I no longer have any idea what we are conversing about — fishfry

Perhaps the paper by Milad Niqui. In that case things may get technical and out of the realm of TPF.

viz,

What I'm proposing is that Niqui arithmetic is more fundamental than the SB-tree. — keystone

Does the axiom of identity mean Ludwig V = keystone ?

Just curious. :smile:

HALFTIME COMMENTARY: For those viewers who might wonder if this thread analogizes everyday discussions in mathematics among its various practitioners, let me assure it does not - at least from my antiquated perspective. Expertise in the "finer" points of logic is rarely required in traditional math, although,I admit, I've lost track of the enormous varieties of the subject over the passing years.

And perhaps I am wrong: checking ArXiv.org I see that in the past week there have been around 25 new logic papers submitted - about the same number as those in my area, complex analysis. And the axiom of extentionality on Wikipedia garners about 60 views per day - a healthy enough following.

Just passing thoughts when reflecting on the current discussion. Kudos to the three or four involved. :clap:

but all of these relations stem from Niqui arithmetic. — keystone

Unknown territory for me. No Wikipedia page I can find (among 26,000+), but perhaps it's under a different heading. You are full of surprises. Are you Niqui? :cool:

Please allow me to respond in the context of the SB-tree. Fractions correspond to nodes. — keystone

Each row of the tree involves medians, which require ratios of integers and arithmetic of these ratios. So, your top down approach always involves bottom up procedures. You cannot correlate rational numbers with nodes without using expressions like a/b. Instead of simplifying, you are complicating something you assume. Just my opinion.

I'm a computational fluid dynamics analyst — keystone

I've wondered about that. Thanks for illuminating.

Sounds like your professor just didn't like foundations — fishfry

You might think so from what I said, but he was young and pretty enthusiastic about teaching the subject. We had numerous worksheets that eventually led to the construction of the exponential function. So, his comment at the end came as a bit of a surprise. :cool:

I agree with this sentiment. Whether it's noncomputable reals, the halting problem, Gödel's incompleteness theorems, or the liar's paradox, they are all screaming at us that there is a potential in mathematics that cannot be fully actualized. But Classical mathematics aims to actualize everything — keystone

Not true. I published papers when I was active that never assumed infinity was actualized. Fryfish and I, sometime back, argued about the use of transfinite math in analysis, particularly functional analysis. He pointed to the use of Zorn's lemma or the axiom of choice as a required tool to prove the Hahn-Banach theorem, and I replied that that was true, but by altering the hypotheses slightly, they were not required. Hahn-Banach was my only very brief encounter with transfinites in my career. But then I sought interesting theory in classical analysis - a far cry from foundations. So, your statement is not entirely correct.

But I admire your tenacity.

Well I didn't become a mathematician! I got to grad school and my eyes glazed. — fishfry

In my first semester as a grad math student I was required to take a course called Introduction to Graduate Mathematics. It was basically naive set theory and at the end the professor said, "You should only continue in foundations if this course really appeals to you. How many of you find that to be the case?" I recall out of thirty students one or two hands went up. The rest of us wiped the glaze out of our eyes and went on into other areas of mathematics.

But this was 1962, back in the dark ages. And at a state university, not a top-notch school, like Harvard. Things have changed since then. When were you in a grad math program?

Three spatial dimensions, one time dimension. Spacetime. Don't try to make time into a fictitious part of space. But who really cares?

Should I be talking about a bijection between the non-dimensional points on a line and the set of integers? — Ludwig V

You can use that term, but only if you are more specific about "points on a line" and specify natural numbers or rational numbers corresponding to these points. That's it.

So if I had said "And when we describe the principle of distinction between non-dimensional points on a line, we find that our counting with natural numbers is endless", you would have agreed? — Ludwig V

No. If "the points on a line" correspond to integers or rational numbers, yes. Way too vague.

Real numbers are uncountable. — jgill

I see. Why can't I count with natural numbers? — Ludwig V

"principle of distinction between non-dimensional points on a line" does not specifically speak of natural numbers. Language play.

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?