God is laughing up His sleeve

[Peter_Plato]

Peter,
I think they do, but science doesn’t; space and time are still continuous quanties.

Yppop

My understanding is that there are three competing theories: 1) discrete, 2) continuous and 3) both.

There is some evidence that the discrete view may be the correct one but it remains an open question.

A… possible effect of discrete spacetime involves very high energy cosmic rays. More than 30 years ago researchers predicted that cosmic-ray protons with an energy greater than 3 × 1019 electron volts would scatter off the cosmic microwave background that fills space and should therefore never reach the earth. Puzzlingly, a Japanese experiment called AGASA has detected more than 10 cosmic rays with an energy over this limit. But it turns out that the discrete structure of space can raise the energy required for the scattering reaction, allowing higher-energy cosmic-ray protons to reach the earth. If the AGASA observations hold up, and if no other explanation is found, then it may turn out that we have already detected the discreteness of space.

From phys.lsu.edu/faculty/pullin/sciam.pdf

 

[JDaniel]

Sure.

Take a piece of paper and a pencil and draw a one inch line segment.

Now count the number of discrete points in it.

“Hold infinity in the palm of your hand…”

Cricket:

I really don’t have enough time for that. But, I have a better idea, which you can do, and you’re probably young enough that you have plenty of time.

Take that same 1 inch line on paper. Very carefully cut each point off of it - carefully maintaining each point’s current dimensions - and stack them, one on top of the other. Then, measure them with a ruler. Do you still get 1 inch? Or, more importantly, do you still get “infinity?” (If you would, could you please hurry it up as I don’t want to be too late for my funeral. )

God bless,
jd

 

[Cricket123]

Last try, then I give up:

Consider this triangle ADE, where segment DE has length 2 and segment BC has length 1

Now draw a line from point A through any point on DE.

This line will pass through one and only one point on segment BC and pair it with one and only one point on DE.

Any line you draw will do this.

Therefore, there is a one-to-one correspondence between the two sets of points. There are the same number of points in each set, even though one set appears as if it should be larger.

 

[JDaniel]

Last try, then I give up:

Consider this triangle ADE, where segment DE has length 2 and segment BC has length 1

Now draw a line from point A through any point on DE.

This line will pass through one and only one point on segment BC and pair it with one and only one point on DE.

Any line you draw will do this.

Therefore, there is a one-to-one correspondence between the two sets of points. There are the same number of points in each set, even though one set appears as if it should be larger.

Cricket:

I’ll try one more time, then I’ll give up. The point I am making is that a point is not a part of a line as it has no dimensionality. NONE. Where does one point leave off and the next point start? It doesn’t; in fact, neither of them do. One can only attribute a point on a line for purposes of delineating the possible location of a line segment. And, a line segment will never reduce to a point! A Planck length is not a point. It is an extremely small line segment.

Physics regards point particles as dimensionless. How do you think God was able to pack a huge, but finite, number of point particles into that singularity (besides his Omnipotence). When you start cutting on that line, you will ultimately come to a smallest “line segment,” but never a point.

My question was, please show me a real (actual) infinity. What you have done is to show me how an infinity might appear - if one were to somehow be able to actualize it.

God bless,
jd

 

[Cricket123]

… And, a line segment will never reduce to a point!

No argument with this, as a line is defined by two unique points.

My point is, once you have defined a line segment, you have created an infinite set (the points between point A and point B). So, if we let point A be zero, there is no “next number.” Any interval, however small, defines an infinite set whose cardinality is equal to the cardinality of R.

By defining that second point, you have defined an infinite number of them.

But perhaps we should just resolve it thus:

A mathematician and an engineer see a beautiful woman at a party. They approach her, each traveling half the distance every time. The mathematician never reaches her. The engineer gets close enough.

Peace.

 

[JDaniel]

No argument with this, as a line is defined by two unique points.

Cricket:

ONLY colloquially and ONLY abstractly. Those so-called “two points” are ONLY called “points” because we use that word equivocally. As to “two unique points,” that means two delimiters, i.e., two loci, one on either end of the line segment.

My point is, once you have defined a line segment, you have created an infinite set (the points between point A and point B). So, if we let point A be zero, there is no “next number.” Any interval, however small, defines an infinite set whose cardinality is equal to the cardinality of R.

Except that, as “points,” they do not belong to a line of any sort. If laid down side by side, where ever they would “touch” each other, they would coincide with each other, and no matter how many points were made tangent, they could never yield an extension, never yield more than the original indivisible character of the first point. (Com. on the Phys., St. Thomas Aquinas, Bk. VI, les. 1, n. 1454) So, you would be admitting that a point is infinite (which you could if you regard infinity as “zero”). But, if that were the case, then any and all existing objects consisting of matter would be infinite. That is not what we see, nor is it the way we have lived.

By defining that second point, you have defined an infinite number of them.

Per the above, there’s no such thing as a line segment that consists of points.

A mathematician and an engineer see a beautiful woman at a party. They approach her, each traveling half the distance every time. The mathematician never reaches her. The engineer gets close enough.

That’s good! :clapping:

Peace and God bless,
jd

 

[GEddie]

if that were the case, then any and all existing objects consisting of matter would be infinite. That is not what we see, nor is it the way we have lived.

That is not the case because although space is infinitely divisible, matter is not.

ICXC NIKA

 

[JDaniel]

That is not the case because although space is infinitely divisible, matter is not.

GEddie:

I am not saying that discrete space is infinitely divisible either. With the caveat, I agree.

God bless,
jd

 

[Ion]

Cricket:

ONLY colloquially and ONLY abstractly. Those so-called “two points” are ONLY called “points” because we use that word equivocally. As to “two unique points,” that means two delimiters, i.e., two loci, one on either end of the line segment.

Nothing in mathematics is equivocally. You did not have contact with axiomatic mathematics, where everything is well defined, ordered and established. Even for the most simple thing, to choose an element from a set, there is an axiom of free choice telling you that you can do it.

 

[Pat_Albertson]

Are you saying that the “visible universe” is merely a part of the “universe”, which is something infinitely large. So even if it expanded, the “visible universe” is still part of “something infinitely large” and hence is infinite because we are wrong to separate and refer to the “visible universe” as the “universe”. Here we run into a problem of definitions,. I will agree that defining the “universe” as that which includes both the “visible universe” and everthing that extends before and beyond, which is infinitely extended in both time and space, then you are correct and the “universe” is infinite. That definition is preferred by atheists, materialistic scientists, and pantheists because it argues against a beginning of the universe in order to imply that there is no God. I prefer the definition of “universe” that refers to the 3-dimensional manifold that contains all that can be observed and which you refer to as “the visible universe”. That which lies before and beyond the visible universe can only be the source and cause of the visible universe, such as the MInd of God.

You are, of course, free to choose either definiton; I will stick with the universe as a finite object and the implication that there is a God… Please correct me if I misinterpreted anything you wrote.

Yppop

Actually, I was making what I thought was a rather prosaic observation about the (presumably infinite) manifold we exist on (or in - you may take your pick of prepositions).

There are (theoretically speaking) supposed to be an infinite number of such manifolds, some very close in nature to our universe and others which are quite different with differing laws of physics.

Here is a wikipedia article discussing Tegmark’s taxonomy of parallel universes. I was making reference to the sort of manifold which would allow a “parallel universe” identical to ours after travelling a far enough distance; i.e. Level I: Beyond Our Cosmological Horizon. Whether or not a manifold which to our best knowledge contained our Hubble volume as a hyperbolic infinitesimal would, when measured more closely, turn out to be some large but finite shape, I don’t think anybody knows. I don’t think measurements have gone past the second derivative with respect to distance. The assumption, therefore, is that our universe is a hyperbolic infinitesimal in the same way that an infinitesimal part of some shapes (like a sphere (finite), saddle (probably infinite) or other shape) contains enough information in the zeroth, first, and second derivatives to determine its final overall shape, or whether the shape is more complex (and either finite or infinite) and needs more derivatives.

With respect to apologetics (and belief in God), I get my idea of God more from what happens when I pray than from (no disrespect meant) games with words and logic. The more mathematical (and logical) a thing is, and the less like standard prose, the better, but it’s still not ironclad, or at least for me it isn’t. Apologetics seems to me more like an entertaining pastime than a source of faith.

 

[Cricket123]

Except that, as “points,” they do not belong to a line of any sort. If laid down side by side, where ever they would “touch” each other, they would coincide with each other, and no matter how many points were made tangent, they could never yield an extension, never yield more than the original indivisible character of the first point. (Com. on the Phys., St. Thomas Aquinas, Bk. VI, les. 1, n. 1454) So, you would be admitting that a point is infinite (which you could if you regard infinity as “zero”). But, if that were the case, then any and all existing objects consisting of matter would be infinite. That is not what we see, nor is it the way we have lived.

Mathematically speaking, this is not correct. Points on a line are not little balls that are somehow tangent. They are continuous. And this is, in fact, what we see and how we live. If I throw a ball through the air, its motion describes a continuous arc.

A single point is, in a manner of speaking, zero; it is pure location. It is certainly not infinite. However, if you define two points, you invariably define an infinite set of them. Just as there is no “next number” after zero, if you have a point on a line, or line segment, there is no next point. Any interval however small defines an infinite set of points with the same cardinality as R.

An example of the implications of this continuity: .999… = 1. The proof is simple: 1/3 = .333…, 2/3 = .666… 1/3 + 2/3 = .333… + .666… = .999… = 1.

Note that this is not .999, but .999… Follow the decimal point with a billion or even a trillion 9s and you will not get to 1. But the infinite string of nines in the decimal expansion will get you there.

Some posts have mentioned the Planck length. Physics is not my area of expertise. Still, there is no reason why, mathematically speaking, one could not have “half a Planck length.” Whether such a thing is actually constructible or measurable is a separate issue.

I did, somewhat briefly, read an article that defined a Planck length as a measure of the oscillation of certain particle strings. Oscillation implies a kind of motion. It would seem that, if a particle were oscillating, it would have to reach a point in its oscillation halfway between the max and the min, no? Not the only defintion I saw, but still…

 

[GEddie]

Mathematically speaking, this is not correct. Points on a line are not little balls that are somehow tangent. They are continuous. And this is, in fact, what we see and how we live. If I throw a ball through the air, its motion describes a continuous arc.

A single point is, in a manner of speaking, zero; it is pure location. It is certainly not infinite. However, if you define two points, you invariably define an infinite set of them. Just as there is no “next number” after zero, if you have a point on a line, or line segment, there is no next point. Any interval however small defines an infinite set of points with the same cardinality as R.

An example of the implications of this continuity: .999… = 1. The proof is simple: 1/3 = .333…, 2/3 = .666… 1/3 + 2/3 = .333… + .666… = .999… = 1.

Note that this is not .999, but .999… Follow the decimal point with a billion or even a trillion 9s and you will not get to 1. But the infinite string of nines in the decimal expansion will get you there.

Some posts have mentioned the Planck length. Physics is not my area of expertise. Still, there is no reason why, mathematically speaking, one could not have “half a Planck length.” Whether such a thing is actually constructible or measurable is a separate issue.

I did, somewhat briefly, read an article that defined a Planck length as a measure of the oscillation of certain particle strings. Oscillation implies a kind of motion. It would seem that, if a particle were oscillating, it would have to reach a point in its oscillation halfway between the max and the min, no? Not the only defintion I saw, but still…

A good illustration of this would be Jose Luis Borges’ “Book of Sand”.

The protagonist in this short story acquires a book of normal size, but having an infinite number of infinitely thin pages. Between any page and the apparent next, there were still more pages. Once having closed the book, there was no way of finding one’s place again.

The story is online; google it if you care to.

ICXC NIKA

 

[JDaniel]

Mathematically speaking, this is not correct. Points on a line are not little balls that are somehow tangent. They are continuous. And this is, in fact, what we see and how we live. If I throw a ball through the air, its motion describes a continuous arc.

Cricket:

If I throw a ball in the air, it is continuously in motion, as a ball-in-motion. If I stack a billion “points” on top of each other, which are not little balls, but are dimensionless, no extension (dimension) can ever be achieved.

A single point is, in a manner of speaking, zero; it is pure location.

Exactly! It is a mathematical abstraction, i.e., there is no sensible matter. A ball-in-motion is matter-in-motion, but the imaginary “points” across which it allegedly passes through, are mental constructs (much like “time”).

It is certainly not infinite. However, if you define two points, you invariably define an infinite set of them.

Again; they are mental constructs. The “points” that (appear to) make up the lattice of space-matter, are, because they are dimensionless, immersed in infinite continuous space, and must not displace that which is infinite - otherwise, that which is infinite is no longer actually infinite. The Higgs boson - postulated 50 years ago - is supposed to show us how mass is amassed.

But the infinite string of nines in the decimal expansion will get you there.

In my mind.

Some posts have mentioned the Planck length. Physics is not my area of expertise. Still, there is no reason why, mathematically speaking, one could not have “half a Planck length.” Whether such a thing is actually constructible or measurable is a separate issue.

Which was my point.

I did, somewhat briefly, read an article that defined a Planck length as a measure of the oscillation of certain particle strings. Oscillation implies a kind of motion. It would seem that, if a particle were oscillating, it would have to reach a point in its oscillation halfway between the max and the min, no? Not the only defintion I saw, but still…

Except: that presumes space, i.e., an extended medium, within which oscillation occurs. Or, it is, upon contact with an adjacent point, one with it. Points have no parts and are, therefore, indivisible. If it were possible to divide a line into “points,” the points could never be put together to make a line.

Your number examples were examples of a potential infinity. That means, it is a finity in the process of becoming an infinity. Which it never will, as “infinity” is not determinable. Remember, the way we define “infinity” is by a via negativa, i.e., by stating what it is not. The words, “…[something] trails off to infinity.” is a euphemism.

God bless,
jd

 

[Alberti_Devoveo]

Some posts have mentioned the Planck length. Physics is not my area of expertise. Still, there is no reason why, mathematically speaking, one could not have “half a Planck length.” Whether such a thing is actually constructible or measurable is a separate issue.

The Planck length is the smallest unit of length using the universal constants G (Newton’s gravitational constant), h-bar (Planck’s reduced constant), and c (speed of light).
Physically, the Planck length is the smallest measurable length, not that it is the smallest possible length. There are some who assert that the Planck length is the point at which quantum effects (i.e., the uncertainty principle) become so large that two adjacent point particles separated by a Planck length would be indistinguishable.

I did, somewhat briefly, read an article that defined a Planck length as a measure of the oscillation of certain particle strings. Oscillation implies a kind of motion. It would seem that, if a particle were oscillating, it would have to reach a point in its oscillation halfway between the max and the min, no? Not the only defintion I saw, but still…

This is false. The Planck length is found from the aforementioned universal constants.

 

[Ion]

“Give me a point, and I will measure the universe…”