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• 3.6k
you have to have a modulus of convergence . . .

I'll bet I've conjectured and proven over a hundred theorems, almost all involving convergence/divergence of sequences and series of one sort or another and have never used this expression.

I go to Wikipedia when I encounter something in math I'm not familiar with to see what the daily average of views is - a very rough idea of how popular the topic is. My own math Wiki site gets about 19 per day, and the topic is way, way off in the margins of mathematics. However, I score higher than the 3 for this topic. But thanks for opening my mind a bit.

Constructive analysis was almost a passing thought until I read about it. I would have called myself something of a constructivist in that I rarely if ever used the excluded middle - if I postulated an entity I constructed it. But reading this description shows how far I am from contemporary mathematics. Once again, I go to Wiki to see how popular this topic is. And I find it scores a 17 - not bad, but still less than my virtually unknown page.

A point of clarity. Thanks. Calculus started with discrete, then moved to infinitesimal, then with technology back to discrete in some sense. — jgill

Is that right?

Probably not. I was thinking of the ancient Greeks breaking apart a sold object and measuring the pieces to approximate the object's volume or whatever. But even Archimedes recognized the infinitesimal.

I'm interested in good ol' real analysis. I just want to place it on a stronger philosophical foundation

And it may be purely philosophical. Given a line segment, points in this object are purely potential, non-existent until a device is used to "isolate" them. Is that about it? If so I doubt any practicing mathematician would be interested. But math philosophers might be. A lot depends upon where you go from here. Just my opinion.
• 3k
I'm sure you can appreciate the problem of substituting rational number approximations of irrational numbers too early in a computation. The best approach is to do all the manipulation first and only perform the substitution at the very end when the computation is required. I would rephrase this as follows:

Step 1: Manipulation of real numbers
Step 2: Computation based on rational numbers (approximations)

Was my mention of constructive analysis helpful?

This is analogous to the 2-step cutting process I outlined in my previous post. In both cases, step 2 is crude and done using computer arithmetic. It's the realm of applied mathematicians and not of interest here. I'm solely concerned with step 1.

You don't believe in the real numbers, how can you manipulate them? You think every real number can be arbitrarily approximated by an algorithm. That's false. But if your approximation only needs to be to the minimum distance in a system of computer arithmetic, then you're doing computer arithmetic.

That's pretty much what I'm saying! But instead of talking about any particular computer (which only becomes relevant to step 2), I want to remain in step 1 and talk in general terms. As such, would you allow me to say that π is (π-ε1,π+ε2), and that the value of ε1 and ε2 only need to be determined in step 2?

Only if you are doing computer arithmetic.

If you say that the above figure makes sense to you, then I can show you a 2D figure, and the benefits and consequences of my perspective will hopefully become clear.

What if none of your figures make sense to me? Your latest uses these epsilon quantities, which you've defined as the minimum possible length in a given physical computer. So you are doing computer arithmetic. Not that there's anything wrong with that! But it seems to me that's what you're doing.

If the noncomputables reals can describe continua it is because below the surface they rest upon a more fundamental scaffolding which can describe continua in and of itself.
]

You don't even believe in the noncomputable reals. And yes, they do rest on a deeper scaffolding, namely (1) Dedekind cuts; or (2) Equivalence classes of Cauchy sequences; or (3) Forget specific constructions, just write down the axioms for the real numbers, which are categorical.

So no, I'm not interested in constructive real analysis. I'm interested in good ol' real analysis. I just want to place it on a stronger philosophical foundation. I think my perspective will become clearer to you when explained in 2D.

Ah. What do you think is wrong with the current philosophical foundation? And why would a mathematician care?

From the outside, it may seem like this conversation isn't progressing, but your reluctance to accept my informal ideas has highlighted areas where I need to strengthen my arguments. So, you are indeed helping me a lot.

You did say something interesting. You are trying to patch up the philosophical basis of the real numbers. Do you understand that this is the first time that you've told me what you're doing? It's a bit of a revelation, for all the times I've said that I don't understand what you're doing.

But the real numbers are categorical. Any two models are isomorphic. So you are not going to be able to produce a "better" model of the real numbers. One representation, construction, or description gives you exactly the same set of real numbers as any other.
• 3k
I'll bet I've conjectured and proven over a hundred theorems, almost all involving convergence/divergence of sequences and series of one sort or another and have never used this expression.

Modulus of convergence is a thing in constructive math. I learned about it arguing with a constructivist here a couple of years back.

I go to Wikipedia when I encounter something in math I'm not familiar with to see what the daily average of views is - a very rough idea of how popular the topic is. My own math Wiki site gets about 19 per day, and the topic is way, way off in the margins of mathematics. However, I score higher than the 3 for this topic. But thanks for opening my mind a bit.

Constructive analysis was almost a passing thought until I read about it. I would have called myself something of a constructivist in that I rarely if ever used the excluded middle - if I postulated an entity I constructed it. But reading this description shows how far I am from contemporary mathematics. Once again, I go to Wiki to see how popular this topic is. And I find it scores a 17 - not bad, but still less than my virtually unknown page.

Andrej Bauer wrote a paper called Five Stages of Accepting Constructive Mathematics. It might be of interest.

https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf

Where do you find these scores? Is that a Wiki feature?

Probably not. I was thinking of the ancient Greeks breaking apart a sold object and measuring the pieces to approximate the object's volume or whatever. But even Archimedes recognized the infinitesimal.

He was way ahead of his time.
• 3.6k
Where do you find these scores? Is that a Wiki feature?

Click on "View History", then "Pageviews". It's a crude estimate of the popularity of a topic. For example, group theory gets 513 views per day and non-standard analysis gets 80.
• 323
You don't believe in the real numbers, how can you manipulate them?

I believe the following:

1) The following two algorithms (written with a finite number of characters as infinite series) correspond to e and pi:

$e = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots$
$\pi = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \cdots$

2) It is possible to compute the partial sums to a finite precision (e.g. π can approximately be represented as 3.14).
3) It is impossible to compute the complete sums to infinite precision (i.e. π cannot be represented as an infinite decimal number).
4) The algorithm itself does not apply any restrictions on the precision (.e. imprecision is only introduced during computation).
5) To prevent imprecision from being introduced, one should work with the algorithm and delay the computation for as long as possible.
6) There are algorithms for performing arithmetic on infinite series (i.e. algorithms on algorithms).
7) It is possible to compute the partial sum corresponding to π+e to a finite precision (e.g. π+e can approximately be represented as
$e = \frac{1}{0!} +4 + \frac{1}{1!} - \frac{4}{3} = \frac{14}{3}$)
8) It is impossible to compute the complete sum corresponding to π+e to infinite precision (i.e. π+e cannot be represented as an infinite decimal number).
9) Such arithmetic algorithms itself do not apply any restrictions on the precision (i.e. imprecision is only introduced during computation).
10) To prevent imprecision from being introduced, one should work with the arithmetic algorithm and delay the computation for as long as possible.
11) One can avoid computation altogether and just speak in terms of algorithms.

But if your approximation only needs to be to the minimum distance in a system of computer arithmetic, then you're doing computer arithmetic.

I'm taking (11) seriously and avoiding computation. By doing so, I'm not approximating anything; By sticking to the algorithms I'm working with perfect precision. While computers can work with algorithms, I'm not talking about the finite arithmetic you are referring to.

You think every real number can be arbitrarily approximated by an algorithm. That's false.

This is false. I think that non-computable real numbers exist but only within intervals. They do not exist as isolated objects. Since numbers are isolated by cuts and cuts are described with algorithm, we cannot even describe how to isolate non-computable real numbers.

What if none of your figures make sense to me?

Then I'll keep trying until you quit. It may be impossible to convince you to adopt my view, but I'll be fully satisfied if, by the end of this discussion, you can at least argue my position, even if you don't accept it.

Your latest uses these epsilon quantities, which you've defined as the minimum possible length in a given physical computer. So you are doing computer arithmetic. Not that there's anything wrong with that! But it seems to me that's what you're doing.

As described above, I care about the algorithms, not the numbers - plans, not the computations. The figure with epsilons illustrates the algorithm defining the cut corresponding to π. As I said earlier, it illustrates the plan, not the execution of the plan. To execute the plan then I need computer arithmetic, but I'm only interested in the plan.

What do you think is wrong with the current philosophical foundation? And why would a mathematician care?

The current philosophical foundation is riddled with actual infinities and paradoxes. Mathematicians have elegant ways of sweeping these paradoxes under the rug (like Russell's Paradox, Riemann's Rearrangement Theorem, the Dartboard Paradox, Zeno's Paradoxes, etc.), but they're still there. However, if you believe there's nothing under the rug, it becomes harder to convince you to care. I see a paradigm shift towards a top-down view having significant consequences across philosophy, especially in the interpretation of quantum mechanics. Such a statement might not seem 'beefy', but let me just say that truth has a history of being useful, even if its utility isn't immediately apparent when it's uncovered.

Do you understand that this is the first time that you've told me what you're doing?

I said things like...

• "I'm familiar with these methods [of building reals from the empty set]. I believe there is a bottom-up and a top-down interpretation of them. I'm not satisfied with the orthodox bottom-up interpretation of them"
• "Pi is just as important in the top-down view as it is in the bottom-up view. However, as with many other things, it just needs a little reinterpretation to fit into the top-down picture."
• "Considering epsilon's role in calculus, let me just say that with some reinterpretation, calculus can be elegantly integrated into the top-down perspective without the need for infinitesimals."

...but I should have explicitly said that I'm trying to patch up the philosophical basis of the real numbers.

But the real numbers are categorical. Any two models are isomorphic. So you are not going to be able to produce a "better" model of the real numbers. One representation, construction, or description gives you exactly the same set of real numbers as any other.

Numbers are the objects of computation, while algorithms are the objects of plans. I aim to shift the concept of reals from numbers to algorithms, from computations to plans. As such, I'm not proposing an alternate number model of the reals. I'm proposing an algorithmic model of reals. This model is structured very different. For one, while the real numbers are used to construct/define the real line, the real algorithms are used to deconstruct/cut the real line. However, I endeavor to show that switching to the top-down view has absolutely no impact on applied mathematicians, even those working with calculus.
• 323
Given a line segment, points in this object are purely potential, non-existent until a device is used to "isolate" them. Is that about it? If so I doubt any practicing mathematician would be interested. But math philosophers might be. A lot depends upon where you go from here. Just my opinion.

At the heart of my view is a simple idea: that infinity is a potential, not an actual. This idea leads to many consequences, one being that points are potential until isolated, as you have noted. However, I hope to show that there are far more interesting consequences to this small idea.

Sometimes a small idea can have huge consequences. For example, at the heart of relativity is the simple idea that the laws of physics are the same for all observers, regardless of their relative motion. The importance of relativity speaks for itself.

And no, I'm not comparing myself to Einstein. I'm just saying that even significant consequences can follow from simple ideas.
• 3k
Click on "View History", then "Pageviews". It's a crude estimate of the popularity of a topic. For example, group theory gets 513 views per day and non-standard analysis gets 80.

That is really cool, thanks! I think I'll be hooked on looking these up now.
• 3.6k
At the heart of my view is a simple idea: that infinity is a potential, not an actual

For those in the profession who do not deal with transfinitisms and set theory or foundations it's likely they would agree. When I say that a sequence converges to a number as n goes to infinity I simply mean n gets larger without bound. I don't think I have ever spoken of infinity as a number of some sort, although in complex variable theory one does speak of "the point at infinity" in connection with the Riemann sphere. But I am old fashioned.
• 3k
I believe the following:

1) The following two algorithms (written with a finite number of characters as infinite series) correspond to e and pi:

I have no problem with you identifying the various computable real numbers with any one of the many algorithms that generate their decimal digits. Or, as a mathematician might do, identify the number with the equivalence class of ALL such algorithms, so that the real number does not depend on which algorithm you choose.

However, you can't do the noncomputable numbers that way. And there are a lot of them.

I'm not sure how your idea of approximation works. If your computer has to truncate the series, then you are only defining an interval around pi or e. You are not being exact enough.

Am I following you?

2) It is possible to compute the partial sums to a finite precision (e.g. π can approximately be represented as 3.14).
3) It is impossible to compute the complete sums to infinite precision (i.e. π cannot be represented as an infinite decimal number).
4) The algorithm itself does not apply any restrictions on the precision (.e. imprecision is only introduced during computation).
5) To prevent imprecision from being introduced, one should work with the algorithm and delay the computation for as long as possible.
6) There are algorithms for performing arithmetic on infinite series (i.e. algorithms on algorithms).
7) It is possible to compute the partial sum corresponding to π+e to a finite precision (e.g. π+e can approximately be represented as
e=10!+4+11!−43=143
)
8) It is impossible to compute the complete sum corresponding to π+e to infinite precision (i.e. π+e cannot be represented as an infinite decimal number).
9) Such arithmetic algorithms itself do not apply any restrictions on the precision (i.e. imprecision is only introduced during computation).
10) To prevent imprecision from being introduced, one should work with the arithmetic algorithm and delay the computation for as long as possible.
11) One can avoid computation altogether and just speak in terms of algorithms.

None of that adds anything. If you are defining pi by some interval around it, I'm sure you can get it all to work. It's like error bars in engineering calculations.

Wait I think I've got it. You are doing Engineering math. In particular, when you have a number x, you also have error bars, so it's really x plus or minus a little wiggle room. And you are taught how to calculate that way, how to calculate with the error bars.

Is that helpful? Is that what you are doing?

https://en.wikipedia.org/wiki/Engineering_mathematics

I'm taking (11) seriously and avoiding computation. By doing so, I'm not approximating anything; By sticking to the algorithms I'm working with perfect precision. While computers can work with algorithms, I'm not talking about the finite arithmetic you are referring to.

You seem to be going off in directions. Those lists, they don't tell me much. I get that you are doing arithmetic with error bars. That's good, that's how they build bridges.

This is false. I think that non-computable real numbers exist but only within intervals. They do not exist as isolated objects. Since numbers are isolated by cuts and cuts are described with algorithm, we cannot even describe how to isolate non-computable real numbers.

Ok. So in your ontology, there are:

* Computable numbers, which have algorithms, or are identified with their algorithms, or are found by executing their algorithsm. Not sure which of those you mean but they're all about the same. But each computable number is the number that WOULD be computed if you finished executing the algorithm, but you can't; so each computable number is a number inside a little interval.

Have I got that right? And

* Inside each of these computable intervals, live all the noncomputable numbers.

I think all that's fine. In fact I could go you one better. Consider for each computable number $c$, the set of open intervals $(c - \frac{1}[n}, c + \frac{1}[n})$, for n = 1, 2, 3, ...

That would give you a countable set of open intervals whose union is the real numbers, including the noncomputables. But you'd never have to "identify" a noncomputable. And in fact each of the endpoints $(c \pm \frac{1}[n}$ are themselves computable.

What do you think of that? Would you say that's a reasonably fair mathematical model of your idea?

Then I'll keep trying until you quit.

Why? The diagrams make it harder for me to read your posts. I think the diagrams are important to YOU. But they are not in general doing me much good. What if the overabundance of diagrams was increasing the likelihood I'd quit? You can see that under that hypothesis, you are acting against your own interests by battering me with diagrams.

It may be impossible to convince you to adopt my view, but I'll be fully satisfied if, by the end of this discussion, you can at least argue my position, even if you don't accept it.

How does battering me with diagrams help? I'm trying to understand your view and your diagrams in general don't reach me and they also make it more difficult for me to read your posts.

I do think the diagrams are very helpful for YOU. So you should do them, just be judicious in how often you include them in posts.

As described above, I care about the algorithms, not the numbers - plans, not the computations. The figure with epsilons illustrates the algorithm defining the cut corresponding to π. As I said earlier, it illustrates the plan, not the execution of the plan. To execute the plan then I need computer arithmetic, but I'm only interested in the plan.

So far I get that your system involves little intervals centered at the computable numbers. I think intervals of 1/n on each side of each computable gives the same topology as your idea of using all the truncations of the series given by the algorithms.

Are we on the same page here? I really feel that we are.

The current philosophical foundation is riddled with actual infinities and paradoxes.

You have infinitely many truncations and infinitely many little open intervals around each computable.

And you surely aren't going to resolve the standard set-theoretic paradoxes with your intervals. I don't see the connection at all.

Mathematicians have elegant ways of sweeping these paradoxes under the rug (like Russell's Paradox, Riemann's Rearrangement Theorem, the Dartboard Paradox, Zeno's Paradoxes, etc.), but they're still there.

Viewing the real numbers as the union of a bunch of open intervals centered at the computable numbers isn't going to resolve those.

Please feel free to show me how to resolve any paradoxes with what you've talked about so far.

However, if you believe there's nothing under the rug, it becomes harder to convince you to care.

The paradoxes have been resolved mathematically over a century ago. The philosophical paradoxes are in fact of interest to me, but not of high interest.

But I am not saying there are no problems. I'm saying that I can't imagine how your computables-and-intervals idea solves anything. Please give me an example of how this would work.

I see a paradigm shift towards a top-down view having significant consequences across philosophy, especially in the interpretation of quantum mechanics. Such a statement might not seem 'beefy', but let me just say that truth has a history of being useful, even if its utility isn't immediately apparent when it's uncovered.

Russell's paradox and QM as well? Please, show me how this is supposed to work.

I said things like...

"I'm familiar with these methods [of building reals from the empty set]. I believe there is a bottom-up and a top-down interpretation of them. I'm not satisfied with the orthodox bottom-up interpretation of them"
"Pi is just as important in the top-down view as it is in the bottom-up view. However, as with many other things, it just needs a little reinterpretation to fit into the top-down picture."
"Considering epsilon's role in calculus, let me just say that with some reinterpretation, calculus can be elegantly integrated into the top-down perspective without the need for infinitesimals."

None of those things have to do with the philosophy of math or with problems thereof. Calculus was formulated without the need for infinitesimals in the late nineteenth century. It's a terrific intellectual achievement, but it's a solved problem.

...but I should have explicitly said that I'm trying to patch up the philosophical basis of the real numbers.

Yes it would have been helpful. Glad we've established that, I find it quite helpful to understand what it is you are doing.

Numbers are the objects of computation, while algorithms are the objects of plans. I aim to shift the concept of reals from numbers to algorithms, from computations to plans. As such, I'm not proposing an alternate number model of the reals. I'm proposing an algorithmic model of reals. This model is structured very different. For one, while the real numbers are used to construct/define the real line, the real algorithms are used to deconstruct/cut the real line. However, I endeavor to show that switching to the top-down view has absolutely no impact on applied mathematicians, even those working with calculus.

Ok. So as far as I get this: The real numbers are made up of a bunch of open intervals centered at the computable reals. Is that right? And FWIW I think your truncated algorithm idea will give the same reals as my plus/minus 1/n intervals.
• 3.6k
How does battering me with diagrams help?

I agree. I keep hoping for an interesting idea to appear, but so far there is nothing novel about the mathematics. If one studies existing mathematics one begins to get a recognition of what has been established. Exploration is the soul the subject, but one does not explore the heart of Africa by strolling around city park. Sorry . Perhaps when you present your ideas in 2D instead of (rather boring)1D (and the mind-numbing SB Tree) something of interest will appear. Philosophically, however, your ideas of potential points may go somewhere, but I don't know what has been done along those lines.
• 323
You are doing Engineering math.

Yes and no.

Yes - The person tasked to execute the cut is an engineer doing engineering math. He knows he'll never be able to cut the line exactly at π so he cuts an interval containing π to give him wiggle room - kind of like a safety factor.

No - The person tasked to generalize all engineer actions is a mathematician. Instead of assuming any particular engineer, the mathematician aims to describe the actions of the 'arbitrary engineer'. Instead of saying that the interval width is any particular value, the mathematician just says that the interval width is ε2-ε1, where ε1 and ε2 can be any arbitrarily positive number.

The cut of (-∞,+∞) at π is generalized as (-∞,π-ε1) U π-ε1 U (π-ε1,π+ε2) U π+ε2 U (π+ε2,+∞)

Computable numbers, which have algorithms, or are identified with their algorithms, or are found by executing their algorithms. Not sure which of those you mean but they're all about the same.

Computable reals are identified with their algorithms.
Computable rationals are found by executing their algorithms.

But each computable number is the number that WOULD be computed if you finished executing the algorithm, but you can't; so each computable number is a number inside a little interval. Have I got that right?

Yes. I would like to distinguish between real numbers and real algorithms. A computable real number WOULD be computed if you finished executing the corresponding real algorithm, but you can't; so, the real algorithm only ever defines an interval within which the real number is inside. No real number can ever be isolated.

• $1.\overline{0}$ is a real algorithm.
• $0.\overline{9}$ is a real algorithm.
• $1.\overline{0}$ - $0.\overline{9}$ = $0.\overline{0}$, which is a real algorithm.
• $1.0$ is a rational number.

That would give you a countable set of open intervals whose union is the real numbers, including the noncomputables. But you'd never have to "identify" a noncomputable. And in fact each of the endpoints (c \pm \frac{1}[n}(c \pm \frac{1}[n} are themselves computable.

I want to avoid talk of the existence of an actually infinite set. We need to frame it in terms of the potential to create an arbitrarily large set. It is very important that the endpoints be rational, otherwise nothing is gained by defining π using intervals.

But they are not in general doing me much good. What if the overabundance of diagrams was increasing the likelihood I'd quit? You can see that under that hypothesis, you are acting against your own interests by battering me with diagrams...just be judicious in how often you include them in posts.

Point taken. I will be more judicious. SB-tree aside, I will grant that I didn't need to use a single diagram for the discussion so far. Interval notation would have been entirely sufficient. I was just hoping that you would warm up to 1D diagrams because when I go to 2D it will be very hard for me to describe what I'm thinking with words. I suppose I'll cross that bridge when we get there.

So far I get that your system involves little intervals centered at the computable numbers.

It almost sounds like you're suggesting that I'm saying that (-∞,+∞) is the union of infinite little intervals. It is not. With the top-down view, we don't construct (-∞,+∞), rather we start with it. Engineer1 may cut (-∞,+∞) five times. Engineer2 may cut (-∞,+∞) five million times. What the mathematician would say is that the 'arbitrary engineer' will make N cuts, where N is an arbitrary natural number. The is no 'privaledged engineer' who has a system that has been cut infinitely many times. Rather, each engineer must work within their own finite system.

Are we on the same page here? I really feel that we are.

I do feel like we're very close to being on the same page now!

Russell's paradox and QM as well? Please, show me how this is supposed to work.

Let's save the paradox discussion for later. I only mentioned it at this point because you asked why a mathematician would care.

Ok. So as far as I get this: The real numbers are made up of a bunch of open intervals centered at the computable reals. Is that right? And FWIW I think your truncated algorithm idea will give the same reals as my plus/minus 1/n intervals.

The real number is interior to the interval defined by the corresponding real algorithm. However, it doesn't necessarily have to be at the center. ε1 and ε2 don't have to be equal. I do think your 1/n values for epsilon works, but I'm not sure if we need to constrain the values of epsilon as such. If we're cutting (-∞,+∞) then it seems to me we should be as general as possible and say that epsilon can be any positive number - even 5 billion.
• 323
For those in the profession who do not deal with transfinitisms and set theory or foundations it's likely they would agree. When I say that a sequence converges to a number as n goes to infinity I simply mean n gets larger without bound. I don't think I have ever spoken of infinity as a number of some sort, although in complex variable theory one does speak of "the point at infinity" in connection with the Riemann sphere. But I am old fashioned.

Well I'd say infinite sets are pretty pervasive in modern mathematics. The reason is that sets are very useful and finite sets are very restrictive - they're not a satisfactory alternative. What I'm proposing is nuanced - arbitrarily big sets. I want to replace 'infinity' with 'arbitrary'...at least most of the time. A sequence converges to a number as n gets arbitrarily big.

I have no problem with the treatment of infinity on the Riemann sphere or in projective geometry.

Exploration is the soul the subject, but one does not explore the heart of Africa by strolling around city park. Sorry ↪keystone

Richard Feynman once said that "the chance is high that the truth lies in the fashionable direction. But, on the off-chance that it is in another direction — a direction obvious from an unfashionable view of field theory — who will find it? Only someone who has sacrificed himself by teaching himself quantum electrodynamics from a peculiar and unfashionable point of view; one that he may have to invent for himself."

My ideas have no relation to field theory, they might not be true, and even if they're true, they might not be interesting. I accept that chances this goes nowhere is high. But you shouldn't discredit my view just because I choose to stroll through unfashionable parks.
• 3.6k
But you shouldn't discredit my view just because I choose to stroll through unfashionable parks.

I look forward to a breakthrough in your quest. But I am very old and have multiple medical conditions, so I may not be around. Smooth sailing, fellow explorer.
• 323
I look forward to a breakthrough in your quest. But I am very old and have multiple medical conditions, so I may not be around. Smooth sailing, fellow explorer.

Thank you! I wish you very many wonderful years ahead.
• 3k
Yes and no.

Yes - The person tasked to execute the cut is an engineer doing engineering math. He knows he'll never be able to cut the line exactly at π so he cuts an interval containing π to give him wiggle room - kind of like a safety factor.

No - The person tasked to generalize all engineer actions is a mathematician. Instead of assuming any particular engineer, the mathematician aims to describe the actions of the 'arbitrary engineer'. Instead of saying that the interval width is any particular value, the mathematician just says that the interval width is ε2-ε1, where ε1 and ε2 can be any arbitrarily positive number.

The cut of (-∞,+∞) at π is generalized as (-∞,π-ε1) U π-ε1 U (π-ε1,π+ε2) U π+ε2 U (π+ε2,+∞)

What? You know, none of this makes any sense. [He's crabby tonight]

Computable reals are identified with their algorithms.
Computable rationals are found by executing their algorithms.

What? There's no difference with respect to algorithms. Consider 1/3 = .333...

Yes. I would like to distinguish between real numbers and real algorithms.

Of course, because they are entirely different things, and there are a lot more real numbers than algorithms.

A computable real number WOULD be computed if you finished executing the corresponding real algorithm, but you can't; so, the real algorithm only ever defines an interval within which the real number is inside. No real number can ever be isolated.

I think I am nearing the end here. You just are not making any sense. You've just made all this terminology up.

... [stuff omitted]

, which is a real algorithm.
1.0
1.0
is a rational number.

If there is a difference between 1.0 and 1.00000... you are off on your own. I can't hold up my end of this. Nothing you write is correct.

I want to avoid talk of the existence of an actually infinite set. We need to frame it in terms of the potential to create an arbitrarily large set. It is very important that the endpoints be rational, otherwise nothing is gained by defining π using intervals.

Not feelin' it tonight.

Point taken. I will be more judicious. SB-tree aside, I will grant that I didn't need to use a single diagram for the discussion so far. Interval notation would have been entirely sufficient. I was just hoping that you would warm up to 1D diagrams because when I go to 2D it will be very hard for me to describe what I'm thinking with words. I suppose I'll cross that bridge when we get there.

I don't see where this is going. I might be doing you a disservice by encouraging you.

It almost sounds like you're suggesting that I'm saying that (-∞,+∞) is the union of infinite little intervals.

You have said so many times, or at least I have understood you to say that. And besides, it is. Just not pairwise disjoint open intervals.

It is not. With the top-down view, we don't construct (-∞,+∞), rather we start with it. Engineer1 may cut (-∞,+∞) five times. Engineer2 may cut (-∞,+∞) five million times. What the mathematician would say is that the 'arbitrary engineer' will make N cuts, where N is an arbitrary natural number. The is no 'privaledged engineer' who has a system that has been cut infinitely many times. Rather, each engineer must work within their own finite system.

Ok whatever.

I do feel like we're very close to being on the same page now!

Ok that's good. Can we turn the page?

Let's save the paradox discussion for later. I only mentioned it at this point because you asked why a mathematician would care.

Then you give me no reason to care. You are not going to "solve QM" with your line of discourse.

The real number is interior to the interval defined by the corresponding real algorithm. However, it doesn't necessarily have to be at the center. ε1 and ε2 don't have to be equal. I do think your 1/n values for epsilon works, but I'm not sure if we need to constrain the values of epsilon as such. If we're cutting (-∞,+∞) then it seems to me we should be as general as possible and say that epsilon can be any positive number - even 5 billion.

Fine.
• 3k
I agree. I keep hoping for an interesting idea to appear, but so far there is nothing novel about the mathematics. If one studies existing mathematics one begins to get a recognition of what has been established. Exploration is the soul the subject, but one does not explore the heart of Africa by strolling around city park. Sorry ↪keystone . Perhaps when you present your ideas in 2D instead of (rather boring)1D (and the mind-numbing SB Tree) something of interest will appear. Philosophically, however, your ideas of potential points may go somewhere, but I don't know what has been done along those lines.

:100: :100: :100: :100: :100:
• 323
What? You know, none of this makes any sense.

I was trying to go along with your idea of engineering math. If the mention of an engineer and a mathematician working together doesn't help, then fine - we'll drop the idea of engineering math. But, I disagree with your statement that none of it makes sense.

What? There's no difference with respect to algorithms. Consider 1/3 = .333...

Consider the follow program which writes the specified fraction in the specified base (NOTE: You can skip over the code):

def fraction_to_base(numerator, denominator, base):
result = "0."
remainder = numerator

while remainder != 0:
remainder *= base
digit, remainder = divmod(remainder, denominator)
result += str(digit)

return result


For 1/3 in base 3, this program returns 0.1.
For 1/3 in base 10, this program returns nothing because the program does not halt. After all, it's trying to compute the sum of an infinite series. Impossible.

Clearly, these are not the same outcomes. They're different because $0.1_3$ is a rational number, while $0.\overline{3}_{10}$ is a real algorithm.

If there is a difference between 1.0 and 1.00000... you are off on your own. I can't hold up my end of this. Nothing you write is correct.

Take $1.0$ and $1.\overline{0}$ literally. Here's what they mean:

$1.0 = 1 + \frac{0}{10}$
$1.\overline{0} = 1 + \frac{0}{10} + \frac{0}{100} + \frac{0}{1000} + \ldots$

If I were to write a program for each of those computations, the former would halt and the latter wouldn't, similar to the 1/3 example above. I understand why you and everyone else think they're exactly the same, but, in the purest sense, they are algorithmically different. Do you not see that?

It's not that I'm incorrect. It's that mathematicians have been so sloppy with the distinction between reals and rationals not realizing that this distinction truly matters, especially from a top-down perspective.

Yes. I would like to distinguish between real numbers and real algorithms.
— keystone

Of course, because they are entirely different things, and there are a lot more real numbers than algorithms.

Allow me to clarify: I want to distinguish between a real number and it's corresponding real algorithm. A real algorithm corresponding to π can be written perfectly with finite characters, such as:

$\underline{\pi} = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \cdots$

or better yet, let me take that infinite series and derive 'a sequence of intervals' version of π:

\begin{aligned}P_N &= \text{Partial sum for first } N \text{ terms of the above series} \\\min_a &= \min(P_a, P_{a+1}) = \begin{cases} P_a & \text{if } P_a \lt P_{a+1}, \\ P_{a+1} & \text{else} \end{cases} \\\max_a &= \max(P_a, P_{a+1}) = \begin{cases} P_a & \text{if } P_a \gt P_{a+1}, \\ P_{a+1} & \text{else} \end{cases}\end{aligned}

Here is the algorithm for the infinite sequence of intervals:

\begin{aligned}\underline{\pi} = (\min_0, \max_0), (\min_1, \max_1), (\min_2, \max_2), \ldots\end{aligned}

The real number π is the only number that lies in \begin{aligned}(\min_{N}, \max_{N+1})\end{aligned} regardless of what N you choose. However, since no interval in the series has zero length, the real number π cannot be isolated. Numerically, π is the only number that lies in any interval described by π, and geometrically, π is the only point that lies in any line described by π.

From the bottom-up view π is equivalent to π.
From the top-down view π is not equivalent to π (any more that an algorithm is equivalent to it's output, or a line is equivalent to a point).

You've just made all this terminology up.

That's because when it comes to reals, mathematicians have been so sloppy with their terminology. I'm trying to make things more precise.

Then you give me no reason to care. You are not going to "solve QM" with your line of discourse.

First off, I'm only claiming to (at least partially) solve the issue of how to philosophically interpret QM. I'm certainly not claiming to have solved quantum gravity or anything like that. Are you saying you want me to jump right to the implications of the top-down view without even explaining the top-down view? I'm certain that without understanding my view you'll just think I'm injecting quantum woo into the top-down view. If you stick with it, what you'll see is that quantum intuitions follow from the top-down view. It is this way because QM is a top-down view of reality whereas classical mechanics (CM) is a bottom-up view of reality. Whether we're talking about mathematics or physics, the bottom-up view has been undoubtedly and demonstrably useful. It's just not correct at a foundational level. The reason why we struggle to interpret QM is because the mathematical top-down view has been neglected. Zeno was the first canary in the coal mine urging us to consider it.

I think I am nearing the end here. You just are not making any sense...Not feelin' it tonight...I don't see where this is going. I might be doing you a disservice by encouraging you...Can we turn the page?

You've been of great help to me so far and I greatly appreciate that. If you ever want to call it quits I will accept that, thank you for your help, and that will be the end of it. Of course, I reallllly hope that doesn't happen...
• 3k
I was trying to go along with your idea of engineering math. If the mention of an engineer and a mathematician working together doesn't help, then fine - we'll drop the idea of engineering math. But, I disagree with your statement that none of it makes sense.

Ok. So far, after all this, what I understand of your idea is that the real line consists of a countably infinite set of overlapping open intervals, each containing a computable number. So far so good? But this is not a very deep idea, there are many ways to express the reals as a union of open intervals.

What? There's no difference with respect to algorithms. Consider 1/3 = .333...
— fishfry

Consider the follow program which writes the specified fraction in the specified base (NOTE: You can skip over the code):

For 1/3 in base 10, this program returns nothing because the program does not halt. After all, it's trying to compute the sum of an infinite series. Impossible.

This is a common misunderstanding of what a halting program means.

The number pi is computable. Clearly no computer program can generate all the digits in one go. But that's not what halting means.

Pi is computable because, given a positive integer n, I can run the program for a finite number of steps, and output the n-th decimal digit.

It's absurd to claim that 1/3 is not computable or not algorithmic (or whatever you're claiming) just because its decimal representation is infinite. Given n, the n-th digit is 3. That's a halting program. Therefore 1/3 is computable. Likewise pi.

If I were to write a program for each of those computations, the former would halt and the latter wouldn't, similar to the 1/3 example above. I understand why you and everyone else think they're exactly the same, but, in the purest sense, they are algorithmically different. Do you not see that?

No. I deleted some text but you were trying to convince yourself that 1 and 1.0 and 1.0000... are different numbers. They are not. And again, it's because you're confused about what a halting program means. Now that you understand it (now that I've explained it to you), you are no longer confused.

All an algorithm is required to do is, given n, output the n-th digit in finitely many steps.

It's not that I'm incorrect. It's that mathematicians have been so sloppy with the distinction between reals and rationals not realizing that this distinction truly matters, especially from a top-down perspective.

You are delusional. Could it be that you are the one who's confused, and not mathematicians?

Does my explanation of what's a halting algorithm to compute a real number cause to you reframe your understanding? Input n, output the n-th decimal (or any base) digit in finitely many steps.

Allow me to clarify: I want to distinguish between a real number and it's corresponding real algorithm. A real algorithm corresponding to π can be written perfectly with finite characters, such as:

π⎯⎯=4−43+45−47+49−⋯

What are you doing that, when I quote your numeric examples, the quote text comes out in a column?

From the bottom-up view π is equivalent to π.
From the top-down view π is not equivalent to π (any more that an algorithm is equivalent to it's output, or a line is equivalent to a point).

Turing defined computability another way. A real number is computable if, given epsilon > 0, there's an algorithm that generates an approximation to the number within epsilon. You can see that this amounts to outputting the n-th digit. The point is that a number can be computable by an algorithm even if the number has infinitely many decimal digits, like 1/3 or pi.

That's because when it comes to reals, mathematicians have been so sloppy with their terminology. I'm trying to make things more precise.

Is that really likely? Or is it more likely that there are some basic things you don't understand, like the definition of halting or the fact that it's trivial to write the real line as a countably infinite union of open intervals?

First off, I'm only claiming to (at least partially) solve the issue of how to philosophically interpret QM. I'm certainly not claiming to have solved quantum gravity or anything like that. Are you saying you want me to jump right to the implications of the top-down view without even explaining the top-down view?

It would help me to understand what you're talking about, and why.

After all, you say your top-down view starts with the real line. But I say, I don't know what the real line is. How do you know there is any such thing unless you construct it from first principles?

I'm certain that without understanding my view you'll just think I'm injecting quantum woo into the top-down view. If you stick with it, what you'll see is that quantum intuitions follow from the top-down view. It is this way because QM is a top-down view of reality whereas classical mechanics (CM) is a bottom-up view of reality.

This sounds very cranky.

Whether we're talking about mathematics or physics, the bottom-up view has been undoubtedly and demonstrably useful. It's just not correct at a foundational level. The reason why we struggle to interpret QM is because the mathematical top-down view has been neglected. Zeno was the first canary in the coal mine urging us to consider it.

Cranky. Grandiose claims not backed by anything coherent.

You've been of great help to me so far and I greatly appreciate that. If you ever want to call it quits I will accept that, thank you for your help, and that will be the end of it. Of course, I reallllly hope that doesn't happen...

I'll slog on a little longer. It would help if you'll engage with my key point tonight, which is that you've been misunderstanding the nature of halting with respect to computable numbers.

Can you see that 1/3 = .333... is computable, because the program "print 3" halts in finitely many steps for an n, giving the n-th decimal digit of 1/3?
• 323
I'll slog on a little longer.

Yay!!

It would help if you'll engage with my key point tonight, which is that you've been misunderstanding the nature of halting with respect to computable numbers. Can you see that 1/3 = .333... is computable, because the program "print 3" halts in finitely many steps for an n, giving the n-th decimal digit of 1/3?

• I agree that the program you describe halts, however, I'll focus on Turing's version, by agreeing that the program that computes 0.333... to an arbitrarily fine precision halts.
• I agree that by definition that's what it means for a number to be a computable number, so by definition 0.333... is a computable number.
• I agree that by that definition pi is a computable number.
• I agree that no program can compute pi to infinitely fine precision.
• We might even agree that no program can compute 0.333... to infinitely fine precision (no shortcuts allowed).

My understanding is that a program halts if it 'terminates'. It's as simple as that. Do you actually disagree with this definition of halt?

You are employing a straw man argument. I'm saying that the program that computes 0.333... to infinitely fine precision does not halt, and you are saying that the program that computes 0.333... to an arbitrarily fine precision does halt. I agree with you, but your argument doesn't address my point.

I believe the term 'computable number' applies to a number which can be represented by an algorithm. That's it. There is no mention of computers or finite resources in this definition. Conversely, the term 'halt' applies to the execution of the algorithm. If it cannot be executed to completion then it does not halt. This distinction highlights one of my central complaints: mathematicians are obfuscating the plan (the algorithm) with the execution (output of the algorithm). And it doesn't help that we call both the algorithm and it's output the same thing: numbers. This is where I'm trying to bring clarity to the situation by redefining terms (like what it means to be a rational vs. real), but it turns out that just makes you think I don't know what I'm talking about.

After all, you say your top-down view starts with the real line. But I say, I don't know what the real line is. How do you know there is any such thing unless you construct it from first principles?

What is an easier starting point - one line or infinite points? The finite or the infinite? I start with Euclid's line which extends to infinity. Everything else in 1D is constructed from that. For example, let me exemplify 1+0+1=2 from Euclid's line.

L(a,b) = length of interval (a,b) = b-a

2) Cut @ 2 --> (-inf,2) U 2 U (2,+inf)
3) Cut @ 0 --> (-inf,0) U 0 U (0,2) U 2 U (2,+inf)
4) Scrap all but (0,2) --> (0,2)
5) L(0,2) = 2
6) Cut (0,2) @ 1 --> (0,1) U 1 U (1,2)
7) L(0,1) + L(1,1) + L(1,2) --> 1 + 0 + 1
8) Assuming conservation from step 5 to 7, 1 + 0 + 1 = 2

Might your issue be that one cannot cut at point 2 if point 2 does not exist prior to the cut? If so, this is a good question for which I have an answer.

Ok. So far, after all this, what I understand of your idea is that the real line consists of a countably infinite set of overlapping open intervals, each containing a computable number. So far so good?

If we must talk in terms of infinite anything then I would say that the real line consists of $2^{\aleph_0}$ points bundled into a single object. When bundled, we can only speak of the bundle - it is simply (-inf,+inf). It is simply a line. It is not the union of anything until we cut it. In step 2 above, we cut it @ 2 so at that step it is the union of 3 objects.

You are delusional. Could it be that you are the one who's confused, and not mathematicians?

Of course that's possible (and likely). After all, that's why these ideas are being discussed in this chat forum and not the Annals of Mathematics. But since any person yelling 'I'm not crazy' only makes them sound crazier, I won't challenge this point further and hopefully the ideas will eventually speak for themselves.

Cranky. Grandiose claims not backed by anything coherent.

I know how it sounds, that's why I'm reluctant to talk about QM and paradoxes at this time. When I communicate the fundamentals, you ask for the implications. When I communicate the implications, you ask for the fundamentals. I'm in a difficult position.

What are you doing that, when I quote your numeric examples, the quote text comes out in a column?

I ask ChatGPT to give me the Latex equivalent of an expression and I insert that Latex string with the math tag.
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