Contingency means that "either P or not-P might not be actual." — aletheist
If you don't know, just admit it! — tom
There is no first real number after 0 with the standard order; there is an uncountable infinity of real numbers between 0 and any arbitrarily small but finite value that one chooses. However, they are all still individual real numbers, thus forming an analytic or compositional continuum, rather than a synthetic or true continuum. — aletheist
I can't really agree with your framing here as my point was that P is only truly actualised to the degree that not-P (as its generic 3ns context) is also actualised. — apokrisis
That is, 2ns is about actualised degrees of freedom - a degree of freedom being a determinate direction of action, or an existent with a predicate. — apokrisis
The purely formal conception that the three affections of terms, determination, generality, and vagueness, form a group dividing a category of what Kant calls "functions of judgment" will be passed by as unimportant by those who have yet to learn how important a part purely formal conceptions may play in philosophy. — CP 5.450, 1905
None of these numbers, except a measure zero fraction, can be represented physically in any way - they are non-computable. — tom
The only reason you can tell they are there is because you know, from the properties of the continuum, that they must exist. — tom
Your claim that indistinguishable numbers are individual is simply a contradiction. — tom
Kudos for quoting Peirce, but I still think that you do not properly understand him. — aletheist
The indivisible present is not a part of time, because time does not consist of indivisible instants; since it is continuous, it is infinitely divisible into durations that are likewise infinitely divisible into durations. An indivisible point is not a part of a line, because a line does not consist of indivisible points; since it is continuous, it is infinitely divisible into lines that are likewise infinitely divisible into lines. — aletheist
Peirce's insight was that time cannot be divided into durationless instants, only into infinitesimal durations; likewise, a line cannot be divided into dimensionless points, only into infinitesimal lines. We can mark time with indivisible instants, such as "the present" or "the primary when" that corresponds to the completion of a change; and we can mark a line with indivisible points. However, those instants are not parts of time, just as those points are not parts of the line. — aletheist
When we divide the line, or divide time between past and future, it is not that we insert a point into the line, or insert "the present" into time, we assume that these points of potential division are within the line or within time itself, and we utilize these points for division. — Metaphysician Undercover
Actually, it wasn't Peirce I quoted, it's a book entitled "The Continuity of Peirce's Thought", by Kelly A. Parker. — Metaphysician Undercover
So I think you agree with me, that it is contradictory, that the indivisible point is within the continuously divisible continuum, and the continuum cannot be divided in this way. — Metaphysician Undercover
But what happens when we divide time in Peirce's way, is that we lose an infinitesimal piece of the order. — Metaphysician Undercover
To claim that two things are the same when it is stated that there is a difference between them, is contradiction. — Metaphysician Undercover
So neither of these proposals, the indivisible point, nor the infinitesimal point, represent an acceptable resolution to the problem of dividing the continuity, they both involve contradiction. — Metaphysician Undercover
But semiotics transcends physics because it can imagine its marks as having zero dimensionality. So we have to recognise the computational aspect of this too. — apokrisis
But in the imagined world of maths - Hilbert's paradise - we can imagine infinitely sharp blades and cuts made ever finer with no issue about the cuts getting mushed or vaguer and vaguer. — apokrisis
Yet while there are two worlds - matter vs sign - in semiotics they are also in mutual interaction. So that gives you the third level of analysis that would be a properly semiotic one ... where sign and matter are in a formal, generically-described, relation. Or pragmaticism in short. The triadicity of a sign relation.
And that is when we can ask about a third, deepest-level, notion of the continuum - one in which the observer, or "memory" and "purpose" are fully part of the picture. It is no longer just some tale about either material cuts or symbolic marks - a bare tale of observables. — apokrisis
Tell you what, read Parker's whole book - or better yet, read some actual Peirce - and then get back to me if you still think that an infinitely divisible continuum is somehow inherently contradictory. — aletheist
How is this different from what I have been saying all along - that there are no indivisible points in a truly continuous line? Why do you suddenly claim to agree with me now, after arguing with me about it all this time? What changed your mind? — aletheist
It is not necessarily a contradiction - I am the same person that I was yesterday, and also different; almost any object that I observe is the same object now that it was a minute ago, but also different. Regardless, the claim in this case is that two things are indistinct, but distinguishable; and this is clearly NOT a contradiction. — aletheist
You are still stuck on the idea of points. Infinitesimals are NOT points of ANY kind, they are extremely short lengths of line. As for your example, all numbers are intrinsically discrete; so the number 2 is an indivisible, not an infinitesimal. Think of it this way - what are the "parts" of the number 2? Mind you, I am not referring to smaller numbers that can be added up to reach 2, but the number 2 itself, as a single "point" on the real number "line." 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. The number 2 cannot be a part of any continuum, because the number 2 itself does not have any parts! — aletheist
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
OK, but we need to relate semiotics to a continuity. — Metaphysician Undercover
The issue with continuity, or the continuum, is whether or not it is something real, or just imaginary. — Metaphysician Undercover
I was using "2" only as an example. It was supposed to represent an infinitesimal value. If you say that it can't, we can use something else to represent the infinitesimal value. Let's use X. X represents an infinitesimal value. — Metaphysician Undercover
We have a continuous order, and we divide it at X. — Metaphysician Undercover
You approach continuity with a stipulated definition. And this stipulation is impeding your ability to understand what continuity really is. — Metaphysician Undercover
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. — Metaphysician Undercover
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
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
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
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
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
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
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
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
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
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
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. — Metaphysician Undercover
I don't agree that you know what it means to be continuous, because your stipulated definition results in contradiction. — Metaphysician Undercover
But further study of the subject has proved that this definition is wrong. It involves a misunderstanding of Kant's definition which he himself likewise fell into. Namely he defines a continuum as that all of whose parts have parts of the same kind. He himself, and I after him, understood that to mean infinite divisibility, which plainly is not what constitutes continuity since the series of rational fractional values is infinitely divisible but is not by anybody regarded as continuous. Kant's real definition implies that a continuous line contains no points.
Now if we are to accept the common sense idea of continuity (after correcting its vagueness and fixing it to mean something) we must either say that a continuous line contains no points or we must say that the principle of excluded middle does not hold of these points. The principle of excluded middle only applies to an individual (for it is not true that "Any man is wise" nor that "Any man is not wise"). But places, being mere possibles without actual existence, are not individuals. Hence a point or indivisible place really does not exist unless there actually be something there to mark it, which, if there is, interrupts the continuity. I, therefore, think that Kant's definition correctly defines the common sense idea, although there are great difficulties with it.
I certainly think that on any line whatever, on the common sense idea, there is room for any multitude of points however great. If so, the analytical continuity of the theory of functions, which implies there is but a single point for each distance from the origin, defined by a quantity expressible to indefinitely close approximation by a decimal carried out to an indefinitely great number of places, is certainly not the continuity of common sense, since the whole multitude of such quantities is only the first abnumeral multitude, and there is an infinite series of higher grades.
On the whole, therefore, I think we must say that continuity is the relation of the parts of an unbroken space or time. The precise definition is still in doubt; but Kant's definition, that a continuum is that of which every part has itself parts of the same kind, seems to be correct. This must not be confounded (as Kant himself confounded it) with infinite divisibility, but implies that a line, for example, contains no points until the continuity is broken by marking the points. 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.
In the calculus and theory of functions it is assumed that between any two rational points (or points at distances along the line expressed by rational fractions) there are rational points and that further for every convergent series of such fractions (such as 3.1, 3.14, 3.141, 3.1415, 3.14159, etc.) there is just one limiting point; and such a collection of points is called continuous. But this does not seem to be the common sense idea of continuity. It is only a collection of independent points. Breaking grains of sand more and more will only make the sand more broken. It will not weld the grains into unbroken continuity. — CP 6.168, c. 1903-1904, paragraph breaks added
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. — Metaphysician Undercover
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
On the whole, therefore, I think we must say that continuity is the relation of the parts of an unbroken space or time.
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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
"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
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. — Metaphysician Undercover
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
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. — Metaphysician Undercover
My claim is that even to say that it consists of parts, is to state a contradiction. — Metaphysician Undercover
I even said it (about Peirce) in the OP (emphasis added):It is not composed of parts, it can only be divided into parts, all of which can likewise be divided into more and smaller parts of the same kind. — aletheist
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.Charles Sanders Peirce likewise took strong exception to the idea that a true continuum can be composed of distinct members, no matter how multitudinous, even if they are as dense as the real numbers. — aletheist
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? — Metaphysician Undercover
We can cut a number line anywhere because its infinite particular members. — TheWillowOfDarkness
The continuum is its own object, not merely a sum of every finite member. — TheWillowOfDarkness
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
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
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
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