you have to have a modulus of convergence . . . — fishfry
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? — fishfry
I'm interested in good ol' real analysis. I just want to place it on a stronger philosophical foundation — keystone
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) — keystone
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. — keystone
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? — keystone
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. — keystone
]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. — keystone
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. — keystone
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. — keystone
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. — jgill
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. — jgill
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. — jgill
You don't believe in the real numbers, how can you manipulate them? — fishfry
But if your approximation only needs to be to the minimum distance in a system of computer arithmetic, then you're doing computer arithmetic. — fishfry
You think every real number can be arbitrarily approximated by an algorithm. That's false. — fishfry
What if none of your figures make sense to me? — fishfry
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. — fishfry
What do you think is wrong with the current philosophical foundation? And why would a mathematician care? — fishfry
Do you understand that this is the first time that you've told me what you're doing? — fishfry
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. — fishfry
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. — jgill
At the heart of my view is a simple idea: that infinity is a potential, not an actual — keystone
I believe the following:
1) The following two algorithms (written with a finite number of characters as infinite series) correspond to e and pi:
— keystone
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. — keystone
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. — keystone
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. — keystone
Then I'll keep trying until you quit. — keystone
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. — keystone
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. — keystone
The current philosophical foundation is riddled with actual infinities and paradoxes. — keystone
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. — keystone
However, if you believe there's nothing under the rug, it becomes harder to convince you to care. — keystone
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. — keystone
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." — keystone
...but I should have explicitly said that I'm trying to patch up the philosophical basis of the real numbers. — keystone
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. — keystone
How does battering me with diagrams help? — fishfry
You are doing Engineering math. — fishfry
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. — fishfry
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? — fishfry
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. — fishfry
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. — fishfry
So far I get that your system involves little intervals centered at the computable numbers. — fishfry
Are we on the same page here? I really feel that we are. — fishfry
Russell's paradox and QM as well? Please, show me how this is supposed to work. — fishfry
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. — fishfry
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. — jgill
Exploration is the soul the subject, but one does not explore the heart of Africa by strolling around city park. Sorry ↪keystone — jgill
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,+∞) — keystone
Computable reals are identified with their algorithms.
Computable rationals are found by executing their algorithms. — keystone
Yes. I would like to distinguish between real numbers and real algorithms. — keystone
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. — keystone
... [stuff omitted]
, which is a real algorithm.
1.0
1.0
is a rational number. — keystone
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. — keystone
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. — keystone
It almost sounds like you're suggesting that I'm saying that (-∞,+∞) is the union of infinite little intervals. — keystone
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. — keystone
I do feel like we're very close to being on the same page now! — keystone
Let's save the paradox discussion for later. I only mentioned it at this point because you asked why a mathematician would care. — keystone
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. — keystone
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. — jgill
What? You know, none of this makes any sense. — fishfry
What? There's no difference with respect to algorithms. Consider 1/3 = .333... — fishfry
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
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. — fishfry
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. — fishfry
You've just made all this terminology up. — fishfry
Then you give me no reason to care. You are not going to "solve QM" with your line of discourse. — fishfry
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? — fishfry
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. — keystone
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. — keystone
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? — keystone
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. — keystone
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−⋯ — keystone
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). — keystone
That's because when it comes to reals, mathematicians have been so sloppy with their terminology. I'm trying to make things more precise. — keystone
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? — keystone
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. — keystone
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. — keystone
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... — keystone
I'll slog on a little longer. — fishfry
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? — fishfry
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? — fishfry
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? — fishfry
You are delusional. Could it be that you are the one who's confused, and not mathematicians? — fishfry
Cranky. Grandiose claims not backed by anything coherent. — fishfry
What are you doing that, when I quote your numeric examples, the quote text comes out in a column? — fishfry
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