By the time you have reached the seventh or eighth article you should be looking at something to do with philosophy. — Sir2u

Two that I am intimately acquainted with went in two steps to the definition of mathematics, as you predict.

The topic remains an opposition between infinitism and foundationalism, with Philosophim taking a foundationalist position. The alternative is an acceptance of infinite complexity, — Banno

An infinite chain of reasons or causation chain may be finite in the sense that as one moves further and further back in time the passage or rate at which time moves (I know, sloppy) possibly changes all the way down to zero.

Infinite complexity to me means all the transfinite stuff that set theory produces. It may never arise in the physical world.

Denver Hospital System may fail due to Treating Migrants

Thank you, Mr Surge the Borders President. :roll:

Blame the Democrats for running a corpse for President. — Hanover

I thought the problem was he was asleep at the wheel. :roll:

I have been looking into what Sir Roger Penrose has to say about transfinite theory in physics, and it appears he thinks it can arise in computational physics as that discipline evolves. He speaks of data fields and defines expressions like ${{\infty }^{r{{\infty }^{q}}}}$.

As for physics outside data fields he doesn't forsee transfinitism. As much as I have read.

Hence you can have people working in science who say they don't care at all about philosophy. — ssu

:smile:

It may not be easy to keep this thread from drifting away from the transfinite.

For all the discussions about the Grandfather Paradox, timelines, alternate histories, etc. I don't think humans will ever travel physically to the past, although in theory we are now able to travel to the future slower and slower via spaceships.

We "travel" to the past when we look at an old photo or watch an old TV show. That kind of technology might develop into a means of recording virtually everything in a particular sphere of activity, allowing those in the future to "travel" into the past without any paradoxes or violations of physical laws. In this sense time travel is dependent upon the present preparing for the future travelers.

If you think physical time travel to the past is possible, begin by explaining how.

That we don't have any use for the larger infinities in physics, at least yet, makes it doubtful that the Cantorian idea of larger and larger infinities is valid. — ssu

I recall from my functional analysis courses the Hahn-Banach theorem, which deals with extending linear functionals on manifolds. This was the only time I encountered transfinitism. Even then a simple change in the hypotheses eliminated the need for going into multiple infinities. It's possible this theorem is the basis for a part of a mathematical process used in quantum theory. We'd have to ask an expert.

Add: Practical Transfinite

The key point is this: we can conceive of an object being non-existent at one moment and existent the next – we do not need to even introduce the notion of “cause” into this thought experiment. — expos4ever

And that's all it is, a mental feat, like flapping my wings and soaring to the moon. We are blessed with the ability to be creative, and much of which we imagine can be brought forth in the real world through our efforts. But not the miracle apple.

I must, however, admit that my examples of causal chains with/without first causes is nothing more than a mathematical illusion, like your apple. :cool:

Yet if there's a Continuum Hypothesis, we clearly don't understand everything about infinity. — ssu

I think much of what we don't understand is a result of definitions in set theory. And when transfinite theory is incorporated into physics, practitioners take notice. I wonder if that has happened?

Anyhow, good luck in keeping this thing going. :smile:

It's a two way street — Count Timothy von Icarus

Of course it is. Virtual particles is fair game for both sides of the intellectual coin. And as I mentioned there seems to be a spectrum between virtual and real, which is fascinating. At some point in this spectrum may be where mathematics is reified.

Thanks for the Floridi remark. I'm looking into it now. New territory for me. :smile:

I was a gymnast in the 1950s and a rock climber up until the early 2000s. Never smoked, never used drugs, but used to have a highball or two every other day up 'til around 2000 when I retired.

Now, at 86, I still do some pull-ups and short hill walking every few days. But I have arthritis all over my body, so I am recently using a cane. I continue minor mathematical explorations. You remember what Sachel said . . . :cool:

What is the specific prior event that caused the decay of that atom at that time? — EricH

That's a problem with virtually all causality chains. There may not be just one cause at each step, there may be many, and so chains interact with one another. The complications are staggering, as your example implies. To all extents and purposes it appears random.

↪jgill

Maybe you can shed some light as to why massed randomness seems so ordered. — mentos987

The Central Limit Theorem may be what you are talking about, although this ancient thread goes all over the place.The CLT concerns arithmetic averages of n observations as the value of n goes to infinity.

This thread might also involve the notion that in an infinite period of time all things that are possible will occur. That's pretty vague.

I haven't used probability theory in over thirty years. @fdrake is more knowledgeable.

But yes in the end everything becomes equally probable. — IP060903

Perhaps someone would offer a proof. :chin:

The question is really about what caused the set of causality to be. If the universe has a finite chain of causality, what caused that to be? If the universe has an infinitely regressive chain of causality, what caused that to be? There is no prior cause in either case. It would be that set without prior explanation; it simply would be — Philosophim

Well, this is certainly a deep issue. Good luck. Nice chatting with you. :smile:

One thing is infinity, that set theory takes just as an axiom — ssu

Not all mathematicians are set theorists. "Without bound" works pretty well for some of us, without transfinite or philosophical overtones.

I have to say I find it pretty annoying, and even irresponsible. — Jamal

I apologize if I have offended you. I seem to have misinterpreted the situation.


Radio Free Europe

Al Jazeera

BBC

The implication being that, were there no law against it.... :yikes: — Wayfarer

Oh, if only that nitwit president had not invited the world to surge the border. Guess Robert Gates is right. Sad state of affairs. Maybe Michelle will rescue our country.

The probability of measuring some part of a system can be computed from the wave function. I've not heard the result of that computation being referred to as a 'wave', but I'm sure it is somewhere — noAxioms

The Schrödinger equation's solution is called a wave function. If one simplifies the equation considerably it has the form dQ/dt=kQ, which has solutions involving e^it=cost+isint, giving it repetitive or wave-like characteristics.

I apologize if I have misinterpreted your comment.

and now Russia but not for much longer! — Jamal

Escape before you're drafted! I hear Putin is even militarizing migrants.

As a mathematician I don't recommend this topic for this forum. There are not many of us here with knowledge of transfinite theory. But maybe some interest will be sparked and the thread will be better populated than I imagine. :cool:

Lets envision an a thought experiment of an actual chain as a visual. — Philosophim

What I have produced in mathematical terms is an actual chain - I can make it more specific with definitions of functions, etc. if you desire. Your actual chain is a complete abstraction.

What I have shown is that first cause is more complicated than what the ancients understood. In my example, n going to infinity, using the same z at each value of n produces an infinite causal chain having that z as a sort of ultimate first cause. I would think this example would stir original philosophical thought rather than a regurgitation of traditional ideas. :chin:

Another causal chain question: Do infinite chains ever end? Can infinite chains have specific values? Think going the other direction, into the future.

I tried to understand cases where numbers are physical entities, but I must admit that I couldn't possibly think of any case number that can be regarded as a physical object — Corvus

:up:

In physics, isn't time just clock-time? Kind of a practical use, rather than a discussion of what it is? — AmadeusD

Seems that way to me. I use time as t, a non-negative real number, in my math. I think most have given up on what time is. But arguments about whether points in time exist or time is durations or intervals persist. See Bergson vs Einstein and Peter Lynds.

Is there's a boil-down source to understand the concept? — AmadeusD

I wish. Time dilation shows that the passage of time is relative to motion, so in a way time is linked to change.

A particle can assume the characteristics of a wave in an experiment and can be shown to exhibit the characteristics of a more conventional particle by changes in the experiment. But the wave form of the particle is not the probability wave of the particle is it?

Though, maybe i'm missing a trick but it seems to be that your suggestion presupposes an 'actual' time, independent of objects passing, rather than time being a description, or set of relations between objects — AmadeusD

I agree with MU here. If it were not for Minkowski spacetime allowing rest frames and thus the "passage" of time with no physical changes I would think time required change.

Thanks for the Brian Greene piece. My only reservation is that unless one is careful in reading it one might think the illustrated wave is the particle, whereas the illustration is the probability wave for the particle, which itself can assume a wave form. Is he saying the probability wave is the particle?

But perhaps I am misinterpreting it.

The institution of philosophy is a bunch of people desperate to justify their own job, all the while pushing people to learn 'the art of publishing' which is not about new ideas, but learning to find what publishers are looking for as well as modern trends. Original ideas that are not forcibly tied to some other famous philosopher are discouraged and rejected. It is not a place of open thought, but stifled institutionalism. — Philosophim

Illuminating. Publishing in math seems to be much more nuanced. Or it was during my time. Still, ArXive.org receiving over 150 math research articles each day of the year makes one wonder.

First, lets stretch a chain from left to right, each link is a prior cause to the next link. The first link in the left is the first cause. It has no prior link of causation — Philosophim

In my example the first cause is on the other end of the chain, the nth function. As one lets n go to infinity, at each step back any z within that disc is an nth cause, I suppose. Infinite means going back in time with no end, for each n computing the accumulated value at the left end of the chain. The further back in time, the more accurate the value at the left end.

Now lets take a chain that's looped together to represent infinite causation. What caused there to be a looped chain? — Philosophim

The correlation between a bank account and the interest it accrues is a simple real world looped chain. A looped chain need not go around like a circle, it is more likely to look like this:

${{C}_{m}}(z)=\left( {{g}_{n}}\circ \cdots \circ {{g}_{1}} \right)\circ \left( {{g}_{n}}\circ \cdots \circ {{g}_{1}} \right)\circ \cdots \circ \left( {{g}_{n}}\circ \cdots \circ {{g}_{1}}(z) \right)$

This time going from right to left. m goes to infinity, as could n. We are still beginning with a "first cause".

7. Because there are no other plausibilties to how causality functions, the only conclusion is that a causal chain will always lead to an Alpha, or first cause. — Philosophim

How about, Originate with an alpha?

:up:

When something is a first cause, it is an uncaused thing which then enters into causality. There is no limitation as to what a first cause could be, as it has no prior explanation for its being. It is unlinked from determinism as to why it exists. However, once it exists, its interactions with other existences then involve causality, or determinism — Philosophim

I've always required examples to flesh out philosophical or mathematical ideas, so I will return to the causal chain ${{F}_{n}}(z)={{f}_{1}}\circ {{f}_{2}}\circ \cdots \circ {{f}_{n}}(z)$ which is defined in a region of the complex plane. My colleague at Trondheim proved the theorem I have mentioned, but later I proved a much simpler version,changing the hypotheses slightly.

Think of a large disc in the plane, full of points,z, and each individual function in the chain taking any such point and producing another point in that disc. Assume that each of these functions draw any two points in the disc slightly closer to one another. Then, when you start the chain you can use any point in the disc as a "first cause". The chain will converge to a single value of z within the disc, but the degree of accuracy of that limit will vary depending upon original choice. That is to say some values of z as first cause will take longer to get close to the ultimate limit. Bigger values of n.

(In fact, in a paper of mine in the Proceedings of the AMS many years ago I showed that if the individual functions approach a specific function, f(z), one may be able to choose a "first cause" that will get the ultimate result the fastest.)

Another recommendation: Take up Zen under the tutelage of a Master. This would give your life a rigid focus, enforced by the physicality it employs. You will change if you take the practice seriously. I've seen it happen.

Why is it always better to cease to exist.

Keep thinking "It is better to exist" You have to corral your thoughts and Zen will do the job if you persevere.

"Or the math department. They don't even want the bins." — Wayfarer

Not true. Housekeeper of a famous mathematician asked what the mathematician does. Answer after deliberation: "He scribbles on paper, then scowls and wads it and throws it in the wastebasket".

The steps in a dynamical system or causation chain occur at time intervals. Are there chains that display instant movement, like instant velocity? This involves smaller and smaller times between steps. Here
are comments related to instantaneous motion in causation chains.

If math is not your thing don't bother.

↪jgill
Wasn't there a gallery of your images on the site somewhere? I'd like to link it. — Banno

There are a few Here, and you can see what the math looks like.

Look at my icon carefully. I could not have planned it and then created the necessary math, in my wildest dreams. — jgill


What are the axes of your drawing? — wonderer1

Like the standard Euclidean plane. Vertical (imaginary) axis in center and horizontal (real) axis across the middle. Sometimes I shift my focus to a small section of the complex plane. I did this to isolate and magnify my Quantum Bug icon.

So in a way of speaking, the images emerge form some, but not all, of the equations? — Banno

There is always an image arising in the manner I have described. Even an empty frame. I never know what will materialize from a mathematical procedure ( I don't do fractals or other well-known imagery). The math procedure I use is not one others use to the best of my knowledge. Mine is intimately connected to cause and effect chains. Not simple iteration of a complex function.

Look at my icon carefully. I could not have planned it and then created the necessary math, in my wildest dreams.

And since all change requires time — Metaphysician Undercover

This depends on the meaning of the word "change". Most usages do require time to complete, but if one thinks of "to change" is "to differ", then time need not be a factor.

dy/dx = 10 vs. dy/dt =0 if x is not a function of time.

So which is it: is the butterfly reducible to the equations, or does it emerge from them...?

Isn't emergence no more than Emperor Reduction in his new clothes?

(@jgill, any thoughts?) — Banno

Not a lot. I can only refer to my own "creations", that are observable in the complex plane. By the way, I program from scratch in BASIC, being old with little time to waste on a plethora of languages, several of which (Fortran, Pascal, . . .) I have used in the distant past.


There first is a defining mathematical procedure giving a point in the complex plane for each point in play. The procedure can be very simple or very complex. In both cases one paints pixels in the plane corresponding to some aspect of the resulting point, like size of its modulus. Point by point the process paints a picture.

Philosophically, one can rather easily go from the procedure to the picture, obtaining a visual account of the feature you are describing, like size. But, it is virtually impossible to start with the big picture and deduce the mathematical procedure. Viz. the image above. So the imagery emerges from the mathematics.

Sometimes the math leads to chaos upon iteration. Other times interesting patterns emerge. So, images emerge from math, but not the other way around (usually). I don't see reductionism in play here.

The Lorenz attractor is a case where patterns are affected by attracting fixed points or sets of such points. In much of the math I have done infinite compositions - infinite causal chains if you wish - take points toward attracting fixed points. There are also repelling fixed points where you can guess what happens by the title of the point. And of more complexity there are indifferent fixed points that both repel and attract.