# Are numbers particulars?

I've often seen numbers be called 'abstract particulars' but as explored in a few of the following questions and answers that I will list, they seem to have the ability to be 'instantiated', does this not stop them being 'particulars'(from a terminology point of view).

Questions:

Mathematical objects existing as different instances

Are mathematical objects a type according to type-theory?

Why can unique mathematical objects have "copies" or exist in multiples?

Why can't numbers be 'used up'?

• Does this answer your question? Mathematical objects existing as different instances There must be a ton more on this subject, already.
– user14511
Aug 2, 2022 at 17:42
• There is, and I have read much of it, this is more a question about the relevant terminology used. Aug 2, 2022 at 21:17
• Philosophers disagree on the status of numbers, some deny that they are abstract particulars. But to those who take them to be that (like Aristotle or trope theorists) "instantiation" is ambiguous, and when used as you describe means concrete instantiation, e.g. number 3 is instantiated in 3 lemons, see SEP, Tropes, and Fisher, Instantiation in trope theory. Aug 3, 2022 at 3:12
• I understand, perhaps we can say the ability to be instantiated in the way we can with numbers, is a property of all abstract objects, as an 'idea' can be seen and used many places and times. Aug 3, 2022 at 7:35
• @Conifold thanks for the links on trope theory, when we perform additions of 'numbers' we do so at type level but if we are adding the instances it will be like combining quantities, '5 apples plus 4 oranges'? Aug 3, 2022 at 19:22

You can't tag every possible thing as a particular or an universal. It all depends on the set/universe/system it refers to.

Particulars are objects that are numerically one, in contrast with universals, which are many -the universe-: if a set is the universe, a member is a particular.

Ergo, being a particular is a relative quality: to be a particular in relation to some universe. In order to know if an entity is a particular, you need to know what is the universe of reference.

If the universe is the set of numbers, the number five (5) is a particular. If the universe is the set of all numbers five (5) that exist in a book, the number five (5) is an universal, the first number five appearing on the book being a particular.

• Who defines particulars to be relative to a universe in this sense? I've never heard of such a thing. The number five does not appear multiple times in a book unless you are some sort of nominalist, which I don't think anyone is these days. Aug 3, 2022 at 5:30
• @DavidGudeman what do you mean when you say 'a number does not appear multiple times in a book'? At least in terms of natural language that would be how most people would say it and view it. Aug 3, 2022 at 7:37
• @user1007028 I mean that there are not multiple instances of a number. "5" is not an instance of the number five; it is a name of the number five, just as "George Washington" is not an instance of the first president of the US, it is the name of the first president of the US. Aug 3, 2022 at 8:33
• @DavidGudeman Most definitions (particular comes from "part", which implies the existence of a whole). E.g. "Particulars are normally contrasted with universals, the former being instances of the latter" (The Oxford Companion of Philosophy). If the number five is an universal, what is an example of a particular in such universe? A "name of the number five"? (your words) Aug 3, 2022 at 10:05
• @rodolfoAP, I don't know of any philosopher who says the number five is a universal, but I'm not well-read in the Scholastics. You would have to read someone who makes that claim to find out what they mean by it. Frege does say it is a higher-order set--the set of all sets of five objects. In this case any set of five objects would be an instance of the number five. Aug 3, 2022 at 12:57

The boundary between general and particular information is not a sharp dichotomy; even if not sorites-theoretically vague, it still occupies a range. I'll quote Kant on the topic first (this is from his assessment of something he calls "amphiboly"):

It is right to say whatever is affirmed or denied of the whole of a conception can be affirmed or denied of any part of it (dictum de omni et nullo); but it would be absurd so to alter this logical proposition as to say whatever is not contained in a general conception is likewise not contained in the particular conceptions which rank under it; for the latter are particular conceptions, for the very reason that their content is greater than that which is cogitated in the general conception.

In the Transcendental Aesthetic, Kant's definition of intuition as delivering particular information, rather than information in the form of discursive general marks, yet allows for (A) forms of intuition (generalizations over intuition) and (B) formal intuitions (particulars, that we cognize an sich no less (Kant does say that we know space and time in themselves, though this is not entirely of a piece with his discourse on the question of an sich knowledge in general(!)).

Now, numbers make individuation possible, so to say; there is one cat on the mat, there are two zebras on the yacht, there are 10666 demons in the throne room, etc. I wasn't aware of it until SE user Mozibur Ullah spoke of it in one of his posts, but Plato did not put Numbers on quite the same level as Forms (or: there at least was a major phase of Plato's thought in which these levels diverged). So consider the following sequence:

1. A = A
2. A + B = B + A
3. 2 + 3 = 3 + 2
4. 10/5 + 27/9 = √9 + 210

(1) is more general than (2), etc., and we could go well beyond (4), particularizing the information further and further.

So for all that, it is not, "Are numbers (or Numbers) particular or general?" so much as, "How particular or general, are numbers?" Now, if there's a number(!) measuring "full particularity," well... (Note also that some numbers might be styled more particular or general than others, e.g. the number 1 might be seen as a very general number, figuring elementally in all positive integers, and inversely in all negative ones; or maybe transfinite numbers can be construed, vs. their closure conditions, as Forms of their closed-off elements, notwithstanding that they (the levels of infinity) are also particular in their own way, on top of representing all the possible particularization measured by their elements.)

• Great answer. I recently was confronted with a claim that said the same: That the universe as we know it can be understood in three primitives: time, space, and scale. I think the notion that generalization and specificity are subsumed by the notion of scale (which is nothing more than normativity applied to measure) is the crux of this response. As in all language, context matters. 1 is general and 2 is particular, except 2 is general and 3 is particular, usw.
– J D
Aug 4, 2022 at 14:42
• Correct me if I am wrong, but what you are basically doing here, is to build a hierarchy or continuum of general-particular entities. Firstly, I understand that the opposite of a particular is not a general, but a universal. And a particular is defined such, that there can be only of it at most, while a universal is defined such that there can be more than one. Secondly, I would formulate that the opposite of a general is not a "particular", but a "specific" and that here does indeed exist a continuum/hierarchy. Nov 25, 2022 at 15:40