You're not joking are you?

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.

The act of dividing something demonstrates that the thing divided is not continuous. — Metaphysician Undercover

Well, you can certainly partition the continuum of the Reals wherever you wish.

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. — Metaphysician Undercover

Only if by "identify" you mean "distinguish." What I thought we were discussing was whether the relation of identity itself is absolute or contextual (more below). An object is not even strictly identical to itself from different points of view, or at different times and places. However, we often treat it as the same object from different points of view, and at different times and places, because doing so suits most of our purposes. Perhaps we agree on this and can move on.

The act of dividing something demonstrates that the thing divided is not continuous. — Metaphysician Undercover

No, the act of dividing something that was continuous causes it to become discontinuous. Not surprisingly, we disagree on whether the infinite divisibility of a line renders it discontinuous, even if it is not actually divided. I am never going to convince you that "x-able" does not entail "actually x-able," and you are never going to convince me that the two are necessarily equivalent; so we might as well just agree not to waste each other's time by going down that road yet again.

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. — Metaphysician Undercover

You made assertions about Peirce's views, so the burden is on you to show that you accurately restated them. The specific language of "a difference that does not make a difference" comes from @apokrisis, not Peirce; and he did not bring it up "to support the proposition that a continuity is divisible," he was talking about identity within existence as contextual, rather than absolute.

Uniqueness would still be defined relatively. Inidividuation or identity is a difference that makes a difference ... We have a difference that is distinctive as part of a context and so can go on to be remembered as changing its developing history. We have the uniqueness of some difference that actually made a difference to the whole. — apokrisis

Actuality is being defined in terms of a difference that makes a difference. This is quite in contrast to a tautology where the actual is simply a difference. — apokrisis

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. — Metaphysician Undercover

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

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.

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?

No, the act of dividing something that was continuous causes it to become discontinuous. Not surprisingly, we disagree on whether the infinite divisibility of a line renders it discontinuous, even if it is not actually divided. I am never going to convince you that "x-able" does not entail "actually x-able," and you are never going to convince me that the two are necessarily equivalent; so we might as well just agree not to waste each other's time by going down that road yet again. — aletheist

What does "dividing" even mean? If you mean that you can take an interval of the Reals, well we learned how to do that at school, so we can do it. We can even write down an expression for taking (countably) infinite number of (finite) intervals of the Reals. So it is actually and trivially x-able.

I leave it as an exercise to the reader to ascertain whether the intervals must be open or closed.

There is simply no way around it. No matter how many individual notes one might string together it will never replicate it come close to describing the sound of a symphony. No matter how many words one strings together (as the modernist novelists attempted to do), it still never describe the sense of duration. No matter how many numbers one pulls together, in any manner one tries, it will never be able to describe the nature of complete and full continuity. Continuity does not live in discreteness, and unfortunately philosophy, for the most part, had chosen, discrete symbolism to describe what is a continuous flow, and the two can never meet.

The only way to understand nature is to fully and completely remove symbolism from the investigation. One must explore music, light, motion, thought, consciousness, dreams, sound, etc. directly. One must use consciousness to directly explore itself and penetrate it deeply.

Admittedly, without telepathic communication available to us, for discussion purposes we must resort to symbolic metaphors that in some way describe the continuous flow (I use the ocean and the symphony as my metaphors) always avoiding any addition of symbols that might allow for discreteness. Much of your arguments fully depend on creating discrete, which may be practical under many circumstance. But when discussing the nature of nature, discrete symbolism is more than impractical, it unleashes all kinds of paradoxes which are sure fire red flags that another mode of analysis is required.

No matter how many numbers one pulls together, in any manner one tries, it will never be able to describe the nature of complete and full continuity. — Rich

I agree with you on this. However, mathematics is not always and only a matter of numbers or other discrete entities. Geometry, especially topology, is an obvious example.

The only way to understand nature is to fully and completely remove symbolism from the investigation. — Rich

Except that symbolism is part of reality, so fully and completely removing it would limit one's overall understanding just as much as focusing on it to the exclusion of the other kinds of experiences that you mention.

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 — Metaphysician Undercover

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.

I do not see a way that mathematics, which relies totally on manipulation of discrete, can describe in any form, continuity.

If course we must rely on symbolism to communicate, since mind to mind communication is not available, but before such communication is performed, one must first probe nature directly and then admit in any use of metaphors that the metaphors are incomplete.

The Dao that is named is not the Dao.

Pretty good though from the ancients.

I do not see a way that mathematics, which relies totally on manipulation of discrete, can describe in any form, continuity. — Rich

As I keep telling you, mathematics does not rely totally on manipulation of the discrete; or at least, mathematics need not rely totally on manipulation of the discrete. The problem is that since the late 19th century, mathematics has largely relied on the manipulation of the discrete, because it has been grounded primarily in set theory. In recent decades, category theory has emerged as a viable alternative that is more general and much more compatible with the concept of continuity. This is evident from subsequent developments like synthetic differential geometry and smooth infinitesimal analysis.

To me it still looks like manipulation of discrete to approximate continuous. I don't see how mathematics can get around this. It is fundamental all about manipulating units as is any language.

If course we must rely on symbolism to communicate, since mind to mind communication is not available, but before such communication is performed, one must first probe nature directly and then admit in any use of metaphors that the metaphors are incomplete. — Rich

But why do you presume the job of the mind is to see reality "as it is"? That makes no evolutionary sense.

So pragmatism/semiosis is a realistic theory of the epistemic modelling relation we have with the world (and then ontologically - the surprising bit - that reality has with itself so as to indeed form "itself").

This flips everything around. Now the job of the symbolising mind is to take as little note of the actuality of the world as possible. To the degree the mind has detached itself from brute actuality, then it is starting to see the world only in terms of its future possibilities.

So for consciousness - as attentive level processing - less is more. The goal is to reduce awareness of the surrounding to the least amount of detail necessary to make successful future predictions, and thus to be able to insert oneself into the world as its formal and final cause. We gain control in direct proportion to our demonstrable ability to ignore the material facts of existence.

This is why science is the highest form of consciousness. It reduces awareness of the world to theories and measurements. We have an idea that predicts. Then all we have to do is read a number off some dial.

The fact that reality might be continuous is the reason why psychological mechanisms evolved to extract semiotic discreteness from it - a tale of distinct signs. The mind is designed to zero in on some single telling point of view in any moment - to attend. And in doing that, everything else can be ignored as noise. The world outside the focus of attention is simply ... vague.

My point is that you, like MU, are arguing from a particular set of presumptions. There is this wrong idea that the mind should see everything exactly as it really is. But that is illogical in evolutionary terms - in terms of the principles of modelling. The mind wants to do the exact opposite - transcend the world, so as to gain the power to re-imagine the world.

Of course the world still exists in brute continuous fashion. It has its recalcitrant being that ultimately acts as a constraint on our desires. Yet that doesn't mean we should just give in and give up. The highest state of consciousness is the one that is most semiotically developed - the best able to impose its own reality on reality through the creativity of a sign relation.

The problem is that since the late 19th century, mathematics has largely relied on the manipulation of the discrete, because it has been grounded primarily in set theory. In recent decades, category theory has emerged as a viable alternative that is more general and much more compatible with the concept of continuity. — aletheist

What? Have you not been paying attention? The continuum was discovered via set theory!

The specific language of "a difference that does not make a difference" comes from apokrisis, not Peirce — aletheist

Bateson must get the actual credit - http://www.informationphilosopher.com/solutions/scientists/bateson/

Thank you for articulating all the reasons that mathematics and science are useless in understanding the nature of nature. The desire "to predict" the totally unpredictable undermines the while process. However, there is nothing one can do or say to those desire this desire advice all else. Such is personality.

The continuum was discovered via set theory! — tom

You need to get out more. The mathematical world is larger than just set theory - https://en.wikipedia.org/wiki/Continuum_(topology)

The continuum made of glued together points is complimented by the continuum made of glued together relations.

And the true continuum of Peirce (or Thom, or Brouwer) goes beyond that duality in being the source of that duality.

The continuum was discovered via set theory! — tom

The continuum was not discovered via set theory, it was (and still is) modeled using set theory. Real numbers merely constitute an analytic continuum; they do not form a true continuum as defined by Peirce - as well as duBois-Reymond, Brentano, Brouwer, and many others.

At the end of the day, your version of understanding nature amounts to you standing there and saying the words "I understand nature". There is nothing more to show.

Science demonstrates its control over existence in everything that in fact makes your own modern existence possible.

So your cry of protest here could hardly sound more feeble.

Thanks for the link. 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.

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

MU's thoughts indeed form an undivided continuum. :)

But why do you presume the job of the mind is to see reality "as it is"? That makes no evolutionary sense. — apokrisis

If I may interject here, it seems to me that the job of the mind (or any organ for that matter) is to provide the organism the necessary nutrients to survive. In the case of the mind (or the brain depending on how you see the relationship between the two), paired with the sense organs, provides the organism a valuable nutrient - information.

Information, of course, needs to be accurate. The mind needs to be able to predict future outcomes, and it does this through trial-and-error learning, habitual behavior and unconscious memory. If everything was perfectly known, there would be no need for a mind. No thinking would be required. Thinking is the process in which we evaluate different sorts of information and construct a path of action. If we wanna go the psychoanalytical meta-psychological route, then consciousness is the (painful) method in which the unconscious satisfies its endless depth of want and need in a temporal world of insufficiency. Without this inherent ambiguity and uncertainty, there wouldn't seem to be any reason for an organism to expend energy on a representation of the world for the sake of representing the world. Certainly the mind cannot be an easy thing to maintain.

So apo is right in that for biological organisms, less tends to be more. Efficiency is what's up. But of course the mind has to be modelling the world somewhat accurately, otherwise theories like apo's wouldn't even make sense themselves. Here we have Plantinga's argument against naturalism, which in my opinion fails but certainly provokes discussion and refinement of naturalism.

The goal is to reduce awareness of the surrounding to the least amount of detail necessary to make successful future predictions, and thus to be able to insert oneself into the world as its formal and final cause. We gain control in direct proportion to our demonstrable ability to ignore the material facts of existence. — apokrisis

The tricky part is to figure out that balance between seeing too little and seeing too much. Curiosity as much as ignorance is a source of many problems. A cultural domino effect.

This is why science is the highest form of consciousness. It reduces awareness of the world to theories and measurements. We have an idea that predicts. Then all we have to do is read a number off some dial. — apokrisis

Why just science, though? Why not soccer? Surely goalies reduce their awareness of the world to the game, its rules and the movement of the players and the trajectory of the ball. Why isn't this the highest form of consciousness?

To denote science (or anything else) as the "highest" form of consciousness is sort of ambiguous in my opinion. Higher than what? What measuring system are we using here?

If anything I would have to say philosophy is the "highest" form of thought, since it deals with abstract concepts in a purely possible modality. Or, hell, even just daydreaming.

This is what happens when one falls in love.

Information, of course, needs to be accurate. — darthbarracuda

Or meaningful in fact.

And this is the semiotic point. Information is not noise but a message when it is a sign connected to our desires.

So there are two views of information. One is that it is a material difference. That makes even noise countable as bits of information.

The other says differences have to make a difference. And that happens when there is a context of purpose which interprets a difference in terms of its meaning. A signal now also has a message.

There is a science of these things you know...

If everything was perfectly known, there would be no need for a mind. No thinking would be required. Thinking is the process in which we evaluate different sorts of information and construct a path of action. — darthbarracuda

I often drive long stretches of road with no conscious memory of a lot of pretty technical and dangerous actions. We are designed to automate our awareness of the world so that we can do everything at the most habitual and inattentive level possible.

So attention and thought are reserved for dealing with the unpredicted, the novel - the things we hope to turn into habits in the future.

That is why the old are wise. They know all the answers already. Correct thought appears effortless.

. If we wanna go the psychoanalytical meta-psychological route, then consciousness is the (painful) method in which the unconscious satisfies its endless depth of want and need in a temporal world of insufficiency. — darthbarracuda

No. Let's not go into the bogus science/dressed-up romanticism of psycho-analysis.

So apo is right in that for biological organisms, less tends to be more. Efficiency is what's up. But of course the mind has to be modelling the world somewhat accurately, otherwise theories like apo's wouldn't even make sense themselves. — darthbarracuda

You just don't get the nuances correctly yet.

The whole notion of "re-presentation" is a psychological fallacy. The mind - as a modelling relation - wants "efficiency" in always knowing the shortest path between its desires and their fulfilment. So it is that shortest path which fills awareness, not the totality of all the world's facts.

To denote science (or anything else) as the "highest" form of consciousness is sort of ambiguous in my opinion. Higher than what? What measuring system are we using here? — darthbarracuda

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.

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.

One-nil, one-nil, onnneee-nilllll-ahh! Comes the happy chant of the home crowd.

If anything I would have to say philosophy is the "highest" form of thought, since it deals with abstract concepts in a purely possible modality. Or, hell, even just daydreaming. — darthbarracuda

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

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.

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"?

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? — Metaphysician Undercover

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.

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.

You are talking about meaningless differences and claiming they are now meaningful. But you can't say why that would be so (except you would then be able to count some points being scored on your private anti-apokrisis metre ... Bing! MU scores another (own) goal!)

A thing' s identity is not the same as what it is identified as. In our previous conversation, the example was Pluto, which had at an earlier time been identified as a planet. 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. Pluto's identity is not affected by what it happens to be identified as, in other words. Identity is always a matter of mere logical stipulation, and it is never actually realized, which I think is pretty much in accordance with something apokrisis had said earlier about the relation between identity and actuality.

What we are inquiring into is ... what it means to be continuous, and what it means to be discontinuous. — Metaphysician Undercover

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.

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

That is apparently your conclusion. Mine is that it is pointless (pun intended) to talk about anything related to this topic with you.

The continuum was not discovered via set theory, it was (and still is) modeled using set theory. Real numbers merely constitute an analytic continuum; they do not form a true continuum as defined by Peirce - as well as duBois-Reymond, Brentano, Brouwer, and many others. — aletheist

And those later objections have been swept aside. Cantor was the first to rigorously define the continuum in 1870s and all the dissenters have been forgotten.

I think you'll find that Peirce got into the act somewhat later than Cantor, after being inspired by Cantor. And, in the history of Real analysis, set theory, etc, Peirce is a dead-end. Cantor's ideas have been extended and developed, Peirce's have been abandoned.

And those later objections have been swept aside. Cantor was the first to rigorously define the continuum in 1870s and all the dissenters have been forgotten. — tom

Developments such as category theory, nonstandard analysis, and synthetic differential geometry or smooth infinitesimal analysis reflect dissatisfaction in some quarters with the dominant paradigm. Only time will tell whether any or all of these supplant set theory and its progeny.

I think you'll find that Peirce got into the act somewhat later than Cantor, after being inspired by Cantor. — tom

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.

And, in the history of Real analysis, set theory, etc, Peirce is a dead-end. Cantor's ideas have been extended and developed, Peirce's have been abandoned. — tom

And yet, here we are, discussing them. There has been quite a revival of interest in Peirce's ideas, both in philosophy and in mathematics, over the last couple of decades. As @apokrisis likes to point out, he was far ahead of his time in many ways, and we are only now catching up to him.