B-T depends upon the Axiom of Choice. This shows how "dangerous" it is to simply add new axioms to ZF. One would think that AOC is obvious and of little consequence, but not so. It takes one away from the world in which we actually live and plunks us down in another universe.

For a more palatable paradox, there is Braess' Paradox (which has been employed in a number of major cities):

For each point of a road network, let there be given the number of cars starting from it and the destination of the cars. Under these conditions, one wishes to estimate the distribution of traffic flow. Whether one street is preferable to another depends not only on the quality of the road, but also on the density of the flow. If every driver takes the path that looks most favourable to them, the resultant running times need not be minimal. Furthermore, it is indicated by an example that an extension of the road network may cause a redistribution of the traffic that results in longer individual running times

You can create the number line with the null set. Let {0} = the null set:

{0}
{0}{0}
{0}{0}{0}
{0}{0}{0}{0}...etc

= 1, 2, 3, 4...: — EnPassant

And then come the fractions . . .

I read that Søren Kierkegaard self-published his books. So don't discard this idea. :smile:

But if they're to be resolved, it must be in and on the ground from which they came . . . — tim wood

One of the weirdest is Banach-Tarski, and that one arises if one assumes the Axiom of Choice. Discard the AOC and it goes away. Another, the Diagonal Paradox, is simply a matter of perspective and magnification.

Physics without math is Philosophy — Gnomon

Possibly. But in the quantum world much of the physics is the math. For example, a virtual particle may simply be a mathematical entity, an yet amateur philosophers may refer to these things popping in and out of physical existence.

I recently read of Feynman's lectures at Cal Tech back in the 1960s. Even there, with very bright physics students, most were unable to understand his attempts at explanation. A few followed the math and comprehended his ideas, but the majority were left in a state of confusion. When he first presented his "sum of all paths" formulation to an audience consisting of prominent physicists, very few initially comprehended what he was saying.

Physics without math is philosophy - if the philosopher is a physicist. That's my opinion, but I'm sure others here will disagree. That's OK. :cool:

Mathematics, properly understood, is the theory of everything. — Syamsu

:cool:

Perhaps the best overall model is that of a quantum wave. — A Seagull

Not quite the same as particle or interference wave. It's more a mathematical device for predicting certain quantum properties.

In mathematics it's easy to find examples of a passage of time with no change, as well as a change at an instant. Is that possible in the physical world? :chin:

A number line is an irrational conflation . . . that's why the idea creates so many problems — Metaphysician Undercover

It makes doing mathematics like walking across a minefield! :fear:

From NOVA online:

One of the missions of the Large Hadron Collider (LHC), the behemoth accelerator straddling the French-Swiss border, has been to test the possibility of unseen extra dimensions . Since the discovery of the Higgs Boson in 2012, completing the Standard Model of particle physics, the idea of looking at such extensions has become more central.

To establish the existence of extra dimensions with the LHC, there are three major avenues of attack. The first involves finding echo versions of existing particles, called Kaluza-Klein states. These would be like the known particles in all respects, except more massive, like overtones in music. At a proton-proton collision energy of 7 trillion electron volts, searches have been made for Kaluza-Klein gravitons, Kaluza-Klein gluons and others, so far to no avail.

Physicists are also using the LHC to search for evidence of gravitons seeping into higher dimensions. Such signals of otherwise unexplained missing energy would have to be sifted from enormous numbers of collision events, carefully ruling out a plethora of more mundane possibilities, such as escaped neutrinos.

Evidence for extra dimensions could also show up at the LHC in the form of microscopic black holes, predicted by certain higher dimensional theories. Famously, before the LHC opened, alarmists raised a fear of such objects destroying the Earth, despite calculations showing they would harmlessly decay within a tiny fraction of a second. Despite the hopes and warnings, miniature black holes have yet to be detected among the collision data of LHC experiments.

Currently, the LHC is switched off and being revamped in preparation for cranking up its collision energy almost twice as high as the previous run. In 2015 it is expected to reopen and collide protons at 13 trillion electron volts, offering the possibility of producing more massive particles and more unusual events. The upgrade will offer a greater chance to detect evidence of extra dimensions.

Where is time in a changless world? — prothero

A photo exists in a changeless environment and freezes time, but examining the photo involves time's passage. Do you really think that a changeless world could exist beyond the confines of a philosophical argument? Could there be places in deep space where there is zero change? And is change dependent upon an observer? We would see a ship approaching a black hole as slowing to a stop, but those on the ship would perceive normal movement.

Our days runnith over with confirmation bias.

See, here it is, the desire to know, right here at the base of mysticism — Metaphysician Undercover

I agree there is the desire to know what may be lying beyond everyday reality, to experience something beyond the normal world. Teresa of Ávila had the burning desire to know Jesus. A Zen student might want to know the truth about one's "I" . But too great a desire may very well be a hindrance to knowledge.

Nice presentation.

A little like the riddle, What is it that has quadrupedal movement, roams grasslands, prone to human domestication, gets hungry, etc etc ?

Would you say this set is NP-complete? — tim wood

Create an algorithm to list all the penguins. Then take the complement in the Universal Set.

This crops up in the Wiki article Talk section:

"Non random probabilistic flow of information in the Planckian order of magnitude"

Whereas, in the article: "In physics, energy is the quantitative property that must be transferred to an object in order to perform work on, or to heat, the object"

Not sure a random physicist could do much better than this.

Yet I am sure that a set that contains itself can be defined - I'm just not clever enough to think of it — tim wood

X={X}

:roll:

"...and among mathematicians only a small fraction are working in or at least interested in foundations." How true!

In the article in Wikipedia:

In 1923, Ludwig Wittgenstein proposed to "dispose" of Russell's paradox as follows: The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself.

In practice this is nonsense as well, IMO, although in some abstract sense it may have weight. In iteration the first application is f(f(z)). So the outer f acts upon a function value and not on the function itself. Fractals arise from these processes. Perhaps f(f) makes less sense. If F is a functional, then neither F(F(z(t))) nor F(F) is normally well-defined.

Matter is empty, so energy fills it up. — Braindead

Goes to show that physics can be both speculative and poetic. :smile:

Wasn't something being proposed about creative writing Baden ? Is that to be limited to Fiction ? — Amity

Obviously not, reading posts in this forum. :smirk:

{a} is a subset of A and A is a subset of X therefore {a} is a subset of X — EnPassant

{A} is a subset of X, not A. Ask fdrake or fishfry to explain this stuff to you. I'm done.

When I write a dynamical systems program to obtain an image, I determine the image. When the program runs, it causes the image to appear. — jgill

This doesn't sound like a very useful program that only displays one image - the one you determined. Computer programs are useful when they can be applied to create various images for different people based on the input from different users. The programmer doesn't necessarily know what images the program will generate because they are aren't aware of all the different kinds of input from different users. We can try to guess, but we can't account for every instance, which is why programs can have bugs. — Harry Hindu

What does "useful" have to do with this example? In fact, the programs I write have a number of different input parameters that I alter frequently to obtain new imagery. I don't design these programs for anyone but myself. But I am so happy you attempt to educate me, explaining why programs may have bugs. I feel so stupid. :yikes:

{a} is in X (because {a} is in A, B, C,...) — EnPassant

No, {a} is not "in" A,B,C,...

IMO this stuff is not worth the effort.

Set A = {a, w}
Set B = {a, x}
Set C = {a, y}

Set X = the set of sets that have {a} as an element. — EnPassant

None of them do. They have "a" as an element. D={{a},z} does.

This stuff is deleterious to mental health. :scream:

I am having some trouble thinking of any well-defined set that does contain itself. Help? — tim wood

x={x}

Nonsense, however.

Set theory can drive a person nuts. :worry:

5). Reversability. Almost all mechanism of physics are reversible and work equally well backwards as forwards. — Benj96

The formulae that involve a time variable seem to have this property. But keep in mind they merely describe phenomena and are not themselves phenomena. The reversibility might just be a quirk of the mathematics. A real physicist should comment. :chin:

When I write a dynamical systems program to obtain an image, I determine the image. When the program runs, it causes the image to appear. Sorry this is such a trite example. :worry:

Sorry. I misread this as "dermatology". :meh:

Yes, but the limit can be defined independently of time. — EnPassant

Of course it can. I merely mentioned a kind of isometry between iteration and time. I deal with infinite sequences whenever I dabble with research, and I rarely consider a correlation with the passage of time.

In mathematics infinity is a set, such as Aleph Null, not a process. Infinity is not 'the biggest number' it is all numbers, together. — EnPassant

More or less true in set theory, a particular branch of mathematics. My area was complex analysis and when I deal with the concept of infinity it is in the sense of unboundedness of sequences or processes. :cool:

Thanks for your comments. :smile:

The discussion is as comprehensible and consequential as highly abstract and modern areas of mathematics appear to be to me, and I was a professional.

Nice presentation. I hadn't thought about your #6 ( the second one) :cool:

Will Biden be able to contend with the pressure exerted by Stacey Abrams? Will he buckle?

But a sequence is not the same as time — Banno

But can be thought of as correlating with linear time, each step separated from the next by a short period of time. Maybe an isometry, but that's not quite right. Unimportant except for philosophers. :smile:

Perhaps fundamental to the way I present my view of "panpsychism" are terms like "mind in nature" or "panexperientialism", "non conscious experience" , "mentality", "psychialism". — prothero

Wiki: Mentality may refer to:
1. Mindset, a way of thinking
2.The property of having intelligence
3. Mental capacity, a measure of one's intelligence, the sum of one's intellectual capabilities

Present your definition of mentality, please. The Stanford article on panpsychism refers frequently to mentality, but I couldn't find a clear definition of the word in that context. Being a math person I prefer an intelligible presentation of basic definitions.

Is this what you mean: "self organizing and self sustaining systems in physics, chemistry and biology." ?

Counting infinity has nothing to do with time. — EnPassant

In a sense. However, as a mathematical analyst, when I iterate w=f(z)=z+1, starting with z=1, the process is unbounded and hence is said to diverge to infinity. In a computer program each iterative step requires a tiny amount of time, so time is tied in with this notion of counting infinity, although in theory the pace is arbitrary. Is it possible to think of a process that counts to infinity and does not require a step-by-step procedural? Certainly the concept of the set of counting (natural) numbers as a theoretical entity is not tied to time. :chin:

Species. :roll:

More so than rocks I wager. But equines in general do reproduce. Is a mule an equine?

I wonder what time seems like to a person suffering from continuous amnesia? No memories from second to second. Does time "pass"?