# Can mathematics actually define 'one'?

Is there is difference between being singular and first? It appears to me that there is - though both notions can be described by the figure of '1'.

Here what I mean by singular - it has no successors, nor any predecesors; to be exact - it is not a part, or a term.

But can mathematics actually describe the singular? Is it always forced to move from the singular to the first?

It appears that whenever mathematics discovers what appears to be a singular notion, it can embed it within a framework where it is no longer singular but a part of a multiple. That is it becomes a part of a sequence of which it is the first term.

An easy example is the integers.

A more sophisticated example are vector spaces.

So can mathematics describe 1 as the singular?

Perhaps one may cautiously advance the trivial group as the embodiment of unity; but this group can be embedded in any number of groups, in fact all. So again its part of a sequence or really a geometry here.

• So I was talking a look at the linguistic origins of "singular" and related terms, and I found an ironic tidbit that should amuse you. Singular derives from Latin 'singulus'("single"). Usage notes for 'singulus': usually only used in the plural ('singuli'). – David H May 13 '13 at 14:12
• I don't think I understand your question. I'm not sure I see the significance of being "singular" vs. being "first". Of course mathematics can describe mathematical structures in which 1 is the only element. But those just aren't very interesting, since they lack structure. Also, what do you mean "not a part or a term"? Why would we care about "1" if it wasn't a term, or care about 1 if it wasn't part of something? – Alex Kocurek May 13 '13 at 15:58
• @DavidH: That is amusing. I'll have to use singuli somewhere or other! – Mozibur Ullah May 15 '13 at 13:28
• @Kocurek: The identity element all by itself is not very interesting, but when in a group its crucial. Similarly in the category of groups its crucial to include the trivial group is for it to have good properties. In fact the idea of a monoid is crucial in category theory where it has all sorts of avatars. If '1' is a term then isn't that more important than its nature as '1'? I'm trying there to gesture towards what I mean by singular. – Mozibur Ullah May 15 '13 at 13:32
• This is unclear. It is obvious that one can define '1' with additional axioms that match what we think about uniqueness. But then you discount this allowing that it could be embedded in a system along with any number of other unique objects. If you allow that, the your '1' object still acts as a '1', but there are other objects in the universe. If you allow augmenting your universe (which you can always do) then of course your '1' is still unique, it is just not alone. So which thing do you want, uniqueness or aloneness? – Mitch Jan 18 '18 at 19:17

Mathematics can certainly describe the singular, for instance, in the instance of singleton sets. Prash's answer describes this perfectly well in terms of predicates:

∃y: P(y) & (∀x: P(x) ⇒ x=y)

If you want to consider a "mathematical universe" in which there is only one element, you can let P be some tautological property (such as P(x) ≡ (x=x) which holds of all objects). Then the above formula can be simplified to

∃y ∀x: x=y

which says that there is an object y, such that the only object in the universe is y.

So: can mathematics define singularity? Yes. Can it do so without proceeding to "first"? Yes again — if you are determined to be lazy. Just because we can take a universe with one element (or a set with one element) and embed it in a larger mathematical context, doesn't mean that you must. But note that "1" is not the "unique first element" (!!) in the standard formulation of set theory. The first constructed element — what in perverse and awkward mathematical parlance, one calls the zeroeth constructed element — is the empty set, or zero. But even this may only be constructed by virtue of there existing an infinite set, per the axiom of infinity. So in fact the "first" element in ZFC is a "dead heat" between an infinite number of elements, of completely unspecified identity or cardinality — we cannot even say if there are more than Aleph-Null of them, nor whether 1 is among them. This goes to show that we may speak of first-ness, uniqueness, and the number 1, but that none of them necessarily "implies" any of the others in the way that you describe.

If you choose to entertain a mathematical universe with more than one element, this doesn't mean that the notion of the singular evaporates, or is subsumed into the notion of "first". The role of "singular" and "first" play significantly different roles in the formulation of mathematics, even in universes where there is more than one object. Qualities such as "singular" apply to elements of sets, whereas "first" applies to items in sequences, which we may construe as functions from the positive integers to a set. For instance, consider the sequence p = (2, 3, 5, 7, 11, ...) of the primes in order, such as one may construct by a sieve technique. In this, the first element is p1 = 2; however, this is not to say that 2 is the one and only prime number. Similarly, one may consider a singleton set {3}, and the (rather boring) sequences one can define over it, e.g. (3, 3, 3, 3, 3). Here, the unique element 3 is the first item in the sequence — and also the second, third, fourth, and fifth.

The number "1" is an object which we use by convention both to describe the idea of something being first and something being unique; but because we use it in different ways when we do so — we apply it in different ways (in particular, we are able to apply it in different ways) — the two ideas may be kept functionally different, just as we may distinguish between the notions of "hot" and "stop" despite the fact that they may both be represented symbolically with the colour red.

• Doesn't a set with one element count as two? That is two concepts, the element and the set? I suggested that 'one' should not be a part, but isn't an element a part of a set? – Mozibur Ullah Jul 13 '13 at 7:07
• I think to answer that question, you will have to develop a formal theory of "coming first". In the traditional development of set theory from constructive principles, one must first describe the element before constructing a set which contains it. In this respect, the empty set is precedes all singleton-sets, as I mentioned above. – Niel de Beaudrap Jul 15 '13 at 13:27
• Do you mean by 'The first constructed element...is the empty set, or zero. But even this may only be constructed by virtue of there existing an infinite set, per the axiom of infinity' - that one cannot construct the empty set without the axiom of infinity in ZFC? So if we drop the axiom of infinity (which means we should drop choice too) we cannot constuct the empty set? But then surely this means we just simply append an axiom to say that the empty set is available? – Mozibur Ullah Jul 15 '13 at 14:39
• @MoziburUllah: the way that the empty set is constructed is to show that it exists as a subset of some other set, even if you know nothing else about that set. In ZFC, the empty set exists by virtue of the fact that there are at least infinitely many sets; but if you eliminated the axiom of infinity, and posited that the empty set exists (or indeed, that at least one set of any sort whatsoever existed), then it would immediately follow that the empty set exists. I'm not saying that the axiom of infinity is necessary for the empty set in every set theory; just in ZF. – Niel de Beaudrap Jul 15 '13 at 15:03

While ∃ indicates that the quantity is potentially unlimited, "one of many" in the way you describe it, ∀ can be used to restrict it. So, we have `∃y(P(y)⋀(∀xP(x) -> x=y))`. This way, the only `x` that has property `P` is the one that's same as `y`. In some notations, that is abbreviated to `∃!1`.

This isn't an entire response, but I wanted to make a point that hasn't been made by the other answers and it ended up being too long to put in a comment. Think of it as an addendum to Niel's reply which had all good points.

Since this isn't the MathSE and many reading this maybe haven't had a dozen math classes to see this, but in math you'll never see "singular" used as a synonym for '1' or '1st' etc., (that's what '1' and '1st' are for!) and instead you'll see it used as a qualitative adjective as opposed to the quantitative usage: singular matrices, singular values of functions, singular solutions to certain differential equations, coordinate singularities in geometries,... Essentially, it usually describes unusual or exceptional cases where standard procedures for solving some mathematical problem break down and have to be handled in a unique way.

The point is, I'd advise against using it as just yet another vague synonym for unity since it serves a very useful purpose in giving mathematicians a way to clearly differentiate between two contrasting intended meaning of "unique": the concept of "unity" and that of "singularity".

• I'd forgotten about that use. I was steering away from unity as thats usually a synonym for identity. Vague notion are important - or rather unformalisable notions. For example, a lot of people think that the idea of infinity has been tamed by Cantor. I don't think it has. I'm just thinking about what one means. – Mozibur Ullah May 15 '13 at 13:44
• But unity simply isn't a synonym for identity. That unity (1) is sometimes the identity element for something like multiplication is simply a coincidence of certain algebraic structures that make use of both concepts. – David H May 17 '13 at 11:42
• true enough - but thats why I said 'usually a synonym'. – Mozibur Ullah Dec 15 '13 at 21:14

From the Seminar on Heraclitus, pg.21

Heidegger: Our German word Ein (one) is fatal for the Greek En (One) To what extent?

Fink: In the relatedness of En (One) and Panta (Cosmos) it is not only a matter of a counterreference, but also of a unification.

Likewise in describing the concept of the Greek En (One), it is easy to be led astray by the mathematical concept of one, as something singular, unique, alone and the generator of succession.

In a sense, these senses are within it, but the sense is better understood if its called the Unifying for it unifies the many wholly together, and the name itself signifies this, as something unified must have implicitly and potentially a manifold of many, even though it is seemingly a one.