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Objectivist Mereology

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So I wonder what, if any, mereology would be implied by Objectivism. Here are a few questions:

Is space-time isomorphic with matter? Are they discrete or continuous?

Is there unrestricted compositionality? I'm pretty sure an Objectivist would think this is mere logical positivism.

Can you have collocated matter in a single space? Can you have overlapping matter in a single space?

Is matter gunky?

Is parthood a transitive relationship, and is it extensional?

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I posted on some related issues pertaining to composition and parthood ages and ages ago - don't waste your time. They think these are scientific questions.

Edited by JMeganSnow

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They are scientific questions--there's no way to determine the answer to any of them without extensive scientific knowledge and experimentation. Hell, you can't even *define most of those terms* without extensive scientific knowledge and experimentation.

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They are scientific questions--there's no way to determine the answer to any of them without extensive scientific knowledge and experimentation. Hell, you can't even *define most of those terms* without extensive scientific knowledge and experimentation.

Mereology in general and the question of unrestricted composition in particular is not a scientific concern. Science will never answer whether an object such that it is the sum of the Queen of England, the Washington Monument and the quarter in my pocket exists. It's an issue of conceptual analysis, how we understand parthood and composition, what sorts of combinations of objects can legitimately be said to exist.

For example, I'm holding my quarter. Does this quarter exist? Sure. Even though that quarter is the mereological sum of a bunch of metallic particles. So how about the funky object I described above? We want to say it doesn't exist because it seems like an arbitrary grouping of objects. But how is it any less arbitrary than the group of metallic particles that constitute the quarter?

The issue goes a lot deeper on both sides of the unrestricted composition debate, but it should be clear that this isn't a problem for SCIENCE! to answer for us.

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I actually agree to some degree, although some are (somewhat obviously) not questions of science, like unrestricted compositionality. That might be a subject of linguistics, perhaps of evolution, but I doubt it.

As for things like isomorphism between matter and space-time, I suspect this is a question of science. It seems like the very term "space-time" is a term in our formal system which describes physical existence or attempts to predict events. I don't think space-time is a concept independent of our creating, and we created it because it seems to fit a lot of things we want fitted. That suggests that there's some "thing" or feature of reality which is isomorphic to it, though, which might mean that there is some such feature of the world which is likewise continuous. (Although doesn't the concept of Plank time imply a smallest measure of time? Would this then suggest a discrete representation of space-time?)

All the same, though, take this hypothetical: Two open spheres approach each other at a constant rate of speed, and exert no repulsive forces on each other (except, perhaps, exchange of force if there is a collision). Do they ever touch?

If yes, then you must assert one of the following:

A point is created out of nowhere to accomodate their touching (because they are open spheres and so do not have any boundary points).

A repellant force is created out of nowhere.

They pass through each other, in which case you have collocation of objects.

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All the same, though, take this hypothetical: Two open spheres approach each other at a constant rate of speed, and exert no repulsive forces on each other (except, perhaps, exchange of force if there is a collision). Do they ever touch?

If yes, then you must assert one of the following:

A point is created out of nowhere to accomodate their touching (because they are open spheres and so do not have any boundary points).

A repellant force is created out of nowhere.

They pass through each other, in which case you have collocation of objects.

This is pure rationalism. How can an open sphere be said to exist at all? With no boundary points, it is an indefinite figure that can be asserted to do anything at all with no risk of contradiction.

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I suspect it's rationalism (or something like it). But I don't think it's metaphysically impossible for an open sphere to exist. What would rule out a table being an open sphere of point-sized atoms ("atoms" in the mereological sense)?

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I suspect it's rationalism (or something like it). But I don't think it's metaphysically impossible for an open sphere to exist. What would rule out a table being an open sphere of point-sized atoms ("atoms" in the mereological sense)?

The Law of Identity. To be is be something definite. A sphere is definite kind of shape, but your open sphere lacks any shape. How could it be said to be a sphere, or to be touching anything if its size is indefinite, or to have any relationship at all with anything? How could one find or define the center of an open sphere?

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An open sphere is an open sphere. How does infinity violate the law of identity, when it is defined appropriately? It has a shape defined by a precise equation over the real numbers. It just cannot touch--but then, we never touch tables or rocks, since we only have repuslive forces between the atoms that keep our hands from passing through solid objects. Pretty neat question, huh?

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An open sphere is an open sphere. How does infinity violate the law of identity, when it is defined appropriately? It has a shape defined by a precise equation over the real numbers. It just cannot touch--but then, we never touch tables or rocks, since we only have repuslive forces between the atoms that keep our hands from passing through solid objects. Pretty neat question, huh?

By definition one cannot define the indefinite. A precise equation describing a hypothetical sphere if it existed is not the same as an actual thing that exists. Just as the meaning of a concept is its referent not its definition, having a definition does not evoke an existent into reality.

We do touch tables and rocks, the repulsive forces between atoms are not identical with the macroscopic entities we interact with. Is using the fallacy of composition and reductionism what mereology is about?

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An open sphere is an open sphere. How does infinity violate the law of identity, when it is defined appropriately? It has a shape defined by a precise equation over the real numbers. It just cannot touch--but then, we never touch tables or rocks, since we only have repuslive forces between the atoms that keep our hands from passing through solid objects. Pretty neat question, huh?

By "open sphere" I take it you mean here a region in three dimensional space of the form {x : |x - a| < r} filled with some homogeneous material?

To talk about such an object, you would need to verify that it has no boundary. To do so would require infinitely precise measurements to demonstrate that your object is really open since one would need more and more precise units of measurement to verify that the sphere consists of material any arbitrary distance less than r away from a, but not exactly equal. It's that "exactly" that gets you into trouble. Since you can never actually verify that it has no boundary it's not a useful thing to consider.

Put differently, other poster here are correct in asserting that an open sphere cannot literally exist as a physical object in reality-- that it has a well defined mathematical description is neither here nor there, since objects like open spheres are concepts of method. Since the law of identity prohibits infinitely precise physical measurements it is inappropriate to use these open spheres rationalistically to gain knowledge about reality. You can't derive knowledge about how things touch from pure mathematics alone without reference to reality.

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By definition one cannot define the indefinite. A precise equation describing a hypothetical sphere if it existed is not the same as an actual thing that exists. Just as the meaning of a concept is its referent not its definition, having a definition does not evoke an existent into reality.

We do touch tables and rocks, the repulsive forces between atoms are not identical with the macroscopic entities we interact with. Is using the fallacy of composition and reductionism what mereology is about?

One can define infinite quite easily: That which is not finite. 'Finite' is perfectly defined, 'not' is a basic logical operator, and their composition is perfectly intelligible and clear. A precise equation, with range and domain over the real numbers, is perfectly defined at every point--I don't know what more you could want, if you are asking for the definition of a thing.

But then you say something about "having a definition does not evoke an existent into reality," which seems like a complete nonsequitor. I never said open spheres exist--I just said they are definable, intelligible, and metaphysically possible. My point is that you don't know they don't exist.

By "open sphere" I take it you mean here a region in three dimensional space of the form {x : |x - a| < r} filled with some homogeneous material?

I'm not sure it has to be homogeneous, but I see no reason why it couldn't. More importantly, though, I don't mean just a region of space-time but an object. (So the notion of "filling" the space is not quite the point.)

To talk about such an object, you would need to verify that it has no boundary. To do so would require infinitely precise measurements to demonstrate that your object is really open since one would need more and more precise units of measurement to verify that the sphere consists of material any arbitrary distance less than r away from a, but not exactly equal. It's that "exactly" that gets you into trouble. Since you can never actually verify that it has no boundary it's not a useful thing to consider.

I'm not so sure. I could very well imagine that an object the boundary of which is discrete should move differently than an object the boundary of which is continuous, and that our theory would be able to predict which behavior belongs to which. So we would not need to have infinitely precise measurements, but merely some way to distinguish what we should expect from physical action involving continuous objects and what we should expect from discrete ones.

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So I wonder what, if any, mereology would be implied by Objectivism. Here are a few questions:

Is space-time isomorphic with matter? Are they discrete or continuous?

I'm going to assume by "matter" you mean "objects", if not please correct. Space cannot be the same as objects because space is conceptually the antithesis of an object, of a thing. Space is an absence, a 0. 0 is not a number, it is strictly a placeholder. An object is a something, a 1. Putting "time" next to the word space does not change this, since time can only be the observation of relative motion, not a thing itself.

Are the fundamental constituents discrete or continuous? Are you asking if the universe is composed of disconnected, separate entities or of interconnected ones?

In this regard it seems the empirical evidence is pretty clear that entities are continuously connected to each other. In particular I have not heard of a rational explanation for most phenomena of light in terms of the discrete corpuscular hypothesis.

Can you have collocated matter in a single space? Can you have overlapping matter in a single space?

In this regard it seems that Nature has indicated the affirmative. Whatever entity is responsible for light does not seem to interact with itself, even if it is colocal. Whatever entity is responsible for magnetism seems also to pass right through other entities, although it does seem to interact with them at least. The most compelling phenomenon of Nature is, I think, that light does not seem to interact with itself.

Is matter gunky?

What do you mean?

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@aleph_0:

I'm not sure it has to be homogeneous, but I see no reason why it couldn't.

True, this would be a simplifying assumption, but I thought I'd assume it without loss of generality.

More importantly, though, I don't mean just a region of space-time but an object. (So the notion of "filling" the space is not quite the point.)

Yes, my description was trying to get at an object composed of some material whose shape was given by the description in my last post. If that isn't what you meant by "open sphere" I'd need to know what you meant by such a thing.

I'm not so sure. I could very well imagine that an object the boundary of which is discrete should move differently than an object the boundary of which is continuous, and that our theory would be able to predict which behavior belongs to which. So we would not need to have infinitely precise measurements, but merely some way to distinguish what we should expect from physical action involving continuous objects and what we should expect from discrete ones.

I need you to define your terms before I can comment. What distinguishes "discrete" boundaries from "continuous" boundaries? Even from a purely mathematical standpoint, the boundary of a set is determined by the set, and doesn't obey special rules as the set changes in time-- if you want to stipulate special rules, fine, but unless you're modeling observed phenomenon you're really just making stuff up, and you aren't really saying things about physical objects.

In any case, my broader point was to object to reasoning about physical objects by assuming that they have some kind of reified mathematical boundary (discrete or continuous) and deriving case conclusions as you did earlier in this thread. Moreover, I believe that speaking about "continuous objects" is vulnerable to the same kinds of objections as "exact measurement" is.

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Mereology in general and the question of unrestricted composition in particular is not a scientific concern. Science will never answer whether an object such that it is the sum of the Queen of England, the Washington Monument and the quarter in my pocket exists. It's an issue of conceptual analysis, how we understand parthood and composition, what sorts of combinations of objects can legitimately be said to exist.

Science will not answer this question because it is fallacious to attempt to verify whether this or that entity exists. An entity exists independent of your belief or attempt at verification.

Additionally groups, groupings, and group hierarchies are conceptual, and concepts are not objects. Each object is an object, you might point to each one, but the association you have between them is entirely conceptual. A grouping or listing of objects is not an object (except trivially insofar as the symbols on the page which refer to those objects and the page itself are, themselves, objects).

It is irrational to talk about combinations of objects "existing". An object is an object is an object. Each entity exists. A combination, group, or listing objects is a list of symbols enumerating entities that exist. You might, conceptually, associate one with another but this concept is of course not an entity. It also makes no sense to talk about "what can be said to legitimately exist". Existence is self-evident and axiomatic. There is no provision for proof, verification, or testimony. I point at something, I recognize that it exists.

For example, I'm holding my quarter. Does this quarter exist? Sure. Even though that quarter is the mereological sum of a bunch of metallic particles. So how about the funky object I described above? We want to say it doesn't exist because it seems like an arbitrary grouping of objects. But how is it any less arbitrary than the group of metallic particles that constitute the quarter?

The problem is you went from simply pointing at the quarter and naming it to describing it. A description is a conceptualization. The moment you point at it and say "quarter" it is an object. Then when you say it is shiny, round, etc. you are conceptualizing it. You're dealing with the *concept* "quarter". When you say it is made of atoms this is a description, and you must first tell us what an atom is, i.e. you will have to point at an atom or a model of an atom. If you can point at an actual atom then now we realize that the quarter was a concept integrated by our brain by the perception of multiple entities, which our brain automatically subconsciously grouped. The atom is an entity, the quarter is a concept. If you cannot point at what you're talking about (an atom), but only a model of it, you're now asking us to assume something like what you're pointing at exists in reality. You will not, and indeed cannot, verify this assumption. This is a hypothesis, an assumption you ask us to take at face value in order for you to explain some phenomenon involving the quarter. At the hypothesis you do not describe the object. You just point to it and name it. Why does it break, bend, etc? When you're done we can choose to believe your explanation, thus believing in the existence of atoms, or we can choose to disbelieve it.

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I'm going to assume by "matter" you mean "objects", if not please correct.

Good distinction, and for now I'm going to go with "matter", though it may be interesting to later consider objects.

Space cannot be the same as objects

You're right, but that's not the claim.

Are the fundamental constituents discrete or continuous? Are you asking if the universe is composed of disconnected, separate entities or of interconnected ones?

Not about connectedness, but about whether you can have space (or matter, or objects) which can be mapped into 3 (or ℝ4). That is: Can there be an object such that the point-sized constituents of it are isomorphic to the points of the graph of x2 + y2 + z2 = 1? If not, then the objects must be mapped onto ℚ, which would be weird, or ℕ.

What do you mean [by “is matter gunky”]?

“Gunk” is a technical term meaning non-atomistic. I.e. infinitely decomposable.

In any case, my broader point was to object to reasoning about physical objects by assuming that they have some kind of reified mathematical boundary (discrete or continuous) and deriving case conclusions as you did earlier in this thread. Moreover, I believe that speaking about "continuous objects" is vulnerable to the same kinds of objections as "exact measurement" is.

I agree, perhaps, with the larger point. (I think smaller points have been addressed above.) My position is that this question employs topological or geometric concepts in order to settle questions about physics or kinematics. That’s just folly. Geometry does not tell you what happens when things interact, it just describes surfaces and solids in a mathematical framework which is supposed to have useful applications to the real world (but not necessarily imply any perfect isomorphism to the real world).

The point against this, though, is that we want to consider what is metaphysically possible, not just physically possible. And so we want an answer to whether we can, by conceptual analysis, rule out the possibility of an object being an open surface, and if not, then what should happen when they approach each other at a constant rate of speed with no repulsive forces? But again, I take the question to be confused, because the concepts that we are using to describe the situation do not give us an answer, so conceptual analysis cannot give an answer. If you build into the situation that it is an open sphere which cannot be collocated, then you are on your way to an answer (vacuously). And so on.

Additionally groups, groupings, and group hierarchies are conceptual, and concepts are not objects.

So a group of pencils is a concept and not an object? Sure, the formation of groups as a mental act may be conceptual, but the groups themselves are just stuff.

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The point against this, though, is that we want to consider what is metaphysically possible, not just physically possible.

I don't understand the distinction-- what is an example of a metaphysically possible object which is not physically possible? This kind of thing usually leads to an analytic/synthetic type of distinction, which I don't accept.

And so we want an answer to whether we can, by conceptual analysis, rule out the possibility of an object being an open surface, and if not, then what should happen when they approach each other at a constant rate of speed with no repulsive forces?

The danger of considering objects introduced by stipulating certain properties (such as a physical open sphere, or a unicorn) is that you may in fact be talking about the empty set. You must show that some objects of the kind you consider actually exist, or you aren't really taking about anything at all. Conceptual analysis is pointless if you don't know whether your concepts have referents.

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My position is that this question employs topological or geometric concepts in order to settle questions about physics or kinematics. That’s just folly. Geometry does not tell you what happens when things interact, it just describes surfaces and solids in a mathematical framework which is supposed to have useful applications to the real world (but not necessarily imply any perfect isomorphism to the real world).

The point against this, though, is that we want to consider what is metaphysically possible, not just physically possible. And so we want an answer to whether we can, by conceptual analysis, rule out the possibility of an object being an open surface, and if not, then what should happen when they approach each other at a constant rate of speed with no repulsive forces? But again, I take the question to be confused, because the concepts that we are using to describe the situation do not give us an answer, so conceptual analysis cannot give an answer. If you build into the situation that it is an open sphere which cannot be collocated, then you are on your way to an answer (vacuously). And so on.

"What is an Entity? A Topological Definition"

http://progressofliberty.today.com/2008/11...t-is-an-entity/

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"What is an Entity? A Topological Definition"

http://progressofliberty.today.com/2008/11...t-is-an-entity/

This makes the same error, I think. In order to show that entities really fit this description one would have to show that it is "closed", which requires showing that it contains its limit points, requiring infinitely precise measurement. Moreover, it allows aleph_0 to object (correctly) that something like a table is in fact a collection of disconnected atoms and so leads to the absurd conclusion that a table is not an entity. Requiring entities to be compact and path connected would effectively involve omniscience about certain aspects of an entity.

The problem behind both of these is that entities and concepts like the boundary of an entity are contextual. A table has a well defined boundary at the macroscopic level of human sense perception, but not so much at the atomic level, and that's OK provided we define our terms correctly.

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I don't understand the distinction-- what is an example of a metaphysically possible object which is not physically possible? This kind of thing usually leads to an analytic/synthetic type of distinction, which I don't accept.

I do accept the distinction, and it probably does ultimately hinge on it, but you presumably take the distinction between metaphysics and physics to be a genuine one (otherwise you would be in the position of thinking that the metaphysics forum should be full of physics formulae and experimental data, which I doubt you want). Take the ancient Greek who was a naive realist. It was at least metaphysically possible that objects existed as he knew them. It was only through much experimentation that we realized that objects were these very complicated bodies that have many hidden features (atoms separated by great distances, exerting repulsive forces, etc.). So while those naively conceived objects are physically impossible, they are metaphysically possible.

Again, the contest between the caloric theory and the mechanical theory of heat was won not by proving what was metaphysically impossible, but by experimentation, and so caloric theory is still metaphysically possible. It is for this reason that scientists were ever able to describe the caloric theory, and to make predictions about what would be true if heat were caloric rather than mechanical. And it is through knowledge of what caloric theory implies (even though the theory describes, as you call it, the null set) that we were able to disprove its reality.

In order to show that entities really fit this description one would have to show that it is "closed", which requires showing that it contains its limit points, requiring infinitely precise measurement.

I don’t see why you insist on this. You don’t need to see each atom of a thing to know that it’s made up of atoms. In fact, it is impossible to see an atom, regardless of the technology you use to visualize it. You must deduce their existence through a theory which implies them.

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This is pure rationalism. How can an open sphere be said to exist at all? With no boundary points, it is an indefinite figure that can be asserted to do anything at all with no risk of contradiction.

I agree. An equation, set of equations, or a word are themselves just symbols without referents to reality. In order to avoid rationalism and absurdity we must start with observation as our first premise. Every entity we observe has shape, i.e. a boundary. We can, then, point at it or at least visualize it.

I suspect it's rationalism (or something like it). But I don't think it's metaphysically impossible for an open sphere to exist. What would rule out a table being an open sphere of point-sized atoms ("atoms" in the mereological sense)?

It is irrational to talk about whether this or that entity exists, or to "verify" it. What kind of sense can it make to verify whether this chair exists?

On the other hand, if you are asking how best to describe an entity, this is a reasonable question. An entity may be best described as round, flat, open, etc. for the purposes of quantitative accuracy or comparison. The entity exists or not. Just because you decide to describe it in a different way does not verify (or fail to verify) its existence.

The Law of Identity. To be is be something definite. A sphere is definite kind of shape, but your open sphere lacks any shape. How could it be said to be a sphere, or to be touching anything if its size is indefinite, or to have any relationship at all with anything? How could one find or define the center of an open sphere?

Agreed. Shape is the most primitive and essential quality of an entity. Without shape we have... nothing. Shape is primitive (i.e. undefinable and self-evident). Similar to how there is no alternative to existence, there is no alternative to shape. To talk about shapeless is to talk about nothing, yet talking about X (or performing any action on X) implies X is a thing, thus invoking a contradiction. Just as talking about non-existence is pointless and absurd, because talking about anything demands existence.

An open sphere is an open sphere.

This is a restatement of identity. Saying something is itself i.e. A=A is an axiom. You are using it to prove that A can have an existent referent. This is a meaningless tautology.

How does infinity violate the law of identity, when it is defined appropriately?

Infinity violates identity and rationality. An object cannot be infinite because then it would lose its most essential attribute, shape. 'An' infinite object is not an object.

It has a shape defined by a precise equation over the real numbers.

Shape is not defined. At best it is described. Furthermore, shape is a static concept. Equations all express dynamic concepts, i.e. they describe motion. The equation of a mathematical line is not the same as the physical entity called a line. It is an itinerary describing the motion of a physical entity. Objects are what we visualize and/or point to. Equations are a way of describing the motion of objects.

It just cannot touch--but then, we never touch tables or rocks, since we only have repuslive forces between the atoms that keep our hands from passing through solid objects. Pretty neat question, huh?

Yes it is! Touch is a "touchy" issue so to speak. When two fundamental entities come together, close enough so that they are no longer separate, do they remain distinct entities? They appear to, now, have a single surface. i.e. two objects appear to have become one. I've more to say on the topic, hopefully I'll remember to return to it.

By definition one cannot define the indefinite. A precise equation describing a hypothetical sphere if it existed is not the same as an actual thing that exists. Just as the meaning of a concept is its referent not its definition, having a definition does not evoke an existent into reality.

Well said. I can visualize entities, but they do not exist (the ones I'm visualizing). They are entities because they have shape, but they do not exist because they lack location.

We do touch tables and rocks, the repulsive forces between atoms are not identical with the macroscopic entities we interact with. Is using the fallacy of composition and reductionism what mereology is about?

Right, of course we do. The repulsive forces can be interpreted as the blending of the electron shells of atoms, i.e. the surfaces of these shells come so close that they become indistinguishable and now possess a single surface.

To talk about such an object, you would need to verify that it has no boundary.

I do not "verify" that this keyboard has a boundary. What kind of sense can it make to prove that *this* keyboard is an object, or to "verify" that it has (or doesn't have) a boundary?

Since the law of identity prohibits infinitely precise physical measurements it is inappropriate to use these open spheres rationalistically to gain knowledge about reality. You can't derive knowledge about how things touch from pure mathematics alone without reference to reality.

Again, what do physical measurements have to do with whether *this* keyboard exists? If someone says "X exists" s/he simply has to point at it. If s/he cannot then s/he can show us what they are visualizing, and we must assume it exists for the purposes of the ensuing discussion. So the more serious issue is that this entity "open sphere" has not been pointed to or illustrated. The entire discussion is non sequitur because "open sphere" is still a placeholder, an arbitrary set of symbols that refer to a set of equations. These equations do not describe objects but rather the motion of objects, they tell us the relative location of one or more object(s) as they traverse a defined path.

One can define infinite quite easily: That which is not finite.

Fallacy, X cannot be defined as "not Y". A dog is not "not a tree". The issue here is that finite is the only term with relevance to reality. Infinite, as an adjective, is supposed to describe objects (and is often incorrectly used as such), but again 'an' infinite object would lack the most essential quality that makes a thing a thing, shape. So the only interpretation of defining infinite as "not finite" is "shapeless" which, at best, means "infinite" is mean to describe a concept and not an object. Since a concept is a relationship amongst entities the word "infinite" is predicated upon some entities, which must be pointed at or at least visualized first, before "infinite" can take on any meaning.

'Finite' is perfectly defined, 'not' is a basic logical operator, and their composition is perfectly intelligible and clear.

Ted: A is perfectly defined.

Bill: What is it?

Ted: It is not B.

Bill: d'oh, but what IS it? Or what's B?

Ted: B is not A.

Bill: What are they though?

Ted: They're not each other...

A precise equation, with range and domain over the real numbers, is perfectly defined at every point--I don't know what more you could want, if you are asking for the definition of a thing.

Things are not what we define. Things are what we point at and/or visualize. After that we may describe them with an equation or by comparing them to other things. But before we can describe or compare, first we must have the thing before us or firmly visualized in our mind.

Edited by altonhare

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Good distinction, and for now I'm going to go with "matter", though it may be interesting to later consider objects.

What precisely is the distinction between "matter" and "objects"? Could you define exactly what you mean by "matter"?

You're right, but that's not the claim.

So, then, why are you talking about the properties of space at all, i.e. how it is related to matter/objects/whatever? If space is a term indicating 0, nothing, a placeholder, i.e. nonexistent, what sense can it make to talk about its relationship or properties? That which exists may have properties and a relationship to other existents.

“Gunk” is a technical term meaning non-atomistic. I.e. infinitely decomposable.

Ah, you mean is there a fundamental constituent, i.e. an object that is just itself, it is not made of other objects (it cannot be broken). It seems that Identity demands it. If A is made of B is made of C... ad infinitum this would seem to preclude A from having identity, since its identity is dependent upon a neverending chain of identities. So it seems there must be some "smallest entity", which is a single piece, and cannot be broken. At some point, we must have an entity that is not made of other entities, an entity with its own identity that is not a result of the relationship of other entities.

The point against this, though, is that we want to consider what is metaphysically possible, not just physically possible. And so we want an answer to whether we can, by conceptual analysis, rule out the possibility of an object being an open surface, and if not, then what should happen when they approach each other at a constant rate of speed with no repulsive forces?

Mathematics can, at best, tell us if one or more entity(ies) can be accurately described in a specific way we refer to as an "open sphere" or not. Mathematics has no power to answer the question of what an entity IS. Math can only describe.

So a group of pencils is a concept and not an object?

Correct.

Sure, the formation of groups as a mental act may be conceptual, but the groups themselves are just stuff.

What's "stuff"? The latter part of your sentence is meaningless unless we know what "stuff" means.

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Take the ancient Greek who was a naive realist. It was at least metaphysically possible that objects existed as he knew them. It was only through much experimentation that we realized that objects were these very complicated bodies that have many hidden features (atoms separated by great distances, exerting repulsive forces, etc.). So while those naively conceived objects are physically impossible, they are metaphysically possible.

Again, the contest between the caloric theory and the mechanical theory of heat was won not by proving what was metaphysically impossible, but by experimentation, and so caloric theory is still metaphysically possible. It is for this reason that scientists were ever able to describe the caloric theory, and to make predictions about what would be true if heat were caloric rather than mechanical. And it is through knowledge of what caloric theory implies (even though the theory describes, as you call it, the null set) that we were able to disprove its reality.

The ancient Greek who was a naive realist should have realized the limits of what his senses could tell him and not assert things he could not know. But then he would not still be naive, which is required for this explanation to work. If we banish naive observers, the metaphysically possible and the physically possible are identical.

The case against the caloric theory of heat you recapitulate can be construed as a modus tollens argument:

If A, then B

Not B

Therefore Not A

You seem to attribute to the premise "If A, then B" metaphysical possibility simply because it can be understood. But observation (Not B) is the standard of truth and the possible. After observation "Not B" is made, conclusion "Not A" follows and it is wrong to allow A any continued possibility, even though it can still be understood.

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Every entity we observe has shape, i.e. a boundary.

I don't see why a thing loses shape when it doesn't contain its boundary.

It is irrational to talk about whether this or that entity exists, or to "verify" it. What kind of sense can it make to verify whether this chair exists?

I don't understand the point. I can talk about whether a chair exists, and I can verify it by observing a chair. I can also talk about whether there are heat calories (in the sense of caloric theory of heat) and then verify or deny.

Agreed. Shape is the most primitive and essential quality of an entity. Without shape we have... nothing.

My thoughts have no shape. Sub-atomic particles may have no shape.

Infinity violates identity and rationality. An object cannot be infinite because then it would lose its most essential attribute, shape. 'An' infinite object is not an object.

This makes no sense, and either contradicts itself or demonstrates a lack of understanding. The set of points on a sphere defined by x2 + y2 + z2 = 1 is infinite yet has shape. The idea of something being “infinitely infinite”, or “infinite in every way,” makes no sense. But infinity itself does not mean indefinite, or indefinable, or without boundary. It means that, in one respect, there is no limit. The universe may have infinite matter, and this would not be self-contradictory. It may be that you cannot find a last bit of matter, and yet it is defined in all the other important ways—this would not mean that there is some thing that is indefinite, or unintelligible. Matter may be infinitely decomposable and this would not be a contradiction. You may never find a smallest unit of matter, and yet you can observe it, see (or know) space that contains it, and space that doesn’t, it can interact with things in a determinate way, and so on.

I have never understood why Objectivists think that infinity somehow lacks definition, or that things which are infinite in some respect, are undefinable.

Equations all express dynamic concepts, i.e. they describe motion.

That’s patently false. 1 + 1 = 2 describes no motion when you’re telling someone how many rabbits are in a field.

Fallacy, X cannot be defined as "not Y". A dog is not "not a tree".

Why can’t you define something negatively, if its essence is negative? How do you define negation? Sure, you can use examples of things which are not defined by negation—but that doesn’t prove that it is, in principle, forbidden. (Hey, “forbidden”! That which you are not allowed to do!)

Things are not what we define.

Who said they were?

What precisely is the distinction between "matter" and "objects"? Could you define exactly what you mean by "matter"?

I take it “matter” is a loaded technical term in physics, while “object” is just anything that exists.

So, then, why are you talking about the properties of space at all, i.e. how it is related to matter/objects/whatever?

I’m not. I’m asking about the metaphysical possibility of the reality of an open sphere.

Mathematics has no power to answer the question of what an entity IS. Math can only describe.

I (tentatively) agree, and have said nothing to contradict this.

I find it simply ridiculous to say that a group of pencils is not an object, or a fleet of ships, or a lamp (which is just a grouping and configuration of plastic, metal, cloth, glass, etc.).

What's "stuff"? The latter part of your sentence is meaningless unless we know what "stuff" means.

It’s a technical term meaning “junk”. It’s the things you can kick and feel.

The ancient Greek who was a naive realist should have realized the limits of what his senses could tell him and not assert things he could not know.

No doubt, but the point is that he could not disprove his naive realism except with investigation. So naive realism was metaphyisally possible though physically impossible.

But observation (Not ;) is the standard of truth and the possible.

How can it be the standard of what is possible? What does that even mean? It was possible that the caloric theory could have been true, it does not contradict any concept of existence, it just turns out to be physically impossible.

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I take it “matter” is a loaded technical term in physics, while “object” is just anything that exists.

Incorrect your confusing existent with entity.

I’m not. I’m asking about the metaphysical possibility of the reality of an open sphere.

When one defines sphere he finds that "open" would contradict its essential charachteristic shape. Unbounded object is the same as saying a= non a

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