Comments

  • Mathematics & Philosophy
    ↪River


    Mathematics does not need philosophy, since pure mathematics is the science of drawing necessary conclusions about strictly hypothetical states of affairs. However, philosophy needs mathematics, since philosophy depends on such reasoning - appropriately adapted to address human experience (phenomenology), the identification and pursuit of ends (esthetics/ethics/logic), and the nature of reality (metaphysics).
  • Turning the problem of evil on its head (The problem of good)
    The existence of evil is insufficient to disprove the reality of an omnipotent, omniscient, and omnibenevolent God. However, by itself, this does not justify the belief that there is such a God.

    Likewise, the existence of good is insufficient to disprove the reality of an omnipotent, omniscient, and omnimalevolent God. Again, by itself, this does not justify the belief that there is such a God.
  • 'Panpsychism is crazy, but it’s also most probably true'
    ↪sime


    I must confess that I am not very familiar with Kant or that particular terminological distinction. Peirce, on the other hand, was very familiar with Kant; but he does not appear to have written anything about that particular terminological distinction. However, this comes from earlier in the same article:

    Self-consciousness, as the term is here used, is to be distinguished both from consciousness generally, from the internal sense, and from pure apperception. Any cognition is a consciousness of the object as represented; by self-consciousness is meant a knowledge of ourselves. Not a mere feeling of subjective conditions of consciousness, but of our personal selves. Pure apperception is the self-assertion of THE ego; the self-consciousness here meant is the recognition of my private self. I know that I (not merely the I) exist. — CP 5.225, 1868

    He seems to be saying that "self-consciousness" pertains to the transcendental ego as "knowledge of ourselves" or knowing that I exist, rather than the empirical ego as "consciousness of the object as represented" or knowing that the I exists.
  • 'Panpsychism is crazy, but it’s also most probably true'
    ↪Wayfarer
    ↪sime

    Rather than a deductive conclusion, Peirce's take was that the initial recognition of one's own existence as a subject of experience is a retroductive conjecture prompted primarily by the unpleasant surprise of being (repeatedly) mistaken:
    In short, error appears, and it can be explained only by supposing a self which is fallible ... At the age at which we know children to be self-conscious, we know that they have been made aware of ignorance and error; and we know them to possess at that age powers of understanding sufficient to enable them to infer from ignorance and error their own existence. — CP 5.234-236, 1868
  • What is the purpose of government?
    We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. — That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, — That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness. — Thomas Jefferson, Declaration of Independence
  • Continuity and Mathematics
    But mathematicians specialize in mathematics, not ontology. — Metaphysician Undercover

    I am only talking about mathematics in this thread, not ontology; maybe you should start your own thread on "Continuity and Ontology."

    Ideals are timeless truths. — Metaphysician Undercover

    I am only talking about ideal states of affairs in this thread, not "ideals"; creations of mathematical imagination, not "timeless truths."

    Oh I'm paying attention, you're just not listening to reason, continually making the same unreasonable assertions over and over again. — Metaphysician Undercover

    Once again - pot, kettle, black.
  • Continuity and Mathematics
    There is no such thing as a mathematical continuity, you are making that up. — Metaphysician Undercover

    That would be news to mathematicians.

    You have claimed that a continuum is both divisible and not divisible. — Metaphysician Undercover

    Not at the same time and in the same respect, hence no contradiction.

    If it is necessary that the continuum is undivided, then it is not possible to divide it. — Metaphysician Undercover

    It is not possible to divide it and still have a continuum.

    But this, acting on the continuum, is not "dividing it" in the mathematical sense of division, it is a change, which constitutes going from continuum to not-continuum. — Metaphysician Undercover

    Dividing it is precisely what causes it to change from a continuum to a non-continuum.

    ... you come up with the idea that the two parts produced are mathematically equivalent to the continuum. — Metaphysician Undercover

    I have never said any such thing.

    The ideal line is defined as consisting of a succession points, and is therefore not continuous, it is discrete. — Metaphysician Undercover

    You are clearly not paying attention at all.
  • Continuity and Mathematics
    But we've already determined that mathematics refers to discrete units. — Metaphysician Undercover

    The first four questions that I posed in the OP were as follows.

    • Is contemporary mathematics inherently discrete, such that it is incapable of accurately capturing the philosophical/ontological notion of real continuity?
    • If so, what specific errors and misconceptions have resulted (and propagated) from thinking otherwise?
    • Is there any way that mathematics could evolve going forward that would enable it to deal with continuity more successfully?
    • If so, what are some specific alternatives?

    Based on the ensuing discussion, my answers are yes; see most of MU's posts; yes; and category theory, in particular smooth infinitesimal analysis. So while I do think that mathematics in accordance with the arithmetic/Dedekind-Cantor/set-theoretic paradigm is intrinsically discrete, I deny that all mathematics is constrained to refer only to discrete units. We certainly have not determined otherwise in this thread.

    So as soon as you describe something as a continuum we are not dealing with mathematics, and therefore I cannot assume that we are dealing with ideal states of affairs. — Metaphysician Undercover

    One more time: I have always and only been dealing with mathematics and ideal states of affairs throughout this thread. Your unwillingness or inability to think of a continuum in mathematical/ideal terms is your own limitation, not mine.

    You want an ideal continuum, so that perhaps you can establish a compatibility with mathematics, but this requires that you can define "continuity" in a way which is not contradictory. — Metaphysician Undercover

    Which is exactly what I have done. You have not demonstrated otherwise; you just keep asserting it over and over, apparently expecting a different result. Where have I claimed that both P and not-P are true at the same time and in the same respect?

    To be divisible, it requires this spatial extension, and this means that to be divided it requires extension outside the mind ... The truly ideal line cannot be divided. — Metaphysician Undercover

    Complete and utter nonsense.

    If it cannot be "so divided", then how is it divided? — Metaphysician Undercover

    Is your reading comprehension that poor, or are you just being obtuse? What I stated is that a continuum cannot be divided into parts that are themselves indivisible; so it can be divided into parts that are themselves divisible, although once that happens it is no longer a continuum.

    Why are the ideal points not part of the ideal line? — Metaphysician Undercover

    Because points are indivisible, and a continuous line cannot be divided into parts that are themselves indivisible. Try to keep up.

    As I said, I do not agree with the way that Peirce dismisses logical principles. — Metaphysician Undercover

    As I said, you are locked into the standard rules of classical logic, which are very useful for most purposes, but not for understanding the nature of true continuity. Failure of excluded middle is not a contradiction in all viable forms of logic.

    Unless it can actually be divided, it is false to say that it is divisible. — Metaphysician Undercover

    I suppose it was inevitable that we would end up right back here, at square one. Cheers.
  • Continuity and Mathematics
    It seems, then, that the last hurdle - as I have already suggested - is your insistence that a continuum cannot have parts of any kind, grounded in your rejection of indefinite parts, such as infinitesimals or Zalamea's "neighborhoods." — aletheist

    I still suspect that this right here is what you are perceiving as contradictory, perhaps because you are locked into the standard rules of classical logic. As Peirce explained, "continuity is simply what generality becomes in the logic of relatives" (CP 5.436, 1904), and "anything is general in so far as the principle of excluded middle does not apply to it" (CP 5.448, 1905). Therefore, the principle of excluded middle does not apply to that which is continuous; and this is all that it means to say that a continuum has only indefinite or potential parts. Intuitionist logic does not uphold excluded middle - or, for that matter, double negation elimination - and thus may be better suited for reasoning about true continuity than classical logic. Excluded middle does not apply universally in smooth infinitesimal analysis, although it does hold for particulars; this is one reason why it seems like a promising candidate for mathematically modeling true continuity.
  • Continuity and Mathematics
    An individual is a physical object and it is divisible (the name "individual" is misleading) ... To begin with, we could recognize that what you have described is a physical object, and it is highly unlikely that any physical object has all parts of the same kind. — Metaphysician Undercover

    As I have repeatedly made clear, I am discussing mathematics here, which has to do with ideal states of affairs; I am not saying anything whatsoever about physical objects. As for "individual," if you look into its etymology, you will find that it has the same root as "indivisible"; one is a noun, the other is an adjective, but they originally meant the same thing - much like "continuum" and "continuous." Nevertheless, since "individual" has come to have a different meaning in common usage, and this seems to be an obstacle for you, we can set that term aside for the sake of clarity and simply substitute "indivisible" as a noun. Restated accordingly: There has to be a way to distinguish a continuum (such as a line) from an indivisible (such as a point).

    There is probably more than one contradiction in this description, but I'll try to sort it out ... We have already determined that it is impossible that there is something real which satisfies your definition of "continuous", because it is contradictory ... You want to take "continuous", and give it an ideal definition which has already been shown to be contradictory. — Metaphysician Undercover

    There is nothing contradictory about my/Peirce's definition, and if you are going to keep insisting that there is, we might as well call yet another impasse and go our separate ways. I get that you disagree with me/Peirce on all this, but I have addressed each of your objections, even if you remain unsatisfied with the result. I have to wonder if you keep saying this over and over because you are still trying to convince yourself.

    This is a collection of discrete individuals. Being described as consisting of parts indicates that it is discrete. — Metaphysician Undercover

    No, a continuum per my/Peirce's definition is not a collection of individuals at all, and having potential parts clearly does not entail that it is discrete; it would only become discrete if it were somehow divided into indivisible parts. But by my definition, it cannot be so divided; therefore, not only is it not discrete, it is not even potentially discrete. A true continuum cannot be composed of discrete elements, and it also cannot be decomposed into discrete elements. We can only introduce indivisible points along a continuous line, and those points are not parts of the continuous line itself.
  • Continuity and Mathematics
    I think that the ideal continuum cannot be divided at all, because it has no parts. — Metaphysician Undercover

    That which cannot be divided at all is an individual, not a continuum - e.g., a point rather than a line. There has to be a way to distinguish these two concepts. What would you call something that satisfies the following definition of a continuum? That which has potential parts, all of which would have parts of the same kind, such that it could be divided (but would then cease to be continuous), and none of the resulting parts would ever be incapable of further division.

    If we want to know what it means to be continuous, we need to look at real examples of continua and determine what they have in common. — Metaphysician Undercover

    This just seems completely backwards to me. How can we identify any real examples of continua without first defining what it means to be continuous? What interests me is whether there is anything real that satisfies my definition of continuity, even if you want to call it something else. That means beginning with space and (especially) time as the framework for our phenomenal experience - examining whether they seem to exhibit the characteristics that I mentioned. If so, we can then move on to whatever spatio-temporal objects we deem the best candidates for also being continuous in that sense.
  • Continuity and Mathematics
    Perhaps it could make sense to me, but to say that a part is indefinite would be to say that this part is unintelligible, it cannot be known. — Metaphysician Undercover

    Not really. To say that a continuum has no definite parts just means that it does not have any distinct, discrete, or indivisible parts. With this qualification, I might even be willing to grant that a continuum has no parts at all, as long as it remains undivided. After all, we agree that the act of dividing a continuum breaks its continuity; so what "infinitely divisible" means in this context is that if we start dividing a continuum, we will never reach the point (literally) of reducing it to an indivisible part. In other words, a continuum is indivisible in the specific sense that if it were divided into parts, and thus made discontinuous, then none of those parts would be indivisible. What do you think?

    However, it is possible that the only real continuities, are those physical, spatial entities which can be divided, but not divided infinitely. — Metaphysician Undercover

    Again, whether there are any real continua is a separate question from what it means to be continuous. Per my definition, something that can be divided, but not divided infinitely (as described above), is not continuous. In any case, I think that the first thing to establish is whether space and time are themselves continuous. If not - if they are discrete - then presumably all spatio-temporal entities are also discrete. However, if we establish that they are continuous, then we can investigate whether anything within space and time is also continuous.
  • The ship of Theseus paradox
    How do you make that distinction - the original and the current? — TheMadFool

    The "original ship of Theseus" is the one made up of the original planks. The "current ship of Theseus" is the one that has had all of its planks replaced over time. Remember, these are just possible names that I suggested people might use to distinguish them; there is nothing philosophically significant about the terms "original" and "current" themselves.
  • Continuity and Mathematics
    ↪Metaphysician Undercover


    In reviewing Fernando Zalamea's paper, "Peirce's Continuum: A Methodological and Mathematical Approach," I came across his explanation of what we have been discussing, which are the properties of a true (Peircean) continuum that he calls reflexivity and inextensibility. Maybe what he has to say will convey the whole idea better than my previous attempts.

    One of the fundamental properties of Peirce’s continuum consists in its reflexivity, a finely grained approach to Kant’s conception that the continuum is such that any of its parts possesses in turn another part similar to the whole: "A continuum is defined as something any part of which however small itself has parts of the same kind." We will use the term “reflexivity” for the preceding property of the continuum since, following a reflection principle, the whole can be reflected in any of its parts.

    As immediately infers Peirce, reflexivity implies that the continuum cannot be composed by points, since points - not possessing other parts than themselves - cannot possess parts similar to the whole. Thus, reflexivity distinguishes at once the Peircean continuum from the Cantorian, since Cantor’s real line is composed by points and is not reflexive. In Peirce’s continuum the points disappear as actual entities (we shall see that they remain as possibilities) and are replaced - in actual, active-reactive secondness - by neighbourhoods, where the continuum flows ...

    We will call inextensibility the property which asserts that a continuum cannot be composed of points. As we mentioned, a continuum’s reflexivity implies its inextensibility (Peirce’s continuum is reflexive, thus inextensible), or, equivalently, its extensibility implies its irreflexivity (Cantor’s continuum is extensible, thus irreflexive). The fact that Peirce’s continuum cannot be extensible, not being able to be captured extensionally by a sum of points, retrieves one of the basic precepts of the Parmenidean One, “immovable in the bonds of mighty chains,” a continuous whole which cannot be broken, “nor is it divisible, since it is all alike, and there is no more of it in one place than in another, to hinder it from holding together, nor less of it, but everything is full of what is.”     — pp. 13-14

    So Zalamea seems to agree with you that a continuum is not divisible, and even quotes Parmenides to that effect; but he also agrees with me (and with Peirce) that it nevertheless has parts, which cannot be points because those do not themselves have any parts. Instead, he calls them "neighborhoods," and he provides an accompanying illustration that is very similar to my microscope example:

    gqf9kh4g6ohhqmmo.jpg

    We have already agreed that a continuum does not consist of points, that it is undivided, and that it is indivisible in the sense that once it is divided, it is no longer continuous. It seems, then, that the last hurdle - as I have already suggested - is your insistence that a continuum cannot have parts of any kind, grounded in your rejection of indefinite parts, such as infinitesimals or Zalamea's "neighborhoods." Do you concur with this assessment?
  • Continuity and Mathematics
    ↪John
    Ah, thanks for the clarification; maybe we are on the same page after all. By "in-finite," do you mean indefinite, or something else?
  • Continuity and Mathematics
    ↪John
    No, consisting of members - no matter how many of them there are - means being discrete, not continuous. There are infinitely many natural numbers, integers, and rational numbers, and yet no one claims that any of these are continuous. Many will claim that the real numbers are continuous, but Peirce (and others) disagreed, for the reasons that I have been citing throughout the thread.
  • Continuity and Mathematics
    Exactly the point, yours and Peirce's concept of true continuum is incoherent and will never be understood ... And this produces the incoherent notion of an indefinite part ... — Metaphysician Undercover

    Your failure to understand it does not render it incoherent. I understand it, I just seem to be unable (so far) to explain it in a way that you will accept. Is this, in the end, the substance of our disagreement here? If you were to wake up tomorrow and decide that the notion of an indefinite part makes sense to you after all, would you have any other objections remaining?
  • Continuity and Mathematics
    I know very well how Peirce defines a true contiuum. — TheWillowOfDarkness

    Apparently not, given your subsequent comments.

    He's failed to understand that a set or collection can be infinite, that it is not defined in a bottom-up manner as a sum of its parts. Consisting of finite members does not mean being finite. — TheWillowOfDarkness

    He understood all of that extremely well. His point was that consisting of members - whether finite or infinite - means being discrete, rather than continuous.
  • Continuity and Mathematics
    Such a "real contiuum" is meaningless. It's a set without any members-- an infinite of nothing at all. — TheWillowOfDarkness

    A true continuum is not a set or collection at all; it does not consist of discrete members.

    Pierce fails to recognise it is a conntuim because he's still stuck trying to account for the infinte by the finite. — TheWillowOfDarkness

    Who is Pierce? If you mean Peirce, it is clear from your comments - especially this one - that you are not familiar with his thought at all.
  • Continuity and Mathematics
    What could it mean for a thing to contain parts but these parts are indefinite? — Metaphysician Undercover

    If and when you ever come to understand this, you will then finally understand what Peirce and I mean by a true continuum.

    There is no point along the continuous line where it is capable of being divided. We already determined, and agreed that there are no points on the continuous line, that would be contradictory. — Metaphysician Undercover

    You are now equivocating on "point." I intentionally did not use that word. Any location along a continuous line is a potential point, but it only becomes a point if we mark it as such - for example, by dividing the line at that location.
  • Continuity and Mathematics
    We can cut a number line anywhere because its infinite particular members. — TheWillowOfDarkness

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

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

    That is the gist of what I have been saying all along. It is a top-down concept, not a bottom-up one.
  • Continuity and Mathematics
    Remember, I pointed out that to define a continuity as a relationship of parts is itself contradictory. To say it consists of parts is to say that it is has separations, is broken, discontinuous. — Metaphysician Undercover

    Remember, I pointed out that this is false, because the concept of separate/broken/discontinuous is not necessary to the concept of parts. In any case, as Peirce stated (and you also quoted), "a continuum, where it is continuous and unbroken, contains no definite parts" (emphasis added). Therefore, its parts are indefinite; or as I have said about infinitesimals, indistinct (but distinguishable).

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

    But Peirce never said that a continuum consists of parts, as if you could somehow build up a continuum from them; and I certainly have never said such a thing, either. In fact, I have said exactly the opposite (emphasis in original):
    It is not composed of parts, it can only be divided into parts, all of which can likewise be divided into more and smaller parts of the same kind. — aletheist
    I even said it (about Peirce) in the OP (emphasis added):
    Charles Sanders Peirce likewise took strong exception to the idea that a true continuum can be composed of distinct members, no matter how multitudinous, even if they are as dense as the real numbers. — aletheist
    The continuum is the more basic concept here, not its parts. You cannot assemble a continuum from its parts, you can only divide it into parts; and once you have done so, even just once, it is no longer a continuum.

    Therefore when you say that the line is divisible, you must mean something other than capable of being divided. What do you mean then by divisible? — Metaphysician Undercover

    A continuous line is divisible if there is no location along it where it is incapable of being divided; but again, once it is divided, it is no longer continuous. So a continuum is undivided but infinitely divisible; you keep focusing on the second characteristic, to which you object, and losing sight of the first, with which you agree. You seem to want to define a continuum as undivided and indivisible, but these are the properties of a discrete point, not a continuous line.
  • Is climate change man-made?
    ↪Pierre-Normand
    As I said before, I am on board with doing what we can, but it will require reaching widespread consensus on both the problem and the solutions.
  • Is climate change man-made?
    Is there any good reason to believe that this would not have an effect on the composition of the atmosphere? — Wayfarer

    Of course not, but it is another matter to claim that this is the only or primary reason why we are seeing detrimental changes to the global climate. Even if I grant that there is "a preponderance of evidence" for this, it does not rise to the level of being "beyond a reasonable doubt."
  • The ship of Theseus paradox
    What something is is not simply a question of its material constitution but of its relationship to other things as well. — darthbarracuda

    This is a very important point. Identity is only possible within a context, where we can distinguish one "thing" from all of the other "things" that are reacting with it in its environment. As I said before, we do so in accordance with our purposes. Sometimes an object's material constitution is what matters most to us, but not always - maybe not even often. As you said, its relationships with other things - including, and especially, ourselves - likely matter more in most cases.
  • Continuity and Mathematics
    If we are talking about a thing which is continuous, and calling it "a continuum", then we are not talking about what it means to be continuous, we are talking about that thing which has been deemed continuous, the continuum. — Metaphysician Undercover

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

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

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

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

    But further study of the subject has proved that this definition is wrong. It involves a misunderstanding of Kant's definition which he himself likewise fell into. Namely he defines a continuum as that all of whose parts have parts of the same kind. He himself, and I after him, understood that to mean infinite divisibility, which plainly is not what constitutes continuity since the series of rational fractional values is infinitely divisible but is not by anybody regarded as continuous. Kant's real definition implies that a continuous line contains no points.

    Now if we are to accept the common sense idea of continuity (after correcting its vagueness and fixing it to mean something) we must either say that a continuous line contains no points or we must say that the principle of excluded middle does not hold of these points. The principle of excluded middle only applies to an individual (for it is not true that "Any man is wise" nor that "Any man is not wise"). But places, being mere possibles without actual existence, are not individuals. Hence a point or indivisible place really does not exist unless there actually be something there to mark it, which, if there is, interrupts the continuity. I, therefore, think that Kant's definition correctly defines the common sense idea, although there are great difficulties with it.

    I certainly think that on any line whatever, on the common sense idea, there is room for any multitude of points however great. If so, the analytical continuity of the theory of functions, which implies there is but a single point for each distance from the origin, defined by a quantity expressible to indefinitely close approximation by a decimal carried out to an indefinitely great number of places, is certainly not the continuity of common sense, since the whole multitude of such quantities is only the first abnumeral multitude, and there is an infinite series of higher grades.

    On the whole, therefore, I think we must say that continuity is the relation of the parts of an unbroken space or time. The precise definition is still in doubt; but Kant's definition, that a continuum is that of which every part has itself parts of the same kind, seems to be correct. This must not be confounded (as Kant himself confounded it) with infinite divisibility, but implies that a line, for example, contains no points until the continuity is broken by marking the points. In accordance with this it seems necessary to say that a continuum, where it is continuous and unbroken, contains no definite parts; that its parts are created in the act of defining them and the precise definition of them breaks the continuity.

    In the calculus and theory of functions it is assumed that between any two rational points (or points at distances along the line expressed by rational fractions) there are rational points and that further for every convergent series of such fractions (such as 3.1, 3.14, 3.141, 3.1415, 3.14159, etc.) there is just one limiting point; and such a collection of points is called continuous. But this does not seem to be the common sense idea of continuity. It is only a collection of independent points. Breaking grains of sand more and more will only make the sand more broken. It will not weld the grains into unbroken continuity.     — CP 6.168, c. 1903-1904, paragraph breaks added

    Notice that infinite divisibility, by itself, is not sufficient to make something continuous; I suspect that I may not have made this clear previously. However, the definition that I have invoked most often - that which has parts, all of which have parts of the same kind - is exactly what Peirce presented here (twice), attributing it to Kant and providing (in my opinion) a convincing case for it. The parts of a continuum, most notably infinitesimals, are not definite; once we create them by the very act of defining them, we have broken the continuity.

    I do not believe that the ideal line is divisible. Once divided, it would no longer be a line, it would be two lines, and two lines is different from one line. — Metaphysician Undercover

    "Once divided" is a different situation from "divisible." Once divided, the line is indeed no longer continuous; but as long as it remains continuous, the line is both infinitely divisible and undivided. Obviously "divisible" in this context does not mean "capable of remaining continuous after being divided," as you seem to be taking it.
  • Is climate change man-made?
    In legal terms, the evidence for global warming is between "a preponderance of evidence" (the low end) and "beyond a reasonable doubt" (the high end). — Bitter Crank

    I think one reason for the shift in terminology from "global warming" to "climate change" is that the latter is less controversial; of course the climate changes over time. The question then becomes the degree to which human activity is the cause of its detrimental aspects.

    Lots of variables go into climate, some human produced, some not. — Bitter Crank

    I agree - my view is that the proposition that human activity has had and is having some negative effect on climate is "beyond a reasonable doubt," but so far there is not "a preponderance of evidence" that human activity is the sole or even dominant reason for allof the worrisome climate changes that we are observing.

    It may be the case that there is little we can do about it. More likely, we can have at least a moderating effect on climate change, and since this is the only place we have, we would do well to get on with whatever we can do. — Bitter Crank

    I am on board with this, but it will require reaching widespread consensus on both the problem and the solutions. For better and for worse, politics will be involved because people will be involved.
  • 'Panpsychism is crazy, but it’s also most probably true'
    Biology understands living beings as active. Physics understands matter as passive, inert. So philosophical speculations may tend toward contriving ways in which matter could be active, living. — Metaphysician Undercover

    Just to add another option to the table, Peirce's version of objective idealism understands mind as living and active, and matter as "effete mind, inveterate habits becoming physical laws," such that it is not entirely dead - only "mostly dead" (for fans of The Princess Bride). There is still an element of spontaneity in the universe that is evident in very slight deviations from the laws of nature. Hence "matter is not completely dead, but is merely mind hidebound with habits"; and "dead matter would be merely the final result of the complete induration of habit reducing the free play of feeling and the brute irrationality of effort to complete death."
  • The ship of Theseus paradox
    This "paradox" simply illustrates how identity is not an intrinsic aspect of macro-scale objects; it is something that we assign to them in accordance with our purposes. Strictly speaking, ship A at one time and place is not identical to ship A at another time and place, even if none of its planks have been replaced yet. Instantaneously replacing all of its planks just makes it more obvious that it is a different object.

    So ship A becomes ship B, but remains "the ship of Theseus" because people continue to call it that, despite the replacement of all its planks. As @apokrisis would say, echoing Bateson, for most people having one new plank - or a lot of new planks, or even all new planks if they are replaced gradually - is not a difference that makes a difference for the purpose of referring to the ship. Hence I suspect that most people would call the reconstructed ship A something like "the original ship of Theseus" to distinguish it from ship B as "the current ship of Theseus."
  • Continuity and Mathematics
    OK, but we need to relate semiotics to a continuity. — Metaphysician Undercover

    First of all, you keep referring to "a continuity" as if it were a thing. Continuity is a property, not a thing; a continuum is a thing that has the property of continuity - i.e., being continuous.

    Take a blank piece of paper and draw a line with arrows at both ends, then draw a series of five equally spaced dots along the line between the arrows. Mark each dot with a numeral from 0 to 4. The drawing itself is not a continuous line with points along it, it is a representation - a sign - of a continuous line with points along it; the latter constitutes the sign's object. What we come to understand by observing (and perhaps modifying) the drawing is the sign's interpretant. All signs are irreducibly triadic in this way - the object determines the sign to determine the interpretant; the sign stands for its object to its interpretant.

    There are three ways that a sign can be related to its object. In simple terms, an icon represents its object by virtue of similarity, an index by virtue of an actual connection, and a symbol by virtue of a convention. As a whole, the drawing is an icon; specifically, a diagram, which means that it embodies the significant relations among the parts of its object. Individually, the drawn line and dots are also icons of a continuous line and points, respectively; but they are symbols, as well, because we conventionally ignore the width and crookedness of a drawn line, as well as the diameter and ovalness of a drawn dot, since they are intended to represent a one-dimensional line and a dimensionless point. The arrows at the ends of the drawn lines are likewise symbols, conventionally suggesting the infinite extension of the line in both directions. The numerals labeling the dots are indices, calling attention to them and assigning an order to them as an actual measurement of the drawn line. They are also symbols, conventionally representing the corresponding numbers.

    What can we learn about continuity from this diagram? We marked five points with dots and assigned numerals to them. Are those dots parts of the line? No, they are additions to the line; we did not draw any dots while drawing the line itself, we came back and drew them later. Likewise, any point along a continuous line is not part of the line; it cannot be, because a continuum must be infinitely divisible into parts that are themselves infinitely divisible. A point is indivisible, so it is not part of the the line; it is added to the line. The middle dot is labeled with the numeral "2". Numbers are like points; they are indivisible. The number 2, which is represented by both the numeral "2" and the dot that it designates, does not have parts. Therefore, no collection of numbers, no matter how dense, forms a true continuum - not even the real numbers, although they serve as an adequate model (i.e., representation) of a continuum for most analytical purposes.

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

    As I keep having to remind you, everything in pure mathematics is "imaginary" - ideal, hypothetical, etc. The question of whether there are any real continua is separate from the question of what it means to be continuous. We have to sort out the latter before we can even start investigating the former.

    I was using "2" only as an example. It was supposed to represent an infinitesimal value. If you say that it can't, we can use something else to represent the infinitesimal value. Let's use X. X represents an infinitesimal value. — Metaphysician Undercover

    This still reflects deep confusion about infinitesimals. They are not "points," and they are not "values." In our diagram, they are extremely short lines within the continuous line, indistinct but distinguishable for a particular purpose.

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

    If you divide it at X, then X is not an infinitesimal, it is a dimensionless point; and by dividing at X, we have agreed that you introduce a discontinuity - you no longer have a continuum. Rather than division, think instead about zooming in on a truly continuous line with an infinitely powerful microscope. No matter how high the magnification, you would never see any gaps in the line; but you would also never find any place along the line where it would be impossible to divide it by introducing a discontinuity in the form of a dimensionless point. This is precisely what it means to be continuous - undivided, yet infinitely divisible.

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

    When it comes to "what continuity really is," there is no "fact of the matter" - it is a mathematical concept, so we can define it however we like. We can then go on to determine whether anything in reality - time, space, motion, etc. - satisfies that definition. This is why it is a mistake to define the possible only in terms of the actual, or the (presumed) actualizable; you block the way of inquiry by ruling out certain kinds of hypotheses before fully explicating them and subsequently examining reality to see whether it conforms to them.

    This method allows us to approach a word like "continuity" without stipulations as to what that word means, and analyze its usage to find out what it really means. — Metaphysician Undercover

    What makes you think that I have any interest at all in how the words "continuum," "continuity," and "continuous" are used in ordinary language? On the contrary, my interest is in a particular concept, not a particular terminology. Telling me that my definition is "wrong" is ultimately beside the point.
  • Continuity and Mathematics
    Actually, it wasn't Peirce I quoted, it's a book entitled "The Continuity of Peirce's Thought", by Kelly A. Parker. — Metaphysician Undercover

    My mistake, I did not follow the link to check your source; everything that you wrote right before the quote implied that it was directly from Peirce himself, so why would I think otherwise? I guess I should have known better. Parker's book was the first one that I read about Peirce, and it is quite good; have you actually read the whole thing? I doubt it, since your link indicates quite clearly that you went Googling for "Peirce divisibility of continuity" and then went with the first reference that came up. Tell you what, read Parker's whole book - or better yet, read some actual Peirce - and then get back to me if you still think that an infinitely divisible continuum is somehow inherently contradictory.

    So I think you agree with me, that it is contradictory, that the indivisible point is within the continuously divisible continuum, and the continuum cannot be divided in this way. — Metaphysician Undercover

    How is this different from what I have been saying all along - that there are no indivisible points in a truly continuous line? Why do you suddenly claim to agree with me now, after arguing with me about it all this time? What changed your mind?

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

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

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

    It is not necessarily a contradiction - I am the same person that I was yesterday, and also different; almost any object that I observe is the same object now that it was a minute ago, but also different. Regardless, the claim in this case is that two things are indistinct, but distinguishable; and this is clearly NOT a contradiction.

    So neither of these proposals, the indivisible point, nor the infinitesimal point, represent an acceptable resolution to the problem of dividing the continuity, they both involve contradiction. — Metaphysician Undercover

    You are still stuck on the idea of points. Infinitesimals are NOT points of ANY kind, they are extremely short lengths of line. As for your example, all numbers are intrinsically discrete; so the number 2 is an indivisible, not an infinitesimal. Think of it this way - what are the "parts" of the number 2? Mind you, I am not referring to smaller numbers that can be added up to reach 2, but the number 2 itself, as a single "point" on the real number "line." As I have stated over and over, in this thread and others, a true continuum is that which has parts, ALL of which have parts of the same kind. The number 2 cannot be a part of any continuum, because the number 2 itself does not have any parts!
  • Continuity and Mathematics
    None of these numbers, except a measure zero fraction, can be represented physically in any way - they are non-computable. — tom

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

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

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

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

    All numbers are distinguishable in principle. That is part of what it means to be a number.
  • Continuity and Mathematics
    I can't really agree with your framing here as my point was that P is only truly actualised to the degree that not-P (as its generic 3ns context) is also actualised. — apokrisis

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

    That is, 2ns is about actualised degrees of freedom - a degree of freedom being a determinate direction of action, or an existent with a predicate. — apokrisis

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

    The purely formal conception that the three affections of terms, determination, generality, and vagueness, form a group dividing a category of what Kant calls "functions of judgment" will be passed by as unimportant by those who have yet to learn how important a part purely formal conceptions may play in philosophy. — CP 5.450, 1905

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