E really does equal mc^2

[Richard]

1 + 1 has to equal 2. If it does not then having definitions for things would be entirely stupid, mean our lives are pointless, and mean everything we have done is completely pointless. That's utterly bullshit if you think that. In order for 1 + 1 not to equal 2 you'd have to change the definition, which would be completely retarded, and brings up a favorite thing of mine: It always seems like the greatest "thinkers" are amazingly stupid and happen upon things by great chance, and even if they can solve the riddles of our existence couldnt figure out how to open the flaps on a damn cardboard box because they'd be spending to much time anazlying if it means the same to open the box here than over there, just plain RETARDED!

Heh, you'd be surprised how much "stupid" contemporary theories can seem. If I asked you what would happen if you step out of your house and cross the street you'd probably say "reach the other side of the road", however quantum theory says there is probability, however infinitesimal, that you do not reach the other side of road but find yourself back in your own porch and there's a percentage you'll end up in pluto and another percentage that says you'll change gender when you get to the other side.

It wasn't that long ago that human beings realised that the angles formed by the three sides of a triangle do not in fact have to total 180 degrees. It's not inconceivable that we may find ourselves in the distant future in a place where 1 + 1 does not equal 2. Definitions, conventions, theories have been challenged since the human brain acquired reasoning capabilities; believing we have solved all the universe's problems and what we know now is a universal truth does strike me as mildly arrogant at best.

In fact, I can tell you right now how 1 + 1 != 2. Change to binary and literally 1 + 1 = 10.

 

[KimB]

Heh, you'd be surprised how much "stupid" contemporary theories can seem. If I asked you what would happen if you step out of your house and cross the street you'd probably say "reach the other side of the road", however quantum theory says there is probability, however infinitesimal, that you do not reach the other side of road but find yourself back in your own porch and there's a percentage you'll end up in pluto and another percentage that says you'll change gender when you get to the other side.

Fortunately for all of these events, the probability will be so low that if you walk across the street once a minute, you will have to do so for longer than the age of the universe to see those effects.

It wasn't that long ago that human beings realised that the angles formed by the three sides of a triangle do not in fact have to total 180 degrees. It's not inconceivable that we may find ourselves in the distant future in a place where 1 + 1 does not equal 2.

Unlikely. You see, a triangle is a physical construct. 1 and 2 are numbers, and by definition 1+1 = 2. It may be possible, of course, to transform into a different space of numbers where unity + unity != 2*unity, but then you wouldn't be using normal numbers, and so it still doesn't break the statement: 1 + 1 = 2.

In fact, I can tell you right now how 1 + 1 != 2. Change to binary and literally 1 + 1 = 10.

Heh, no. You've just changed what you call 2. I could say 1 + 1 = beljworks, and you would know that beljworks = 2 in decimal.

 

[K.I.L.E.R]

In fact, I can tell you right now how 1 + 1 != 2. Change to binary and literally 1 + 1 = 10.

00000001 + 00000001 = 00000010

It's a less misleading representation.

 

[winterbird@nerdshack.com]

It wasn't that long ago that human beings realised that the angles formed by the three sides of a triangle do not in fact have to total 180 degrees.

Ok, you have me very interested.

How is that possible?

 

[KimB]

Ok, you have me very interested.

How is that possible?

It's not terribly difficult, really. Draw a triangle on the surface of a sphere. You'll find that the angles of the edges add up to greater than 180 degrees. For example, if you draw an equillateral triangle starting at one of the poles, going down to the equator, moving along the equator one quarter way around the globe, then back to the starting point, you will have drawn a triangle whose angles add up to 270 degrees.

A surface that does the opposite, where the angles of a triangle add up to less than 180 degrees, is a saddle.

This is an easy way to visualize it, as you are embedding a curved two-dimensional surface in three-dimensional space. But, it turns out that it is entirely possible to describe a curved two-dimensional surface only using two-dimensional mathematics. This is the basis of differential geometry, and its use in general relativity, where one describes the 3 space and 1 time dimensions of the known universe as being curved in a general sense without adding extra dimensions.

 

[arjan de lumens]

Ahem, only if you accept that 1 + 1 always equals 2 everywhere in the universe, which if going by the scientific method is only a theory and never a certainty, as such even Planck units are inherently anthropologic because they are derived from human experience.

Mathematics as a field is generally considered to be essentially disconnected from the empirical sciences; sometimes the methods and results of mathematics may be extremely useful in science, but mathematics itself is not inherently dependent on it. 1+1=2 is a result of applying a given set of definitions and axioms; its correctness depends only on the definitions/axioms used and not on its usefulness in modelling real-world phenomena.

The 1+1=10 example is IMO overused, boring, unenlightening etc as it is just a very simple symbol replacement, no more profound or enlightening than translating a sentence from English to German; for some slightly more profound rule changes, try any of:

• 1+1=1 (Boolean arithmetic)
• 1+1=0 (modulo-2 arithmetic)

 

[Richard]

Fortunately for all of these events, the probability will be so low that if you walk across the street once a minute, you will have to do so for longer than the age of the universe to see those effects.

Of course.

Unlikely. You see, a triangle is a physical construct. 1 and 2 are numbers, and by definition 1+1 = 2. It may be possible, of course, to transform into a different space of numbers where unity + unity != 2*unity, but then you wouldn't be using normal numbers, and so it still doesn't break the statement: 1 + 1 = 2.

Numbers are physical constructs in the sense they were created by human beings (that is the first of my points in jumping into this thread). Btw, how do you know that when we find a space/time/dimension/cookie jar where the very foundation of adding doesn't work that we won't be using "normal numbers"? (That's by second point for posting here).

Heh, no. You've just changed what you call 2. I could say 1 + 1 = beljworks, and you would know that beljworks = 2 in decimal.

Of course, I'm being facetious in the face of the "real life skepticism of science". Didn't you have to explain the enlightment of non-euclidian geometry? Doesn't it bother you that a scientific discovery roughly 200 years old is unknown to so many? Most importantly, doesn't it say a lot about us when it took us over 2000 years to "discover" it all the while mankind was building boats which clearly demonstrate it in practice?

 

[Basic]

Numbers are physical constructs in the sense they were created by human beings.

That doesn't make them physical constructs. They are designed by human beings with a completely theoretical framework.

Btw, how do you know that when we find a space/time/dimension/cookie jar where the very foundation of adding doesn't work that we won't be using "normal numbers"?

1+1=2, and that's not a corollary, it's by definition. If 1+1 is anything else, then you're not working with the "normal" number system.

You might find a paralell world where some physics behave such that putting two thingys next to each other make them look like 1.5 thingy. But that's strange physics, and doesn't change math so that 1+1=1.5.

 

[nutball]

You might find a paralell world where some physics behave such that putting two thingys next to each other make them look like 1.5 thingy. But that's strange physics, and doesn't change math so that 1+1=1.5.

No, because then you have to question your definition of "thing". If you put two "atoms" next to each other and they act like 1.5 atoms, it tells you something about atoms, it doesn't tell you something about two.

 

[KimB]

No, because then you have to question your definition of "thing". If you put two "atoms" next to each other and they act like 1.5 atoms, it tells you something about atoms, it doesn't tell you something about two.

That's what he was saying

 

[Crazyace]

just to be pedantic..

It's not terribly difficult, really. Draw a triangle on the surface of a sphere. You'll find that the angles of the edges add up to greater than 180 degrees. For example, if you draw an equillateral triangle starting at one of the poles, going down to the equator, moving along the equator one quarter way around the globe, then back to the starting point, you will have drawn a triangle whose angles add up to 270 degrees.

A surface that does the opposite, where the angles of a triangle add up to less than 180 degrees, is a saddle.

What you have described isn't a triangle, but instead a shape generated by connecting 3 points with 3 curves..

 

[Neeyik]

What you have described isn't a triangle, but instead a shape generated by connecting 3 points with 3 curves..

Which is a triangle by its very definition - ie. a three sided closed plane figure. The only reason why the sides are curved is because the plane is curved.

 

[K.I.L.E.R]

Let me get this straight:
This can actually be considered a triangle?

 

[Xmas]

No, the edges of a triangle are "straight" connections between the vertices, as in "minimum distance". On a curved plane however, the path of minimum distance can be curved, and so angle rules don't hold true any more. But most people just don't have to care about curved planes.

 

[KimB]

Which is a triangle by its very definition - ie. a three sided closed plane figure. The only reason why the sides are curved is because the plane is curved.

Nope. Because in curved space you have to generalize what you mean by a straight line. But, if you think about it, if you move in a "straight line" on the surface of a sphere, logically you'd go all the way around the globe, in what is called a "great circle" that is the intersection between the sphere and a plane that touches the center of the sphere. The triangle I described previously is made up of such great circles, and thus is a triangle in the proper sense.

So straight lines are still perfectly well-defined in curved space-time. You just ask what sort of path an object that is under no forces other than gravity would take. In flat space-time, this is what is what people usually think of as a straight line.

But because matter curves space, "straight lines," and thus the trajectories of objects under the influence of gravity, take parabolic paths near the surface of the Earth.

In a more general sense, you produce a straight line by asking the question: what is a path such that the direction of movement stays parallel to itself for the entire path? Mathematically this turns out to be a simple distance-optimization setup, where you use variational calculus to find the optimal path.

 

[Neeyik]

So in other (and less pedantic) words, you've basically just said what I did - the object in question is still be classed as a triangle because the sides are the minimum distance on a curved surface, which is what you would get if you took a triangle on a flat plane and then applied it to something like a sphere.

 

[KimB]

So in other (and less pedantic) words, you've basically just said what I did - the object in question is still be classed as a triangle because the sides are the minimum distance on a curved surface, which is what you would get if you took a triangle on a flat plane and then applied it to something like a sphere.

I'm sorry, I misread what you were saying in your previous post.

But it turns out that what you've said in this one is incorrect: you can't actually take a triangle on a flat plane and apply it to a sphere. Attempting to do so would require that you start removing space from the flat plane (i.e. you can't keep the paper smooth: it'll have to have folds). So you won't necessarily have a triangle any longer.

 

[Neeyik]

But it turns out that what you've said in this one is incorrect: you can't actually take a triangle on a flat plane and apply it to a sphere. Attempting to do so would require that you start removing space from the flat plane (i.e. you can't keep the paper smooth: it'll have to have folds). So you won't necessarily have a triangle any longer.

Quite right and anyone looking at a map of the Earth will understand this, but I think the basic point of what I (and yourself) were making is clear enough!

 

[Crazyace]

A "triangle" on a sphere would be a shape drawn so that it's projection viewed from a infinite point ( orthographic ) would be a triangle..

Any curved space is just a function in a higher dimension space - as such you can't talk about shortest lines on the curve, as the shortest lines only exist in the higher dimension... Overwise you are just playing games with shadows...

( It's only a pedantic point... If you redefine the terms you can make up rules as much as you like... but you've still changed the definitions : You say a triangle doesn't always have 180degree internal angle and give an example - I say they always do, and what you have defined is not a triangle in the strictest sense )

 

[KimB]

A "triangle" on a sphere would be a shape drawn so that it's projection viewed from a infinite point ( orthographic ) would be a triangle..

Again, nope. Doesn't work that way. No matter how far away you are, the projection of the triangle I described wouldn't look like a triangle.

Any curved space is just a function in a higher dimension space - as such you can't talk about shortest lines on the curve, as the shortest lines only exist in the higher dimension...

Nope, not at all. General Relativity is formulated in such a way that you describe the curvature without speaking of any extra dimensions. In fact, you really have to talk about curvature in this way if you don't want to use an inordinate number of extra dimensions for more complex curved systems.

( It's only a pedantic point... If you redefine the terms you can make up rules as much as you like... but you've still changed the definitions : You say a triangle doesn't always have 180degree internal angle and give an example - I say they always do, and what you have defined is not a triangle in the strictest sense )

Not really. You're describing a specific kind of triangle: one drawn in flat space. A more general triangle, one drawn in curved space, would have angles that add to a different number. This is, in fact, an experiment that cosmologists are attempting to perform on a cosmic scale.