What does [that a line is not composed of points, but instead points emerge from lines] mean? — fishfry

Imagine that lines are fundamental, not composite objects. Take a string and mentally label the two endpoints -∞ and ∞. In my world, this string only has two points - the endpoints (I'd actually call them pseudo-points but that's not important here). Cut the string somewhere and mentally label the cut (i.e. the gap) 42. This string now has an additional 'point'. It doesn't matter where this cut was along the string as long as you follow this rule: if you cut the string somewhere between 42 and ∞ then that cut will be mentally labelled with a number greater than 42 (and if you cut the string somewhere between 42 and -∞ then that cut will be mentally labelled with a number less than 42).

The only object in my example was a string, which I'm using to describe a line. There are only a finite number of points on the string and they correspond to the cuts...the absence of string. The points only emerge when you make cuts and the numbers and their proper ordering are contingent on you (a 'computer') actively thinking about them. If you forget about the mental labels, then the numbers no longer exist. You are left with a string...a line...a continuum....and nothing more.

This way of building the parts from the whole is not new. It was (loosely) how Aristotle resolved Zeno's Paradox. I believe that this construction is paradox-free, but I'd love to hear your criticisms.

The real number line is composed of real numbers. How can you disagree with that? — fishfry

What if instead of 'the real number line', I defined the object that's constructed from all of the real numbers 'the real number point'? Does my definition make it so? It may seem like I'm giving a silly example but I truly think you'd have a tough time explaining why your definition is better than mine, and I think this because I believe that the real numbers are singular. But since you are not convinced by Wildberger's arguments, what are the chances that an untrained engineer could convince you with informal reasoning. Perhaps your time is better spent telling me why my 'parts-from-whole' view is wrong rather than hearing me informally complain about why I think your 'whole-from-parts' view is wrong...

we also can draw the unit square and its diagonal and try to measure it 'perfectly' or 'ideally' and discover irrational numbers. — norm

We can perhaps use Newton's method or some other algorithm to produce better and better approximations of sqrt(2) but trying to measure a 'perfect' value doesn't imply that you've discovered it. Perhaps all that you've discovered is an algorithm...and not an irrational number.

You were dismissive of my question without answering it and apparently not even getting the substance of it. — tim wood

I said your hand is not at absolute rest because the earth is moving. And you asked me how do I know this. So I answered that. How am I supposed to know that you were asking me something other than what you were asking me?

If I am at rest in any sense whatsoever, then on your account any acceleration I'm subject to must be in the instant infinite. And that is absurd. — tim wood

It isn't absurd, it just shows that the idea of an "instant", a zero point in time is absurd. The conclusion of an infinite acceleration is only produced from the idea that there is a point in time when a thing goes from resting to moving. Obviously then, what is absurd is the idea of a point in time, not the idea of rest. And the falsity of this idea (of a point in time) is borne out by special relativity which describes simultaneity (being how we determine a point in time) as frame dependent. There are no real points in time, they are arbitrarily assigned according to a frame of reference.

I don't have a problem with saying 'x approaches infinity' in the context of a potentially infinite process. I interpret it as 'the value of x is continuously growing'. 'x approaches infinity provides some information about the journey, even if we never can arrive at some final destination. — Ryan O'Connor

I don't like this phrase "approaches infinity" because it reifies infinity as a thing which is approached. Either the process is designated as infinite, or it is not. Suppose we don't know whether it is or it is not infinite, then we might say that it is potentially infinite, because we don't know. Perhaps, for some reason we are not ready to commit to infinity, like if someone worked out pi to hundreds of decimal points, and was still not convinced that it would go forever. That person would say that it appears to approach infinity, and it is potentially infinite, but I think it might still reach an end at some point, so I won't admit that it's actually infinite.

Don't you think that we have enough evidence to make the judgement, it is infinite? Suppose we say that it appears to be infinite, but it's still possible that it is not. How is this compatible with "approaches infinity"? Then you'd be claiming that there is some actual thing called "infinity", and the values appear to be heading in that direction. But how does this make any sense? What sort of thing could that be, which is called "infinity"?

This is not a typical graph in that it spans all possible values of x and y. Think of it topologically in that it is a single system which maintains its topological properties when undergoing continuous deformations. In this plot, there is a point at (1,1) and a pseudo-point at (∞,0). In this context, it makes sense to say that we're starting at (1,1), travelling along y=1/x and heading towards the pseudo-point at (∞,0). By plotting Gabriel's Horn like this, (∞,0) is no longer out of sight, it's right there in front of us. And because of that we have the ability to use it in some contexts without requiring infinite measuring capacity. — Ryan O'Connor

But what is the meaning of that point which you label as (∞,0)? How can ∞ represent a point? You say it's a "pseudo-point". I assume that this means that it's not a valid point. What's the point in having a non-valid point? I can see how it's useful in practice, but this is an exercise in theory.

an infinitely long horn, were you to pour paint into it, could be filled with a finite amount of paint — tim wood

How do you figure this? It's impossible. The integral of any function that is between zero and plus infinity and the curve of which never reaches the X axis, is infinity. And if the curve reaches the X axis, but on the opening end it is constantly widening, as a horn would be, then the definite integral is still infinity.
I am obviously wrong in this opinion.

if someone worked out pi to hundreds of decimal points, and was still not convinced that it would go forever. That person would say that it appears to approach infinity, and it is potentially infinite, but I think it might still reach an end at some point, so I won't admit that it's actually infinite. — Metaphysician Undercover

When I say potential infinite, I don't mean 'something that might be infinite'. I mean a process that certainly goes on to no end. We are certain that if you begin to write out the digits of pi that that process would never end.

But what is the meaning of that point which you label as (∞,0)? How can ∞ represent a point? You say it's a "pseudo-point". I assume that this means that it's not a valid point. What's the point in having a non-valid point? I can see how it's useful in practice, but this is an exercise in theory. — Metaphysician Undercover

I call it a pseudo-point because it's not an actual point on the graph but instead a potential point. It's a point that you can approach but never arrive at. By 'never arrive at' I mean you can't plug one of its coordinates into a calculator to compute the other.

How can ∞ represent a point? — Metaphysician Undercover

The north pole of the Riemann sphere. But carry on.

We can perhaps use Newton's method or some other algorithm to produce better and better approximations of sqrt(2) but trying to measure a 'perfect' value doesn't imply that you've discovered it. Perhaps all that you've discovered is an algorithm...and not an irrational number. — Ryan O'Connor

Well, yes! I'm certainly open to that view. Irrational numbers are fictions, constructions. The Cauchy sequence construction sorta-kinda says that a real number just is a streaming approximation.
For instance: 1, 1.4, 1.41, 1.414, 1.4142, .... just 'is' the square root of 2, if only we could ignore the extra complexity of equivalence classes. We can't, because any subsequence of the one above also represents root(2), and that's just the beginning of equivalent sequences.

Have you looked into Dedekind cuts? Consider the set of all rational numbers q such that q < 0 or q^2 < 2 for q >= 0. That set of rational numbers just is root(2), and we don't have to worry about equivalence classes. There are something like 15 constructions of the real numbers that I've heard of and looked into (some quite briefly, because some are quite complex and strange.)

If you want to use algorithms (an idea I like), it seems you need to either use mainstream computability theory or rebuild that too. But the computable numbers have measure 0, so you'll have to rebuild measure theory or stick with early analysis.

But to your original point: I'm happy with the word 'invented.' My point is that people wanted to connect symbol math to pictorial math and discovered that our two basic forms of intuition (discrete and continuous) don't play well together. (Consider Zeno's paradoxes in this light.)

The north pole of the Riemann sphere. But carry on. — jgill

Amazing, the things which mathematicians will come up with, in an attempt to solve their problems, instead of simply recognizing that the dimensional representation of space is wrong.

Amazing, the things which mathematicians will come up with, in an attempt to solve their problems, instead of simply recognizing that the dimensional representation of space is wrong. — Metaphysician Undercover

An infinite acceleration is required to go from rest to moving. — Metaphysician Undercover

Nothing more need be said. But it will be said anyway.

Incidentally, GR and SR use dimensional representations of space. This includes with GR the use of Penrose diagrams, which have multiple infinities.

When I say potential infinite, I don't mean 'something that might be infinite'. I mean a process that certainly goes on to no end. We are certain that if you begin to write out the digits of pi that that process would never end. — Ryan O'Connor

Why call this "potential infinite" then? If you are certain that the process goes without end, then you are certain that it is actually infinite.

We might however use terms like potential and actual to distinguish between things which have real existence in the world, and things which are completely imaginary. In this case, there is no such infinite process actually going on, so we say that if someone endeavoured to carry out that process they would find it unending, therefore infinite,. But since they would never arrive at "the infinite", we'd say that such an infinity only exists potentially. One could never prove it to be infinite by reaching infinity.

This is why I didn't like your use of "infinity". You used it as if it signified something with actual existence, which one could be approaching. It seemed as if you thought that if you carried out such an activity, then after a designated point you could be said to be "approaching infinity", when in reality you know that you would never be approaching infinity, because you clearly recognize that the process would never end without ever reaching infinity. Therefore that is a deceptive us of words, when you claim to be approaching something which you clearly acknowledge you can never reach.

Incidentally, GR and SR use dimensional representations of space. This includes with GR the use of Penrose diagrams, which have multiple infinities. — InPitzotl

Exactly, look at the problems which envelope modern cosmology and quantum physics, due to the use of inadequate spatial representations, and a massively complex and fractured numbering system with imaginary numbers etc., required to cope with these problems.

The problem with Platonism (Pythagorean Idealism), is that once we assign to mathematics the status of objects, the objects obtain the status of unchangeable, and therefore eternal truth. Then it appears impossible that these mathematical constructs could be wrong. So when a problem emerges, something which cannot be figured out under the current mathematical system, a fix must be created, axiomatized, and added to the system. These fixes are nothing other than exceptions to the fundamental rules, causing the whole mathematical structure to get extremely top heavy with fix after fix. The fundamental rules which ought to be seen as faulty if they require such exceptions, cannot be apprehended as such, because they are already axiomatized as objects, eternal truths of Platonism.

Some time ago, at this forum, there was a discussion concerning the validity of Euclid's fifth postulate, the parallel postulate. It was argued that the geometry required by GR rendered the parallel postulate invalid. To any rational human being, this would indicate that the fundamental postulates of dimensional space need to be revisited and recreated. But human laziness inclines us to avoid this task, and simply make exceptions to the rules in an attempt to establish compatibility between what science now tells us about spatial existence, and what the rules created thousands of years ago say about spatial existence. Then we rationalize our laziness by saying that mathematical principles are eternal Platonic objects, and couldn't be changed even if we wanted to change them.

In case you didn't quite get it, what you call "multiple infinities" is what I called "fix after fix". Instead of addressing the problem which is the question of what causes the appearance of an infinity, the mathematicians create a "fix" to deal with the infinity.

Instead of addressing the problem which is the question of what causes the appearance of an infinity, — Metaphysician Undercover

...and what is that problem?

the mathematicians create a "fix" to deal with the infinity. — Metaphysician Undercover

...and how are you fixing that? And why should we trust a guy whining about lack of reality when it's the same guy who claims it takes an infinite amount of acceleration to move an object at rest?

I am still waiting for your explanation of your claim that you know that nothing is at absolute rest. And this not a claim there is such a thing or place, but instead how it is that you know that there is not.

the real question is why brilliant people have pokéballs tattooed on their arms.....

Perhaps your time is better spent telling me why my 'parts-from-whole' view is wrong rather than hearing me informally complain about why I think your 'whole-from-parts' view is wrong... — Ryan O'Connor

I don't understand your idea at all. Suppose the position of a particle at time $t$ is given by $f(t) = t^3 - 5 t^2 + 9t - 6$. Find the acceleration of the particle at t = 47.

How do you do that problem after you've thrown out 350 years of calculus and our understanding of the real numbers? What happens to the whole of physics and physical science? Statistics and economics? Are you prepared to reformulate all of it according to your new principles? And what principles are those, exactly? That there aren't real numbers on the real number line?

For instance: 1, 1.4, 1.41, 1.414, 1.4142, .... just 'is' the square root of 2 — norm

I find this example unsatisfying given that everything important is contained in the ellipses. You are no better just writing "For instance: ... just 'is' the square root of 2". And so that equivalence class could just as well correspond to 42. The only way to give it meaning is to state the algorithm used for generating the sequence, which is why I think non-computable numbers are questionable since there is no algorithm behind them.

Have you looked into Dedekind cuts? — norm

Here's a dumb question for you: how can the rational numbers (of which there are only aleph-0) can be cut in c unique ways? For example, if there are 2 numbers, then there's only 1 unique cut. If there are 3 numbers, then there are only 2 unique cuts. If we approach the limit, how do we end up with more cuts than numbers?

If you want to use algorithms (an idea I like), it seems you need to either use mainstream computability theory or rebuild that too. But the computable numbers have measure 0, so you'll have to rebuild measure theory or stick with early analysis. — norm

I think our problem is that we're using numbers to model a continuum. As I'm discussing with fishfry in this thread, I think we should do the opposite and instead use a continuum to model numbers. I think flipping this on its head avoids the paradoxes, allows objects to have non-zero measure, and does not require us to decide between the discrete and continuous because they actually do play well together.

Why call this "potential infinite" then? If you are certain that the process goes without end, then you are certain that it is actually infinite. — Metaphysician Undercover

This is standard terminology. Check out this wikipedia page

This is why I didn't like your use of "infinity". You used it as if it signified something with actual existence, which one could be approaching. — Metaphysician Undercover

Who says only actual things be approached? I can certainly approach a mirage.

the real question is why brilliant people have pokéballs tattooed on their arms..... — TaySan

One day they may reveal the pokémon stored in that ball and the answer will be clear.

I am still waiting for your explanation of your claim that you know that nothing is at absolute rest. And this not a claim there is such a thing or place, but instead how it is that you know that there is not. — tim wood

What I said is that I know your hand is not at absolute rest. And, I also said that if there is something at absolute rest, that thing would have to go through infinite acceleration to start moving. You, and some others here maybe, are the ones claiming infinite acceleration is absurd. I suggest to you, that instead of thinking about the absurdity of infinite acceleration, you simply apprehend absolute rest as absurd.
Then there is no need to think about infinite acceleration because it is only the idea of absolute rest which produces that idea. Make sense?

...and what is that problem? — InPitzotl

That problem is, the deficiency in our capacity to measure. Obviously, if the thing appears to be infinite, this means that we do not have adequate capacity to measure it.

...and how are you fixing that? — InPitzotl

I told you already. We need to revisit our spatial representations, from the beginning. Go back to the fundamental principles, armed with what we now know about space and time, derived from modern science, and rework them all, from the very beginning, starting with the most basic relationship between the non-dimensional point, and the dimensional line. The fundamental (Euclidean) axioms of geometry provide us with inadequate modelling principles which incapacitates our attempt to understand a large portion of spatial existence. Ryan gets it:

The real issue is with our assumption: that time can be broken down into a collection of instants. Or more generally, that a line is composed of infinite points. — Ryan O'Connor

And you even admit that your view is shrouded in mystery. Why not consider the alternative...that a line is not composed of points, but instead points emerge from lines? Why won't you consider my...line...of thought? — Ryan O'Connor

I have a slightly different way of looking at this issue. I give priority to non-dimensional existence, represented as points. Points can be related to each other through lines, but these are still non-dimensional, meaning that these lines do not correspond to real physical, dimensional existence, they are ideals, by which we relate other ideals, points. To establish a correspondence between the non-dimensional, ideal, the point, and the dimensional, real physical existence, we need to bring time in as the first "dimension". Time must be the most fundamental mathematical representation as simply order. Order can be expressed without spatial reference. An understanding of the passing of time will determine the relationships which points can have with each other in real dimensional existence, and geometrical figures must be produced in accordance with the principles derived.

What I said is that I know your hand is not at absolute rest. — Metaphysician Undercover

Yes, and now for the third time you have - I have to presume - deliberately evaded the question. Which is unfortunately par for the course for you. Which earns for you a change of tone. How the F do you know, you ******* ******?

I don't understand your idea at all. Suppose the position of a particle at time tt is given by f(t)=t3−5t2+9t−6. Find the acceleration of the particle at t = 47.

How do you do that problem after you've thrown out 350 years of calculus and our understanding of the real numbers? What happens to the whole of physics and physical science? Statistics and economics? Are you prepared to reformulate all of it according to your new principles? And what principles are those, exactly? That there aren't real numbers on the real number line? — fishfry

Because I'm lazy, I'm going find the velocity of the particle instead of acceleration. First, I want to show you three valid graphs of y=t3-5t2+9t-6 (valid in my construction, that is).



You have to remember that these plots are topological so even though it looks linear in the first image I could have just as well drawn it with squiggles. The key thing is that I've only made a finite number of 'cuts' (exactly like cutting the string in my last post to you, but this time I'm cutting a 2D continuum). All of the information contained in these three graphs is correct. It's just that the graphs have the potential to be cut infinitely more times. The more cuts I make, the more points will emerge and I (a 'computer') will make sure that the points have coordinate values consistent with the functions.

First, I need to point out that (in my view) velocity at an instant is meaningless. It's equivalent to 0/0. Velocity applies, not to points, but to curves. In the third graph I've highlighted a section of the function between x=47 and x=48. As normal, I can calculate the average velocity across this interval using the coordinates as follows (99498-93195)/(48-47)=6303.

But obviously we're not satisfied with that. We want to shrink this interval as much as possible. And we can do so by making cuts closer and closer to x=47 and finding the average velocity across those shrinking intervals. This is what the limit describes (in my construction), it is a potentially infinite process.

What calculus does is describe the potential of that process. And I believe that when calculus was made rigorous by going from numbers (infinitesimals) to processes (limits) some 'infinite-like' numbers (irrational numbers) were left behind. I believe that to complete the job, we need to reinterpret irrational numbers as irrational processes. Calculus is the study of potentially infinite processes. In my view, the math is the same, dy/dt=3t2-10t+9. It's just that the philosophy is different.

This may seem like a trivial difference, but I believe that with this continuum-based view (as opposed to the standard points-based view) many paradoxes are no longer paradoxical. In fact, I can't even think of a paradox with this view (especially given our refined intuitions developed through quantum-mechanics).

I hope I was clear in my explanation. I'd love to hear your feedback, especially if you have criticisms. Thanks!

I find this example unsatisfying given that everything important is contained in the ellipses. You are no better just writing "For instance: ... just 'is' the square root of 2". And so that equivalence class could just as well correspond to 42. The only way to give it meaning is to state the algorithm used for generating the sequence, which is why I think non-computable numbers are questionable since there is no algorithm behind them. — Ryan O'Connor

Consider though that ellipsis are just shorthand that lazy mathematicians use for one another. In this case, it should be obvious how the sequence proceeds. Lots of different algorithms can give the exact same sequence, and that's why equivalence classes are necessary.

Personally I think non-computable numbers are questionable, but root(2) is of course not one of them and is ultimately a finite object (since there's a finite Turing [actually many!] machine that gives arbitrarily good approximations. )

Just to be clear, I think metaphysics in general is a bust. To talk about numbers is (for me) to talk about the talk about numbers. So I'm just quoting some mainstream math, pointing out some of the issues.

Here's a dumb question for you: how can the rational numbers (of which there are only aleph-0) can be cut in c unique ways? For example, if there are 2 numbers, then there's only 1 unique cut. If there are 3 numbers, then there are only 2 unique cuts. If we approach the limit, how do we end up with more cuts than numbers? — Ryan O'Connor

Don't forget the jump between finite and infinite sets! A typical limit operation doesn't make sense here. I do like your question, because it highlights the strangeness of math. (The diagonal proofs, when I first saw them, were lots of what seduced me away from engineering toward pure math and proofs. I've come up with some of my own to prove smaller subsets of R uncountable, where the diagonal is not perfect...fun stuff, engineering with infinity, in pure thought.)

I think our problem is that we're using numbers to model a continuum. As I'm discussing with fishfry in this thread, I think we should do the opposite and instead use a continuum to model numbers. I think flipping this on its head avoids the paradoxes, allows objects to have non-zero measure, and does not require us to decide between the discrete and continuous because they actually do play well together. — Ryan O'Connor

I'm all for bold ideas. I don't know of any paradoxes. I think 'discomforts' is better. AFAIK, mainstream math works, is correct (even if we can't prove it.) The only problem is that it offends lots of peoples intuition here and there.

You have to remember that these plots are topological so even though it looks linear in the first image I could have just as well drawn it with squiggles. — Ryan O'Connor

I know what you mean here, but reading it makes me uneasy.

Imagine that lines are fundamental, not composite objects. Take a string and mentally label the two endpoints -∞ and ∞. In my world, this string only has two points - the endpoints (I'd actually call them pseudo-points but that's not important here). — Ryan O'Connor

I like this string. It's maybe how we first think of the line. This idea would require a radical change in the foundations, sounds like even set theory is jettisoned. If you could rewrite a calculus textbook so that calculations come out the same (so as not to clash with mainsteam math in applications), it could be presented as a pedagogical alternative. The vibe I'm getting is pre-proofs applied mathematics, because there's no mention of axioms. You are offering a nice intuitive foundation. If you want a 'normal discourse' (an objective discipline where disputes can be resolved somehow) you'll have to come up with rules, something like axioms. It could be a calculus in the old sense of the world, with rules like arithmetic except in a calculus context. (I think non-math people experience calculus this way already.)

Epistemologically speaking, why should people trust the results? If you get mainstream results, all is well. You could just present Newton's calculus with different metaphors. What would you do with limits? Infinite sums?

To sum up, nice basic metaphor, but you'd probably have your hands full fleshing it out (which is not meant to be discouraging)

But obviously we're not satisfied with that. We want to shrink this interval as much as possible. And we can do so by making cuts closer and closer to x=47 and finding the average velocity across those shrinking intervals. This is what the limit describes (in my construction), it is a potentially infinite process.

What calculus does is describe the potential of that process. — Ryan O'Connor

If you look at the definition of a limit, it's actually timeless. For all epsilon > 0 , there exists a delta > 0 such that ETC. So there is a leap from the intuition of the potentially infinite approximation process. The fundamental question is something like: what are we approximating? A limit is a real number, a point, and not the process (in the mainstream view). Different processes can converge to the same point. (Subsequences make this obvious, but it's not only subsequences.)

We can draw the symbol root(2) confidently because we can prove that it exists from the axioms, (IVT) entirely without pictures. The desire to free math from pictures should perhaps be addressed here. Can your system free itself from pictures? A theory of continua would presumably have to be symbolically established. Would classical logic work? Would you still find the system charming if the pictures were secondary and only props for the intuition? (Just trying to ask productive questions. Hope they inspire you!)

What calculus does is describe the potential of that process. And I believe that when calculus was made rigorous by going from numbers (infinitesimals) to processes (limits) some 'infinite-like' numbers (irrational numbers) were left behind. I believe that to complete the job, we need to reinterpret irrational numbers as irrational processes. Calculus is the study of potentially infinite processes. In my view, the math is the same, dy/dt=3t2-10t+9. It's just that the philosophy is different. — Ryan O'Connor

This is very good insight. My belief is that we need to go one step further, and apprehend an infinite process, or "irrational process" as actually impossible. But since this process is a potential process, as you describe, this means that it is a possible process which is actually impossible. Therefore the infinite process must be rejected as logically invalid, because it's contradictory.

Rejection of the impossible needs to be recognized as the greatest epistemological tool which human beings possess. It is the basis for certainty in knowledge. In our quest for an understanding of "what is" we narrow the field of possibilities by rejecting any proposed possibility which can be determined as impossible.

In the case of the infinite process, we have a very difficult judgement. It appears to be a reasonable and logically valid possibility. But for some reason it's extremely repugnant to intuition, and the reason why it is repugnant cannot be properly identified, so as to prove logically that it is impossible. It appears as absolutely impossible to show, or demonstrate the impossibility of infinite possibility, because it cannot be done with an empirical demonstration. There is however a logical demonstration commonly known as the cosmological argument, in Aristotle's Metaphysics, and I assume that other forms may be produced.

This may seem like a trivial difference, but I believe that with this continuum-based view (as opposed to the standard points-based view) many paradoxes are no longer paradoxical. In fact, I can't even think of a paradox with this view (especially given our refined intuitions developed through quantum-mechanics). — Ryan O'Connor

I believe that this distinction between the continuum perspective and the points perspective is a very good start, but I don't think it's an either/or question. We need to allow for both. It is the application of both, the two being fundamentally incompatible, which leads to infinity, and the appearance of paradoxes. However, we cannot simply exclude one or the other as unreal, and unnecessary, because there is a very real need for both non-dimensional points, and dimensional lines. You cannot remove the points because this would invalidate all individual units, therefore all number applications would be arbitrary.

What I propose is a fundamental division between numerical arithmetic and geometry, which recognizes the incompatibility between these two. Numbers refer to discrete units, and geometrical constructs refer to a continuum. We ought to recognize that reality consists of both these aspects, and the metaphysical question which we are faced with is to determine in which circumstances each system is applicable. Of course the need to establish a system of correlation between the discrete and continuous will never go away, this is mathematics, but we need to get a firm handle on which aspects of reality are discrete, and which are continuous, before we can axiomatize that correlation in a way which might serve us adequately.

If you look at the definition of a limit, it's actually timeless. For all epsilon > 0 , there exists a delta > 0 such that ETC. So there is a leap from the intuition of the potentially infinite approximation process. The fundamental question is something like: what are we approximating? A limit is a real number, a point, and not the process (in the mainstream view). Different processes can converge to the same point. (Subsequences make this obvious, but it's not only subsequences.) — norm

The problem I see here, is that a process is fundamentally continuous. If it were not continuous we would identify it as a number of different processes. We apply a point, to limit or individualize a process, marking a beginning or ending. But if these points are arbitrarily applied, i.e. not in reference to real points in the apparently continuous world, the numbering system will not provide us with truth (in the sense of correspondence).

When we look at the physical world empirically, it appears like we find true limits in spatial extensions, the boundaries of objects. The reality of these spatial limits have always justified individuals, units, numbers, and quantity, with distinct spatial objects forming discrete entities. However, now that we've come to look closer at these entities, the boundaries have become vague, and when we take the physical object right down to the micro level, those boundaries are not at all valid. So we find that the empirical evidence has really been misleading us, we think that there are spatial boundaries, and distinct entities when there really is not, spatial existence is continuous.

Likewise, if we look at time rationally, (because we cannot look at it empirically), we find the exact opposite situation. Our intuition tells us that time is continuous, that's how it seems to be, time is continually passing. However, we know that there is a big difference between past and future. Since there is such a difference, it is necessary to assume a boundary between past and future. If such a boundary exists, (and it is logically necessary that it does, to maintain a difference between past and future), then this boundary will provide the necessary principles for discrete units of time.

Yes, and now for the third time you have - I have to presume - deliberately evaded the question. Which is unfortunately par for the course for you. Which earns for you a change of tone. How the F do you know, you ******* ******? — tim wood

Do you understand the principle, that if you have two premises which you do not know whether they are true or not, and they lead to a logical conclusion which is obviously false, then one or both of the premises must be false? The two premises are, rest may be absolute, and there is motion. These two premises lead to the conclusion of infinite acceleration. The conclusion is obviously false. The second premise, "there is motion" looks true, so the first must be false.

Thank you. I think we can pin this down, here.

P1: "Rest may be absolute."
P2: "There is motion."
Therefore: There is "infinite acceleration."

The conclusion is obviously false. — Metaphysician Undercover

Small point. I have asked you how you now something, and you have just exhibited that you do not know it, but instead accept it as the consequence of an argument, which is neither an answer to my question nor, if the argument is otherwise flawed, defensible in that way either.

It isn't absurd, it just shows that the idea of an "instant", a zero point in time is absurd. The conclusion of an infinite acceleration is only produced from the idea that there is a point in time when a thing goes from resting to moving. Obviously then, what is absurd is the idea of a point in time, not the idea of rest. And the falsity of this idea (of a point in time) is borne out by special relativity which describes simultaneity (being how we determine a point in time) as frame dependent. There are no real points in time, they are arbitrarily assigned according to a frame of reference. — Metaphysician Undercover

*sigh*

Is velocity a component of motion? After all, if something is not at rest, then it is in motion and has velocity, yes? And at rest would be zero velocity? But velocity is with reference to some framework, yes? So velocity depends on the framework. If your framework is a railway car that you're sitting in, what is your velocity with respect to the framework of the railway car?

Further, acceleration is a change in velocity? Yes? As such there is always some rate of change. Let's imagine any ordinary circumstance of acceleration, and let's suppose that the rate of acceleration is constant, at least for the time under consideration.

Nowhere in this is zero velocity ruled out as a special case. Why or how could it be, given there is always at least one framework where velocity is zero? As a direct consequence, if "at rest" yields infinite acceleration, then that's your and everyone's experience on the tram at every change in velocity. And that is absurd.

But we're at rest all the day long (in at least one framework). And while we are at rest we are, it follows, at rest for all of the moments that constitute the time we're at rest. Acceleration, then is always from a state of rest, and this, and in every case, always from/at some moment in time.

But because there are those moments, and infinite acceleration does not happen, what is absurd is your argument, which seems to be based in combined misunderstandings of motion, rest, acceleration, time, and the uses of logic.

Small point. I have asked you how you now something, and you have just exhibited that you do not know it, but instead accept it as the consequence of an argument, which is neither an answer to my question nor, if the argument is otherwise flawed, defensible in that way either. — tim wood

I give up. I've tried numerous times to answer your question, and it appears I have no idea what you're asking. It seems like you're having the same difficulty with the word "absolute", which you had the last time we discussed the use of that word. You just carry on as if the word isn't there, and insist that it has no meaning. So, onward with your diatribe if it makes you feel good, tim.

It seems like you're having the same difficulty with the word "absolute" — Metaphysician Undercover

Actually, it seems you're having problems with the word "acceleration". If an object at rest remains at rest, it is ipso facto not accelerating. Likewise, if an object at rest accelerates, it ipso facto starts moving. But somehow in MU land, absolute means something can stay at rest and still accelerate, so long as it's not accelerating infinitely. Whatever that means.

I give up. I've tried numerous times to answer your question — Metaphysician Undercover

But in fact you have made no effort. And this we recognize as part of a consistent pattern of evasion of yours. You make a claim of knowledge and cannot or will not answer how you know that as a piece of knowledge. It's a fair question ,and evading it is patently unfair. It is one thing to say as a matter of theory there is no such thing as absolute rest, and no sane educated person disputes it. But it is altogether a different claim to say that you know it, Especially when that claim is the ground for another more outrageous and ridiculous - and ignorant - claim.

Consider though that ellipsis are just shorthand that lazy mathematicians use for one another. In this case, it should be obvious how the sequence proceeds. Lots of different algorithms can give the exact same sequence, and that's why equivalence classes are necessary. — norm

It's only obvious to me because I only know a handful of real numbers so I assume you're talking about sqrt(2). But it's not a matter of laziness, no finite amount of terms would have allowed me to eliminate any possibility. From this view (when there is no algorithm) it seems like the only important number in a Cauchy sequence is the last one...and there is no last one! Anyway, sorry for putting you in a position having to defend a position you don't support!

Don't forget the jump between finite and infinite sets! — norm

Yes, something magical happens at infinity...

I'm all for bold ideas. I don't know of any paradoxes. I think 'discomforts' is better. AFAIK, mainstream math works, is correct (even if we can't prove it.) The only problem is that it offends lots of peoples intuition here and there. — norm

There are a lot of infinity-related paradoxes which offend students' intuitions. Given that it drew you in, perhaps the strangeness is a strength (not a weakness) of infinity. I have no doubt that math works, it's 'why' that has puzzled me for many years.

This idea would require a radical change in the foundations, sounds like even set theory is jettisoned. If you could rewrite a calculus textbook so that calculations come out the same (so as not to clash with mainstream math in applications), it could be presented as a pedagogical alternative. — norm

I haven't worked it out, but IF this is a valid perspective I don't think the math would change much. I think that (in a way) this is consistent with how we've been thinking all along (as I described to jgill below). We'd write the same equations and draw the same pictures, we'd just think about it differently. I think it's a matter of philosophy, not math. But you're totally correct, this is only an intuition...an idea...a formal theory is a whole different thing. And you're right...I'd have my hands full fleshing it out. And to be honest, I don't have the necessary skills to flesh it out.

What would you do with limits? Infinite sums? — norm

I believe that my view is in agreement with limits and infinite sums as long as you think of them as descriptions of unending processes. For example, 1+1/2+1/4+1/8+... corresponds to the unending process of computing larger and larger partial sums which approach but never arrive at 2.

The fundamental question is something like: what are we approximating? — norm

I think this question is very important. In my view, the topological graphs that I drew actually exist. The geometric graphs that we imagine imagining don't exist, but they are incredibly convenient approximations of what we could do in reality to topological graphs.

A limit is a real number, a point, and not the process (in the mainstream view). — norm

True, but what if we reinterpret real numbers as real processes which describe continua, not points? Wouldn't we be able to keep the same math? Can't we just say that our algorithms for calculating the 'number' pi can never output the number completely and that pi actually corresponds to those (potentially infinite) algorithms? Why do we need the number pi anyway? We have never precisely used it as a number anyway.

We can draw the symbol root(2) confidently because we can prove that it exists from the axioms, (IVT) entirely without pictures. The desire to free math from pictures should perhaps be addressed here. Can your system free itself from pictures? A theory of continua would presumably have to be symbolically established. Would classical logic work? Would you still find the system charming if the pictures were secondary and only props for the intuition? (Just trying to ask productive questions. Hope they inspire you!) — norm

These are good questions indeed, thanks!! I don't know the answers to them, but here are my thoughts. There's a convenience to the completeness of the real numbers. We get a one-to-one correspondence between our numbers/equations and our graphs. If in our graphs, lines cross at a point, then we can always find the coordinates of that point (and most often they'll be irrational numbers). If we calculate the derivative of a function at a point, then we can always draw a tangent at that point. With my view, we cannot always do these as they would amount to completing an infinite process. We need to be clever to avoid actual infinity and I think the way to do it is to use both numbers/equations and graphs. Sometimes numbers/equations are needed to describe a system, sometimes graphs are. We can't avoid the pictures.

As for logic, this puts me even further out of my comfort zone but I'm guessing some hybrid would be required: classical logic for the discrete points and fuzzy logic for the continua. This may seem like a patchwork solution to connect the discrete with the continuous but it has parallels to what we do with quantum mechanics to describe our universe (measured objects have definite states and unobserved objects remain in a superposition of potential existence).

I know what you mean here, but reading it makes me uneasy. — jgill

I appreciate why it makes you feel uneasy, but consider this claim: Every** graph that you have ever seen was topologically precise and geometrically approximate. For example, if you and I were to independently sketch out a graph of y=x^2-2, our graphs would share few geometric properties. They would be of different sizes and neither would perfectly capture the curvature of the parabola. But when we compare our graphs we still see them being the same because we continuously deform the graphs in our mind to see that they correspond to the same system. And this goes for computer generated plots as well. A computer plotting y=x^2-2 does not place infinite points on your screen but instead calculates the rational coordinates of a finite set of points and imprecisely connects the dots. But once again, we have no problem with it because it is topologically precise. The only difference is that in my plots I continuously deformed the graphs to a state where it was clear that they were not geometrical.

With 'parts-from-whole' constructions, the systems are topological and described perfectly (without approximation). There's no need to imagine a cloud of infinite points, what you see is what you get. It's just that these plots have infinite potential to be cut.

**When I said 'every graph' in the first sentence of this response I was exaggerating a little because some systems cannot be drawn with perfect topological precision. For example, y=sin(1/x) and y=1 cannot be plotted in the same graph because we'd have to draw infinite points where they interest and that's impossible. In such cases I'd just conclude that that pair of functions cannot be plotted simultaneously. I know it's an odd claim, but it's not too far from the complementarity principle in quantum mechanics which holds that objects have certain pairs of complementary properties which cannot all be observed simultaneously.

I see that you're a retired math professor so I'm especially keen to hear your feedback, especially if you see a flaw in my argument. Thanks!

My belief is that we need to go one step further, and apprehend an infinite process, or "irrational process" as actually impossible. But since this process is a potential process, as you describe, this means that it is a possible process which is actually impossible. Therefore the infinite process must be rejected as logically invalid, because it's contradictory. — Metaphysician Undercover

We can do so much with potentially infinite processes. Not only can we interrupt them to produce rational numbers, but we can work with the underlying algorithms themselves. For example, the following program to outputs the entire list of natural numbers. This program can never be run to completion, but it still is a valid program...I'm talking about it after all and it makes sense even though I've never run it. The same can be said about irrational processes. We need to embrace potential infinity for what it is, not reject it.

while ( i > 0):
    print (i)
    i = i + 1

I believe that this distinction between the continuum perspective and the points perspective is a very good start, but I don't think it's an either/or question. We need to allow for both. It is the application of both, the two being fundamentally incompatible, which leads to infinity, and the appearance of paradoxes. However, we cannot simply exclude one or the other as unreal, and unnecessary, because there is a very real need for both non-dimensional points, and dimensional lines. You cannot remove the points because this would invalidate all individual units, therefore all number applications would be arbitrary. — Metaphysician Undercover

In my continuum-based constructions there are still points, it's just that there are only ever finitely many of them and they are not fundamental. Can you expand on a situation where points need to be fundamental?

What I propose is a fundamental division between numerical arithmetic and geometry, which recognizes the incompatibility between these two. — Metaphysician Undercover

I think we might be on the same page regarding this issue. I mentioned to norm above that I think numbers/equations and graphs are complementary, not equivalent. The problems occur when we think there is a one-to-one correspondence between the two.

I see that you're a retired math professor so I'm especially keen to hear your feedback, especially if you see a flaw in my argument. — Ryan O'Connor

What makes me uneasy is using a concept like topological equivalence and then discussing slopes and derivatives, which, as you know, do not carry over in that way. As for your continuum ideas, almost twenty years ago Peter Lynds wrote a paper appearing in Foundations of Physics Letters that postulated time having no instants and instead being composed of intervals. Something like Bergson's notions from a century ago. Here it is.