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.

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.

So MU, you quote Peirce in a way that directly contradicts you and directly supports me.

Interesting argumentative strategy. — apokrisis

Of course it supports you, that's the point. My claim is that this is Peirce's mistake. And, if you follow it, it is also your mistake. The mistake is to say that if the difference between two things is infinitesimal, then the two things are the same. Clearly, there is a stated recognition of difference, which indicates a recognition that the two are not the same. Then the claim is that since the difference is infinitesimal, we can just say that the two are the same. It's blatant contradiction. We recognize that the two are different, but we're just going to overlook that fact, and say that they are the same, because the difference is so small.

We could never possibly take account of every difference, and even if we actually had taken account of every difference we would have no way of knowing we had, because it would always be possible that there could be differences we had missed.

So identity is always something stipulated, not something logically proven or empirically demonstrated. — John

I do not completely agree. Identity is also something which we stipulate that an object has, whether or not we are capable of determining it. We do stipulate the identity of the thing, as you say, when we say it is "X" or whatever. This seems to be the way that apokrisis speaks of identity, we give a thing an identity relevant to the purposes at hand.

On the other hand though, we say that a thing has its own identity, it is what it is, independently of our efforts to identify it. This is the basis of Aristotle's law of identity, a thing is the same as itself. This puts the real identity of a thing within the thing itself, rather than the identity which we stipulate. Apokrisis appears to be saying that there is no use in assuming such a principle of identity. But I think it is very important. It is important because if we cannot identify every little difference, as you say, we will still respect that those differences are there even though we incapable of identifying them. Therefore we have respect for a difference between the identity we stipulate and a thing's true identity. And of course, respect for the possibility of mistake.

From the other perspective, the thing's identity is the identity which we give it, regardless of any other differences. So this does not take into account the fact that we might be mistaken when we say that one instance is the same as another. For example, we may say that instance X and instance Y are two occurrences of the same object. This serves our purpose, so we have no reason to doubt that. Therefore we conclude that there is a continuity of existence between them X and Y, and they are the same object. The continuity of existence is true by the fact that we have identified them as such and this identification serves our purpose. Only if we allow that the object has an identity proper to itself, independent of the identity we stipulate for pragmatic purposes, do we allow that we may have made a mistake in this determination.

I knew that you did not invent it; you are just the one who introduced it to this thread. MU wrongly attributed it directly to Peirce and claimed that the latter relied on it to support the proposition that a continuum is divisible. — aletheist

Going back to this point now. The difference which Peirce claims does not matter, which enables him to divide the continuity, is the difference in order. Consider the example of dividing at 2. On one side of 2 the order goes lesser, and on the other, the order goes greater. And there is a place, occupied by 2. Peirce considered time to be a primal continuum, and dividing time creates this same problem with order. Any division in time creates this same issue of a different order on each side of the division, and a place occupied by the inserted divider.

Here's the reference you requested:

The point that is cut is not afterall a point. It is a place, an infinitesimally small part of a continuum, and so is itself a continuum capable of infinite division.
The parts A and B can be considered different in their location on a line (because a specifiable ordering relation) but the difference is infinitesimally small. They may be thought of as 'overlapping' so that they occupy different places. The difference is infinitesimal, however, so it is in principle indiscernible. If the difference is indiscernible then we might easily say that A and B are the same.

https://books.google.ca/books?id=iy76kUCZYb0C&pg=PA88&lpg=PA88&dq=peirce+divisibility+of+continuity&source=bl&ots=tyuF0tOcKH&sig=BeofMNRZHX7Uu_58YstLlE4Bk5g&hl=en&sa=X&ved=0ahUKEwj99tn0h7TSAhWb8oMKHciMAOsQ6AEIOzAF#v=onepage&q=peirce%20divisibility%20of%20continuity&f=false

The issue is well explained in BK. 6 of Aristotle's Physics. After stipulating that anything continuous, including time, is divisible, and necessarily infinitely divisible, he proceeds to determine "the present" as indivisible. Then he describes a "primary when" as indivisible also. This creates a problem, because these indivisibles which are used in the act of dividing, are inconsistent with the infinite divisibility which has been assigned to the continuous time. To divide the continuity requires that there is an "indivisible" within the divisible. Failure to reconcile these two, the continuous and the individisible, produces infinite regress in all change and motion.

By those preceding principles, all change and motion must be infinite. But infinite regress in change and motion is contrary to the doctrine that all change is finite, that change is from something to something, and motion is from here to there. So at the end of the book 6 it is demonstrated that all motion except circular motion, is in fact, finite. Now we must reflect back on the primary assumption, that continuity is infinitely divisible, because this assumption produces the unacceptable conclusion that all change and motion is infinite.

But how did it make a difference to you that you ate one and not the other? And how even did it make any difference to the world, if the world had any discernible interest in the matter. — apokrisis

How is this relevant? If I chose one over the other, for a reason, then it made a difference to me. If I flip a coin, then it doesn't make a difference. But all this is just a distraction, because we are discussing identity, so the issue is whether or not there is a difference between the two. Clearly, this makes a difference to me, because if there is really no difference between them, they are one and the same, and I have no choice.

Now, the issue with continuity, and divisibility, is whether or not there is a difference between what is on the two sides of any proposed division. If part A is identical, the same as part B, then there is no problem with division, we keep dividing infinitely. However, if part A is really identical to part B, then in what sense can we say that they are two distinct parts. They are, by the identity of indiscernibles, one and the same. But if we claim that there is a difference between part A and part B, in what sense were they ever continuous in the first place? There would be a change, a discontinuity, between a and B.

So - as has been repeated ad nauseum by both me an altheist now - it is not that there isn't a difference, but there needs to be a difference that makes a difference ... which is the difference that makes a difference in this discussion. — apokrisis

The difference is the difference which makes one burger #1, and the other burger #2. If there was no difference between them, then by the identity of indiscernibles, I would have no choice, there would be only one burger.

It's easy to see that its identity cannot logically be the same as was it is identified as, because Pluto is the entity which had previously been identified as a planet and is now not identified as a planet. — John

If I understand you correctly, we are assuming an object which has the identity "Pluto". Now, let's say that we remove this object from sight, and then we bring an object, which also has the identity "Pluto". We want to know if they are both the same object, so we consider differences. Unless we consider all possible differences, we cannot jump to the conclusion that they are both the same object.

If that were true, then you would not be arguing with me, because it is simply a fact that - going back at least to Aristotle - "continuous" means being infinitely divisible though actually undivided. In any event, this is what I mean by continuous, and your insistence on your idiosyncratic definition is not going to change that. — aletheist

OK, this is what you mean by "continuous". Now are you ready to face the problems with this definition? That's what I've been trying to bring to your intention, there are problems inherent within this definition. First, address the issue of my last post. It is impossible that anything divisible is infinitely divisible. That's what I explained in the last post. Do you agree? If not, why? If so, then you need to change your definition. Either the continuous is not divisible at all, or the continuous is divisible but not infinitely so.

No, the act of dividing something that was continuous causes it to become discontinuous. — aletheist

That may be true, but what we are discussing is continuity itself, as an identified thing. This is like if we were discussing 'red" as an identified thing, not the objects which are red. You refer to "something that was continuous". So you are deferring now to an underlying substance, a thing which is continuous, and you are saying that this substance could be continuous, or it could be divided to be discontinuous. It's analogous to if we were talking about a liquid. It is incorrect to say that a liquid could become a solid, because it is the underlying substance, water, H2O, which changes form, from being a liquid to being solid. The property of being liquid is negated, for the property of being solid. Likewise, if something continuous is divided, and becomes discontinuous, it is the underlying substance which changes its form from being describable as continuous to being describable as discontinuous.

What we are inquiring into is not the nature of the underlying substance, but what it means to be continuous, and what it means to be discontinuous.

Not surprisingly, we disagree on whether the infinite divisibility of a line renders it discontinuous, even if it is not actually divided. — aletheist

Now you are talking about the infinite divisibility of a line. But if that line has substantial existence, as if it were written on a paper or something like that, it is impossible that it is actually infinitely divisible. We could only cut up the paper into so many pieces. So I assume that you are talking about an ideal line, in the mind, and assuming that the ideal line is infinitely divisible.

This is our substance now, an ideal line, and I will assume that it is continuous. How is that ideal line divisible? If you divide it up into sections it is no longer the ideal line which it was. Either the idea is of a continuous undivided line, or it is an idea of a discontinuous divided line. It cannot be both because this is contradictory. It makes no sense to say that the ideal continuous line is divisible, because you don't actually divide it. You replace the idea of a continuous line with the idea of a divided line, each having a different definition. You do not divide the ideal continuous line in your mind, so it makes no sense to say that this ideal line is divisible. The line on the paper is divisible, but it makes no sense to say that it is infinitely divisible.

The conclusion is that it is nonsense to talk about "the infinite divisibility of a line".

If I offered you the choice between two McDonalds cheeseburgers, would it make a difference which one you picked? — apokrisis

Yes of course it would make a difference. The one I chose would be the one that I eat, the one I didn't choose would not be eaten by me. Do you think the difference between being eaten and not being eaten is not a difference?

If there are differences that don't make a difference, then there are differences that do. And on that logical distinction would hang the pragmatic definition of a principle of identity. — apokrisis

In the case of the identity of indiscernibles, as I explained, every difference matters. We cannot claim to have properly established identity unless every difference is accounted for.

If in some circumstances, we can establish a usable identity without accounting for all of the differences, then that is fine for those circumstances. But as a formal "principle of identity", upon which one would base a logical structure, it is unacceptable to allow that there are differences which do not make a difference, because this allows that there may be two distinct things which have the same identity.

You may insist on your own unpragmatic definition. It would be interesting to hear what it might be. How does difference end for you? What makes something finally "all the same" for your impractical point of view? — apokrisis

You are going in the wrong direction here. It seems like you have some sort of backward notion of identity. The purpose of a principle of identity is to ensure that each thing has it's own identity, that it is identifiable as the thing which it is. The goal here is not to make things "all the same", it is to make every thing different, thereby allowing that everything is identifiable as the thing which it is, and not confused with anything else. That is why there will be no success to any principle of identity which does not seek to determine every last difference.

I have disagreed with you on this point previously, and clearly showed you that identity is not the same as identification, and yet you continue to repeat this mistaken thought. Things are not identified by means of their identity, that is absurd; they are identified because they stand out, and they stand out on account of their differences from, and similarities to, other things. — John

I really don't see your point. A thing's identity is established according to its individuation, and this means its difference from other things. And, it is identified according to its identity. So I really do not know what you mean when you say that things are not identified by means of their identity. It doesn't make sense. What division are you trying to create between "identity", and "identify"?

This sentence makes no sense to me. Differences that do not matter enable us to treat two things that are not identical as if they were identical, for a particular purpose; this is the opposite of claiming that two identical things are not, in fact, the same thing. If our purpose is to distinguish two things, then obviously more differences will matter. — aletheist

Correct, now reflect on what you have said. If our purpose is to identify things, which is what we are discussing here, identity, then allowing that there are differences which do not matter, defeats our purpose. This is because it allows two things which are not identical to be treated as if they are identical. This mean, explicitly, that we will confuse one with the other, and we will fail in our efforts at identity.

Again, this is backwards. The point is not to claim that there is a difference that does not matter in order to distinguish two things that are really identical, it is to treat two things as identical because the real differences between them do not matter within the context of a particular purpose. — aletheist

Let me reiterate, the particular purpose which we are discussing is identity, identification. Under no circumstances would we want to treat what we know as "two distinct things" as identical, when our purpose is identification, because this explicitly defeats the purpose.

It defeats that particular purpose, but it can be useful for other purposes. By acknowledging that the law of identity has a particular purpose, rather than being an absolute and intrinsic feature of the universe regardless of the context, you are effectively agreeing with the point that we have been discussing. — aletheist

Sure, it is useful to treat distinct things as similar, for certain purposes, such as generalization, and so consider that some differences do not matter. But in doing this, we respect the fact that similarities do not render two distinct things as the same, we simply produce a generalization. So when we overlook differences which do not matter, we do this with the intent of looking at things as similar, not with the intent of looking at distinct things as having the same identity. We overlook differences which do not matter, for the purpose of saying that two things are similar, not for the purpose of identity, or saying that two things are the same.

No one is disputing that actually dividing a continuum introduces a discontinuity. However, that discontinuity is not there until we break the continuity by that very act of division. — aletheist

The act of dividing something demonstrates that the thing divided is not continuous. The claim that it was continuous prior to being divided needs to be justified. If a continuity can be divided then the logical conclusion is that it cannot consist of indivisible parts, this would deny continuity, so it is necessarily infinitely divisible. To prove that the thing divided was in fact continuous then, requires that it be divided infinitely. This produces an infinite regress, and is a simple denial of the fact that it is impossible to divide something infinitely. The fact that it is impossible to divide something infinitely implies that it is impossible to divide the continuous. Therefore the thing which you divide, was never continuous in the first place and the continuous is actually indivisible.

Indeed, but what you still refuse to acknowledge is that a continuum does not contain any points at all. — aletheist

Isn't that exactly what I said? A continuum necessarily has no points, and this is why it is inherently indivisible. You, on the other hand, by assuming that a continuum is divisible, assume that there are points of possible division. If we deny that there are any points to the continuum, then it is necessarily indivisible because there are no points where it could potentially be divided. How do you propose that the continuum is divisible if there are no points of possible division?

Again, citations please. As far as I can tell, you have no clue about what Peirce had to say regarding these matters. — aletheist

I've read enough Peirce, and secondary sources, to know what he was talking about. If you think that what I said is wrong, then please correct me with more accurate information, I would welcome a chance to upgrade my understanding.

A couple of recent quotes from Republicans on press freedom. — Wayfarer

There appears to be a very simple strategy if one is against free press. First, fill the media with lies and fake news, not hard to do for someone with money and power. Then use the abundance of abuse in the media, to justify restricting the media.

I don't see a problem. Nor does one appear when we make a second cut at >2. We now have three pieces: <2, 2, >2.

Nor is a problem introduced when considering continuity. My simple understanding is that a line is continuous if it is differentiable. Well, the limit of <2 as it approaches 2 is 2. It does not seem problematic. — Banno

The problem with this representation, is that 2 here is not part of the continuity, it is a point of difference, differentiating one part from the other.. There is a supposed continuity which goes right through 2. If we divide that continuity at 2, then 2 is not part of the continuity, but a point of difference. A continuity cannot have a point of difference because this would make it discontinuous.

Yes, back to the op. Peirce employs this notion, of a difference which doesn't matter, to support the proposition that a continuity is divisible. If we can divide a continuity, at 2 for example, such that we have <2 and >2, then there cannot be any real difference between <2 and >2 or else that difference would indicate that there was no continuity here in the first place. Peirce proposes that we can assume a difference which does not matter, such that <2 and >2 may be identified as different, but because this difference doesn't matter, <2 and >2 can be said to be the same, so that there is no real difference between them, and there is continuity through 2.

Peirce appears to face the problems which are associated with the idea widely accepted in mathematics, that a continuity is divisible. But instead of following through to where his investigations lead, and coming to the proper conclusion, that a true continuity is truly indivisible, he compromises and proposes this principle of a difference which does not matter. That is a mistake. The proper decision would have been to accept the metaphysical principle that continuity is indivisible, because to divide it proves it to be discontinuous, regardless of what mathematicians want to do.

The very thing of a purpose defines its own epistemic boundaries - the point at which differences don't make any difference. And if you can't follow that argument, then that's your problem. — apokrisis

The purpose, as you yourself, described in that passage, is identity. If you are prepared to say, that two things with the exact same identity, are not in fact the exact same thing, (according to the identity of indiscernibles), because of some differences which do not matter, then you only defeat the purpose of identity, which is to distinguish one thing from another.

Read more carefully - in the comment that you referenced, ↪apokrisis did not say anything about a difference not being a difference; he was talking about a difference not making a difference. — aletheist

Correct, and whether or not a difference matters, depends on the context. And the context in which this was stated was in relation to a point where the "identity of indiscernibles", is supposed to no longer be relevant.

So the laws of thought presume the brute existence of the indiscernible difference that secures the principle of identity. And Peirceanism flips this to say indiscernability kicks in at the point where some 3ns ceases to have a reason to care, and so 1ns is left undisturbed. — apokrisis

But in this situation there is no such thing as a difference which doesn't make a difference. Consider identity A and identity B. If these two identities are the same, then according to the principle of identity of indiscernibles, they are one and the same thing. If you deny that principle of identity, and say A and B are really not the same thing, because of some difference between them which does not matter, and is therefore not part of the identity, (the identity being one and the same), then how is it true to say that this difference does not matter? It is only by claiming that there is a difference between them, which does not matter, that you can say they are two distinct things, rather than necessarily one and the same thing, as stipulated by the "identity of indiscernibles". So it is false that this difference does not matter, because it is the only difference which makes them two distinct things.

Therefore, apokrisis' claim, from Peirce, is that two distinct things can have the very same identity, if we allow that there are differences which do not matter. But of course these differences really do matter, because these are the differences whereby we distinguish the two things as distinct. And it is simple contradiction to say that these differences do not matter.

Why would this always be a mistake? Standardization and mass production are all about minimizing unimportant differences, such that we can treat different things as effectively identical. When I select a particular section for that beam, I am counting on the fact that it is irrelevant which mine produced the iron ore, which cars and washing machines provided the scrap metal, which mill melted all of that together to make the steel, which service center stored it after rolling, which fabricator assembled it, or which erector installed it. None of those differences make a difference in the finished product, as long as it meets certain minimum specifications - i.e., there are no differences that would make a difference - and that is a good thing! — aletheist

The point is, that with respect to the principle of identity these minimal differences are the differences which really are important. If you do not respect this fact, then you allow that all mass produced items are in fact, the very same entity, because you are insisting that they are identical. My car is the same object as your car, because they are mass produced and identical. Your desire is to claim that the factors which differentiate them (the differences of the particular) do not actually differentiate them, and identify them as distinct, as those differences are unimportant. So you will claim that they have the very same identity, yet you will also claim that they are two distinct objects. They are distinct objects not by being different though, because those differences don't matter, they have the same identity. What would justify the claim that they are different then? Or is it the case that my car and your car might really be the very same object?

The purpose of the law of identity is so that we can distinguish one object from another, and come to know that object as the thing it is. To claim that we can overlook some minor differences such that numerous objects may have the same identity only defeats this purpose. We simply deny ourselves the capacity to tell these objects apart.

But what if it turns out that vagueness is a fundamental and ineliminable aspect of reality? What if the truth is that vagueness constitutes an actual limitation on our ability to determine the truth? In that case, your dogmatic insistence on assuming that every difference matters hinders your ability to determine the truth about vagueness. — aletheist

You should consider that perspective as rather nonsensical. Even if vagueness is real and fundamental, we will not know this until it is proven. And we cannot prove its reality without identifying it.

The very act of distinguishing one thing from other things already involves neglecting differences that do not make a difference. Why do we pick out this chair or that table or this book or that door as individual objects, rather than always and only referencing them at a molecular, atomic, or even quantum level? Because the difference between one particle and those adjacent to it within the object is irrelevant to our purpose in picking out that object as a single object. You do this all the time, but it comes so naturally that you do not realize it. No one is capable of paying attention to every single difference among phenomena, because there are far too many of them to do so - even just within your field of vision during the passing of one second. — aletheist

Yes of course, distinguishing one thing from another usually involves neglecting differences which do not matter. But here, we are in the context of the principle of the identity itself, the idenity of indiscernibles. So we are, according to that defined context, dealing with things which appear to be the same. We can conclude, as apokrisis implies, that we cannot tell them apart, because they have the very same identity, yet they are not really the very same thing, due to some differences which do not really matter. Or, we can uphold the principle of the identity of indiscernibles, and conclude that if they are distinct entities, then there must be real differences, which matter, by which we can tell them apart.

If you want to be taken seriously, talk sense. — apokrisis

Oh, that's a gas, coming from the one who's reply to my post was: nature's going to frustrate you.
If you're at all serious, then address the points of my post, and quit making a joke of yourself by saying that I'm the one who's not being serious.

In case you've forgotten, I'm waiting to hear justification for overruling the "identity of indiscernibles" with the claim that some differences don't matter. Obviously, if overlooking these differences allows you to deny the "identity of indiscernibles", then they do matter.

Yeah, so you will be with those who feel that nature frustrates you with its fundamental quantum indeterminism and general relativity. You want existence to be exact and totallly knowable, even if that has already been discovered to be a kind of mania. — apokrisis

Wow, that's your counterargument? You think I am going to be frustrated by not be able to figure out specific things which I believe are actually knowable in principle? Meanwhile, you deny the fundamental principles of logic, which might be used to figure these things out, and you satisfy yourself with the claim that such things can't be known. Relax, you're absolutely right, of course they can't be known, when you deny yourself the capacity to know them. Ha, ha, ha, humour me some more, it relieves me of my frustration. So, go ahead then, pour yourself a nice drink, and congratulate yourself, you seem to have convinced yourself that you already know all that it is possible for you to know.

Mind you, if you claim that everything actually does matter to you, excuse me if I think that is patent bullshit. Does it make any difference to you if I wear a red or blue shirt tomorrow? Do you need that to be another determinate fact ... or do you believe in free will in contradiction to your what you just posted? — apokrisis

If we are talking about identity, and the overriding purpose is, that we want to know the truth about the matter, then of course every difference matters. That's no bullshit, it's reality. If we allow that some differences do not matter, then we allow that two distinct things can have the same identity. Since giving two distinct things the same identity is a mistake, then in relation to identity, there is no such thing as a difference which does not matter.

You claim that the "identity of indiscernibles" cannot be upheld, but this is only supported by the claim that some differences do not matter. If we allow into our principle of identity, the notion of "some differences do not matter", then we have a compromised law of identity. Failure to hold fast to strong logical principles allows vagueness to creep into the logic. Such vagueness hinders our ability to determine the truth. Therefore, if our purpose is to determine the truth, we must uphold the principle of identity to the strongest of our capacities, and assume that every difference matters.

Your "Peircean flip" is this act of compromise. It takes identity from the particulars of the individual, and loses it into the vagueness of the general. By claiming some differences don't matter, you claim that we can disregard accidentals to focus on what is essential, so what is identified is a generality. But since the principal purpose of identification is to identify the particular, distinguishing it from other similar things, you negate the capacity to fulfill this fundamental purpose of identity, with that process, the flip.

Well, isn't that just the way life is? Grow some culture on a petri dish and it will flourish, until it uses up all the nutrients, then it will die off. Death is not absolute though, some stragglers will persist, living off of God knows what, the spores will go into long term suspended animation, and other species will move in to live off the dead. Evolution waves its magic wand.

What if you perceive something, and you call it a table, but you are really perceiving a desk, and you just called it by the wrong name by mistake.

But where the reductionist thinks that the differences that make a difference are atomistically unbounded - there is no reason why we could ever in principle cease the pursuit of further detail, chase down the last decimal of the expansion of pi until we are exhausted - the Peircean system offers principled relief. We can stop when the differences cease to matter to our over-riding purpose. — apokrisis

When we seek the truth, differences never cease to matter. A difference, by its very nature, as a difference, is a difference, and therefore it must be treated as a difference. If one adopts the perspective that a difference may be so minute, or irrelevant, that it doesn't matter, and therefore doesn't qualify as a difference, then that person allows contradiction within one's own principles ( a difference which is not a difference), and the result will be nothing other than confusion.

While it is debatable that the table that I perceive is real or an illusion, the undeniable fact is that I perceive a table. — Samuel Lacrampe

Why is this the "undeniable fact"? What if what we thought you were perceiving was a table, but what you really were perceiving was a desk? How can you say that it's undoubtable that what you perceive is a table when you could be perceiving something else, and incorrectly calling it a table?

As you can see, with this model there is no discreteness. — Rich

I suppose this means that there can be no beginning point of a wave. Such a beginning would be a discrete occurrence.

The image that would analogue this would be the Ocean Wave with Gravity embedded within it to create movement. — Rich

Can you describe the "Ocean Wave"? In your model, is there a wave which initiates from a point, like when you drop a pebble in water, or is there just perturbations in existing waves? If there is such a wave, which initiates from a point, what would cause this wave?

So assume that I am sitting, nice and still, meditating. I decide that it's time to get up. So I suddenly stand up and walk away. How does my memory suddenly cause the waves which are necessary to move my body? The cause is not some surrounding waves, or non-local activity, because it comes from right within my mind. A particular, separate, and independent wave must be created right at this very locale, and this wave spreads outward into the surrounding area as I get up and walk away.

The bank robber in your example needn't have used arithmetic to conclude that he would have more money after a successful robbery than he had before. It's his moral reasoning that is wrong, not his rough quantitative judgment. Getting hung up on the culprit's correct use of quantitative reasoning distracts from the real problem in this case. — Cabbage Farmer

OK, his "moral reasoning" is wrong. But that's the whole point, that mathematics cannot be used for moral reasoning. The issue here is how can one use mathematics in performing moral reasoning. I think that it can't be done. And the further point is that if one does think that there is a way to use math in moral reasoning, that individual could very easily have a wrong answer (because you actually can't use mathematics in moral reasoning), and also be convinced that it is right answer because math was used.

Notice that in such examples, mathematics cannot determine moral models or judgments all by itself. We'd still have to rely on moral agents to supply moral values, moral intuitions, and so on. — Cabbage Farmer

Well this is my whole point. We cannot use mathematics to make moral judgements. You seem to be arguing that we can. But now you've qualified that to say that we would have to have moral agents, to supply moral values. This implies that the moral judgement has already been made by the moral agent. So the mathematics is not going to be used to make any moral judgement, this is already supplied by the moral intuition of the moral agent. What is the mathematics to be used for then? If the moral agent supplies the moral values, then the moral questions of what is right and wrong, has already been answered, prior to applying the math.

Moving an object would be analog to one wave in an ocean moving another. Ocean and waves provide the basic analog of nature (it is a mirrored manifestation). The only thing missing is the impetus behind the movement. — Rich

But it is this, the missing impetus which demonstrates that the parts are really separate. The free willing act can move the object any which way, so the wave in the ocean analogy is not really adequate to explain this motion.

This would be Consciousness or Bergson's Elan Vital. With this image (that can only be intuited based upon many manifested patterns) one can begin to understand the nature of nature without paradoxes (any unit derived symbol will muddy the waters :) ). What you refer to as parts are simply wave perturbations. — Rich

So how would the conscious, free will act move one particular object independently of the other objects? It cannot be by means of the wave perturbations which you describe, because these are not independent. It's easy to make the claim that Bergson's Elan Vital solves this problem, but until you explain how one object (a living being) moves itself independently of all the surrounding objects, your description of an ocean with waves remains incompatible with this reality.

You imply that the belief of everything being infinite doesn't make sense because we perceive boundaries, but what about what we can't perceive due to our human nature? — Aucellus

If there is "that which we can perceive", and "that which we cannot perceive", don't you think that there is necessarily a boundary between these two? They are described as opposing, "can" and "cannot" perceive, so surely there must be some sort of boundary between these two. It is when we dissolve these boundaries, such as is the case when we dissolve the boundary between being and not being, to say that all exists as varying degrees of becoming, that the infinite appears to us.

So your example of "what we cannot perceive due to our human nature" can be taken in two ways. We can assume that whatever it is that we cannot perceive, maintains the same type of existence as what we can perceive, and this is boundaries between individual objects. Or, we can assume that what we cannot perceive exists in some completely different way from what we can perceive, in which case we need a boundary between what is perceivable and what is not perceivable. In each case we have boundaries.

I found what you say about mathematics really interesting, that it's only applicable to the point where it approaches infinity, because it ignores the real limits, you say that we should search for new limits, but how can we find them? do we have to theorize them? how can we know what's the real delusion in that case? — Aucellus

It's not so much that the use of mathematics ignores the limits, but certain limits are assumed, taken for granted, prior to applying the mathematics. What is at issue is the relationship between the assumed limits, and the real limits. When these two do not correspond, then we have the appearance of infinity in the mathematics when it is applied to what is real. Reality continues on, where the assumed limits do not allow, or vise versa, the mathematics continues, where reality doesn't allow, and this is where infinities arise.

you say that we should search for new limits, but how can we find them? do we have to theorize them? how can we know what's the real delusion in that case? — Aucellus

Yes, I think we find the limits through theorizing, and confirming through experimentation, the scientific method. But we have to analyze the places where infinities arise in the mathematics because the occurrence of those infinities indicates that our understanding of the boundaries which are assumed to lie there, is inadequate.

Because they may come to America from said countries? Duh. — Thorongil

Oh, so it makes sense to ban travel from Turkey, Greece, Germany, France, Britain, and Canada, and every country, because terrorists could come to America from those places? We already know that terrorists have come from Canada to the U.S., why not ban travel from Canada first and foremost?

Take the travel ban on what was it, 7 countries or so, which was immediately framed as a Muslim ban. That's fake news. There was no Muslim ban. — Agustino

Fact is that the claimed motive behind the proposed travel ban is completely inconsistent logically, therefore we can conclude that the true motive remains unrevealed. Since it is unrevealed, we can only assume that the reasons for not revealing it is that the true motive is something untoward. So we are left to speculate as to what that untoward motive is.

I am saying that as far as empirical evidence exists at this time, there is no evidence of full and total separation. — Rich

I apprehend "full and total separation" as a rather useless concept. Things always exist in relation to other things. To not have a relation to something else (full and total separation) is to not exist. Unless you conceive of a whole which consists of all existing things (the universe), and this whole, by definition would not have a relation to anything else, because it is everything, full and total separation is impossible. But what kind of separation is that? It's just the logical separation between what is, and what is not. So is this the "full and total separation" you refer to, the separation of logic, between being and not being?

There seems to be more evidence to the contrary. You are speaking of separation (the concept of isolated particles) for which there is no empirical evidence and never was. The idea of somehow separate particles is a belief system, which one is free to embrace, but then one must explain what is in-between. — Rich

As I said, we move objects around, in different directions relative to each other. Does this not indicate a separation between them to you? It's nonsense to insist that this separation must be absolute, such that there is not even any relationship between the two, because then one object would have to be existing and the other not-existing. So you requirement for "separation" is to completely annihilate the object. You will not admit that one object is separable from another unless it can be completely annihilated, removed from any relationship to the other. Why do you not allow that moving one object in all sorts of different directions relative to another, constitutes a real separation between them? What we assume as "in-between", which allows for such movement, is "space". Why do you hold such a strong propensity to reject this idea?

The wave in the above description is not part of anything, it would be the fabric of the universe. Consciousness, movement (energy) and memory are all sewn into this fabric and are everywhere just as an image is sewn into every part of a hologram. It is waves that make this all happen. — Rich

I do not see why you claim that this idea is "rather simple". Do you not recognize that waves require a medium? So all you are doing is reducing the "substantive matter", and taking for granted a new substance which necessarily underlies the waves. I assume a "space" between substantive objects, you assume "waves", which are necessarily in a substance, then you have to account for the appearance of objects, so one is not more simple than the other. You've replaced my lack of substance, "space", with substance, "waves". Now you still must account for what I call substance, objects. The difference, is that my position allows for the real separation between objects, which we utilize daily, to move objects in different directions relative to each other. You deny this real separation. So how is it that we move objects like this then? How do you account for our capacity to freely move objects this way and that way in relation to each other, if there is not real separation (space) between them?

quote="Rich;57691"]The wave in the above description is not part of anything, it would be the fabric of the universe. Consciousness, movement (energy) and memory are all sewn into this fabric and are everywhere just as an image is sewn into every part of a hologram. It is waves that make this all happen.[/quote]

Don't get me wrong, I am not denying the need to refer to wave activity, it as well as objects, is observable, and waves are empirically verified. The point though is that it doesn't get us any further ahead, to deny the reality of the independent activity of objects, for the assumption that all reality is a "whole" consisting of waves. What is needed is to establish compatibility, not to choose one over the other by excluding the possibility of the other.

As we are enormous to a unicelular being, we are insignificant compared with the sun, i'm saying that this kind of comparisons are infinite (there will always be something bigger and something smaller), i also like to believe in this because it makes free will something much more real, since there in not a basic structure that supports matter, as i like to see it, there is not such thing as a possible complete predictability of the future based on the observation of the physical behavior of this basic particle, as there is not such thing. — Aucellus

When we do not apprehend the limit, it appears like the sizes may just keep getting bigger and bigger, or smaller and smaller. Therefore the numbers involved go to infinity. When we apprehend the limit, that limit is an end to this process of the numbers getting bigger and bigger, because it serves as a starting point, where the numbers can be applied according to new principles.

I really do not believe that these comparisons you refer to are limitless. That doesn't makes sense, because it would indicate the likelihood that all things are limitless, when in reality what we perceive is boundaries, limits. Once we perceive limits, then we can apply mathematics. But the mathematics is only applicable to the point where it approaches infinity again, then we need to find new limits. If we do not look for the real limits, we will not find them. So if all we want to do is sit around and apply mathematics, instead of seeking the real limits in the world, we'll suffer from the delusion that all is infinite.

But why limit travel from countries where terrorists have not been coming to America from?

That is why, when challenged in the courts, it was immediately suspended, and, note, it has now vanished altogether from the public discourse.. — Wayfarer

But Hanson is right. It makes sense to place some sort of a moratorium on immigration from countries whose populations we have little to no information about and which house large numbers of terrorists. These same countries were being watched by the Obama administration as particularly dangerous. — Thorongil

The thing is, no terrorists have been coming to America from these countries. Most terrorists in America come from America, what a surprise. If you want to stop terrorism in America, then focus on American terrorists.

Is you saying this is your belief or are you saying there is empirical evidence? — Rich

You don't think that there is evidence that the area of your field of vision is made up of separate objects, separate parts? Isn't the fact that I can pick up a chair and move it to the other side of the room, or move the dishes from the cupboard and use them, then wash them, evidence that they are individual parts, able to move independently of the others? Isn't the fact that water boils and evapourates evidence that it is made of separate parts, molecules? Aren't chemical reactions evidence that the molecules are made of parts, atoms? What more evidence do you need.

As far as I understand there is no evidence one way or the other, but there is evidence of persistent entanglement and fields that extend forever. — Rich

Fields are mathematical formulae. You are just entering a fictional fantasy like aletheist, referring to some ideal, a fiction of how you think reality should be, then you will describe things to match this ideal, instead of shaping your ideal to the way thins really are..

I know of no evidence for separation of waves into distinct particles. — Rich

Waves are something distinct from particles, but they clearly are a pattern of movement of particles, like sound waves and water waves.

Models should not be confused with nature and there is no splitting of atoms. — Rich

I believe the atom has been split, in the nuclear reaction. And electrons are commonly separated from atoms in electrical practises.

One way to picture this would be the shaping and reshaping of waves in an ocean. There is never separation. The "parts" we carve out (waves) are simply different shapes within the whole. — Rich

A wave can only exist within an assembly of parts, it is a particular type of activity of particles.

Things are separable from their surroundings because each one is a part of the overall whole. If it were not a part, it would not be separable from the whole. Any individual object, like a rock, is divisible itself because it consists of parts, molecules. A molecule is divisible because it consists of atoms, etc.. If a whole did not consist of parts it would not be divisible. And because it consists of parts, the whole is not continuous.

We could assume the existence of a continuous whole, but it would be false to say that this continuity consists of parts. It is equally false to say that it is divisible.

You cannot actually divide a continuous line without introducing a discontinuity (point), but it is potentially divisible without limit, as the SEP article explains. — aletheist

This is where you stray from observed empirical reality. Empirically proven principles demonstrate that anything which is divisible is such because it consists of parts. The points which provide for division are already existent within the divisible thing, or else it could not be divided. You are assuming that a continuous thing, a thing which exists without such points for potential division, can still be divided. This is an unsupported fantasy.

Mathematically, infinitesimals likewise have no ends; they are indistinct, such that the principle of excluded middle does not apply to them. — aletheist

And this is nonsense. Such entities have no individual identity.

Just answer a couple quick questions for me, if you really believe that you have a tenable position. How can you have short lines unless they have ends? And how can you have a continuity which has ends inherent within it?

Why is it so hard for you to understand that there are no points in a continuous line, only shorter lines? Positing points of division makes the line discontinuous. — aletheist

If they are shorter lines, they must end. Where they end, there must be something which signifies the end or else there is no end and therefore no shorter lines. If you don't want to acknowledge that "end" as a point, then call it something else, but the fact is that this "something else" interrupts any supposed continuity.

quote="aletheist;57607"]NO! There are no intrinsic boundaries between the parts of a continuum; in this case, between the smaller lines within a continuous line.[/quote]

Then what ends the short lines, making then short lines? How can you not see the contradiction? How can there be short lines if there is nothing to end these lines, making them short lines.

I asked you for sources, not a rationalization; and in any case, it should be quite clear by now that I reject your unwarranted stipulation that a "part" is necessarily "individuated" or "separate." — aletheist

If it's not individuated, or separated from the whole, how can you say that there's a part? All you have is a whole.

The problem with your metaphysics, is as I described earlier in this thread. You have some idea of the way things should be described, an ideal, then you define your terms according to this ideal. But this ideal is just a fantasy, a fiction, and you have no respect for reality, for the way that things actually are, according to empirical observation. So you continue your metaphysics based in some fictional ideal, rather than in solid principles of how things actually are.

If you want to stick to your guns and claim that Aristotelian logic somehow contradicts Aristotle's own explicitly stated views ... well, good luck with that. — aletheist

Aristotle says many different things in many different places, often contradicting himself. That's not odd, he has a lot of material. Pretty much the entirety of "On the Heavens" has been discredited, proven wrong.

"It is a well known metaphysical principle, that the continuous is indivisible"; but you provided none, which is telling. — aletheist

It's simple Aristotelian logic. Anything divisible necessarily consists of parts. Every part is individuated, or separate from every other part. A continuity has no such separations. Therefore a continuity is indivisible.

You seem to take exception to the opening premise, assuming that something which does not consist of parts (continuum) is divisible. But then you contradict yourself by describing that thing as consisting of parts.

Before you posit point B, it does not actually exist; if anything, it is merely potential. — aletheist

I agree.

Furthermore, the "two distinct continuities" that you get by assuming the point B are not "parts" of the original continuity in the relevant sense, since the point B itself is not part of the original continuity at all. — aletheist

Right, the two distinct continuities are not parts of the original continuity, they are produced by division.

Remember, the parts of a continuous line are not points - they are shorter lines. — aletheist

Now why do you go and contradict yourself? There are no such shorter lines until you posit some points of division. If there were such shorter lines, there would not be a continuity, because the shorter lines would be already separated out. There would be a series of shorter lines in contiguity. Do you understand the difference between continuity and contiguity? I really don't think that you do because you keep describing a contiguity, and claiming that it is a continuity. They are not the same. If the long line consists of shorter lines, then it is necessary that there is a boundary between the shorter lines, so that it actually consists of shorter lines. But these boundaries contradict "continuity", you have only a contiguity.

By the way, according to your view, which "part" contains B - the one from A to B, or the one from B to Z? — aletheist

This is not my view, I was offering you a compromise, to allow for your insistence that a continuity can consist of parts. Parts imply separation, so I offered points in the continuity as separations. You seem insistent that the continuity consists of such parts, without any separations, but this is purely contradictory. Without the separations there are no parts.

On the contrary, here is what the SEP article on "Continuity and Infinitesimals" has to say (italics in original, bold mine). — aletheist

Your SEP article appears to have a very shallow and unmetaphysical explanation of continuity. Suppose we assume that a continuum is in principle divisible, how do you avoid the problem of my prior post? It is necessary that the continuum does not actually consist of the parts which it will be divided into, or else it is not, at that time a continuum.

Consider this example aletheist. Let's assume a continuity denoted by A to Z. Everything between A and Z is continuous. This continuity is not absolute by any means because it begins at A and it ends at Z. So it is a limited continuity. Now if we assume a point in between A and Z, say B, such that we have A to B, and B to Z then we no longer have a continuity of A to Z, we have two distinct continuities, A to B, and B to Z. So if we assume that A to Z consists of such parts, we are not assuming that A to Z is a continuity.

I've never seen a definition of continuous, which refers to parts, but I'm sure you can find one, even if you just make it up yourself. My OED defines continuous as unbroken, uninterrupted, connected throughout space and time. Any philosophical definition I've seen describes continuous as unbroken, uninterrupted. And that's exactly what division does, it breaks, or interrupts. It is necessary therefore that the continuous is indivisible, because if it is divided it is not continuous. To divide the continuous would make it no loner continuous, so to describe it as consisting of parts is to describe it as non-continuous. It is a well known metaphysical principle, that the continuous is indivisible.

You want to describe the continuous as consisting of contiguous parts. But then you are describing a contiguity rather than a continuity. Do you recognize the difference between contiguous and continuous?

There are, sadly, quite a few people in this bad state. "Self-help" doesn't always deliver salvation. Major life changes which I didn't engineer gave me new life circumstances which solved my problem. — Bitter Crank

I tend to believe that self-help has a limited success. What is really successful is what you describe, major life changes. And, most often these cannot be engineered because it is far too complex to determine what is needed, and even if that were properly assessed, to bring it about would require a large amount of manipulating others. Even if one could determine the required major life changes to rescue oneself from the "bad state", most I think, would not choose these changes.