General Relativity Quantum Mechanics and the new String Theory

[Robh (Rob)]

Originally Posted by sjastro

Rob,

Consider geometrical theorems/conjectures. The catalyst for geometrical concepts is through measurement. In a subtle way a geometrical theorem is analogous to a scientific theory except that individual measuresurements themselves do not prove the theorem.

As an example consider the Pythagorean theorem for right angle triangles c^2=a^2+b^2.
Did Pythagoras invent this geometric property for right angle triangles? Clearly no.
Mathematicians before Pythagoras knew of the relationship on the basis of measurement but where not able to prove it for all "a" and "b". Before Pythagoras the geometrical property was based on conjecture.

The geometrical property has always been around, measurement has simply confirmed it's existence.

The element of invention is through proof. There are at least 8 ways to prove the Pythagorean theorem. Each method is a product of logical processes at work but the ultimate objective of each proof is to confirm the existence of the mathematical property.

This example can be extended to mathematics in general.

In essence the theorems are "already there", how we proceed to prove them is where invention comes into the picture.

Regards

Steven

I've had a long, hard think about that one. Up to a point, I think we are in some agreement.
Interestingly, as you mentioned, Pythagoras' Theorem (like a lot of the early maths) was formulated from practical observations.
It stands as a provable theorem from defined axioms in the plane (flat geometry). However, the Earth is close to a sphere and Pythagoras' Theorem is decreasingly accurate for points at greater distances. At greater distances, spherical geometry must be applied. On a larger Universe scale, Pythagoras' Theorem only applies where the Universe is relatively flat. So the theorem is not a logical result of the Universe itself- it just happens to be true where its is locally flat.
Now consider this hypothetical. On planet X in some far away galaxy, a right-angle has no significance. 60 degrees is the angle of importance. All dwellings are hexagonal pyramids. The great mathematician Alpha has developed a theorem for triangles with one angle 60 degrees- c^2=a^2+b^2-ab with c being the side opposite the angle of 60 degrees. Like Pythagoras' Theorem the expression was first hinted at from early measurements.
My question is- was Pythagoras' Theorem or Alpha's Theorem there to be discovered or was it just an invention of the planet's conscious beings. Sure, they are both provable from defined axioms of plane geometry, but is not the theorem a logical result of the path of development taken by each planet civilisation.
However, having said this, either culture could prove the other's theorem from certain basic axioms. So the question I'm asking myself is what is it that is universal in the maths? Is it the logic itself? Theorems can be constructed but are not necessarily there to be found. The mathematical path taken will determine whether a conjecture is made or a theorem is constructed. Conscious beings from different Universes, with entirely different mathematical theorems and constructs, should be able to navigate each other's mathematics from some logical base.
As a digression, if we were to communicate with an alien civilisation, maths would be a logical starting point but what would be in that set to get the ball rolling? It is unlikely they would count in base ten, but is base 2 (numbers made up from zeroes and ones) a logical starting point?

Regards, Rob.

[sjastro]

Originally Posted by Robh

I've had a long, hard think about that one. Up to a point, I think we are in some agreement.
Interestingly, as you mentioned, Pythagoras' Theorem (like a lot of the early maths) was formulated from practical observations.
It stands as a provable theorem from defined axioms in the plane (flat geometry). However, the Earth is close to a sphere and Pythagoras' Theorem is decreasingly accurate for points at greater distances. At greater distances, spherical geometry must be applied. On a larger Universe scale, Pythagoras' Theorem only applies where the Universe is relatively flat. So the theorem is not a logical result of the Universe itself- it just happens to be true where its is locally flat.

The observable Universe is flat.
The geometry of the Universe is governed by the amount of mass and energy present.
A geometrical measurement therefore only confirms the underlying geometry, it doesn't determine it.
Pythagoras' theorem (or spherical geometry for the Earth's surface) only reflects the nature of the underlying geometry.

Now consider this hypothetical. On planet X in some far away galaxy, a right-angle has no significance. 60 degrees is the angle of importance. All dwellings are hexagonal pyramids. The great mathematician Alpha has developed a theorem for triangles with one angle 60 degrees- c^2=a^2+b^2-ab with c being the side opposite the angle of 60 degrees. Like Pythagoras' Theorem the expression was first hinted at from early measurements.
My question is- was Pythagoras' Theorem or Alpha's Theorem there to be discovered or was it just an invention of the planet's conscious beings. Sure, they are both provable from defined axioms of plane geometry, but is not the theorem a logical result of the path of development taken by each planet civilisation.
However, having said this, either culture could prove the other's theorem from certain basic axioms. So the question I'm asking myself is what is it that is universal in the maths? Is it the logic itself? Theorems can be constructed but are not necessarily there to be found. The mathematical path taken will determine whether a conjecture is made or a theorem is constructed. Conscious beings from different Universes, with entirely different mathematical theorems and constructs, should be able to navigate each other's mathematics from some logical base.

Let's take the analogy further. Suppose the inhabitants of Planet X have been building these structures for centuries before Alpha came on the scene. Let's suppose they have built the same shape but at different scales. The formula c^2=a^2+b^2-ab applies for all scales and represents a limitation or mathematical constraint. For example the inhabitants can't independently alter dimension c without changing the overall shape of the structure.

When Alpha comes along he discovers the limitation. He may have used logic, lateral thinking etc to derive the formula but it is still a discovery. If he invented the theorem he has also invented the limitation, in which case there was no limitation prior to the invention which of course is a logical contradiction.

There is no right or wrong in these type of discussions always a perpetual grey, it's why this debate has gone on for centuries.

Regards

Steven

[bojan]

Originally Posted by sjastro

When Alpha comes along he discovers the limitation. He may have used logic, lateral thinking etc to derive the formula but it is still a discovery. If he invented the theorem he has also invented the limitation, in which case there was no limitation prior to the invention which of course is a logical contradiction.

There is no right or wrong in these type of discussions always a perpetual grey, it's why this debate has gone on for centuries.

Regards

Steven

The flaw in this logic is that you (like many others before you) are (quietly) assuming a priori that there is something going on in the background and then you are trying to deduce that very thing using logic, which IMHO is wrong - with logic deductions, you can go forward starting from assumptions, but not backwards, trying to deduce assumptions. I can not see the contradiction that you are pointing at.

Pythagoras theorem is not a discovery, it is mathematical construct which is applicable to reality, but it does not necessarily follow from reality (like most of advanced mathematics). It does not exist without mind that invented it... Just because the idea is started by observing the reality does not mean it existed before it was formulated and proven as theorem.

[sjastro]

Originally Posted by bojan

The flaw in this logic is that you (like many others before you) are (quietly) assuming a priori that there is something going on in the background and then you are trying to deduce that very thing using logic, which IMHO is wrong - with logic deductions, you can go forward starting from assumptions, but not backwards, trying to deduce assumptions. I can not see the contradiction that you are pointing at.

Oh come now!
If you want to go down this line of argument about logic flaws be very careful your own comments don't get caught up well.
Your comments are also based on assumptions.
Unlike you I don't see this discussion as a black or white issue but I do tend towards discovery instead of invention.

Pythagoras theorem is not a discovery, it is mathematical construct which is applicable to reality, but it does not necessarily follow from reality (like most of advanced mathematics). It does not exist without mind that invented it... Just because the idea is started by observing the reality does not mean it existed before it was formulated and proven as theorem.

The reason why I used the Pythagorean Theorem was to draw parallels with scientific theory such as gravity, given that both were formulated on observation.

Did gravity exist before it was theorized?

Steven

[bojan]

Originally Posted by sjastro

Did gravity exist before it was theorized?

Steven

Of course it did.
But the existence of gravity has nothing to do with mathematical model of gravity.
Model was developed based on observation, of course.

Newtonian formula F=m1*m2*g/r^2 is a model, applicable to the one behavioural aspect of gravity.
The same formula can be applied to many other natural phenomena but this does mean they are the related at all.

Originally Posted by sjastro

When Alpha comes along he discovers the limitation. He may have used logic, lateral thinking etc to derive the formula but it is still a discovery. If he invented the theorem he has also invented the limitation, in which case there was no limitation prior to the invention which of course is a logical contradiction.

I am careful with my statements.... And I still do not see where you see the the contradiction in above.. maybe I am just dumb.. Please explain more clearly I am sure there are others who do not see it either but are afraid to ask

[sjastro]

Originally Posted by bojan

Of course it did.
But the existence of gravity has nothing to do with mathematical model of gravity.
Model was developed based on observation, of course.

So was the Pythagorean theorem. So why should a mathematical theorem be treated any differently in this context.

Newtonian formula F=m1*m2*g/r^2 is a model, applicable to the one behavioural aspect of gravity.
The same formula can be applied to many other natural phenomena but this does mean they are the related at all.

That's not correct. F=m1*m2*G/r^2

G is the Gravitational constant. =6.67x10^-11 m^2/kg-sec^2.
g = acceleration = 9.8m/s^2.

I'm refering to g. How did apples fall off trees before Newton?
g is a mathematical vector field.

I am careful with my statements.... And I still do not see where you see the the contradiction in above.. maybe I am just dumb.. Please explain more clearly I am sure there are others who do not see it either but are afraid to ask

c^2 =a^2+b^2-ab

c is a dependant variable depending on a and b.
a and b are independant variables.

They can only build to the correct shape (but also at different scales) if the above equation is true. In other words if they vary the scale by having a and b at different dimensions, c will vary accordingly.

Even though the builders do not know what equation is because it hasn't been discovered yet (or your case invented) there is still a relationship between c, and a,b in order to build to the correct shape.

Alpha comes along and discovers (or invents) the equation.

If he discovered the equation there are no problems, he has simply derived the equation that shows the relationship between c and a,b.

If he has invented the equation there is a problem. By definition prior to the invention, the equation never existed. If the equation never existed then neither did relationship between c and a,b. If the relationship between c and a,b never existed then how could the builders build to the correct shape?

So therefore if the equation is invented then it was impossible for the builders prior to the invention to build to the correct shape. To suggest it is possible is a logical contradiction.

Steven

[bojan]

OK, mate, we are not cathching the typos here (g and G).. the shape of the formula is what counts. But let it be... 1:0 for you here, since I was not careful enough in that detail.

However, in my view, Pythagorean theorem did not exist before it was formulated, because it is just a model which describes how angles and sides of a right-angle triangle behave. This behaviour is one property of Euclidean space (mathematical model again) which, as it seems so far, is applicable to what we perceive as the space we live in, or at least to a small part of it. But, it is not a real thing, it is just a description we are using in our minds to try to understand the reality.
This reality, or rather, perception we have about reality is formed in our minds based on available inputs (sight, hearing touch etc), and it is again more or less accurate and self-consistent model (perfectly useful for our survival as species, but VERY limited.. I was always wandering what it would have been if we were living in completely different environment.. and what we would have been capable of understand or not in such a case), which we are trying to make more accurate and easier to communicate, using mathematics. But it is certainly not a reality.
So I can not accept your argumentation here, despite being apparently self-consistent (and I am still looking into this, it is not as simple as you presented it, something does not seem to be right about starting assumptions), because it does not fit the wider context, sorry.

[Robh (Rob)]

Originally Posted by sjastro

Alpha comes along and discovers (or invents) the equation.

If he discovered the equation there are no problems, he has simply derived the equation that shows the relationship between c and a,b.

If he has invented the equation there is a problem. By definition prior to the invention, the equation never existed. If the equation never existed then neither did relationship between c and a,b. If the relationship between c and a,b never existed then how could the builders build to the correct shape?

So therefore if the equation is invented then it was impossible for the builders prior to the invention to build to the correct shape. To suggest it is possible is a logical contradiction.

Steven

Consider this. As a pure mathematician, I derive theorems for right-angle triangles that allow me to find c (opposite the right-angle) for a plane, spherical and hyperbolic geometries. To derive these, I need have no recourse to the real world i.e I need not physically measure anything.
I transport to planet X on the other side of the Universe and, by luck, find the local Universe is flat and my measurements confirm the results of the plane geometry created on Earth. Alpha, on planet X, says he can beam me into parallel Universe U2. In U2, I find the geometry is spherical and my measurements for right-angle triangles confirms this. Getting back to planet X, Alpha tells me that there seems to be no hyperbolic parallel Universes.
So, on Earth, Pythagoras' Theorem is confirmed from measurement and was, truth be told, originally formulated from measurements. On Alpha, Pythagoras' Theorem is unknown but I can also confirm the theoretical geometry from measurements I make there. Back on Earth, one could say that the relationship described by Pythagoras' Theorem was discovered by measurement. However, the proof of the theorem is not so much a discovery but an exercise in logic.
But did I discover spherical geometry or create it- I can certainly confirm it in U2? But the mathematics for a hyperbolic Universe exists even though it seems such a Universe does not. Is it a creation or a discovery? It is not a physical discovery of the world we live in.

Regards, Rob.

[sjastro]

Originally Posted by Robh

But did I discover spherical geometry or create it- I can certainly confirm it in U2? But the mathematics for a hyperbolic Universe exists even though it seems such a Universe does not. Is it a creation or a discovery? It is not a physical discovery of the world we live in.

To take a leaf out of the book of relativity, an observer makes a measurement in his own frame of reference. The frame of reference doesn't have to be the Universe itself.

Mathematicians refer to objects being embedded in higher dimensional space. A triangle drawn on a sheet of paper is an example of a two dimensional object embedded in three dimensional space.

If I measure the dimensions of this triangle, my frame of reference is the surface the of paper not the Universe.

Therefore even if a hyperbolic Universe doesn't exist I can still construct a triangle on a hyperbolic surface (much like a horse saddle).

So I believe the hyperbolic geometry is a discovery.

Regards

Steven

[bojan]

Originally Posted by sjastro

Therefore even if a hyperbolic Universe doesn't exist I can still construct a triangle on a hyperbolic surface (much like a horse saddle).

So I believe the hyperbolic geometry is a discovery.

Discovery of something that does not exist?
It looks like contradiction to me.

We have to first (re)define terms "discovery" and "invention" it seems...

[sjastro]

Originally Posted by bojan

Discovery of something that does not exist?
It looks like contradiction to me.

We have to first (re)define terms "discovery" and "invention" it seems...

I thought I made it perfectly clear about objects embedded in space.
We live in a flat observable Universe but spheres exist. Correct?

[bojan]

Correct.
And so does contradiction (or I am really dumb).

[sjastro]

Originally Posted by bojan

Correct.
And so does contradiction (or I am really dumb).

What exactly are you struggling with?

A triangle drawn on the surface of a sphere takes on a spherical geometry, not the geometry of the a Universe.

Or is it because the observable Universe the flat, a sphere and the resultant geometry must be an invention?

Steven

[bojan]

I am struggling with your assumptions and definitions of discovery and invention.
Starting from there, the end conclusion does not sit well into my understanding of what mathematics is all about (to me it is just a tool).
Lets say, the philosophical aspects or your conclusions are not acceptable to me.

[Nesti (Mark)]

Ah yes, when the dialog reduces 'Détente' is near.

[sebastien]

thanks guys, some very interesting posts

Regards,
Sebastien.

[sebastien]

Originally Posted by orestis

Hi everyone,
Very interesting discussions.Hi sebastion i too am 14 yrs old and love these topics on physics and maths.Thanks for starting thread.
The elegant universe series is very good.
Why do people always say parellel universes what about if there were unique universes which are completely diferent but have charachteristics which are the same.

Heres a very good topic-Albert Einstein said that nothing in the known universe can excell the speed of light but Theoretically WARP DRIVE can go faster than the speed of light. warp drive is a simple concept which uses gravity amplifiers to expand and contract space time at the rear and at the front of the spacecraft respectively. What do you think about warp drive.

If you like philosophy and science go to www.symphonyofscience.com
and watch the music videos .There great.

orestis
Warp drive is awesome.

Hey Oretis

did you get my pm?? If so please tell me, if not well....tell me as well

Thanks again,
Sebastien.

[orestis]

Yes sebastien i have and replied..

[Vanda (Ian)]

"Warp drive" - and I was/am a great fan of Startrek - remains a science fiction concept at the moment. We don't know yet if we will be ever able to create gravity by a machine let alone warp space with it.

This is very problematic for our species. We need gravity to stop our bones dissolving etc. We can mimic the effect of gravity (1G) by accelerating/breaking or by rotating a spacecraft at the correct speed - centrifigal force.

Sadly without some form of warp drive humankind is unlikely to see much of our immediate stellar vicinity within 10 light years let alone anything more distant. We will likely need to find the target planet before we leave. A good starting point for research.

Just because science fiction wants warp drive does not mean it will be possible some day. I think the 2nd law of thermodynamics becomes involved and may not be breakable. I hope it is though!

[orestis]

Hi vanda,

From what i know warp drive does not create gravity but alters it to their advantage.I don't how these kind of machines could do this but i've read that they use both negative and positive energy to amplify gravity waves which in turn warp spacetime.Also to create the warp bubble to protect the space ship.

Your centrifugal force idea is great but can't we just make artificial gravity 1G. How does the ISS make 1G?

One problem that i have found is that it will compress anything in front of the space craft and expand anything behind thus destroying everything In its path.
Unless we can make a clear highway..Which is virtually imposible with our technology now.

It might just be posible one day.''If we do not destroy ourselfs we will one day venture to the STARS" [Carl Sagan]

orestis