Following on, this also means particular continuities don't have a beginning or end. Yes, any given object has a start and end — TheWillowOfDarkness

Impossible to do so this. Everything is in a constant change as much as at the edges as anywhere else, where we have clouds not edges.

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.

I'm not talking about energy here, I'm talking about the physical things in the room. — Metaphysician Undercover

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? And what do believe surround us? The energetic form is simply moving within an energy field as a wave moves in water. There is no separation.

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.

To describe a thing, and to describe the activity of a thing, is two distinct description. — Metaphysician Undercover

As a discussed earlier, the problems involved in accurate descriptions is not at issue here, e.g. how to describe things that are separate from their environment. Usually approximate descriptions are sufficient for practical purposes. What is at issue is a precise description of the nature of nature. Physical objects, as normally understood, in themselves have degrees of substantiality, e.g. energy, air, water, fire, humans, rocks etc.. The continuum between it all cannot be broken though substantially most certainly can be sensed to a certain degree.

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.

An individual is a physical object and it is divisible (the name "individual" is misleading) ... 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. — Metaphysician Undercover

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. As for "individual," if you look into its etymology, you will find that it has the same root as "indivisible"; one is a noun, the other is an adjective, but they originally meant the same thing - much like "continuum" and "continuous." Nevertheless, since "individual" has come to have a different meaning in common usage, and this seems to be an obstacle for you, we can set that term aside for the sake of clarity and simply substitute "indivisible" as a noun. Restated accordingly: There has to be a way to distinguish a continuum (such as a line) from an indivisible (such as a point).

There is probably more than one contradiction in this description, but I'll try to sort it out ... We have already determined that it is impossible that there is something real which satisfies your definition of "continuous", because it is contradictory ... You want to take "continuous", and give it an ideal definition which has already been shown to be contradictory. — Metaphysician Undercover

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. I get that you disagree with me/Peirce on all this, but I have addressed each of your objections, even if you remain unsatisfied with the result. I have to wonder if you keep saying this over and over because you are still trying to convince yourself.

This is a collection of discrete individuals. Being described as consisting of parts indicates that it is discrete. — Metaphysician Undercover

No, a continuum per my/Peirce's definition is not a collection of individuals at all, and having potential parts clearly does not entail that it is discrete; it would only become discrete if it were somehow divided into indivisible parts. 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.

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." — aletheist

I still suspect that this right here is what you are perceiving as contradictory, perhaps because you are locked into the standard rules of classical logic. As Peirce explained, "continuity is simply what generality becomes in the logic of relatives" (CP 5.436, 1904), and "anything is general in so far as the principle of excluded middle does not apply to it" (CP 5.448, 1905). 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. Intuitionist logic does not uphold excluded middle - or, for that matter, double negation elimination - and thus may be better suited for reasoning about true continuity than classical logic. Excluded middle does not apply universally in smooth infinitesimal analysis, although it does hold for particulars; this is one reason why it seems like a promising candidate for mathematically modeling true continuity.

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.

But we've already determined that mathematics refers to discrete units. — Metaphysician Undercover

The first four questions that I posed in the OP were as follows.

• Is contemporary mathematics inherently discrete, such that it is incapable of accurately capturing the philosophical/ontological notion of real continuity?
• If so, what specific errors and misconceptions have resulted (and propagated) from thinking otherwise?
• Is there any way that mathematics could evolve going forward that would enable it to deal with continuity more successfully?
• If so, what are some specific alternatives?

Based on the ensuing discussion, my answers are yes; see most of MU's posts; yes; and category theory, in particular smooth infinitesimal analysis. 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.

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. — Metaphysician Undercover

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.

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. — Metaphysician Undercover

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?

To be divisible, it requires this spatial extension, and this means that to be divided it requires extension outside the mind ... The truly ideal line cannot be divided. — Metaphysician Undercover

Complete and utter nonsense.

If it cannot be "so divided", then how is it divided? — Metaphysician Undercover

Is your reading comprehension that poor, or are you just being obtuse? 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.

Why are the ideal points not part of the ideal line? — Metaphysician Undercover

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

As I said, I do not agree with the way that Peirce dismisses logical principles. — Metaphysician Undercover

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.

Unless it can actually be divided, it is false to say that it is divisible. — Metaphysician Undercover

I suppose it was inevitable that we would end up right back here, at square one. Cheers.

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

There is a break between any object and a plan about an object. A plan, as an object, is a different finite state. I make a plan-- it's a state that begins and then ends. Then I have a created object, another state that begins and ends; a plan has ended, the planned state begun.

Niether of these objects are a whole, either of the plan or the object. The wholes are indivisble and so remain untouched. 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. If one could break a contiuum, it not indivisible. The infinite nature of a contiuum means it must be unaffected by beginnings and ends.

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.

There is no such thing as a mathematical continuity, you are making that up. — Metaphysician Undercover

That would be news to mathematicians.

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

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

If it is necessary that the continuum is undivided, then it is not possible to divide it. — Metaphysician Undercover

It is not possible to divide it and still have 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. — Metaphysician Undercover

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

... you come up with the idea that the two parts produced are mathematically equivalent to the continuum. — Metaphysician Undercover

I have never said any such thing.

The ideal line is defined as consisting of a succession points, and is therefore not continuous, it is discrete. — Metaphysician Undercover

You are clearly not paying attention at all.

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? — Metaphysician Undercover

I didn't.

I said the beginning and end, or any other point of an object, was not the whole of an object. I never said an object wasn't whole. By it's own right, as an individual entity, a whole, we can say it's a whole. In other words, there are only perfect wholes and one is expressed by every single object.

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. — Metaphysician Undercover

The point is it does suffice just be a unity. Since this is the case, the problem you present is nothing more than a red-herring-- 5, 8, 12 all have their own unity, as does 1 and the set of real numbers. All are compete and indivisible. When picks out a number from the set of real numbers, it doesn't make the set of real numbers divided. When we divide 8 by 2, it doesn't undo the unity of 8. And so on. They are all perfect in their wholeness.

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

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. — Metaphysician Undercover

Yes.

No, all wholes or continua are perfect.

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.

Impossible, all wholes are perfect. "More perfect" is an oxymoron. Perfection means "the best."

still a lot more to read in this thread but Kant published a less than popular work entitled, Negative & Positive Sums, at least in referenxe to my education. 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. Fichte expanded way more on the topic but there is something to consider in regarding this exostence as The World of Zero.

In reference to quantum mechanics, I am right on your sids. The minute truths of existence express themselves through their larger appearances. But they are two sides of the same coin. A big problem with the way a lot of people will approach this is due to Kant. Most believed he killed metaphysics, but nothing of the sort. He simply seperated metaphysical discource from emprical discource because the terminologies seldom overlapped. But in regards to your comment, there is a tretise called Pascal's Infinite Indefinitism that states, no matter how far up we look,and no matter far down we reach, we always reach an indivisble point. A point of pure energy. The discrete can only express a contiuum by reference to its connection to that pantheistic layer of reality. The greeks termed it, "Han Kai Pan", the one that is all.

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. They c are far too abbreviated and limited by the symbols they use. The poor modern novelists would use pages upon pages of words to describe a single experience and still fall short of describing the fleeting memory of such experience and the experience is interwoven with so many others.

First, before any description, one must penetrate deeply with all faculties. Forget about all the symbolic tools learned in school. And once one behind to fanthom the actual experience, then and only then it's one ready to attempt to create a metaphor that may describe that experience.

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 way to say this is where it begins and this is where it ends, yet it is all there.

The discrete can only express a contiuum by reference to its connection to that pantheistic layer of reality. The greeks termed it, "Han Kai Pan", the one that is all. — Wolf

This would even go deeper, and it certainly shares much with Daoism. If the nature that we are connected with is all continuity and waves, is there an opposite of Singularity from which it all began - the Dao?

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.

But mathematicians specialize in mathematics, not ontology. — Metaphysician Undercover

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

Ideals are timeless truths. — Metaphysician Undercover

I am only talking about ideal states of affairs in this thread, not "ideals"; creations of mathematical imagination, not "timeless truths."

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

Once again - pot, kettle, black.

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?

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? — Metaphysician Undercover

Descriptions are necessarily limited, inaccurate, imprecise, and provide no avenue to understand the nature of nature in themselves. They are simply a tool for communication which may or may not help two explorers to better understand. To this end, I have always felt metaphors to be far more helpful.

Have you ever tried describing duration in words or mathematics? Just your own memory as it flows continuously and unceasingly and never stops evolving. You should try it. Direct observation of what you are suggesting is possible. Stop duration, create a state, and describe it while still observing your efforts to describeit in the same duration. with such an attempt you should witness the impossibility of what you are suggesting as should anyone who believes that mathematics, words, logic, or any symbol is adequate to describe the nature of experience in duration.

ignoring the things which are passive, or unchanging — Metaphysician Undercover

A single example?

Descriptions are necessarily limited, inaccurate, imprecise, and provide no avenue to understand the nature of nature in themselves. They are simply a tool for communication which may or may not help two explorers to better understand. To this end, I have always felt metaphors to be far more helpful. — Rich

I don't understand how a metaphor is a better means for understanding the nature of nature than a description is.

Have you ever tried describing duration in words or mathematics? — Rich

OK, this is how I describe duration. I recognize a difference between past and future by means of memory and anticipation. This gives me a sense of being present. As I am aware of being present, I notice that things are changing while I am present, and I can refer to duration through describing these changes which occur.

Stop duration, create a state, and describe it while still observing your efforts to describeit in the same duration. with such an attempt you should witness the impossibility of what you are suggesting as should anyone who believes that mathematics, words, logic, or any symbol is adequate to describe the nature of experience in duration. — Rich

So at the same time that I am noticing changes, which enable me to describe duration, I also notice things which are not changing. I can describe these things as not changing, for the entire duration of the change which I am describing.

A single example? — Rich

So for example, I have the blueprint for the layout of my kitchen, and this is a description of the things which are not changing in my kitchen. It describes where the cupboards, counters, the sink, the stove, and the fridge are, and this is a static description which persists and remains true through this day. I can stand in my kitchen, frying eggs in the frying pan, and this is a change which I can focus on to give me a sense of duration, because it always takes about the same amount of time to fry the eggs. But at the same time I can notice that the location of all the cupboards, counters, sink, stove and fridge, remain static, in the same place, throughout that duration of time.

I don't understand how a metaphor is a better means for understanding the nature of nature than a description is. — Metaphysician Undercover

Metaphors are more of a holistic, active image that two people may share. Something like one picture is worth a thousand words.

OK, this is how I describe duration. I recognize a difference between past and future by means of memory and anticipation. This gives me a sense of being present. As I am aware of being present, I notice that things are changing while I am present, and I can refer to duration through describing these changes which occur. — Metaphysician Undercover

I do not mean give a brief definition of duration. I am suggesting that you actual describe an actual experience in duration as you are describing it. This provides an actual observation of your own duration and the impossibility for you to describe it. I am asking for a more direct experience.

So at the same time that I am noticing changes, which enable me to describe duration, I also notice things which are not changing. I can describe these things as not changing, for the entire duration of the change which I am describing. — Metaphysician Undercover

If you are attempting to describe your own duration directly you may notice that your act of describing is melting into what you are trying to describe. There is no "state" . There is a continuous flow of one into the other. Observe closely that one memory that you are attempting to describe is flowing directly into the description itself, continuously and unceasingly. It cannot be stopped long enough for you to describe it. In other words, your act of describing is within that which your are attempting to describe.

I am not asking you to describe some past memory, which will be as complete as you may remember and subject to change, I am asking you to describe duration as you are experiencing it.

So for example, I have the blueprint for the layout of my kitchen, and this is a description of the things which are not changing in my kitchen. — Metaphysician Undercover

So, this is s metaphysical viewpoint that I cannot argue because it is something you believe very strongly. However, if I was to be put in the same kitchen, I would observe everything changing on the macroscopic level (the dust in the air, the deterioration in the wood, your life itself, the ink on the paper), and at the microscopic level (the energy of all quanta). This is why I say, philosophers need to be constantly exercising their observation skills via the arts. I first learned of the skill in the art of observation when I studied photography many years ago. A philosopher must always be exercising and refining the art of observation.

This provides an actual observation of your own duration and the impossibility for you to describe it. I am asking for a more direct experience. — Rich

If that's not an example of my experience of duration then I don't know what you are asking. To me it's an example of my experience of duration. What more are you asking for?

However, if I was to be put in the same kitchen, I would observe everything changing on the macroscopic level (the dust in the air, the deterioration in the wood, your life itself, the ink on the paper), and at the microscopic level (the energy of all quanta). — Rich

As I said already, I don't deny that some things are changing, but I also don't deny that some things are staying the same. You, for some reason seem to be intent on denying that there are some things around you which are staying the same in time, and that we can describe these things, and notice that the things described remain the same.

This is why I say, philosophers need to be constantly exercising their observation skills via the arts. I first learned of the skill in the art of observation when I studied photography many years ago. A philosopher must always be exercising and refining the art of observation. — Rich

If you cannot see that there are things around you which remain the same through a duration of time, then I don't think that you are very good at the art of observation.

Yes, I understand. You really can't see everything in constant motion. This would certainly affect anyone's metaphysics.

I believe that the earth is moving, but that fact is irrelevant to the fact that the layout of my kitchen remains the same. That's the point, we have to have respect for what is staying the same, as well as what is changing.

I believe that the earth is moving, but that fact is irrelevant to the fact that the layout of my kitchen remains the same. That's the point, we have to have respect for what is staying the same, as well as what is changing. — Metaphysician Undercover

It is quite clear to me that everything is changing in one manner or another all the time. There is nothing I can say or do to convince you of this. It must come to you by your own personal observations. Until then, your metaphysics will be determined by your belief that some things don't change some of the time. For me, it is an impossible chasm to cross until I observe something that is not changing. At this same point, physics itself will come crashing down along with my own personal beliefs.

It is quite clear to me that everything is changing in one manner or another all the time. There is nothing I can say or do to convince you of this. — Rich

If this were true, then how do you explain the fact that the layout of things in my kitchen is the same still as it was ten years ago? Everything is in its proper position. It's easy for you to assert that everything is changing, even if what you say is false, because people state untruths all the time. So if you want me to take you seriously, you should be prepared to explain to me how there are all these relationships around me which appear to stay the same. I can take a tape measure, and measure things, minute after minute, hour after hour, day after day, and show you that they are staying the same. Do you really believe that the distance between my fridge and my stove changes from one minute to the next? Why are you so convinced that these things aren't really staying the same for any length of time? If you could explain to me how the distance between the fridge and the stove changes from one minute to the next, when my tape says that it stays the same, then perhaps I might believe you.