They're called virtual particles — Philosophim

A mathematical convenience that cannot be observed through instruments.

Let think of an inch by inch square of space. Anything could appear in that square of space at any moment right? Right — Philosophim

Wrong. And I think you mean an inch cube in 3-space? Or an inch square in my favorite, the complex plane.

But divide the square in half. Anything could appear in that square at any moment, and not in the other half. Right? Right. — Philosophim

Wrong. Where do you come up with these flights of fancy?

What does this mean? For every one square inch we see that has one chance out of the infinite, we have a square that subdivides down into magnitudes smaller, meaning in the comparative likelihood of one square inch, its much more likely that something appear very small. I don't want to math this out, maybe someone else could. — Philosophim

Please don't. And don't ask a mathematician to do so. And something would appear very small if it is very small.

So over time its not surprising that we would see extremely small 'things' forming and unforming as they enter into existence, interact, and wink out — Philosophim

Since they can't be seen or observed, winking out is problematic. But there is a scale going from unobservable to observable, I believe. Where on this scale a virtual particle "becomes" "real" is interesting, though.

I love it when philosophers dabble in physics and math. Especially quantum physics. :cool:

I think in the long run, climate stories would be #1 — Mikie

Afraid Greta, Gore and many others have defeated their agenda with the American people by overzealous preaching and ridiculing. "Stupid people, wake up!! before you die of climate catastrophes"

But I'd move it up to #6.

I don't want to turn this thread into one regarding ontologies. — javra

Does a causation chain have being? It does if there is a dog at its end.

Assuming the Romney/Manchin ticket does not materialize. If it does, all bets are off.

I would vote for them.

Does an inch exist on a ruler without someone looking at it? — jgill

An inch no more exists without anyone contemplating it than does any word (such as the word “money”) exist without anyone contemplating it. — javra

I'm speaking of a ruler with inch markings sitting on the table in front of you. You turn away to ask your wife for a glass of wine, then turn back and look again at the ruler. Did it exist for that short period you glanced away? You know, Einstein and the moon.

Rabbit hole country.

A formatting question - sometimes I get a line feed before the math expression, other times, not - I don't see any obvious reason why. Any suggestion? — Banno

I use MathType, set to Wikipedia format, then change < to [ and > to ] at either end. It usually works but sometimes not, and I have no idea why.

The measures simply are. — jgill

That's a mistaken idea. Measurements need to be made, and measurement is an act which requires time. — Metaphysician Undercover

Does an inch exist on a ruler without someone looking at it?

the causal order A --> B --> C comes equipped with a dual order in the opposite direction, C --> B --> A. — sime

A-->B-->C-->A => C-->A-->B ?

It is really this phrase, "It simply is, there is no prior explanation for its being." that is ultimately true in any causal relationship. Do we call that a first cause? An uncaused cause? What do you think? — Philosophim

Mathematically, I would not. If an infinite composition exists it is the limit of a process that requires at each step a variable upon which that segment is defined.

${{F}_{n}}={{f}_{1}}\circ {{f}_{2}}\circ \cdots \circ {{f}_{n}}$ versus

${{F}_{n}}(z)={{f}_{1}}\circ {{f}_{2}}\circ \cdots \circ {{f}_{n}}(z)$

As n increases without bound one can look at the entire structure as a mathematical entity that has the value $\alpha =\underset{n\to \infty }{\mathop{\lim }}\,{{F}_{n}}(z),z\in S$

Now, outside the bounds of the theorem I have been discussing, alpha may not exist and the entire structure may oscillate or go to infinity, or whatever. Then the causation chain exists as a mathematical enterprise but cannot be associated with a particular value. It simply is. (My attempt at philosophy) :cool:

Pardon the intrusion into a less than delightful conversation. First, take the calculus out of the picture by considering ratios of distances, $\frac{\Delta y}{\Delta x}$. If the measure of y depends upon a measure of x, then for a small measure of the latter, like looking at the space of one inch on a ruler, there will be a small measure of the corresponding y. Time does not enter into the discussion. The measures simply are. One inch on the x-axis may correspond to two inches on the y-axis.

Now, if you throw in a quantum-like perspective ( a favorite tactic on this forum), the time it takes to look at each measurement counts. And you have complicated something very simple. Good show.

On the other hand to introduce time set $\frac{\Delta y}{\Delta x}=\frac{\Delta y/\Delta t}{\Delta x/\Delta t}$. Now we consider the passage of time in determining these distances.

An example might be the change in height of a hill with regard to distance from the peak. — Banno

The distance from the hill as one walks towards it grows smaller, and the line of sight distance to the peak also diminishes, but the height of the hill remains constant. The angle of line of sight grows also.

I mean, obviously, live people are more likely to succeed and reproduce than dead ones — Vera Mont

:lol:

Roughly? — Banno

Bull's eye, buddy :up: :cool:

The point is that if you take the entire set of the infinite regress and ask, "What caused it to be an infinite regress?" you realize that's the finite end. It simply is, there's no prior explanation for its being. — Philosophim

There are various ways to form analogues of causal chains in mathematics. One, that I initiated in a publication over thirty years ago concerns composition chains like

${{F}_{n}}(z)={{f}_{1}}\circ {{f}_{2}}\circ \cdots \circ {{f}_{n}}(z)$

In which one desires to formulate the individual functions to produce limits


$F(z)=\underset{n\to \infty }{\mathop{\lim }}\,{{F}_{n}}(z)$


A colleague of mine from the University of Tronheim discovered what is the most relaxed requirements to achieve this result, and went further to prove that as n grows larger and larger all values of z in a region of the complex plane will lead to the same result, that is to say $F(z)=\alpha$

Thus, we have a regression that at each stage begins with a specific z but ultimately all z's in a region give the same ultimate value. Looking at limits here.

"What caused it to be an infinite regress?": The design of the structure.

"you realize that's the finite end": No. That makes no sense, alpha is a limit, not a finite end. The further back one goes the more accurate the observed present day value of alpha. There is no finite end to this regression, only partial ends.

"It simply is, there's no prior explanation for its being" Yes.

If this thread continues I'll describe other analogues of infinite regression, for good or bad.

Various quantum effects, for a start — Banno

Probability and statistics can weave a tale as the story of quantum physics unfolds.

We know some things don't. — Banno

Huh. Please elucidate.

I own an Encyclopaedia of Philosophy (MacMillan) published in 1960s, and it is still very useful — Corvus

Yes, I can understand philosophy and history retaining their value over years. Science and math change much more.

The reasoning demonstrates that even an infinite regress falls into a finite regress of causality. — Philosophim

Why is that? I'm a little slow today. :chin:

Why do you believe so? — Corvus

They can't be continually updated, like Wikipedia. They cost $.

True, some knowledge doesn't change, but Wikipedia has that too.

As a captain in the reserves in the early 1960s, having done my obligation supporting ADC and SAC at a base near the Canadian border, I was asked to go active reserves (i.e., Vietnam) or resign my commission to make room for some other junior officer. I resigned my commission (being a math grad student at the time and married). But the USAF treated me very well while I was active.

How many degrees do you have, if I may ask? — frank

Three, plus post-grad certification for the USAF (and USWB) 1958-59 as a meteorologist.

It is easiest to simply imagine that all the [general] things known to humans that can be written down in language have already been written down. Now we have the {body of analytic knowledge}. — PL Olcott

So if listed, the listing might have to be refined as new knowledge is accrued. Still way to vague for me, but others may feel differently. I admire your tenacity on the subject.

A huge amount of progress has been made, but there are still problems, like the cloud problem — frank

In the late 1950s we had virtually no computer access when I was a post-grad meteorology student at the U of Chicago. (In 1962 at the U of Alabama there was a giant computer filling the wall, floor, and ceiling that was an ordeal to use.)

We learned to classify cloud formations. In atmospheric physics we studied droplets.

The body of all analytical knowledge — PL Olcott

Unless you can describe this vague notion as it might appear in a computer program - that is to say a list with #1, #2, . . . - I can't get beyond it to the conclusions you draw. @TonesInDeepFreeze is recognized as a go-to source on these kinds of subjects.

Will the textbooks and Encyclopaedia will still be in demand? — Corvus

The print textbooks will remain a primary source of specialized information. The print book encyclopaedia are dead ducks. Wikipedia is the best thing on the internet IMO.

Schmidt says climate models have come a long way from the simple energy balance and general circulation models of the 1960s and early ‘70s to today’s increasingly high-resolution and comprehensive general circulation models

Back then it was mostly statistical studies. Then after that period atmospheric physicists joined in and made it a real science.

The problem though, as I've read, is that this "moving apart" can be much faster than the speed of light. And since the motion of objects is limited by the speed of light in relativity theory, this "moving apart" cannot be categorized as motion, in order to avoid contradiction — Metaphysician Undercover

Yes. I stand corrected. The limits due to the speed of light seemed contradictory. It's difficult to imagine "nothing" expanding. It's an age of discovery and conjecture where our intuitions - formed by everyday experiences - must give way to a deeper reality in which math replaces direct sensations. And perhaps a newer, emerging math replaces that which has served so well up to this point.

Points and continuums, space and time . . . . . remain beyond complete understanding, although we manipulate them confidently. When I asked an old friend, an analytic number theorist, what he thinks of real analysis, he says, "It's very, very complicated and it starts with a metaphysical notion, points."

Years ago when George Bush was asked who his favorite philosopher was, he replied (after a bit of thought), Jesus Christ.

One can draw all sorts of conclusions from George's choice, negative or positive, but His teachings provide a way of life that could mitigate all those inequalities. Just a thought.

During the Pandemic the government (USA) gave out a lot of money, particularly to parents of small children. I remember thinking, I hope those parents don't use this for booze and drugs. I like to think that most didn't, but who knows?

The Meek shall inherit the Earth — jgill

what's left of it, once the Bold are done striving — Vera Mont

One of my old outdoor colleagues once said of my rock climbing,
He's not bold . . . but he doesn't need to be.

The meek are not necessarily the weak.

but denying that a trivial problem is a real problem, turns a trivial problem into something substantial — Metaphysician Undercover

Whether a thing is a problem or not is a societal decision. If the vast majority do not consider it a problem, it likely is not.

AI:

So, while Zeno's paradoxes, including the arrow paradox, are not considered unsolved problems in mathematics or physics, they do continue to inspire ongoing philosophical discussions¹³.

My understanding is that two objects move further apart with time; space itself (whatever it is) doesn't change. — jgill

However, I don't think it is proper to call this "motion" because the activity known as spatial expansion is not consistent with our conceptions of "motion", and the physical laws which describe "motion". — Metaphysician Undercover

From BingAI:

Yes, space does expand. The expansion of the universe is the increase in distance between gravitationally unbound parts of the observable universe with time¹. This is an intrinsic expansion; the universe does not expand "into" anything and does not require space to exist "outside" it¹.
. . .
However, it's important to note that this is not a generally covariant description but rather only a choice of coordinates. It is equally valid to adopt a description in which space does not expand and objects simply move apart while under the influence of their mutual gravity¹. Although cosmic expansion is often framed as a consequence of general relativity, it is also predicted by Newtonian gravity¹.

Does "simply move apart" imply motion in the common sense? Can something move without motion?

Not a finite set of axioms, rather a countably infinite set of axioms. — TonesInDeepFreeze

I hoped you would chime in. :cool:

But Godel was speaking of a small finite collection of axioms, not an axiomatic system that continues to increase without end. At what point does one initiate the drawing of conclusions? Tacking on the axiom of choice took math into new dimensions, as did infinity axioms. BOAK seems bewildering rather than enlightening, imo.

Everyone desires some advantage, some way to be better, smarter, faster, stronger, more talented, more charming or more beautiful than others of of our species. But we're not all willing to pay the same price or make the same amount of effort or take the same risks to achieve it — Vera Mont

:up: :up:

But there is a way to deal with the inequalities and be peaceful & honest. — YiRu Li

The Meek shall inherit the Earth

Undecidability cannot possibly occur. — PL Olcott

Not undecidability. Rather potential facts. I know my Corgi could not understand analytic geometry, and that is a general assumption for Corgis. But there is the faint possibility that one will come along and understand the math. Then that fact becomes a member of the set of "axioms" in your system. But it is not at present. Thus your system continues to grow, and with each new axiom there is the possibility of contradicting a previously established axiom. So, when you ponder Godel what system are you talking about?

My question remains: show how exactly all axioms can be listed for reference. What is axiom #1?, #2?, . . .

The Cyc project has {Thing} at the root of its knowledge tree — PL Olcott

The objective of the Cyc project was to codify, in machine-usable form, the millions of pieces of knowledge that compose human common sense.

A far cry from listing all facts (axioms).

The body of all current analytical general knowledge is not only countable it is finite. — PL Olcott

So, the system of axioms is constantly increasing. Proof it is finite at a particular time?

What I mean by axiom is any expression of language that has been stipulated to be true — PL Olcott

"True" by what measures? What of potential inferences not realized?

Season's Greetings

Merry Xmas, Lads.

But the space itself is expanding within any duration of time — universeness

My understanding is that two objects move further apart with time; space itself (whatever it is) doesn't change. Some of this stuff is assumed axiomatically.

All of the basic facts of the model of the current world are stipulated to be necessarily true, thus are the axioms of BOAK. — PL Olcott

OK, this means an uncountable collection of "axioms". How could you organize these axioms in such a fashion they represent a data set in CS? What is axiom #1 ?

Some time back we had a promising theory of everything that started with the premise all facts could be catalogued within a program. But when asked "how?", things began to fade.

and we should accept that in mathematical terms, the distance between A and B cannot be traversed unless 'infinite acceleration' is a real thing? — universeness

:lol:

But, in Minkowski spacetime it seems progression in the time variable requires no movement in space. I think of time as being in some ways linked to movement - a philosophical perspective. Just a feeling. JS might be able to address this issue, which is probably a triviality on my part.

There was another quantum physics guy on the forum a couple of years ago, @Kenosha Kid. He said he left physics to make his fortune playing the guitar. Well, actually he chuckled at the "fortune" bit. He was a transactionist.

In my naive fantasies I wonder if our macro world is indeed a simulation wherein the creative mechanisms are intentionally hidden from us in the quantum realms. With math we are able to manipulate results down there to some extent but remain puzzled at non-locality - a feature of that realm where scales of measurements and dimensions might be easily manipulated by our overlords. :roll:

In relativity theory it seems there is possible the passage of time without any physical change. Time being completely independent of change seems peculiar. This, unfortunately, puts me somewhat in MU's camp: we don't truly understand either time or space. But I am old and have lost brain cells. :chin:

, ,

As I sit on the sidelines I find this conversation fascinating, and I am learning some math as well. I've never worked with formal integral transforms and am surprised at how many there are. I now understand a bit more about the uncertainty principle.

(IMO Zeno should be dead and buried)

My theory about his attraction to dictators and facists is that it’s not grounded in political theory, but the simple fact that they wield the kind of power that he craves — Wayfarer

I agree. But his following continues to grow in a world turned upside down.

Also, I only just learned that this environment is tex-enabled too, so that's nice. Can I ask how you did that? I can't seem to figure it out. — Jaded Scholar

I use MathType, what you see is what you get. Then copy and paste to this forum, changing "<" to "[" and so on. But there is a tutorial here on MathJax I think. Enter the word in the search box.

Thank you for your comments about the use of series in physics. A power series is one kind of infinite composition of complex functions. I got started in the subject with another, continued fractions. Then there are infinite product expansions. Finally, examples I came up with after a colleague pointed the direction, elementary functions

The Bogoliubov transformation is a new one for me. Not quite sure how to decipher it in Wikipedia. Some symbols probably physics related.

When I got my degree fifty years ago, a professor told us that we would never know as much math as we did then. He was correct. As the years flow by we mathematicians get more and more entrenched in our specialty, and don't get off that path very much, allowing progress in math to surge past us. At my age I'm lucky to remember elementary topology or group theory. :worry: