It never glows--even in bright light--and you have to hold it over your head yourself. — Bitter Crank

Even while driving? Seems dangerous. :D

I am not looking for a pat on the back; just saying that I am not blithely neglecting the kinds of concerns being expressed here.

I think 'avoiding burning fossil fuels' should unquestionably be a major policy goal. — Wayfarer

I worry about anything that is characterized as "unquestionable," especially in the realm of public policy. But I also drive a hybrid vehicle, and believe that we should be good stewards of the planet - after all, so far it is the only one we have.

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.

If we covered the entire world in windmills I don't know what the effect would actually be. — VagabondSpectre

This is really the only point that I wanted to make. We do not (yet) know the global effects of widespread implementation of various "alternative" energy sources. We just assume that since they do not involve burning fossil fuels, they are automatically better/cleaner.

that kind of expression plays right into the hands of petro-chemical industry scare-mongering. — Wayfarer

The point is that "clean" and "renewable" are buzzwords implying that anything that avoids burning fossil fuels is inherently and unquestionably less damaging to the environment. I am not convinced that we know this to be a fact at this moment in history. For one thing, no form of energy is literally renewable; it is different energy that we capture over time from the sun, wind, waves, etc.; not the same energy over and over.

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.

This brings up a question that I have long had about allegedly "clean" or "renewable" energy sources. There is no free lunch, so we are always redirecting energy that otherwise would have gone somewhere else. Solar, wind, waves, whatever - taking that energy out of the environment must have some kind of effect. What (if any) studies have been done to gauge this and confirm that the overall outcome it is net positive relative to burning fossil fuels?

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.

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.

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.

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.

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

Marking a point on a line, or breaking it into two lines that have points at their separated ends.

Peirce employs this notion, of a difference which doesn't matter, to support the proposition that a continuity is divisible. — Metaphysician Undercover

He does? Where? Please cite his writings to support this claim. Did Aristotle also employ this notion, since he likewise held that a continuum is (infinitely) divisible, though actually undivided?

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. — Metaphysician Undercover

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.

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. — Metaphysician Undercover

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

A continuity cannot have a point of difference because this would make it discontinuous. — Metaphysician Undercover

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

Measurements are always approximations and it is why measurements specifically and mathematics in general (because of its discrete nature) are very poor tools for understanding nature. — Rich

I continue to be skeptical of your claim that all mathematics is inevitably discrete. In the last several decades, category theory - as a more general alternative to (discrete) set theory for the foundations of mathematics - has facilitated developments like synthetic differential geometry or smooth infinitesimal analysis, which show great promise for more faithfully representing the continuous as continuous.

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. — Metaphysician Undercover

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.

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". — Metaphysician Undercover

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.

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. — Metaphysician Undercover

Again, it depends entirely on our purposes, which depend entirely on the context. For the most part, the difference that makes a difference in this example is that one car is yours and the other is mine - a human convention, not something intrinsic to the objects themselves. If they were sitting side-by-side on a dealer's lot - same year, make, model, trim, colors, options, condition, mileage, price, etc. - then there would be no differences that make a difference, until you (arbitrarily) choose which one to buy.

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. — Metaphysician Undercover

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.

For the purpose of understanding the nature of nature, we need precision otherwise we miss the boat. — Rich

I am surprised that you would say this, considering that we started the thread with your comments to the effect that discrete mathematics cannot properly represent the continuity of nature. Precision is a matter of measurement, and measurement is a matter of discrete mathematics; but the continuous is indeterminate.

It is alright to say that a book, for practical purposes, has the same identity before and later. But it is more precise to say that the book has changed and continuously changes so that it is never is the same in duration. — Rich

I made this point earlier; the contextuality of actuality entails that it is not necessarily true that this object from one point of view, or at one place and time, is identical to this object from another point of view, or at another place and time. Perhaps we can agree that it is more precise (in your sense) to recognize the imprecision (in my sense) of reality.

God has infinite attributes. To only be one would be a contradiction with God's very nature. — TheWillowOfDarkness

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.

God is infinite. God cannot be said to begin or end at any point. It's anything but vague. — TheWillowOfDarkness

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

Realising the necessity of potential, Spinoza also points out potential cannot be "firstness." Why? Well, because it never begins nor ends. — TheWillowOfDarkness

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.

When we seek the truth, differences never cease to matter. — Metaphysician Undercover

What matters to someone is always a function of that person's purposes. Surely you can agree that some differences matter to you more than others; and since everyone has finite resources (including time), we have to prioritize which differences - as well as which similarities - are significant enough within a given context to warrant our focused attention.

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. — Metaphysician Undercover

Read more carefully - in the comment that you referenced, did not say anything about a difference not being a difference; he was talking about a difference not making a difference. Do you see the difference (pun intended)? There are times when a difference really is so minute, or irrelevant, that it does not matter in that context; i.e., it is not worth taking into account, given one's purposes. It still is a difference, but it does not make a difference. I care not one whit about the color of paint that is going to be applied to a steel beam when I am analyzing it by means of a mathematical model to determine whether it can carry the forces that the building code says it must be able to withstand as part of an actual structure. The architect has a different purpose, and therefore might have a different stance - although typically he or she just wants to hide everything above the ceiling anyway.

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. — Metaphysician Undercover

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!

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. — Metaphysician Undercover

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.

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. — Metaphysician Undercover

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.

I think what you mean to suggest is that our experiences of mental and physical phenomena seem fundamentally different, so common sense says that that they are fundamentally different.

But the difference is that now I have made the natural relativity of the question of identity explicit. — apokrisis

So the idea is that the context of x is not-x, and defining the identity of x as not-not-x recognizes this, rather than making it a contextless tautology? "x is x" does not apply to the contextual, but "x is not-not-x" does apply as an apophatic alternative?

While I am at it, do you agree or disagree with my other "first cut" definitions of "contextual" that parallel what Peirce wrote about "vague" and "general"? — aletheist

But contextuality leaves it open whether the further possibility is 1ns or 2ns. It could a future condtional (the coming battle with the Persian fleet) or it could be some event already fixed by a determination (what will I discover when I finally check my ticket for the lottery drawn last week?). — apokrisis

Did something get accidentally deleted from your post? I do not see how your response here addresses my question.

Of course then along came relativity to demonstrate all this classical definiteness was relativistically contextual and quantumly indeterminate. That is why Peirce gets credit for foreseeing the physical revolutions about to come. — apokrisis

Indeed, "frame of reference" has been in my mind throughout our discussion of contextuality.

What is "necessary reasoning"? — Cabbage Farmer

Deriving conclusions from information that is already present in the premisses. Also known as deductive reasoning.

What sort of necessary reasoning is commonly associated with quantity? — Cabbage Farmer

Arithmetic is an obvious example, such as 2+2=4.

What sort of necessary reasoning is not commonly associated with quantity? — Cabbage Farmer

Syllogisms are an obvious example, such as "All men are mortal, Socrates is a man, therefore Socrates is mortal."

In what broad sense is mathematics "the application of necessary reasoning"? — Cabbage Farmer

I am following Charles Sanders Peirce in suggesting that all necessary reasoning is fundamentally mathematical reasoning. He defined mathematics as the science of drawing necessary conclusions about ideal states of affairs by means of diagrams, which are representations that embody the significant relations among the parts of their objects.

Can we apply reasoning to real states of affairs, or only to hypothetical and ideal states of affairs? — Cabbage Farmer

We can apply reasoning to real states of affairs, but typically we do so by modeling them as ideal states of affairs. We have to identify the significant parts and relations of the actual situation and create a diagram accordingly within an appropriate representational system, whose rules govern our transformations of the diagram.

Can we apply necessary reasoning to real states of affairs, or only to hypothetical and ideal states of affairs? — Cabbage Farmer

Only to ideal states of affairs, since we can never be absolutely sure that real states of affairs are completely deterministic.

Do you mean to say ... — Cabbage Farmer

Your paraphrase seems about right.

What is the difference between the "model" and the "representational system" that governs subsequent transformations of the model? How are each of these terms related to the definition of "hypothetical or ideal states of affairs"? — Cabbage Farmer

The representational system is a set of rules, such as Euclid's postulates for geometry. It is ideal because it may or may not accurately capture aspects of reality; for example, non-Euclidean geometry is more appropriate in certain cases. The model is a diagram constructed and manipulated in accordance with those rules, such as a sketch of a triangle and any auxiliary elements that must be added in order to carry out a particular proof. It is ideal because the actual drawing includes features that are irrelevant to the problem at hand, such as the thickness of the lines and their deviation from being perfectly straight.

How do we isolate "assumptions" that guide the definition of model, representational system, and states of affairs? Is it the assumptions, or the whole package, that determines the aptness of the conclusions obtained? — Cabbage Farmer

Isolating assumptions can be quite a challenge, especially for more complex situations, such as a computer model of a structure that I analyze in accordance with the principles of mechanics in order to ascertain whether all of the members and connections are adequately designed for the forces to which they might be subjected. It is the whole package that validates the conclusions - the representational system and its assumptions, the individual model and its assumptions, and their correspondence (in some sense) to the actual state of affairs. As I like to put it, engineers solve real problems by analyzing fictitious ones, which involves simulating contingent events with necessary reasoning.

What role does measurement play in your account? Or more generally: How are "aspects of reality" translated into formal signs in the model, or into bits of "necessary reasoning"? — Cabbage Farmer

The representational system is often grounded in past inductive investigations; i.e., science. We have learned from collective experience that making certain assumptions and applying certain rules generally produces results that are useful. Learning how to create appropriate models is part of the personal experience that is required to develop competence in a particular field, since it often involves exercising context-sensitive judgment, not just following prescriptive procedures. Again, the modeler must be able to ascertain which parts and relations within the actual situation are significant enough to warrant inclusion in the model.

It's not clear how this follows from anything you've just said about "necessary reasoning". Isn't human behavior part of nature? Aren't human behaviors "natural phenomena"? — Cabbage Farmer

The behavior of matter much more closely conforms to exceptionless laws of nature than the behavior of people, even taking their habits into account. As such, necessary reasoning is much more likely to be useful and effective in modeling and predicting the behavior of material things than the behavior of intelligent and willful people, who are quite capable of deviating from their habits at any time.

Have you signed up for special troubles associated with dualism? — Cabbage Farmer

No, Peirce vigorously rejected both dualism and materialism/physicalism; he wrote, "The one intelligible theory of the universe is that of objective idealism, that matter is effete mind, inveterate habits becoming physical laws."

Accordingly, I'm not sure what difference you've gestured at here, and what relevance it may have for our conversation about the uses of mathematics. — Cabbage Farmer

That is fine. Hopefully these additional responses have helped clarify my thoughts for you.

Again, much to ponder. Thanks for taking the time. For the moment, I can offer just a few initial responses.

If we are now talking cosmology, it is the Universe that is indifferent to any difference that doesn't make a difference in being beyond the needs of its genenralised purpose. — apokrisis

Which is what, in your view? According to Peirce, "the universe is a vast representamen, a great symbol of God's purpose, working out its conclusions in living realities … The Universe as an argument is necessarily a great work of art, a great poem - for every fine argument is a poem and a symphony - just as every true poem is a sound argument" (CP 5.119, 1903, emphasis added). The dynamic object of this sign is God Himself, and its immediate object is His purpose, the development of reason - i.e., the growth of our knowledge about God and His creation. As an argument, the interpretant is its conclusion, the living realities that the universe is constantly working out.

1ns can't be the true initial conditions of existence as Peirce's own logic makes necessary. — apokrisis

My interpretation is that 3ns is the true initial condition of reality - which is prior to both possibility (1ns) and existence (2ns) - as Peirce's own cosmology makes necessary (Ens necessarium).

So again, 2ns in Peirceanism is about the emergence of crisp possibility or determinate degrees of freedom. — apokrisis

This is one point at which I am having consistent trouble tracking with you. I understand 2ns in Peirceanism to be about brute reaction/resistance, the absence of freedom (1ns) and reason/purpose (3ns).

This of course is what I deny. There is only relativity, never the absolute. — apokrisis

How is this inconsistent with my suggestion that everything is general (i.e., indeterminate) to some degree? Perhaps I just need to clarify that this is generality in the broad sense, both negative (vagueness) and positive. It is a corollary of the thesis that all three categories are present and irreducible in every actual phenomenon.

X is being made its own context. That is the tautology here. The assertion is being made that the context is crisply existent too - thus bringing out that which Peirceanism would seek to deny. — apokrisis

By "here," do you mean the standard application of the principle of identity as x=x and/or x=not-not-x? Who is making the assertion "that the context is crisply existent too"?

The upshot then is that the statement is true only to the degree that either term is true. — apokrisis

Which statement? Which terms? I want to make sure that I clearly understand what you are saying here. Also, how would you fill in the blank with some formalized version of the principle of identity?

If x is contextual, then it is not necessarily true that _____.

While I am at it, do you agree or disagree with my other "first cut" definitions of "contextual" that parallel what Peirce wrote about "vague" and "general"?

A sign is objectively contextual, in so far as, leaving its interpretation indeterminate, it relies on some aspect of the actual situation to complete the determination. "That house is on fire." "What house?" "That one over there."

The contextual might be defined as that to which the principle of identity does not apply. This object from one point of view, or at one time and place, is not the same as this object from another point of view, or at another time and place.

There is much to ponder here, but you still have not explained - at least, not in a way that "clicks" for me - what you mean when you say that the principle of identity (x = x, or perhaps x = not not-x) demands discontinuous actualization by employing the context to derive specificity via a dichotomy. If you could spell this out, I would be grateful.

Another way to put it is that if generality and vagueness are real yet not actual, then the actual would be the not real. — apokrisis

Peirce's definition of "real" is that which has characters regardless of what anyone thinks about it. He came to realize by about 1896 that all three categories are real in this sense, which is why Max Fisch characterized him as a "three-category realist" from that point until the end of his life. Consider also this passage:

Existence, then, is a special mode of reality, which, whatever other characteristics it possesses, has that of being absolutely determinate. Reality, in its turn, is a special mode of being, the characteristic of which is that things that are real are whatever they really are, independently of any assertion about them. — CP 6.349, 1902

Of course, here we once again encounter the notion that existing things are "absolutely determinate," which - as we have been discussing - is really (pun intended) just an ideal limit. Strictly speaking, nothing "exists" in this sense. Is that what you are getting at by suggesting that the actual is the not real?

Existence can approach but not reach the perfection of discontinuous actualisation that the principle of identity demands. — apokrisis

Again, please elaborate. How does the principle of identity demand discontinuous actualization?

I don't think it is essential to arrive at one perfect word. — apokrisis

I agree, but I think that it will be helpful to clarify the distinctions that we are trying to draw if we can assign a term to 2ns that goes along with "vague" for 1ns and "general" for 3ns.

But if vagueness is the best term for 1ns, and generality the best for 3ns, then another term for 2ns (after hierarchy theory) would be specificity. — apokrisis

I am not very familiar with hierarchy theory, but I know that you refer to it a lot, and I might do some reading about it once I finish my current exploration of category theory and smooth infinitesimal analysis. Although I can see how "specific" is an antonym for both "vague" and "general," it strikes me as too much of a synonym for "determinate," such that the principle of identity would apply. Furthermore, we can "specify" something by description, rather than requiring an index to pick it out within a particular context.

I mean 2ns looks the most like the regular reductionist notion of the atomistically and mechanically determinate - in simply being Newtonian action and reaction. — apokrisis

I agree, especially since part of Peirce's point in describing vagueness/1ns as the inapplicability of PC and generality/3ns as the inapplicability of PEM is to define 2ns as that to which both PC and PEM do apply. However, he also wrote those two definitions of "individual" that I quoted a while back (CP 3.611-613, 1911). The first requires determinacy with respect to every general character, and thus - as he wrote elsewhere (see below) - can only be an ideal limit; while the second makes individuality a matter of reaction, and therefore existence. Both effectively deny the identity of indiscernibles, the first by virtue of the different "hecceities" that two distinct individuals must have, and the second because no two reacting things can have the same spatial (or, I would add, temporal) relations.

The latter is what I had in mind when I suggested as an example of contextuality, "This object from one point of view, or at one time and place, is not the same as this object from another point of view, or at another time and place." I was also thinking of this passage:

The logical atom, or term not capable of logical division, must be one of which every predicate may be universally affirmed or denied ... Such a term can be realized neither in thought nor in sense ... In thought, an absolutely determinate term cannot be realized, because, not being given by sense, such a concept would have to be formed by synthesis, and there would be no end to the synthesis because there is no limit to the number of possible predicates. A logical atom, then, like a point in space, would involve for its precise determination an endless process. We can only say, in a general way, that a term, however determinate, may be made more determinate still, but not that it can be made absolutely determinate. Such a term as "the second Philip of Macedon" is still capable of logical division - into Philip drunk and Philip sober, for example; but we call it individual because that which is denoted by it is in only one place at one time. It is a term not absolutely indivisible, but indivisible as long as we neglect differences of time and the differences which accompany them. Such differences we habitually disregard in the logical division of substances. In the division of relations, etc., we do not, of course, disregard these differences, but we disregard some others. There is nothing to prevent almost any sort of difference from being conventionally neglected in some discourse ... This distinction between the absolutely indivisible and that which is one in number from a particular point of view is shadowed forth in the two words individual {to atomon} and singular (to kath' hekaston); but as those who have used the word individual have not been aware that absolute individuality is merely ideal, it has come to be used in a more general sense. — CP 3.63, 1870

This is the basis on which I have elsewhere suggested "extreme realism" as the view that reality consists entirely of generals, or at least that everything real is general to some degree. In other words, there are no absolute individuals/singulars that are determinate in every conceivable respect. Hence "general" here encompasses both the "positive generality" of 3ns as "conditional necessity" and the "negative generality" (i.e., vagueness) of 1ns as "the merely potential," since they both constitute forms of indeterminacy.

I remain intrigued by the prospect of identifying a third kind that pertains uniquely to 2ns, and still think that "contextuality" is the most promising candidate. It is that which precludes any individual from being absolutely singular, because it cannot strictly satisfy the principle of identity while occupying different places at different times and/or being referenced from different points of view.

"That is why I said the contradiction lies in the genesis of specificity. Peirceanism says it is a contextual deal. The laws of thought say it is brutely tautological. — "apokrisis

I guess you have been talking mainly about how the genesis of individuality/determinacy/identity is contextual, rather than brute, because it involves the ongoing interaction between vague freedoms and general constraints. I have been talking mainly about how the nature of individuality/determinacy/identity is contextual, rather than absolute, because nothing is exactly the same as anything else, including itself.

Or x = not not-x is true. That employs the context to derive the specificity via a dichotomy. — apokrisis

I am not sure what you mean here. Could you please elaborate?

As a second cut: If x is contextual, then it is not necessarily true that under all circumstances, x = ¬¬x.

This version appeals to me because it seems to parallel better the "failures of distribution" of the other two principles for the vague and the general. Peirce stated within his first definition of "individual" that "the principles of contradiction and excluded middle may be regarded as together constituting the definition of the relation expressed by 'not'" (CP 3.612, 1911). Hence if one or the other does not apply, negation is left undefined. The same is true if this formulation of the principle of identity does not apply, and it also eliminates the (classical) logical equivalence of the other two principles.

Love the argument, Tom; but I have to say I agree with Pierre-Normand that what you have shown is that the totality of facts is uncountable, not that it is impossible. — Banno

But if something is uncountable, then by definition, counting it is impossible; and if counting it is impossible, then by definition, it is infinite; and if it is infinite, then by definition it is impossible. Just ask Metaphysician Undercover. :-}

The more I think about it, the more I really like "contextual" as a candidate for the 2ns counterpart to "vague" for 1ns and "general" for 3ns. If I am tracking with you properly, it expresses the specific kind of indeterminacy that is characteristic of the actual, especially as manifested in Peirce's semeiotic concept of the index. Consider this passage:

A sign is objectively general, in so far as, leaving its effective interpretation indeterminate, it surrenders to the interpreter the right of completing the determination for himself. "Man is mortal." "What man?" "Any man you like." A sign is objectively vague, in so far as, leaving its interpretation more or less indeterminate, it reserves for some other possible sign or experience the function of completing the determination. "This month," says the almanac-oracle, "a great event is to happen." "What event?" "Oh, we shall see. The almanac doesn't tell that." — CP 5.505, c. 1905

As a first cut: A sign is objectively contextual, in so far as, leaving its interpretation indeterminate, it relies on some aspect of the actual situation to complete the determination. "That house is on fire." "What house?" "That one over there."

Peirce used a very similar example to illustrate the indexical nature of pronouns, which "call upon the hearer to use his powers of observation, and so establish a real connection between his mind and the object; and if the demonstrative pronoun does that - without which its meaning is not understood - it goes to establish such a connection; and so is an index. The relative pronouns, who and which, demand observational activity in much the same way, only with them the observation has to be directed to the words that have gone before" (CP 2.287, c. 1893). Proper names are also indices, and rely entirely on the interpreter already being familiar with whom or what they reference.

Now back to the other passage:

The general might be defined as that to which the principle of excluded middle does not apply. A triangle in general is not isosceles nor equilateral; nor is a triangle in general scalene. The vague might be defined as that to which the principle of contradiction does not apply. For it is false neither that an animal (in a vague sense) is male, nor that an animal is female. — CP 5.505, c. 1905

As a first cut: The contextual might be defined as that to which the principle of identity does not apply. This object from one point of view, or at one time and place, is not the same as this object from another point of view, or at another time and place.

As I noted before, Zalamea - right after quoting the same passage - formalizes generality and vagueness as "failures of distribution" of the principles of excluded middle and contradiction, respectively. (Robert Lane provides a helpful explanation of the important differences between these principles, as defined and used by Peirce, and the modern laws of excluded middle and non-contradiction.) He associates the general with the universal quantifier and the vague with the existential quantifier, so it seems like the contextual should be associated with a singular proposition. After wrestling with his notation for a while - I am still not sure that I am interpreting it correctly - I came up with these formulations:

• If x is general, then it is not necessarily true that for any predicate P, ∀xP ∨ ∀x¬P
• If x is vague, then it is not necessarily true that for any predicate P, ¬(∃xP ∧ ∃x¬P)

As a first cut: If x is contextual, then it is not necessarily true that under all circumstances, x = x.

What do you think?

What you do is important in Protestantism, as well - it just does not contribute to salvation; rather, it is the natural outcome of salvation. In the very next verse after what I quoted from Paul earlier, he added, "For we are his workmanship, created in Christ Jesus for good works, which God prepared beforehand, that we should walk in them."

It is a common misconception among Protestants that Roman Catholicism teaches salvation by works. This is not true, as I came to realize when I was seriously considering it as an option for myself. The two traditions simply have a different understanding of how God goes about saving people. Protestants view saving grace as a disposition of God by which he freely gives us what we do not deserve. Roman Catholics view saving grace as a substance that God distributes to people in various ways, including good works and the Sacraments. I am not as familiar with Eastern Orthodox theology, but I gather that it stresses becoming one with God ("theosis") over the course of one's life.

By the way, Calvin got much of his "total depravity" doctrine from Augustine. Solomon was right - "There is nothing new under the sun."

Anyway 2ns would stand in relation to the law of identity as this same kind of protest - I am not constrained by that constraint which is said to be required to produce the brutely particular. — apokrisis

The law of identity expressed in what way, either verbally or formally (or both)? What exactly is this constraint with which 2ns "refuses" to comply? Would "contextuality" be a good descriptive term for this characteristic, as the second member of a trichotomy with vagueness and generality? What about "substance" to go along with matter and form?

It is perhaps more emphasized in this particular formulation by certain kinds of Protestants, but I honestly think that most Roman Catholics and Eastern Orthodox would affirm the same basic idea - God saves us, we cannot save ourselves.

I explained this already, multiple times and in various ways. A continuous line has no ends, so by definition its parts also have no ends. You cannot actually divide a continuous line without introducing a discontinuity (point), but it is potentially divisible without limit, as the SEP article explains. Mathematically, infinitesimals likewise have no ends; they are indistinct, such that the principle of excluded middle does not apply to them. In Aristotle's words, "the extremities of things [i.e., parts] that are continuous with one another are one [i.e., not two or more] and are in contact [i.e., not separate]."

Based on our past encounters, I expect you to respond by insisting that a line is "divisible" only if someone can actually divide it. I see no use in going back down that road, so again, cheers.

I see believing in God and being ethical as two different things. Ultimately it is not about anything that I do, it is only about what God has done for me in Christ and through the Holy Spirit. " For by grace you have been saved through faith. And this is not your own doing; it is the gift of God, not a result of works, so that no one may boast."

Also, belief in God is not "one size fits all." We have to be sensitive to context when deciding what approach to take when someone asks for the reason for the hope that we have. Ultimately it all comes down to relationships - we have to know the other person, and of course the whole point is that we want them to come to know God.

Your obstinate dogmatism would be quite impressive if it were an admirable trait. You simply refuse to accept the established definitions - as quoted from a standard dictionary, an online philosophy encyclopedia, and the writings of Aristotle - of what it means for something to be continuous. Once again, engaging with you has been a waste of my time. Cheers.

I think I get what you mean when you say that the principle of identity does not apply to the actual - it is a limit that existing things can approach, but never fully achieve. But in what sense, then, is this distinctive of 2ns, in the same way that the inapplicability of the principles of contradiction and excluded middle are distinctive of vageness/1ns and generality/3ns, respectively? Again, can you state and/or formalize exactly what you mean by the principle of identity in this context?

Any thoughts on the excerpts that I posted as possible clues to why Zalamea claims that the synthetic continuum is recovered fully by category theory via synthetic differential geometry or smooth infinitesimal analysis?

It is possible to consider it so, but for me, it's more an ethical commitment. — Agustino

People can be - and are - ethical without believing in God. People can be - and are - unethical despite believing in God.

It's the same as I asked above - do you think there is a necessary link between philosophical/metaphysical commitments and theism, or can one be a theist pretty much regardless of their other philosophical commitments ... — Agustino

I suspect that the vast majority of theists worldwide do not have, or at least do not recognize, many (if any) other philosophical commitments. Most people are just not wired to approach issues in the way that we typically have in mind when we call it "philosophical thinking." That is not necessarily a bad thing, though; Jesus wanted his followers to have the faith of a child, and Paul warned against being taken captive by "philosophy and empty deceit."

But obviously this entails that it's very difficult, if not impossible, to bring someone to God by yourself - through your own work - it will ultimately have to be God who brings them. — Agustino

I believe that it is, in fact, impossible. Apologetics is not about convincing people to believe in God, it is about preparing Christians to be ready to give an answer - if and when someone asks for the reason why they believe what they do. Only the Word and the Spirit can do the real work of changing hearts and minds.

So I personally don't believe in the effectiveness of "arguments" for God ... — Agustino

Like most reasonings, they are more effective after someone already believes the conclusion, by serving as a way to reinforce that belief and/or explain it to someone else.

There are no such shorter lines until you posit some points of division. — Metaphysician Undercover

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.

Do you understand the difference between continuity and contiguity? — Metaphysician Undercover

Yes, contiguity only applies to discrete things; so that is obviously not what I am describing.

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. — Metaphysician Undercover

NO! There are no intrinsic boundaries between the parts of a continuum; in this case, between the smaller lines within a continuous line.

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. — Metaphysician Undercover

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." Besides, what did Aristotle himself have to say about this matter? Warning - you are not going to like it!

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. — Physics VI.1

Now a continuum is that which is divisible into parts always capable of subdivision ... — On the Heavens I.1

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.

Interesting. I'm a theist as well, but I've always found it hard to stake belief in God on metaphysical commitments. — Agustino

That is why I asked for clarification of what falls within the scope of one's metaphysics for the purposes of the OP. Arguably, belief in God is a metaphysical commitment; and for many, including myself, it is a central metaphysical commitment, such that contrary metaphysical views are effectively ruled out. I engage in philosophy for self-enrichment, but theism is part of my core identity.

Are you a Christian theist? How do you view the theism-metaphysics connection? And if belief in God is related to your metaphysics, do you ever fear that you may find something which will shake that belief? — Agustino

Yes. Not sure what you mean by "connection" here. Not really, since I believe that even my own belief in God is itself a supernatural gift from Him, so I am content to leave it in His infinitely capable hands.

Based on what are you making the connection between behaviour and metaphysics - or even belief? — Agustino

I am really just affirming a central tenet of pragmaticism - a belief just is a habit of feeling, action, or thought; nothing more, nothing less. In other words, what we actually believe is manifested in what we do, not in what we claim to believe. "Actions speak louder than words," as the saying goes.

As Paul says in the Bible - "I do not do the good I want to do. Instead I keep on doing the evil I don't want to do" — Agustino

Any honest Christian can relate to that. We all too often do things that we know are wrong, and thus contrary to our professed beliefs. We typically rationalize doing those things before, during, and after the commission of the acts. As Paul says at the end of the passage, "Wretched man that I am! Who will deliver me from this body of death? Thanks be to God through Jesus Christ our Lord!"

"My" SEP article? I certainly did not write it, I just referenced it. I asked you for sources to justify your claim, "It is a well known metaphysical principle, that the continuous is indivisible"; but you provided none, which is telling. See above for my response to your example.

You already acknowledged that this is not a true continuum, because it has points at the ends, which are discontinuities. When you add a third point, you indeed break the continuity yet again; in fact, that is precisely the nature of all points on a line - they are discontinuities that we introduce by the very act of marking them. Before you posit point B, it does not actually exist; if anything, it is merely potential. 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. Remember, the parts of a continuous line are not points - they are shorter lines.

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