Usually the concept of work relates to a change of energy, kinetic or potential. When an object follows a path through a force field, if that field is conservative, the path the object takes from point A to point B is immaterial regarding work; all such paths produce the same work. This idea aligns with Cauchy's Theorem in complex analysis.

The analogy I recall is two people about the same weight standing before a mountain. One takes off directly to the summit over a series of cliffs, and the other follows a trail that winds round and round the mountain, finally reaching the top. Both have done the same work.

A sequence of smooth contours in the complex plane I devised that start at zero and end at one grow longer and longer without bound while converging uniformly to the straight line path along the x-axis from zero to one. Set in the backdrop of a conservative force field an object moving from zero to one along the x-axis does the same work as on a contour that could be unraveled and stretched from Earth to Alpha Centauri and beyond.

:up:

Richard Muller, physics professor emeritus at UC Berkeley, states that energy is the most difficult concept to understand in the basic physics curriculum. It will be interesting to see what people say.

In what sense inaccessible? Do you mean that generalisation actually ends up cutting its connection to the particular? That shouldn’t happen if it is being done right — apokrisis

Mathematical Schemes is an example of current levels of abstraction. If I were an algebraist or topologist I would probably see the values therein. This entity aided in solving Fermat's Last Theorem. But its value in real or complex analysis is debatable. Here's an example that, for me, is vague - which you value. For others that vagueness is merely the ectoplasm of math.

You speak often of systems theory, and in math that begins with dz/dt=f(z,t) in the complex plane. Here the levels of vagueness are low, and chaos may grow out of these scenarios. What scheme theory has to say about this is a question for experts in these areas. But it appears schema theory means something else in biology.

Schema Theory and the Dynamical Systems Theory are the predominant behavior theories that address how the nervous system produces a movement.

The Generalized Motor Program Theory (GMP) or Schema Theory and the Dynamical Systems Theory are the predominant behavior theories that address how the nervous system produces a movement. The debate of movement scientist and the contrasts of these theories centers on whether movement is created through hierarchical control in the nervous system (i.e., cortical control) or if movement control is distributed throughout cortical, subcortical, spinal, and even musculoskeletal levels of the nervous system. While compromise between these two theories may be possible, each theory has its respective adamant supporters who will argue for the support of one over the other. In this assignment, you will evaluate these theories to determine which theory you believe is the more plausible explanation.

At the lower level of technicians perhaps, but I doubt it since thinking involves both hemispheres. Advances in science require imagination, which certainly does.

So that for me is the meaning of metaphysics. The move from the particular to the universal. From the concrete to the abstract. From that which is true of some things to that which is true of all things. — apokrisis

Well, this has happened in mathematics as specifics have given way to greater and greater generalities, an approach that has brought together various ideas under broad umbrellas, and is certainly a popular trend (with virtually every grad student knowledgeable of category theory), but it leaves lower level intricacies inaccessible - particularly in real and complex analysis, the latter being very important in QT.

I wish I knew more about QT so I could assess how beneficial this has become (beyond Hilbert spaces, etc.) :chin:

Einstein had taught himself differential and integral calculus by age 15, and had a teaching diploma in math and physics before the patent office job. His PhD may have been awarded while he still held that job (1905).

Ramanujan had studied mathematics for some time, both on his own reading and in school, before he interested Hardy in his original results.

In both cases these geniuses had backgrounds in mathematical thought before they became celebrated.

In the movie, however, the janitor had no previous math experiences - except a reference to being self-taught - and simply picked up an advanced topic (requiring a background) by simply looking at notes left on a blackboard. One of my relatives asked me my opinion about this, and I told him "possible but unlikely".

That shows in your support of Tonioni for example — apokrisis

Probably you mean Giulio Tononi. His Phi function is untenable.

It could be that a non-specialist somehow cracks the problem — Manuel

Reminds me of the movie Good Will Hunting in which a janitor solves a ridiculously difficult mathematics problem while erasing a blackboard each day. Possible, I suppose, but extremely unlikely since prodigies are so rare. A pretty good film nevertheless. :smile:

A non-differentiable manifold. Live with it. :brow:

Spoken like a mathematician but not a physicist or metaphysician? — apokrisis

True enough. I don't have a Professional Degree in metaphysics. But I respect those who do. :cool:

If mathematicians need sets, for example, even if we're not happy about it, we'll deal with the philosophers who say they can't have them. — Srap Tasmaner

Bless you, young man! :fear:

Plenty of philosophers ignore aesthetics, for instance, or ethics, but I always thought Quine didn't so much ignore them as exclude them. Do you read him differently? — Srap Tasmaner

I only meant I could not find where he might have said this. Nit picking. :wink:

But they smell blood in the water, and won't go away — Gnomon

They still search for fire they can breathe.

We simply never were interested in what might “breath fire” into our equations? I really was wasting my time? :up: — apokrisis

Breathing fire is vastly overrated. Exploring the math can do that job. No need for unicorns.

Quine had this idea that philosophy is the handmaid of science, — Srap Tasmaner

Really? I can't seem to find a reference. Do you have one handy?

On the other hand, among those physicists who are aquatinted with philosophical accounts of realism and anti-realism, most consider themselves philosophical realists. — Joshs

That would be my guess. Were I a physicist I would be in that camp. Actually, I probably wouldn't care one way or the other.

Was it Halmos who said our mathematics is not ready for the Collatz conjecture? — Srap Tasmaner

Paul Erdős

Did you have this sort of thing? $\sum\limits_{{}}^{{}}{\prod{{}}}$ (I use that all the time)

Have fun. It looks dreadfully unappealing.

But the general answer I get from those who know this stuff is NO. It doesn't say anything like that. When I get into the philosophy about it I get stuff like "well that depends what you mean by reality", after that I pretty much tune it out. — Darkneos

If you search for "real" in your Schaum's Outline of Quantum Mechanics you will find nothing, save mentions of the real number system. "reality" is in the domain of speculation by both experts and quantum mysticists.

It looks like progressives vs regressives. I guess I don't see the purpose of it.

Well yes. It’s the difference between tossing a classical coin to discover if it lands head or tails, and knowing that if you toss one of a pair of quantum entangled coins — apokrisis

I was thinking only of a more mundane application of S's equation, with wave packets and probabilities of a particle being at a particular place at a particular time. But I have zero knowledge of the experimental mechanisms involved. So I should avoid talking about things I do not understand. :confused:

Breathe fire :cool:

How do you mean? The act of measurement that picks out a solution is the tricky issue. — apokrisis

From a purely mathematical perspective, when one solves a kind of partial DE there may be many linear combinations of solutions. Question: which one applies to a particular experiment? Answer: the one correlating with an observed measurement. It seems that these are just possible solutions to the problem being investigated, not "possible worlds" or "superpositions" in quasi-mystical senses.

Like a simple problem in d=rt, where one gets two possible solutions, one negative, one positive, and the physical situation determines the positive solution is appropriate. There are not two possible worlds, just two possible answers, one of which correlates with observations. I suspect you are saying this kind of distinction is much trickier in QM?

Thanks for taking the time to answer these questions.

Who wrote the "laws" limiting how far amateur philosophers can speculate, beyond the "revealed Word" of physical Science? — Gnomon

There are few restrictions here. But when views are presented others are free to poke at them.

You keep looking for the "stuff" that breathes fire into the equations.

But I'm not. I'm saying its a mistake to presume it. — noAxioms

Yes, I agree. The mistake is to assume the universe was created to raise human emotions.

What would you say about the idea that there is happening no collapse at all.But we just think that we "spot"one ,cause we are condemned from our own consciousness to see it like that? — dimosthenis9

Good point. , to what extent is the "collapse" simply the experiment being resolved by identifying one of the many solutions of the Schrödinger equation, all of which "exist" together?

Here is a graph summarizing the results of a survey of physicist's opinions of the correct interpretation of quantum mechanics: — T Clark

It's interesting that the survey showed zero percent in favor of a transactional approach. I seem to recall that one of the few actual physicists on TPF, Kenosha Kid, argued for that.

Compossibility

Euler diagrams

If you have a program on your computer you need to pay and be a subscriber to upload an image to a post.

to make someone understand what you yourself don't understand? — Agent Smith

I see this attempted on TPF when someone starts talking about quantum theory or relativity. Now, there are a few individuals on the site who know what they claim to know, but others shovel out the terminologies and key words from stockpiles only a few centimeters thick, thinking they do understand while not knowing the extent of their misconceptions. To some extent this is due to books and articles attempting to convey information to the masses in ways palatable to those masses.

Sean Carroll is now attempting to remedy this unfortunate practice by going deeper into the mathematical basis of ideas in physics and cosmology in a book describing popular science notions.

One deplorable example I see over and over here is "curvature of space", exemplified by the ubiquitous image of the Earth sinking into a basketball net of gravity.

which is that the behavior of a small percentage of the world population is responsible for most problems. — Xtrix

Spare me clichés — Xtrix

OK

which is that the behavior of a small percentage of the world population is responsible for most problems. — Xtrix

I assume you mean by "small percentage of the world population" the highly civilized nations (HCN), to which millions flee, desperate to be admitted, for the promises of a better life. Are you saying the HCNs both create most problems but solve many problems?

Were it not for the HCNs life would be barbaric with early deaths from disease and injuries. Look at the American Indians.

Therefore, is what is needed for better philosophy actually a fasting and detoxification of thought? — Xtrix

I like this. In order to improve philosophy, don't do philosophy.

Non-thinking is relevant to mathematics as well. Long retired, I still conjure up math notes and post them as a hobby. But in doing so I may be harming the subject. arXiv.org receives hundreds of papers a day, every day of the year. Much of this activity is polluting thought and contaminates what is pure, creating a swamp of diversity that drags the discipline into incomprehensible sludge.

You have a lot of patience. The "vagueness" principle apokrisis advances might work better. It would parallel the actual history of math.

The interval would be made so small that the length of the line was the same size as the width of the line - both being now effectively zero. — apokrisis

A line in turn would be arrived at as the constraint on the quality of “plane-ness”. Squish the 2D plane from either side and the limit of its compression becomes how a 1D line arises. — apokrisis

The sequence of contours of the form ${{Z}_{n}}(t)=t+i\tfrac{1}{\sqrt{n}}Sin\left( 2\pi tn\sqrt{n} \right)$ demonstrate convergence to both an infinitesimal straight line (on the x-axis) and the origin, 2D down to 1D down to a point. $t\to \tfrac{1}{\sqrt{n}}$ and $n\to \infty$ with ${{Z}_{n}}(t)\to 0$.

Oh oh, I almost forgot $\text{Length }{{Z}_{n}}(t)\to \infty$. You can ignore that I suppose :cool:

Whoops, mistake.

Space is a concept (opinion) that has reality existence as an immaterial existent. — val p miranda

It seems like you are just playing with words. Suppose your idea is accepted. What difference does it make besides a brief discussion in a faculty lounge? Is it really worth the effort?

Time, however, is a concept and it can be defined as what clocks measure, but time is not limited to this definition. — val p miranda

As a concept, time is the measurement of motion. — val p miranda

See the difference between these two concepts of "time" val? — Metaphysician Undercover

In math, time = t. So simple.

So in this view, you start from a material vagueness or everythingness - a quantum foam of possibility - and this then reacts with itself to become a more limited and stable arrangement of somethingness. Existence evolves in a least action or path integral fashion where everything cancels down to whatever definite form can stabilise the situation and make for an orderly Universe unfolding in dissipative fashion in an emergent spacetime.

I find this somewhat hard to understand, but it seems sensible enough. — noAxioms

It's poetic as well.

In the philosophy of mathematics, I am not a scholar, but I have read many books and articles. In mathematical logic and set theory, I'm not a scholar, but I have a good handle on the basics through taking courses and careful study of several textbooks — TonesInDeepFreeze

Hey, I'm a fan. You have my respect. :cool:

I used to pick up bits of modern math knowledge from @fishfry before he departed.

Why do we need to order the curves? — keystone

If you want to start with curves (or continuous objects) in order to derive points you need a systematic way of talking about them. If you plan to assemble contours or curves by gluing together tiny straight lines, then you are doing nothing more than is done when one calculates the length of a contour in complex analysis.

But I guess what you are really interested in is potential infinity and its kin. So my comment is what difference does it make how you deal with that? We've had posters here who spend years working up what they consider astounding revelatory articles, only to be more or less ignored. They become so enraptured with their ideas they get caught in that spiral in which the more effort you exert the more you think your product is of value, losing their objectivity.

But if you enjoy your project as a personal challenge, have at it. I have certainly been in that boat!

If what exists exists in space and if space is an existent then space exists in itself. If it exists in itself it cannot be the same as itself. — Fooloso4

Maybe it is. A possible definition of space. :snicker:

All bijections are injections. So you're confused to begin with

The reason old-fashioned terminology is not so bad. One-to-one onto, etc. Bourbaki may be to blame? Just nit-picking, ignore me.