apply this to pens? And what do we say, then, about particles such as electrons?

You correctly note that if the pens had different colours, you could use those colours to distinguish them, and that if you have several pens of the same make and colour, subtler means are required to distinguish them (perhaps the level of ink in them, or the pattern of wear on them). Of course, there may be in some cases that the only differences are ones that would be hard even to measure: what does one do in that case? You could identify them by their positions, at least; it's not an intrinsic property, but if you're not concerned about the individual pens so much as just enumerating the collection, their different positions certainly allows you to distinguish them as elements of a collection.

What to say then about electrons? I've said that electrons are indistinguishable, but certainly this isn't the same as saying that there is only one electron. Electrons can still be in different states, and have different positions; it's just that interchanging "some electron" with "another electron" in the field equations makes no observable difference. (The wave-function changes by a 'global phase' of -1, but this is a technicality, and certainly not an observable one.) But in reality talk of individual electrons is a little silly; in quantum field theory, electrons are in fact excitations in a single matter field, in the same way that photons are excitations in the electromagnetic field — or in the same way on the macroscopic level that water waves are an 'excitation' of the ocean. Does it even mean anything to "replace" a wave on the ocean with another? A small wave in a large body of water is just an energy impulse transmitted by trillions of water molecules; the water itself doesn't really move, it just transmits a pattern of energy and information onward. Similarly, an electron is an impulse of energy and information in a field of matter, which moves in simple ways unless it interacts with other matter.

If we say that there are four electrons in a region of space, we are making a statement not about distinguishable objects, but distinguishable pieces of information. In an un-ionised atom of Beryllium, it isn't important to distinguish the electrons which have various energies, angular momenta, or spin from each other, except inasmuch as they have different energies etcetera; and then it is the properties such as the energy which are important. The electrons are the notional objects which give Beryllium its chemical properties; an ion which had more or fewer electrons than four would change how the Beryllium interacts chemically, just as filling a bottle with more or less water changes the pitch of the sound when you blow across it's opening.

What does all this have to do with mundane objects such as pens? When we teach children to count — four apples, for instance — we de-emphasize the difference between the apples. We take for granted that the child can see that the objects, even if very similar, are distinct. They are distinct in several different ways in practice, but in principle we only use things such as position to distinguish them, which is not an intrinsic property of the apple but only a property of the system — the relationships between the apples. And we teach addition by gathering more apples near to each other, to suggest two collections which are made one; but the other apples haven't changed except to move close to one another. Counting is a process of identifying systems and subsystems, and the largest bulk property of those systems.

In the end, apples and pens are made of electrons (and protons and neutrons), which are all just matter fields in the same way. A pen is an excitation in multiple interacting fields of matter; and a very stable excitation as well, one which is very unlikely to appear or decay by chance. The different properties of the pen, such as the colour of its ink, the amount of ink inside of it, the pattern of wear on it, and so forth, are all questions about the pen as an aggregate system of the same sorts of particles; they are all information about the system. To distinguish the pen from others is to make statements about the energy and distribution of other physical quantities, just as it is for individual electrons. But even if the pens were made perfectly identical (except that they lay in four different locations), it would still be reasonable to talk about having four pens, because the collection of the pens itself still has the property of having four equivalent "excitations in the matter field".

In conclusion, distinguishability of objects is sufficient, but not necessary, to identify them as distinct "objects" (signals of information which may be more or less stable), and thus allow them to be considered as discrete subsystems in the world, and counted.

In formal logic, at least, a subscript is often preferred. Here's the set of all possible pens:

{ p1, p2 ... pn } ∈ P

Sometimes, primes are used (you can also mix and match, but staying consistent is always a good idea). Here's the set of all blue pens possible:

{ p′, p″, p‴ ... } ∈ Pblue