Philosophy says that a point is a negation of a line — Gregory

Please clarify this.

Sure. (Remember I'm coming at this as a philospher). Pi specifies a a certain precision between the numbers 3 and 4. When it comes to a segment, the precision is at every spot of it. However these points are nothing but precision. They're "not-space" and therefore negative to the segment. But if we turn the segment upside down the finite nature of the points become a positive length of finitude in the segment. The negative can be positive as double negative and therefore we have a segment of a line. This same process happens as we go from dimension to dimension. But remember I think this is probably more philosophy than mathematics. There is not a true supertask in motion because there is not an infinite set of actual lengths being covered. But the division OF the length covered by motion has no end, and this is true of anything spacial
I really hope this was helpful. Space has a strange relationship to itself.

Just out of curiosity though, how do you develop a theory of the real numbers without infinite sets? Even the constructivists allow infinite sets, just not noncomputable ones......The problem with finitism is that you can't get a decent theory of the real numbers off the ground. — fishfry

I think this is the wrong question to ask. We know what we want (a consistent foundation for calculus) and we think we know how to get it (with real numbers and infinite sets). But real number might not be the answer.

I think a much better question to ask is 'can we build a consistent foundation for calculus without real numbers and infinite sets?' I believe the answer is yes. Now, this doesn't imply that the alternative requires scrapping everything about real numbers. For example, I believe the alternative would have still have the area of a unit circle being pi, it would just mean something different.

If I see a coherent one presented I'll engage with it. In the past I've engaged extensively with constructivists on this site and learned a lot about the contemporary incarnations of that viewpoint. I've also studied the hyperreals of nonstandard analysis. So in fact I'm very open to alternative versions of math, but I don't see that you've presented one. — fishfry

Like the real numbers, the fundamental objects of the hyperreals are points/numbers. A continuum is constructed by assembling infinite points, each with a corresponding number. And so the hyperreals gets no closer to answering the question 'how can a collection of points be assembled to form a line?' I haven't investigated the constructivists' methods to produce an informed comment but from what I've seen it looks like more of the same. Points/numbers are treated as fundamental.

I don't have a formal theory but I do have an intuition on how the alternative would look. In line with Aristotle's solution to Zeno's Paradox, my view has continua (not points) being fundamental. I think the mathematics of calculus would be almost entirely unaffected in moving to a continuum-based view, but before we could even talk about it, you'd have to first be open to (at least temporarily) shedding your point-based biases and look at graphs in a new light. I would love to hear your criticisms on my earlier post to you where I drew the polynomial in an unorthodox way.

I really hope this was helpful — Gregory

Thanks. I would need a greater knowledge of philosophical theory to really understand. :smile:

We measure the car at 60mph and maybe that's accurate to within a small margin of error. But at no point during its 60 mph run is its speed zero. — tim wood

There are two ways to describe the car's motion.
1) Over any non-zero interval of time the car's average velocity is 60mph.
2) At any instant the car's instantaneous velocity is 60mph.

I believe that the latter description does not make sense since velocity is a measure of change over time. But this doesn't imply that I believe that there is some interval for which the car's average velocity is 0mph.

It seems to me that with any line I look at I'm looking at an infinite number of points. Not potential points, but actual points - that is, to the degree that one, or any, point is actual. — tim wood

This is how we were taught to think, but is it possible that you're just looking at a line? Do you think it's even possible to see a single point? If not, what about 2 points? 3? If something magical happens at infinity when they form a continuum, why not skip the magic and just start with a continuum?

There is an issue of truth here. There is something there causing the form, and the concept of "field" attempts to account for whatever it is. If the concepts employed are inadequate, then it's not true to say that this is what is there. — Metaphysician Undercover

Sure, physicists could be wrong but that doesn't mean you should stick with outdated information (which could also be wrong).

If the process is terminated then it is untrue to say that it is potentially infinite. — Metaphysician Undercover

A program written to spit out the natural numbers one at a time is potentially infinite, regardless of whether it's been executed or interrupted.

And if we know that in every instance when such a process is useful, it is actually terminated, then we also know that it is false to say that a potentially infinite process is useful, because it is only by terminating that process, thereby making it other than potentially infinite, that it is made useful. — Metaphysician Undercover

If you have ever seen π as the solution to a problem (instead of, say, 3.1415) then the process hasn't been terminated, it hasn't even been initiated. It's incorrect to say that potentially infinite processes are only useful when prematurely terminated.

In line with Aristotle's solution to Zeno's Paradox, my view has continua (not points) being fundamental. I think the mathematics of calculus would be almost entirely unaffected in moving to a continuum-based view, — Ryan O'Connor

Why isn't Aristotle's solution just circular because it makes the results of a mathematical construction prior to the construction itself? Plato starts with a line boundless in both directions, then designates bounds to derive segments. Think of it this way, first mark any point on the line as the Origin to divide the line into a half-open dichotomy, then designate any other point as the unit marker to construct a fixed continuous interval. If I give up points as bounds, then how would I have anything but an endless line?

Why isn't Aristotle's solution just circular because it makes the results of a mathematical construction prior to the construction itself? — magritte

I'm not a physicist but this is my understanding of how our world works. A wave function of the universe (describing potential/probabilities) exists spanning all of time. When a measurement is made, that wave function locally collapses to a distinct state. So we start with a probabilistic description of the universe which exists prior to the individual moments.

The construction itself certainly must come first. What we disagree on is what that construction is. You see the individual measured points in time being fundamental while I see the continuum wave function being fundamental. Zeno's paradox highlights the weaknesses in your point-based view. In short, we can't build a timeline from points in time.

If I give up points as bounds, then how would I have anything but an endless line? — magritte

We don't need points to bound the open interval (-∞,+∞).

.



In the transitions of time, motion covers space as if it is fixed ( "discrete") and as if it is undifferentiated continuity. It's two sides of a coin

I think this is the wrong question to ask. We know what we want (a consistent foundation for calculus) and we think we know how to get it (with real numbers and infinite sets). But real number might not be the answer.

I think a much better question to ask is 'can we build a consistent foundation for calculus without real numbers and infinite sets?' I believe the answer is yes. Now, this doesn't imply that the alternative requires scrapping everything about real numbers. For example, I believe the alternative would have still have the area of a unit circle being pi, it would just mean something different. — Ryan O'Connor

This is a bit speculative without some sort of indication or hint of specifics. For example as I mentioned, I've looked into constructivism a bit, which is enjoying a modern resurgence due to the influence of computer science and computerized mathematical proof assistants. But you would reject even that.

Like the real numbers, the fundamental objects of the hyperreals are points/numbers. A continuum is constructed by assembling infinite points, each with a corresponding number. And so the hyperreals gets no closer to answering the question 'how can a collection of points be assembled to form a line?' I haven't investigated the constructivists' methods to produce an informed comment but from what I've seen it looks like more of the same. Points/numbers are treated as fundamental.

I don't have a formal theory but I do have an intuition on how the alternative would look. In line with Aristotle's solution to Zeno's Paradox, my view has continua (not points) being fundamental. — Ryan O'Connor

Sounds like a Peirceian viewpoint, about which I don't know much. But why am I supposed to be burdened for finding a mathematical viewpoint other than the standard one accepted by almost all the world's mathematicians except for those pesky constructivists and rare finitists and ultrafinitists? Why is this my problem, or math's problem?

I think the mathematics of calculus would be almost entirely unaffected in moving to a continuum-based view, but before we could even talk about it, you'd have to first be open to (at least temporarily) shedding your point-based biases and look at graphs in a new light. — Ryan O'Connor

You're just waving your hands.

I would love to hear your criticisms on my earlier post to you where I drew the polynomial in an unorthodox way. — Ryan O'Connor

I suppose I owe you that. I'm going to find it depressing and frustrating but I'll take a run at it sometime.

But this doesn't imply that I believe that there is some interval for which the car's average velocity is 0mph. — Ryan O'Connor

That leaves the question, when there is no interval of time, is it meaningful to speak of time or anything that requires time. And this moves us towards the idea of a limit. Not as the thing itself, but as the thing approached.

If it makes sense to speak of a time as a "time" when there is no motion, then that same notion applied to a car implies that it's spending most of its time, even while in motion, not in motion.

Leaving that absurdity behind, we're left with continuous motion, and with that, Zeno's paradoxes disappear. And as well it makes perfectly good sense to describe the car's velocity, keeping in mind that the description has a generalized quality.

I've looked into constructivism a bit, which is enjoying a modern resurgence due to the influence of computer science and computerized mathematical proof assistants. But you would reject even that. — fishfry

I certainly like the idea of constructivism in that it is necessary to construct a mathematical object to prove that it exists, and my understanding is that constructivists reject actual infinity. However, I haven't investigated the details of constructivism to know where my views differ. It's on my to do list....and as of now, so too is the Peirceian viewpoint.

But why am I supposed to be burdened for finding a mathematical viewpoint other than the standard one accepted by almost all the world's mathematicians except for those pesky constructivists and rare finitists and ultrafinitists? Why is this my problem, or math's problem? — fishfry

This is a great question. Nothing is your problem unless you want it to be. But here's my response to your 'all the world's mathematician's' comment: the world's mathematicians are all climbing up the 'point-based' mountain (and certainly doing good work) but it's a competitive space and (I think) most contributions these days are in highly specialized niche areas. I'm here pointing out that there's another place to climb. I can't promise that it's a mountain (but it might be) but I can say that very few people are climbing it. What's the harm in taking a quick look?

I suppose I owe you that. I'm going to find it depressing and frustrating but I'll take a run at it sometime. — fishfry

...ah okay, so the harm is potential depression and frustration. I'm sure it is frustrating talking math with a non-mathematician, but don't be so certain about it being depressing. There's a chance that you'll like the ideas.

A program written to spit out the natural numbers one at a time is potentially infinite, regardless of whether it's been executed or interrupted. — Ryan O'Connor

We already discussed the difference between the rule ("program" in this case) which sets out, or dictates the process, and the process itself. If the process is interrupted, it ends, and is therefore not infinite. The rule ("program") is never infinite, nor is it potentially infinite, it's a finite, written statement of instruction, like "pi", and "sqrt (2)" are finite statements, even though they may be apprehended as implying a potentially infinite process.

If you have ever seen π as the solution to a problem (instead of, say, 3.1415) then the process hasn't been terminated, it hasn't even been initiated. It's incorrect to say that potentially infinite processes are only useful when prematurely terminated. — Ryan O'Connor

I don't see how you can say this. Pi says that there is a relationship between a circle's circumference and diameter. This information is totally useless if you do not proceed with a truncated version of the seemingly infinite process, such as 3.14. "The solution to the problem is pi" doesn't do anything practical, for anyone, if you cannot put a number to pi.

What about the halting problem?

That leaves the question, when there is no interval of time, is it meaningful to speak of time or anything that requires time. — tim wood

We can certainly talk about a system at a particular instant of time. For example, we can talk about the time on our clock, and the position and shape of objects at that instant. What we can't talk about is rate of change.

If it makes sense to speak of a time as a "time" when there is no motion, then that same notion applied to a car implies that it's spending most of its time, even while in motion, not in motion. — tim wood

Let's say that every instant has a corresponding number on my stopwatch. I believe that the numbers on my stopwatch can only ever be computable numbers. Since traditionally we say that the computable numbers are countably infinite, the duration at which the car is not in motion has measure 0. Therefore the car is spending all of it's time in motion. What you must appreciate is that there is 'something' in between the instants and it is during this 'something' when the car moves.

When time is suspended in indeterminacy space is fluid and discrete. When space is seen as knowingly infinite is its parts it is suspended against the discrete nature of time. Motion has an aspect that is spatial and one that is temporal. We do talk of spacetime now, but we must speak of time and space separately in these examples. Saying we have points and instances which are infinite (which must be crossed by motion) is to forget that time is not space, space is not pure mathematics, and that even pi can only be understood as part of a finite number (3)

We already discussed the difference between the rule ("program" in this case) which sets out, or dictates the process, and the process itself. If the process is interrupted, it ends, and is therefore not infinite. The rule ("program") is never infinite, nor is it potentially infinite, it's a finite, written statement of instruction, like "pi", and "sqrt (2)" are finite statements, even though they may be apprehended as implying a potentially infinite process. — Metaphysician Undercover

Hmm, maybe I wasn't consistent, let me try again.
-The aforementioned program is written with finite characters.
-The execution of the program is potentially infinite.
-The complete output of the program would be actually infinite (if it existed).
-The program spits out numbers as it is being executed, so it doesn't need to be terminated to get something useful from it.
-We can discuss the execution of the program without ever running it (e.g. we can say 'if I executed the program, it would be potentially infinite)

In the end, I think you're splitting hairs here. What's your point?

I don't see how you can say this. Pi says that there is a relationship between a circle's circumference and diameter. This information is totally useless if you do not proceed with a truncated version of the seemingly infinite process, such as 3.14. "The solution to the problem is pi" doesn't do anything practical, for anyone, if you cannot put a number to pi. — Metaphysician Undercover

π is often written as the solution to a problem - for one it's what they say is the volume of Gabriel's Horn. And there are many cases where π gets cancelled out, for example cos(2π)=1, so there's no need to evaluate it. Also, who said math had to be practical?

even pi can only be understood as part of a finite number — Gregory

I'm not sure how to respond to most of your post, but as for pi we typically understand it precisely using some algorithm described with finite characters.

I certainly like the idea of constructivism in that it is necessary to construct a mathematical object to prove that it exists — Ryan O'Connor

It is a lot more satisfying to do this than to argue indirectly, IMO. I have told this little story before, but it bears repeating: There was a PhD math student who spent considerable time on his culminating research project creating a mathematical structure about a particular set of functions, until one day he was asked to provide an example of one of these functions. It turned out he was working with an empty set. :sad:

Your ontology is weak. You say a table is one, yet it has 4 legs and a top piece. What number of parts do these have? This process is infinite and it takes a delicate balance to understand all it's intricacies. Motion passes through infinity and the finite, but you want to reduce the question to Aristotle's lame argument,: namely that parts are only potentially there. Bringing in QM isn't going to help your case mr. idealist. The world is real. "Ignore the world and the world will come to you"

-The program spits out numbers as it is being executed, so it doesn't need to be terminated to get something useful from it. — Ryan O'Connor

But spitting out numbers is not something useful. Useful is the application of the numbers towards counting or measuring, or something like that. If the computer is tasked with counting something and does not complete the task it hasn't been useful.

-We can discuss the execution of the program without ever running it (e.g. we can say 'if I executed the program, it would be potentially infinite) — Ryan O'Connor

But what good is that?

n the end, I think you're splitting hairs here. What's your point? — Ryan O'Connor

I can't even remember now, but I believe I said it would be good to rid the system of infinities and you said there is no problem with working with infinities so long as we recognize that they are merely potential.

But what's the point to working with infinities? If an infinity represents an uncompleted tasked, then isn't it better to complete the task before proceeding. After a while the unfinished tasks start to pile up and become a little overwhelming. And if it is a task which is impossible to complete, then to give oneself an infinite task is to set oneself up for failure, so we ought to address the conditions by which this happens so that we can avoid it.

π is often written as the solution to a problem - for one it's what they say is the volume of Gabriel's Horn. — Ryan O'Connor

Obviously, that's not a real solution.

Also, who said math had to be practical? — Ryan O'Connor

This is probably the crux. "Math does not have to be practical". There's a fundamental element of free choice which lies at the base of all of our understanding of everything. "Has to be" is thereby excluded. And so, we do not have to do anything, nor do we have to figure anything out, or anything like that. One can refuse to move and die if one wants. However, we choose to try and figure things out, we choose to try and understand the nature of reality, and mathematics plays a very big role here. So we need to choose the appropriate mathematics.

Of the mathematicians, the people who dream up axioms, and produce elaborate systems, some might have the attitude that math does not have to be practical, and others might have the attitude that math ought to be practical. But the idea that mathematics does not have to be practical is just an illusion. Each such mathematician will choose a problem, or problems to work on, as that's what mathematics is, working on problems. And problems only exist in relation to practice, as that's what a problem is, a doubtful aspect of practice which needs to be resolved. Without the influence of practice, the need to resolve the issue does not arise, therefore there is no problem. So the reality of the situation is that since mathematicians work on resolving problems, and problems only exist in relation to practice, math is always fundamentally practical, and this fact cannot be avoided. That's why math is classified as an art rather than a science. Despite the huge amount of theory which goes into it, it is theory which is always directed toward solving problems. Therefore, despite the fact that math doesn't have to be practical, it always is practical. If the people who dreamed up axioms and other systems weren't doing something practical (resolving problems), they would have come up with something other than mathematics.

In line with Aristotle's solution to Zeno's Paradox, my view has continua (not points) being fundamental. — Ryan O'Connor

Sounds like a Peirceian viewpoint, about which I don't know much. — fishfry

I have not had the time, energy, or patience to jump into the substance of this discussion so far, but I can offer a couple of reading suggestions based on these two comments.

The first is John Bell's book, The Continuous and the Infinitesimal in Mathematics and Philosophy. It provides an excellent historical overview followed by chapters specifically on topology, category/topos theory, nonstandard analysis, constructive/intuitionistic mathematics, and smooth infinitesimal analysis/synthetic differential geometry.

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.307.7578&rep=rep1&type=pdf

The second is my own recent paper, "Peirce's Topical Continuum: A 'Thicker' Theory." Its thesis is that a collection--even an uncountably infinite one, like the real numbers--is bottom-up, such that the parts are real and the whole is an ens rationis; while a true continuum is top-down, such that the whole is real and the parts are entia rationis. Accordingly, five properties are jointly necessary and sufficient for the latter: rationality, divisibility, homogeneity, contiguity, and inexhaustibility.

https://www.jstor.org/stable/10.2979/trancharpeirsoc.56.1.04

Jon, thanks for the link to Bell's work. My area was analysis, so I will enjoy reading it. :cool:

The consensus among Peirce scholars seems to be that SIA/SDG comes the closest among modern mathematical developments to capturing his notion of a true continuum. Bell also has an excellent primer specifically on SIA, and Sergio Fabi wrote something similar for SDG.

https://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf
https://pages.physics.ua.edu/staff/fabi/InvitationSDG.pdf

I think a much better question to ask — Ryan O'Connor

Ryan, I only quoted this to mention you. I've noticed that when I hit the @ button and enter Ryan, your handle doesn't come up. Do you know why that is? Moderators, any clues?

Anyway apropos of our convo re video, I thought you might enjoy this. It's a pretty cool video regardless. It's a helicopter taking off with its rotor speed exactly synced to the video frame rate, making the rotors look motionless. So even if things look motionless maybe they're moving. For all we know, still photos are moving like crazy but the lights in our room are blinking. Reality is not what it seems, or something like that. The comments below the video are amusing too.

https://www.youtube.com/watch?v=yr3ngmRuGUc

The second is my own recent paper, — aletheist

Thanks for the info. I was not able to get past the JSTOR paywall even though I have a public JSTOR account (the kind they deign to give to us unwashed non-academics). Do you know why? Usually I can read JSTOR articles on the website even if I can't download them.

Your ontology is weak. You say a table is one, yet it has 4 legs and a top piece. What number of parts do these have? This process is infinite and it takes a delicate balance to understand all it's intricacies. Motion passes through infinity and the finite, but you want to reduce the question to Aristotle's lame argument,: namely that parts are only potentially there. Bringing in QM isn't going to help your case mr. idealist. The world is real. "Ignore the world and the world will come to you" — Gregory

Newton's world was one where the present took the center stage, time marched forward instant by instant, and everything always held a definite state.

Einstein's world is one where simultaneity and the present are relative. This view challenged presentism, making eternalism and the block universe (i.e. the whole universe spanning all time and space) an attractive view.

Bohr's world is one where objects only have definite states when they're measured.

My views does not ignore reality, rather is agrees with the latest developments in physics. Perhaps you're stuck in a classical world, a world plagued by singularities. It at least seems that way given your affinity for infinity. As it is when we grapple with singularities, I doubt that you have a compelling solution to Zeno's Paradox.

Your question about a table seems silly. If you want to talk about 'the whole' you have to talk about the wave function of a whole system, not a table. If you don't like the word 'potential existence', would 'superposition' be more appealing to you?

Consider this: in between any two distinct states of a system lies an unobserved wave function. If we continue to observe the system it will not evolve to a different state (Quantum Zeno Effect). Motion happens when we are not looking. Why are you assuming (as Newton did) that everything always has a definite state?

The greatest error of modernity is saying that the world is information and is not as it appears to us. The world we see transcends any interpretation of QM and psychological studies on mind-matter interaction. What you see is what there is. There is more there, but not less. Any other position is insanity. Zeno's paradox will never have a complete solution, but it is a sign of a healthy position to be comfortable with a paradox

But what good is that? (in reference to talking about a potentially infinite process) — Metaphysician Undercover

I've said this many times, but I'll say it one more time. We can talk (using finite statements and operations) about potentially infinite processes without ever initiating (let alone completing) the potentially infinite process. That's what calculus is all about (in my view). If you avoid infinity altogether (i.e. actual and potential) then there is no way to get calculus off the ground. Let go of executing the potentially infinite process and embrace the finite statements that talk about the potentially infinite process.

This is probably the crux. "Math does not have to be practical". — Metaphysician Undercover

Maybe we are using 'practical' differently. I mean practical as having an applied benefit, like engineering better gadgets. If you consider the proof of Fermat's Last Theorem 'practical', then by your definition I think all of math is practical.

---

I find your philosophy of mathematics to be quite disjoint from the act of doing mathematics. Your view does not explain equations or geometry or axioms, it doesn't provide any explanatory power. It only attempts to delegitimize powerful and proven tools (e.g. calculus).

I am going to talk on the phone tomorrow with my cousin, who is a high profile computer programmer in LA. I will tell him about your claim that infinities play no role in programs and see what HE has to say about that. I feel like you come at these questions from a very limited philosophical perspective and make broad claims about stuff you apparently aren't very familiar with. Expect a reply on this thread tomorrow night