No, that is not what I am saying. I am not really talking about physics at all, just a hypothetical/mathematical conceptualization that might have phenomenological and metaphysical applications. — aletheist
Peirce came before Brouwer, and my interest in SIA/SDG has nothing to do with intuitionism or computers. — aletheist
If Peirce had followed through on his skepticism of excluded middle and omitted what we now (ironically) call "Peirce's Law" from his 1885 axiomatization of classical logic, then he would have effectively invented what we now (unfortunately) call "intuitionistic logic" and it might be known instead as "synechistic logic"; i.e., the logic of continuity. — aletheist
Maybe not hopeless, but I suspect that there is a "curse of knowledge" aspect here on my part, given my immersion over the last few years in Peirce's writings and the secondary literature that they have prompted. — aletheist
Thanks for the attempt, sorry for the resulting effect. — aletheist
Peirce would say that there is no point missing, because there are no points at all until we deliberately mark one as the limit that two adjacent portions of the line have in common. — aletheist
If we make a cut there, then the one point becomes two points, since each interval has one at its newly created "loose end." — aletheist
In this thread I find jgill and @fishfry, who I believe are or were professional mathematicians, — tim wood
I mean continuum in the context of the geometrical objects of extension studied in elementary calculus, the objects that we typically describe using the cartesian coordinate system. — keystone
I'm talking about the mathematical world. The two sentences in this quote are quite different. The first sentence essentially states that it passes through infinite intervening points. The second sentence states that it passes through all intervening locations where there could be points. I actually agree with the second sentence. — keystone
What I'm trying to convey is that no matter where Atalanta's mathematical universe lives (whether in an infinite computer or the mind of God) — keystone
it is impossible to construct Atalanta's journey from points because that would amount to listing the real numbers. — keystone
The only way to build her universe is to deconstruct it from a continuum, working your way down from the big picture to specific instants. — keystone
When an engineer tries to solve Zeno's Paradox (of Achilles and the Tortoise) they ask questions about the system as a whole, specifically 'What are the speed functions of Achilles and the Tortoise from the beginning to the end of time?' With that information we don't have to advance forward in time, instant by instant. We just find where their two position functions intersect and conclude that Achilles passes the tortoise at that instant. — keystone
And if this mathematical universe lives in that engineer's mind, that's the only actual instant that exists. Sure, the engineer could calculate their positions at other instants in time, but the engineer isn't going to calculate their positions at all times. That would be unnecessary...and impossible. — keystone
I'm sure you agree with the above paragraph — keystone
(and perhaps are a little offended that I'm positioning it as the engineer's solution...hehe) — keystone
but my point is that knowing a function doesn't imply that we can describe it completely using points. — keystone
Any attempt to do so would be akin to listing the real numbers. — keystone
I like when you earlier said 'every intervening location where there could potentially be a point'. It is worth creating a distinction between actual points and potential points. — keystone
If we make that distinction, then I agree with you that there are only (actual and potential) points between a and b. What I would disagree with is the claim that there are only actual points between a and b. Actual points are discrete while potential points form a continuum. — keystone
So instead of saying that there are finite actual points and infinite potential points between a and b, I think it is much better to say that there are finite actual points and finite continua between a and b. For example, in the image below, there are 3 actual points and 4 continua between 0 and 1.]/quote]
Nonsense. You keep repeating this and I keep calling it nonsense (last time I called it silly) but I'll soon run out of adjectives and also of patience. This isn't going anywhere. I disagree with your view and don't find there to be any meaningful content in it.
— keystone
If we start with continua, the actual points only exist when we make a measurement. It seems like you agreed with aletheist on this. — keystone
With a continuum-based view, when we make a measurement, we are not labeling points that existed all along, we are bringing them into existence (i.e. actualizing them). — keystone
Until then they are potential points and can only be described as a part of a collection (i.e. a continuum), which I described using an interval. I am totally serious about this argument. — keystone
My view is only silly when seen from a point-based view because you assume that all we can talk about are actual objects...an infinite number of them. — keystone
I don't see that either of us has said anything new in a long time. — fishfry
I agree with this. I need to study more to either accept that it's nonsense or find a way to better communicate it. Until then, we're just wasting our time. Let's not waste any more time. I really appreciate your patience sticking this out with me on this up until now. Thanks! — keystone
I appreciate this and would be content to leave it at that.I do understand Peirce's point that the real line isn't a continuum because it's made up of individual points. — fishfry
I guess it comes down to the meaning of the concept of continuity. Someone immersed in modern mathematics, where the real numbers are routinely called a continuum, is understandably satisfied with that definition. Someone like Peirce who objects to finding any discrete parts whatsoever in something that is supposed to be continuous can never accept it. He was motivated primarily by logical considerations rather than mathematical ones.How can I agree or disagree with that statement, without sharing your inner visions on the nature of the true continuum? — fishfry
According to a paper by Conor Mayo-Wilson, "Peirce and Brouwer seemed to have no knowledge of each other's work." However, they were indirectly linked through Lady Victoria Welby, with whom Peirce exchanged a fair amount of correspondence including some of his most important writings about semeiotic, and whose ideas about significs were later adopted by a group of Dutch thinkers that eventually included Brouwer.Was Brouwer familiar with Peirce? — fishfry
Yes, Dedekind's name appears in a bunch of his writings, and his most fundamental disagreement with him was about the relationship between mathematics and logic. For Peirce, logic (generalized as semeiotic) depends on mathematics, as does every other positive science; while for Dedekind, mathematics is a branch of logic.Surely Peirce must have been familiar with Dedekind. — fishfry
Peirce would not talk about "sets"--or "collections," his usual term--when referring to a continuum at all. By definition, a collection consists of discrete parts, which are ontologically prior to the whole ("bottom-up"); while in a continuum, the whole is ontologically prior to the parts ("top-down").Are you saying Peirce would make sqrt(2) both the largest of the smaller set and the smallest of the larger set? — fishfry
Because the only points at all are the ones that we create by marking them. When we mark a line without separating it, we create one point. When we separate the line, we create two points, one at the discontinuous end of each resulting portion. When we put them back together, we have only one point again. The points are never parts of the line itself, because they are of lower dimensionality. Every part of a line is one-dimensional, but a point is dimensionless. SIA seeks to capture this with its infinitesimal segments that are long enough to have "direction" but too short to be curved.How can one point become two points? — fishfry
Indeed, I believe that his use of "principle" rather than "law" for excluded middle is very deliberate. As he wrote elsewhere, "Logic requires us, with reference to each question we have in hand, to hope some definite answer to it may be true. That hope with reference to each case as it comes up is, by a saltus [leap], stated by logicians as a law concerning all cases, namely, the law of excluded middle. This law amounts to saying that the universe has a perfect reality."What Peirce questions is not the LEM, but instead the applicability of it as referenced. To be sure, he calls it the "principle of the excluded middle," and in my opinion the substitution of "principle" for "law" makes all the difference. — tim wood
This is strange. Do you understand it? Because I do not. Try reading it closely and see if it doesn't begin to seem to you that the writer is confused about his subject.and real (so that what is true or false of it is independent of any judgment of man or men, unless it be that of the creator of the universe, in case this is fictive)." — aletheist
in the land of Ps and Qs, that being the place where the game of logic is played by certain rules. Therein truth or falsity determined exactly by those rules, that is to say, recognized by people who follow the rules. Even God can play if he likes, but even he must obey the rules. But what (exactly) does this have to do with the world?classical logic is strictly applicable only — aletheist
gives me a faint hope that Zeno's paradox could be solved, — Gregory
The ancient Chinese new about this: "One of the few surviving lines from the school, 'a one-foot stick, every day take away half of it, in a myriad ages it will not be exhausted,' resembles Zeno's paradoxes." — Gregory
Can God divide a soccer ball into infinite parts? — Gregory
Consequently, classical logic is strictly applicable only where "a recognized universe [of discourse] is ... real (so that what is true or false of it is independent of any judgment of man or men, unless it be that of the creator of the universe, in case this is fictive)." — aletheist
It makes perfect sense to me. Again, the basic definition of real is being such as it is regardless of what anyone thinks about it. If we are talking about a fictional universe, then what is true or false of it depends entirely on what its creator decides about it, but not on what anyone else thinks about it. In Shakespeare's "Hamlet," the title character is the prince of Denmark and kills Claudius because Shakespeare says so; but no one can now truthfully claim that within the universe of that play, Hamlet is the king of Spain and spares Claudius. That is why there are objectively right and wrong answers on tests that students of English literature have to take after reading it.This is strange. Do you understand it? Because I do not. Try reading it closely and see if it doesn't begin to seem to you that the writer is confused about his subject. — tim wood
This strikes me as merely shorthand for your own characterization of logic as a game with certain rules. In the case of classical logic, one of those rules is excluded middle--every constituent of the universe of discourse must be treated as "individual (so that any assertion is either true or false of it)," which "amounts to saying that the universe [of discourse] has a perfect reality."And I see "logic requires us." Logic does no such thing, nor can. Hmm. — tim wood
I believe so, as this seems to be simply what it means for a world to be smooth. As Bell says on p. 276, "If we think of a smooth world as a model of the natural world, then the Principle of Microstraightness guarantees not just the Principle of Continuity--that natural processes occur continuously, but also the Principle of Microuniformity, namely, the assertion that any such process may be considered as taking place at a constant rate over any sufficiently small period of time." For me this is reminiscent of the following passage.In other words, all functions defined in the system are continuous by definition. Am I correct? — jgill
Accepting the common-sense notion [of time], 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. Not only must any given instantaneous value, s, implied in the change be itself either absolutely unchanging or else always changing continuously, but also, denoting an instant of time by t, so likewise must, in the language of the calculus, ds/dt, d^2s/dt^2, d^3s/dt^3, and so on endlessly, be, each and all of them, either absolutely unchanging or always changing continuously. — C. S. Peirce
How can this be when the "true or false of it" is exactly dependent on people and not otherwise?so that what is true or false of it is independent of any judgment of man or men, — aletheist
How do you characterize classical logic? Are you caviling about "game"? It's a way of doing certain well-defined things according to well-defined rules. That's not "my characterization." It's a fair statement of what classical logic is. Accept it, refine it, or correct it.This strikes me as merely shorthand for your own characterization of logic as a game with certain rules. — aletheist
I do not understand the question. What is true or false of a fictional universe is only dependent on what its creator thinks about it. It is such as it is regardless of what anyone else thinks about it.How can this be when the "true or false of it" is exactly dependent on people and not otherwise? — tim wood
It codifies how we can properly draw necessary conclusions about states of things that are definite, thus conforming to non-contradiction; individual, thus conforming to excluded middle; and real, in the sense that they are such as they are regardless of what we think about them.How do you characterize classical logic? — tim wood
No. What he says about it.What is true or false of a fictional universe is only dependent on what its creator thinks about it. — aletheist
so that what is true or false of it is independent of any judgment of man or men, — aletheist
I do not know why you resort to fictional universes. Indeed, if it's a fictional universe, then no proposition about it is true - except the proposition that it is a fictional universe. But you have evaded the point. Truth and falsity are assigned to propositions. If no propositions, or, if no one to assign truth or falsity, then no truth or falsity. You need the assigner.How can this be when the "true or false of it" is exactly dependent on people and not otherwise?
— tim wood
I do not understand the question. What is true or false of a fictional universe is only dependent on what its creator thinks about it. It is such as it is regardless of what anyone else thinks about it. — aletheist
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