Hi Jon S, List
It looks like we are barking up the same trees. As Peirce points out in the 8th Cambridge Conferences Lecture in RLT, the self-returning character of a space or time manifold is a topological character of unbounded manifolds generally. We don't need to add in postulates concerning straightness and a line called the absolute needed for a projective geometry for the point about the self-returning character of hyperbolic manifolds to hold. Hyperbolic manifolds come in different shapes. Some have an odd number of twists (i.e., cross-caps) in them. Others have an even number or no twists at all. Some manifolds, for instance, have the intrinsic character of a torus with no twists. If a torus has two or more holes, then it is hyperbolic in character. If it has one hole it is parabolic. If it has no holes, then it is elliptical. Roughly, a similar point holds for the number of cross caps found in a manifold. Peirce makes this point when he suggests that the first question we should ask about our experience of time is its Euler characteristic or Listing number. On my reading of Peirce, it is important that we start by asking these kinds of questions about the topological character of our experience of time before turning to questions of how time is ordered--projectively or metrically. That is, we need to ask these phenomenological questions about our experience of time before turning to metaphysical questions about its real nature. By asking these phenomenological questions about the character of our experience, we put ourselves in a better position to analyze the surprising observations that are calling out for metaphysical hypotheses. For example, we ask: why does our experience of space seem have three dimension while time has only one, and why is time ordered in a manner that space is not? In turn, we hope to put ourselves in a better position to measure the data that are being used to test those explanations. --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: Jon Alan Schmidt <jonalanschm...@gmail.com> Sent: Monday, March 9, 2020 9:23:49 AM To: peirce-l@list.iupui.edu Subject: Re: [PEIRCE-L] The Reality of Time Jeff, List: How can time be hyperbolic, yet return into itself? The answer is found in projective geometry, which introduces a "line at infinity" such that the different conic sections are distinguished only by how many times they intersect it--zero for an ellipse, which is finite; one for a parabola, which extends to infinity in only one direction, either positive or negative; and two for a hyperbola, which extends to infinity in two directions, both positive and negative. All three can then be conceptualized as closed curves such that the line at infinity never intersects an ellipse, is tangent to a parabola at one point, and crosses a hyperbola at two points (see attached "Projective Conics.jpg"; read its source<https://cre8math.com/2017/07/10/what-is-projective-geometry/> to learn more). These two points are the limits that divide a hyperbolic continuum into two portions, which have both of those limits in common. As Peirce explains, we can then proceed to measure each portion using numbers such that one limit corresponds to positive infinity and the other to negative infinity. CSP: A hyperbolic quantity is one which varies from zero through all positive values to positive infinity (a "logarithmic" infinity, which is not equal to negative infinity), through that to a wholly new line of quantity where it descends through positive values to a sort of zero upon another line and thence through negative values through negative infinity to ordinary negative quantities, and so back to zero. (NEM 2:266; 1895) CSP: A circuit of states is a line of variation of states which returns into itself and has no extreme states ... The order of states in a line of variation may be shown by attaching to sensibly different states different numbers. For if the line of variation forms a circuit, its states are related to one another like the real numbers, rational and irrational, positive and negative, including ∞ [infinity] ... The numbers may occur in every assignable part of the circuit [parabolic], or may be contained between two limits [hyperbolic], or a part of the series of numbers may cover the whole circuit [elliptical]. In the last case [elliptical], we suppose the remaining numbers to be assigned to the circuit taken over and over again in regular arithmetical progression. In the second case [hyperbolic], we are at liberty to fill up the vacant part of the circuit with a second series of numbers which will be distinguished by having a quantity not a number added to it ... [Measurement is] Hyperbolic, when the entire line of [real]* numbers occupies but a portion of the circuit of variation, and leaves a portion vacant … In hyperbolic motion there are just two firmamental states, and in both the regions into which they sever the circuit the state-numbers increase toward one of these and away from the other; and the quantity of the whole circuit is infinite.** (CP 7.287-304; c. 1895) *Peirce mistakenly wrote "finite" in the manuscript **Peirce mistakenly wrote "not infinite" in the manuscript Although "there must be a connection of time ring-wise," nevertheless "Events may be limited to a portion of this ring" (CP 1.498; c. 1896) in such a way "that evolutionary time, our section of time, is contained between those limits" (CP 6.210; 1898). They correspond to the infinite past at negative infinity and the infinite future at positive infinity. We "reckon" our portion between them by assigning real numbers to hypothetical instants relative to an arbitrary unit interval, usually based on a "cycle" for which we can detect regular recurrence; e.g., a year for one revolution of the earth around the sun, or a day for one of its rotations about its axis (cf. NEM 2:250; 1895). As for the other portion, "on the further side of eternity" (CP 8.317; 1891), it is "vacant." However, we can still assign numbers to its hypothetical instants such that they differ from the numbers on our side by "a quantity not a number," which Peirce derives--in a section of the second manuscript quoted above (R 254) that was omitted from CP 7.304--as "i = +∞ - ∞. That is i is a possible value of this indeterminate expression." The corresponding circuit of states "evidently proceeds by contraries" from positive to negative, "from the infinite future to the infinite past" (CP 8.317). What might it mean to describe this "region" of time as "vacant"? There is a possible hint in one additional passage. CSP: There are two distinct questions to be answered concerning time, even when we have accepted the doctrine that it is strictly continuous. The first is, whether or not it has any exceptional instants in which it is discontinuous,--any abrupt beginning and end ... There is no difficulty in imagining that at a certain moment, velocity was suddenly imparted to every atom and corpuscle of the universe; before which all was absolutely motionless and dead. To say that there was no motion nor acceleration is to say there was no time. To say there was no action is to say there was no actuality. However contrary to the evidence, then, such a hypothesis may be, it is perfectly conceivable. The other question is whether time is infinite in duration or not. If it has no flaw in its continuity, it must, as we shall see in chapter 4, return into itself. This may happen after a finite time, as Pythagoras is said to have supposed, or in infinite time, which would be the doctrine of a consistent pessimism. (CP 1.274; 1902) Here Peirce offers the conceivable alternative of an abrupt beginning of time, when "velocity was suddenly imparted to every atom and corpuscle of the universe; before which all was absolutely motionless and dead." The latter description is consistent with the state of things at the end of time in the hyperbolic diagram, perhaps suggesting that it persists throughout "the further side of eternity." CSP: If time returns into itself, an oval line is an icon of it. Now an oval line may be so measured as to be finite, as when we measure positions on a circle by an angular quantity, θ, running up to 360º, where it drops to 0º (which is the natural measure in the case of the circle); or it may be measured so that the measure shall once pass through infinity, in going round the circle, as when we project the positions on the circumference from one of them as a centre upon a straight line on which we measure the shadows by a rigid bar, as in the accompanying figure, here. (CP 1.275) Peirce next explains how an entire finite circle can be mapped to a single infinite line, and provides an accompanying illustration (see attached "Circle-Line.jpg" from R 427). If we take a slightly different approach, locating the center between the two points where the circle representing a hyperbola intersects the line at infinity, we can similarly map two portions of it to two parallel infinite lines, which are understood to "meet" at positive and negative infinity (see attached "Circle-Lines.jpg"). This further suggests how a hyperbolic continuum returns into itself. CSP: The question, however, is, What is the natural mode of measuring time? Has it absolute beginning and end, and does it reach or traverse infinity? Take time in the abstract and the question is merely mathematical. But we are considering a department of philosophy that wants to know how it is, not with pure mathematical time, but with the real time of history's evolution. This question concerns that evolution itself, not the abstract mathematical time. We observe the universe and discover some of its laws. Why, then, may we not discover the mode of its evolution? Is that mode of evolution, so far as we can discover, of such a nature that we must infer that it began and will end, whether this beginning and this end are distant from us by a finite number of days, hours, minutes, and seconds, or infinitely distant? (CP 1.276) After posing these additional questions about real time as distinguished from mathematical time, Peirce goes on to discuss three additional diagrams of time, which as far as I know are the only other ones that he actually drew (see attached "Spiral-1.jpg," "Spiral-2.jpg," and "Spiral-3.jpg" from RS 16). They are neither hyperbolic nor elliptical, but instead spirals corresponding to equations intended to suggest different ways of conceiving "the character of time as a whole." In each case, one revolution around the origin corresponds to "the lapse of a year," and the radius corresponds to "the measure of the degree of evolution in the universe." * Spiral-1 is for a universe that "had an absolute beginning at a point of time in the past immeasurable in years," and whose "stage of evolution ... constantly increases ... until its final destruction in the infinitely distant future." * Spiral-2 is for a universe that also began "in the infinitely distant past," but whose "evolution does not stop" in the infinitely distant future; instead, it "continues uninterruptedly" for an infinite series of infinite series of years. * Spiral-3 is for a universe that "was created a finite number of years ago ... and will go on for an infinite series of years approximating indefinitely to a state ... after which it will begin to advance again, and will advance until after another infinite lapse of years it will then in a finite time reach the stage ... when it will be suddenly destroyed." Peirce concludes, "This last spiral is much the most instructive of the three; but all are useful. The reader will do well to study them." Regards, Jon S. On Sun, Mar 8, 2020 at 3:30 PM Jon Alan Schmidt <jonalanschm...@gmail.com<mailto:jonalanschm...@gmail.com>> wrote: Jeff, List: JD: Focusing on the points made in 3 and 4, how might we understand the contrast being made between our side of things, and the part of time that is on the further side of eternity? A helpful approach, I think, is to start with a mathematical diagram. What kind of diagram might we use to clarify the hyperbolic evolution from the infinite past to the infinite future? Using this diagram, what is the contrast between our side of things and the further side of eternity? Answering your questions is not an entirely straightforward matter, because Peirce made some seemingly inconsistent remarks about which kind of mathematical diagram best represents real time as a whole. CSP: At present, the course of events is approximately determined by law. In the past that approximation was less perfect; in the future it will be more perfect. The tendency to obey laws has always been and always will be growing. We look back toward a point in the infinitely distant past when there was no law but mere indeterminacy; we look forward to a point in the infinitely distant future when there will be no indeterminacy or chance but a complete reign of law. But at any assignable date in the past, however early, there was already some tendency toward uniformity; and at any assignable date in the future there will be some slight aberrancy from law. Moreover, all things have a tendency to take habits. (CP 1.409, EP 1:277; 1887-8) CSP: Time as so defined is a "hyperbolic" continuum; that is to say, the infinitely past and the infinitely future are distinct and do not coincide. This, I believe, accords with our natural idea of time. (W 8:134; 1892) CSP: Observation leads us to suppose that changing things tend toward a state in the immeasurably distant future different from the state of things in the immeasurably distant past ... It is an important, though extrinsic, property of time that no such reckoning brings us round to the same time again. (NEM 2:249-250; 1895) These passages are all basically consistent with the hyperbolic diagram that he described in the 1891 letter to Ladd-Franklin that you quoted. CSP: The triadic clause is that time has no limit, and every portion of time is bounded by two instants which are of it, and between any two instants either way round, instants may be interposed such that taking any possible multitude of objects there is at least one interposed event for every unit of that multitude. This statement needs some explanation of its meaning. First what does it mean to say that time has no limit? This may be understood in a topical or a metrical sense. In a metrical sense it means there is no absolutely first and last of time. That is, while we must adopt a standard of first and last, there is nothing in its own nature the prototype of first and last. For were there any such prototype, that would consist of a pair of objects absolutely first and last. This, however, is more than is intended here. Whether that be true or not is a question concerning rather the events in time than time itself. What is here meant is that time has no instant from which there are more or less than two ways in which time is stretched out, whether they always be in their nature the foregoing and the coming after, or not. If that be so, since every portion of time is bounded by two instants, there must be a connection of time ring-wise. Events may be limited to a portion of this ring; but the time itself must extend round or else there will be a portion of time, say future time and also past time, not bounded by two instants. The justification of this view is that it extends the properties we see belong to time to the whole of time without arbitrary exceptions not warranted by experience. (CP 1.498; c. 1896) CSP: But now, a continuum which is without singularities must, in the first place, return into itself. Here is a remarkable consequence. Take, for example, Time. It makes no difference what singularities you may see reason to impose upon this continuum. You may, for example, say that all evolution began at this instant, which you may call the infinite past, and comes to a close at that other instant, which you may call the infinite future. But all this is quite extrinsic to time itself. Let it be, if you please, that evolutionary time, our section of time, is contained between those limits. Nevertheless, it cannot be denied that time itself, unless it be discontinuous, as we have every reason to suppose it is not, stretches on beyond those limits, infinite though they be, returns into itself, and begins again. Your metaphysics must be shaped to accord with that. (CP 6.210; 1898) By contrast, these longer passages both appear to be saying that time is elliptical--in order to be truly continuous, it cannot have any limits and must instead be connected "ring-wise," such that it "returns into itself." Nevertheless, it might be the case that events are "limited to a portion of this ring," which is "evolutionary time, our section of time"; and if so, then the limits of that portion are still somehow in the infinite past and future. CSP: It may be assumed that there are two instants called the limits of all time, the one being Α, the commencement of all time and the other being Ω, the completion of all time. Whether there really are such instants or not we have no obvious means of knowing; nor is it easy to see what "really" in that question means. But it seems to me that if time is to be conceived as forming a collective whole, there either must be such limits or it must return into itself. This is an interesting question. At any rate, it is a help and no inconvenience for the present purpose to assume such limits. (NEM 3:1075; c. 1905) Here Peirce simply acknowledges that time as "a collective whole" either has two limits or "must return into itself," calling this "an interesting question" and choosing to assume that it does have initial and final instants for the sake of what follows. That subsequent exposition is where he identifies four different classes of states of things--momentary, prolonged, gradual, and relational--and describes diagrams for the first three, although he does not draw them. I have taken a stab at it myself and anticipate sharing the results in a future post. For now, though, the question is whether and how we can reconcile these seemingly incompatible descriptions. In ordinary geometry, an ellipse and a hyperbola are two different conic sections. An ellipse is a single closed curve, and the simplest equation for one is x2 + y2 = 1, producing a circle. A hyperbola consists of two separate curves that approach certain lines called asymptotes without ever actually reaching them; the simplest equations are xy = 1, whose asymptotes are the x-axis and y-axis, and x2 - y2 = 1, whose asymptotes are the lines defined by x + y = 0 and x - y = 0. Peirce characterizes his cosmology as "hyperbolic" because it likewise posits ideal limits that the universe never actually reaches--an absolutely indeterminate state of things in the infinite past, and an absolutely determinate state of things in the infinite future. What are we then to make of his statements about time "returning into itself," which suggest instead an elliptical cosmology? I will address that in another post, rather than making this one any longer than it already is. From previous exchanges, I suspect that you (Jeff) already know where this is headed; but for me and others who are not very familiar with projective geometry, it comes across as counterintuitive and even paradoxical, at least initially. Regards, Jon S. On Fri, Mar 6, 2020 at 11:32 AM Jon Alan Schmidt <jonalanschm...@gmail.com<mailto:jonalanschm...@gmail.com>> wrote: Jeff, List: JD: At the beginning of the post, you note that Peirce engaged in "mathematical, phenomenological, semeiotic, and metaphysical" inquiries concerning time. Do you have any suggestions about how we might tease out the different threads? Each seems to involve somewhat different methods. I agree that each involves different methods, and I have made several attempts (so far unsuccessful) to start writing a paper (or two) with the goal of teasing out those different threads. Peirce himself seems to think that we can "harmonize" them (his word) by recognizing the continuity of time; in fact, our direct perception of the continuous flow of time in phenomenology is what prompts our retroductive hypothesis of a true continuum in mathematics, which we then explicate deductively and evaluate inductively in other sciences. CSP: One opinion which has been put forward and which seems, at any rate, to be tenable and to harmonize with the modern logico-mathematical conceptions, is that our image of the flow of events receives, in a strictly continuous time, strictly continual accessions on the side of the future, while fading in a gradual manner on the side of the past, and that thus the absolutely immediate present is gradually transformed by an immediately given change into a continuum of the reality of which we are thus assured. The argument is that in this way, and apparently in this way only, our having the idea of a true continuum can be accounted for. (CP 8.123n; c. 1902) Logic then provides a plausible explanation for the so-called "arrow of time." Peirce initially wrote the following in one of his notebooks. CSP: 1. A time is a determination of actuality independent of the identity of individuals, and related to other times as stated below. According to the present proposition we may speak of the state of different things at the same time as well as of the states of the same thing at different times and, of course, of different things at different times and of the same thing at the same time. 2. At different times a proposition concerning the same things may be true and false; just as a predicate may at any one time be true and false of different things. Time is therefore a determination of existents. (NEM 2:611; c. 1904-5) A few years later, he offered a correction on the opposite page, which is otherwise blank. CSP: I can hardly now see how time can be called a determination of actuality. It is certainly a law. It is simply a unidimensional continuum of sorts of states of things and that these have an antitypy is shown by the fact that a sort of state of things and a different one cannot both be at the same time. And in consequence of this antitypy a state of things varies in one way and cannot turn round to vary the other way. Or to state it better a variation between state A and state B is limited to occurrence in one direction, just as the form of a body in space is limited to one or other of two perverse positions in space. (NEM 2:611; 1908 Aug 13) Peirce here maintains the mathematical characterization of time as "a unidimensional continuum," but describes its parts as "sorts of states of things" and affirms the phenomenological fact that time flows in only one direction. Turning to metaphysics, one thing that occurred to me just this week is that these different threads at least loosely correspond to the three main theories about time in the current philosophical literature. 1. Eternalism - past, present, and future all exist. 2. Presentism - only the present exists. 3. Growing Block - only the past and present exist. If we substitute reality for existence, these correspond respectively to Peirce's mathematical, phenomenological, and logical/semeiotic conceptions of time--a one-dimensional continuous whole, isomorphic to a line figure (cf. CP 1.273; 1902); an indefinite moment that involves memory, confrontation, and anticipation (cf. CP 7.653; 1903); and an ongoing process by which the indeterminate becomes determinate (cf. CP 5.459, EP 2:357-358; 1905). I have come to believe that #3 is closest to his overall view and can incorporate the insights of the other two. It is unfortunate that there is not a more formal name for it; one recent dissertation suggests "accretivism," but I doubt that this will catch on. My tentative name for Peirce's version of it is temporal synechism. It seems noteworthy that the basic idea of the "growing block" is that reality itself is getting "larger," which is reminiscent of a passage in Kelly A. Parker's book, The Continuity of Peirce's Thought. KAP: The dynamical object in each successive representation in the process [of semeiosis] is necessarily different from that of its predecessor. The dynamical object of the first representation is the real universe at that time, and the immediate object is an abstraction consisting of some aspects of this reality. The next representation, however, cannot have exactly the same dynamical object. The real universe is at that point populated by at least one additional entity--the first representamen itself. Every successive representation in the semeiotic process thus has as its dynamical object not just the universe which the first representamen represented, but that universe plus the first representamen itself. (p. 148) The object that determines the sign is different from the object that determines the interpretant, because the interpretant's object includes the sign itself. Likewise, the past that determines the present is different from the past that determines the future, because the future's past includes the present itself. Moreover, the object affects the sign and interpretant, but not vice-versa; and likewise, the past affects the present and future, but not vice-versa. As ongoing and continuous processes, both semeiosis and time are irreversible because they conform to Gary R.'s vector of determination (2ns→1ns→3ns, object→sign→interpretant, past→present→future); and once the universe as a vast quasi-mind becomes more determinate, it cannot become less determinate again. This leads us to the passage that you quoted in your second post. CSP: [1] I may mention that my chief avocation in the last ten years has been to develop my cosmology. This theory is that the evolution of the world is hyperbolic, that is, proceeds from one state of things in the infinite past, to a different state of things in the infinite future. [2] The state of things in the infinite past is chaos, tohu bohu, the nothingness of which consists in the total absence of regularity. The state of things in the infinite future is death, the nothingness of which consists in the complete triumph of law and absence of all spontaneity. [3] Between these, we have on our side a state of things in which there is some absolute spontaneity counter to all law, and some degree of conformity to law, which is constantly on the increase owing to the growth of habit ... [4] As to the part of time on the further side of eternity which leads back from the infinite future to the infinite past, it evidently proceeds by contraries. (CP 8.317; 1891) The cosmological basis for the "arrow of time" is Gary R.'s vector of process (1ns→3ns→2ns). The universe is evolving from an absolutely indeterminate state of things at the hypothetical instant corresponding to "the commencement of all time" (NEM 3:1075; c. 1905), when everything would have been in the future, toward an absolutely determinate state of things at the hypothetical instant corresponding to "the completion of all time" (ibid), when everything would be in the past. As I said at the end of my initial post, what is always realized in the present is an indefinitely gradual state of change, and this terminology conveniently lends itself to another categorial analysis--the present is an indefinitely gradual state of change in its 1ns, an indefinitely gradual state of change in its 2ns, and an indefinitely gradual (i.e., continuous) state of change in its 3ns. Returning to mathematics, in a List post<https://list.iupui.edu/sympa/arc/peirce-l/2019-09/msg00055.html> last September I proposed five properties that are jointly necessary and sufficient for a true Peircean continuum. (Incidentally, I am pleased to report that my essay based on that and several related List discussions, "Peirce's Topical Continuum: A 'Thicker' Theory," has been accepted for publication in Transactions of the Charles S. Peirce Society.) The first was regularity, which I now prefer to call rationality--every portion conforms to one general law or Idea, which is the final cause by which the ontologically prior whole calls out its parts (cf. CP 7.535; 1899 and CP 7.535n6; 1908). I now suggest that time is a real Peircean continuum, and that an indefinitely gradual state of change is the one general law or Idea to which every lapse of it conforms; i.e., every moment when it is present. Since this has gotten quite lengthy, I will try to take up your specific questions in a later post. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Professional Engineer, Amateur Philosopher, Lutheran Layman www.LinkedIn.com/in/JonAlanSchmidt<http://www.LinkedIn.com/in/JonAlanSchmidt> - twitter.com/JonAlanSchmidt<http://twitter.com/JonAlanSchmidt> On Thu, Mar 5, 2020 at 1:56 AM Jeffrey Brian Downard <jeffrey.down...@nau.edu<mailto:jeffrey.down...@nau.edu>> wrote: Jon, List, Consider what Peirce says about his cosmological conception of time in a letter to Christine Ladd-Franklin. For the sake of clarity, I'll separate and number the points he makes. 1. I may mention that my chief avocation in the last ten years has been to develop my cosmology. This theory is that the evolution of the world is hyperbolic, that is, proceeds from one state of things in the infinite past, to a different state of things in the infinite future. 2. The state of things in the infinite past is chaos, tohu bohu, the nothingness of which consists in the total absence of regularity. The state of things in the infinite future is death, the nothingness of which consists in the complete triumph of law and absence of all spontaneity. 3. Between these, we have on our side a state of things in which there is some absolute spontaneity counter to all law, and some degree of conformity to law, which is constantly on the increase owing to the growth of habit. 4. As to the part of time on the further side of eternity which leads back from the infinite future to the infinite past, it evidently proceeds by contraries. 8.316 Focusing on the points made in 3 and 4, how might we understand the contrast being made between our side of things, and the part of time that is on the further side of eternity? A helpful approach, I think, is to start with a mathematical diagram. What kind of diagram might we use to clarify the hyperbolic evolution from the infinite past to the infinite future? Using this diagram, what is the contrast between our side of things and the further side of eternity? --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: Jeffrey Brian Downard Sent: Wednesday, March 4, 2020 11:37:06 PM To: peirce-l@list.iupui.edu<mailto:peirce-l@list.iupui.edu> Subject: Re: [PEIRCE-L] The Reality of Time Hello Jon, List, At the beginning of the post, you note that Peirce engaged in "mathematical, phenomenological, semeiotic, and metaphysical" inquiries concerning time. Do you have any suggestions about how we might tease out the different threads? Each seems to involve somewhat different methods. --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354
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