Jeff, List: JD: 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.
I have not dug into RLT on this topic yet, since I only have a hard copy rather than a searchable PDF. Which specific pages do you have in mind? JD: 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. I have already admitted that projective geometry is a conceptual stretch for me, and topology is even more so. Is there a relatively simple primer anywhere online for hyperbolic/parabolic/elliptical toruses in topology, like the one that I found and linked for hyperbolic/parabolic/elliptical circles in projective geometry? JD: 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. "Topological character" is mathematical, while "our experience of time" is phenomenological. How would you suggest that we translate back and forth between the two sciences? JD: 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. I agree, and so does Peirce. CSP: The only important thing here is our metaphysical phenomenon, or familiar notion, that the past is a matter for knowledge but not for endeavor, that the future is an object that we may hope to influence, but which cannot affect us except through our anticipations, and that the present is a moment immeasurably small through which, as their limit, past and future can alone act upon one another. Whether this be an illusion or not, it is the phenomenon of which the metaphysician has to give an account. (CP 8.113; c. 1900) Our *phenomenological *experience of time prompts our *mathematical *hypotheses about time. We then employ *logical/semeiotic* principles in order to ascertain the *metaphysical *reality of time. JD: 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. The Peirce quote above explains how our phenomenological experience requires something like the "arrow of time" to account for the undeniable difference between our memory of the past and our anticipation of the future. Elsewhere he suggests that this is precisely what *requires *time to be one-dimensional, which is obviously not the case with space. CSP: For example, every-day experience is that events occur in time, and that time has but one dimension. So much appears necessary. For we should be utterly bewildered by the suggestion that two events were each anterior to the other or that, happening at different times, one was not anterior to the other. But a two-dimensional anteriority is easily shown to involve a self-contradiction. So, then, that time is one-dimensional is, for the present, necessary; and we know not how to appeal to special experience to disprove it. But that space is three dimensional involves no such necessity. We can perfectly well suppose that atoms or their corpuscles move freely in four or more dimensions. (CP 1.273; 1902) Along similar lines, a manuscript that was presumably an early draft of some ideas for RLT, "Abstracts of 8 Lectures" (R 942), begins with this interesting passage. CSP: We thus see that the bare Nothing of Possibility logically leads to continuity. For the first step a unidimensional continuum is formed. Logically, this step is of the nature of induction. Now induction arranges possible experience after the type of logical law. But the logical law *par excellence* is that of logical sequence. Hence, the first dimension of the continuum of quality is a sequence. A sequence is a unidimensional form in which there is a difference between the relation of A to B and of B to A. Mathematically considered, in one dimension it is a progress from a point A to a point B, where A and B are different or A and B may coincide, or they may both vanish [see attached "Sequences.jpg"]. Of these three forms of sequence, the first is distinctly that of logic since the ultimate antecedent and the ultimate consequent are different in logic. You cannot proceed from antecedent to consequent till you reach again your original antecedent (as in the 3rd kind of sequence, the elliptical), nor do you *tend *to such a return (as in the second, or parabolic sequence), but the two are distinct. It follows that the first dimension of the continuum of possible quality* had to be of the nature of a hyperbolic sequence. That is to say, there is one general mode of relation, which we may name *coming after*, defined by these conditions: 1st, of any two qualities which are not entirely alike in their relations of *coming after*, one *comes after* the other; 2nd, whatever *comes after* another comes after whatever that other comes after; or otherwise stated if N comes after M then whatever, say P, comes after N also comes after M; 3rd, nothing comes after itself ... Now the logical sequence itself is essentially unidimensional, because it is a purely internal law, and unity and interiority are inseparable. (NEM 4:127-128; 1898) *Peirce mistakenly wrote "quantity" in the manuscript Peirce is discussing the continuum of possible quality here, rather than time, but it is not much of a stretch to recognize the parallel between *logical *sequence--which he confirms to be a *hyperbolic *sequence, rather than an elliptical or parabolic sequence--and *temporal *sequence. Both involve "one general mode of relation, which we may name *coming after*"; and as he elaborates in a subsequent paragraph, both proceed from an ideal beginning toward a *different *ideal end. CSP: Let me say, by the way, that there is in the logical law this difference between the absolutely first antecedent and the absolutely last consequent, both of which are unattainable limits. The last consequent is the very reality itself. That is our very conception of reality, the essence of the word, namely, what we should believe if investigation was carried to its furthest limit where no change of belief further was possible. That is of the nature of an infinite, a true singularity of the logical continuum differing *toto caelo* [by the entire extent of the heavens] from every intermediate step however near to it. I mean that it thus differs, not merely in its logical relations as leading to no consequent other than itself, but also and more particularly, as being a radically different kind of consciousness, a consciousness which is the very reality itself and no mere image seen *per speculum in aenigmate* [through a glass darkly]. But the absolutely first antecedent is simply the blank ignorance, the *zero *of knowledge, although in its logical relations it is singular in leading to nothing, as a needle precisely balanced on its point will never fall, yet as a state of mind it differs indefinitely little from other states near it. Hence, though a limit as to the advance of logical development, it is not so as a mode of consciousness. (NEM 4:134) Just as temporal sequence has an initial state that is absolutely indeterminate and a final state that is absolutely determinate, logical sequence starts with "blank ignorance" and ends with "the very reality itself," what Peirce sometimes called the ultimate opinion. A decade later, he further connected both kinds of sequence to the concept of *negation*. CSP: Indeed, so far is the concept of *Sequence *from being a composite of two Negations, that, on the contrary, the concept of the *Negation *of any state of things, X, is, precisely, a composite of which one element is the concept of Sequence. Namely, it is the concept of a sequence from X of the essence of falsity ... The question will here pop up, Why does not this show that the concept of Sequence is a composite of three concepts; that of some antecedent state, that of some consequent state, and between them, that of a state of Heraclitan Flux? ... Your question answers itself ... your supposition assumes that there is what we conceive of as Time ... For we never think at all without reasoning; and if we try to do so, the attempt merely results in our reasoning about reasoning. Now reasoning takes place in Time; and so far as we can understand it, in a Time that embodies our common-sense notion of Time. But this common-sense notion of time implies that every state of things that does not endure through a lapse of time is absolutely definite, that is, that two states, one the negation of the other, cannot exist at the same instant; which, by the way, necessarily follows, if negation be but a particular sort of sequence; though it would be to no purpose to stop to prove this here. Accepting the common-sense notion, then, I say that it conflicts with that to suppose that there is ever any discontinuity in change. That is to say, between any two instantaneous states there must be a lapse of time during which the change is continuous, not merely in that false continuity which the calculus recognizes but in a much stricter sense. (R 300:52-55[51-54]; 1908) Sequence is a *simpler *concept than negation, which is why Peirce defined a *cut *in existential graphs as a *scroll *with its inner close containing the *pseudograph *and reduced to infinitesimal size (cf. CP 4.454-456; 1903 and CP 4.454n; c. 1906). Moreover, logical sequence is a *simpler *concept than temporal sequence, such that we can prescind the former from the latter, but not the latter from the former. The upshot is that the negation of a prolonged state of things requires "a lapse of time during which the change is [strictly] continuous"; i.e., a general determination of time at which an indefinitely gradual state of change is realized. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Professional Engineer, Amateur Philosopher, Lutheran Layman www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Mon, Mar 9, 2020 at 1:54 PM Jeffrey Brian Downard < jeffrey.down...@nau.edu> wrote: > 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 > >>
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