Let's go back to the beginning.

a = a + 1

1) a is NOTHING — TheMadFool

Context means everything:

(1) a=a+1 (no finite solution. In complex analysis a would be the point at infinity - corresponding to an actual point at the north pole of the Riemann sphere)

(2) a=0
for k=1 to 100
a=a+1
next
(now what is a?)

I'm still mulling over "bad habits" in math. Sloppiness; jumping over points in a proof assuming they are true; assuming a hypothesis and then proving it; muddling a proof so badly other mathematicians can't verify it; etc. Using infinity or infinitesimals are the least of our concerns. :nerd:

and it is manifest to reason that nothing can actually extend for infinity, or be composed of an actual infinity of parts. — Bartricks

Manifest=clear or obvious to the eye or mind. And does the eye or mind unravel every detail of the universe? If there are features of reality we cannot fathom does that negate their existence? Your hotel example is not terribly convincing, IMO.

I'm not aware of any real applications along these lines, meaning predictive power. And keep in mind, economics is called the "dismal science" because it most frequently looks back in time and not forward.

However, I have dabbled in dynamics of this nature for many years, and only recently has one of my theorems been employed in a predictive sense in a social context. Even then the authors overlooked the mathematical setting for the theorem, assuming it would apply to social reality. Me, I don't know.

Probably, I should not have used the word "ergodic" and would not have done so were it not Lem's appellation. It's a bit of a fuzzy notion and means slightly different things in different contexts. The Ergodic Theorem is a sophisticated result in math and for two forms of averages to coincide requires the time dependent form involve a highly restrictive function. One that is reduced to virtual triviality in the aspect of complex analysis in which I explore.

The abstract mathematical playground was in use long before its application to the natural empirical one. — Mww

I'm not so sure of this, but prefer not to argue the point. When I taught college algebra courses some of the word problems went back very far in time. For instance, the problem of determining how long it would take for two workers to plow a certain field working together if it is known how long it would take for each individually. Cuneiform tablets 5K years ago.

I like Chalmer's definitions of strong and weak emergence, particularly the weak variety which he explains in terms of computer programs. Speculation, of course, is that consciousness or mind may be the only example of strong emergence. We discussed much of this on SuperTopo, a climbers forum, in the thread "What is Mind" - over 25K posts I recall.

No conclusions.

As I stated earlier, every time that "the infinite", "infinity", or the mitigated "infinitesimal", occurs in a mathematical application, this can be judged as a bad effect of the mathematical habit. — Metaphysician Undercover

OK, might want to jettison calculus then . . .and all the technology we use as a result.

(This thread demonstrates why one of my profs sixty years ago advised his class of grad math students not to take a course in mathematical logic.)

But it looks like everyone is having fun!

Human bodies age at different rates according to genetics and other factors. In high school at the age of 17 I could barely grow enough facial hair to shave, while a friend of the same age had a thick black beard. Nothing to do with advanced physics I assume. In a recent note I explained chronological age vs "imaginary" (i^2=-1) age (goofy mathematical speculation). I wonder if imaginary age would also be affected by gravitation and acceleration?

not a dimension. — Bartricks

And yet it can be dealt with as a dimension in mathematics and physics and predict observed results.

yet nothing that is infinitely divisible can exist in reality — Bartricks

Why not? Just curious.

$\int\limits_{0}^{1}{\phi (z,t)dt={{\lambda }^{2}}}$

Aha! The Wikipedia cut/paste on Mathtype with square vs angle "math"

It's true. These forums require some sort of balance between intellect and babble. I was a climber for many years and on a great climbers site called SuperTopo. ST had a large membership with ordinary folk and academics and a wealth of threads anywhere from "How do I fix my Ford axle" to "What is Mind." A critical number of academics (well-known physicist, neuroscientist, organizational professor, professional philosopher, etc.) were active on the latter. Then ST went dark and several new forums appeared, none of which have generated the scope and depth of ST. That's primarily the reason I have joined this site, which I find quite impressive - even a little intimidating with the lengthy and articulate posts I've seen. That's OK, I'm a retired math prof who lacks depth in philosophy.

Now this is equal to 1/2 + sum of the positive fractions - sum of negative fractions ok.

This gives 1/2 = sum of positive fractions - sum of negative fractions.
Now the sum 1/4 + 1/8 + 1/16 etc sums to a 1/2
I hope you would agree that no matter how I reorder this sum then it always = 1/2 ok — Umonsarmon

Just looking at this it appears you are rearranging a conditionally convergent series and expecting the same sum. If this is the case you can draw no logical conclusion.

https://en.wikipedia.org/wiki/Riemann_series_theorem

But I haven't followed the discussion so I may be misinterpreting your argument. If so, I apologize.

Please tell me how you post mathematical expressions on this forum. Thanks.

Can you explain the "and/or" part? — Wallows

Complex time, t=x+iy , is productive in certain settings in physics, and its metaphysical interpretations seem immaterial to those who simply want answers that agree with observed reality and are predictive. If one posits time in infinitesimal increments then both x and y are infinitesimals. I am not aware of any applications involving these tiny critters, but who knows what's around the metaphorical corner? Most mathematicians in analysis don't use the Non-standard Analysis approach although textbooks on calculus have been written in that venue.

Strong attractors are fixed points (FPs) of functions that, upon iteration, draw a given initial point towards the FP. But there are other FPs that either repel the iterations or are neutral. In the example f(z)=2(z-a)+a all points close to z=a are pushed away from that point. What is the historical analogue of these points? And the most interesting FPs are indifferent fixed points, where the iterative behavior around the point is difficult to analyze. f(z)=(3z-4)/(z-1) has an IFP at z=2. This case, it would appear might be the most prevalent in evolutions of societies. Both attracting and repelling.

And then there are Strange Attractors (no, not a Netflix series) which generally are sets or clusters of points in C (the complex plane) that iterations are attracted to. Enough.

As I understand it chaos theory basically claims that negligible differences in initial conditions lead to chaotic behavior — TheMadFool

I think the word "negligible" should not be used in the context of chaotic dynamical systems. Clearly if a slight variation at the beginning of an iteration process leads to bizarre behavior that variation, no matter how small, is not negligible. Just the contrary. Your notion of "cumulative contributory causation" is well put.

It's also called a phase space. — Pfhorrest

That I am familiar with as I play with dynamical systems in C that involve velocity, etc. But configuration space looked a little strange. Just me. Thanks for the explanation.

"In my conception of time . . " I like this. Well thought out IMO. Of course it depends upon the idea of "moment" - I am sure this has been chewed over in this forum sufficiently. In real and complex analysis in mathematics most practitioners work with the real number system as it's described by Cantor and others and time is simply a real variable. But set theorists and math logic people go off to abstractions quite readily. At the end of an introductory course in set theory in grad school in the early 1960s the professor said, "If you don't intend to spend your careers in mathematical logic, I recommend you never take a course in that subject." I heeded the advice.

Associated with ideas of Time, the concepts of infinitesimals, however, goes back in history. These are numbers in a sense that are both positive and smaller than any real number. Both Newton and Leibnitz had their ideas along these lines. And in the last century a mathematician named Robinson developed a mathematical model, making these strange little critters legitimate. It's called Non-Standard Analysis. One can teach a course in calculus from this perspective, avoiding the epsilons and deltas commonly employed. Time, it seems, can be Complex and/or infinitesimal. Go figure.:cool:

I'm not sure what a configuration space is, but your idea is intriguing. I had a friend, a physics prof, years ago who would take a cigar box with a lid to class and open it, revealing a set of neat stacks of coins. The he would shake the box, reopen it and say, "That's entropy!" When notions are a tad vague and it's difficult to put them in some sort of numerical context, I'm out of my depth.

Thanks for the links. I hadn't heard of some of those ideas. "Ergodic" is difficult to pin down, but the statistical idea in a dynamical system is not hard to see. Roughly it can be explained as taking averages of a phenomenon two or three different ways and getting more or less the same result. But it's not entirely clear cut.


This is a mathematical concept, so I'll give a simple example. Start with the complex plane (corresponds to the XY plane) where z=x+iy, x and y real numbers and i^2=-1. An attracting fixed point is a point in the plane (or complex number) that attracts for some function f(z). It doesn't exist as an attractor by itself. The simplest and most powerful attractor is f(z)=a, where a is some point in the plane. Then no matter what value you use for z, the function takes you instantly to a.

A slightly more complicated example is f(z)=.5(z-a)+a. Starting with a particular value of z, let z1=f(z), z2=f(z1), z3=f(z2), .. . . .Then this process of iteration, with let's say each step taking one second, moves the point z as close as we wish to a in a finite period of time. This limiting process is the heart of the branch of mathematics called analysis. And when analysis concerns complex numbers it is complex analysis, my specialty.

I pulled it up and looked at it again, and found a misprint. But don't worry, I don't require readers to find them as well! It's actually pretty trivial stuff compared to most of modern math.

"Math equations that express a behavior as a function of time, are misleading when the time is interpreted as a causative factor. The time of an event must be assigned after the event occurs, i.e. after awareness!"

Not quite sure of what you are getting at. Newtonian physics gives distance as a function of time, but this is purely descriptive and time is not interpreted as causative. Probably just me, so ignore. I haven't read through all the posts on the subject.

I haven't read that, but thanks for the suggestion. I'm going to start a new thread in the next day or so on theories of history that relate to these existing time topics. Very easy to understand, although I have created mathematical analogies which are of no consequence. :roll:

This might be of interest: Towards the end is a section on perception of time passage corresponding to advancing age. Nothing serious.

https://www.researchgate.net/publication/321554337_An_Elementary_Note_Playing_With_Complex_and_Distorted_Time_in_C

:cool:

Here is a mathematical note I wrote a while back that demonstrates how a math person might view the passage of time. It's meant to be amusing as well.

https://www.researchgate.net/publication/321554337_An_Elementary_Note_Playing_With_Complex_and_Distorted_Time_in_C

:cool:

Not yet. It will take awhile! Thanks.

Just joining the discussion. I'm a retired math professor interested in the nature of time.