Is Mathematics An Object?

[Bluemist]

Rocks and past light cones are of no concern whatsoever unless they directly and noticeably disturb me or us.

I think a realist would say the rock exists whether or not it directly concerns us. OK, I'm not that guy, but I wouldn't have said it your way either. Saying' 'something is of no concern' is an incomplete statement. Saying it is of no concern to me might give a different answer than saying it is of no concern to the rock or to the bug getting squashed by it.

I should have said that a rock and bug are of less concern to me because they are further away in my consciousness therefore they are less real to me than my family and friends are.

Even for mathematical objects, the numbers 1,2,3 and numbers I can count on my fingers are closer to me in familiarity and have more individuality than 1753 or the square root of -1. The familiar numbers are more real to me than more remote or more abstract mathematical expressions. Only my self is fully real everything else gradually loses reality with distance and time.

A thoroughgoing Protagorean would leave the numbers, rocks and bugs to the mathematicians, physicists, and biologists.

[Consul]

Seriously I do not think even a realist would fall into the elephant trap of pretending that mathematical object are real.

Mathematical platonists do believe in the reality of abstract mathematical objects: https://plato.stanford.edu/entries/plat ... thematics/

We line in a 3 dimensional world which is temporal. How can a point exist when I has no substance? A point is just a reference, a line just a distance between two points. Squares have no depth, being 2 dimensional and so cannot be real.

If there are space(time)-points or point-particles, they are concrete, physical points.
Points (0D objects), lines (1D objects), and surfaces (2D objects) can be concrete things by being boundaries of physical 3D objects (including regions of space).

[Sculptor1]

Seriously I do not think even a realist would fall into the elephant trap of pretending that mathematical object are real.

Mathematical platonists do believe in the reality of abstract mathematical objects: https://plato.stanford.edu/entries/plat ... thematics/

I think you might want to ACTUALLY read the first paragraph of what you linked here.
"Independent" of us is not the same as "real". And "abstract" is not way near "real".

We line in a 3 dimensional world which is temporal. How can a point exist when I has no substance? A point is just a reference, a line just a distance between two points. Squares have no depth, being 2 dimensional and so cannot be real.

If there are space(time)-points or point-particles, they are concrete, physical points.

Don't be ridiculous. Concrete is a hard material that requires 3 dimensions and time to exist. A point Has none of these this and is arbitrary according to human interests.
If points are "concrete" as you say then you ought to be able to say how many points there are in a cubic centimeter!
Give it a try!

Points (0D objects), lines (1D objects), and surfaces (2D objects) can be concrete things by being boundaries of physical 3D objects (including regions of space).

Blah Blah - all abstractions

[Halc]

I think a realist would say the rock exists whether or not it directly concerns us. OK, I'm not that guy, but I wouldn't have said it your way either. Saying' 'something is of no concern' is an incomplete statement. Saying it is of no concern to me might give a different answer than saying it is of no concern to the rock or to the bug getting squashed by it.

I should have said that a rock and bug are of less concern to me because they are further away in my consciousness therefore they are less real to me than my family and friends are.

While I'd definitely agree about the sliding scale of concern about these objects, relative existence is pretty much a binary relation, so I'd not have said that thing A is more real (to something) than is thing B. Each is or is not.

[Consul]

Mathematical platonists do believe in the reality of abstract mathematical objects: https://plato.stanford.edu/entries/plat ... thematics/

I think you might want to ACTUALLY read the first paragraph of what you linked here.
"Independent" of us is not the same as "real". And "abstract" is not way near "real".

"Abstract" and "real" certainly aren't synonyms; but according to ontological platonism, reality contains some kind(s) of abstract entities.

The thin concept of reality is synonymous with the concept of existence; and the thick concept of reality is synonymous with the concept of independent existence, with independence being relative to some kind of things (e.g. mental or conceptual representations).

"There are two general aspects to realism, illustrated by looking at realism about the everyday world of macroscopic objects and their properties. First, there is a claim about existence. Tables, rocks, the moon, and so on, all exist, as do the following facts: the table's being square, the rock's being made of granite, and the moon's being spherical and yellow. The second aspect of realism about the everyday world of macroscopic objects and their properties concerns independence. The fact that the moon exists and is spherical is independent of anything anyone happens to say or think about the matter. Likewise, although there is a clear sense in which the table's being square is dependent on us (it was designed and constructed by human beings after all), this is not the type of dependence that the realist wishes to deny. The realist wishes to claim that apart from the mundane sort of empirical dependence of objects and their properties familiar to us from everyday life, there is no further (philosophically interesting) sense in which everyday objects and their properties can be said to be dependent on anyone's linguistic practices, conceptual schemes, or whatever.

In general, where the distinctive objects of a subject-matter are a, b, c, … , and the distinctive properties are F-ness, G-ness, H-ness and so on, realism about that subject matter will typically take the form of a claim like the following:

Generic Realism: a, b, and c and so on exist, and the fact that they exist and have properties such as F-ness, G-ness, and H-ness is (apart from mundane empirical dependencies of the sort sometimes encountered in everyday life) independent of anyone's beliefs, linguistic practices, conceptual schemes, and so on."

Realism: https://plato.stanford.edu/entries/realism/

If there are space(time)-points or point-particles, they are concrete, physical points.

Don't be ridiculous. Concrete is a hard material that requires 3 dimensions and time to exist.

The material called concrete is ontologically concrete, because it's physical; and everything physical is ontologically concrete.

If points are "concrete" as you say then you ought to be able to say how many points there are in a cubic centimeter!

If space is continuous, with every point of it being mathematically represented by a triple of real numbers, then there are infinitely many points in a region of space with a volume of 1cm³.

Blah Blah - all abstractions.

What exactly do you mean "abstraction"?

[Sculptor1]

I think you might want to ACTUALLY read the first paragraph of what you linked here.
"Independent" of us is not the same as "real". And "abstract" is no way near "real".

"Abstract" and "real" certainly aren't synonyms; but according to ontological platonism, reality contains some kind(s) of abstract entities.

The thin concept of reality is synonymous with the concept of existence; and the thick concept of reality is synonymous with the concept of independent existence, with independence being relative to some kind of things (e.g. mental or conceptual representations).

As we have discussed elsewhere, Gandalf exists, as an idea. Platonists assert that the concept of a circle exists as an abstract idea. Neither Gandalf nor circles are "REAL" in the sense of the word used by realists. Where reason leaves the room is the assertion by Platonists that the abstract notions of mathematics exist regardless of the human minds necessary to generates these ideas; but this is easy as Plato existed in a world in which Gods existed to sustain these ideas.

"There are two general aspects to realism, illustrated by looking at realism about the everyday world of macroscopic objects and their properties. First, there is a claim about existence. Tables, rocks, the moon, and so on, all exist, as do the following facts: the table's being square, the rock's being made of granite, and the moon's being spherical and yellow. The second aspect of realism about the everyday world of macroscopic objects and their properties concerns independence. The fact that the moon exists and is spherical is independent of anything anyone happens to say or think about the matter. Likewise, although there is a clear sense in which the table's being square is dependent on us (it was designed and constructed by human beings after all), this is not the type of dependence that the realist wishes to deny. The realist wishes to claim that apart from the mundane sort of empirical dependence of objects and their properties familiar to us from everyday life, there is no further (philosophically interesting) sense in which everyday objects and their properties can be said to be dependent on anyone's linguistic practices, conceptual schemes, or whatever.

In general, where the distinctive objects of a subject-matter are a, b, c, … , and the distinctive properties are F-ness, G-ness, H-ness and so on, realism about that subject matter will typically take the form of a claim like the following:

Generic Realism: a, b, and c and so on exist, and the fact that they exist and have properties such as F-ness, G-ness, and H-ness is (apart from mundane empirical dependencies of the sort sometimes encountered in everyday life) independent of anyone's beliefs, linguistic practices, conceptual schemes, and so on."

Realism: https://plato.stanford.edu/entries/realism/

Don't be ridiculous. Concrete is a hard material that requires 3 dimensions and time to exist.

The material called concrete is ontologically concrete, because it's physical; and everything physical is ontologically concrete.

If points are "concrete" as you say then you ought to be able to say how many points there are in a cubic centimeter!

If space is continuous, with every point of it being mathematically represented by a triple of real numbers, then there are infinitely many points in a region of space with a volume of 1cm³.

Agreed. The number of points in a cubic centimeter are literally infinite. This means that points cannot be "concrete" in any sense.

[Consul]

As we have discussed elsewhere, Gandalf exists, as an idea. Platonists assert that the concept of a circle exists as an abstract idea. Neither Gandalf nor circles are "REAL" in the sense of the word used by realists. Where reason leaves the room is the assertion by Platonists that the abstract notions of mathematics exist regardless of the human minds necessary to generates these ideas; but this is easy as Plato existed in a world in which Gods existed to sustain these ideas.

To say that Gandalf "exists as an idea" is to say that the idea of Gandalf exists and Gandalf doesn't exist. The existence or reality of Gandalf is one thing, and the existence or reality of the Gandalf-idea is another. (The name of the Gandalf-idea isn't "Gandalf" but "the Gandalf-idea".)

Mathematical platonism isn't about the existence/reality of conceptual, linguistic, or other symbolic representations of mathematical objects, but about the existence/reality of mathematical objects. For example, mathematical platonism about numbers isn't about numerals (numerical signs), but about numbers.
(There's another platonism: platonism about linguistic or semiotic types: Like numbers, types of numerals are abstract objects.)

Agreed. The number of points in a cubic centimeter are literally infinite. This means that points cannot be "concrete" in any sense.

A space-point isn't a material object (in the narrow sense), but it's a physical object and thus a concrete one.
(If physical properties such as mass are directly ascribed to space-points rather than to point-particles occupying them, then a space-point with physical properties such as mass can well be regarded as a material object.)

[Sculptor1]

As we have discussed elsewhere, Gandalf exists, as an idea. Platonists assert that the concept of a circle exists as an abstract idea. Neither Gandalf nor circles are "REAL" in the sense of the word used by realists. Where reason leaves the room is the assertion by Platonists that the abstract notions of mathematics exist regardless of the human minds necessary to generates these ideas; but this is easy as Plato existed in a world in which Gods existed to sustain these ideas.

To say that Gandalf "exists as an idea" is to say that the idea of Gandalf exists and Gandalf doesn't exist. The existence or reality of Gandalf is one thing, and the existence or reality of the Gandalf-idea is another. (The name of the Gandalf-idea isn't "Gandalf" but "the Gandalf-idea".)

Mathematical platonism isn't about the existence/reality of conceptual, linguistic, or other symbolic representations of mathematical objects, but about the existence/reality of mathematical objects. For example, mathematical platonism about numbers isn't about numerals (numerical signs), but about numbers.
(There's another platonism: platonism about linguistic or semiotic types: Like numbers, types of numerals are abstract objects.)

Agreed. The number of points in a cubic centimeter are literally infinite. This means that points cannot be "concrete" in any sense.

A space-point isn't a material object (in the narrow sense), but it's a physical object and thus a concrete one.

Yet you said it was concrete. The same problem exist with all other mathematical concepts.

(If physical properties such as mass are directly ascribed to space-points rather than to point-particles occupying them, then a space-point with physical properties such as mass can well be regarded as a material object.)

There is a difference between mass in the shape of a circle, and a mathematical circle.
There are no actual circles in nature; only rough shapes that we apply the concept to in order to describe it.

[Bluemist]

I should have said that a rock and bug are of less concern to me because they are further away in my consciousness therefore they are less real to me than my family and friends are.

While I'd definitely agree about the sliding scale of concern about these objects, relative existence is pretty much a binary relation, so I'd not have said that thing A is more real (to something) than is thing B. Each is or is not.

Subjective reality is a personal affair for me, or for my family, or for my company, or for my municipality, or for my country, or even for humanity as a whole. The outlook on reality is always centered on me, here, and now. My reality spreads in ever more fading and more distant concentric circles to more remote concerns.

My reality is not binary, nor even discrete, but continuous and ever changing. Because it is not fixed, I can act to change myself, my environment, and my world, hopefully for the better. This proactive attitude is essential to this sort of philosophy.

On the other hand, absolute existents either are or are not, a thing or property either exists or it does not. Existence is apparently binary, which means existence has to be unchanging and fixed in a timeless or eternal world.

Relative existence needs to be more complicated because there are many varieties of relativism. But, as you say, in each instance existence should be a binary state.

Whether numbers and mathematics are objects will depend on either metaphysical (in a fundamental Aristotelian sense) or ontological (what sort of things exist?) considerations.

Metaphysical relativism means that there are many potential logically different philosophical systems, and to decide which of these philosophies we choose to apply will depend either on the differing logic of each system, or on the different intended domains of applicability.

In an absolute sense, Harry Potter does not exist. In a relative sense, in the books and only in the books, Harry Potter existed at the fictional time of the fictional events. For mathematicians, numbers are existing objects of mathematics, because they are defined to exist or are derived from existents in math.

[Halc]

Subjective reality is a personal affair for me,

Again, I agree, but I'm also not talking about anything subjective. A rock has no (or very little) subjectivity, and yet things exist or not in relation to the rock. That itself is a subjective statement since the 'rock' as a cohesive system is a human designation, and the universe has no such objective boundaries such as 'the rock' and 'everything not that rock'. The line can be drawn anywhere, and is implicitly done so when I refer to 'the rock', or to 'me'.
That sort of reality is binary, but I agree that the subjective reality you describe is not.

On the other hand, absolute existents either are or are not, a thing or property either exists or it does not. Existence is apparently binary ...

That's the sort of thing I'm talking about, except I define it as a relation, not absolute/objective. Yours is a more realist definition.

Relative existence needs to be more complicated because there are many varieties of relativism. But, as you say, in each instance existence should be a binary state.

I find myself trying to take a relative stance in almost anything. There are some things that are just plain properties. The proper mass of a rock is a property of that rock, independent of the relation the rock has with anything else, including its existence. E.g. a 1kg rock is a 1kg rock whether or not it exists in relation to thing X. Existence is not necessary for a system to have whatever properties it has. Square root of 5 is irrational regardless of the answer to the question asked by the subject line of this thread.

Whether numbers and mathematics are objects will depend on either metaphysical (in a fundamental Aristotelian sense) or ontological (what sort of things exist?) considerations.

If you follow the last statement above, I basically say it doesn't matter if they're objects or not.

In an absolute sense, Harry Potter does not exist.

That seems to be a realist statement. I exist. Harry does not. An idealist (a different sort of realist) might assert that Harry does indeed exist.

In a relative sense, in the books and only in the books, Harry Potter existed at the fictional time of the fictional events.

It isn't fictional to Harry. Both Harry and I exist at the times of our events, pretty much by definition. Harry exists in relation to Ron. I exist in relation to you, but not in relation to Harry.

For mathematicians, numbers are existing objects of mathematics, because they are defined to exist or are derived from existents in math.

Sounds sensible.

[Bluemist]

I'm also not talking about anything subjective. A rock has no (or very little) subjectivity, and yet things exist or not in relation to the rock.

I really don't know if I understand what you are saying here. To my mind, 'relational' is just a more complicated twist on objective realism. Whether we talk of objects, properties, relations, or facts, as long as these describe a fixed world of discrete entities with identity, we must be talking realism.

In the philosophically interesting article "Relations" in the Stanford Encyclopedia of Philosophy (SEP), Fraser MacBride wrote the following :

... Properties are “one-place” or “monadic” or “unary” because properties are only exhibited by particulars or other items, e.g., properties, individually or one by one.
Relations are “many-place” or “n-adic” or “n-ary” (where n>1) because they are exhibited by particulars only in relation to other particulars. ...

To me, among others one thing that makes relations confusing is that there are many types of relations to consider, including seemingly subjective ones of the self and us in relation to you and us, and also to him and to them and to everyone else, all managed as if these are timeless objects.

But then, as I said, I don't understand this approach.

[Halc]

Whether we talk of objects, properties, relations, or facts, as long as these describe a fixed world of discrete entities with identity, we must be talking realism.

Well, I'm not describing a fixed world, and the sorts of things that populate it don't seem to have obvious identities, so I'm not talking realism.
MWI for instance is a realist stance, but still doesn't give identity to persistent things. RQM is similar, but discards the realism.

To me, among others one thing that makes relations confusing is that there are many types of relations to consider, including ...

In this universe, the relation that matters (as far as existence is concerned) seems to be measurements. X exists to Y if Y has measured X. There's nothing special about people, concepts, perception, or whatever. X and Y can be anything. The relation is not commutative. If X has measured Y, then Y cannot have measured X. If they have, then at least one of them lacks a defined identity. So for instance, we measure each other, thus that wording doesn't really relate two things with identities.

I don't seem to have an identity, but I can assign an identity to a state at an event with a past worldline leading up to and terminating at that event. That has an identity that satisfies the definition above.

[Bluemist]

Whether we talk of objects, properties, relations, or facts, as long as these describe a fixed world of discrete entities with identity, we must be talking realism.

Well, I'm not describing a fixed world, and the sorts of things that populate it don't seem to have obvious identities, so I'm not talking realism.
MWI for instance is a realist stance, but still doesn't give identity to persistent things. RQM is similar, but discards the realism.

To me, among others one thing that makes relations confusing is that there are many types of relations to consider, including ...

In this universe, the relation that matters (as far as existence is concerned) seems to be measurements. X exists to Y if Y has measured X. There's nothing special about people, concepts, perception, or whatever. X and Y can be anything. The relation is not commutative. If X has measured Y, then Y cannot have measured X. If they have, then at least one of them lacks a defined identity. So for instance, we measure each other, thus that wording doesn't really relate two things with identities.

I don't seem to have an identity, but I can assign an identity to a state at an event with a past worldline leading up to and terminating at that event. That has an identity that satisfies the definition above.

I'll have to rethink if any of the interpretations of quantum mechanics actually make any philosophical sense, whether they attempt to force QM into realism, like various theories of everything, or trying for an interactive interpretation like Rovelli. Sadly, all science but especially quantum physics has become so compartmentalized that only insiders can understand their own language, and in the other direction of the conversation, physicists appear to be totally ignorant of philosophical reasoning and historicity yet insist on telling us that we are dead.

The world in itself cannot be relational because everything interacts with everything else all at once, not just one to one, or even one to many, as relational might have it. The mistake is to ignore the unknowable, the uniquely different context of each actual instance of application of some physical law. Unfortunately, since the context is always unique, it cannot be stated absolutely. We can only axiomatically assume that it is constant, just so physics can become possible.

[Halc]

I'll have to rethink if any of the interpretations of quantum mechanics actually make any philosophical sense, whether they attempt to force QM into realism, like various theories of everything, or trying for an interactive interpretation like Rovelli.

Heh.. Rovelli is my interpretation of choice. Maybe not exactly the way published. Haven't studied it in depth.

The world in itself cannot be relational because everything interacts with everything else all at once, not just one to one, or even one to many, as relational might have it.

I'm not aware of that. I suppose it can be characterized that way.

[Bluemist]

Heh.. Rovelli is my interpretation of choice. Maybe not exactly the way published.

Rovelli's relational quantum mechanics (RQM) would be my favorite as well. That is, as long as RQM is interpreted as relations between physical systems seen in a fully relativistic setting. The problem is the observer, not the physics. If there is no Newtonian universal observer to take the measure of 'reality' so that each and every unique observer sees a measurably different world, then to presume a single scientific reality becomes a hopeless cause.

In physics measurement is all there is -- only mathematical objects are really real.

Physical measurement doesn't just tell us about some supposedly underlying reality, measurement is reality. In one piece of Einsteinian weirdness, the world distances are shorter in the direction of motion, so that I can see farther in the cosmos in the direction I am walking than when looking backwards in a mirror. When I turn around, the opposite is seen. This is not an illusion, the world is really like that for each one of us relatively at the same time.

Quentin Ruyant, "Can we make sense of Relational Quantum Mechanics?" (2017)

The relational interpretation of quantum mechanics proposes to solve the measurement problem and reconcile completeness and locality of quantum mechanics by postulating relativity to the observer for events and facts, instead of an absolute view from nowhere".