So, you replace pi with a tiny line segment whose length depends upon a computer. So, changing computer affects this small interval. — jgill

I am one of those and I doubt your claim, but there may be others who find it of interest. I don't see anything of substance here so far, but I may be missing the point. — jgill

Well if we're doing computer arithmetic and some variant of discrete calculus, that's interesting to know. What do you think? — fishfry

• π is the familiar irrational number
• ε1 and ε2 are placeholders for positive numbers which can be as small as your computer allows
• (π-ε1,π+ε2) describes a line with rational upper and lower bounds. This line can be as small as your computer allows. Ultimately, I want to call this line pi. To distinguish it from the point/number, π, I'll call this line π.

• With the top-down view, the plan and it's execution are distinct steps such that π remains a line, no matter how powerful your computer is.
• With the bottom-up view, the plan and it's execution are equivalent such that π collapses to a point. I believe this is an unacceptable and an unnecessary leap of thought akin to claiming that there is a last term in a Cauchy Sequence.
• Although step 2 is incredibly useful for applied mathematics, that's not what I'm concerned with. I'm solely concerned with step 1 and I believe step 1 is what is of interest to pure mathematicians.
• Counter to standard belief, I believe calculus is about plans (not their execution) and I believe it's unknowingly been this way all along.
• For example, when a mathematician describes π they always describe the algorithm, they rarely talk about the algorithm's execution...unless referring to a Pi Recitation Contest...

What is an isolated real number? — jgill

Show us elementary calculus from the top down. I am curious. — jgill

You claimed completeness. Do you now retract that? Or have a private definition? — fishfry

If executed, such a program will eventually output the same number over and over, until its computing resources run out. You are factually wrong and I hope you can see why. — fishfry

Even so, there is no smallest positive real number, and you have not provided an argument. — fishfry

• 1) Assume that any computer running the program will eventually overflow and stop printing numbers.
• 2) If we limit our scope to computers that print at least two numbers, for any given computer there will always be a last number printed (let's assign this property of a given computer the label e).
• 3) For a computer with few resources, e might equal $2^{-5}$.
• 4) For a computer with more resources, e might equal $2^{-1,000,000}$.
• 5) In our observable universe, there is a smallest possible value for e.
• 6) However, it is possible that in an existence outside of our observable universe e can be even smaller.
• 7) Instead of placing a finite upper bound on the size of a computer, it is convenient to assume that a computer can be arbitrarily large but must necessarily be finite. To be clear, I'm not claiming that there is an infinite computer. I'm simply remaining agnostic on the bounds of finite computers.
• 8) Thus, e can be arbitrarily small but must necessarily be a positive number. To be clear, I'm not claiming that there is a smallest positive number. I'm simply remaining agnostic on the bounds of finite computers.
• 9) e is not a property of the program, but instead a property of the computer executing the program.
• 10) To describe the program, we must talk in general terms. Let ε be a possible value for e.
• 11) ε necessarily lies on the open interval (0,1).
• 12) I illustrate this description of the program as follows:

But even so. I have repeatedly asked you to give me the big picture. Give me something. — fishfry

• 1) Get you to agree to my use of ε in the computer example (including understand the illustration).
• 2) Get you to agree to my use of ε in the pi example (including understand the illustration).
• 3) Progress to 2D, where the Cartesian Coordinate system is replaced with a top-down alternative, and the zeros of y=x^2-2 have a very different meaning. Illustrations become important here which is why we need to get past 1) and 2) first.
• 4) Top-down interpretation of calculus.

So I will stipulate that you have a construction of the real numbers. — fishfry

Well if you have the intermediate value theorem and the least upper bound property -- ie, completeness -- then what you have, whatever it is, is isomorphic to the standard real numbers. — fishfry

No, I don't need to. If calculus works, then you have the standard real numbers. — fishfry

No such thing as an arbitrarily small positive real number. — fishfry

def small_number_generator():
    n = 1
    while True:
        print(n)
        n /= 2

But then, so what? I keep asking you that. — fishfry

Well then the intermediate value theorem is false. Calculus would collapse. — fishfry

Thankfully, ↪fishfry is there to help guide you. — jgill

That's why the Amish communities weren't hit hard — fishfry

Why do you insist on the one decomposition that we can't do? — fishfry

...it certainly is the pairwise disjoint union of ONE interval, and so what? — fishfry

Conversely, if the whole precedes the parts, then I should be capable of bisecting the whole into smaller sections, continuing to do so until I have arbitrarily small parts. This approach is feasible.
— keystone

You surely can't do that with countably many cuts. — fishfry

And your bisection idea doesn't work, you can't get any irrationals that way. But I believe you've already agreed with that. — fishfry

When it comes to the real numbers, I do think building the parts from the whole is difficult, because you'd need uncountably many cuts. — fishfry

But Dedekind has already built the reals from cuts of rationals, so it can be done. But there are uncountably many cuts. — fishfry

I asked you earlier: Suppose that rather than snipe line by line at this paragraph, I just accept it for sake of discussion. Can we move forward? — fishfry

But of course your whole approach is pointless (that's a pun) so maybe I'm getting it. — fishfry

Not bird flu I hope. Jeez the medical propaganda is everywhere these days. Are we all doomed? Like not eventually, but as soon as next week? — fishfry

Correct, but why does that matter? (0,1) is already the disjoint union of open intervals, namely itself. — fishfry

But points and numbers are entirely synonymous in this context. The "real line" is just the set of mathematical real numbers. — fishfry

I literally have no idea what we've been talking about the past several weeks. Which makes me feel foolish sniping at it. — fishfry

It's a very complicated game requiring perseverance and dedication. Are you in it for the long haul? — jgill

The inability for dimensionless points to be reconciled with the continuum is what motivated Whitehead's point-free geometry, a precursor to the field of Pointless Topology, as for instance formalised using Locales whose distributive law characterizes the meaning of a "spot". (It might be useful to test this law in relation to the SB tree, for both the truncated and infinite version). — sime

No worries, as they say. Get well soon. — fishfry

You can see that if x is in (0,1), then x is in a least one (actually all but finitely many) of the sets (1/n, 1 - 1/n). — fishfry

I would have to give this some thought. Would it make progress if I stipulate to your metaphysics? I don't know what to say anymore. — fishfry

Yes ok, so if you have an alternate way of getting to the same real numbers, what does it matter? — fishfry

Not a line, a nested collection of lines. The point zero is (-1, 1), (-1/2, 1/2), (-1/3, 1/3), etc. — fishfry

Every interval containing a given real number, necessarily contains other real numbers. That's the definition of (not) being isolated. — fishfry

(0,1) is the union of (1/n, 1 - 1/n) as n goes to infinity. I just wrote (0,1) as the union of infinitely many open intervals. — fishfry

• n=1 (1,0)
• n=2 (1/2,1/2)
• n=3 (1/3,2/3)
• etc.

Plans are far more general than algorithms...Chaitin's Omega is one such noncomputable number that can be specifically defined. — fishfry

You're right that most of the noncomputables have no unique definition and can't be "isolated," but so what? — fishfry

A point can never be perfectly represented using a line, no matter how small that line is. — keystone
Did I say the contrary? I don't recall doing that. — fishfry

No real numbers are isolated. — fishfry

Whatever. This is depressing me a bit. I no longer know what we're talking about. — fishfry

The nested interval construction can be explicitly written down. I perhaps am not sharing your vision here. — fishfry

Finitely many cuts won't get you enough of the points. Your continuum will be full of holes. The set of real numbers approximable by finite sequences is only countably infinite. — fishfry

The sequence is defined as pi. And thereafter, it might as well be taken for pi since, by suitably defining arithmetic on the set of sequence stacks, it will have all the required properties of pi. — fishfry

• We can sensibly devise a plan for arithmetic on rationals, AND we can completely execute arithmetic on rationals.
• We can sensibly devise a plan for arithmetic on irrationals, but we cannot completely execute arithmetic on irrationals.
• There is a distinction here that gets lost when you give rationals and irrationals the same status.

The reals are logically constructed from the rationals. If you have the rationals you get the reals for free. — fishfry

You haven't mentioned algorithms — fishfry

But now you're saying that just because you can't express something, it doesn't exist. — fishfry

Well, some of them can be isolated, if by that you mean defined. Most can't. — fishfry

Yes. Agreed. But they can ALSO be taken to be nested stacks. And then there is no difference in status between the rationals and the irrationals. — fishfry

If that's true, then you are saying that supertasks are a formalism or a concept that let you reproduce standard math, while pretending that you reject parts of standard math. — fishfry

Bundles is is. Should I think of them as tiny little wriggly micro-continua? ... Ok, You have all the intervals, but no individual points. — fishfry

I'm trying to clarify ideas about mathematics, and trying to frame your ideas in the context of what's already known about mathematics. — fishfry

Nothing showed up underlined so I don't know what you are referring to. But if you agree that a descending stack of intervals can be taken as the definition of a point, that's a major agreement between us. — fishfry

What, now you believe in irrationals? You know the S-B tree is not the only kind of tree structure that represents the real numbers. I don't know why you are fixated on it. — fishfry

Well, irrationals are downward nested stacks of intervals. That's the next best thing. Can we agree on that? — fishfry

But ... so are the rationals [downward nested stacks of intervals]! Right? — fishfry

But if you mean that a point has length 0, and an interval has a positive length, the unsigned difference of its endpoints, we agree. — fishfry

Mathematicians in general have no interest in supertasks. They're mainly a curiosity for the computer scientists as I understand it. — fishfry

I don't believe I'd take well to getting up at 5am to milk the bull. — fishfry

I like the modern world, but I don't think that applied mathematicians are universally engaged in creating good. — fishfry

I think they are chainsaws, not to be trifled with by the untrained masses. — fishfry

There are fiber bundles in math. A hairbrush with bristles sticking out is a fiber bundle. Off topic but reminded me of the name. — fishfry

Ok, it's an aggregate price where the components haven't necessarily been priced. So you have aggregate lengths, but no individual ones. Something like that? — fishfry

I honestly think that what you are doing is coming to understand, in your own way, the nature of the real numbers. — fishfry

Well sure, every irrational can be identified with a descending sequence of open intervals. I can locate pi in the sequence (3, 4), (3.1, 3.2), (3.14, 3.15), (3.141, 3.142), ... I mean that the sequence itself IS the number pi... Does that idea resonate with you? — fishfry

You just have an ... ahem ... irrational prejudice against irrational numbers. — fishfry

I'm with you descending down to points via sequences of open intervals. — fishfry

And if you don't believe pi is really there, then no problem. You just define pi as the sequence of nested open intervals and you've got an object that, if it's not the "real" pi, is just as good. — fishfry

This bit about planning and execution is a little off the mark. In math when we conceive a thing it's automatically done. Would that the rest of the world were so simple! — fishfry

Does the world seem improved to you? — fishfry

Why on earth would you think that? It clearly has no sum, since the sequence of partial sums has no limit. — fishfry

Needs explanation. — fishfry

Cuts as in Dedekind cuts? If you already have continuum-many points, why do you need cuts? — fishfry

I don't know what an "arbitrarily small cut] means. It conflicts with your previous use of cut. — fishfry

• Potential points in (0,1.5) = $2^{\aleph_0}$
• Actual points in [1.5,1.5] = 1
• Potential points in (1.5,2) = $2^{\aleph_0}$

• Length of (0,1.5) = 1.5
• Length of [1.5,1.5] = 0
• Length of (1.5,2) = 0.5

• $\epsilon_1$ and $\epsilon_2$ are arbitrarily small positive real numbers such that $\varphi - \epsilon_1$ and $\varphi + \epsilon_2$ are rational.
• $\varphi$ is the golden ratio number

• The essence of my perspective (top-down) remains the same, although it requires some minor adjustments.
• Having to reformulate my view underscores the significant value I've derived from our conversation—thanks once more!

How do you propose to pass from a finite line to a circle, say? If you are considering topological transformations, how can you express them? Sorry for butting in, but I remain curious. — jgill

I'll stipulate to your non-rigorous conception of a continuum of being made of tiny little continua "all the way down," with no need for actual points, if that's your idea. I think this is what Peirce is getting at. — fishfry

The line contains a frothing sea of tiny little micro-continua that are not points. Is that about right? — fishfry

Well here you are in trouble. If you allow "cuts" then à la Dedekind we have the real numbers. But you don't want to go there so ok. There are cuts but not so many as to allow the reals. — fishfry

Of course all mathematical entities are fictional, so I can't see what the difference is between and actual and a fictional point. — fishfry

You are saying the exact same thing, but changing the name of irrationals to "fictionals." I don't see how that changes anything. You just changed their name but they're the same irrationals. — fishfry

You correctly note that the sum of the lengths of the points is 0. But then you say that the sum of the lengths is 1, and I'm not sure how that follows. — fishfry

Didn't I ask you about this several posts ago? Ok, Euclid's line. — fishfry

And by the way, what is this "+" symbol? Have you defined it? Is this the standard + of the rational numbers? — fishfry

what does the notation (0,1) mean? — fishfry

• No points exist on lines, including the unit line (0,1). To put it another way, there are no 'actual points' present on that segment. (Actual vs. potential is discussed below).
• Cutting line (0,1) in two will introduce an 'actual point' between the two resulting line segments. That point will have a rational coordinate between 0 and 1.
• In my last post, I noted that -inf and +inf are not 'actual points' but rather are used as helpful shorthand. I should have called them 'potential points'.
• With a similar shorthand, we can say that on line (0,1) exist $2^{\aleph_0}$ 'potential points', which have real number coordinates between 0 and 1.
• The rational 'potential points' can become 'actual points' through cuts.
• The irrational 'potential points' are permanently confined to their 'potential point' status.
• I want to reiterate that 'potential points' don't actually exist. They're just a fiction that may help us comprehend the potential in continua. If you don't think potential points are a useful concept we can just drop.
• The interval "(0,1)" describes the potential of the corresponding unit line.

Since your intervals are entirely made up of rationals, the total length must be 0. Where is the extra length coming from? — fishfry

I'm lost and dispirited. It's not my role in life to feel bad about myself for endlessly sniping at your heartfelt ideas. — fishfry

unless you mean the original line of Euclid, "A line is breadthless length." — fishfry

What is a line? What does the notation [0, 0.5] mean? — fishfry

I'm not qualified to discuss the philosophy of the continuum with you, except as it relates to the standard mathematical real numbers. — fishfry

That directly contradicts what you said earlier. — fishfry

But you are the one saying that you only have rationals. — fishfry

In standard set theory, the only thing that sets can contain is other sets. We can call them points but that's only a word used to convey geometric intuition. Actually sets don't contain points, they contain other sets. — fishfry

I don't know anything about set theory with urlements. — fishfry

You only believe in rationals. Where are you getting these things? — fishfry

By bounds you mean endpoints? So now you are backing off entirely from your last half dozen points, and saying that your ontology consists of intervals with rational endpoints. But the real numbers are indeed present inside the intervals after all? Is that what you are saying? — fishfry

But now only the endpoints are rational, leaving me baffled as to what those objects are. — fishfry

A forum member once pointed me to the ideas of Charles Sanders Peirce (correct spelling) who said that the mathematical idea of a continuum could not be right, since a true continuum could not be broken up into individual points as the real numbers can. — fishfry

The length of that union is zero, if the intervals are restricted to rationals. Do you agree with that point? — fishfry

No points. So they're all the empty set? I'm not supposed to push back on this? — fishfry

Why on earth do you troll me into arguing with your points, then admitting that you agree with me in the first place? — fishfry

I very much believe in the continuum, which is pretty well modeled by the standard real numbers. — fishfry

I am at an utter loss as to what you have been getting at all this time. Can you get to the bottom line on all this? So far I get that your "continua" are either empty or have length 0. Or that they somehow have length 1, despite being composed of only rationals. — fishfry

How can it be if it contains only rationals? I have challenged you on this point several times already without your providing satisfactory explanation. — fishfry

there are 2^ℵ0 real numbers — fishfry

MU has written a similar notion about continua and points. Perhaps you can put some meat on the table. — jgill

not referencing the real line or numbers. — jgill

avoiding the real line entirely at first — jgill

Please stop talking about the S-B algorithm. — jgill

do your thing and persist until the thread dries up and vanishes — jgill

Leave the realm of real numbers at first. — jgill

It looks like you simply move the point [.3,.3] down the line segment to different (faulty) positions. — jgill

I don't know. — jgill

But you haven't got a continuum if your intervals contain only rational numbers. — fishfry

But the nature of a continuum is pretty deep, way beyond my knowledge of philosophy. — fishfry

Do you believe in the number 1/3 then? — fishfry

Consider one of your rational intervals [0,1]. What is its length? — fishfry

It looks like you simply move the point [.3,.3] down the line segment to different (faulty) positions. How does this affect your metric? — jgill

Very very few contemporary mathematicians give a fig leaf about Platonic vs non-Platonic arguments or similar discussions about whether math is embedded in nature or in the mind. — jgill

What would be a homeomorphism of [0,0]U(0,.3)U[.3,.3]U(.3,.5)U[.5,.5] ? — jgill

Well, if you were to avoid both metric spaces and variations of the word "topology" it might mitigate what seems to be a questionable attempt to employ legitimate mathematical notions within a somewhat murky mix of ideas. — jgill

In any event at some point you must present a clear and detailed description of your ideas that mathematicians might have reservations about but can follow the logic. — jgill

• 1) You can follow it and you think it's wrong - Great. I won't waste any more time on it.
• 2) You can follow it and you think I'm onto something - Great. I'll invest the needed time and money to make it better and hopefully learn a few tips from the discussion.
• 3) You can't follow it so you have no idea what to think - Bummer, I don't know what to do now.

Slowly work your way through this book and you will see why we ask so many questions. — jgill

And don't mix philosophy of mathematics with the real deal. — jgill

So, a continuous deformation takes path A to path B, but inside the ms of path A? Or a new ms of path B? You might illustrate this. I'm curious about these continuous deformations in the contexts of your ideas. — jgill

You accept some rational numbers. Not much of a continuum you have there. You understand that, right? — fishfry

There's no difference between an algorithm and the number it generates. 1/3 = .3333..., an infinite decimal, but 1/3 has a finite representation, namely 1/3 — fishfry

A view that has near universal mindshare, but ok, I'm a brainwashed mathematical sheep if you like. — fishfry

I think you are an intuitionist. — fishfry

You reject the algorithm given by the Leibniz series pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...? — fishfry

If you have a continuum but disbelieve even in the set of rationals, the burden is on you to construct o define a continuum. — fishfry

Can we move on please? — fishfry

How did this become so important to you? — jgill

Is this a legitimate "path" ? — jgill

So you do not compare "points" from one path to another. Altering the path, even slightly, places it in another metric space. But a ms could be a subspace of a bigger ms. Just talking to myself, here. — jgill

You spoke earlier of an "elastic band". where does that come into the picture? Especially with regard to metric spaces? Can a path be circular? — jgill

Can a path be circular? — jgill

So, do you believe in the rational numbers? Is that the number system we're working in? — fishfry

You're the one with some notion of enclosing set. A metric space is a set with a distance function. If it lives in a larger ambient set, then you have to say what that is. — fishfry

In which case I have to echo jgill's excellent question as to whether you accept intervals like [pi, pi + 1], and if not, why not. — fishfry

So you are willing to start with the Peano axioms? Is that your starting place?...But at least after all this you agreed to stipulate the Peano axioms. That's a start. A start from classical, bottom-up math. — fishfry

Although the rational numbers are tragically deficient as a continuum. You know that, right? They're full of holes. They're not continuous in the intuitive sense. — fishfry

You speak of a metric space. Precisely what are the "points" in such a space? Then explain the metric you have created giving "distances" between these points. — jgill

So you believe in the rational numbers? But then the reals are easily constructed from the rationals as Dedekind cuts or equivalence classes of Cauchy sequences. If you believe in the rationals you have to believe in the reals. — fishfry

You are the one who started at 0, then got to (0, .5), and then magically completed a limiting process to get to .5. I ask again, how is that accomplished?
You are the one who started at 0, remember? — fishfry