Intelligent Design

In number theory, 2 + 2 = 4. Period.
In physics, velocity is a unit of distance divided by a unit of time.
Or V = d / t.
Of course you already know that, but what makes the simple formula so subtle is that there is a screwy relation between Einstein's space and time in that they are two different aspects of the same measure. Distance and time measurements depend on the motion of the observer as you point out in your lightening and train example. In this context the only constant in Relativity is generally denoted as S, which is the distance between two events. But any further discussion of that of course lies beyond the scope of Intelligent Design.
Exactly. Intelligent Design, so far as anyone has shown so far, is proved thusly: "'*poof*' he explained"
 
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In number theory, 2 + 2 = 4. Period. .

While much of the discusson of 2+2=4 is tongue in cheek there is more to be said.

I agree that in number theory you can end that with a period.

But the original discussion is about science which always starts with an observation of reality.

I did poke around on the net and found that I have been intending to discuss Euclidean Geometry, not Newtonian, and Einsteinian theories.

Do we agree that in Euclidean Geometry (EG) that the angles inside a triangle always add up to 180. using various proofs we could determine that if the angles inside a triangle add up to 180 then 2+2 must equal 4. It would take a few steps but if one is true then the other must be true and if one is false then the other must be false. Right? After all if we take a triangle in which the inside angles do add up to 180 and one side is 2 units long and another side is 2 units long then the sum of these two sides must be 4.

But in an Einsteinian world we learn that things are just not that perfect. due to the warping effects of gravity a triangle made of of three rays of light would not have inside angles that add up to 180.

And here is what others have said:

"For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Einstein's theory of general relativity is that Euclidean space is a good approximation to the properties of physical space only where the gravitational field is not too strong.[4] "

Since EG is only an approximation of reality then 2+2 is only approximately 4
due to the various effects of gravity and near speed of light phenomena in any observations.
 
While much of the discusson of 2+2=4 is tongue in cheek there is more to be said.

I agree that in number theory you can end that with a period.

But the original discussion is about science which always starts with an observation of reality.

I did poke around on the net and found that I have been intending to discuss Euclidean Geometry, not Newtonian, and Einsteinian theories.

Do we agree that in Euclidean Geometry (EG) that the angles inside a triangle always add up to 180. using various proofs we could determine that if the angles inside a triangle add up to 180 then 2+2 must equal 4. It would take a few steps but if one is true then the other must be true and if one is false then the other must be false. Right? After all if we take a triangle in which the inside angles do add up to 180 and one side is 2 units long and another side is 2 units long then the sum of these two sides must be 4.
That is correct. The kind of geometry that Euclid practiced (compass and ruler) was found to be in a one to one correspondence with analytic geometry (using real numbers in a plane.)
But in an Einsteinian world we learn that things are just not that perfect. due to the warping effects of gravity a triangle made of of three rays of light would not have inside angles that add up to 180.

And here is what others have said:

" ...... "

Since EG is only an approximation of reality then 2+2 is only approximately 4
due to the various effects of gravity and near speed of light phenomena in any observations.
To make it simple, suppose the non-Euclidean geometry is on the surface of a sphere of radius R. Suppose each angle of a triangle is a, b, and c. Suppose the area of the triangle is A.

According to Girard's Theorm, the sum of the angles is, 180 + area / R squared. Or, algebraically,

a+b+c = 180 + A/(RxR)

This can be interpreted to say if the radius is huge, or the triangle area is small, the sum is close to 180. This makes sense -- the triangle is small compared to the size of the sphere. E.g. a small triangle drawn in the sand on the planet Earth.

In one sense this formula is analogous to saying something like 2+2 is slightly larger than 4, but it is also gives a very explicit formula for deciding what the 2+2 actually is. That formula itself uses regular real numbers in the 2+2 = 4 sense. So the formula above is a way of relating the non-Euclidean geometry to the usual simple arithmetic properties of numbers.

If you want to think of this whole thing as 2+2 not= 4, that's fine, but a mathematician would prefer to think in terms of Girard's formula that uses arithmetic as we generally know it.

I think the discussion earlier was in terms of scientists disagreeing on what is "fact". The above discussion illustrates that people might want to look at 2+2 from two different perspectives, but understanding why and how the perspectives differ is the important factor that says the two perspectives are not in conflict.
 
Dr Who, Lagboltz, et al,

I must admit, I am enjoying the conversation; the lurker that I am.

That is correct. The kind of geometry that Euclid practiced (compass and ruler) was found to be in a one to one correspondence with analytic geometry (using real numbers in a plane.)

But in an Einsteinian world we learn that things are just not that perfect. due to the warping effects of gravity a triangle made of of three rays of light would not have inside angles that add up to 180.

And here is what others have said:

" ...... "

Since EG is only an approximation of reality then 2+2 is only approximately 4
due to the various effects of gravity and near speed of light phenomena in any observations.

To make it simple, suppose the non-Euclidean geometry is on the surface of a sphere of radius R. Suppose each angle of a triangle is a, b, and c. Suppose the area of the triangle is A.

According to Girard's Theorm, the sum of the angles is, 180 + area / R squared. Or, algebraically,

a+b+c = 180 + A/(RxR)

This can be interpreted to say if the radius is huge, or the triangle area is small, the sum is close to 180. This makes sense -- the triangle is small compared to the size of the sphere. E.g. a small triangle drawn in the sand on the planet Earth.

In one sense this formula is analogous to saying something like 2+2 is slightly larger than 4, but it is also gives a very explicit formula for deciding what the 2+2 actually is. That formula itself uses regular real numbers in the 2+2 = 4 sense. So the formula above is a way of relating the non-Euclidean geometry to the usual simple arithmetic properties of numbers.

If you want to think of this whole thing as 2+2 not= 4, that's fine, but a mathematician would prefer to think in terms of Girard's formula that uses arithmetic as we generally know it.

I think the discussion earlier was in terms of scientists disagreeing on what is "fact". The above discussion illustrates that people might want to look at 2+2 from two different perspectives, but understanding why and how the perspectives differ is the important factor that says the two perspectives are not in conflict.
(COMMENT)

I am quite amazed at how our perspectives mold our perception and understanding as outside observers. I quite understand the differences you discuss; because I was taught to evaluate based on the definitions in the language of the local conditions versus cosmic application. But I know that not everyone has the same troubleshooting outlooks. I see gifts in both pieces to your approaches.

(QUESTION)

Where does that lead us in the application of these perspectives, relative to the discussion? (Intelligent Design)

Most Respectfully,
R
 
Dr Who, Lagboltz, et al,

I must admit, I am enjoying the conversation; the lurker that I am.

I didn't think anyone else was reading this.
(COMMENT)

I am quite amazed at how our perspectives mold our perception and understanding as outside observers. I quite understand the differences you discuss; because I was taught to evaluate based on the definitions in the language of the local conditions versus cosmic application. But I know that not everyone has the same troubleshooting outlooks. I see gifts in both pieces to your approaches.
Interesting observation. Dr Who is looking at things as a local real world observer where apparent paradoxes may occur. This perspective is the area of human understanding. In science the observations are modeled using math by an outside observer, and hopefully the paradoxes are explained. The paradoxes of relativity are a little bit more straightforward than Quantum Mechanics where the paradoxes are very persistent.
(QUESTION)
Where does that lead us in the application of these perspectives, relative to the discussion? (Intelligent Design)

Most Respectfully,
R
This thread was not intelligently designed, but evolved into peripheral topics. But if you are looking for relevance, I suppose you could say the ancients looked at their world locally and saw a creator in the flora and fauna. Darwin toured the world and got a global perspective of many species and found a creator wasn't a necessary explanation.
 
This thread was not intelligently designed, but evolved into peripheral topics. But if you are looking for relevance, I suppose you could say the ancients looked at their world locally and saw a creator in the flora and fauna. Darwin toured the world and got a global perspective of many species and found a creator wasn't a necessary explanation.

I agree and have not been at all worried about adding to the derailment of this thread.

Yes perspective is important. An earlier perspective was that there was a designing force then Darwin had a newer perspective in which he stated that there was no designing force. Which he quickly abandoned as it was not scientific. (yet just like the ID'ers who really believe in God and hide behind science Darwin really did not believe in pure science and when he removed his statement that there was no God he was also hiding behind science) (perhaps as people of all sorts hide behind science the science will influence them to be more accurate and truth will win out)

A more refined perspective may yet reverse the first reversal. (though again it may be too strong to state that reversals have even occurred since the number of believers has probably not really changed that much)
 
I don't think that the formula for looking at a triangle on a sphere applies here.

Consider three points in space. They will form a plane. Now consider three rays of light each going through two points. They will form a triangle that is on a plane and not on a sphere.

The effects of gravity will warp the space they are in such that the sum of the internal angles will not be 180 even though in pure math it would be. In the real world the sum of the internal angles does not equal 180. And since we could prove that if the sum of the internal angles equals 180 that 2+2 must equal 4 then the corollary would be true: If the sum of the internal angles does not add to 180 then 2+2 must not equal 4.

As I tried to state early but relied on Newtonian math by mistake: using Euclidian math if a particle forms a line as it travels at light speed then doubling the length of the line does not result in a particle that is traveling at twice light speed. In other words 1C +1C does not equal 2C which of course means that 2C + 1C does not equal 4C. Um, I think:)
 
I don't think that the formula for looking at a triangle on a sphere applies here.
You are right strictly. The formula that should be applied is Einstein's "metric tensor".
http://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)
"The metric (tensor) captures all the geometric and causal structure of spacetime, being used to define notions such as distance, volume, curvature, angle, future and past."
Consider three points in space. They will form a plane. Now consider three rays of light each going through two points. They will form a triangle that is on a plane and not on a sphere.
Yes, only if there is no gravitational fields.
The effects of gravity will warp the space they are in such that the sum of the internal angles will not be 180 even though in pure math it would be.
Gravity will warp a plane into a surface in three dimensions that we cannot see or measure directly. A two dimensional warped plane cannot be fully understood in two dimensions. It is a three dimensional surface where all activity is totally constrained to the surface. A straight line in a warped space is called a "geodesic" and is the shortest "line" between two points where the line is constrained to the 3-D surface. (Our warped 4 dimensional universe can be described mathematically as a 5 dimensional sphere according to Richard C. Tolman.)

For example the shortest distance from NY to Sidney Australia is a line through the center of the earth. However if we are constrained to the 2-D surface, an airline will choose a "great circle" on the spherical earth. Girard's formula is based on triangles embedded in a warped plane which is a spherical surface. The sides (geodesics) are great circles since there can't be a straight line on the surface of a sphere.
In the real world the sum of the internal angles does not equal 180. And since we could prove that if the sum of the internal angles equals 180 that 2+2 must equal 4 then the corollary would be true: If the sum of the internal angles does not add to 180 then 2+2 must not equal 4.
A beam of light will appear to go in a straight line; indeed it is the definition of a physical straight line. If you set up surveying equipment on top of three mountains and measure the three angles, the sum will be greater than 180 degrees. (However in practice, imprecision and air turbulence will make this impossible to see.) You must go to a 5 dimensional description and then see that the light is actually traveling in curved geodesics because of gravitational distortion by Earth. But, again, from our perspective the lines look straight.
As I tried to state early but relied on Newtonian math by mistake: using Euclidian math if a particle forms a line as it travels at light speed then doubling the length of the line does not result in a particle that is traveling at twice light speed. In other words 1C +1C does not equal 2C which of course means that 2C + 1C does not equal 4C. Um, I think:)
Velocity is a different phenomenon than gravitation, but it is another example of how unintuitive our universe is.
 
Yes, only if there is no gravitational fields.
.

Where is that place in the universe?

If no such place exists then there is no such thing as a triangle. Would you agree with me a mathematician could no doubt prove that if there is no such thing as a triangle in the universe then there is no such thing as 2+2=4 in the universe?

In fact if the universe is so unstable that it cannot even form a triangle from any three rays of light passing through three points in space then it must not be stable enough for a line segment to persist at the same length. Every line segment in the universe must be constantly "stretching" and "shrinking" depending on the gravity field(s) it is in. The concept of 2 would be, in an absolute sense, meaningless when applied to any part of the universe. Said 2 would always be 2 plus something or 2 minus something.
 
Where is that place in the universe?
Only on paper.
If no such place exists then there is no such thing as a triangle.
I think you mean there is no such thing as a triangle whose angles add up to 180 degrees exactly in the physical universe. I would agree with that.
Would you agree with me a mathematician could no doubt prove that if there is no such thing as a triangle in the universe then there is no such thing as 2+2=4 in the universe?
"2+2=4" exists in number theory. You are talking about the physical universe. When you talk about "2" in physics, you must use units, like 2 grams, 2 meters, 2 degrees. A physicist would agree that there are no such triangles whose angles add up to 180 degrees exactly.

In a similar way the summation of velocities is not 2+2, it is 2 mph + 2 mph which is slightly less than 4 mph.

You can't generalize that summing quantities with units (that don't add up as expected) lead to conclusions about quantities without the units.
In fact if the universe is so unstable that it cannot even form a triangle from any three rays of light passing through three points in space then it must not be stable enough for a line segment to persist at the same length. Every line segment in the universe must be constantly "stretching" and "shrinking" depending on the gravity field(s) it is in. The concept of 2 would be, in an absolute sense, meaningless when applied to any part of the universe. Said 2 would always be 2 plus something or 2 minus something.
As I said, you can form triangles in the physical universe with sides consisting of geodesics. And yes this stretching and shrinking happens. GPS systems use 3 or 4 satellites signals to fix your location, but the satellites are in constant motion and the stretching and shrinking of their relative positions must be accounted for in order to get accurate ground location readings. The effects of earth's gravity means the clocks on satellites run 45 microseconds faster per day than on earth. It doesn't sound like much, but it is when GPS systems need nanosecond precision.

Again, it is always wrong to say 2+2=3.9999956, however it could be true that 2 mph + 2 mph = 3.9999956 mph.
 
Lagboltz, Objective Voice, et al,

One must remember to avoid the Paradox of the undefined equation.

Where is that place in the universe?
Only on paper.
(COMMENT)

Actually, in cosmology, we refer to it all the time.

When we speak of the "universe," what are we saying. Well, in a nut shell, we are speaking of the maximum extent to which the energy of the original "Big Bang" (under that theory) might effect or exercise influence.

When we speak of the "expanding universe," we are speaking of an "undefined region" beyond the influence of the point of origin (The Big Bang). That is, what is the universe expanding into?

(QUESTION)
But, there is a related question: Relative to the "fabric of space?"

When we think of the "fabric of space," are we thinking for a three-dimensional uniform construct? Or are we thinking of a spherical construct, like a globe, where the lines in the fabric grow more distant as it expands (inverse square law type stuff)?

  • What is the "fabric of space?"
  • How does the "fabric of space" change the proportions in physics?

Does the "speed of light" change in distance the closer you get to the point of origin; and increase the farther away it is from the point of origin?

Most Respectfully,
R
 
Lagboltz, Objective Voice, et al,
One must remember to avoid the Paradox of the undefined equation.
(COMMENT)

Actually, in cosmology, we refer to it all the time.

When we speak of the "universe," what are we saying. Well, in a nut shell, we are speaking of the maximum extent to which the energy of the original "Big Bang" (under that theory) might effect or exercise influence.

When we speak of the "expanding universe," we are speaking of an "undefined region" beyond the influence of the point of origin (The Big Bang). That is, what is the universe expanding into?
Lofty questions. There are lots of theories about what the universes is in the context of a bigger space -- what was it that banged, and how did it bang. Often these involve multiverses that may embody many other universes, all of them expanding bubbles in an 11 dimensional space. Of course that answer answers nothing and breeds more questions.
(QUESTION)
But, there is a related question: Relative to the "fabric of space?"

When we think of the "fabric of space," are we thinking for a three-dimensional uniform construct? Or are we thinking of a spherical construct, like a globe, where the lines in the fabric grow more distant as it expands (inverse square law type stuff)?

  • What is the "fabric of space?"
  • How does the "fabric of space" change the proportions in physics?

Does the "speed of light" change in distance the closer you get to the point of origin; and increase the farther away it is from the point of origin?

Most Respectfully,
R
It's not clear what the fabric of space means. Certainly we don't want to go back to the earlier theory of ether. Tolman's work of the 1920's considered the galaxies as inertial systems. I.e. there are no forces causing a particular galaxy to move anywhere. The forces of all the surrounding galaxies cancel out so our galaxy (and all the others) for the most part stays still, twirling around some point.

Yet these separate inertial systems are getting further apart. Tolman defined a co-moving coordinate system which simply moves along with the galaxies, and that defines where a point in space is. That coordinate system is what is expanding. The laws of physics in each galaxy are the same as any other galaxy. Or I should say, there hasn't been any indication that the constants of physics are changing for remote galaxies, so the speed of light is still constant.

The speed of light isn't just the speed of photons, it is fundamental to the geometry of space-time. The speed of light is just a conversion constant relating space and time. It is really interesting how it comes about, and can be understood with high school algebra. But squares and square roots are too hard to type within the constraints of the forum reply box.

The fabric of space
It is a four dimensional construct because it must include time as one of the dimensions. The huge mass of all the galaxies causes a warp so that the space turns in on itself similar to how a black hole is self contained.

It is easy to think of drawing triangles in a warped 2-D plane by transforming it to a 3-D curved surface. However it is impossible for us to visualize the warp of the 4-D space of the universe. Our universe can be mathematically represented as existing on the infinitesimally thin surface of an expanding 5-D hypersphere.
 
Only on paper.

I think you mean there is no such thing as a triangle whose angles add up to 180 degrees exactly in the physical universe. I would agree with that.

"2+2=4" exists in number theory. You are talking about the physical universe. When you talk about "2" in physics, you must use units, like 2 grams, 2 meters, 2 degrees. A physicist would agree that there are no such triangles whose angles add up to 180 degrees exactly.

In a similar way the summation of velocities is not 2+2, it is 2 mph + 2 mph which is slightly less than 4 mph.

You can't generalize that summing quantities with units (that don't add up as expected) lead to conclusions about quantities without the units.

As I said, you can form triangles in the physical universe with sides consisting of geodesics. And yes this stretching and shrinking happens. GPS systems use 3 or 4 satellites signals to fix your location, but the satellites are in constant motion and the stretching and shrinking of their relative positions must be accounted for in order to get accurate ground location readings. The effects of earth's gravity means the clocks on satellites run 45 microseconds faster per day than on earth. It doesn't sound like much, but it is when GPS systems need nanosecond precision.

Again, it is always wrong to say 2+2=3.9999956, however it could be true that 2 mph + 2 mph = 3.9999956 mph.

I believe we are in agreement. I have always meant my statements to by in the physical universe and to be about numbers with units.
 
I believe we are in agreement. I have always meant my statements to by in the physical universe and to be about numbers with units.
I think we are only sort of in agreement. My final statement is not strictly true:
"Again, it is always wrong to say 2+2=3.9999956, however it could be true that 2 mph + 2 mph = 3.9999956 mph."
I should have said it could be true that 2 mph augmented by a 2 mph velocity could be could be 3.9999956 mph.

The more accurate formula was given in post #136, V =
img198.gif
 
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I think we are only sort of in agreement. My final statement is not strictly true:

I should have said it could be true that 2 mph augmented by a 2 mph velocity could be could be 3.9999956 mph.

The more accurate formula was given in post #136, V =
img198.gif
Now I am curious.

Am I understanding this correctly?

2 bodies traveling at 2mph and crashing into a wall object will release less energy than if one of those bodies collides with a wall at 4 mph?
 
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