What are the properties of Mathematical Objects?

Question

I have been thinking a lot about how one knows when an observation contains mathematical elements. Many years ago when I was in school, I found that there was often little time taken out to discuss what makes a new concept being taught mathematical. I suppose this may be a difference between teaching how to complete a calculation vs. teaching students to be capable of recognizing whether a given observation has mathematical properties, what they are, and of course what conclusions one may be able to draw from them. So in an effort to improve my own abilities in this area, I have been trying to piece together a few things. First, one can start by looking at the use of deductive reasoning as one of the key properties of mathematics. But then I asked, at what point can one characterize deductive reasoning as mathematical deductive reasoning? So then I compared the below in an attempt to shed some light on the question. Please note, I am by no means well versed in mathematics so there may be inaccuracies in the examples below, but I think one can still get the gist of where I am going.

A. General Deductive Reasoning Form:

 All 'A' are 'B'.

 'C' is 'A'.

 Therefore, C is 'B'.

B. Non-mathematical example of the Form:

 All men are mortal.

 Socrates is a man.

 Therefore, Socrates is mortal.

C. Mathematical example of the Form:

 All polygons that contain the properties of a rectangle are also  
 rectangles.

 A square contains all of the properties of a rectangle.

 Therefore, a square is also a rectangle.

So I have interpreted (potentially incorrectly) section B as being non-mathematical because the objects being referred to do not seem to have properties that have and will remain the same forever and always, wheres the objects in section C seem to have properties that are, more or less, eternally unchanging (well at least compared to the changes that mankind undergoes). Thus, this approach in answering my question seems to boil down to being knowledgeable of the properties that make an object mathematical, especially the ones that differ from non-mathematical objects.

I am not sure whether there is general agreement on the properties of mathematical objects (especially in the context of making distinctions from the properties of non-mathematical objects), but I would appreciate thoughts from others on the above. I am also interested in hearing from those who believe that perhaps this is not the most useful path to go down in trying to become capable at identifying when an observation has mathematical elements to it.

Answer 1

I believe that the majority of mathematicians would take this view :

A mathematical object is a set of abstract entities together with the relationships between them. According to this view, the word property is synonymous with relation.

For example, the set of integers is a mathematical object. The only properties of integers are those present in the relations between them.

We do not invent mathematical objects, we only invent the notations we use to identify them and study their properties. Key to this view is that mathematical objects are identified and defined by humans in a purely abstract way, without any human baggage.

There are many philosophical objections to this view.

Deductive reasoning is not, as you suggest, a property of mathematics. It is a method humans use to explore the properties of mathematical objects. Logic and mathematics are not the same thing.

Answer 2

That's a good question and I have no definitive answer, but deduction might not be the good start, since as you observe deduction applies to any domain.

Here are some ideas: mathematical objects are purely abstract (they don't exist in space-time. Their representations or symbols do, but the representation is not the object.). They're not perceived through the senses. They are formal. They're based on purely logical axioms and definitions, without any reference to the external world. They have no qualitative aspects.

Answer 3

One of the most fundamental debates in the philosophy of mathematics is a debate about what mathematics is. Your question ("What makes an object mathematical?") might be understood directly from understanding, "What is mathematics?" I would therefore suggest you begin by reading the SEP's list of the four main schools of thought. Each school answers the question differently.

For example, to a logicist, an object is mathematical if it is simply a statement in some logical system. Thus, the first and third of your examples are certainly mathematics, and the second might be also if "socrates", "mortal", and "man" were axiomatically defined. A statement is mathematical if its content (meaning) can be defined in purely logical terms. Bertrand Russell defined mathematics thus:

Pure mathematics is the class of all propositions of the form "p implies q", where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants.

In practice, however, doing mathematics is much more complicated, and logicists will have to concede that it is only after a lot of work that a field of math is reduced to purely logical terms. Intuitionism, formalism, and predictavism, as well as the more ancient platonism, have their own things to say.

My own opinion is that an object is mathematical in the same way an object is physical: it is observed as a pattern in the real world. Thus I would say that all three of your examples are mathematical, the second and third merely being special cases of the first generic scheme. To clarify, mathematicians usually abstract away irrelevant details and focus in on very specific properties of things; so "Socrates" and "Mortal" are replaced by general concepts, and so "pile of rocks", "stack of coins", and "organism of cells" are all reduced to the abstract entity called a "set" (and, in these cases, a finite set).

Answer 4

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 😊.

Answer 5

What are the properties of Mathematical Objects?

First some examples of mathematical objects: Numbers, sets, topological spaces, vector spaces, differentiable functions from calculus, manifolds, complex spaces ...

All these objects are ideas.

Most of these ideas are well-defined. E.g. in a course on calculus one defines: A differentiable function is a map from an open subset of the real numbers to the set of real numbers, which has a derivative at each point of definition. A derivative is ...

Other objects like "set" cannot be defined. They do not reduce to other mathematical objects. Instead, these objects are introduced as undefined basic notations and are characterized by axioms, which often fix the relation between such objects: For each two sets the union exists as a set, such that the elements of the union are exactly the elements from the two original sets.

Definitions and axioms are phrased in a formal language which avoids any kind of ambiguity. Mathematicians do not quarrel with each other on the meaning of their statements. That's one big difference between mathematical objects and other ideas, e.g. concepts from philosophy, not to speak about concepts from theology or even religion.

It is an open question why mathematical objects and mathematical theories are a suitable tool to formulate laws of nature. The classics is "Paul Wigner: The Unreasonable Effectiveness of Mathematics in the Natural Sciences (1960)"

See http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

Answer 6

Thanks again to everyone who has contributed to the answers. I thought I would take what I have read and learned from various sources to date and attempt to answer my own question. Not intended to be formal, necessarily free from contradiction or redundancy, or complete but here is an answer currently swimming around in my head.(Note: I tend to use the terms property, characteristic, and quality interchangeably throughout the answer)..

Any and all things, physical and non-physical can be characterized with the same general label. Humans commonly use the term "object" as a label that can be used to reference any and all things. Humans can identify a particular object by scoping (or at least attempting to scope) the nature of the parts or boundaries for any aspect of an observation (physical or non-physical). All objects have some qualities or characteristics that are tied to the object (even a primitive term can be viewed as an object that has a characteristic of having no intrinsic relationships with other objects), so humans are often able to identify a previously observed object based on qualities or characteristics of a current observation.

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Many objects that have proven to be useful to human beings can be identified by the process of abstraction, whereby common characteristics observed from a collection of objects are labeled as an object in and of itself. Some of the objects that can be identified from the abstraction process have a a pattern of qualities that are consistent with what is known as Abstract objects. Abstract objects have a pattern of demonstrating at least the following qualities:

• They are eternal. But, knowledge or significance of them to any form of existence may not be eternal.

• They are not required to occupy any dimension known to humans.

• They are not required to be observable by some form of existence. In fact, there may be no collection of existence(s) that is capable of becoming aware of all possible abstract objects.

• A collection of abstract objects is also an abstract object.

• Processes that abstract objects are exposed to are also abstract objects.

• A characteristic associated with an abstract object may also be viewed as an abstract object (e.g. A cube can be constructed using squares, which in turn can be constructed using lines, etc.)

• An abstract object may have all of the properties of another abstract object (e.g. A square is a special case of a rectangle), but it may also have other properties shared by additional abstract objects.

• Abstract objects can map to other abstract objects.

Although the identification of Abstract objects may not require the process of abstraction, it seems plausible that the reason the word "Abstract" came in to use with the word "object" was because so many objects with the above qualities have/can been identified via some form of an abstraction process.

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Mathematics is the collection of patterns associated with abstract objects whereby the abstract objects appear to posses at least the following quality:

• The state of their patterns is eternal. That is, both the state of their characteristics and the state of their relationships with other abstract objects are eternal. Put simply, their patterns have and will never change.

The type of abstract objects that posses this quality are often labeled Mathematical Objects.

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Of further note, it also seems that:

• Patterns identified in physical objects can always be mapped to / represented by abstracts objects, however it is unclear as to whether there exist any physical object that can have properties that are perfectly identical to that of its associated abstract object.

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So, then we have mathematicians. Mathematicians identify and analyze mathematics. In pursuit of their studies, mathematicians generate a "collection" of accepted methods, tools, techniques, standards, and best practices that are useful in advancing the body of knowledge associated with mathematics. Both the "collection" and the body of knowledge are shared and modified as the field matures as a set of highly connected mathematical realizations.

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And so how do humans recognize mathematical objects? Well, it occurs when one is able to identify objects that contain the qualities listed above for mathematical objects, usually done while studying mathematics or pondering interesting questions that contain mathematical elements. This will likely be done in the context of the human version of the "collection" and existing body of knowledge. By doing so, human mathematicians are able to leverage widely accepted approaches and conclusions against the mathematical object being observed.