In mathematics, existence is logical consistency, so everything that is consistent exists, and exists necessarily (because it cannot be inconsistent) and truths about it are necessary truths.

We must tack on a tensor field to specify some energy density at every point in this spacetime. We have to tell Lorentzian spacetime how it should actually curve. A literally material constraint must be glued to the floppy Lorentzian fabric to give it a gravitational structure. — apokrisis

Energy density is a quantity (number) that is related via Einstein's mathematical equation to spacetime curvature. Pure mathematics.

And even then, the quantum of action - how G scales the interaction between the energy density and the spatiotemporal curvature - remains to be accounted for. — apokrisis

Maybe G can be derived from some general principles and maybe its value is specific just to the spacetime in which we happen to live and may have different values in other spacetimes.

Within mathematics in general, there are numerous contradictions such as Euclidean vs. non-Euclidean geometry, — Metaphysician Undercover

Not sure if you missed my reply:

https://thephilosophyforum.com/discussion/comment/219667

The problem I see with this is that if a mathematical "object", say the number five, has no existence apart from its concrete representations, then it cannot qualify as an object at all, since its representations are potentially infinite in number. — Janus

Abstract objects and their particular representations are inseparable. There cannot be one without the other. Representations cannot exist without that which they represent, and that which is represented cannot exist without its representations. If there is no number 5 then there are no 5 objects.

I think it is better to think of a "mathematical object" as a way of thinking or speaking, so the sameness consists in the human action. It's like, for example, traveling by train from one station to another; the journey is both always the same and yet different every time, just as each instantiation or representation of fiveness is. There is no perfect form of fiveness, just as there is no perfect form of the train journey. The sameness in both cases is the result of the human process of abstraction. — Janus

The sameness is also a fact about the external world.

So formal truths have a sort of... transcendality? Transcendency? Whatever. They go above and beyond possible worlds, basically. — MindForged

If you mean truths that hold in different possible worlds, then these truths constitute a more general/more abstract/higher-order possible world.

Maths is spatial, or at best, spatiotemporal, and doesn't speak to energy or action in any basic way. It about the logical syntax of patterns and structures, and not about whatever breathes physical fire into those equations. — apokrisis

Spaces are indeed traditionally studied mathematical objects and time is treated in theory of relativity as a special kind of spatial dimension, completing a more complex mathematical object - spacetime. Now, this spacetime in which we live is not empty/uniform but it contains additional structure, like embroidery in a sheet of cloth, in the form of objects extended in space and time (which we perceive as extended in space and moving through space, changing and enduring in time), and there are also certain regularities in these extensions that we call laws of physics, laws that govern how spatially extended objects move and change in time and in relation to each other, how they interact with each other, and how these attractive and repulsive interactions define their boundaries. We may use concepts like "force" or "energy" to describe these movements, changes and interactions. Force is the product of an object's acceleration and mass, where mass is a quantity related to spacetime curvature in the place where the object is located. Energy is an object's ability to exert force over space, that is, a quantity that determines how the object accelerates another object in interaction.

So spacetime with its complex structure seems to be a specific mathematical object. One of all possible mathematical objects, and one whose structure allows the existence of what we call living conscious objects - like us.

Your comparison isn't completely right. Finite mathematical objects exist both like the uncarved material and the carved material. I suppose you compared the carved material to the representations of mathematical objects in human consciousness.

So sure, when maths is understood as just a realm of everything that unconstrained syntax will produce - a Borges library - then it seems to bear no real relation to a reality in which limitation or finitude is apparent everywhere. — apokrisis

The relation between such an unconstrained world of math and a limited finite world is that the limited finite world is a part of the unconstrained world of math.

In any case, the sense in which Platonism says that numbers (etc) are independent of particular minds, is simply based on the observation that they are the same for anyone capable of counting. But at the same time, they're only perceptible by a rational intelligence. So they're 'intelligible objects', or the objects of reason, which are fundamental to the operations of rational thought, and indeed to science itself. — Wayfarer

If they are only perceptible by a rational mind it doesn't necessarily mean that they are inside the rational mind. They may be perceptible only by a rational mind and still be outside the rational mind - that's what Platonists/realists are saying. They are saying it because it seems inconceivable that truths about numbers didn't hold before someone perceived them or will stop holding when there is no one around to perceive them. Maybe a more accurate word would be "to infer" than "to perceive", since abstract objects like numbers don't seem amenable to sensory interaction. But to infer the existence of an object doesn't mean that the object only exists in the mind that is doing the inference. Scientists make inferences (predictions) about the external world and test them.

Not only is this mathematical realm full of junk, but it's also full of contradictions. Go figure. Because of such contradictions, mathematics is clearly not logical. So, which is more reliable, mathematics or logic? — Metaphysician Undercover

Only consistently defined objects can be part of the mathematical world.

Axioms are properties of an object (also called axiomatic system). Axioms like "The continuum hypothesis is true" and "The continuum hypothesis is not true" would be contradictory if they were properties of the same object but they are not contradictory if they are properties of different objects.

So magnitude allows for opposite orders. It also allows for any other order that one might like to use, counting by tens by twenties, odd numbers, even numbers, Fibonacci order, subtracting magnitudes, dividing or multiplying magnitudes, any possible order. Since it allows for the possibility of opposite orders, and any other order, it really doesn't define order at all. — Metaphysician Undercover

No, it defines all those orders you mentioned.

Your condition was "all possible groups of points". If you restrict this to some groups, then we no longer have that initial condition. And if you restrict the group of points, to the definition of a line, then clearly we are not talking about all possible groups of points in a given space, we are talking about a defined line. — Metaphysician Undercover

But the group of points that define a line is contained in the group of all possible groups of points, which is the space itself. So the line exists in the space, together with other lines and curves.

Objects that can be sensed are parts of spacetime. But what is spacetime? Theory of relativity treats spacetime as a mathematical structure, a kind of metric space where the dimension of time is a special kind of spatial dimension. And there are many other possible mathematical structures, for example pure spaces, without a time dimension. So objects that are parts of such spaces cannot be sensed, yet they are not in our heads either. And then there are mathematical structures that are not even metric or topological spaces, for example functions.

Existence, in its most general sense, is identity: any object that is identical to itself, exists. And it exists in the way in which it is defined. (note however that it must be defined consistently in relation to everything, otherwise its identity would be violated)

You still haven't attempted to answer the hard question I posed to you. — Janus

I gave you examples of objects that may exist outside of our minds and unable to be sensed: abstract objects and objects in worlds without time.

No, you experience many actual collections of objects; trees, dogs, parks, cities, people, etc, etc, but you only imagine or think of arbitrary collections of totally unrelated objects. — Janus

Then I infer that even objects that I can't sense together as a collection, in fact constitute a collection. I infer it from what collections have in common. It seems arbitrary, without any ontological relevance to say that some objects constitute a collection and some don't. Why do you think a city is an objective collection, for example? Because when you see it from a plane it seems to form a relatively compact object? And when you are inside that city, do you still see it as a collection?

I believe our very idea of real existence comes from the idea of the existence of those objects we can sense. — Janus

And I find it absurd to believe that something only exists objectively when someone can sense it. Did the Moon exist objectively before anyone sensed it?

Yes, but what exactly is that "objective existence" if it is not concrete material existence and yet is something more than the merely ideal existence of the contents of thought? — Janus

It may be an object in a different universe which we may never be able to sense. There may or may not be other conscious beings in that universe that can sense it. It may be an object in a world that only has space and no time - and no life, so it cannot be sensed by anyone. And then there are abstract objects like numbers, which don't have any particular position in space or time and so can't be sensed, and yet the truths about numbers seem to be objective truths, independent of humans, and also reflected in our physical world.

I should note though that by "set theory" I don't necessarily mean ZFC. As I clarified here, by set theory I mean all consistent versions of pure set theory. ZFC is just one version of pure set theory (it also appears to be consistent, though we may never know for sure because of Godel's second incompleteness theorem).

What all versions of pure set theory have in common is the concept of set as a collection of objects that can be defined by listing those objects (set members) or by specifying their common property. As it turned out, not every definition via a common property is consistent, and since it would not be very useful to define sets only via listing of members, one must also specify with axioms what common properties can be used to define sets. Which gives rise to uncountably many axiomatic versions of pure set theory.

Too technical for me, I am no mathematician.

Stanford Encyclopedia of Philosophy says:

"The axioms of set theory imply the existence of a set-theoretic universe so rich that all mathematical objects can be construed as sets. Also, the formal language of pure set theory allows one to formalize all mathematical notions and arguments. Thus, set theory has become the standard foundation for mathematics, as every mathematical object can be viewed as a set, and every theorem of mathematics can be logically deduced in the Predicate Calculus from the axioms of set theory."

https://plato.stanford.edu/entries/set-theory/
(from year 2014)

What objects cannot be represented through set theory? And what does it have to do with your link to small categories?

I don't understand. Are there mathematical objects that cannot be expressed in set theory? I heard that set theory can express all mathematical objects - as sets. That's why it is considered a foundation of mathematics.

So you are saying that there are mathematical objects that cannot be represented as sets?

But they can be instantiated in sets.

But how mathematics looks as a category theorist is quite a lot different from how it looks like under the aspect of set theory. Say, to a category theorist, natural numbers don't look like the names of individual objects, they look like isomorphism classes of sets. Set theory was built out of intuitions about composite objects of multiple elements, category theory was built from intuitions of transformation and symmetry. — fdrake

Yes. This is how I understand it: more general (more abstract) mathematical objects are instantiated in more specific mathematical objects (e.g. "geometric object" is instantiated in "triangle") and ultimately in concrete mathematical objects (e.g. in concrete triangles), which are not instantiated in anything else. (Those objects that can be instantiated in other objects are also called properties.) All concrete objects are concrete collections, that is, collections of concrete objects, so all mathematical objects are ultimately instantiated in concrete collections. This fact is used in set theory, where every mathematical object is represented as a collection (set) and that's why set theory can be a foundation of mathematics.

The collections referred to in set theory are not concrete collections though but abstract collections (generalized collections), because differences between concrete collections of the same kind are not relevant for mathematical purposes. So for example, set theory does not refer to concrete empty sets but to one abstract empty set (which is instantiated in all concrete empty sets).

The approach of category theory is not to represent mathematical objects as collections but to study similarities (morphisms) directly between mathematical objects themselves. Collections, then, are treated just as one of many kinds of mathematical objects.

I have no idea how you took the main thrust of my post to be about beauty or truth. The main thrust is simply that most mathematical objects aren't worthy of study, and agglomerating them all together; producing the final book and the final theorem, far from the ideal vision or ultimate goal of mathematics - produces a writhing mass of irrelevant chaos. — fdrake

Well, it depends on how you define mathematics. In the context of set theory, traditional mathematical topics can be extended to the study of all relational structures. This covers the whole relational aspect of reality and so is relevant for metaphysics/ontology.

But anyway, the thrust of the argument is: if we took the results of all possible axiomatic systems, agglomerated them into one giant object, then granted that object independent existence - what would it look like? It would contain all kinds of bizarre crap, navigating through this world you'd hardly ever find an axiomatic system which resembled anything like our own. — fdrake

Yes, most of it might not be beautiful or useful but we are talking about metaphysics, which I don't think depends on subjective notions of beauty or usefulness.

There are other foundations of mathematics which are currently in use. — fdrake

Yes, there are various approaches to the study of relational structures. Maybe some are less comprehensive than others. Set theory seems to be the most popular foundation of mathematics and it seems to me that it is an exhaustive study of relational structures (of course it can never be completed even in principle, due to Godel's first incompleteness theorem).

in category theory, the category Set is a subcategory of the category of relations, — fdrake

In set theory, all relations are defined as sets.

Of course, the reality of an abstraction would only depend on its utility to us if the abstraction were not independently real to begin with. — Pneumenon

Mathematics is a feature of the external world, a consequence of the fact that there are differences and thus more than one object in the external world.

As long as there are any objects in the external reality, there are also relations between them, in the external reality. Relations and the objects between which they hold are inseparable.

Sure, but what sorts of things are structures and relations? Do they exist in themselves rather like intelligible forms in Platonic heaven? If you assume that they are universals that exist by themselves, quite independently from the constitutive roles of our practices of reasoning and discussing about them, then, in that case, you are begging the question in favor of mathematical Platonism. — Pierre-Normand

Relations are objects that hold between other objects (those other objects may be relations or non-relations). Relations are inseparable from the objects between which they hold.

I don't believe you. You don't experience and arbitrary collection of objects. — Janus

Why not? I experience any collection of objects as a collection.

The actuality of the collection consists in its being able to be viewed. — Janus

You mean sensed? Why would the objective existence of anything depend on whether some creature can sense it?

As to the purported existence of mathematical objects: what kind of existence do they have? We know that things exist for us materially (things we can sense) and also ideally (things we can think or imagine); what other kind of existence can you think of? — Janus

We can also infer the objective existence of things we can't sense from things we can sense and think. I think that's the case with some sets/collections and other mathematical objects.

Why wouldn't the bigger be prior to the smaller? — Metaphysician Undercover

It can - it is the reversed order to "smaller prior to bigger". Magnitude defines both orders.

All possible groups of points does not make a line, nor does it make a curve. — Metaphysician Undercover

Some of those groups do.

M is the Platonic world of math. The problem, though, is that this world is essentially full of junk. The vast majority of it is simply useless, and of no interest to anyone whatsoever. — StreetlightX

So his argument is that the Platonic world of math doesn't exist because it is... uninteresting? :lol:

The most general definition of mathematics I know is that it is a study of structures/relations. If there is more than one object, these objects form relational structures and mathematics studies these structures. The most general relation is "similarity" (also known as "difference"), because it is a relation that holds between any two objects. It means that the two objects have some different properties and some same properties. Which gives rise to another general relation called "instantiation", which is the relation between a property and its instance. The instantiation relation is a special kind of the similarity relation but less general than similarity since it doesn't hold between arbitrary two objects. Finally, any objects can define a collection of them (for example based on their common property, as long as such a definition is consistent), which gives rise to another general relation called "composition", which is the relation between a collection and its part. The composition relation, too, is a special kind of the similarity relation but less general than similarity since it doesn't hold between arbitrary two objects.

So, the similarity relation, together with its special kinds - instantiation and composition, defines all possible relational structures. All these three relations come together in set theory, the foundation of mathematics. In other words, the world of mathematics is a world defined by set theory (more accurately, by all consistent versions of pure set theory).

The fluid intelligence diffused over the Jupiter-like planet, could have developed mathematics without ever thinking about natural numbers. — StreetlightX

As long as there are any differences on Jupiter, you can start counting.

My experience is that collections exist even when they are loosely connected. I don't feel the need to deny their objective existence.

And I am far from being alone in this. Most mathematicians think that mathematical objects objectively exist, and these objects may not even be physical things.

Reality does not care what we find special or significant. It just is.

With respect to collections, I don't see how you can restrict their objective existence in any way. Whether they are "tightly connected" or "loosely connected", they are out there. You can talk about how tightly or loosely they are connected but you can't deny they are out there.

No, you're not paying attention; the criterion is physical connection. — Janus

A physical connection of arbitrary kind and to arbitrary degree. Everything in the universe is physically connected in some way. And Sun, Earth and Moon are gravitationally connected, yet you deny that they constitute a group.

I cannot see any reason to impute extra-mental existence to arbitrary collections of objects whether those objects are themselves real or merely imagined. — Janus

But that's what you are doing. You are imputing extra-mental existence to collections of objects based on arbitrary criteria.

No. the magnitude does not determine the order. There is nothing inherent within magnitude which says that 100 is before or after 200, or 50, or whatever. — Metaphysician Undercover

The magnitude says that 100 is smaller than 200 and thus orders the numbers from smaller to bigger.

You're being ridiculous again, claiming "all lines and all other possible curves in that space are defined", without the existence of any definitions. — Metaphysician Undercover

All points in space exist and thus they constitute all possible groups of points, that is, all possible lines and curves in that space.

Because we can arbitrarily think of any old selection of objects as a collection. Do we really need to go over this again? — Janus

Just because we can think of something doesn't mean it only exists in our thoughts.

Yes, and a good example of that is the realization that arbitrary collections exist only in thought. — Janus

Why would they only exist in thought? A collection is constituted by the objects it is a collection of. If Sun, Earth and Moon exist outside our thoughts why should the collection they constitute exist inside our thoughts?

Then I don't know what you are saying or how it differs from what I have been saying. — Janus

I am not denying that our understanding and experience reflect something of objective nature. I am talking from my understanding and experience too.

I disagree because your position disqualifies any talk about the objective existence of anything. — Janus

First, objective existence doesn't depend on whether anybody talks about it. Second, even if we accept that everything exists we can still talk about the ways in which this or that object exists.

We are a part of nature, which means that our experience and understanding is also a part of nature, so why should we not think that our understanding and experience reflects something of objective nature? — Janus

I didn't say that.

I wouldn't impose any criteria on which groups objectively exist because every restrictive criterion would be arbitrary to some extent - the more restrictive, the more arbitrary; the less restrictive, the less arbitrary (more universal/general).

Sure, but with these arbitrary, subjective criteria you want to decide what objectively exists?