What does [that a line is not composed of points, but instead points emerge from lines] mean? — fishfry
The real number line is composed of real numbers. How can you disagree with that? — fishfry
we also can draw the unit square and its diagonal and try to measure it 'perfectly' or 'ideally' and discover irrational numbers. — norm
You were dismissive of my question without answering it and apparently not even getting the substance of it. — tim wood
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
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
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
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.an infinitely long horn, were you to pour paint into it, could be filled with a finite amount of paint — tim wood
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
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
How can ∞ represent a point? — Metaphysician Undercover
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
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. — Metaphysician Undercover
Nothing more need be said. But it will be said anyway.An infinite acceleration is required to go from rest to moving. — 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. — Ryan O'Connor
Incidentally, GR and SR use dimensional representations of space. This includes with GR the use of Penrose diagrams, which have multiple infinities. — InPitzotl
...and what is that problem?Instead of addressing the problem which is the question of what causes the appearance of an 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?the mathematicians create a "fix" to deal with the infinity. — Metaphysician Undercover
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
For instance: 1, 1.4, 1.41, 1.414, 1.4142, .... just 'is' the square root of 2 — norm
Have you looked into Dedekind cuts? — norm
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
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 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
the real question is why brilliant people have pokéballs tattooed on their arms..... — TaySan
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
...and what is that problem? — InPitzotl
...and how are you fixing that? — InPitzotl
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
What I said is that I know your hand is not at absolute rest. — Metaphysician Undercover
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
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
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
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
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
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
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
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 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
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
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
The conclusion is obviously false. — Metaphysician Undercover
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
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
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.It seems like you're having the same difficulty with the word "absolute" — 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.I give up. I've tried numerous times to answer your question — Metaphysician Undercover
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
Don't forget the jump between finite and infinite sets! — norm
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
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
What would you do with limits? Infinite sums? — norm
The fundamental question is something like: what are we approximating? — norm
A limit is a real number, a point, and not the process (in the mainstream view). — norm
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
I know what you mean here, but reading it makes me uneasy. — jgill
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
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
What I propose is a fundamental division between numerical arithmetic and geometry, which recognizes the incompatibility between these two. — Metaphysician Undercover
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
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