Why do you think that it has been annihilated, rather than just broken? Is it just an exaggeration? Or something else? Although, if it was just an exaggeration, then you wouldn't really mean what you say, and you wouldn't really think that it has been annihilated. — Chief Owl Sapientia

When something is destroyed we can say that it is annihilated. The drinking glass no longer exists, it has been broken, destroyed, therefore I say it has been annihilated. There is no exaggeration. If something comes to its end without being destroyed we might not say it has been annihilated, such as when a human being simply dies we wouldn't say that person was annihilated. But if the human being is blown to bits, like what happens to the drinking glass, we can say that it was annihilated.

Then let's see if we can do that, because I want to understand your point of view, which seems to differ from my own. It seems to me like you might be moving the goalposts by referring to a "drinking glass". The drinking function ceased, but that is just a subject's way of seeing the object. The drinking part is not a part of the object. — Chief Owl Sapientia

"Drinking glass" has a particular meaning, that's why I used it instead of just saying "glass", so there would be no ambiguity. There is no moving the goalpost, because I intentionally used "drinking glass" from the beginning to avoid such ambiguity. A drinking glass is a glass which is used for drinking out of. Annihilate means to destroy. When the drinking glass is broken to bits, why do you not agree that it has been annihilated?

If a certain structure is an essential part of the existence of an object, and that certain structure is destructed, then the object would cease to exist. Is that your thinking? — Chief Owl Sapientia

I don't think it's a case of being a "certain structure", it's a case of fulfilling the conditions of the definition of the word. We have certain expectations of what any word refers to, and it could be a certain structure, but in many cases, like "drinking glass" it is mostly a purpose, a use. If the object no long fulfills the conditions expected of the word, it should no longer be referred to by that word. And since the object was broken to bits, we can say it was annihilated.

But one can identify the object in different ways, so as to identify a particular. We're talking about this cup, not that cup, or any other cup. I could point to it or give it a unique name or specify its time or location with enough precision to differentiate it from others. — Chief Owl Sapientia

In many cases one can point to the object, though right now it would do me no good if you pointed to the object. But what is at issue here is if it is still the same object which you pointed to, after you point to it. Say you point to the object, and this acts as our defining of the object, if we leave and come back later, how do we know that it is the same object we are looking at?

I am only talking about mathematics in this thread, not ontology; maybe you should start your own thread on "Continuity and Ontology." — aletheist

Then what is that continuous thing you are always referring to as a continuum, which you are attempting to understand with mathematics?

I think it would take a global catastrophe, or a long and resisted period of global depression, to actually put population growth into the negative. — VagabondSpectre

Isn't that what we're talking about here, global catastrophe?

Fertilization boosts efficiency but it's not necessary. — VagabondSpectre

Have you ever tried harvesting garden crops off a piece of land for decades with out putting anything back?

It's not so much about description, but about how identity is assigned. If a group of people think of it in the way that you describe above, and assign identity accordingly, then that is the practical meaning of identity for that group of people. — Sapientia

What I said early, is that identity in the sense of the temporal continuity of an object, is something which is simply assumed. It has never been proven that an object continues to exist as itself, through temporal duration, so there are no firm principles whereby we determine that an object is the same object through a period of time. However it is a very useful practise, and that practise is shaped by convention, just like language use.

There can be no annihilation of an object and a rebuilding of it. That is impossible. It would have to be something else which is annihilated, or something else which has ceased. Your example of the glass is not an example of an object being annihilated and rebuilt. The object is just broken, melted and reformed, not annihilated. — Sapientia

As I said, it's a matter of convention. I think that when a drinking glass has been broken into a whole bunch of bits, it has been annihilated, you don't.

Problems like this are the reason why we developed a second sense of identity, which I call logical identity. In this sense, we have a defined term, and as long as the object meets the conditions of the definition, it has been identified accordingly. So we could agree on a definition of "drinking glass" and we could agree on a definition of "annihilate", and determine whether or not the drinking glass has been annihilated.

Logical identity has its own problems though. The object is identified through a definition, so it is more likely that a "type" is identified, and unless the description is extremely thorough, it doesn't properly single out a particular identified object. There could be more than one object which fits the description. Following the Leibniz principle, identity of indiscernibles, some effort has been made to define temporal continuity, such that we could determine logically whether something maintained its identity as the same thing through a period of time, but we have no such principles.

Any attempted solution which priorities one way of speaking and rules out others will always have to compete with the opposing proposed solutions which it rules out, rather than reconcile these proposed solutions under a broader encompassing theory which avoids the kind of problems that these proposed solutions face. There are more sophisticated and less problematic proposed solutions to this problem. — Chief Owl Sapientia

Yes I agree, it's a matter of convention, so there will always be disagreement. however, if we could determine the principles which constitute temporal continuity, and we could all learn these principles, and refer to them when deciding whether or not an object continued to be the object which it is, then we'd have much better agreement.

If experience can’t be the object of knowledge, then you can’t make any claims about it. Full stop. There’s no middle ground here that I can see, but perhaps you could take a stab at explaining how your claims about experience are possible if experience literally can’t be known. — Aaron R

Isn't it possible to make subjective claims about experience? So I wouldn't go so far as to say "you can't make any claims about it". That conclusion is uncalled for. Would you agree that we can have subjective knowledge concerning experience?

That would be news to mathematicians. — aletheist

I know it's news to mathematicians, that's what we've been arguing. Many mathematicians believe in the contradiction of an infinitely divisible continuity. But mathematicians specialize in mathematics, not ontology. So it's more likely that a metaphysician would know these principles better than a mathematician

Not at the same time and in the same respect, hence no contradiction. — aletheist

You don't seem to understand what an ideal is. Ideals are timeless truths. "The same time and in the same respect" is irrelevant to an ideal, 2+2=4 regardless of what time it is. We cannot attribute to a thing, that it is continuous, and that it is not continuous at the same time, by the law of non-contradiction. We agree with this. But as you keep saying, we are concerned with the ideal here. The ideal continuum is indivisible regardless of what time it is. To say that the ideal continuum might at some time be divided is like saying that 2+2 might at some time not equal 4. To say that the ideal continuum is divisible is contradiction, false. That's the problem here, you want to compromise the ideal, with principles of temporal existence, so that the thing which is called a continuum can be at one time not-divided, but at a later time divided, thus it is divisible. But ideals are timeless principles they are not things which can change. So all you do is introduce contradiction into a timeless ideal. A continuum, when the thing which is continuous is ideal, rather than a physical object, is necessarily indivisible. If it's a physical object, it's divisible but not infinitely divisible.

It is not possible to divide it and still have a continuum. — aletheist

An ideal is a timeless truth. And that a continuum cannot be divided is such an ideal. So are you arguing that there will be a time after the ideal continuity is divided and then there would no longer be such an ideal? But since an ideal is a timeless truth, if there will be a time when there is no longer an ideal continuity, then there must not be an ideal continuity even now.

Dividing it is precisely what causes it to change from a continuum to a non-continuum. — aletheist

An ideal cannot change to what it is not. That's the thing with an ideal, it must always remain the same. 4 will always be 4, it cannot change to non-4. A point will always be a point, it will not change to a non-point. A circle will always be a circle, a square always a square. These things, ideals, do not change. It doesn't make sense to say that the ideal continuity will change to be non-continuous. You are just committing category error, trying to introduce characteristics of physical existence, "change", into the ideal. If you want to talk about a continuum which can change to a non-continuum, then we are talking about a physical object, not an ideal. And physical objects cannot be divided infinitely.

You are clearly not paying attention at all. — aletheist

Oh I'm paying attention, you're just not listening to reason, continually making the same unreasonable assertions over and over again.

I didn't. — TheWillowOfDarkness

Sorry then, I misunderstood.

Since this is the case, the problem you present is nothing more than a red-herring — TheWillowOfDarkness

Not a red-herring, just a misunderstanding. I was trying to make a point to you, but I guess I didn't realize that you already agreed with that point.

What you suggest as a problem is just a category error, a mistaken assumption that unity is given by other things. — TheWillowOfDarkness

But I still think that unities have a beginning and an ending, therefore they must be caused, ("given" by something else). You seem to think that all unities are infinite, but I don't see any examples of infinite unities, and I don't see how anything other than the ideal unity could be infinite.

No, the unity of an object does not exist in the first place and has no end, so there is no end to the particular continuum-- hence the dead and fictional are still whole, despite being broken apart or never existing. — TheWillowOfDarkness

We create objects, which are unities, continuums, we bring them into existence, and annihilate them, so I don't see the basis for your claim that a particular continuum has no end.

Claimed numerical continuum must be an illusions because this, the electromagnetically regulated reality we perceive, is almost always the world of zero. Meaning no matter what we find via the senses and our sensory based instruments we find only +1 & -1 of discrete objects. — Wolf

I don't quite get what you mean by "world of zero". What do you mean by this, and, all we find is "+1 & -1 of discrete objects"?

Descriptions are descriptions and in all situations must be taken as some attempt but in no way or manner so they ever come close to the actual experience. — Rich

Are you saying that descriptions are absolutely false? If not, then there must be some truth to a description. Just because it doesn't describe every aspect of the scene which it is describing, doesn't mean that it is false. So if a description describes some things which are unchanging during a period of time, then don't you think that there are some aspects of reality which are unchanging during that period of time, corresponding to the description?

Labeling with descriptions are useless. What is useful, is watch the ocean and the waves and observe closely what is actually happening as forms come and go in seamless never ending pattern of becoming and going. No states, no division, no difference between the whole and the parts, no way to divide, no b way to say this is where it begins and this is where it ends, yet it is all there. — Rich

Concentrating on the active parts of the world, and complete ignoring the things which are passive, or unchanging, is no better of a way to produce an ontology than concentrating on the unchanging aspects and ignoring the activities.

So while I do think that mathematics in accordance with the arithmetic/Dedekind-Cantor/set-theoretic paradigm is intrinsically discrete, I deny that all mathematics is constrained to refer only to discrete units. We certainly have not determined otherwise in this thread. — aletheist

Now what I've been trying to explain, is that dealing with continuity is not an issue of developing different, or better, mathematical principles, it is an issue of getting a proper definition of "continuity" one which renders real continuity intelligible to mathematics. What I've been trying to demonstrate is that your idea of continuity, or definition of "continuity" does not properly represent real continuity, and this is why mathematics has difficulty with your sense of "continuity".

One more time: I have always and only been dealing with mathematics and ideal states of affairs throughout this thread. Your unwillingness or inability to think of a continuum in mathematical/ideal terms is your own limitation, not mine. — aletheist

There is no such thing as a mathematical continuity, you are making that up. We can apply mathematics toward analyzing or understanding something which is assumed to be continuous, a continuum, but this does not mean that the mathematics itself is a mathematical continuity. The only mathematical continuity there is, is the consistency of all mathematical principles together, if there were no contradictions, which would form one continuous whole, that is called "mathematics".

Which is exactly what I have done. You have not demonstrated otherwise; you just keep asserting it over and over, apparently expecting a different result. Where have I claimed that both P and not-P are true at the same time and in the same respect? — aletheist

You have claimed that a continuum is both divisible and not divisible.

What I stated is that a continuum cannot be divided into parts that are themselves indivisible; so it can be divided into parts that are themselves divisible, although once that happens it is no longer a continuum. — aletheist

You circle around the contradiction, without addressing it, as if by circling it it will be obscured. You have a continuous thing which you call a continuum. It is a continuum because it is undivided, by definition. It is necessarily undivided, or else it cannot be said to be a continuum. If it is necessary that the continuum is undivided, then it is not possible to divide it. It is indivisible by that definition. Clearly, it is contradictory to say that a continuum can be divided. You seem to think that by the power of you statement alone, "a continuum can be divided into parts", it actually can be divided into parts. It's blatantly contradictory, and you're just in denial of that fact.

Look at it this way. There is a thing which is being called continuous. Because of this it may be called a continuum. That thing must remain undivided or else it is no longer a continuum. Let's say that something can act on that continuum and change it into something else. This is what you call "dividing it", and this is the premise for your claim that a continuum is divisible. Something can act on the continuum and change it into something other than a continuum. But this, acting on the continuum, is not "dividing it" in the mathematical sense of division, it is a change, which constitutes going from continuum to not-continuum.

By your premise, that this is "dividing" the continuum, in the mathematical sense of "dividing", you come up with the idea that the two parts produced are mathematically equivalent to the continuum. But this is not the case at all, in reality. And this is why it is wrong to assume that when a continuous thing is divided, it can be divided infinitely. When a continuous thing is divided, what is produced is two things which are not mathematically equivalent to the original continuum. So we must respect a difference between mathematical division, and "dividing" a continuum, which is not really an act of division at all, in the sense of mathematical division. What we call "dividing" the continuum, is really annihilating that continuum to produce something new. What is produced may be a number of new continuums. But it is wrong to believe that the new continuums produced are mathematically equivalent to the original continuum.

Because points are indivisible, and a continuous line cannot be divided into parts that are themselves indivisible. Try to keep up. — aletheist

But this is plainly wrong. The continuous line, the one which exists on the paper can be cut up, but eventually there will be parts that are indivisible, too small to cut. The ideal line is defined as consisting of a succession points, and is therefore not continuous, it is discrete. You seem to have no respect for the fact that the ideal line is defined as consisting of points, and is therefore discrete, because you want to work with a continuous line. By ignoring this reality, you put yourself into your contradictory position.

As I said, you are locked into the standard rules of classical logic, which are very useful for most purposes, but not for understanding the nature of true continuity. Failure of excluded middle is not a contradiction in all viable forms of logic. — aletheist

Yes, standard logic is very good for understanding the nature of true continuity, as was demonstrated by Parmenides. The problem is that you define "continuity" in some absurd, contradictory way, so you must resort to some absurd logic in an attempt to understand this absurd notion of continuity. All you get is lost in vagueness. What is required to understand this subject is firm adherence to fundamental principles.

Any object has a start and end... but these can only be finite states. They are never a whole in the first place. — TheWillowOfDarkness

Why would you say that an object is not a whole? Sure it is not a whole in the perfect sense, like in the sense of a unity of everything is a the perfect whole, but by its own right as an individual unity, can't we say that it's a whole?

Niether of these objects are a whole, either of the plan or the object. — TheWillowOfDarkness

How would you define "whole" then? To be a whole, doesn't it suffice just to be a unity? A unity doesn't need to be a perfect unity in order to be a whole. So numeral such as 5, 8, 12, signify wholes, but since they are each not the complete whole of all the numbers, nor the primary unity, 1, they are not perfect in their wholeness.

My plan doesn't suddenly become "not whole" because it began and ended. Nor does the created object. The point is starts and ends do not amount to a breaking of a contiuum. — TheWillowOfDarkness

Do you agree that a continuum is a whole? And do you agree that there are wholes, continua which are less than perfect in their nature? If an object which is a unity, a whole, ceases to exist, isn't that the end of that particular continuum? But if that object is described as part of a larger, more perfect whole, then that larger, more perfect continuity would persist, and the annihilation of that smaller whole, which was really just a part of the more perfect whole, would just be a slight change to that more perfect continuum.

If one could break a contiuum, it not indivisible. The infinite nature of a contiuum means it must be unaffected by beginnings and ends. — TheWillowOfDarkness

You seem to be assuming that all continua are ideal, infinite. But I don't see why you shouldn't consider that any existing object, as a whole, a unity, is a continuum. And surely these objects can be annihilated, so the continuum which is that object must be broken. I wouldn't call this dividing the continuum though, as I explained to aletheist, because "divide" implies a mathematical division. That's why I believe that a continuum must be capable of beginning and ending, but this is not properly called dividing.

As I have repeatedly made clear, I am discussing mathematics here, which has to do with ideal states of affairs; I am not saying anything whatsoever about physical objects. — aletheist

But we've already determined that mathematics refers to discrete units. So as soon as you describe something as a continuum we are not dealing with mathematics, and therefore I cannot assume that we are dealing with ideal states of affairs. You want an ideal continuum, so that perhaps you can establish a compatibility with mathematics, but this requires that you can define "continuity" in a way which is not contradictory. I can, "continuity" refers to an indivisible whole, and this is consistent with mathematics.

There has to be a way to distinguish a continuum (such as a line) from an indivisible (such as a point). — aletheist

Sure, they are different, the point is defined as non-dimensional, and the line has a specified dimensionality.. As non-dimensional, the point is purely ideal. The "line" in its definition is purely ideal, but it describes a spatial extension so what is described is not completely ideal. To be divisible, it requires this spatial extension, and this means that to be divided it requires extension outside the mind. It is only by means of this non-ideality, looking at the physical thing which the description describes, do we get divisibility. So when you say that a line is a divisible continuum, you are appealing to a non-ideal line, a physical representation to say that it is divisible. It is by means of this assumed spatial extension, which makes it non-ideal, that we say it can be divided. If there is a truly ideal line, it exists by definition only, and cannot be divided because then it would not be a line, it would be a line segment or something like that. The truly ideal line cannot be divided.

There is nothing contradictory about my/Peirce's definition, and if you are going to keep insisting that there is, we might as well call yet another impasse and go our separate ways. — aletheist

If you insist, then have fun with your crossing back and forth from the ideal to the physical thing, with all the contradictions which that entails. Look, you want to be able to divide the ideal line, but that's clearly impossible, and contradictory. The line is defined as a specific form of spatial extension, and you think that because of this spatial extension you should be able to divide it. But it's an ideal, you can't divide it, because that would render it other than its definition, and this is contradictory. So you must face the logical conclusion that the line as an ideal, is an indivisible entity. What distinguishes it from the point is to be found in its definition, of a particular form of spatial extension. But that is its definition only, it doesn't make it have real spatial extension, such that you can divide it, it's an ideal.

But by my definition, it cannot be so divided; therefore, not only is it not discrete, it is not even potentially discrete. A true continuum cannot be composed of discrete elements, and it also cannot be decomposed into discrete elements. We can only introduce indivisible points along a continuous line, and those points are not parts of the continuous line itself. — aletheist

OK, so you say that a continuum "cannot be so divided". Why do you keep insisting, in a contradictory way, that a line is infinitely divisible. In one sentence you'll say that a continuum is infinitely divisible, then you insist that you are not contradicting yourself, and then you say "it cannot be so divided". If it cannot be "so divided", then how is it divided? You talk as if there is some magical way of dividing something which doesn't actually involve dividing it. What could that be? Thinking of something continuous as being divisible doesn't actually divide it, nor does it make it divisible. Unless it can actually be divided, it is false to say that it is divisible.

We can only introduce indivisible points along a continuous line, and those points are not parts of the continuous line itself. — aletheist

Why are the ideal points not part of the ideal line? This is completely consistent with common geometrical principles, and consistent with mathematics as well. It is only your desire to do the impossible, define a divisible continuity, which makes you reject the standard definition that a line is a collection of points.

Therefore, the principle of excluded middle does not apply to that which is continuous; and this is all that it means to say that a continuum has only indefinite or potential parts. — aletheist

As I said, I do not agree with the way that Peirce dismisses logical principles. It is unwarranted. He does this in order to compromise, where compromise is unnecessary. As I've been trying to explain, we can stick to the principles which keep the ideal separate from the non-ideal, and proceed toward a much more comprehensive understanding, than Peirce's compromised understanding. It should be evident from the above passage, that Peirce's move only plunges us into an unnecessary vagueness, by failing to maintain the difference between that which has parts, and that which does not have parts. Thus he compromises the principles with "indefinite" parts.

As oxymorinic as it sounds it could be a case of reason-based convention, just not arbitrary convention. — TheMadFool

If arbitrary means without reason, then I don't believe there is any truly arbitrary convention, as human beings always act with intention in there somewhere, so there is always reasons for why they are doing what they are doing. The problem is that we are often incapable of determining those reasons, so it's easier just to say that things are arbitrary.

The point being that if the paradox has any worth i is the exposure of our poor understanding of identity. — TheMadFool

I agree with this, and I suggest that it is not really a paradox. Do you recognize that there are two distinct senses of "identity", two distinct meanings of "the same"? One is identity according to a continuity of existence, which we have been discussing. If we can identify a continuity of existence, then despite numerous changes to the identified thing, (even taking it apart and rebuilding it, or changing all the parts of it in repair), we say it remains as "the same" thing. This is the common sense meaning of "the same".

The other is a sense of identity which is more proper to logic. In this sense, identity is a formal description, or formal definition, not a continuity of existence. Identity is determined by a statement of "what" is being referred to. So we could have a description of ship A, and so long as the thing referred to fit that description, it is ship A. But it must fit that description. In this case, then, as soon as we removed one plank from ship A, we'd have to refer to the definition of ship A, and see if it allows for this change. If not, then the ship is no longer ship A.

The appearance of a paradox is created by mixing up these two distinct senses of 'the same". There is an illusion created that "the ship" has a formal definition, in the logical way, that there is a definition of ship A, and we must adhere to that definition, in order that the object is ship A. But no formal definition has been provided, so we are left only with the continuity of existence as our only form of identity. We take this default sense of identity naturally because it is the common sense form of identity. This allows for change to the ship. Change to the ship implies that we may contradict the formal definition of "the ship". But there is no formal definition of "the ship", so there really is no contradiction and no paradox, because we are simply left to distinguish ship A according to prevailing conventions, not any principles of logic. And there is always discrepancies between such conventions. The apparent paradox is just a matter of these discrepancies, not a failure of logic.

If we assume that two descriptions describe the very same thing, then there is an assumed continuity between those two descriptions provided for by the belief that the two descriptions describe the very same thing. But if one description refers to a state, and the other description refers to an activity, then I don't think that such an assumption is justified.

They are one and the same. It is a continuum. The is no discontinuity between that which physical and that which creates it. I am bewildered at how you are able to separate the two. If we aren't energy, then what are we? The energetic form is simply moving within an energy field as a wave moves in water. There is no separation. — Rich

To describe a thing, and to describe the activity of a thing, is two distinct description. So I am bewildered at how you do not recognize this. To describe my car as a physical object, and to describe what my car is doing, is two very distinct descriptions. You can insist all you want, that there is no difference between the description of the car, and the description of what the car is doing, but that doesn't change the fact there is a difference between these two.

Actually, it is constantly changing. Some quite overtly others very subtly. But everything is constantly changing in one manner or another. Energy never stands still. Heraclitus was right and my guess is that he intuited it. If you were correct, then a whole new problem is created, like how does all quanta stop long enough, in concert with each other, to create your state. That would be interesting. — Rich

I'm not talking about energy here, I'm talking about the physical things in the room. Clearly something must be staying the same, as I can describe the room, write everything down, and then come back later, to find that it is the same. You want to describe the room in terms of energy, but that is a completely different description. If there is an incompatibility between the two descriptions, then this indeed is a problem, and there might be the need to work out some principles to reconcile this.

That which cannot be divided at all is an individual, not a continuum - e.g., a point rather than a line. There has to be a way to distinguish these two concepts. — aletheist

An individual is a physical object and it is divisible (the name "individual" is misleading). It is also a continuum, that's how we can call it a whole, because of its continuity, as one unit. A point may be an ideal individual, being dimensionless it is indivisible, but we don't generally call a point an individual. The difference is that "individual" refers to a physical object, but "point" refers to an ideal.

What would you call something that satisfies the following definition of a continuum? That which has potential parts, all of which would have parts of the same kind, such that it could be divided (but would then cease to be continuous), and none of the resulting parts would ever be incapable of further division. — aletheist

There is probably more than one contradiction in this description, but I'll try to sort it out. This is a collection of discrete individuals. Being described as consisting of parts indicates that it is discrete. A "potential part" meaning nothing more than a potential parting, indicating that the point of potential division exists within the so-called continuum. What you have described is a physical object which is capable of being divided. That each part will again be created of parts, infinitely, creates the contradiction we already discussed. That all of the parts are of the same kind will probably result in a contradiction, as well, if we follow it through to analyze what this means. But how we might resolve "each part is of the same kind" depends on how we might resolve infinite divisibility. The two contradictions would play into each other, how one is resolved would depend upon how the other is resolved. To begin with, we could recognize that what you have described is a physical object, and it is highly unlikely that any physical object has all parts of the same kind.

The problem though, which results in the contradictions, is that you take as your premise, your starting point, an individual thing, a continuum, which is divisible, and this is a description of a physical object, and then you try to turn it into an ideal. You cross categories. We haven't found a way to have an ideal continuum, except as Parmenides' indivisible whole, so when you start with a divisible continuum, you are starting with a physical object, not an ideal. But then you want to assign ideal qualities to this physical object, such as infinitely divisible, and having all parts of the same type..

This just seems completely backwards to me. How can we identify any real examples of continua without first defining what it means to be continuous? What interests me is whether there is anything real that satisfies my definition of continuity, even if you want to call it something else. — aletheist

This is the point I described to you earlier, concerning Platonic dialectics. The words "continuous", and "continuity", are commonly used. We look and see what kind of things are described by these names, and see what they have in common, why people call them continuous, and from here we can say what it means to be continuous. We have already determined that it is impossible that there is something real which satisfies your definition of "continuous", because it is contradictory. So we can conclude that somehow, along the way, some people got mixed up, and produced a faulty definition of continuous, and this got accepted and used. Peirce worked at determining the faults, and attempted, with no success in my opinion, resolution.

Just like In my example of Plato's Theaetetus, "knowledge" was defined as necessarily being true, i.e. excluding the possibility of falsity. But all the instances of knowledge in the world, what people were referring to as "knowledge", could not be shown to necessarily exclude falsity. So this was a faulty definition. It defined an "ideal" knowledge, but knowledge as it exists, and what we call "knowledge", doesn't have that ideal character. So when we have confused concepts like "continuous", we need to straighten things out by referring to how the word is used, we cannot just accept a definition which has been shown to be defective.

You want to take "continuous", and give it an ideal definition which has already been shown to be contradictory. The reason this definition is contradictory can be understood like this. "Continuous" already has an ideal definition, as described by Parmenides, indivisible, whole. It also has a definition which we use to refer to physical things, a whole which is divisible. The two are clearly incompatible, but some people have wanted to create a single definition, which encompasses both, the ideal indivisible whole, and the physical divisible whole. So they compromise in one way or the other.

You might notice that in the above description, the two definitions of "continuous" do have something in common. They both refer to a "whole". The ideal continuum is an indivisible whole, and the physical continuum is a divisible whole. So they do have a point of compatibility, and if we want to produce a definition which encompasses both, we should start with this, "a continuum is a whole". Do you agree with this definition, a continuum is a whole, whether it is an ideal continuum or a physical continuum?

The whole doesn't get divided in instances where we cut up an object. In such an instance, we are destroying a particular state of the world. When we cut a carrot, we don't target the whole. The knife doesn't split a whole into two halves, such there is a division of the whole.

If I try and say: "Here is half the whole carrot," my statement is incohrent. Since the whole is indivisible, I can't split it such that I have half the whole here and the other half of the whole over there.

In a sense we could say I destroy the whole. In cutting, I take a state expressing an infinite of continuity out of the world. Where one the whole was expressed in the world in front of me, now it is only done so in logic. There's never a split in the whole though, such that we end up with seperate parts of it. We are only destroying an object which expesses the whole. — TheWillowOfDarkness

Yes, I can, in principle, agree with this. We don't actually divide the whole, we destroy it, cause its existence to end. And in doing so though, we cause the beginning of existence of the wholes which we have created, what we call the parts, as they are now actually wholes. We do end up with separate parts, but each part is not a part, it is itself a whole. Prior to cutting the object, we can describe our intended action in terms of "parts", describing the parts we will cut, which will each become separate wholes after the act of cutting. In this way we can say that any whole is indivisible, just like the ideal whole of Parmenides, because we never really divide the whole, we just destroy it. The physical object wholes, are describable in terms of parts but these wholes, physical objects, have a beginning and an ending to their existence, and this is why we can describe them as parts, unlike Parmenides' ideal whole. Parmenides' whole which is defined as without a beginning or an end cannot be described in terms of parts, because this would imply that it could end.

Following on, this also means particular continuities don't have a beginning or end. Yes, any given object has a start and end, but this is not the unity expressed by it. Whether we are talking about a rock, a person or bacteria, it doesn't take existence for them to be whole-- imagined objects are no less whole than existing ones. In the birth and death of states, there only presence in time, as divided moments. It is only those divided moments, expressing a whole, which are lost and formed. Wholes themsleves are neither created or destroyed. — TheWillowOfDarkness

I don't follow your logic here though. If an object has a start and an end, doesn't this imply necessarily that the continuity of that object is broken? How can you assume that the continuity continues through the end, or prior to the beginning of the object. Let's say that prior to an object's physical existence there is an intended existence, an idea, plan, formula, or blueprint for that object, and after the object's physical existence there is the memory of that object. Aren't these two distinctly different from the physical object itself? Isn't there a break in the continuity between the plan and the physical object, and between the physical object and the memory of it? This being the difference between being in a mind and being independent of a mind.

There are no states in nature. — Rich

I disagree. I can look around my room and describe the positioning of the objects, and this will stay the same until it is changed, therefore it is a state that naturally persists. That's what Newton's first law describes, the state which things are in, will persist until a force causes that to change.

Not really. To say that a continuum has no definite parts just means that it does not have any distinct, discrete, or indivisible parts. With this qualification, I might even be willing to grant that a continuum has no parts at all, as long as it remains undivided. — aletheist

Yes, that is what I was arguing, if the parts are not distinct, or discrete, it doesn't really make sense to think of them as parts. So why not just say that the continuum has not parts?

After all, we agree that the act of dividing a continuum breaks its continuity; so what "infinitely divisible" means in this context is that if we start dividing a continuum, we will never reach the point (literally) of reducing it to an indivisible part. In other words, a continuum is indivisible in the specific sense that if it were divided into parts, and thus made discontinuous, then none of those parts would be indivisible. What do you think? — aletheist

I think that the ideal continuum cannot be divided at all, because it has no parts. And to give it parts would deny its existence as a continuum, so the ideal continuum must be indivisible. Therefore it makes no sense to talk about dividing an ideal continuum. But now I've been talking to the others about real existing continuums, and these are physical objects which display the quality of continuous existence. We can divide an object, or change an existing state, but in doing so we end its continuity, and start new continuities, as the divided parts are now objects which display continuous existence. But I believe these objects are not infinitely divisible. Infinite divisibility seems like an impossibility.

Again, whether there are any real continua is a separate question from what it means to be continuous. — aletheist

This is the same type of point that we are always disagreeing on. If we want to know what it means to be continuous, we need to refer to real things which are continuous. You seem to think that we can just stipulate the meaning of a word, regardless of whether there are any real examples of this. What good does that do us? If we want to know what it means to be continuous, we need to look at real examples of continua and determine what they have in common.

I think that the first thing to establish is whether space and time are themselves continuous. If not - if they are discrete - then presumably all spatio-temporal entities are also discrete. However, if we establish that they are continuous, then we can investigate whether anything within space and time is also continuous. — aletheist

OK, if we are going to venture into this subject we need to agree on some fundamental principles in order that we can understand each other. I believe that there are real existing things in the world, physical objects. And I also believe that we have concepts of space and time, and that these concepts have been produced to help us deal with, and understand the physical objects. So if we are to understand what the concepts of space and time refer to, we must relate them to physical objects, because that is where these concepts are abstracted from, the assumed existence of objects. They are not abstracted from the observations of real space, and real time. We cannot start to talk about space and time, as if they are real things in the world, because they are just concepts derived from our study of objects. Therefore if we assume that there is a real space, and a real time, because we have concepts of these, our understanding of these things is limited by our understanding of objects, so that any misunderstanding of objects which we have, will also manifest as a misunderstanding of space and time. Our only means to understanding real space, and real time, is through our understanding of objects.

So if we assume that discrete objects are continuous, as things, perhaps beings, we can proceed to analyze how this relates to space and time. First, I would suggest that an object appears to be discrete in relation to other objects. They may overlap, as gravity overlaps, and substances overlap in solutions, but the objects appear to be generally discrete in space. Therefore space appears to be discrete. However, objects appear to obtain their continuity from having continuous temporal existence. So we might consider that time is continuous.

To describe to you what I mean by the indivisibility of the continuous, consider that time is continuous. We can mark points in time, and durations in time all over the map. any where we want. But these are marked on the map, they are not within time itself. These points and segments are not part of time itself, they are part of the grid, the map, or marking system, which is independent of time itself.

If there's any convention in the paradox we're discussing it is not of type 1 (arbitrary). Rather ''identity'' is a reason-based convention. We have to reason out what ''identity'' means and then, much later after rigorous analysis, we establish the convention that ''identity'' means so and so.

Hence, we can't simply brush aside the problem by saying it's just a matter of convention. — TheMadFool

OK, I agree very few conventions, if any are purely arbitrary. So even in the symbols, 1 and 2, etc., there are some reasons why these are the symbols which became used and not something else. So I assume that there are reason why, when the kid takes the toy apart and rebuilds it, we think of that as part of the identity of the toy, yet if we remolded the broken drinking glass to remake the glass, we would identify it as a new glass. Conventions come into existence for different reasons, so they are not truly arbitrary. Why do you think it's not a matter of convention?

The description changes, not the intrinsic continuity. — Rich

The problem is, that according to the laws of logic, when the description changes the continuity which was described, ends, and a new continuity begins. That is why the laws of logic are not compatible with "becoming", and becoming is not compatible with continuity. So logic forces us toward discrete units, states of existence, and becoming occurs somehow in between. The position aletheist was arguing, the one derived from Peirce, takes continuity from the individual things with beginnings and endings, and hands continuity to the "becoming", the spatial temporal continuum. But this may render the laws of logic as useless.

As with mathematics, descriptions (for communication purposes only) is symbolic. Symbols are not that which is being described. Just because I describe two different events in my life does is constantly starting and stopping. Duration is continuous when observed directly. Symbolics only are necessary for communication or as a tool for manipulation. — Rich

The symbol must symbolizes something, and intelligibility depends on the assumption that what is symbolized remains the same. This is the continuity which I am referring to. The description, being symbols, describes something, and what it describes must remain the same. So that which is being described, remains the same, as a continuity. If the description is no longer applicable, then the state intended to be described, has changed, and that continuity no longer exists.

The important point is that no continuity is ever lost and no symbolic, which is intrinsically formed by individual units can possibly capture this continuity. — Rich

The problem with this perspective is that what exists between the described states, is activity, becoming. And since becoming is change, it cannot be understood as continuity, which is a remaining the same. That is why we have had so much difficulty in this thread describing space and time as continuous. These are the principles of flux, and flux is contrary to the continuity of being. I have now opted to describe individual existing objects as wholes, continuities. So "continuity" must be used to refer to each described state which continues to exist without change, and "becoming" is something other. Therefore continuity is always being lost into becoming, as things change.

This thread is basically about the ability for symbolics to adequately describe continuity. It can't. In fact, the description they yield is pretty much totally contrary to experience. The waves never, ever, ever break the continuity of the ocean. The objects are formed and reformed out of the continuity. — Rich

What is described by the symbols, what the symbols refer to is something which doesn't change in time, therefore the thing described is a temporal continuity. But if we want to describe "continuity" itself, what it means to be continuous, that is something different from what the symbols refer to, it is this unchangingness. There is nothing "contrary to experience", about describing what it means to be continuous, as a state which doesn't change in time. We observe all sorts of objects to be like this. We also observe that these continuities can be ended, as is the case when we divide up an object, and when an object is constructed, a continuity begins. The aspect of reality which is difficult to describe with symbols is the "becoming" the means by which these various continuities which may be described, begin and end. That is because our symbols are intended to have a static, fixed meaning, while becoming is a changing.

Absolutely not. When a wave in an ocean transforms in two or more or even dissolves in the ocean, no continuity is lost whatsoever. — Rich

Of course there's a loss of continuity. The description of the wave as one wave applies no longer, and the description of two waves applies. Therefore the continuity of the one wave ends, and the two distinct continuities begin. You might be thinking that the continuity of energy is not lost, but if you are talking about the energy involved, this is something different than talking about the waves involved. The continuity of energy is a different continuity than the continuity of the wave.

Forms of substance are nothing more than waves in the fabric of the universe. They are just more solid by degrees. How does one break continuity in the universe, in space, in duration? With a very fine knife? Exactly how fine? Finer than Planck's constant? Continuity can never be broken. It can only be reformed, as waves reform in oceans. — Rich

As I explained in my post, it's quite obvious that continuity is broken, this we know. It is not broken in the sense of being divided though, it begins and it ends. But this is not "ideal continuity" this is the continuity of existence of various different things, they have beginnings and endings. And as I said, I don't think we know the cause of continuity. What you call "reforming" of the same continuity, is really the beginnings and endings of various different continuities. We must allow that there are various different continuities to account for the fact that there are various different, individual objects.

Are you arguing that there is no such thing as individual objects?

As children it is common for us to play games. One of these games involves breaking apart toys into its components and then rebuilding. We've all done it and we've seen others do it too. In such cases we never think that the process of annihilation - reconstruction yields a different toy. Are you saying this common sense intuition is wrong? — TheMadFool

I think I explained this. Identity, based on continuity of existence, is an assumption only. As such there are no real objective standards for assigning such "identity" to an object, such "identity" is somewhat arbitrary. There are however, conventions, but the conventions are informal, and vary depending on many different factors, just like our conventions for using words. So if we describe the activity of taking the thing apart and rebuilding it, as a continuity of existence of the thing, then we believe that it maintains its identity according to that convention. But if we describe the activity as an annihilate of the thing, and a rebuilding of a new thing, then we believe that the thing does not maintain its identity, according to that convention.

It is my opinion that the effects which human activity have had, and continue to have, on atmospheric ozone, has a far greater influence on climate change than does CO2 emissions.

Atmospheric ozone O3, has come about due to the presence of free oxygen O2. O2 is not native to earth's atmosphere, it was put there by the activities of living creatures and it's an important catalyst for the evolution of higher life forms. In its interaction with sunlight, some O2 becomes O3, and O3 is very effective at absorbing certain wavelengths of UV radiation into the stratosphere. This energy is intercepted and prevented from heating the earth's surface, and is later radiated to space. Fluctuations in solar UV radiation are considerable, so O3 plays an important role in stabilizing the earth's surface temperature, and therefore the climate in general.

Overall, the intensity fluctuations of solar radiation are small. In long-term average they amount to only the fraction of a percent of the total irradiance. The ultraviolet radiation, however, shows greater fluctuations and is also regarded as particularly climate-effective. Since the Earth's atmosphere absorbs this radiation to a large extent, it influences critical chemical reactions in the upper layers of the atmosphere. Indirectly, these processes can also affect the temperature at the Earth's surface.

http://www.mps.mpg.de/4017144/PM_2015_07_09_UV-Schwankungen_der_Sonne_unterschaetzt

Our "cutting" of a whole is merely picking out something specific. — TheWillowOfDarkness

I don't think I agree with this part of your statement. An object, as a whole, is something specific. The wholeness, the unity of the object, is something real, and something which is destroyed when we divide that object. Dividing an object is a real activity which has a real effect on the world, one object becomes two objects, and this is a considerable difference. It is not simply a case of choosing some specific parts, over the whole, it is a case of making those apprehended parts, into wholes themselves, through the act of dividing.

So we must have respect for what really happens in division, and that is that one whole becomes two wholes. And we know that it is different to be a whole than it is to be a part, because unity is attributed to the whole, not the part. So when we divide a whole, we take the unity away from it, and give unity to the parts. It is not the case that the unity of the whole is given to the parts, the unity of the whole is destroyed, and a new unity is given to each of the parts. A unity, or continuity is never divided, it simply has a beginning and an end.

If we associate continuity with the whole, with the unity, then each time we divide something, we destroy a continuity, and create new continuities. From this perspective, continuities have temporal beginnings and ends, so we need to be able to determine a cause of continuity. If nothing comes from nothing, there must be some actuality, some act, which causes the beginning of any particular continuity. Likewise, there must be an act which causes the end of a particular continuity.

There isn't a conflict. Ideal continuity is present. Any infinite can't be divided such that it said to begin or end. — TheWillowOfDarkness

I agree that this would be the ideal continuity, an infinite continuity, without beginning or end. We could assign this continuity to "the present" of time, as a working premise, and then we apprehend the whole of existence as one infinite continuity. But when we look at the nature of individual continuities, as I described above, we see that they all begin and end, so by induction, it is probably the case that there is no such ideal continuity, the infinite continuity.

This may be where mathematics misleads metaphysics or ontology, by allowing the possibility of the infinite. It is necessary that we allow infinity in mathematics in order that we can understand the most vast expanse which is possible. We cannot imagine the most vast expanse, because we don't know what it is, so we must allow that mathematics is limitless in order that the most vast expanse can be apprehended. But if we allow that the infinite has real existence within the world which we are trying to apprehend, then we deny ourselves the capacity to understand that world.

Your failure to understand it does not render it incoherent. I understand it, I just seem to be unable (so far) to explain it in a way that you will accept. Is this, in the end, the substance of our disagreement here? If you were to wake up tomorrow and decide that the notion of an indefinite part makes sense to you after all, would you have any other objections remaining? — aletheist

Perhaps it could make sense to me, but to say that a part is indefinite would be to say that this part is unintelligible, it cannot be known. Unless it can be demonstrated that such a part actually exists, why would I accept this assumption? To arbitrarily designate something as unknowable is contrary to the philosophical nature of human beings, which is the desire to know.

Instead, I know of a much more reasonable way to understand the ideal continuity, and that is as indivisible. Maintaining that the ideal continuity is indivisible, avoids this problem of having to assume unintelligible parts. So I see a reasonable approach, which is that the ideal continuity is indivisible, and an unreasonable approach, which is that the ideal continuity consists of unintelligible parts. So until it is demonstrated to me that there is something wrong with this notion that the ideal continuity is indivisible, why would I be inclined toward an unreasonable ideal, which contains unintelligible parts?

We have already agreed that a continuum does not consist of points, that it is undivided, and that it is indivisible in the sense that once it is divided, it is no longer continuous. It seems, then, that the last hurdle - as I have already suggested - is your insistence that a continuum cannot have parts of any kind, grounded in your rejection of indefinite parts, such as infinitesimals or Zalamea's "neighborhoods." Do you concur with this assessment? — aletheist

Yes I think I concur. But there is still another option which we haven't yet explored, and this is what I stated in my first post, that it is possible that an ideal continuity is just a fiction. The only way I can conceive of an ideal continuity is as Parmenides did, as an indivisible whole, without parts. However, it is possible that the only real continuities, are those physical, spatial entities which can be divided, but not divided infinitely. The fact that we divide continuous things, objects, wholes which we can cut up, inclines us to believe that there is some sort of thing (space or time for example) which may be capable of being divided infinitely. From this, we create the notion of an ideal continuity, one which could be divided infinitely. But this idea proves to be logically unsound, and we are forced toward the realization that the only ideal continuity is the indivisible whole, having no parts. Now we started with the fact that a continuous thing is divisible, and have ended with an opposing conclusion concerning the ideal continuity, that it is indivisible. Is it the case that we need to include divisibility in the ideal continuity? Perhaps we are just chasing an impossible dream, and an ideal continuity is just a logical impossibility. Maybe continuity is just not a real thing, and that's why it's impossible to understand.

If and when you ever come to understand this, you will then finally understand what Peirce and I mean by a true continuum. — aletheist

Exactly the point, yours and Peirce's concept of true continuum is incoherent and will never be understood. As I said, Peirce was proceeding in the proper direction, but didn't follow through. Instead of adhering to his logical conclusion, that defining parts into a continuum negates its essence as a continuum, and therefore a continuum is necessarily an indivisible whole, he follows his desire to have a divisible continuum. And this produces the incoherent notion of an indefinite part, the part which doesn't exist as a part until it is defined, but this definition renders it as an individual whole itself, rather than as a part because the thing which it was supposed to be a part of is negated by the so-called part's very existence. If he would have only adhered to the logic, he would have discovered Parmenides' indivisible, continuous, whole, and there would have been no need for this incoherent "indefinite part".

In any case, as Peirce stated (and you also quoted), "a continuum, where it is continuous and unbroken, contains no definite parts" (emphasis added). Therefore, its parts are indefinite; or as I have said about infinitesimals, indistinct (but distinguishable). — aletheist

No, if something is known to contain no definite parts, the logical conclusion is that it contains no parts. That it contains "indefinite parts" is illogical. What could it mean for a thing to contain parts but these parts are indefinite?

The continuum is the more basic concept here, not its parts. You cannot assemble a continuum from its parts, you can only divide it into parts; and once you have done so, even just once, it is no longer a continuum. — aletheist

The quote from Pierce is: "I think we must say that continuity is the relation of the parts of an unbroken space or time." This clearly implies that parts are a necessary aspect of the continuity. It doesn't make sense turn this around, and say that the parts only come about through division.

But let's assume that we can do this. Let's say that the continuity has no parts, that is consistent with what I say, I think that to define continuity as containing parts is contradictory. So, how do the parts come about? We cannot define the continuity as having parts, we've already denied that. So if we "define into existence" some parts, these are not parts of the continuity, they are parts of something else. Don't you agree?

A continuous line is divisible if there is no location along it where it is incapable of being divided; but again, once it is divided, it is no longer continuous. — aletheist

There is no point along the continuous line where it is capable of being divided. We already determined, and agreed that there are no points on the continuous line, that would be contradictory. Therefore it is impossible that there is a point on the line where it is capable of being divided, as this assumes a point on the line. So your claim that the continuous line is divisible, cannot be supported in this way.

Aletheist, notice that Peirce claims the act of defining the parts breaks a continuity, but the continuity somehow consists of indefinite parts. Remember though, that we are dealing with the ideal here, and indefiniteness is incoherent within the ideal. So I claim that even describing a continuity as consisting of parts is to negate its essence as a continuity.

How do you deal with facts like, for example, a motor vehicle consists of parts, it is divisible, it can be disassembled and its various parts sent all over the world? — John

The continuum which we are talking about is an ideal. I do not deny that things like cars are continuous in our common manner of using continuous, and these things are divisible, like the line on the paper is divisible. But I deny that these things are infinitely divisible.

Let us imagine a scenario which hopefully will make you see my POV.

Ship A needs to be transported from city x to city y. However, it has to be done by land and also it becomes necessary to disassemble it for easier transport. These kind of situations are quite common. So nothing difficult in imagining it.

After the parts of ship A reach city y they are reassembled in the original exact configuration. In this case annihilation is present but the ship A hasn't lost its identity. There is nothing grossly wrong in holding such a belief. — TheMadFool

It all depends on how you describe the act of dismantling and assembling. If this is described as a continuity of existence of the object, then it becomes part of the object's identity, as per the description. That is the case in your example. If the act is described as an annihilation, and the rebuilding of a new object, then there is no such continuity of existence, as per the description. There is a description of one object ceasing to exist, and a new one coming into existence. So for instance, if you break a drinking glass, and collect the pieces, melt and remold them into a drinking glass, we would describe this as one object being annihilated and a new one coming to be.

The point is, that identity, as a continuity of existence is something which is assumed. Continuity of existence has never been proven, so you hear things like people wondering if an object continues to exist if it is not being looked at. Since continuity of existence, and therefore the "identity" which is associated with it, is just an assumption, then what constitutes continuity of existence, in our beliefs, depends on how we define it.

Notice that infinite divisibility, by itself, is not sufficient to make something continuous; I suspect that I may not have made this clear previously. However, the definition that I have invoked most often - that which has parts, all of which have parts of the same kind - is exactly what Peirce presented here (twice), attributing it to Kant and providing (in my opinion) a convincing case for it. The parts of a continuum, most notably infinitesimals, are not definite; once we create them by the very act of defining them, we have broken the continuity. — aletheist

Actually, from your quoted passage, Peirce is dismissing infinite divisibility as the defining characteristic of continuity, and that's why I said earlier, he was on the right track. However, as I said, I don't think he follows through with the principles he implies, he compromises, and this is his mistake.

On the whole, therefore, I think we must say that continuity is the relation of the parts of an unbroken space or time.

...

In accordance with this it seems necessary to say that a continuum, where it is continuous and unbroken, contains no definite parts; that its parts are created in the act of defining them and the precise definition of them breaks the continuity. — CP 6.168, c. 1903-1904, paragraph breaks added

So we are back to where we were on the last thread. Remember, I pointed out that to define a continuity as a relationship of parts is itself contradictory. To say it consists of parts is to say that it is has separations, is broken, discontinuous. If we talk about space or time as a continuity, an unbroken whole, what justifies the claim that the unbroken whole consists of parts? As Peirce indicates, If it is continuous, it cannot consist of "definite parts". Peirce claims that the mere act of defining the parts breaks the continuity, I am more strict than that, adhering firmly to the underlying principles. My claim is that even to say that it consists of parts, is to state a contradiction. And if it cannot be said to consist of parts, it cannot be divisible.

"Once divided" is a different situation from "divisible." Once divided, the line is indeed no longer continuous; but as long as it remains continuous, the line is both infinitely divisible and undivided. Obviously "divisible" in this context does not mean "capable of remaining continuous after being divided," as you seem to be taking it. — aletheist

The point though is that the line cannot be divided unless it is no longer what it is said to be, a line. To divided it once is to deny that it remains a line, so clearly it cannot be infinitely divided. Therefore when you say that the line is divisible, you must mean something other than capable of being divided. What do you mean then by divisible?

Our posts seem to have gotten long and convoluted,. So I'm going to get right to the point of where I think our difference of opinion lies, and if you're willing perhaps we could work it out. If you do not want to, don't reply to the last post, but consider this point.

We agree to the difference between a line on a paper, and the ideal line. Also, the line on the paper is divisible, we can cut the paper, or do whatever is necessary to divide it. We cannot divide it infinitely though, that is impossible.

But ideals have a different type of existence from things in the physical world. Our difference is with respect to the ideal line. I do not believe that the ideal line is divisible. Once divided, it would no longer be a line, it would be two lines, and two lines is different from one line. This cannot be accepted as an ideal, because it allows that you can do something to the ideal line, divide it, which would make it no longer a line. You cannot negate ideals in this way. Allowing that the line can be divided allows that you can make the ideal line what it is not. And in the realm of the ideal, i.e., what is and is not, this is contradiction. So to allow that the ideal line is divisible, is to allow within the ideal, contradiction.

First of all, you keep referring to "a continuity" as if it were a thing. Continuity is a property, not a thing; a continuum is a thing that has the property of continuity - i.e., being continuous. — aletheist

This is philosophy, it is common, and accepted practise to refer to particular abstracted properties as things. We take a concept such as "blue", "infinite", "continuity" and treat it as a thing. In this way we analyze what it means to have that property. So when I talk about "a continuity", as a thing, I am talking about what it means to have that property of being continuous.

If we are talking about a thing which is continuous, and calling it "a continuum", then we are not talking about what it means to be continuous, we are talking about that thing which has been deemed continuous, the continuum. What happens if we were mistaken in designating that thing as continuous? Then if we come to an understanding about "continuity" according to that particular thing designated as a continuum, we will actually be misunderstanding "continuity". This may be the root of our difference, I want to talk about what it means to be continuous (continuity), you want to talk about a particular thing which is continuous (a specific continuum).

Don't you think that we need to first determine what it means to be continuous, before we can designate a particular thing as being a continuum? The problem I am having, is that you already presume to know what it means to be continuous, so you presume that you can move along and designate something as a continuum. I don't agree that you know what it means to be continuous, because your stipulated definition results in contradiction.

Take a blank piece of paper and draw a line with arrows at both ends, then draw a series of five equally spaced dots along the line between the arrows. Mark each dot with a numeral from 0 to 4. The drawing itself is not a continuous line with points along it, it is a representation - a sign - of a continuous line with points along it; the latter constitutes the sign's object. What we come to understand by observing (and perhaps modifying) the drawing is the sign's interpretant. All signs are irreducibly triadic in this way - the object determines the sign to determine the interpretant; the sign stands for its object to its interpretant. — aletheist

OK, I agree with your terminology here, but what is this "object" you refer to here. The line on the paper is a sign which represents an object. I assume that the object is an ideal. This ideal is a line which has the property of being continuous. From my perspective, we need to determine what it means to be continuous, (what it means to be a continuity) if we want to understand this ideal, the line which has the property of being a continuum.

There are three ways that a sign can be related to its object. In simple terms, an icon represents its object by virtue of similarity, an index by virtue of an actual connection, and a symbol by virtue of a convention. As a whole, the drawing is an icon; specifically, a diagram, which means that it embodies the significant relations among the parts of its object. Individually, the drawn line and dots are also icons of a continuous line and points, respectively; but they are symbols, as well, because we conventionally ignore the width and crookedness of a drawn line, as well as the diameter and ovalness of a drawn dot, since they are intended to represent a one-dimensional line and a dimensionless point. The arrows at the ends of the drawn lines are likewise symbols, conventionally suggesting the infinite extension of the line in both directions. The numerals labeling the dots are indices, calling attention to them and assigning an order to them as an actual measurement of the drawn line. They are also symbols, conventionally representing the corresponding numbers. — aletheist

To me, this is irrelevant because you are talking about how continuity is signified, not what it means to be continuous.

What can we learn about continuity from this diagram? We marked five points with dots and assigned numerals to them. Are those dots parts of the line? No, they are additions to the line; we did not draw any dots while drawing the line itself, we came back and drew them later. — aletheist

I agree here, the dots are not part of the line, in this representation, they are added signifiers.

Likewise, any point along a continuous line is not part of the line; it cannot be, because a continuum must be infinitely divisible into parts that are themselves infinitely divisible. — aletheist

Now I think you are making an unwarranted assumption. You are assuming that a continuum must be infinitely divisible. Let's drop that assumption for now, and wait until it is proven necessary, before we are forced to accept it as a logical necessity. We have an assumed continuous line, with no reason to deny that there are points assumed as part of that continuum. The line is composed of contiguous points, and this produces a continuity of parts. It is only when we want to make the point a defined ideal point, saying that it is zero dimensional, and indivisible, that the nature of the point becomes inconsistent with the nature of the line and therefore we are forced to conclude that the point is not a part of the line. But we need to respect the fact that just because the ideal point is inconsistent with the nature of the line, this does not mean that the continuous line is infinitely divisible.

It is only when we assume an ideal line, that we can designate it as being infinitely divisible. But we have choices when defining the ideal line. We might also define the ideal line as continuous. My argument is that these two are incompatible. If we define the ideal line as continuous, it is impossible that it is infinitely divisible, and if we define it as infinitely divisible it is impossible that it is continuous.

So, you are talking about an ideal line. Now you need to define your ideal line. You cannot define it as continuous and also infinitely divisible until you demonstrate that these two are compatible. I've already demonstrated to you that they are incompatible. No matter how you attempt to show that the continuous is infinitely divisible the result is contradiction, so I suggest you choose a new way to define your ideal line.

As I keep having to remind you, everything in pure mathematics is "imaginary" - ideal, hypothetical, etc. The question of whether there are any real continua is separate from the question of what it means to be continuous. We have to sort out the latter before we can even start investigating the former. — aletheist

The issue here is not one of simple imagination. I agree that pure mathematics, and ideals are imaginary. The issue is whether we can produce this image, the concept of continuity, the idea of what it means to be "continuous", while avoiding contradiction. If we cannot produce an ideal continuity without contradiction, then the ideal of "continuity" is logically unsound, fictitious, impossible, false. And this type of ideal is one which should be rejected.

This still reflects deep confusion about infinitesimals. They are not "points," and they are not "values." In our diagram, they are extremely short lines within the continuous line, indistinct but distinguishable for a particular purpose. — aletheist

I was referring to "values" because we were using "time" as our continuum. the same objection holds with your continuous line. If we remove a short section of line, we have not divided the continuous line, we have removed a section. No matter how infinitesimally small that short section is, you have removed a section, so this must be respected. you have created your division by removing a section. If it's a finite, continuous line, you cannot remove an infinite number of infinitesimal section, so you do not have infinite divisibility, in this way.

No matter how high the magnification, you would never see any gaps in the line; but you would also never find any place along the line where it would be impossible to divide it by introducing a discontinuity in the form of a dimensionless point. This is precisely what it means to be continuous - undivided, yet infinitely divisible. — aletheist

You do not seem to understand that if the line is continuous as you describe, inserting a dimensionless point does not divide it. The dimensionless point is completely invisible, it does not itself divide the line. And for all you know there could be an infinite amount of such dimensionless points already along the line, you wouldn't see them no matter how much magnification. So inserting a dimensionless point does nothing to divide the line, there could already be an infinite number of them there, and the line still be undivided. But you are not talking about an ideal line here, it is an observed line. If we are talking about an ideal line, one might refer to the existence of those dimensionless points to deny that the line is truly continuous. But this depends on how one understands "continuous".

When it comes to "what continuity really is," there is no "fact of the matter" - it is a mathematical concept, so we can define it however we like. — aletheist

That's not true at all. You cannot define a mathematical term in any way that you want. It must be defined in a way which is consistent with the existing conceptual structure. The problem here is that "continuity" is not a mathematical concept as you claim, it is rooted in the ontology of Parmenides. That, I thought was the topic of your op, whether the ontological concept of continuous could be consistent with mathematical concepts. Mathematicians want to apply mathematics to assumed continuums, so they give "continuity" a definition which is consistent with what they desire to do with the mathematics. But this definition ends up being inconsistent with ontological understandings of "continuity".

On the contrary, my interest is in a particular concept, not a particular terminology. Telling me that my definition is "wrong" is ultimately beside the point. — aletheist

Why are you not interested to know whether or not your definition is inherently contradictory? Your op clearly demonstrates that you are interested in the relationship between the discrete, the continuous, and mathematics. When someone demonstrates to you that your definition of "continuity" actually involves contradiction, why say that's "beside the point", I am only interested in my particular concept regardless of what you think? You should either defend your concept, or accept the demonstration of contradiction and move on toward a different conception. But to just keep stipulating your definition, and reasserting that this is the only concept of "continuity" which I am interested in, is rather pointless.

You remember how all this started, right? A contrived example. — Baden

Is a contrived example sufficient to demonstrate exclusivity, or is it a case of deception?

They are different, but unless you contrive them, they are probably not exclusive.

Your meaning is exclusive. — Baden

No it is the scientific definition which is exclusive. It reduces "seeing red" to a particular sort of seeing red, whereas the common understanding of seeing red includes the scientific instance as well as others.

Continuing from there, we collect ALL the atoms that were replaced in person A and reconstitute it as another body in its original configuration. Wouldn't you say this is person A? Isn't ship A the ship of Theseus? — TheMadFool

No, as I explained, I don't agree. Why would you think that completely annihilating an object, and then completely rebuilding a copy of the original object, with the same parts, constitutes having the same object?

... about your daughter’s missing teeth?

Then it's probably not the case that one is right and the other is wrong, there is just a difference. Since these are two very distinct ways of using "coulour", or "red", then what would be wrong would be to equivocate.

There's two ways of looking at it. I made it clear in my first post. I'm not saying either is nonsensical on its own terms. However, if you claim that to be red is just to look red, that's equivalent to saying there can be no science of colour. But there is. So, you're wrong. — Baden

What may be indicated here is that the "science of colour" is inconsistent with "colour" as we commonly use the word. If the scientist says that "red" refers to a very specific range of wavelengths, yet we see "red", and refer to a thing as "red" under all sorts of different conditions, then there is such an inconsistency.

In the vast majority of instances of perceiving colour, what is perceived is a combination of different wavelengths, not a single restricted range of wavelengths. The eyes have ways of dealing with those combinations. So speaking of a particular colour as a particular range of wavelength, is not very realistic.

No one doubts that A is a referent of ''the ship of Theseus. — TheMadFool

But you have two very distinct descriptions of A. You have the original ship, and you have a later ship which is built out of salvaged materials. The original ship is clearly "the ship of Theseus", because you have bestowed this identity upon it in your description. Later, you have built another ship out of salvaged lumber. Why would you give this ship the same name as the other ship? That makes no sense unless you are doing this intentionally to create ambiguity. This, other ship, which is built from salvaged material is what you should call ship B.

Furthermore, there is no "ship B" as per your description. Ship A can go through as many changes and repairs as you want, and we still identify it as ship A. To change its name to ship B because it has been repaired X number of times, is unnecessary and unwarranted. Identity is maintained through a continuity of existence, so ship A does not become ship B just from going through numerous repairs.

To explain, imagine if ship A was torn down at one go while simultaneously replacing the parts to build ship B. Permit me to use the word ''instantaneous'' here. So, if the whole exercise was done instantaneously there would be no doubt that A is the ship of Theseus. Ship B is just a copy of A.

Therefore, the gradual replacement of ship A's parts counts as a relevant factor in the paradox. But is it truly relevant? — TheMadFool

You should respect the fact, that to completely destroy something is to annihilate it, and deny the continuity of its existence and therefore identity. To rebuild a copy, is as you say, to rebuild a copy. But this is different from your original description which has ship A being repaired, and its continuity of existence maintained, and its identity maintained.

Good observation! — Wayfarer

The principal difference between panpsychism and substance dualism is how they each attribute the source of activity of living creatures. Biology understands living beings as active. Physics understands matter as passive, inert. So philosophical speculations may tend toward contriving ways in which matter could be active, living. The dualist way is to posit a soul, which is a separate, active, immaterial substance, that has the capacity to affect matter at its most fundamental level, causing matter to move in ways unknown to physics. The panpsychic way is to posit the source of such activity as inhering within matter itself.

As I have stated over and over, in this thread and others, a true continuum is that which has parts, ALL of which have parts of the same kind. — aletheist

There is a very real problem, which Plato demonstrated in numerous dialogues, with working in philosophy using stipulated definitions. What happens, is that if the stipulated definition of the word is not consistent with how the word is actually being used in society, then when we try to find the real existence of the thing referred to by the word, we get lost, incapable of finding that object, mislead by the stipulated definition. A very good example is found in the Theaetetus. The participants in the dialogue approach "knowledge", with the preconceived notion (stipulation) that knowledge excludes falsity, and mistake. So when they proceed to look at all the different ways in which knowledge could exist, in actuality, they are stymied because none of these is capable of excluding mistake. So it appears like there is no such thing as knowledge in the world, because no human process can exclude the possibility of mistake. At the end of the dialogue, Socrates points out that perhaps the problem is that they have approached knowledge with the wrong stipulation. They themselves were wrong to approach "knowledge" with this preconceived notion, because the thing which we actually call "knowledge" in the real world, doesn't exist like this, knowledge doesn't exclude the possibility of mistake.

This, aletheist, is what I think you are doing with "continuity". You approach continuity with a stipulated definition. And this stipulation is impeding your ability to understand what continuity really is. And just like Plato's stipulated meaning of "knowledge", it doesn't matter how many thousands of others have utilized this stipulation, you just blindly follow them down a mistaken pathway. This is why Plato introduced the dialectical method. This method allows us to approach a word like "continuity" without stipulations as to what that word means, and analyze its usage to find out what it really means.

Perhaps, but if we are discussing why we see blue strawberries as red, we might want to remind ourselves that we are not looking at strawberries, but a picture of strawberries.

It should be noted that seeing strawberries and seeing a picture of strawberries, is not the same thing. I'm near sighted, and wear glasses to see things far away. But when I'm shown a picture of things far away, I have to take off my glasses to see well, those objects in the picture. So clearly it is a distinct process by which you see a picture of an object from actually seeing that object.