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.

However, my primary focus is on the objects themselves, such as the Cartesian coordinate system. I believe this system needs a parts-from-whole, continuum-based reinterpretation, as the current understanding relies heavily on the notion of actual infinity. — keystone

Not sure what you mean by actual infinity. Are you speaking of infinity as a sort of number that can be arithmetically manipulated, or infinity as unboundedness? I have always used the concept of the latter rather than the former. But set theorists use both I think. Please provide an instance of "actual infinity" in the Euclidean plane. A projection onto a sphere is not allowed.

Sounds interesting. Life in the complex plane. By the way have you seen much of the modern graphing software that's so good at representing complex functions and Riemann surfaces and the like? Don't you wish you'd had that back in the day? I wish they'd had LaTeX, I always had bad writing. — fishfry

No I haven't. And I have little interest in Riemann surfaces. I have used MathType for years with Microsoft Word for writing purposes. For imaging, I have found BASIC is excellent for what I want to do, and have written many math programs. The image of the Quantum Bug on my info page was done with a simple program. Higher, more sophisticated languages seem to be directed toward what is popular in math, and what I do is virtually unknown.

Reminds me of Tim Gowers's distinction between problem solvers and theory builders. — fishfry

Somewhat similar, but not quite the same thing. I've never particularly enjoyed solving problems, but rather exploring where certain specific ideas in classical analysis lead. The celestial aviators can cruise the heavens taking us ground troops on ethereal adventures.

Thus the make up isn't at all close to the natural 50/50 divide, hence mathematics is male dominated. — ssu

Here's a personal anecdote that may be telling: My PhD class had several women. One dropped out for health reasons, and another was the top student, by far. Shortly after graduation she married a forest ranger and became a housewife.

In the international research clique I joined there were several women, but more men than women. A fairly close colleague, a European woman who had left behind a role as housewife, became the holder of an endowed chair at a major Scandinavian university.

wouldn't this mean then that mathematics is different from being just a social construct of our time? — ssu

Mathematics can be thought of as a structure or system or whatever, but certainly it is not "just a social construct". Sociologists are a little irritating.

Yet today, set theory is a basic part of the undergraduate math major curriculum — fishfry

Perhaps at the more prestigious schools, and maybe at less elite institutions as well. I think I checked on this for Harvard and found such a course, but the branch of the state university where I taught doesn't seem to offer such a course, although set theory is mentioned in a couple of conglomerate classes.

Math is very much a social process. — fishfry

Absolutely. Although many mathematicians work alone much of the time, with social contact allowing critiques by colleagues. It gets very social when one publishes a paper, with regard to referees.

Isomorphisms have everything to do with structuralism. An isomorphism says that two things are the same that are manifestly not the same. That's structuralism — fishfry

I might comment that whereas isomorphisms are very important in mathematics, not all practitioners are heavily involved with them. I have written many papers and notes without mentioning the word. However, the trend for the past how many years, maybe 70 or 100 or so, has been to rise above the nitty gritty of much of classical material and look for generalities that show how one subject in one area is "isomorphic" (use here in a more general sense) to another subject in another area. Or create generalities that when applied specifically to a lower level collection of results show them to be instances of one higher outcome.

I didn't go in this "modern" direction, and my late advisor would speculate that where he and I explored (ground troops, not aviators) would at some future time return to fashion. These days there is such a plethora of subject matter I'm not sure what is fashionable.

So, I am really really interested in whether or not mathematical objects exist in a mind-independent way. Would there be numbers even if we weren't here? I want to say yes. I think that numbers would be here — Pneumenon

Was it Einstein that said something to the affect that God gave us the natural numbers, all else of mathematics are mans' ? Or something like that. When I think of one of my theorems about the indifferent fixed point of an infinite composition of LFTs that converge to a parabolic LFT I never stop and wonder if all that preexists in a timeless realm beyond the thoughts on man. I do believe it exists in a potential way.

a societal phenomenon and a power play that a group of people (read men) do — ssu

In America women make up 25-30% of PhD students. 15-20% of math faculties. Not entirely men.

More like that the truths in mathematics are tautologies: a statement that is true by necessity or by virtue of its logical form. Wouldn't that description fit to mathematics? — ssu

To me this seems like a word game. Describing a theorem, if A then B, requires specific terms, going beyond its "logical form". The idea of categorizing math as a tautology contributes nothing to its practice.

when it's just what mathematicians declare doing. — ssu

Years ago the maid for a prominent mathematician was asked what her employer did. She replied, after some thought, "He scribbles on pieces of paper, grumbles, then wads the papers up and throws them in the trashcan."

perhaps the old idea of math being a tautology comes to mind — ssu

Statements devoid of content? (Frege)
I think not.

The only reason you don't want math to fully apply to reality is because you suspect a problem with infinite divisibility, right? — Gregory

I've never known a fellow mathematician who would have agreed with this. A mathematical philosopher perhaps. Let's see what @fishfry has to say. I always enjoy his commentaries.

Perhaps a very stupid question: why isn't Math referred simply to being a system? — ssu

Math itself is full of systems, like the system of the natural numbers, or the system of addition. So, in a sense, you might say it is a system itself, but a very complex system. @fishfry will give a much more sophisticated answer to your question.

If I had to guess where you are headed, I might say that taking a continuum (a line,say) as axiomatic somehow you are cutting it into a fine mesh using the S-B Tree. But how this has a bearing to elementary calculus is a bit foggy. Perhaps Farey sequences to partition Riemann integrals. Just guessing.

Why is it that the intro to calculus/analysis textbooks I’ve read never mention topology? — keystone

Over the years colleges have designed their curricula to suit the levels of abstract thought students can bring to the classroom. Calculus is taught in high schools at minimal levels of sophistication and during the first year or two at college with a bit more rigor. But even there the emphasis is on understanding applications of the subject. Most students in these classes are not math majors. Analytic geometry is part of this instruction. The student progresses to a higher level at the junior or senior years. At this point they are usually capable of the sort of abstract thinking that underlies calculus in an advanced calculus course, or an introduction to real analysis course. As for topology, I taught an introductory course at the junior/senior levels.

All of the above may not be true at a sophisticated university with high entrance standards.

And the above are generalizations. Individual students may be more or less proficient than I have indicated. For example, a number of years ago a young freshman at my school registered for my advanced calculus course - and received an A (he was also a talented climber and we became friends). Then I have had seniors who barely passed.

(I began college at Georgia Tech in 1954, and was fortunate to be one of the few incoming students who scored high enough on the entrance exam for me to start anywhere I wished, so I was placed in an experimental class of beginning calculus, immediately taught with epsilon-delta precision, with rigorous proofs. For the first half of the semester I had hardly a clue what was going on, while some of my classmates seemed to understand the material. Then halfway through all of it suddenly made sense. After that introduction, when I got into the regular curriculum for the next semester it seemed almost trivial)

I will be interested in what younger and more agile brains make of this. :chin:

How is that any less of a leap than starting with curves, which are inherently continuous? — keystone

And then straighten out a (continuous) curve and you have a continuum, which the OP argues does not exist. Perhaps you should start a thread entitled "The Continuum does exist".

I've never been able to see where it is you are going. Maybe it's just me, old and weary of days.