How could triangles exist? [closed]

Question

I do not fully understand why there is a debate in the philosophy of mathematics about to what extent mathematical objects "exist inherently". I think the main reason I am confused is because I'm not sure what "exists" means.

I have seen Stewart Shapiro make a statement a number of times that he thinks the biggest problem with the Platonic notion of mathematical objects existing in their own atemporal and causally inert world is how humans can then gain knowledge of such objects.

Maybe a good naive place to start to think about what we mean by 'exist' would be "to occupy time and space". This is a very physical view of existence. However, if the mind and consciousness is made of bits of matter interacting with each other, then all mental phenomena, like thinking about a unicorn, is indeed existing and taking place in time and space, simulated by atoms, electrons, etc.

It feels like the question of "but does an equilateral triangle with an angle of such-and-such degrees actually exist?" is not problematic. It certainly does not exist, in that idealized mental form, in physical reality. In physical reality, what exists are subatomic particles. If we then say, "Yes, but insofar as you can define and conceive of such a notion as that triangle, surely it exists, albeit in some other mode of existence, under some other definition of 'exist'?", we could respond, "Sure, it's logically definable." But, so what? That second kind of "existence" isn't similar to the first kind at all. We aren't saying there is a location that it resides in, like a second dimension of space lying above the universe's one. I feel almost like there's no problem. I'm curious what leads us to need to ask if triangles "actually exist" or not.

Answer 1

This is sometimes known as the Problem of Epistemic Access, and it's a problem I agree is a serious concern for the Platonist school of mathematical realism. If the things that our mathematical statements are supposed to refer to are based in a realm that is separate from the realm of our ordinary sensory experience, in what sense can we talk about them at all?

But Shapiro is coming at this from a slightly different angle, and one informed at least in part by the work of Michael Resnik and others on Structuralism in the philosophy of mathematics. Shapiro et al aren't seeking to try to connect to an ideal infinite higher plane of reality, but rather to explain how maths doesn't need it in order to work for what it is we want mathematics to do for us.

So, what is it that maths is doing for us, what are the norms and conventions and truths in the language that we use to discuss mathematical structures, and how do we connect the way that we work with mathematics to things in the world that we want to make sense of through mathematical methods?

For Willard van Orman Quine, this is not a problem specific to mathematics but is part of the wider general question of how we come to acquire knowledge about the world, how we gather evidence and discover truths and form complex theories that we subject to the truths of experimentation. The sense in which we gather evidence about the existence of prime numbers is not that radically different than the sense in which we gather evidence about the existence of photons - we are engaged in a lifelong project of discovery of the world, of forming and reformulating theories of it.

And, in as much as our best theories of this are scientific and make effective and novel predictions about the world, they often do so using the language of number, geometry, topology and other forms of mathematical framing. We find out where to look in building our models of reality by looking at the data, following the calculations, and confirming what we find there.

For Quine, this gives rise to an argument for the existence of numbers in our current best practice theory of the world which is known as the Indispensibility argument. Numbers, shapes, surfaces, sets etc. are a necessary part of the descriptions of our theories of reality - you can't do science without them, so in your overarching theory of what stuff there is in the world, a comprehensive classification will include the stuff needed to do mathematics. A realist about Science - someone who says that the stuff our scientific methodology says exists really does exist - is committed to also say the same about abstract mathematical objects, on pain of a massive methodological hole in their model of reality.

(for Quine, it's not so much that there's a hole, as much as that the whole model of scientific theorising engages with reality as a web of interrelated meanings, and stands or fails in as much as our account of the whole connects well with the test of evidence, but the basic weakness of a theory ungrounded is the same! This is sort of where Shapiro and the structuralists are also coming from, and why they like the idea that "the things" in maths are whole structures rather than individual objects, but that's jumping ahead a bit.)

So, what numbers/sets/abstract objects are we committed to by virtue of our scientific practice? Well, the simple answer is to say that we are certainly committed to the things that our theory says exists. This is part of the virtue of First Order Classical Logic, which is the form of logic that uses the existential quantifier ∃ in its grammar (e.g. ¬∃x.KingOfFrance(x)) The logic of our working mathematics talks quite openly about the existence of prime numbers, solutions, limits etc. and in as much as our mathematical models that we use to frame our scientific theories is engaged in this kind of talk, so too is our science.

HOWEVER! The interesting question about Quine's theory of the indispensibility of mathematical objects now becomes "how much mathematical abstraction is needed to ground our theories, and can we do more with less? Is all of it indispensible, or can we maybe dispense of some of it in formulating a suitably parsimonious theory of reality?"

For example, we often talk about Representation in abstract mathematics. We can create representations of Groups in the model of Sets, so Groups don't need to be considered concrete objects in themselves. We can do our mathematics of group theory without presupposing that groups are things that exist in themselves over and above the world, because everything you do in group theory you can do just fine in Set theory; Group theory is just a nice (and very fruitful!) way of packaging and applying ideas that are already contained in set theory.

This is a very different theory of the existence of objects like groups - groups do exist, demonstrably, because sets that represent them exist, but they don't have any kind of unique, special existence as groups. Group theory is a useful language and set of theoretical tools to deal with the things that are groups, but those things are things of other types (some might say more grounded types) in our theory of reality before they are groups (and this is part of what makes Group theory so interesting, because it can talk about things of many kinds rather than just talking about abstract objects that are always and only groups).

David Hilbert proposed that perhaps most of mathematics worked this way. We might have some finite core to our mathematical theories, like number theory, symbolic logic, or a specialist inner model of Sets, that can be used to ground the rest of our mathematical theorising. And, in fact, we might go further still, and say that even this finite core is just a way of talking abstractly about real physical things, such that it is contained in our ability to use language rather than a distinctly mathematical capability.

This is the Formalist position in the philosophy of mathematics - that the stuff of number and shape exists because we are using general linguistic forms and terms to talk about the real things that are examples of the qualities and quantities that maths works with. It accepts Quine's suggestion that we have to integrate our mathematical commitments into our scientific theories by making the ontology of mathematics a question entirely of Semantics - what is the minimum that we need to do the mathematically rich language games needed to ground our scientific practice and perform mathematical proofs?

However, this isn't trivial, and has fallen afoul of many traps facing this ideal! It's often thought that Kurt Godel disproved formalism by showing that first order number theory coded its own contradiction, and naive attempts to do mathematics entirely in finite number theory certainly do struggle with contradictory semantics. But it's an active research programme to ask whether some key subset of mathematics necessary for empirical science might suffice to account for the rest of our open applied mathematical questions, and their applications to live R&D problems.

So, are we committed to things like triangles existing? For the moment, we can probably say that we have to acknowledge that things that are triangles exist, and in absentia a better foundation for the language of geometry we might as well work with abstract triangles and save ourselves the headache of constantly translating into real stuff. But in the grander scale of things, this is just an operational working theory, and we may eventually come to settle on a better one, when it turns out we no longer need the abstractions of platonic heaven.

Answer 2

I think that it depends on our analysis of existence. There are at least two types of analysis.

First, we can linguistically analyze existence. In other words, we can explain what the word "existence" means. For example, a linguistic analysis of water might yield: "water" means "clear, odorless, tasteless liquid that falls from the sky and nourishes our body".

Perhaps "existence" expresses a simple concept, i.e. a concept not composed of any other concepts. If so, then it's conceptually coherent to say that non-spatiotemporal objects exist, and so to ask whether non-spatiotemporal triangles exist.

Of course, if "existence" means "being spatiotemporal", then it's conceptually incoherent to posit the existence of non-spatiotemporal triangles. But we shouldn't take this linguistic analysis for granted, I think. It needs much more argumentation.

Second, we can metaphysically analyze existence. In other words, we can explain what existence is in more incisive terms. For example, a metaphysical analysis of water might yield: water is just H2O, which in turn is just a composite of prime matter and a certain substantial form.

Philosophers have offered metaphysical analyses of existence, such as:

1. to exist = to be spatiotemporal
2. to exist = to resemble God in some respect
3. to exist = to have some character or nature
4. to exist = to be a Form or participate in one
5. to exist = to have causal dispositions or powers
6. existence can't be incisively identified in other terms

If (1) is correct, then it's not metaphysically possible for non-spatiotemporal triangles to exist. However, (2)--(6) don't immediately rule out the existence of non-spatiotemporal triangles. And we can't take (1) for granted; (1) needs to be argued for.

Answer 3

First of all, let's be clear that the existence of physical objects is not all that solid. The reason you think physical objects exist is because you perceive them, but you often perceive things that don't exist in dreams, in movies, in illusions, and in imagination (if you have a vivid enough imagination). Many people experience hallucinations which is another way of perceiving things that don't exist. Furthermore, there is the Matrix scenario in which everything you perceive could be false (note, the point of this scenario is not that it might be true, but that if it were true, you wouldn't know it and everything you perceive would be fake).

In addition, there is the Ship of Theseus paradox, which I'll give in a modified form. Suppose you take a car apart into many pieces and then re-assemble it in Dean Wormer's office. Most people would say that it is still the same car. Now imagine a different scenario. You have a car, A, in your garage and you replace the tires. Is it still the same car? Now you take off all the body panels and replace them; still the same car? Replace the engine, the drive train, the braking system, he axles, the frame--all one piece at a time so that most people would say it's still the same car (yes, I know that legally, the frame is the car, but this isn't about legalities; it's about intuitions, if you want to get picky about that, make it a car with no frame).

If you did this replacement in small-enough pieces, most people would say that at each step it's still the same car. If you deny this, then you need to explain when the initial car, A, changes to a different car, B. At what part does A become B? There is no principled non-arbitrary point at which this happens.

If you accept that A is the same car as B, then you have another problem. I secretly saved all the parts and now I assemble them into another car C. We already decided from the Dean Wormer example that A and C are the same car, so if you say that A and B are the same car, then B and C are the same car--two distinct objects existing at the same time that are the same object.

What this example shows is that identity of physical objects is not as simple and clear an idea as we intuitively think it is. But if something exists, then there should be clear criteria for saying when two of those things are the same object. That doesn't seem to be the case for physical objects. To a large extent, the existence of physical objects seems to be merely a matter of human psychology: we have perceptions, and part of the perception is the intuitive conviction that the thing we perceive exists, yet we know that sometimes that conviction is wrong, so why should we ever think it is right?

Now, onto abstract objects like numbers. Unlike physical objects, there are some pretty clear criteria for saying when two numbers are the same, and that's an important marker for existence. If x is the same as y but neither x nor y exists, what does it mean that they are the same?

Also, abstract objects, like physical objects, are objective (hence the name "object"). If numbers were just part of a mental state, there would be no reason to think that my mental idea of three would have anything to do with your mental idea of three. But instead, numbers are something we can share. When we talk about three, we are talking about the same thing, and how can this be unless three exists?

There is also an important contrast between abstract objects and fictional objects. Fictional objects don't exist. Why? Well, in part because there is nothing that is true about them beyond what we say is true. You may think John McClaine was a good guy saving the occupants of Nakatomi Plaza from death, but I think he was actually trying to rob the place himself and had to stop the other robbers so he could rob it. There is no objective way to settle this dispute because John McClaine doesn't really exist. By contrast, if we have a disagreement about numbers, in almost all cases, one of us is right and one of us is wrong. How could this be the case if numbers are fictions?

Furthermore, there are true statements in mathematics about whether something exists. There exists a least prime number. There exists an integer solution to a certain polynomial, etc. What does "there exists" in mathematics mean if not that something exists?

If physical reality were completely unproblematic then there might be a good argument for taking physical reality as the criterion for existence, but physical reality is not unproblematic. It is at least as problematic as the existence of abstract objects (and in the case of identity, more problematic). Given this, it is hard to justify saying that physical objects exist and mathematical objects don't exist.

Answer 4

The identification of mathematical realism with mathematical platonism confirms the notion that, ”If mathematical objects exist (in any sense of exist), it must be a platonic existence”. That’s because triangles can’t exist physically like you said. And since they lack time and spatial extent by not being physical, they are platonic.

With that notion, when we ask, how or do triangles exist, we are thus also asking: “are there platonic objects”? That’s why we can keep asking whether triangles exist. It’s because we want to know if platonic objects exist in any sense—are there platonic objects.

Answer 5

Welcome to physicalism.

You touched on the reason physicalists are physicalists. We have no coherent sense of existence beyond existence in space and time. Our conceptualisation of existence is heavily tied to space and time. We are, after all, spatial-temporal beings ourselves, and all we can observe is a spatial-temporal world. To say that something exists nowhere at no time is incoherent and self-contradictory. We have no frame of reference to make sense of what that even means.

Any claim that such things exist would, at best, require an entirely different definition of existence, which in no way resembles how we currently understand and define existence.

* This isn't a firm assertion that no such thing exists. But it is saying that humans cannot have justified belief in any such thing.

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you often perceive things that don't exist in dreams, in movies, in illusions, and in imagination

Right, things that "don't exist"... don't exist. Why is "things that don't exist" the start of an answer trying to argue that things DO exist?

the Ship of Theseus paradox

The Ship of Theseus paradox demonstrates that our concept of identity is flawed (not that our concept of existence is flawed). This is because atoms (and other physical parts) don't have identity markers, but people treat them as if they do. We look at some clumped-up group of atoms, and we label that as "Ship of Theseus". But then after replacing every atom, we want to keep the "Ship of Theseus" label for the same group of atoms, even though none of the original atoms remain. And the original atoms are now in a new group, which seems like it should also/instead have the "Ship of Theseus" label. None of this suggests that atoms don't actually exist. It just suggests that our labelling is flawed.

if something exists, then there should be clear criteria for saying when two of those things are the same object

No.

What possible basis could you even have for saying this, when all that we can reasonably say exists are physical things, which directly violates this inference (as per the Ship of Theseus paradox)?

if something exists, then there should be clear criteria for saying when two of those things are the same object. That doesn't seem to be the case for physical objects

You make an entirely-unjustified and counter-factual presupposition, and your desired conclusions follow directly from that presupposition. This is basically just presupposing one's conclusion.

there are some pretty clear criteria for saying when two numbers are the same, and that's an important marker for existence

This also follows from the entirely-unjustified and counter-factual presupposition. If anything, the fact that this is so clear for numbers suggests that they don't exist.

If x is the same as y but neither x nor y exists, what does it mean that they are the same?

x and y both reference the same concept in your mind. I can describe a unicorn in 2 ways, both reference the same concept of a unicorn. But that doesn't mean a unicorn exists. I can reference a fictional character in multiple ways, referencing the same fictional character, but that doesn't mean said fictional character exists.

Maybe one can say that the concept of the fictional character exists, but what does that even mean? We're just back to the original problem, except not quite. If I reference the number 3 in the same way I reference a fictional character, and in the same way I reference a real object, we've established that I can reference something that may or may not exist. This strongly undermines the claim that number concepts objectively exist from the fact that references to them exist.

* Certainly there might exist drawings and such of a fictional character. But that doesn't mean the concept itself is a thing that exists.

If numbers were just part of a mental state, there would be no reason to think that my mental idea of three would have anything to do with your mental idea of three

Incorrect. We can both perceive a collection of 3 objects in the real world. With that, we can both create a mental concept of the number 3. We have good reason to think that's the same concept because we can both point to that collection and say "there are 3 objects".

For a mathematical system more generally, it follows from axioms (which have referents in the real world) and logic rules, which gives us reason to think we share mental concepts of that mathematical system as a whole.

if we have a disagreement about numbers, in almost all cases, one of us is right and one of us is wrong. How could this be the case if numbers are fictions?

Because a mathematical system follows logic rules. We can say something is wrong when it violates those rules. If we have if A then B, and someone says not B and A, then we can say that violates logic rules regardless of whether A and B actually exist.

There exists an integer solution to a certain polynomial, etc. What does "there exists" in mathematics mean if not that something exists?

It means there is a value or concept that is consistent with our logic rules.

You may think John McClaine was a good guy saving the occupants of Nakatomi Plaza from death, but I think he was actually trying to rob the place himself and had to stop the other robbers so he could rob it. There is no objective way to settle this dispute because John McClaine doesn't really exist

We can't settle that dispute not because John McClaine doesn't exist, but because unjustified claims about someone's intent are unfalsifiable. This is true whether we're talking about a real or fictional person. People often make such claims about real people. For example, group A says "don't bomb children" and group B responds by accusing group A of just saying that because they're antisemitic or pro-Hamas.

But we can objectively settle the dispute of HOW John McClaine saved the occupants of Nakatomi Plaza, in said fictional story, because we can check the reference material for that story. At least we can "objectively settle" that in as far as we can objectively settle what happened in historical events, for which we have a similar amount of reference material.

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In conclusion, we still have no good reason to think concepts exist. That's still incoherent and self-contradictory.

No space magic is required to make sense of us being able to do math.

Answer 6

I think I have been drinking too much this week and my usually hyper-analytical mind has gotten a bit softened. Nonetheless I will try to put forward a couple assertions, reframe the problem in my own terms, and see if I can bring the question forward by identifying, "What is actually the problem?"

First of all, let's begin with the naive (the etymology of this word is related to the words 'natural' and 'native' rather than connoting 'foolish') point of view that what exists is what occurs in space and time. Without needing to study advanced modern physics, I can still describe my native conception of reality, which is something like 3 dimensions of space and 1 of time, and that there are things inside space. Maybe we can call this existence-1. I claim that there are things, you can call them particles or pieces of matter, and they are endowed with certain properties like mass, charge, spin, position, volume, etc. If we can imagine a set of all conceivable things, we would say a thing has the property exists-1 if and only if it is in the set of particles just described. Therefore, the only thing that exists-1 are these particles. The entire course of the history of the universe, human history, etc., is just a sequence of descriptions of the state of those particles: at any given instant, what their location, energy, force, velocity, acceleration, etc. was. And maybe those particles can also pop in and out of existence during the course of that history.

In this picture of the world, in what sense does my bank account exist? Of course I am inclined to say I really do have a bank account, at the International Bank of Money; it's really there. Well, what is my bank account? Is it made of particles? I think the mental notion I have of my bank account is like a data structure or schema I have in my mind. When I think of my bank account, I can tell you the routing number, the current balance, the date it was opened, which mobile app to download to interact with it, the logo and color scheme of my bank, and so on. Just as this information can be encoded and stored in computer memory, and it can be retrieved or modified by actions on some stateful system (a computer), information can also be encoded in the neural system of my mind, and the stateful system of my brain can be acted on and changed to encode new information, retrieve information that's already there, update pre-existing information, etc. This all corresponds to changes in the stateful properties of the particles, which are the only thing that exists-1. So, I still don't see that there's any problem. When I walk up to a physical location of my bank branch, particles (photons) from the physical mass of that bank (walls, windows, posters, cloth overhangs, which are all made of particles) travel across space into my eyes. My eyes are also made of particles. The photon particles make contact with particles that are in the optic nerve inside my eye. The particles in the optic nerve send other particles - electrons - traveling in a certain direction. When those particles come to a neuron, also made of particles, that neuron is able to trigger more electrons (particles) to travel in a certain direction; and so on. The result is that the system of my mind experiences an image, encodes it in neurons and synapses, etc. So "my bank account" - at least, the concept that I think of when I contemplate those words - is something like a pattern of information encoded in the configuration of particles in my brain.

A lot of the time in philosophical inquiry, I find that the goal is to problematize something. I don't know off the top of my head what problems there are that lead us to need to debate if triangles really exist or not, but I can try to brainstorm some. I think the key question here is, "What is at stake?" If triangles do or do not exist, what are downstream problems that will be affected by this thesis?

I have been studying a little bit of Basic Formal Ontology, and I might try to bring in some concepts from there.

In Basic Formal Ontology, it appears that they acknowledge the distinction between classes of things and instantiations of those classes. As of right now, I think of this as a result of how humans cognize the world, and not at all something inherent to the world itself. The world-engine itself (the system defining and simulating the world) doesn't know what chairs are (it doesn't need the definition of a chair to be able to operate as it currently does). The "theory of the world" only acknowledges primitive entities like subatomic particles in its 'theory'.

If we are to claim that a triangle is a "thing", it seems pretty natural to claim that it also carries the "thing-ness" property of having both a class form and an instance form. Instances tend to be "flesh-and-blood", to speak metaphorically: there is some domain that we emphasize is where "real existence" occurs; instances actually pop up there; that is their locus. Classes do not. Classes are often something above or separate from the plane of "actual performative reality".

We might say the class of all triangles is defined by some properties that every instance of a triangle shares. If we want to use set theory (which I think brings in plenty more weird ontological questions like "what actually are sets?"), maybe we can define 2-dimensional space, then the notion of a line, then talk about how a triangle is more or less the set of points that are amongst the points constituting the line segments that are respectively defined as the part of a line between the intersection points of 2 other lines. So, in pure set theory, what's a triangle? I think a problem is that pure set theory is ontologically dissatisfying. It claims the only thing that exists are "abstract sets", which are "collections of things", except the only things they can be collections of are other such "abstract collections". So maybe this is one small, winsome realization: when we try to formally define any mathematical object, is it still making use of a formal language which still asserts the existence of certain ontological primitives, at the start? I know that the older, original versions of set theory actually assumed the existence of "urelements", which I find a lot more philosophically satisfying: sets are collections of things, and even if we don't need to say what things, there is always the opportunity to look and see what actual thing we are talking about, in any given situation.

Maybe a next point we can consider is, even if we assert certain ontological primitives, like that the world is made of fundamental-element-1, or fundamental-element-2, we might argue that the abstractness of set theory and math makes it clear that whatever a triangle is, it is not going to be bound to a specific ontological realization. It will probably be just as valid regardless of the things it is made of. So here, we encounter the idea that "there is an abstract thing called 'structure'".

In Basic Formal Ontology, there is a notion of a dependent entity. A dependent entity is a thing that does not exist on its own. It requires a host. This also leads to some hard philosophical questions. Is "redness" a thing, in and of itself, or do we only ever regard things which have the property of being red? Maybe there are some examples which make the difference more apparent: an object has a mass (a weight), but we cannot have mass on its own. There is only mass if there is some thing, endowed with true thing-in-itselfness, which is the bearer of that property of having such-and-such mass.

If a triangle is a dependent entity, then it is actually a property of some actually-existing-in-the-world things.

Even if this is the case, I am still not flummoxed by the question "Do mathematical entities exist inherently?" I think when we talk about whether certain mathematical objects exist or not, there are examples of this where sure, it definitely 'matters'. Is there a general formula to solve any quintic polynomial equation? There is not. Such a formula doesn't exist. But maybe we can call this existence-2. Perhaps this is structural existence, or something. A structure requires a host to be realized in - something with real-world ontological existence. But we can still mentally conceive of structures, and talk about which structures are logically feasible or not.

This is pretty amateur and probably superficial, but it's what I came up with today as I continue to engage with this question.

Answer 7

We could try the following heuristic to decide existence (by nature, hand wavy): Something exists, if it is

(a) reported by several, preferably geographically or culturally separated sources/groups

(b) in a way that hints at uniform characteristics.

In this way, a 19th century explorer might become convinced of the existence of a certain mountain or disease on reports of several distinct tribes, or an astronomer of the existence of a comet that had a close fly-by in ancient times. On the above criterion, the existence of the objects and relations of mathematics is exceptionally compelling: multiple civilizations seem to have discovered the exact same objects (e.g., triangles) with the exact same properties (e.g., Pythagoras Theorem).

(Not a philosophical answer)

Answer 8

As most words in every day language, the verb "to exist" has no clearly defined meaning. It is "defined" by a set of examples and counter-examples, which means that there a lot of edge cases where people do not agree whether the "object" exists or does not exist.

Mathematical objects, especially yet undiscovered ones, fall into this category. This is more a linguistic problem than a philosophical one.

Answer 9

Clearly the abstract concept of a triangle does not exist in the same way as a specific triangular wedge of cheese does, and if you are happy to leave it at that there is no problem- you are simply recognising that the word “exists” has a range of meanings.