Interesting to see if anyone who was actually a grunt posts here. I was in the USAF but far from being a grunt, so I can't comment from experience. Helloooo . . . anyone out there in the aether?

So metaphysically, this is quite complex. Some history of constraints has to drive the system to the point that it is in a state of poised perfection — apokrisis

You do write the most interesting posts. This is related to catastrophe theory as well. A frightened guard dog teeters on the edge of attack or retreat, like a pencil balanced on its point, or a married couple at the edges of each other's nerves, the slightest provocation and a serious collapse of wave functions.

As with cellular automata, the mathematician sees a pattern emerge from the algorithm and finds it striking because it is a pretty pattern. Maybe even a suggestive pattern. Possibly even what looks like a pair of butterfly wings that might seem to stand as a good model of bistability in a natural system. It all gets very exciting - a la Wolfram — apokrisis

He had such high hopes with what he considered a new science. Like almost everyone who attempted to read his massive book, I gave up after a few hundred pages.

But it is then so easy to skip over the many steps needed to start using these sparkling new toys as actual scientific models — apokrisis

I have always enjoyed mathematics as a completely abstract playground, never having illusions, nor wishes, that my modest research would have applications. A method of accelerating convergence of certain function expansions as continued fractions years ago, that may have had use in computations in QM. And more recently, a surprising application of convergence of infinite compositions in a paper on decision making within groups. But all the rest unicorns in a velvet sky.

I've wondered whether fixed points (attracting, repelling, indifferent) have any metaphysical properties. Stanislaw lem's ergodic theory of history presents a counterpoint to the butterfly effect in Chaos theory: certain social movements are so strong that minor fluctuations have little to no effect on large scale outcomes.

The Lorentz strange attractor caused excitement as a model for that reason. — apokrisis

Most of those systems iterate a single complex function. My approach has been infinite compositions of differing functions, producing imagery like the one I use for my icon on TPF.

It seems to me what we have discussed here regarding math and chaos is that along one line of thought math describes chaos and along another line of thought it creates chaos.

You have mentioned symmetry breaking several times in posts. I know practically nothing of it, but it seems to somewhat parallel the fundamental notion of chaos theory, sensitive dependence on initial conditions. How do you perceive it? Does it resonate in metaphysics?

What is the general definition of chaos? — apokrisis

Mathematical models then track the growth of wildness as constraints are systematically removed — apokrisis

OK. Your presentation focuses on real world chaotic behaviors that can be approached probabilistically or statistically, not with a more precise iterative tool. I am not familiar with that approach. Thanks for the link.

The theory of chaos in a purely mathematical setting is somewhat more exact:

Chaos theory is an interdisciplinary scientific theory and branch of mathematics focused on underlying patterns and deterministic laws, of dynamical systems, that are highly sensitive to initial conditions, that were once thought to have completely random states of disorder and irregularities.

wiki

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space.

wiki

chaos must have structure as free possibility becomes its own system of constraints. — apokrisis

Math aside, this part looks interesting. I'd like to see it expanded upon.

↪jgill
https://onlinelibrary.wiley.com/doi/10.1111/j.1420-9101.2009.01775.x — apokrisis

I don't see how you conclude

Because maths tells us that chaos must have structure . . . — apokrisis

The lengthy and frankly overwhelming article is about biology and probability as far as I can tell without reading it carefully - If you can find "chaos" in there please point it out. The author alludes to chaotic behavior when he speaks of sensitive dependence on initial conditions in the terminology of the science, but I don't find anywhere, glancing over the paper, a reference to mathematical chaotic behavior.

One speaks of chaos mathematically in the contexts of certain dynamical systems, those having SDIC properties. Frequently these involve the complex plane, and I am familiar with this environment.

Because maths tells us that chaos must have structure . . . — apokrisis

No it doesn't. :roll:

But the Lorentz factor is always positive, so how can that be? :chin:

Nevertheless, you're on an amazing roll. :clap:

but we may be able to determine a tendency or process, and speculate from there. — Ciceronianus

Increasing entropy, I suppose.

It's about not chucking in technical terms that lack technique - tools that don't work - or don't exist. — Cuthbert

With my scant knowledge of philosophy (or metaphysics) I can't tell whether Bob is out on the cutting edge or is being cleverly deceptive, ala Sokal affair. Has he taken simple ideas of generational derivations and convoluted them on purpose, or am I just failing to appreciate his insight?

Whatever. He has stirred up a conversation. :cool:

Light moves slower than your minivan

:rofl:

Which one? When?

I thought it could be fun to post an excerpt from a paper I'm publishing — Enrique

In a journal?

How do we know that light can only travel at exactly 1 speed? — TiredThinker

It doesn't. It depends on the medium through which it travels. It can be made to creep very slowly indeed in the laboratory.

Now, imagine that contextual relation continued forever. — Bob Ross

And complicated by the fact that different contexts could mean different superordinate circles.

Circling back to "grid parity" these added costs need to be included in the cost of solar / wind which, when done honestly, demonstrate the immense scale of our predicament. Energy storage on a large scale is a very large infrastructure project that requires decades to build even if the technologies required were cost-competitive with subsidised fossil. Add in some disruption to the global system as resource competition heats up, a few material bottle necks ... and ... its gone — boethius

:up: Says it in a nutshell. Things Greta Thunberg didn't think of when she became aware of the problem at the age of eight.

I am happy you have finally found a number to use in place of infinity. You could show us how that works with the Lorentz factor. :cool:

This has to be exciting news for theoretical physics!

I want to emphasize that the above example is incredibly over-simplified — Bob Ross

Thanks. It's a matter of chains of encompassing superordinate categories with possible overlaps, I suppose. If so, I can see where you deal with infinities.

superordinate category

Thanks.

Regarding {0,1} graph {x = 0.1 + 0.01 + 0.001 + 0.0001 + 0…nth.1 < 1}.

Above is my attempt to show a counting series from 0 towards 1 for values of x that graphs as an asymptotic progression — ucarr

An asymptotic relationship requires a function g(x) where Lim f(x)/g(x) =1 as x becomes infinite. Or something similar.

The “gravitational” force of an infinite volume curves its own graphic progression to such an extreme it never achieves “escape velocity” to the next whole integer. — ucarr

A reference for this would help an awful lot. :roll:

Banned Jackson for low quality posts and continually showing no interest in discussion. — Xtrix

Did you warn Jackson at least once, or did you simply ban him?

Lets look at the true infinite as all possible numbers. Within that infinite, you can have bounded infinites. For example, all numbers that end on the tenth's place is a bounded infinite within the true infinite. A bind is a limit. To speak of an unbounded infinite, is to speak to something without limits. — Philosophim

This seems arguably non-mathematical. "all numbers that end . . ." is still unbounded in the traditional sense. If, on the other hand, you mean positive numbers less than or equal to 1, this set contains 10 elements. I think the real problem here is in not using existing math concepts for infinities, boundedness and unboundedness The notions of things being derived from preceding things and participating in further derivations, some of those preceding things being essential and others not, is pretty simple.

I admit, I did not read the entire essay. :sad:

Note: had to correct myself. Easy to put foot in mouth.

You got me wondering if anyone does this line of research anymore. My conclusion, rarely. Here are a couple of papers, the second being more a survey.

Summing Series

Summing Series

Is mathematics inconsistent? — Agent Smith

The expression on the right is a way of summing a divergent series. (assuming we know what you mean with the notation)

we've struck gold, oui monsieur? — Agent Smith

Fool's gold.

You seem fascinated by this anomaly. It's a kind of summation result for series that don't converge in the mathematically acceptable manner. Here's a link that should keep you occupied until bedtime:

1 + 2 + 3 + ...

The slightest somethingness is already a pointer towards the two ultimate anchoring bounds of nothingness and everythingness - the two distant limits that show the somethingness to be what it is in terms of what it is not ... which is either a nothing or an everything — apokrisis

Profound or not? Something or nothing? :roll:

It all breaks down to computing the path integral of the symmetry of quantum foam, so that a vacuum results. :chin: — jgill

I appreciate you playing along with my bullshit. :smile:

There is an example in the essay which I think explains it well: did you find it to be confusing as well? — Bob Ross

Sorry, Bob. See if you can parallel what he did in a short paragraph. A clear example with less abstraction. Give a clear example of the principle of regulation as well. Or just ignore me and continue on - I would not take offense. :smile:

Notice however that the ∞ sum has a finite value (−1/12). That's the killer move! — Agent Smith

When one tries to do analytic continuation of the Riemann Zeta function where it is not warranted this kind of nonsense results. What makes it useful in physics is beyond me.

I would love to have a conversation about infinities (e.g., set theory) if you are interested: my knowledge of it is by no means expert level and would love to hear what you think of it — Bob Ross

Tones-in-a-deep-freeze is more an expert in this area. I'm from the generation of naive set theory. Your use of infinite is a philosophical excursion beyond my experience.

Positing an infinite value (unspecifiable volume) within bounds is tricky because, in my opinion, territorial limit takes on a special meaning such that limit transforms into asymptote. — ucarr

The set [0,1] is uncountably infinite with no asymptotes. Clueless what you mean.

Perhaps curiously, an infinite value "warps" a (conceptual) boundary into a "curved space" that functions as an unspecified boundary in that it is a boundary that is never reached. — ucarr

Give an example, please.

I fail to see where any of this is going. I got lost somewhere around a principle of regulation. Some time back Philosophim gave a short but clear example of where he thought this was going. Bob, I recommend you do the same.

↪jgill
:smile:

Isn't what I said implied by finitism? — Agent Smith

Finitism simply avoids infinity. There is no such thing as the "largest number" as far as I know.

Here's where you are going (Wiki):

Ultrafinitism (also known as ultraintuitionism) has an even more conservative attitude towards mathematical objects than finitism, and has objections to the existence of finite mathematical objects when they are too large.

one which to my reckoning involves substituting ∞ whenever and wherever it occurs with an appropriate finite number; this could really be a big help in my humble opinion. — Agent Smith

:lol:

The ultimate question of metaphysics is, "What is the ultimate question of metaphysics?"

It all breaks down to computing the path integral of the symmetry of quantum foam, so that a vacuum results. :chin:

So here we make the usual mistake of attributing issues that arise in the mind to nature, that are not a problem for nature. — Manuel

:up:

are you contending that I ought to remove the unbounded vs bounded distinction because it is not highly disputed amongst mathematicians? — Bob Ross

Not at all. I just pop in now and then whenever math is mentioned and provide my perspective. Most in my profession are not in foundations. In the most recent 24 hour period only 1 in 58 papers submitted to ArXiv.org were in that subject (logic, set theory, etc.).

I appreciate your friendly attitude! :cool:

Are all the infinities that appear in physics calculations [aleph naught]? — Agent Smith

Infinities can appear at singularities. This is the notion of infinity as unboundedness and/or non-infinite but bizarre behavior at such points. What you are talking about are ways of categorizing infinite sets in set theory. The terms of the series S=1+2+3+... form a set having cardinality aleph naught, but the series is unbounded. If Tones tunes in he could explain the details. In my research I never needed anything more.

Why don't you send Tegmark a question about you OP? Then report back. :cool:

Our friend, AS, inquired about an upper bound on numbers used in physics formulas. I assume both theoretical and practical. If strictly practical perhaps the inverse of one of Planck's constants.

As an example, A -> B. But also, C -> B. If we removed A from the derivation, we would still have C. So neither A, nor C, are a sqn. If however we had A -> D, and in the removal of A, it is no longer possible to ever derive D, we have a sqn. Does this approximate the idea fairly? — Philosophim

That being the case, we can create superordinate clauses that work, but do not negate the subordinate when removed. — Philosophim

Nice :smile:

I think any attempt to predict the particulars of a recession is a fool's errand. I'm talking more about how "what has been the case," is erroneously seen as "what must be the case in the future." — Count Timothy von Icarus

The dismal science. :cool:

the debate, philosophically and mathematically, is between actual and potential infinities. In other words, the valid form or forms of infinities is highly disputed, regardless of them all being limitless in content. — Bob Ross

"Highly disputed" within a certain, relatively small, subset of mathematicians.

Dispute Over Infinity Divides Mathematicians

With the hypothesis unresolved, many other properties of cardinal numbers and infinity remain uncertain too. To set theory skeptics like Solomon Feferman, a professor emeritus of mathematics and philosophy at Stanford University, this doesn’t matter. “They’re simply not relevant to everyday mathematics,” Feferman said.

Sorry to have gotten off on this tangent. I guess I don't understand your philosophical argument. To me "derivation" means putting together certain things, and this can involve the passage of time. Hence, a kind of reverse iteration of causations.

Renormalizations & Regularizations