All, I it seems that I don't have enough points to respond to your post individually as a comment, so I will respond through an answer format.
@Bob, on why I am interested in being capable of recognizing the mathematics of an observation. I suppose it goes to the basic question of how do we know when we are engaging in mathematics? We go to grade school and sit in a class room where a teacher says,today's we are going to talk about integers, or addition, or, shapes. Yet, there is often no mention as to why these concepts belong to a discussion on what is called mathematics. This is not only true only of mathematics classrooms, but I have personally found it particularly difficult with mathematics because mathematics also shows up in chemistry, biology, physics, economics, psychology, etc., and so it has been difficult for me to realize a satisfactory answer on on my own. But, I would also believe a generally agreed upon explanation to such a question might turn out to be useful in helping students be better equipped at seeing the connection to mathematics in many aspects of their lives, and perhaps spark more interest in the subject. I read a book once that stated that when learning about a subject like mathematics,one should learn what it means to think mathematically, so you can get into the practice of thinking like a mathematician. Seems like reasonable advice to me for perhaps any field of study, but when I pass this advice on to my children, I hope they will ask good questions like: But how do I know (measure) when I am thinking like a mathematician, or how do I know when I should think like a mathematician, or simply how do I know when I am doing mathematics. I would be nice to be able to respond with some guidelines that are both generally agreed on, and truly useful for them when they navigate such questions.
Nick, I think you make a good point regarding deductive reasoning as being a method for studying mathematics but not a fundamental requirement for studying mathematics. I suppose an alien race could come down and communicate some body of knowledge that would not include methods similar to deductive reasoning (from the perspective of both humans and aliens 😊), yet it offers similar capabilities that mathematics has offered to humans. Although I would be curious as to whether humans would accept it into the field of what we call "Mathematics", or instead give it a new label to distinguish the two. On the view you offered on what a mathematical object is, it appears that you are saying that it is not the abstract object itself that is the key to what makes something a mathematical object, but it is the relations defined for those abstract objects that gets you there. (Although, back to the conversation on deductive reasoning, one might consider the use of "relationships" as also being a method humans have created to explore the concept of mathematics...in fact on might ask where in mathematics does something not map back to some employed "method"? I will have to give that one more thought). But, if the word "property" is synonymous with "relation", and I were to substitute the word property with the word relation (or by extension, properties with relations), then one of your subsequent sentences might read:
The only relations of integers are those present in the relations between them.
Can you help clarify how I should interpret this observation? Also, if relationships are the key, then what would you consider to be the distinction between a mathematical relationship and a non-mathematical relationship? And if it is not the relationship that needs a distinction, then once again, what are those elements of mathematics that make it mathematical? If it is not the abstract object alone, not the relationships alone, and not how we approach the reasoning process alone, then is there something unique about how all three of these things interact? If so, what label have we given this interaction so that we all know what we are talking about when we want to refer to and study it?
@Quen-tin, I have a question on what you means when you say mathematical objects have no reference to the external world? What do you mean by external world? Do you mean mathematical objects are not required to be physical objects that are observable to humans? Do you mean humans have no means of measuring whether any physical object perfectly aligns with the characterization of mathematical objects? Similarly, what do you mean when you say they have no qualitative aspects?
@C-S, thanks for the reference for literature on the topic...it is interesting that there are different schools of thought on this matter. Now that I have started to think about it, I can see why that would be. It is interesting that you see examples B and C as mathematical. I guess B has elements to it commonly associated with mathematics, such as the use of formal reasoning. If I think about the concept of "justice" as an abstract object, I suppose that even if I reason formally about it, I don't know that I would be able to ever draw conclusions from doing so that would ever be considered mathematical. As I think about the discussion with Nick (see above), perhaps I am unable to provide an unambiguous description for the abstract notion of justice in terms of relationships on a set.
In general, I once listened to a Utube video on Möbius strips and the questions they were posing did not jump out to me as mathematical. At a later time, I recognized that they were certainly finding interesting ways to abstractly represent the patterns being observed and they certainly applied deductive reasoning to draw conclusions on the object of study. But, initially I was like...how is this mathematics? It was once again a moment where I was surprised that the course of study was considered mathematics. And once again I realized that I do not have a good foundation for recognizing the mathematical potential of an observation. But, thanks to kind people like yourselves, I do feel like I am starting to understand the points of discussion around the topic better. Thus, many thanks for sharing your thoughts! I look forward to reading any further comments you may have 😊.