Cantor was the first to rigorously define the continuum in 1870s and all the dissenters have been forgotten. — tom

You really do live in your own private Idaho. Underground like a wild potato.

I'm liking Brouwer's "two-oneness" intuitionistic approach to the continuum that Zalamea mentions. It gets the pan-semiotic point that existence depends on memory (or 2ns on 3ns).

So the first mark (or cut of the line) anchors the second mark by becoming the memory - the context in which a difference can now make a difference.

This introduces the direction of time, and hence energy and dissipation, into axiom-level mathematical thinking. The past is spent (gone to synechic 3ns), and yet the future is still open (still primal or tychic 1ns).

So the continuum does have this neighbourhood property - this extra hidden dimension - which is its memory. The first cut becomes the context for a second cut (and together they underwrite an endless repetition of cuts). And this is where counting and even ordinality gets justified. Effort has to be spent in constructing a history of what has happened. But the future extends to infinity and beyond - underwritten by its own past success.

In appreciating the intuitionistic approach to the continuum, we can see what the set theoretic approach simply freezes out and takes for granted. The Cantorian infinity is timeless and effortless - and thus patently unreal on that score.

If maths wants to speak more truly of nature, we can see how memory and action must be added back into the mix.

The point is the difference is meaningful, no matter how much you might pretend otherwise-- a thing's identity is not found in what it is to you (i.e. your experience of it, semiotics, your "epistemic cut" ), but rather itself. There is a difference ( "This one dies, not the other" ) no matter if you care about it. Your generality is a myth, a dishonest story you tell yourself to eliminate subjects in the contexts of your "practical" concerns.

How is "itself" to be "found" if not in experience and thought?

That's the trick-- it is found in experience and thought... but it is not the experience and thought.

Consider the knowledge of you posting on this forum. How would anyone know about this? Well, there are only their experiences, be they of the empirical world (i.e. observations of your posts) or experiences of logical comprehension (e.g. their identity, your identity, the logical discintion of a forum and posts, etc.). We really do know objects themselves-- else we would be merely talking about ourselves rather than other things or people.

But... as apo demonstrates here, our experiences are their own. They are distinct from the thing-itself, such that, on some occasions, we make mistakes about things. Apo's experience shows these two states to be identical, even though they are not. He does not know the identity of these states itself. His experience has missed it.

And yet the traditional/classical conception of God is that He is absolutely simple; His attributes are not discrete in the way that you seem to be suggesting. — aleteiest

Which is exactly the problem with the traditional/classical God. There is no way an infinite of attributes can belong to such a being. Such a God is simple and empty. No thing belongs to them because it would mean owning the discrete. The continuum of God is left with other at all, merely an empty set that has nothing to do with the world.

Spinoza's point is both the continuum and it's members are discrete-- the former as the continuum itself, the latter as each particular member. To be a continuum or category is discrete, not to lack identity or discintion. The infinite is its own discrete truth-- it's the infinite itself, a set which contains endless discrete members.

With all due respect, that seems rather ... vague to me. — aleteiest

Only because you don't recognise the infinite as its own thing. It seems "vague" because you are trying understand it though a finite lens. You want us to say what is in the world or what it does to the world, how does it manifests in our observations of the world.

The point is the infinite is not vague at all. It means a set without beginning nor end, a distinction and identity all of its own, which is never any member of itself.

This seems like a case where Peirce's attempt to use generic terminology for his categories may have been misleading. They are not called 1ns, 2ns, and 3ns because they always and only come about in that order; on the contrary, my interpretation of his cosmology is that in the hierarchy of being, 3ns is primordial relative to the other two. In any case, 1ns/possibility does not "end" where 2ns/actuality "begins," they are both - along with 3ns/necessity - indispensable and irreducible ingredients of ongoing existence. — aleteiest

A hierarchy is an order. No doubt he is talking about logical terms, but hierarchy or order is an inherently a conception of the finite and time. It results in a leakage into causality (as we see in apo's argument which treat "vagueness" as the origin of the states of our reality), where 2ns (actuality) is considered to be born of 3ns (semiotics, necessity) and 1ns (possibility).

Possibility is necessary. In Peirce's terms, it is also 3ns, along with logical distinction forms (semiotics), possibility is necessary. The hierarchy collapses. Since possibility is necessary and infinite, it never begins or ceases. It is just as "primordial" as semiotics. Logical distinction has always been. Possibility has always been. Neither came first. Peirce's triad collapses into the necessary (semiotics, possibility) and the contingent (actual states).

Furthermore, the hierarchy between the necessary and contingent does not make sense. For a state to be actual, it takes more than the presence of either semiotic discintion(i.e. form, meaning) and possibility. If I say: "There is a logical form of me being the US president and the possibility of me being the US president," it doesn't birth the actual state of me as the US president. Only an actual state can do that. In terms of definition, actual state are self-defined, not given by the necessity of semiotic and possibility. Actuality becomes just as "primordial" as semiotics or possibility.

If I am to be president of the US, there can only be a concurrence of possibility, semiotics and actuality. Only when those are all at once, of themselves, am I US president. There can be no hierarchy.

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.

The point is the difference is meaningful, no matter how much you might pretend otherwise-- a thing's identity is not found in what it is to you (i.e. your experience of it, semiotics, your "epistemic cut" ), but rather itself. There is a difference ( "This one dies, not the other" ) no matter if you care about it. Your generality is a myth, a dishonest story you tell yourself to eliminate subjects in the contexts of your "practical" concerns. — TheWillowOfDarkness

If you could define meaningful in a meaningful fashion here - ie: in a way that makes a quantifiable difference - then there might be something to talk about.

So how do you define meaningful exactly?

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. — Metaphysician Undercover

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.

I agree the things are not our experiences and thoughts about them. This is tautologously true. But from that it does not follow that the things have any existence which is indentical to their experienced and conceived existence, independently of their being experienced and conceived.

Your understanding seems not incorrect. — apokrisis

It took me a while, but I finally figured out Zalamea's notation, and thus his point about "failures of distribution": the concepts of vagueness and generality are manifested in the non-distributive nature of the universal quantifier for disjunction and the negated existential quantifier for conjunction in classical predicate logic.

• Vagueness: ¬(∃x)(Px ∧ ¬Px) does not entail ¬[(∃x)(Px) ∧ (∃x)(¬Px)]
• Generality: (∀x)(Px ∨ ¬Px) does not entail (∀x)(Px) ∨ (∀x)(¬Px)

Of course, the first expressions correspond to the standard laws of non-contradiction and excluded middle, respectively; and they are equivalent to each other if double negation elimination is permitted. The second expressions are also equivalent to each other, but regardless of whether double negation elimination is permitted; and rather than spelling out that they do not follow from the first expressions, Zalamea simply presents them as non-tautologies. As such, vagueness means that "both P and not-P are possible," while generality means that "neither P nor not-P is necessary." No big revelation here - in fact, Zalamea says this quite plainly - but I am only now finally connecting the dots.

Although the first expressions do not entail the second ones, it does work the other way around - the first expressions follow from the second ones. If only P (or not-P) is possible - i.e., P (or not-P) is necessary - then LNC and LEM are trivially true. Now, what you were saying about identity is that it is likewise a non-tautology when it comes to the actual as unavoidably contextual. I wonder if it then makes sense to say this:

• Contextuality: ¬¬Px → Px does not entail Px = ¬¬Px

This is parallel to the other two in that the first expression requires double negation elimination and follows trivially from the second one. So perhaps contextuality means that "P is whatever is distinguished from not-P."

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.

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

Interesting argumentative strategy.

Following on from the point about Brouwer, memory and action, I am reminded of Landauer's erasure principle - https://en.m.wikipedia.org/wiki/Landauer's_principle

This ties computation to physical actuality in a helpful fashion here.

Memory or history is irreversible symmetry breaking created by some expenditure of energy. So 3ns can be neatly defined in information theoretic terms as the erasure of 1ns. While 2ns is a still locally reversible state - the dynamics can be read off in either direction until fixed by a 3ns context.

Hence the continuum does model time as memory and action. The past is 3ns - the information fixed by irreversible acts of information erasure. What was possible as 1ns is now decided with counterfactual definiteness one way or the other.

The present is then 2ns, the instant when there is just an event that could be read in either direction. Which is action, which reaction? All we know is that there is an event - a symmetry breaking - that could be about to be fixed as something definite and 1ns erasing in the 3ns memory of a developing history.

Then the future is 1ns or the vague. It is open possibility. It is the freedom waiting to be dissipated by acts that steadily rob the system of the energy to locally distinctive rather than globally generic.

So the continuum can't just be freely divided or counted without limit. Computation has been defined by Landauer in terms which spell it out as a game of diminishing returns. The Brouwerian requirement that the actual numberline needs a memory (context being primal) means that constructing the memory is dissipative. It costs to erase possibility.

In our universe, this is captured with complete precision by quantum mechanics. There is a holographic limit on computation. Try to do too much of it and the resulting heat would even melt spacetime, turn it locally into a black hole.

Of course maths can simply ignore all these issues - imagine numberlines as spatial things with no time, no memory, no action, no dissipation.

But while that might make a paradise for Hilbert, mathematical physics might believe that it wants mathematical conceptions much more in line with reality as it is observed. Which is where a Peirce comes in.

So perhaps contextuality means that "P is whatever is distinguished from not-P." — aletheist

Or formally, each is the other's context. As in the logic of a dichotomy - that which is mutually exclusive and jointly exhaustive.

So mathematically, it is a reciprocal or inverse relation.

X = 1/not-x. And not-x = 1/x.

This is why I say the degree each secures the other contextually is strictly relative. Each is only as precise as its alter ego allows it to be. If not-x is vaguely defined, then x remains just as vague too.

This ties identity directly to the strength of the answering context. And so put together, it allows for a controlled way to go from 1ns to 3ns via 2ns. You have every degree of mutual definition on the way from one limit (the 50/50 completely vague state of not knowing which is which), to the 1/0 limit which would represent absolute counterfactuality or completely secured identity).

I defined it - going the furthest in reducing awareness of reality to a matter of signs - that is, the theory we create and then the numbers we read off our instruments. — apokrisis

But why should I see this as the "height" of consciousness? Are you saying that this is consciousness at its most effective, as a well-oiled cog in the dynamic of the world?

The soccer goalie does just the same in the end. Success or failure is ultimately read off a score board ticking over - the measurement of the theory which is the rules of a game. — apokrisis

I mean, soccer isn't the only example available. What about artists who paint pictures blind or compose pieces deaf? Or the taxicab driver who doesn't need to look at the speedometer to know how fast he's going? Or the laborer who pounds stakes in the ground in a perfect repetition?

You are forgetting the role of measurement. Ideas must be cashed out in terms of impressions. — apokrisis

So, Hume? mkay

Science is the metaphysics that has proven itself to work. It is understanding boiled down to the pure language of maths. And so measurements become actually signs themselves, a number registering on an instrument. — apokrisis

It's really not that romantic, though. What if the instrument isn't working properly? What if you messed up in the calculation? What looks like understanding can easily be an error propagating through a system.

You say it boils down to the pure language of mathematics. Yet surely not all science rests on mathematics. Unless you wanna go all Meillassoux on us.

He was indeed inspired by Cantor, but he also achieved some of the same results and reached some of the same conclusions at least semi-independently. In the end, he became disenchanted with Cantor's whole approach; as Rich has been emphasizing, you cannot adequately represent true continuity with something that is discrete. — aletheist

What is discrete in the Reals? What aspect of the Reals is being inadequately represented by this discrete thing?

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.

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 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. — Metaphysician Undercover

This is called infatuation with winning an argument vs. truly interested in understanding nature. Translation: The difference is small but big enough for me to admit that I will lose the argument, so let's ignore it.

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

The issue is well explained in BK. 6 of Aristotle's Physics. — Metaphysician Undercover

That would be the same Book VI of Aristotle's Physics that I quoted at some length in the thread on "Zeno's paradox," which you immediately dismissed because "Aristotle says many different things in many different places, often contradicting himself." Since you brought it up here, let me quote it again:

Now if the terms 'continuous', 'in contact' [i.e., contiguous], and 'in succession' are understood as defined above - things being 'continuous' if their extremities are one, 'in contact' if their extremities are together, and 'in succession' if there is nothing of their own kind intermediate between them - nothing that is continuous can be composed 'of indivisibles': e.g. a line cannot be composed of points, the line being continuous and the point indivisible ...

Again, if length and time could thus be composed of indivisibles, they could be divided into indivisibles, since each is divisible into the parts of which it is composed. But, as we saw, no continuous thing is divisible into things without parts. Nor can there be anything of any other kind intermediate between the parts or between the moments: for if there could be any such thing it is clear that it must be either indivisible or divisible, and if it is divisible, it must be divisible either into indivisibles or into divisibles that are infinitely divisible, in which case it is continuous.

Moreover, it is plain that everything continuous is divisible into divisibles that are infinitely divisible: for if it were divisible into indivisibles, we should have an indivisible in contact with an indivisible, since the extremities of things that are continuous with one another are one and are in contact. — Aristotle, Physics VI.1, emphases added

So that which is continuous must be divisible into parts, and those parts cannot themselves be indivisible; in fact, they must be infinitely divisible, because they are likewise continuous.

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. — Metaphysician Undercover

Yes, but he already resolved this paradox in Book IV:

For what is 'now' is not a part: a part is a measure of the whole, which must be made up of parts. Time, on the other hand, is not held to be made up of 'nows' ... obviously the 'now' is no part of time nor the section any part of the movement, any more than the points are parts of the line - for it is two lines that are parts of one line. In so far then as the 'now' is a boundary, it is not time, but an attribute of it ... — Aristotle, Physics IV.10-11

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. 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.

What is discrete in the Reals? What aspect of the Reals is being inadequately represented by this discrete thing? — tom

Numbers are intrinsically discrete; and it is not a matter of whether this discrete thing adequately represents the real numbers, but whether it adequately represents true continuity.

Another thought.

• Vagueness: ¬(∃x)(Px ∧ ¬Px) does not entail ¬[(∃x)(Px) ∧ (∃x)(¬Px)]
• Generality: (∀x)(Px ∨ ¬Px) does not entail (∀x)(Px) ∨ (∀x)(¬Px)
• Contingency: (∀x)(Px ∨ ¬Px) does not entail (∃x)(Px ∨ ¬Px)

Vagueness means that "both P and not-P are possible."
Generality means that "neither P nor not-P is necessary."
Contingency means that "either P or not-P might not be actual."

Numbers are intrinsically discrete; and it is not a matter of whether this discrete thing adequately represents the real numbers, but whether it adequately represents true continuity. — aletheist

But surely you are aware that the set of real numbers is complete?

Right, there are no "missing" numbers; but that still means that the set of real numbers consists of individual numbers. A true continuum does not consist of individuals.

Apokrisis appears to be saying that there is no use in assuming such a principle of identity. — Metaphysician Undercover

This seems to be the way that apokrisis speaks of identity, we give a thing an identity relevant to the purposes at hand. — Metaphysician Undercover

You seem uncertain that this is my actual position for some reason. Is that because you know you're just making up things I never would say?

Example of discrete: 1 min., 2, min., 3, min .... into the impossible infinite.

Example of continuous: duration (time) as it is actually experienced by consciousness.

Example of discrete: 1 ft., 2, ft, 3 ft etc.

Example of continuous: space as we actually experience it as memory.

Right, there are no "missing" numbers; but that still means that the set of real numbers consists of individual numbers. A true continuum does not consist of individuals. — aletheist

What is the first number after 0?

Please just make your point, if you have one. The real numbers constitute an analytic continuum, not a true continuum as defined by Peirce (and others).

Please just make your point, if you have one. The real numbers constitute an analytic continuum, not a true continuum as defined by Peirce (and others). — aletheist

What is the first number after 0 according to Peirce?

What is the first number after 0 according to mathematics - i.e. Cantor/Dedekind/Cauchy et al?

If you don't know, just admit it!

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.

In Peirce's terms, the real numbers between 0 and any arbitrarily small but finite value form a collection with an abnumeral multitude, which has an even larger power set (in Cantor's terms). However, the potential points between any two actual points marked on a truly continuous line exceed all multitude, and thus have no power set.