One hears a lot about mathematical entities being abstract or at least spoken about as such. Now abstract is usually presented as non spatio-temporal, for example its said that number 2 is present [in whole] in a set of two birds and in a set of two planets at the same time, something that is not possessed by concrete objects, since they cannot present in whole in two distinct places at the same time. Similarly sets are said to be abstract.

In some sense I see this to be strange given the modern axiomatic development of number theory and set theory, like in Peano arithmetic and in ZFC set theory. I don't see anything in the exposition of these theories that refers to the objects of their discourse (i.e.; numbers and sets) being necessarily outside of space and time? Actually it is imaginable that those axiomatic systems can have models with domains that are composed entirely of concrete objects (real or fictional).

I think there is a confusion between number 2 which is an object in the domain of discourse of say Peano arithmetic, and between duality which is the property of a set having two members, truly the last is, as any property, an abstract entity, but the former need not be abstract at all, it can be a concrete object that serves as a name for that property.

To go further one can even say that number 2 is not even a name for duality, it can be used to name duality of some sets, but that is only a function that number 2 is assigned to play rather than a reality about number 2 itself. Number 2 is simply the successor of number 1, or what results from addition of 1 to 1. We can use it as an index of cardinality of some sets yes, but that need not be the whole picture. Actually given the modern set theory of ZFC, there would always be two objects a,b such that the duality of those objects is not indexed by 2, for example any two models of ZFC. So duality itself cannot be fully captured in sets [unless we use other weird versions of set theory like NF], so one cannot really capture 2 as an index of duality in any complete sense. On the other hand one can capture number 2 as a successor of 1 and define it as such in a rigorous axiomatic system like PA, and that would serve as a full meaning of number 2, the other meaning attached or imposed on number 2 can serve as a connotation of number 2 and not an inherent property of it.

According to that number 2 is the successor of 1 in a linear structure that begins with 0 [or 1], and it only have the characteristics that number 2 relates to other members of that structure. In this sense number 2 can be perfectly reasoned as a "concrete" object that bears the successor relation to a concrete object called number 1.

To be noted that when I'm speaking about such concrete objects I don't mean they really exist in time and place of the real physical world, I'd rather mean a fictional world but its objects do possess the same relations to time and place of that fictional world as ordinary concrete object possess to time and place of the real world, much as Oliver Twist behaves as a concrete object in the world of that fictional story.

What I'm saying is that one can imagine Peano arithmetic [or any arithmetic] as a game played with fictionally concrete objects. And for the playing of that game to be significant, the assumption of consistency is added, as long as there is no proof of inconsistency of that game at our hands.

The same applies to sets, which one can find a concrete model for them, this can be encoded in sets being concrete labels of mereological totalities of concrete labels, and the set rules can be easily paraphrased in the language of mereology and labels. On can build up the whole cumulative hierarchy through labeling going in stages from the empty sets going up through powering, then replacement and union, and of course with the stipulation of having an infinite stage among the stages of that hierarchy. All sets can be modeled by that recursive concrete labeling of mereological totalities of concrete labels. So the end result is a world of concrete objects that are called sets.

This also can be generalized to all mathematical theories about Categories, Groups, Topology, Mereology, and of course Geometry, etc...

Mathematics as a whole can even be defined as the "investigation of playing games with fictionally concrete objects in a consistent manner"!

Of course "significant mathematics" occurs when those games find applications, or at least foreseen as having the potential to find application, like how the fictional syntactical game of arithmetic found application in differentially indexing the quantity of members in sets, especially sets of objects in the real world. So one needs to add the word "that are potentially applicable in the real world" to the above definition of mathematics in order to get a definition of "significant mathematics". Sometimes applicability can come in an indirect manner of course, like being useful in understanding and developing other areas of mathematics that have applications or that are themselves also useful for understanding other areas of having applications, and so on...; so there can be levels of indirect contribution to application.

My point here is that the real world is composed of concrete objects the discourse about which does encounter speaking about properties of those objects, for although those properties are in themselves not concrete, but they function to provide a discourse about the concrete objects. If mathematics is a kind of remote simulation of this world, then it better be imagined as playing games with similarly concrete objects, albeit fictionally, yet they'd bear closer resemblance to what's going on in the real world, than when claiming them to be outside of space and time; and this nearer approximation can in some sense promote finding applications.

The point is that I don't see how claiming that mathematical objects are "abstract", or "outside of space and time", plays an essential role in understanding the nature of those objects? or in even understanding their contribution to their various applications? Hence my question presented in the head post!

  • The main issue IMO with "concrete" (i.e. spatio-temporal) objects as the domain of discourse of mathematics regards the infinite. When we have theories, like PA and ZF, that ask for infinite models, how we "match" them with current spatio-temporal experience ? Jul 1 '19 at 8:44
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    The question seems to be rhetorical with your own answer already given in the post. How mathematics is "better imagined" is a matter of personal preference, and plays no role in what theorems are proved, or how. So this is not a question for SE.
    – Conifold
    Jul 1 '19 at 8:46
  • @MauroALLEGRANZA, spatio-temporal doesn't mean the "real" spatio-temporal, I'm speaking about a "fictional spatio-temporal" more strictly the property of not existing in whole at two distinct places at the same time. Numbers as postulated in peano arithemtic and in set theory aren't properties at all, they are OBJECTS of the universe of discourse of those theories. I'm speaking of course of FIRST order theories, which are the main ones.
    – Zuhair
    Jul 1 '19 at 12:37
  • @Conifold, No that is not correct. Truly I've put what I think it to be an answer, but it runs AGAINST the known current expression about numbers and sets, and mathematical objects in general, that they are "abstract" entities. That's why I'm looking for what kind of evidence for that philosophical stance and against my own stance. It is not about matters of opinion as you say. It is about philosophically convincing arguments about which direction in that subject.
    – Zuhair
    Jul 1 '19 at 13:22
  • There is no "known current expression", abstract entities are controversial, and there are plenty of fictionalist accounts, among others, that reject them, see Fictionalism in the Philosophy of Mathematics. If you are looking for a defense of abstract entities then most of your post should be deleted and the title changed accordingly.
    – Conifold
    Jul 1 '19 at 16:14

Disclaimer: I am not familiar with advanced set theory, which you have talked about a lot in your question, and have no advanced knowledge of mathematics.

However, I would be happy to provide a deductive argument for the abstractness of numbers.

Firstly, I admire your knowledge and research, but your question is much too complicated and unstructured to address each point (for me at least). As Leonardo da Vinci said: Simplicity is the ultimate sophistication. Of course, you might respond as Mark Twain: I apologize for such a long letter - I didn't have time to write a short one. and I would excuse you.

Secondly, the distinction of the number 2 and duality seems to me purely superficial, as in, we do use the terms in slightly different ways, but they are both (yet another term to denote duality) referring to the same quality.

Finally, numbers are units of measuring quantity, quantity is a property. All properties are abstract, and a property of an abstract property surely can't be concrete?

And, of course, if mathematical objects aren't abstract but spatiotemporal, how and when did they come into existence? Any and all spatiotemporal objects have a simultaneous beginning in space and time.

The Stanford Encyclopedia of Philosophy states in its entry on Abstract Objects:

There is a great deal of agreement about how to classify certain paradigm cases. Thus it is universally acknowledged that numbers and the other objects of pure mathematics are abstract (if they exist), whereas rocks and trees and human beings are concrete.

Quantity is a property, it is expressed with mathematical objects, they cannot be part of the spatiotemporal universe since there is an infinite number of them. Some of them infinitely long (such as pi). How would a number manifest itself materially?

I hope you will excuse my answer consisting, along with arguments, of multiple counter-questions, but I feel like that is the best way for people to realize something, to conclude it is true for themselves. So, try to answer these questions, and you will quickly find the answers negate the concrete numbers theory.

  • I'm speaking about numbers as presented to us by PA and ZFC. Those are formalized there as OBJECTs, and I'm speaking of the first order versions of them, which is the main version of these theories. I think all of what you said applies to the properties those numbers are associated with, rather than the inherent properties that they have due to the structure they are in. A number in those theories is NOT a property of a property, actually in Peano arithmetic it is not even an index of quantity, you can call it in some sense an index of succession. But it is not a property of succession, it is
    – Zuhair
    Jul 1 '19 at 13:28
  • more like milestones, but of succession. A kind of markers of iteration of successions. So they look like OBJECTS. And indeed they are in the first order formalization of them. With set theory the problem is bigger, for any implementation of numbers in them is clearly incomplete, and so they cannot act as complete indices of quantity of elements in sets. And also there they are OBJECTs of the universe of discourse, they are not presented as properties at all.
    – Zuhair
    Jul 1 '19 at 13:30
  • Unfortunately, this additional information isnt useful to me since I am not familiar with number theory. The best I can do is repeat my arguments for the abstractness of numbers, and appeal to the authority of SEP if you care about that.
    – Glorius
    Jul 1 '19 at 15:17

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