What could it mean for a thing to contain parts but these parts are indefinite? — Metaphysician Undercover
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. — Metaphysician Undercover
Such a "real contiuum" is meaningless. It's a set without any members-- an infinite of nothing at all. — TheWillowOfDarkness
Pierce fails to recognise it is a conntuim because he's still stuck trying to account for the infinte by the finite. — TheWillowOfDarkness
If and when you ever come to understand this, you will then finally understand what Peirce and I mean by a true continuum. — aletheist
I know very well how Peirce defines a true contiuum. — TheWillowOfDarkness
He's failed to understand that a set or collection can be infinite, that it is not defined in a bottom-up manner as a sum of its parts. Consisting of finite members does not mean being finite. — TheWillowOfDarkness
Exactly the point, yours and Peirce's concept of true continuum is incoherent and will never be understood ... And this produces the incoherent notion of an indefinite part ... — Metaphysician Undercover
One of the fundamental properties of Peirce’s continuum consists in its reflexivity, a finely grained approach to Kant’s conception that the continuum is such that any of its parts possesses in turn another part similar to the whole: "A continuum is defined as something any part of which however small itself has parts of the same kind." We will use the term “reflexivity” for the preceding property of the continuum since, following a reflection principle, the whole can be reflected in any of its parts.
As immediately infers Peirce, reflexivity implies that the continuum cannot be composed by points, since points - not possessing other parts than themselves - cannot possess parts similar to the whole. Thus, reflexivity distinguishes at once the Peircean continuum from the Cantorian, since Cantor’s real line is composed by points and is not reflexive. In Peirce’s continuum the points disappear as actual entities (we shall see that they remain as possibilities) and are replaced - in actual, active-reactive secondness - by neighbourhoods, where the continuum flows ...
We will call inextensibility the property which asserts that a continuum cannot be composed of points. As we mentioned, a continuum’s reflexivity implies its inextensibility (Peirce’s continuum is reflexive, thus inextensible), or, equivalently, its extensibility implies its irreflexivity (Cantor’s continuum is extensible, thus irreflexive). The fact that Peirce’s continuum cannot be extensible, not being able to be captured extensionally by a sum of points, retrieves one of the basic precepts of the Parmenidean One, “immovable in the bonds of mighty chains,” a continuous whole which cannot be broken, “nor is it divisible, since it is all alike, and there is no more of it in one place than in another, to hinder it from holding together, nor less of it, but everything is full of what is.” — pp. 13-14
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
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
There isn't a conflict. Ideal continuity is present. Any infinite can't be divided such that it said to begin or end. We don't divide continuous things, such as objects and wholes at all. Our "cutting" of a whole is merely picking out something specific. It doesn't affect continuity. If I pick out a rock, it doesn't make the whole of the world go away. The whole remains, uncut and indivisible, no matter how many times we might suggest we separate it, an all together different object to the individual states we pick out.
A whole has or is no parts, even as parts belong to it. When we pick out a part, the whole remains and is undivided (and is indivisible).
The mistake people make is thinking wholes as defined by parts in the first instance, such selecting a part would somehow divide and destroy the whole. It hides the indivisible nature of whole from us and sees us misread the wholes we do encounter as failed continuity. — TheWillowOfDarkness
Our "cutting" of a whole is merely picking out something specific. — TheWillowOfDarkness
There isn't a conflict. Ideal continuity is present. Any infinite can't be divided such that it said to begin or end. — TheWillowOfDarkness
If we associate continuity with the whole, with the unity, then each time we divide something, we destroy a continuity, and create new continuities. — Metaphysician Undercover
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
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
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. — Metaphysician Undercover
The description changes, not the intrinsic continuity. — Rich
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 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
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
The problem with this perspective is that what exists between the described states, is activity, becoming — Metaphysician Undercover
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. — Metaphysician Undercover
However, it is possible that the only real continuities, are those physical, spatial entities which can be divided, but not divided infinitely. — Metaphysician Undercover
However, if we establish that they are continuous, then we can investigate whether anything within space and time is also continuous. — aletheist
There are no states in nature. — Rich
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
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
Again, whether there are any real continua is a separate question from what it means to be continuous. — aletheist
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
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. — Metaphysician Undercover
I think that the ideal continuum cannot be divided at all, because it has no parts. — Metaphysician Undercover
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. — Metaphysician Undercover
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