You dismiss my analysis just because I didn't utilize decimals? — Enrique

Not a decimal person myself. But equations do not determine reality.

If you look at the arrow at a particular instant it's not moving. How does it know what to do next in terms of direction and speed? — fishfry

The arrow may be momentarily stationary, but it has momentum.

Look up Shota Kojima and infinite compositions. A lot of stuff out there on fractals and simple iteration, but composing different functions endlessly not very much, especially if one considers complex functions that are not holomorphic, which I have done. They are far more interesting IMO.

When teaching almost thirty years I would publish a paper each year just to avoid collapsing intellectually, and all were readily accepted and published, but after I retired in 2000 I decided to do the research for me and anyone else who might be interested, avoiding the formalities of publishing. Hence a collection of informal notes on researchgate. It's amazing the audience one reaches there. Virtually every civilized nation has cropped up with reads of my stuff. But fewer than half read the entire note. Still, that can be 20-30 a week that do. And it's free, unlike most journals - and that irritates me since few journals pay the referees, at least when I was active. (in all fairness, institutions take up the slack with fewer teaching hours, etc.)

I consider the major journals corrupt in their financial practices. But that's just me. Pay no attention.

This notation was suggested by a Japanese mathematician. I was starting to use something else, but switched to his.

What's the meaning of L? — fishfry

Here's what's going on, with a simple example:

$\begin{align}& {{g}_{2}}\circ {{g}_{1}}(z)={{g}_{2}}\left( {{g}_{1}}(z) \right) \\ & {{g}_{1}}(z)=2z+1,\text{ }{{g}_{2}}(z)={{z}^{2}}-3\text{ }\Rightarrow \text{ }{{g}_{2}}\circ {{g}_{1}}(z)={{\left( 2z+1 \right)}^{2}}-3=4{{z}^{2}}+4z-2 \\ & \underset{k=1}{\overset{2}{\mathop{L}}}\,{{g}_{k}}(z)={{g}_{2}}\circ {{g}_{1}}(z)={{g}_{2}}\left( {{g}_{1}}(z) \right) \\ \end{align}$

$\underset{k=1}{\overset{\infty }{\mathop{L}}}\,{{g}_{k}}(z)=\underset{n\to \infty }{\mathop{\lim }}\,\underset{k=1}{\overset{n}{\mathop{L}}}\,{{g}_{k}}(z)$

The process arises using techniques from functional equations. For example,

$\begin{align}& f(z)={{e}^{z}}-1\text{ }\Rightarrow \text{ }f(2z)=f(z)\left( f(z)+2 \right) \\ & f(z)=z(z+2)\circ f(z/2)=\left( \frac{{{z}^{2}}}{4}+z \right)\circ 2z\circ f(z/2)=\left( \frac{{{z}^{2}}}{4}+z \right)\circ 2f(z/2) \\ & \text{ =}\left( \frac{{{z}^{2}}}{4}+z \right)\circ \left( 2{{z}^{2}}+4z \right)\circ f(z/4)=\left( \frac{{{z}^{2}}}{4}+z \right)\circ \left( \frac{{{z}^{2}}}{8}+z \right)\circ 4z\circ f(z/4)=\cdots \\ & \text{ =}\underset{k=1}{\overset{n}{\mathop R}}\,\left( \frac{{{z}^{2}}}{{{2}^{k+1}}}+z \right)\circ {{z}_{n}},\text{ }{{z}_{n}}\to z \\ \end{align}$

But here
$\underset{k=1}{\overset{3}{\mathop R}}\,{{f}_{k}}(z)={{f}_{1}}\circ {{f}_{2}}\circ {{f}_{3}}(z)$
etc. Going the other way involves rules for inverses of compositions.

This sort of thing appears in a general study of infinite compositions, a topic of practically no interest in the larger mathematics community. Abstractions and generalizations are more attractive.

So what does the L mean in your equation earlier? Not familiar to me. — fishfry

Give me a few moments. See the Gabriel's thread.

When you consider the predictive power of maths, the fact that through it you can discover things about reality that you otherwise could never know - how is that reconcilable with the idea that it's a 'useful fiction'? — Wayfarer

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences"

My former and late father-in-law's good friend Eugene Wigner raised this question years ago in a famous paper. I agree - I don't perceive it as a useful fiction.

A quarter counterclockwise turn in the plane. That's the simple meaning — fishfry

That's what happens when multiplying a+bi by i.

I play in the complex plane all the time, and I have always visualized figures and imagery and motion. Even created what might be considered art in the process. :nerd:

Please return to metaphysics. The real deal is a challenge.

Would the fully corroborated evidence of supernatural events necessarily lead one to believe in God, or at least in the supernatural;? — Jacob-B

I'll let you know if it happens. :roll:

The only group that interests me is SU(2). :chin:

↪jgill

That's pretty rad! — norm

One of my inventions (probably! You can't tell in mathematics.) :smile:

hat's been the case in my experience too. For applications, though, the dual numbers are actually important today. Some of the autodifferentiation powering machine learning use the forward method, employing dual numbers to great effect (and even hyperdual numbers.) This allows one to compute f(x) and grad(f(x)) at the same time at low cost. — norm

See, I learned something from your post! Thanks, norm.

but felt that she entirely missed the meaning of complex numbers — fishfry

What would you say the meaning is? Just curious. :cool:

Pi only encodes a finite amount of information — fishfry

Bet you haven't seen this:


$\begin{align}& ArcTan(z)=\underset{k=1}{\overset{\infty }{\mathop L}}\,\frac{2z}{1+\sqrt{1+\tfrac{1}{{{4}^{k}}}{{z}^{2}}}},\text{ }\underset{k=1}{\overset{n}{\mathop L}}\,{{g}_{k}}(z)={{g}_{n}}\circ {{g}_{n-1}}\circ \cdots \circ {{g}_{1}}(z),\text{ } \\ & \pi =4ArcTan\left( 1 \right) \\ \end{align}$

so as a basic dude, the more my body breaks down and my reasons for living become less accessible and then clouded by annoyances like poor sleep, physical limitations and ailments, the balance sheet starts to sway towards I'd rather go out on a high.. — dazed

I've seen this play out in the sport of rock climbing. Sometimes those afflicted follow through. Becoming obsessed with physical performance and not capable of gradually adjusting one's attitude over time is a recipe for termination.

There are times when I think back on my life in the sport, reflect on my performances when young, but I see all that from a different perspective now, and it seems distant and pleasurable but not all that significant. Young, energetic climbers frequently anticipate themselves in old age retaining that fascination that compels them. But it ain't necessarily so. Give yourself a little slack.

I have not looked into SIA — Ryan O'Connor

Weird stuff, IMHO. Low priority in the world of mathematics.

. . . that one person should not both write the textbook, and teach from it, and test the students on whether they know their stuff. But, arguing for or against that is going to be a topic of one of those later threads — Pfhorrest

Depends on that person. Is it a good text? Does the prof profit from its sale to his students? Is the prof fair? Not an easy yes or no here. But carry on.

I see that you're a retired math professor so I'm especially keen to hear your feedback, especially if you see a flaw in my argument. — Ryan O'Connor

What makes me uneasy is using a concept like topological equivalence and then discussing slopes and derivatives, which, as you know, do not carry over in that way. As for your continuum ideas, almost twenty years ago Peter Lynds wrote a paper appearing in Foundations of Physics Letters that postulated time having no instants and instead being composed of intervals. Something like Bergson's notions from a century ago. Here it is.

A separation of research, teaching, and testing branches of education
is therefore analogous to a separation of legislative, executive, and judicial branches of government — Pfhorrest

Questionable analogy. In academia one person may have responsibilities in all three, in fact usually does. Not so in government.

Old age is not for everyone. And you never know what's going to go wrong for you or right for you in life. At age 84, my only advice is keep an open mind and stay physically and mentally active as long as possible. I recently returned to my fitness gym, after getting both vaccine shots, and it was a pleasurable experience. But then I have been athletic since the age of 17 and have kept it up. Don't boo hoo when you begin to decline and can't kick the soccer ball like you used to. If you keep at it you may be pleasantly surprised in old age, for your perspective will change. I too have scoliosis and severe arthritis. But four ibuprofen tablets and an acetaminophen taken an hour before exercise can work wonders.

If you make it there, that is. It's a crap shoot. :cool:

You have to remember that these plots are topological so even though it looks linear in the first image I could have just as well drawn it with squiggles. — Ryan O'Connor

I know what you mean here, but reading it makes me uneasy.

LoL. I'm sure the philosophy students in the room were aghast. — Ryan O'Connor

That would be the empty set, Ryan. We were all math majors and the course was taught in the math department. :smile:

You don't know anything about it.

Yet you have persistent critiques of it.

How do you do it? — GrandMinnow

With a certain aplomb. I admire his spirit while avoiding his critiques. :cool:

To me it is concerning that the foundations are so disconnected from the applications — Ryan O'Connor

It's unfortunate the word "foundations" is used in mathematics. Foundational set theory (that intersects with philosophy significantly) and all that revolves around it is a marvelous intellectual area, but a great deal of mathematics flows unimpeded by its pronouncements. Do most seasoned mathematicians worry about the existence of irrational or non-computable numbers? Or non-measurable sets? Not likely.

I've mentioned my own experience before: as beginning grad students sixty years ago we were required to take an introductory course in set theory (foundations). Afterwards the prof - a young energetic fellow - made the statement, "Unless you are really fascinated with the material in this course, I recommend you never take another foundations of mathematics class." I wasn't and I didn't. But I ultimately went into classical complex analysis. Had I gone into a more abstract realm of math I might have needed it.

Could this be an indication that further foundational work is required? — Ryan O'Connor

It depends on which mathematician you ask. Let's hope not. It makes little sense to ask anyone outside the profession.

I'm no expert, but my impression was the math moved toward being totally mechanized, totally formal, totally computer-checkable — norm

And I'm sure Norm would agree, that movement would drive most mathematicians out of the profession. It can't be emphasized enough how much mathematics depends on intuition, imagination, inventiveness, and a spirit of exploration. Devising and proving theorems is an art form.

How can ∞ represent a point? — Metaphysician Undercover

The north pole of the Riemann sphere. But carry on.

Unfortunately, there are math teachers who criticize a student who solves a problem in a novel way.

Before I retired twenty years ago there were discussions in the math department about race and subject matter. We agreed that whenever a topic came up in which a minority had made a significant contribution we would mention that fact. But twisting the subject matter around into some sort of ethnomathematics was not considered.

Hi ! I'll private-message you about that. — norm

Oh boy they're gonna gossip about the rest of us! — fishfry

Not so, my friend. Norm is mathematically authentic, as are you and fdrake, and I will probably learn something from his posts, as I have from the two of you.

I'm of two minds about revealing anything about the expertise of math people on this forum. I realize the knowledge may intimidate some others and dissuade them from contributing their ideas. Or it might have the opposite effect of encouraging attacks on academia. Oh well, not a big deal.

After years of formal study (proof writing), I still would argue that intuition is primary and that math is a language — norm

What's your background, Norm? Another math proof on the forum? If so, welcome. :smile:

Connected with the ideas of "complex" and "complicated" in proofs is the word "elementary", which, when used by a mathematician, usually does not signify "simple" or "easy", but refers to the level of a mathematical argument, specifically that that argument involves only basic concepts in the specific subject area. Some elementary proofs are very complicated. Others, not so much.

"Non-elementary" usually means the argument involves more advanced concepts and results, and may actually be fairly straightforward and uncomplicated - or not. :cool:

In your professional research, did you have the feeling that you were investigating aspects of truths outside yourself that you were trying to find out about? — fishfry

I'm more or less continuously involved in light, unimportant research these days. As an example, I've defined a variation of linear fractional transformations in which the normally constant fixed points are themselves functions of the underlying variable. Then I search for criteria that produce convergence upon iteration. I definitely look for "truths" outside myself, and it's this exploration that's fascinating. Knowing that such truths depend ultimately on axiomatic structures has no bearing on my minor investigations, and the very thought of moving around symbols in a formal game is anathema.

I was once a rock climber and it was the delight of exploration that was intensely compelling. :cool:

One can take the viewpoint that symbolic math is a human endeavor; and that a thing has mathematical existence whenever a preponderance of mathematicians agree that it does. — fishfry

It's true, math is a social activity, but I bet a lot of it exists without a preponderance of mathematicians even being aware of it, much less agreeing it exists. A single practitioner can have an original idea, one that assumes mathematical existence at that moment.

Wiki: "Existence is the ability of an entity to interact with physical or mental reality."

On the other hand, it does take that kind of recognition to establish the importance of a mathematical idea in the mathematical community.

Once again, a topic of limited interest to practitioners. :smile:

I'm lost on why the problem is such a big deal — Darkneos

It's not. Seems like common sense.

Is this thread intended to find something akin to the p vs np problem (related to complexity in computing)? — emancipate

Good question. I think there is confusion regarding complex in a technical sense and complicated in a more general sense. I know of no "scale" describing levels of complication in a mathematical proof, although when one conjectures a proof path it may be apparent that low or high levels of complication might ensue. And maybe there are practitioners who speculate "Why this looks to be about an eight on the scale of complexity!"

Again, a discussion of mathematics that mathematicians pay little attention to. But fun for amateur philosophers. :smile:

I have no use for category theory, but it does attempt to generalize areas of math that have similarities. I fear your knowledge of mathematics is so minimal that we are not getting anywhere here. Since I have been a mathematician (over fifty years) the subject has grown so dramatically and is so complex now I understand little of it myself.

A lot of people are baffled by this topic for some reason, where I see this in a sense very apparent in mathematics nowadays to talk about the sort of lack of categorization, that can even at all begin or take place! — Shawn

Category Theory

(and perhaps you would even agree that the greatest value which y never reaches is 0) — Ryan O'Connor

There is no such value. Might it be 100? Or 1000?. y=1/x, x>1 has a greatest lower bound, 0, which it never reaches. Your attempts at the philosophy of mathematics may never bear fruit if you consider this a cogent statement.

Try this one: https://www.reddit.com/r/PhilosophyofMath/

Of course, Stack Exchange in mathematics is sort of a forum, and the participants include very knowledgeable professionals, although not so many mathematical philosophers. Very few on TPF have any graduate math experience. Exceptions include fishfry and fdrake, both of whom are up to speed on modern math topics including foundations, which overlaps philosophy. I retired over twenty years ago and am not as reliable. :sad:

Sure, but this is quite a conundrum towards the notion that everything in mathematics should or is determinate. — Shawn

The question of whether mathematics is created or discovered has been around for a very long time. This, perhaps, is not precisely what you are asking, but close. An act of imagination may lead to interesting results that are determinate, but the original thought might never arise. Then all that follows will not exist. An original thought is required to set the train in motion. Is that thought determinate?

Let's suppose a math guy has that original thought, and a process unfolds in a determinate manner, then someone else has an original idea that influences the direction of that process. The second guy is part of the process, but his thought may not be determinate.

Most mathematicians pay little to no attention to issues like this. I never did. :cool:

Is this the result of one of your mathematical formulas ? — Pop

Yes. I've developed the elementary theory of infinite compositions of complex functions in the complex plane - a subject of very, very low importance in mathematics. It's a kind of dynamic systems. The image arises from a BASIC program I wrote some time ago.

David Chalmers, a philosopher and social scientist, wrote a nice article on weak and strong emergence a few years ago.