For example, when I write "n∈ N", I don’t mean that n is an element of the actual infinite set of natural numbers. Rather, I mean that, it is a natural number according to the SB tree (details omitted). — keystone

Good luck with that. Probably of more interest to CS people.

The function x(n) — keystone

A sequence of rationals I assume. If you had two functions on Q then a suitable metric would be the supremum.

An actual curve is an indivisible, one-dimensional object with length but no width or depth. It extends continuously between two actual points but excludes the endpoints. — keystone

How do you define "continuous"? Are you sure it is indivisible?

Sorry, but your list of definitions is mind-numbing. Your top down is becoming way more complicated that bottom up, IMO. And the irrational numbers have yet to appear.

Your best bet would be to find a mathematician willing to deal with your arguments and pay him/her a fee to do so.

Too bad that the basics of category theory aren't taught in school. But then again, the educational system doesn't care much about the philosophy of mathematics or the foundations of mathematics. — ssu

I assume you are speaking of "school" as in "university". It is taught at some elite institutions and some not so high on the scale as well. If you mean elementary or high school the very thought is laughable.

I remember being in a discussion with several mathematicians fifty years ago when category theory came up. The consensus was that it "doesn't do anything". It may not prove any theorems other than those about itself. With set theory one can start from scratch and build a logical system, but CT requires knowledge of the various facets of the category. It's mostly an outgrowth of abstract algebra.

Computer science may be another matter.

I agree that calculus can work quite well with the concepts of unboundedness and potential infinity, but 'actual' infinities are implicitly assumed throughout the standard treatment. — keystone

I was speaking of ordinal numbers beyond the naturals. Our definitions of "actual" infinities differ. No big deal.

As I have said before, I have written many papers and notes without ever becoming transfinite. — jgill
Have you written calculus papers/notes that are not (implicitly or explicitly) built upon infinite sets like R? — keystone

Of course I have used R, but not a transfinite number. Unless I occasionally use the "point at infinity" in complex analysis. Which I rarely do since it is a projection upon the Riemann sphere. It might appear that you are moving in the direction of Discrete calculus. But go ahead. I am curious.

But that program (even in an infinite world*) cannot actually output a set with a cardinality of ℵ0. Potential is important and I feel like it's been forgotten in our Platonist world. — keystone

Elementary calculus does not require "actual" infinities. It gets along quite well with unboundedness, or what you might call potential infinity. As I have said before, I have written many papers and notes without ever becoming transfinite.

I think the whole idea grossly overestimates people's interest in having an opinion on every political subject all the time — Benkei

:up:

By moving the focus from the destination to the journey the need for actual infinity vanishes. — keystone

By "actual infinity" I suppose you mean a kind of number that can be manipulated by arithmetic processes.

In my recent posts, I have been establishing that real numbers instead describe potential k-curves, which can be thought of as yet to be constructed k-curves which when constructed have the potential to be arbitrarily small (but always retain a non-zero length). — keystone

This is either very deep - or shallow gobblygook.

So far I'm not seeing anything beyond a line segment between two points that converge to one. From a continuum to a point. Why should one care about this?

Reading the masters (Kant, etc) sometimes is confusing since they might not express themselves in the best ways. Read commentaries of them if you encounter problems.

Of course it is true you're not obliged to find that interesting. — Wayfarer

Unfortunately, I will be long gone when they sort this out. You may be around :cool:

The challenge for scientific realism is the concept of superposition — Wayfarer

Wiki:

Quantum superposition is a fundamental principle of quantum mechanics that states that linear combinations of solutions to the Schrödinger equation are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a linear differential equation in time and position. More precisely, the state of a system is given by a linear combination of all the eigenfunctions of the Schrödinger equation governing that system.

I'm not a physicist, merely an old mathematician who prefers this definition. So I don't see the magic.

He ends up advocating (maybe just "showing the benefits of" is a better term) of an approach grounded in category theory. — Count Timothy von Icarus

Category theory is beyond my pay grade. It's quite popular (the Wiki page has over 600 views per day - people want to know what it's all about). So far it seems not to have included classical complex analysis. When I look at diagrams what is familiar is composition of functions, of which I am fairly proficient.

I see a mistake in your last figure, typo probably. And I assume -1/0 (meaningless) designates negative infinity, however you define that. I see nothing of interest so far.

From Wiki:

Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system.

I have mentioned before that fundamentally the S. equation reduces to a simple calculus concept: the instantaneous change in a thing is proportional to the amount of that thing at that time. Think of continuous compounding of interest in the financial realm. Nothing magical.

Move on to 2.

Question from the stands: are individual numbers considered objects? — Wayfarer

Sure.

What would be the mathematical object behind/described by the "well ordering theorem" — ssu

This brings up an interesting question: If two things are equivalent, A<->B, does that mean they represent the same math object? In the example I gave there are two ways of representing the object, Mobius transformation, analytical or matrix, without a non-trivial equivalence argument. The WOT and AofC require a logical argument to verify. Representation theory, in general, includes equivalences.

All of this gets technical and may not be suitable for TPF. Also, set theory quickly moves beyond my levels of proficiency. More appropriate for and

I think the one issue in mathematics is that defining a "mathematical object" can be difficult if there are equivalencies, multiple ways of representation of the "object". There's a whole field of mathematics just looking at these similarities, category theory. — ssu

Yes, the similarities don't define the object, however. Is an "object" its' representation picked at random? Or, is there a more metaphysical meaning of the one object having representations? Is there an Object Theory? Just thinking of a way this thread might proceed.

1. should be interesting. You have density, but then continuity is next. Intuitionism math perhaps.
I thought you were defining these lines as continuous. Fundamental objects.

I don't think you will get a reaction from anyone but me until you produce a plan moving forward from your images of edges, vertices and surfaces. What is your goal and how do you plan to proceed? So far it appears everything you have given is uninteresting from a math perspective.

Now that you've moved into graph theory I suppose I see some sort of a way to move forward by taking a lattice graph over an area and allowing the number of vertices and edges to increase without bound leading to a countable number of points in the area. But this would be inadequate regarding the reals. But you might be able to push into the irrationals some way. Speculation. You need to actually start moving beyond your pictures. I am not familiar with graph theory, but perhaps @fishfry and @Tones are. And some on the forum who are or were CS professionals.

You have done your imagery very well. I will wait and see what comes next.

Your second figure is bewildering. Maybe go back to 1D and explain the real numbers as you see them. Expressions like k-vertex instead of point are confusing.

What's wrong with a democratic nation deciding how much immigration it wants to let in? — Philosophim

I recall that Sweden allowed large numbers in, then several years later changed its mind.

What can the head of a mathematics department say when accused that there are too few if any women or minorities represented in the staff? Stop hiring your male buddies and follow the implemented DEI rules! — ssu

One more slight digression from the original topic. I have been in this position. Rules of Affirmative Action applied and the dean asked for the top three candidates. There was a woman, but no minorities. The dean then placed a minority in with our recommendations. When the time came to decide to make an offer the dean picked the minority. It did not work out well in the long run.

I would like to go back to the actual topic of this thread. — ssu

May I suggest focusing on math objects having several representations (like my example four days ago) and speculating on what the object "really" is or looks like. Or where it lies in a metaphysical sense. To say math is one object is absurd IMO.

Is the "parole" plan in 3. above a reasonable policy? I think not. — jgill

Can I ask why you think this and what you think should be done differently? — Samlw

If the government insists on flying in "inadmissible" immigrants, then they should be carefully chosen to
benefit the nation in some manner. Doctors and nurses, scientists, engineers, might well be encouraged to apply. That does not appear to be the case.

I appreciate the graphs you have drawn. You have 2D surfaces that are defined by interiors of edge figures. The surfaces, edges and vertices seem to constitute fundamental objects. Time for a few axioms.

How all this simplifies normal calculus is questionable.

Framing the conversation in terms of preserving the state or nation.

1. "A failed state is a state that has lost its ability to fulfill fundamental security and development functions, lacking effective control over its territory and borders." (Wikipedia)

2. Immigrants to the USA should "Surge the border"

3. Flying in inadmissible aliens.

And here we are three years later.

In the context of humanitarian issues, there are limits on resources that dictate that the nation establish rules and laws of immigration so that those who follow those rules be given an opportunity to present their cases. Is the "parole" plan in 3. above a reasonable policy? I think not.

The only objects in these graphs that can be 'cut' are the edges — keystone

Why resort to graph theory and call a simple line an edge? Is this an effort to enhance an almost trivial concept of line and point? Again, why not go to 2D? Maybe your ideas will make more sense in that context.

I find myself defending a hill that I'm definitely not willing to die on. If it made a difference to anyone, I'd gladly deny, renounce, disavow, and forswear my earlier claim that "Math is what mathematicians do." It was a throwaway line, a triviality, a piece of fluff — fishfry

Exactly what it was intended to be. How about my previous statement about a mathematician is one who scribbles on paper, curses, then wads the paper up and throws it into the trash. Nobody seemed offended by that. Folksy I guess.

Yes I know these people. How bad has it gotten when Scientific American, of all outlets, publishes Modern Mathematics Confronts Its White, Patriarchal Past. — fishfry

That is a pretty bad article. It paints a picture of an entire profession based on a few incidents.

Does make you wonder. If this is the real Grinin he may have signed up, then upon reading some of the threads decided to move on. His Wikipedia page gets 4 views per day.

..the first step is to accept that k-curves are indivisible. k-vertices in these graphs cannot be partitioned — keystone

OK, you have a line that is indivisible. But it has k-vertices that "cannot be partitioned". Can a vertex be partitioned? Like saying a point can be partitioned. Concise language is very important in math, not so much so in philosophy.

Although I don't subscribe to mathematics being one object, when one looks at specific areas of the subject one can say that one object prevails, and there might be representations of that object that appear disparate. I have dabbled with Möbius transformations in the complex plane for years and still do not understand all the nuances of the subject. They form a group under composition, a procedure described below. Here are three representations: (a) geometric - rotations etc. of the Riemann sphere (b) the analytic form (c) the matrix form.
Question: what is the one object being represented?

Sorry, it looks like you are taking a line segment and dividing it into two smaller segments. Then comparing. If you think there is something significant here you had better present a philosophical argument supporting it. There is virtually no mathematics so far. Except for interpreting vertices and edges from graph theory, which only complicates a vacuous scenario.

A good idea often begins with some handwaving as it's being formed, but through refinement and rigorous thought, it can mature into a precise and well-supported explanation. — keystone

True. I hope there is something of interest coming from this discussion. But we've been through metric spaces and topology and now are venturing into graph theory with some sort of hope of connecting that with calculus. I have my doubts, but am trying to keep an open mind.

You seem not to understand how the mathematical method of handwaving works — TonesInDeepFreeze

True enough. But I keep hoping there is something profound I am missing in all this. :roll:

BaDbE can be unified into BaC because DbE can be treated as a whole that captures all the structure of C. In other words — keystone

What structure? A line segment has structure? One line segment has the same "structure" as another? You must see something there that eludes me. But I am old and a lot gets past me.

I have two continua described by Graph 1 and Graph 2, respectively. — keystone

I seem to lack your insight in this example. It appears you simply take a real line and divide it into several line segments by inserting "k-vertices". You are assuming the existence of these points on the line. Indeed, the line segments are continua. In the example A-B-C what if instead you used the square root of two as the dividing k-vertex? You seem to be assuming the common notion of the real line. Maybe if you extend your ideas into 2D they will seem to be more than trivia? As a constructivist, what are you constructing other than a few line segments?

Why don't you jump right into calculus concepts in 2D instead of dwelling on the trivial, incredibly boring 1D case. Either that or make the 1D case something interesting, to capture the attention of a reader. Just a suggestion.

But you seem to be using visualization software in your images. They didn't have that stuff when I was in school — fishfry

It's just fairly simple BASIC programming that I enjoy creating. I tried Pascal, Fortran, Mathematica, C++, and one or two others, but by the mid 1990s I returned to BASIC. I use Liberty Basic now. Microsoft's Visual 6 was excellent, but one morning I turned on my computer and it was gone. Instead Microsoft tried to get me to use some new programming language you had to employ at their servers. I've never quite forgiven them. I've written 3D programs, but haven't been happy with them. I'm a 2D guy.

Oh. Interesting — fishfry

Retired for 24 years. Lots of things slip by. Hard enough to persist along the lines of mathematical thought I know about.

If this is true, then we can assume that there is quantum entanglement between the brains of related individuals in nature — Linkey

A big jump in credulity. But OK for the Lounge I suppose.

I wonder if Calculus on Finite Weighted Graphs is the direction you are headed? This is a topic even less popular than mine, with a scant 8 views per day on Wikipedia.

The article mentions several applications connected to data processing and CS. But calculus approached this way is obscure and unlikely to replace elementary calculus as it it is currently taught. Just my opinion. You are probably not pursuing this line of thought.

The biggest hurdle for an intelligent but amateur mathematician is rediscovering a result established some time ago. Hence, my words of caution.