Contingency means that "either P or not-P might not be actual." — aletheist

So to complete the pattern, 2ns would need to be characterised by a failure of distribution of the principle of identity. And you are saying 2ns is really to be labelled contingency rather than actuality?

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.

Returning to my original remark about a "tension", vagueness is pure contingency while 2ns is constrained or contextualised contingency. 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.

So it is confusing to call 2ns contingency when 1ns is usually regarded as the maximally contingent. Really, 2ns is contingency limited, regulated, contextualised.

My argument has been that the principle of identity makes a claim that a thing is the same as itself by definition - it appears no context or larger relation is needed and no contingency or uncertainty could be involved.

The laws of thought make identity the concrete and completely uncontingent starting point for then reasoning about the particular. Particularity is claimed as an atomistic fact and so off logic can merrily trot to derive its further two laws.

Peirceanism then stands against that with its holism. Now the concrete particularity of identity - the category of the actual - is instead the emergent intersection of the possible and necessary, the local potential and a wider context of constraint.

So actuality or 2ns becomes the transition zone. It reflects the mixing of the polar extremes of being - vague possibility and crisp generality. Or total freedom vs total constraint. Actuality is actualisation - the process of coming to be framed by the limits on being. A developmental arc is being described that (for me, if not for you) goes from vague 1ns to generic 3ns via the concrete foothold or symmetry breaking which is bald 2ns - a difference that can make a difference in that it does serve to construct, or at least continue to reinforce, some large state of 3ns habit.

The emergent nature of 2ns or the concretely particular is what makes for ambivalence. We are talking about actuality - but the concreteness is secured by the 3ns it anticipates. Context is what gives the particular its definite character, what allows it to be seen and remembered as an occasion that is the same as itself/different from aught else. And this in turn means the particular has been sharply formed by the pruning away of all unnecessary possibilities. Identity is arrived at apophatically.

So there is a pattern to be completed. The law of identity ought to have an exact apophatic definition in the "true meaning" of 2ns, or actuality.

And what does identity presume most? It presumes brute existence instead of emergent development. It presumes a pure state rather than a mixed state. It presumes it stands at the beginning rather than arriving at the end.

Yet then "actuality" in semiosis requires the wholeness of 3ns (the 3ns that incorporates the 2ns and 1ns). So the identification of 2ns as actuality - or better yet, actualisation - has to be understood in that light.

Thus in terms of your logical formalisation - "contingency means that "either P or not-P might not be actual" - it seems to me rather that we are talking not about contingency but about actualisation. So the category of contingency reduced to its deterministic minimum by the constraint of a generality - ie: a freedom that has a direction.

You simply seem to be re-stating the fact that the PNC does not apply to the vague (the vague being the radically contingent).

So Vagueness means that "both P and not-P are possible." = Vagueness means that "either P or not-P might not be actual."

Although, as I say, vagueness defined directly is the degree to which P and not-P are co-jointly not actual. This captures the anticipated 3ns which is the further rule that actuality is irreducibly contextual. So it takes a matching degree of P and not-P for actualisation. And a matching absence of P and not-P for there to the maximum indistinctness or lack of identity.

In vagueness, P is indistinguishable from not-P. In actualiity, they are as distinct from each other as possible. And in generality, that actualised counterfactuality is not merely a one-off event but a habit, a law, a routine state of affairs, an irreversible fact of history.

If you don't know, just admit it! — tom

You need to try harder to keep up with the thread. You are still wanting to construct your numbers, and yet the point being made is that the continuum needs to be cut or divided - which is an act of primal constraint, not construction.

So every cut of the continuum must leave behind a continua that is capable of being cut again. Thus every naming of some "first number" must allow the naming of yet still earlier numbers ... as the continuous can never be computationally erased. Constraint isn't just subtraction or negative addition. It is what it says, a limitation marking continuity. And the divisibility of the continuum is inexhaustible. Every named number - in attempting to cut a part of the line away from the whole - still leaves a bounded line segment.

So the answer in terms of a constraints-based understanding of number is just obvious.

And if your own constructive viewpoint actually could account for the numberline, then you would be granting a zero dimensional point some actual size. Which is why the paradox implicit in your constructive viewpoint was also the bleedingly self-evident since Zeno first put stylus to wax.

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

Sure, an uncountable infinity of real numbers exist within any finite interval, but you can't identify them and can't distinguish them.

None of these numbers, except a measure zero fraction, can be represented physically in any way - they are non-computable. The only reason you can tell they are there is because you know, from the properties of the continuum, that they must exist.

Your claim that indistinguishable numbers are individual is simply a contradiction.

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 fine, I was just playing around with another angle. I like how you stated this.

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

I finally found where Peirce did make a trichotomy with vagueness (1ns) and generality (3ns), but not with a third type of indeterminacy; rather, his third term was determination (2ns).

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

Nevertheless, consistent with your approach, something is determinate only to the extent that it is distinguished from its context. Identity in this sense, as (∀x)(Px = ¬¬Px), defines double negation elimination, rendering LNC and LEM equivalent for anything to which it applies.

None of these numbers, except a measure zero fraction, can be represented physically in any way - they are non-computable. — tom

Are you now taking up the argument that MU always insists on making? Pure mathematics has nothing to do with what is actual, physical, or computable.

The only reason you can tell they are there is because you know, from the properties of the continuum, that they must exist. — tom

That is all it takes for them to fail to qualify as a synthetic/true continuum - "that they must exist" as (individual) numbers.

Your claim that indistinguishable numbers are individual is simply a contradiction. — tom

All numbers are distinguishable in principle. That is part of what it means to be a number.

Yes, I agree that we do impute individual identity to entities in the absence of complete knowledge of the said entities, and that is the point of the 'Pluto' example.

And I also agree that we think that identity as being independent of our thinking of it. So, logically, the entities have their identity completely independently of our imputations of identity. In fact it is more or less taken for granted that we impute identities, whether correctly or incorrectly, on account of the actually independent identities of entities. But we are thinking all this from the standpoint of our imputations and their logical corollaries.

It is assumed that we are led to impute identity by the independent nature of things, and that our imputations therefore reflect, or better express, the independent nature of things. But we cannot give an account of that process of expression from the independent things to our expression of them, because we are 'inside' our expression of them. We thus cannot say how identities could 'be' in an absolutely independent actuality. This is why it has been said that identities are in God or substance, and that our logical expressions of identity are finite expressions of infinite identity.

That this discussion had become so convoluted and opaque is just further evidence of what a mess mathematics makes of nature.

There are no numbers in nature. Numbers are symbols that we share with each other for some practical applications. All of mathematics for that matter is a bunch of highly limited symbols, which are forever changing, depending the application. It is simply a tool. Nothing more. And certainly should never be used or recognized as anything more that that.

If someone wishes to understand nature, as a philosopher might (but not necessarily), one needs to observe nature directly and experience it directly via music, art, sports, etc. Direct experience is what is required not a constrained set of incomplete and inadequate symbols.

I think I'll never see,
A poem as lovely as a tree.

This is philosophy of nature.

Direct experience is what is required not a constrained set of incomplete and inadequate symbols. — Rich

Ah. Direct experience. Good luck with that. :)

It's been extremely successful. Don't watch someone dance. Dance! There is no substitution for direct experience, observation, and increased awareness.

So why are you hanging around a philosophy forum instead of being out there jiggling your booty. Is dance ... not enough?

Believe it or not, there are other philosophers who are like me. But most are more likely to believe they can understand nature by manipulating different symbols on paper. Different strokes for different folks.

Kudos for quoting Peirce, but I still think that you do not properly understand him. — aletheist

Actually, it wasn't Peirce I quoted, it's a book entitled "The Continuity of Peirce's Thought", by Kelly A. Parker.

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

OK, now I think we satisfactorily understand each other's terms, that we can approach the problem. As Aristotle indicated, the continuity (continuum), is divided by the means of the indivisible point. But the continuum, if it is divisible, must be infinitely divisible, and therefore cannot consist of any indivisible points. The indivisible point would produce a discontinuity Do you agree with me, that this is a problem? 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. Once we remove the present, or the indivisible moment, from time, there is no apparent means for dividing time. That is why Peirce turns to the infinitesimal. 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.

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

Now Peirce's proposal is that the continuum consists of infinitesimal durations, like you say. But what happens when we divide time in Peirce's way, is that we lose an infinitesimal piece of the order. So according to the book I quoted there is an infinitesimal difference in the order between part A, and part B. The Peircean procedure is to say that this difference doesn't matter, and claim that the end of part A is the same as the beginning of part B, despite the acknowledgement that there is an infinitesimal difference between these two.

To claim that two things are the same when it is stated that there is a difference between them, is contradiction. So all that Peirce has done, is replaced the contradiction of having indivisible points within the divisible continuum, with another contradiction of saying that two different things are the same.

In case you are not understanding, refer to the example of "2". If "2" represents an indivisible point, within the infinitely divisible continuity, we have a contradiction. There cannot be an indivisible 2 in the infinitely divisible continuum. Now, if "2" represents an infinitesimal part of the continuum, then there is an infinitesimal difference in value between less than 2 and greater than 2. To claim that the highest value of "less than 2" is the same as the lowest value of "greater than 2", when it is stipulated that 2 is an infinitesimal part of the order, is also contradictory. 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.

What I tried to explain in the other thread, is that to truly resolve this problem we need to turn to some deeper metaphysical principles. Parmenides placed much importance in the principle of non-contradiction. He proposed a unity, one, continuous, indivisible, whole. In this instance the continuous is indivisible. In Aristotle's physics, and hylomorphism, the continuity of existence is provided for by the existence of matter. Matter is what persists, continues existing, through time. This is temporal continuity, which matter gives us. It is the form of an object which changes, and which is divisible, not the matter which is indivisible and continuous. In modern science, mathematics, with all its operations of additions and divisions, is applied to forms, it is not applied to the matter itself.

In reply to the op, I submit to you, that it is the mathematician's desire to represent the continuum as divisible, which is what leads to the afore mentioned problems, and ultimately contradiction. A true continuum must be as Parmenides describes, whole and indivisible. The claim by mathematicians, that the continuum is divisible can only result in infinite regress, and contradiction. That is because the continuum can only be understood as indivisible. Defining it as divisible is what causes the problems. To start with the assumption that it is divisible, is to start with a contradictory premise.

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

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.

For the hardware of the computer, a bit - the state of switch - is a purely physical thing. It has materiality and thus a cost involved in switching it back and forth. Eventually it will even wear out.

Yet then the same switch, the same bit, can also be a symbol, a sign, within a software's system of interpretance. The programmer can encode some model (representing a purpose) in a syntactical structure (a logical form), then run it on the machine. The switch flips back and forth, doing its entropic or material thing. Meanwhile - in a place with zero material constraints, as the hardware doesn't care if what it computes is meaning or noise - a system of signs does its thing, crunches away to some symbolic end.

So when talking about a mathematical model of the continuum, we have to allow for this fundamental distinction between the real world (which is materially dissipative) and the sign world (which can pretend what it likes, so long as it costs the hardware nothing extra to switch in one direction instead of the other).

Thus in the real world, cutting a material line quickly gets messy. Our knife eventually gets too blunt and starts mushing when the cuts are getting fine. And there is no such thing as an infinitely sharp blade.

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.

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.

Actually, it wasn't Peirce I quoted, it's a book entitled "The Continuity of Peirce's Thought", by Kelly A. Parker. — Metaphysician Undercover

My mistake, I did not follow the link to check your source; everything that you wrote right before the quote implied that it was directly from Peirce himself, so why would I think otherwise? I guess I should have known better. Parker's book was the first one that I read about Peirce, and it is quite good; have you actually read the whole thing? I doubt it, since your link indicates quite clearly that you went Googling for "Peirce divisibility of continuity" and then went with the first reference that came up. 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.

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

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?

But what happens when we divide time in Peirce's way, is that we lose an infinitesimal piece of the order. — Metaphysician Undercover

Remember, to say that something continuous (like time) is infinitely divisible is NOT to say that we can actually divide it, without breaking its continuity.

To claim that two things are the same when it is stated that there is a difference between them, is contradiction. — Metaphysician Undercover

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.

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

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!

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

OK, but we need to relate semiotics to a continuity.

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

This doesn't negate the problem. If the continuity is cut with indivisible points, like Aristotle suggested, or if it is cut with infinitesimal points as Peirce suggests, the result is contradiction, like I explained.

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

The issue with continuity, or the continuum, is whether or not it is something real, or just imaginary. If it is real, we need to describe what type of existence it has. Is it of the nature of matter, or of sign, or of the interaction? To tell me that it has to do with memory and purpose suggests that it is just imaginary. That was my first reply to the op. It is possible that continuity is just imaginary, fictional, and if so then it really doesn't matter how mathematics relates to it.

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

As I said, I've read enough Peirce and secondary sources already to know what he's talking about. I made a statement about how Peirce deals with dividing the continuum, utilizing the principle of "the difference that doesn't matter". You did not believe me that this was true of Peirce's division, and asked for a reference. So I googled the subject and found someone else who made the same statement about Peirce, Parker. Are you suggesting that Parker misunderstands Peirce as well?

If you have a different opinion about how Peirce claims that the continuum is divisible, then bring it forward, so we can discuss it. If not, then why not just accept what Parker and I have said, and discuss that. Unless you can point to how my understanding of Peirce is really a misunderstanding, why do you insist that I should read more Peirce. I've exposed Peirce's mistake, so why should I inquiry further down that mistaken road? If you believe that Peirce's procedure is not mistaken, then present your argument.

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

This is where we've always agreed, that saying a continuity is divisible with indivisible points, is a mistake. Where we proceed from there, is in different directions. I claim that a true continuity is indivisible, you claim that it is divisible by Peirce's means.

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

Here, you are describing yourself in terms of continuity, such that you are the same person despite having changed. And of course this is not contradictory, because of that assumed continuity. Aristotle assumes the existence of "matter" to justify that continuity. But in the case of Peirce's claims, the continuity is distinctly broken. That is what Peirce is doing, dividing the continuity. So you cannot refer to a continuity between the two different things, to justify any claim that they are the same here, despite being different, because the continuity between them is what has been divided.

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

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. We have a continuous order, and we divide it at X. The value on one side of X is not the same as the value on the other side of X, because X signifies an infinitesimal value. Therefore there is an infinitesimal difference from one side of X to the other. To say that the two values, on either side of X are the same, is contradictory, because we've already stated that there is an infinitesimal difference between them. You cannot deny the contradiction by claiming that they are still the same in the way that you are still the same person after changing an infinitesimal amount, because the identity of your person is justified by an assumption of continuity. With X, the continuity is what has just been divided.

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.

OK, but we need to relate semiotics to a continuity. — 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.

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.

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.

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. 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. A point is indivisible, so it is not part of the the line; it is added to the line. The middle dot is labeled with the numeral "2". Numbers are like points; they are indivisible. The number 2, which is represented by both the numeral "2" and the dot that it designates, does not have parts. Therefore, no collection of numbers, no matter how dense, forms a true continuum - not even the real numbers, although they serve as an adequate model (i.e., representation) of a continuum for most analytical purposes.

The issue with continuity, or the continuum, is whether or not it is something real, or just imaginary. — Metaphysician Undercover

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.

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

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.

We have a continuous order, and we divide it at X. — Metaphysician Undercover

If you divide it at X, then X is not an infinitesimal, it is a dimensionless point; and by dividing at X, we have agreed that you introduce a discontinuity - you no longer have a continuum. Rather than division, think instead about zooming in on a truly continuous line with an infinitely powerful microscope. 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.

You approach continuity with a stipulated definition. And this stipulation is impeding your ability to understand what continuity really is. — Metaphysician Undercover

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. We can then go on to determine whether anything in reality - time, space, motion, etc. - satisfies that definition. This is why it is a mistake to define the possible only in terms of the actual, or the (presumed) actualizable; you block the way of inquiry by ruling out certain kinds of hypotheses before fully explicating them and subsequently examining reality to see whether it conforms to them.

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

What makes you think that I have any interest at all in how the words "continuum," "continuity," and "continuous" are used in ordinary language? 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.

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.

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.

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

Why are you so adamant about imposing your terminology on any discussion here? I was just trying to improve clarity by distinguishing "continuity" as the property of being continuous from "continuum" as any object (real or imagined) that possesses that property. It really does not matter what words we use, it is the concepts that are at issue.

I don't agree that you know what it means to be continuous, because your stipulated definition results in contradiction. — Metaphysician Undercover

Having quoted The Princess Bride in one thread already today, I might as well do so again: "You keep saying that word; I do not think it means what you think it means." I know you believe that my/Peirce's concept of continuity results in contradiction, but contrary to your repeated assertions, you have yet to demonstrate this. Instead, you keep revealing over and over that you still have not properly grasped the concept, no matter how many different times and in how many different ways I have tried to express it.

Maybe this is because I have done a terrible job of explaining myself. Maybe it is because you cannot get past your own alternative concept of continuity. Maybe it is because you cannot set aside your dogmatic insistence that "x-able" must always and only mean "actually x-able." Whatever the reason, we just keep going around and around, wasting each other's time and energy. So rather than respond directly to your other comments, I will simply quote a somewhat lengthy passage from Peirce that summarizes the matter to my satisfaction. These are marginal notes that he wrote by hand in his personal copy of the 1889 Century Dictionary, next to its definition of "contintuity," which he himself had provided to its editor in about 1884.

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

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.

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

"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.

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?

The point is contiuum is a thing. Indivisble things made up of seperate finite states. We can cut a number line anywhere because its infinite particular members. Whether we cut at 2, 3,50,12445,9765564 or788956765677, we select a member of the contiuum.

But how does the contiuum change when we do this? It doesn't. 3 still comes after 2 in the contiuum of the number line. And so on in both directions. In selecting 3, we haven't destroyed the contiuum of the number line at all. We've just stop talking about it because we are interested in a particular member.

The notion of dividing the contiuum is a red herring. It confuses what we are talking about (3) with everything for that moment. We confuse ourselves into thinking the contiuum has been lost or divided when we are talking about one of it members. In truth, it's still there, infinite and undivided, just as it was before we were talking about one of its members.

The continuum is its own object, not merely a sum of every finite member.

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?

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.

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

Remember, I pointed out that this is false, because the concept of separate/broken/discontinuous is not necessary to the concept of parts. 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).

My claim is that even to say that it consists of parts, is to state a contradiction. — Metaphysician Undercover

But Peirce never said that a continuum consists of parts, as if you could somehow build up a continuum from them; and I certainly have never said such a thing, either. In fact, I have said exactly the opposite (emphasis in original):

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

I even said it (about Peirce) in the OP (emphasis added):

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

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.

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

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. So a continuum is undivided but infinitely divisible; you keep focusing on the second characteristic, to which you object, and losing sight of the first, with which you agree. You seem to want to define a continuum as undivided and indivisible, but these are the properties of a discrete point, not a continuous line.

In that sense, the motor vehicle doesn't consist of parts. One doesn't have a motor vehicle when they have a wheel or dashboard. A whole is not a sum of parts. It's its own thing.

That's why one cannot use the loss or absence of parts to identify when there is no longer a car. Do I no longer have a car if I lose a door? What about a steering wheel? An engine? Most of the time those are still cars, only missing a part they are expected to have. Sometimes a car has hardly anything at all-- consider someone building a car who refers to a half finished frame as "their car".

We can cut a number line anywhere because its infinite particular members. — TheWillowOfDarkness

As stated in the OP and several times since then, the real number line is not a true continuum as defined by Peirce, nor is anything else that consists of discrete members - even if there are infinitely many of them.

The continuum is its own object, not merely a sum of every finite member. — TheWillowOfDarkness

That is the gist of what I have been saying all along. It is a top-down concept, not a bottom-up one.

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.

But that's exactly the problem. Such a "real contiuum" is meaningless. It's a set without any members-- an infinite of nothing at all.

Pierce fails to grasp the nature of contiuum here. Supposedly, when finite members belong to a contiuum (e.g. a number line), it cannot be said to be a proper contiuum. He treats having finite members as if it meant a beginning and end, as if it meant the given contiuum wasn't infinte.

This isn't true. A number line is really infinite and a genuine contiuum. It doesn't begin or end. It is indivisible. (picking out a number from the line doesn't destroy the line).

Pierce makes the mistake of treating such a conntuim as if it were made, in a bottom up manner, by the finite members that belong to it. It is not. The infinite set of finite members is its own. Pierce fails to recognise it is a conntuim because he's still stuck trying to account for the infinte by the finite.

You no longer have a car when you no longer have something that can function as a car.