There can be no counting to begin with. — jgill
I'm surprised. Could you explain why? — Ludwig V

Real numbers are uncountable.

Having studied psychology for years — Mikie

Were you a professional? Just curious. — jgill

Guess not. That's OK, I never became a professional in the outdoor activity I alluded to. And I spent countless hours at it.

The problem arises when people believe that the infinite convergent series is the necessary outcome of the problem of infinite divisibility instead of seeing it as one possible representation. — Metaphysician Undercover

Although I don't agree there is a problem with "infinite divisibility", another procedure you might mention is described by Tannery's theorem, which concerns series in which each term changes as the series progresses. In the extreme case, a series in which each term converges to zero as described will itself converge to zero. I.e., infinite summable to zero.

(I extended this idea to composition theory some time back Generalizations . . .)

Although you and I don't agree on the soundness of established mathematics, I do enjoy reading what you have to say.

And when we describe the principle of distinction between non-dimensional points on a line, we find that our counting is endless. — Ludwig V

There can be no counting to begin with.

Or, one could say that one doesn't do things formally. That's fine, but then a comparison with mathematics is not apt since mathematics rises to a challenge that informal quasi-mathematical ruminations do not — TonesInDeepFreeze

:up:

Having studied psychology for years — Mikie

Were you a professional? Just curious.

But you shouldn't discredit my view just because I choose to stroll through unfashionable parks. — keystone

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.

How does battering me with diagrams help? — fishfry

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.

At the heart of my view is a simple idea: that infinity is a potential, not an actual — keystone

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.

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

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.

you have to have a modulus of convergence . . . — fishfry

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? — fishfry

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 — keystone

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.

A lot of leadership and innovation reduces to being at the right place at the right time. I'm going to briefly describe my own experience, at the risk of appearing narcissistic.

I got into an outdoor activity in the 1950s that was just beginning to become popular in America. After a year or so, for various reasons, I began to see this activity in a different light, and began practicing it from this perspective. Without an effort to influence, some others began, slowly, to see the activity as I did. As the years passed my version of the activity gained considerable popularity and attracted those far more fit than I, and my achievements were substantially eclipsed.

This was an activity going back to the late 1800s at least. It is now an Olympic event. I was in the right place at the right time for a normal person to influence the future.

have Felon-1 sit in a Washington DC jail until his "January 6th Conspiracy" trial begins — 180 Proof

Would his SS protection be placed in his cell, also? :cool:

Still reading Royal Robbins - The American Climber by David Smart. Royal was a friend BITD.

↪RogueAI
Probably, but I am conservative. — Bob Ross

Go to a Republican fundraiser and see what happens. Do not mention anything philosophical. Complement a young lady on her MAGA cap.

Things get much more interesting in 2D — keystone

I recommend you move on to 2D. Just a thought.

Thomson's Lamp and similar supertasks can be placed in an alternate context simply by using time dilation. Assume the lamp goes on and off at increments of $\frac{1}{{{2}^{n}}}$ but the experiment is on a spaceship that travels at ${{v}_{n}}=c\sqrt{1-\frac{1}{{{4}^{n}}}}$. Then, on board the ship, as time quickly ticks down to zero, on Earth each such tick corresponds to 1, so that the Earthbound observer recognizes a tick each constant unit of time and the task goes on forever.

Just a passing thought on a thread that behaves in a similar fashion.

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

I believe step 1 is what is of interest to pure mathematicians. — keystone

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.

It's ok, I have my second wind I think. Especially now that I know you're just doing computer arithmetic, fixed or floating point. — fishfry

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

I acknowledge that for the bottom-up view, calculus requires the complete set of isolated real numbers, the intermediate value theorem, and the least upper bound property to "work"...I use quotes because it also requires some mental gymnastics. However, that's just not the case for the top-down view. It works perfectly in absence of all of the above...including the mental gymnastics — keystone

What is an isolated real number?

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

This is not a paradox, but a confusion of concepts (like the number 1 or infinitely) with actual things (like a one step down one stair, and never reaching the bottom or doing so in a minute). — Fire Ologist

:up:

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

Why do you want to know? — keystone

I find their concepts a bit challenging to grasp just by skimming. It seems like a thorough reading might be required to truly understand these ideas, something I'm not quite ready to dive into — keystone

What you are doing seems to me to be more metaphysics than mathematics. And that's OK. But without studying what is accepted mathematics you have a rough road ahead if you wish to contribute to that discipline. However, there have been amateurs within the last half century who have made significant discoveries. Marjorie Rice. Thankfully, is there to help guide you.

My point is that once we've entered the realm of speculative fantasy, where do we stop? — fishfry

Pretty much sums up this thread.

I found this paper that adopts intervals instead of points in its framework, which is quite relevant. — keystone

It also demonstrates the difficulty of trying to do something original and noteworthy in mathematics. It's a very complicated game requiring perseverance and dedication. Are you in it for the long haul?

Thanks for introducing math topics that I was only barely aware of into the discussion. Good to learn something.

Something flashing on and off at a constant rate is not comparable, because the description is of a rapidly increasing rate. And the rate increases so rapidly that the prescribed rate becomes incoherent even to the mind, as well as the senses. This is just an example of how easy it is to say something, or even describe a fictional scenario, which appears to make sense, but is actually incoherent — Metaphysician Undercover

In fact, one could simulate the on/off lamp so that at a certain rate you would see what appears to be a constant light. Flashing 0 and 1 cards would seem to be a zero with a one through it. These are models of the supertasks in a rough sense.

Admittedly not the real deals which are metaphysical fantasies.

The rules of this (language-game) still make no sense to me — Ludwig V

If one watches the lamp in a dark room, at some point it will appear to be on continuously.

I think that (1) is a tautology whereas no evidence has been offered in support of (2) — Michael

What is "evidence" in a metaphysical realm?

It is not built from anything. (0,1) is one object - a line — keystone

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.

It's length is 0.3 for all 3 paths depicted below because all 3 are homeomorphic. — keystone

Unfortunately, your approach is a muddled mixture of traditional ideas and speculative continua. I think you need to go back to the beginnings of your efforts and truly start with continua and develop an original approach to math in which points arise from these continua, avoiding the real line entirely at first. MU has written a similar notion about continua and points. Perhaps you can put some meat on the table.

Define a continuum as an abstract entity and not in terms of the real line. As a matter of fact, use another word for your creation. State the properties of the continuum, again not referencing the real line or numbers. This is a tall order. Metric spaces and topological functions are perhaps inappropriate in this regard. I don't know. You will be going into unexplored territory.

Please stop talking about the S-B algorithm. Both my colleague and I would rather not contemplate this thing. Leave the realm of real numbers at first.

Or, do your thing and persist until the thread dries up and vanishes. Good luck.

Rather, that interval description describes paths which can be transformed into each other via stretching and compressing, such as the following 3 paths: — keystone

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

you and fryfish are having a really tough time — keystone

:cool:

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


I don't understand why you would say this — keystone

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.

Do you think I'm using the term topological incorrectly? — keystone

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. However, I applaud your enthusiasm. I used to teach point set topology and metric spaces, so I am biased toward their traditional interpretations. 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.

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

The late George Simmons of Colorado College wrote a marvelous book many years ago, perhaps the finest introduction to modern analysis and topology ever written: Introduction to Topology and Modern Analysis. Slowly work your way through this book and you will see why we ask so many questions. And don't mix philosophy of mathematics with the real deal. Just an I idea.

Intuitionism is closely related to constructivism, the idea that mathematical objects only exist if there's an algorithm or procedure to construct them. Intuitionism is like constructivism with an extra bit of mysticism that I can never quite grasp. — fishfry

On those very rare occasions in which the subject arises I have felt the two to be more or less alike. But, here is what Wiki has to say:

Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity. Other forms of constructivism are not based on this viewpoint of intuition, and are compatible with an objective viewpoint on mathematics.

Ok. My eyes glaze a little more every time you mention the S-B tree — fishfry

Ditto.

the metric space is topological — keystone

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. A topology, on the other hand . . .

Formally, let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:
Both the empty set and X are elements of τ. Any union of elements of τ is an element of τ. Any intersection of finitely many elements of τ is an element of τ.
If τ is a topology on X, then the pair (X, τ) is called a topological space.

Wikipedia

The saddest words of tongue or pen: it might have been.

I have a strong affinity for the Stern-Brocot Algorithm — keystone

This is perhaps the second time this oddity from number theory has cropped up on this forum. I knew a tiny bit about it since it can involve elementary continued fraction theory. How did this become so important to you?

Is this a legitimate "path" ?. A linear arrangement of points and intervals.

Ordered pair 1: (3,3)
Ordered pair 2: (3,3.7)
Ordered pair 3: (3.7,3.7)
Ordered pair 4: (3.7,4.2)
Ordered pair 5: (4.2,4.2)
Ordered pair 6: (4.2,5.1)
Ordered pair 7: (5.1,5.1)

So, a "metric space" for this path consists of "points" (a,b) within this structure. For example, d((3,3.7),(4.2,5.1))=|(3+3.7)/2 - (4.2+5.1)/2| = 1.3. 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.

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?

d([2,3],[1,4])=0 ? [2,3] not equal to [1,4] — jgill

Returning to your example, (2,3) and (1,4) cannot both be elements of a continuous set so the set you are considering is not included in the enclosing set — keystone

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.

From the Bleachers: Does

$[\pi ,\pi ]\cup (\pi ,\pi +1)\cup [\pi +1,\pi +1]\cup (\pi +1,5)\cup [5,5]$

exist in your system? Or are you assuming rational numbers only?

Is

$[5,5]\cup (5,5+\varepsilon )\cup [5+\varepsilon ,5+\varepsilon ]\cup (5+\varepsilon ,5+2\varepsilon )\cup [5+2\varepsilon ,5+2\varepsilon ]\approx 5$

for small epsilon?

Step 5: Arrive at point 0. — keystone

?

Returning to your example, (2,3) and (1,4) cannot both be elements of a continuous set so the set you are considering is not included in the enclosing set — keystone

I don't know what you are talking about. You and @fishfry can sort all of this out if he is willing. Good luck.

"In a metric, the distance between two distinct points as always positive."


d([2,3],[1,4])=0 ? [2,3] not equal to [1,4].

According to some interpretations of irrational Infinity though, an infinite-sided die is not impossible, only supernatural, in the sense that you can imagine it, as an ideal concept --- e.g. a perfect multidimensional sphere --- but never reach-out and grasp it, in the real world. In what sense does that set of one "imaginary die" exist? :joke: — Gnomon

What is "irrational infinity"? Infinite sided die seems like a sphere in 3D.

Mathematics has a similar structure to certain conceptions of magic. It requires years of studying something entirely incorporeal, it seems to exist independent of the physical realm, it’s very powerful and has the ability to predict and influence the world around us, and it’s practitioners are BIZARRE — Gnomon

Does magic influence the world around us? Wow, what bizarre powers I wield! :cool: