Ratzinger
If an object goes from A to B it passes infinitely many points. How can the object pass an infinite number of points in a finite amount of time? If the object was at every moment (infinitley small time span) at a point (infinitely small distance), then how is motion possible?
So that's the paradox of motion that mathematical thinking deals with.

But then you got physics where when you go down in size you hit the quantum world.

So what implication has physics for the paradox of motion? Is there no infinite small in physics, but small means quantum world here?

?

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Ratzinger said:
If an object goes from A to B it passes infinitely many points. How can the object pass an infinite number of points in a finite amount of time? If the object was at every moment (infinitley small time span) at a point (infinitely small distance), then how is motion possible?
So that's the paradox of motion that mathematical thinking deals with.

But then you got physics where when you go down in size you hit the quantum world.

So what implication has physics for the paradox of motion? Is there no infinite small in physics, but small means quantum world here

?

Quantum physics has observation built right into it. So one time you observe the particle HERE, and some other time you observe it THERE. What happens in between is described as the evolution of the wave function, which doesn't happen in spacetime, but does behave classically (i.e. in accordance with measure theory).

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If an object goes from A to B it passes infinitely many points. How can the object pass an infinite number of points in a finite amount of time?

Why couldn't it?

In general, these questions are pseudoparadoxes -- they're statements that go against (some people's) intuition, but they are lacking in any actual reason why there's a problem.

Since it seems that you

(1) find it self-evident that objects do appear to pass an infinite number of points in a finite amount of time,

and since you

(2) have not given a reason why objects should not appear to pass an infinite number of points in a finite amount of time,

(3) objects can pass an infinite number of points in a finite amount of time.

Ratzinger

Vague talk? That's 'the arrow' version of Zeno's paradox. For Bertrand Russell "a plain statement of an elementary fact". For me, too.

Quantum physics has observation built right into it. So one time you observe the particle HERE, and some other time you observe it THERE. What happens in between is described as the evolution of the wave function, which doesn't happen in spacetime, but does behave classically (i.e. in accordance with measure theory).

Yes, but wave functions ain't no classical trajectories of particles. What is motion, dynamical variables in the quantum world?

Why couldn't it?

In general, these questions are pseudoparadoxes -- they're statements that go against (some people's) intuition, but they are lacking in any actual reason why there's a problem.

Since it seems that you

(1) find it self-evident that objects do appear to pass an infinite number of points in a finite amount of time,

and since you

(2) have not given a reason why objects should not appear to pass an infinite number of points in a finite amount of time,

(3) objects can pass an infinite number of points in a finite amount of time.

In the real world, of course, things do move. But taking this self-evident fact together with Zeno's considerations makes it paradoxical and doesn't refutes it.

Do you mind taking a look at this?
http://www.mathpages.com/rr/s3-07/3-07.htm

Staff Emeritus
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Yes, I'll say vague -- I've never been impressed by any presentation of Zeno's arguments.

Compare with, say, the Liar's paradox, in which I can formally derive a contradiction:

If P is the statement "P is false", then we have:

P = T or P = F
P = T --> P = F
P = T --> P = T and P = F
P = F --> P = T
P = F --> P = T and P = F
P = T and P = F

This is a real paradox. It, and other similar paradoxes, are the reason why the usual formal logic is designed in such a way that statements cannot refer to themselves (even indirectly!)

I've never seen anyone do anything remotely similar with Zeno's paradoxes. I'm assuming, I suppose, that you're not taking as an postulate the statement:

"objects cannot pass an infinite number of points in a finite amount of time"

since it seems to me a rather silly argument to say:

"Well mathematically, objects pass an infinite number of points in a finite amount of time... but I'm assuming objects cannot pass an infinite number of points in a finite amount of time... therefore there's a paradox!"

but, I guess this would be an actual paradox if you are actually making this assumption.

The problem is that people rarely give a reason for that assumption, and seem much more willing to discount hundreds of years of successful physics and mathematics instead of some postulate for which they cannot even begin to defend.

I have seen it defended in terms of "tasks", but that approach begged a different question -- it assumed that in any "doable" collection of tasks, there must be a first and a last task. (Which is equivalent to assuming that only finite collections of tasks are "doable")

Ratzinger
Alright, I see your point here. Very interesting. I have to take this in.

What is about getting smaller and smaller? Conceptually and mathematically I can go down in scale as deep as I like. If there is a distance, area, volume or a period, I can keep on enlarging or slicing it up as much as I want. Never any dramatic happens, I can go on forever.
But different story in the physical world. When going down here, dramatic things do happen.

So what does that mean for math and physics and their relationship?

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Ratzinger said:
Alright, I see your point here. Very interesting. I have to take this in.

What is about getting smaller and smaller? Conceptually and mathematically I can go down in scale as deep as I like. If there is a distance, area, volume or a period, I can keep on enlarging or slicing it up as much as I want. Never any dramatic happens, I can go on forever.
But different story in the physical world. When going down here, dramatic things do happen.

So what does that mean for math and physics and their relationship?

Well, I assume you are talking about quantum mechanics? Or quantized spacetime? Neither one postulates discrete space or time (I assume you are still with Zeno). "Quantized" is not the same as "discrete". In quantized systems what we observe, through the reduction of the wave function is a possibly discrete set of OUTCOMES, for example in LQG they have such "eigenvalues" of area and volume. Quantum theories do not contain an account of what the quantized object is or does between observations. And they are all dependent on the continuous (indeed analytic) behavior of the wave function.

Besides which your imagination is fine as far as it goes, but if you study topology a little more you'll be able to imagine a much richer class of spaces.

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But different story in the physical world. When going down here, dramatic things do happen.

I suppose you mean the behavior of matter -- but that is a question about matter, not space-time itself. To date, there has not been one piece of evidence for space-time itself to look or behave any differently on small scales than on large scales.

On the small end, as we continue to improve the resolution of our scanning devices to peer into smaller and smaller realms, it still looks like a continuum.

On the large end, we don't see the blurring of incredibly distant objects that is predicted by many discrete theories.

The only evidence we have is very indirect -- some of the work in attempting to develop a theory of quantum gravity results in mathematics, for which it might be reasonable to interpret as a discretized space-time. Of course, some other work suggests that what's going on is incredibly complicated, rather than something as nice as simple as discreteness.

As for the relationship beween physics and mathematics, from some viewpoint, Physics is just the science of trying to select, among all possible mathematical models, which best represents the universe. If a continuum turns out to be a bad choice, then you just adopt some other mathematical model.

sd01g
Hurkyl said:
Yes, I'll say vague -- I've never been impressed by any presentation of Zeno's arguments.

Compare with, say, the Liar's paradox, in which I can formally derive a contradiction:

If P is the statement "P is false", then we have:

P = T or P = F
P = T --> P = F
P = T --> P = T and P = F
P = F --> P = T
P = F --> P = T and P = F
P = T and P = F

This is a real paradox. It, and other similar paradoxes, are the reason why the usual formal logic is designed in such a way that statements cannot refer to themselves (even indirectly!)

")

It seems to me that the statement 'This statement is false' is not the same as 'The statement is false'. 'THIS statement is false' is a paradox while 'THE statement is false' refers to a statement that may or may not be false. I was wondering if you agree or disagree?

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You mean P(Q) is the statement "Q is false"? It still leands to paradoxes in naive logic...

Consider the statement G = P(G)

If P(G) is true, that means G is false, which means P(G) is false.
If P(G) is false, that means G is true, which means P(G) is true.

In (the usual) modern approach, we have a strict hierarchy of formulae. A first-order formula is never allowed to take a formula as input. A second-order formula is allowed to take first-order formulae as input. A third-order formula is allowed to take first-order or second-order formulae as input, et cetera.

This way, it is impossible to construct a literal self-reference like the liar's paradox, or the above variant.

El Hombre Invisible
I've never read a particularly satisfactory solution to Zeno's paradox, but then I've never read a particularly satisfactory expression of it. Most solutions take the vantage that Zeno was unaware that an infinite series can sum to a finite number, and maybe that's where Zeno's problem lay, but this is basically the same as saying the particle DOES move through an infinite number of co-ordinates in a finite time. Zeno's point was to break the process down into more and more fundamental steps - an infinite sum just ignores the problem entirely by integrating it back to total distances and time intervals. I think Zeno's question was more profound than this.

Zeno's question presupposes that Achilles and the tortoise have definite positions at any given time (cross-refer to his arrow paradox) and definite momenta, but does not consider the dimensions of the runners, suggesting he was thinking of them as particles. QM now teaches us that particles do not propagate in this way, but do so as waves. However, Zeno's posing of the question leads to incompatibilities with QM, such as the fact that it does not consider the possibility of NOT finding a particle at a given position and time, and does not foresee observation as having an effect on those particles, as well as HUP problems.

On a classical, macroscopic level, would the following solution (that does not avoid the problem of increasingly fundamental intervals) be appropriate? Model the progress of Achilles and tortoise as six particles, two representing the legs of Achilles and four representing the legs of the tortoise. An observation of position at a given time is an observation of each of these six particles as being in contact with the ground, or not. This does two things: it removes the problem of an infinite number of points while honouring Zeno's approach to increasingly fundamental steps, but it also introduces the possibility of NOT finding a particle at a given time and, in the case of Achilles, not finding any of the particles at a given time. These non-observations simply have to be not counted when marking the progress of the runners as a whole and at least one further observation taken. There will be a minimum of one interval, possibly two or three, at the beginning of which Achilles is behind the tortoise and at the end of which Achilles is in front of it.

The reason there may be three intervals is that you might have measurement 1 as both Achilles particles behind the closest tortoise particle, measurement 2 as either a) one Achilles particle level with/in front of any given tortoise particle and the other behind the closest, or b) one or no Achilles particle observed, and measurement 3 as one Achilles particle level with any given tortoise particle and the other in front of the foremost. Problem solved.

Treating these particles as quantum particles, would it be fair to say that Zeno's paradox, and in my opinion the actual crux of the problem (how do particles propagate through space and interact with observers) is similar, insofar as we would ignore anything that is not measured and only count observations that yield results? And treating Achilles and the tortoise as complex quantum systems, this question is still pending?

Of course, this entire post takes the point of view that the infinite sum solution missed the point, about which I could be wrong.

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The big thing missing from statements of Zeno's paradox, as I've said, is any reason to think that a problem has arisen.

Some people just don't seem to like the "no problem has been demonstrated" response, even if it is accurate. *sigh* So, to get through to those people, a responder has to make their best guess at the missing reason why people think there is a problem.

The typical guess is that someone is thinking "how can you accomplish infinitely many tasks in finitely much time?" and then one demonstrates how adding up the time it takes to do each individual task yields a finite result.

The next most popular thought, I think, is to look at the "last task". Of course, the problem with that thought is that there isn't any particular reason to think that, in some collection of tasks, there must be a final task. (Except for hastily generalizing one's intuition about finite sequences)

Of course, if someone is thinking there's a problem for yet another reason, yet a different response would be needed.

Ratzinger
I would like to come back to getting smaller and smaller in the physical world vs. in math.

Two pillars of contemporary physics support the expectation that as we resolve the fabric of spacetime with an imaginary microscope at ever smaller scales, spacetime will turn from an immutable stage into the actor itself. First, due to Heisenberg's uncertainty relations, probing spacetime at very short distances is necessarily accompanied by large quantum fluctuations in energy and momentum - the shorter the distance, the larger the energy-momentum uncertainty. Second, according to Einstein's theory of general relativity, the presence of these energy fluctuations, like that of any form of energy, will deform the geometry of the spacetime in which it resides, imparting curvature which is detectable through the bending of light rays and particle trajectories. Taking these two things together leads to the prediction that the quantum structure of space and time at the so-called Planck scale must be highly curved and dynamical.

So what is with highly curved and dynamical at the Planck scale? And again:

What is about getting smaller and smaller? Conceptually and mathematically I can go down in scale as deep as I like. If there is a distance, area, volume or a period, I can keep on enlarging or slicing it up as much as I want. Never any dramatic happens, I can go on forever.
But different story in the physical world. When going down here, dramatic things do happen.

So what does that mean for math and physics and their relationship?

moving finger
El Hombre Invisible said:
Most solutions take the vantage that Zeno was unaware that an infinite series can sum to a finite number, and maybe that's where Zeno's problem lay,
That's your solution - the rest of your post is (with respect) redundant

MF

moving finger
Ratzinger said:
What is about getting smaller and smaller? Conceptually and mathematically I can go down in scale as deep as I like. If there is a distance, area, volume or a period, I can keep on enlarging or slicing it up as much as I want. Never any dramatic happens, I can go on forever.
But different story in the physical world. When going down here, dramatic things do happen.
So what does that mean for math and physics and their relationship?
I don't understand where you are having a problem. Are you trying to compare a continuous function (the mathematical concept of the real number line) with a possibly discontinuous function (the quantum physical world)? If so, why on Earth would you expect the same story, why would you expect a perfect relationship?

MF

ps - Zeno's paradox implicitly assumed a continuous function

dubmugga
Ratzinger said:
I would like to come back to getting smaller and smaller in the physical world vs. in math.
So what is with highly curved and dynamical at the Planck scale? :

pardon my wandering thoughts but...

...my guess would be swirling bubbles of light surrounding a specific type of vacuum peculiar to our universe or section thereof ?

this brings to mind exactly what is light made of does it have mass and therefore gravity and how does it travel in spacetime or what if it is spacetime itself ?

I was also thinking, time must warp at the Planck scale which suggests a blipping in and out either to another place in the megaverse or an alternately negatively charged negaverse as part of a multiverse...

In that scenario of a negatively charged symmetric partner to our whole 3+1d universe does that imply background dependence and what of the ramifications for no locality?

...nothing moves only the background changes shape but because we are locked in the system and the universe reconstitutes itself at superluminal speed we can't tell the difference nor measure it

but anyway back on topic...

Why are there an infinite amount of points within a finite distance between A and B or is B an infinite length away from A in which case the time taken to travel between the 2 points would be infinite ?

...does non locality and blipping in and out of our universe mean that time is meaningless at superluminal speed and also at distances less than Planck scale distances and greater than the radius of which the universe is inflating ?

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Why are there an infinite amount of points within a finite distance between A and B
Assume that there exists a line segment with only finitely many points on it.

Then we can call that line segment L.

Since L has only finitely many points on it, and we can order the points on L, we can find two points with no other points between them.

Then we can call such a pair of points P and Q.

We know from high school geometry that we can construct the midpoint between any two points.

Then, there is a point that is half-way between P and Q.

Let's call that point R.

R cannot exist, because there are no points between P and Q. But we've proven R exists. This is a contradiction, so our initial assumption is incorrect.

Therefore, all line segments have infinitely many points lying on them.

dubmugga
^^^surely the shortest distance between 2 points is zero if the 2 points are touching, making it impossible to add another point between finite points on a line...

...so you can't get an infinite number of points on a line !

If 2 points are touching they become 1 as there is no distance between P & Q to have a halfway point between?

does that make sense ?

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Before I respond, do you understand the concept of a proof by contradiction, also known as reducto ad absurdum? (If I spelled it right)

dubmugga
^^^if by that, you mean that by taking arguments to an extreme where it becomes absurd and beyond the confines of logic just to prove a point then yeah i think so...

...kinda like if you can't dazzle em with brilliance baffle them with bull**** ?

Staff Emeritus
Gold Member
Then no, you don't.

A proof by contradiction is when:
(1) You make an assumption.
(3) You conclude the assumption was false.

Why can you make that conclusion? Because the assumption you made obviously cannot be true. (Because it leads to an absurd conclusion) Therefore, it is false.

When you make the assumption that there are two (Euclidean) points that are "next to each other", it leads to the absurd conclusion that we cannot construct the midpoint of these points, despite the fact we know full well how to do so.

Therefore that assumption cannot be true: it cannot be true that there exist two points that are "next to each other".

Suppose there are n points between A and B, where n is any number from 0 to infinity. As n goes to infinity, the distance between two consecutive points becomes infinitely small, hence the time needed to traverse this distance (given some velocity bigger than 0 ) also becomes infinitely small. The sum of an infinity of infinitely small values in this scenario must be finite, hence i can traverse the infinite number of points between A and B in finite time.

To go further, saying that we can't go from point A to B in finite time because there are infinitely many points between A and B is like saying that since a metal rod of any finite size, say 1 Meter, can be divided into infinitely many pieces, then the metal rod must have infinit length. In this scenario it's easy to see that though we may divide the rod into an infinity of pieces, these pieces are infinitely small so that if i glue them back together the rod will have the same length as it had before we touched it. Hence the sum of infinitely many infinitely small values as described above must be finite.

Applying this strategy to time, we say that just because i divide a time interval into an infinity of infinitely small time intervals, does not make the time interval infinitely large. Hence the total amount of time necessary for me to traverse all infinite points from A to B given a velocity bigger than 0 is very finite.

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dubmugga
yeah thanks for that Hurkyl but you're trying to baffle me instead of dazzle me...

...of course points on a line are "next to each other" where would you say they are in relation to each other ?

Is the shortest distance between 2 points zero or infinity on a small scale ?

and going back to what Ratzinger suggested are we talking imaginary numbered points that take up no space in our 3+1d or Planck size points which are supposedly indivisible cos that 1m metal bar will hit the wall of a singularity meaning at some stage meaning it's make up is of a finite nature unless...:)

My answer would be, the shortest distance between 2 points is zero given that the 2 points occupy the same space. This can only happen if they don't occupy it at the same time...non-locality

...so in effect, if one point blips out while another blips in in the same space at superluminal speed, no distance has been traveled it has just gone beyond the confines of our universe in such a time as to make time itself irrelevent

but coming back to the initial example of points on a line...

...would it be fair to say there is no line only a collection of finite points all so close to each other as to appear touching and giving the appearance of a line ?

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Gold Member
yeah thanks for that Hurkyl but you're trying to baffle me instead of dazzle me...
I'm trying to do neither. I'm trying to explain a proof to you, as well as the (important!) concept of a proof by contradiction.

of course points on a line are "next to each other"
Why "of course"? What would lead you to believe such a thing?

where would you say they are in relation to each other ?
Separated.

I really can't make any sense of the rest of your post.

dubmugga
I'm sure you can make and have made sense of the rest of my previous post...

...but anyway, take your 2 points on a line and reduce the distance between them to zero

they are now next to each other and touching, separate but together...

...individually they are points but together they form a line. A finite line of finitely 2 points and because they are touching you can't half the distance between them as half of zero is zero

so please tell me of the importance of proof by contradiction and where it applies mostly, can you apply it to non locality ?

If the distance between two points is 0 then their coordinates must be the same, the same point. If we see points as having a size and not being non-entities as is traditional then we'll run into the problem of points having endpoints, and these having endpoints as well and so on. We need points as non-entities.

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dubmugga
thanx for explaining that Job...

...unfortunately metal bars and all such physical objects are not made of non entity points

it seems we only need non entity points to make absurd assumptions seem true or false by contradiction...

...but do we really need them ?

so what do you think of point entities occupying the same space but not the same time and changing shape to accommodate the perception of movement ?

A system using points as entities can be implemented in a system that uses points as non-entities, we would just call your entity points something different (to avoid confusion) like "paint", and say paints are entities which are bounded by these points.
Then, if you want to ignore points and use paints you can easily do it. So points can describe anything paints can but have other worthwhile abilites, like, for instance, in math, the function f(x) = 1/x^2 as x goes to infinity gets infinitely small which is hard to conceive using paints and not points because f(x) gets smaller than a paint. Also, suppose we establish the area of a paint, then, we are establishing that no particle will be smaller than a paint, and although it may be true that if we keep zooming into matter: molecules, atoms, protons and electrons... and so on, that eventually there will be a particle that is smaller than the rest, right now that's not a safe assumption to make, and if we were to make it, what value would we use for the area of a paint? If we knew these values we might learn something about the nature of movement, like whether movement is continuous [a particle in movement in the time interval t1, t2 no matter how small t1,t2 is (>0) always moves a little in that interval] or periodic [between t1 and t2 the particle moves, and between t2 and t3 it is stopped, even though the particle is generally in movement]. Suppose movement is continuous then paints aren't enough, otherwise they may be.

mathphys
Hurkyl said:
Assume that there exists a line segment with only finitely many points on it.
Then we can call that line segment L.
Since L has only finitely many points on it, and we can order the points on L, we can find two points with no other points between them.
Then we can call such a pair of points P and Q.
We know from high school geometry that we can construct the midpoint between any two points.
Then, there is a point that is half-way between P and Q.
Let's call that point R.
R cannot exist, because there are no points between P and Q. But we've proven R exists. This is a contradiction, so our initial assumption is incorrect.
Therefore, all line segments have infinitely many points lying on them.
Hi Hurkyl
Ejmm, excuse me but am i missing something? There is something disconcerting with your example here. Because you can start with 'high school geometry is true ' (implicitely you are). Then you know that there are infinitely many points in a line. Then it makes no sense (to me) to talk about a line with a finite number of points and then to try to prove that this is a contradiction using the middle point thing and then conclude that your initial premise is true. Ab initio you know there are infinitely many points in between of two points of a line, so what's the logic behind the example?

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Staff Emeritus
Gold Member
I've certainly fogotten the context of this thread, and the point of your response is lost upon me, so I will make a generic answer, and hope I cover it. I was sketching the proof that, in Euclidean geometry, that any line segment has infinitely many points.

When talking about lines, the application of Euclidean geometry is fair game, because, by default, that is what one means when speaking about lines. I seem to recall this thread was plagued by some fairly erroneous reasoning, rather than an attempt to offer up and discuss some different concept.

mathphys
I see your point. It was just that your proof put in my mind the idea of the real line (because of the 'we can order ' (<--specially this one , it is and interesting issue how we can ) and 'midpoint') and Dedekind cuts. Althought that is not high school euclidean geometry. Last edited:
Eyesaw
Hurkyl said:
Assume that there exists a line segment with only finitely many points on it.
Then we can call that line segment L.
Since L has only finitely many points on it, and we can order the points on L, we can find two points with no other points between them.
Then we can call such a pair of points P and Q.
We know from high school geometry that we can construct the midpoint between any two points.
Then, there is a point that is half-way between P and Q.
Let's call that point R.
R cannot exist, because there are no points between P and Q. But we've proven R exists. This is a contradiction, so our initial assumption is incorrect.
Therefore, all line segments have infinitely many points lying on them.

Your starting assumption is that there is a line segment with finite
points on it. Then you introduce an assumption (high school
geometry of constructing the midpoint between any two points)
starting assumption is contradictory. That is a fallacy.

That is, your definition of a line segment in the first assumption
is one that has finite points. Your "high school geometry's
construction of the midpoint between any two points" relies
on a definition of a line segment as one that has infinite
points on it. If you use two opposite definitions for a line
segment, you will obviously prove the two definitions contradict
each other.

Staff Emeritus
Gold Member
Your starting assumption is that there is a line segment with finite
points on it. Then you introduce an assumption (high school
geometry of constructing the midpoint between any two points)
starting assumption is contradictory. That is a fallacy.
No, it's called a proof by contradiction. Even if you think it's a trivial proof, it is still not a fallacious proof.

Incidentally, there are geometries whose lines have finitely many points, and yet any pair of points has a midpoint.

And finally, I have never seen a setting where the fact that a line has infinitely many points was part of the definition, rather than proven as a theorem from the axioms.

mathphys
You are right, it is just that in the context of euclidian geometry the premise that there is always a midpoint between two arbitrary points P, Q on a line segment implies that there are infinitely many points between them. That is what i found disconcerting as part of the proof.

Nevertheless, lines and segments are abstract undefined objects in eucliden geometry as you mention. If you take the existence of the mid point as an axiom for objects called lines (or a consequence of more fundamental axioms), a property they have to satisfy, then it is clear(by your proof) that and object with a finite number of points cannot satisfy this 'line axiom', and hence is not a line.
That was the argument or intention of your post, i guess. But notice that just because you proved this way that an object with a finite number of points is not a line, that does not imply that a line has infinitely many points, that is independently implied by the midpoint premise.

Just starting from the midpoint argument , one may conclude that there are infinitely many points on a line, and this stablishes a necessary condition for sets of points to be 'considered' as lines, in euclidian geometry. Although not a sufficient one. That is way i associated inmediately with the real line.Nevertheless, again, when we said things as sets, and order, we are going out of the scope of euclidian geometry.

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Dmstifik8ion This is my first post so please feel free to chastize me and put me on the straight and narrow path; but

I sense a wide spread problem here involving the human perspective.
Reality (space/distance) is not made of points. Points are attributed to specific places (locations) in reality to enable our conceptual grasp and understanding of aspects and relationships in reality.

Might my insight and perpsective be better applied elsewhere and/or better appreciated in another area of this forum. Thank's for any comments and assistance.

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