[jdonn]

Here is a lecture on the beginning of time by Stephen Hawking. It is extraordinarilly interesting.

http://www.hawking.o...ctures/bot.html

[mike777]

Here is a Hawking article on the end of time.

Hawking: Humans Must Spread Out in
Space
AP Jun. 14, 2006
*************************
The survival of the human race
depends on its ability to find new
homes elsewhere in the universe
because there's an increasing risk
that a disaster will destroy Earth,
world-renowned physicist Stephen
Hawking said Tuesday. "Life on Earth
is at the ever-increasing risk of
being wiped out by a disaster, such
as sudden global warming, nuclear...
http://www.kurzweilai.net/email/newsRedire...ml?newsID=5634&

[luke warm]

jdonn, on Jun 15 2006, 04:16 PM, said:

Here is a lecture on the beginning of time by Stephen Hawking. It is extraordinarilly interesting.

http://www.hawking.o...ctures/bot.html

amazing coincidence, i just read that article at work... notice (again) how, to get around even the possibility of a creator, hawking et al have to rely on imaginary numbers... otherwise, a beginning... 'from nothing nothing comes', while scoffed at by some, is still a philosophical idea held in high regard by many

Quote

"Life on Earth is at the ever-increasing risk of being wiped out by a disaster, such as sudden global warming, nuclear..."

yes, he thinks such things have happened many times in the past already, how else to explain man's present level of existence (after however long we have supposedly been here)?

[helene_t]

Jimmy, imaginary numbers are not difficult at all and it's convenient to use them in the description of several physical phenomenas (oscilating voltage, sound waves, quantum fields). Some phycisist believe in God and some don't but they all use imaginary numbers.

[luke warm]

my interests lean more toward philosophy and not mathmatics, but the statement "the square root of -1=the square root of -1" is tautologous... i see a definite internal contradiction when a person says that i represents a number for which no value exists other than the -1 one supposedly gets from squaring it

i don't dispute that imaginary numbers are used, i've just never read a satisfactory explanation as to why, or seen *any* example with an answer ...

[helene_t]

luke warm, on Jun 16 2006, 12:49 PM, said:

my interests lean more toward philosophy and not mathmatics, but the statement "the square root of -1=the square root of -1" is tautologous... i see a definite internal contradiction when a person says that i represents a number for which no value exists other than the -1 one supposedly gets from squaring it

i don't dispute that imaginary numbers are used, i've just never read a satisfactory explanation as to why, or seen *any* example with an answer ...

Where have you been looking? Google on
"complex numbers" voltage
for example.

There's realy no problem with defining the number i as the square root of -1. You just have to get used to the idea that numbers don't have to be confined to the real axis. I don't see why they should be. You can also have four-dimensional numbers if you want but they are less usefull in physics.

[david_c]

helene_t, on Jun 16 2006, 12:54 PM, said:

There's realy no problem with defining the number i as the square root of -1.

Well, actually I think that defining i as the square root of -1 and then defining the rest of the complex numbers from that is the wrong way round. It's better and less confusing to start by defining the complex plane as a whole:

We define a complex number to be a pair (a, b), where a and b are real numbers. So immediately it has an obvious interpretation as a point in a plane. We call a the "real part" and b the "imaginary part". A complex number with zero imaginary part can be thought of as being a real number. That is, we associate the complex number (x,0) with the real number x.

Addition is defined by (a, b) + (c, d) = (a+c, b+d).

Multiplication is defined by (a, b) x (c, d) = (ac-bd, ad+bc).

Having done this, now we say we will use i as a shorthand for the complex number (0,1). Then it follows from our definitions of addition and multiplication that the complex number (a, b) can also be written as a + ib, and that the square of i is -1.

[helene_t]

Well you could do that, but it looks artificial (why would one want numbers to be two-dimensional?) and it leaves the question if some alternative construction could be used instead.

The whole purpose of the exercise is to provide solutions to algebraic equations. So I think it's more elegant to to develop the complex numbers like this:

Let's assume that there is a solution to the equation x^2 = -1 and lets call the solution i. Now it can be proved that all algebraic equations have solutions. Everything that can be deduced from this automatically holds for any construction that provides solutions to all algebraic equations.

Later on, we can introduce the following construction as a model for the complex number: Associate the number 1 with the matrix

1 0
0 1

and associate the number i with the matrix

0 1
-1 0

[kfgauss]

david_c, on Jun 16 2006, 01:16 PM, said:

Well, actually I think that defining i as the square root of -1 and then defining the rest of the complex numbers from that is the wrong way round. It's better and less confusing to start by defining the complex plane as a whole:[snip]

helene_t, on Jun 16 2006, 02:06 PM, said:

Well you could do that, but it looks artificial (why would one want numbers to be two-dimensional?) and it leaves the question if some alternative construction could be used instead.

The whole purpose of the exercise is to provide solutions to algebraic equations. [snip]

And we have the algebraist vs geometer distinction .

Not to come down on one side or the other, but there are many purposes for complex numbers, not one "whole purpose of the exercise," unless you mean understanding them in order to better understand Hawking's lecture. In that case, he takes a rather geometric viewpoint.

Andy

[david_c]

kfgauss, on Jun 16 2006, 03:36 PM, said:

And we have the algebraist vs geometer distinction  .

lol, yes I was thinking that! My work is in dynamical systems, can you tell?

[jdonn]

luke warm, on Jun 16 2006, 05:49 AM, said:

my interests lean more toward philosophy and not mathmatics, but the statement "the square root of -1=the square root of -1" is tautologous... i see a definite internal contradiction when a person says that i represents a number for which no value exists other than the -1 one supposedly gets from squaring it

i don't dispute that imaginary numbers are used, i've just never read a satisfactory explanation as to why, or seen *any* example with an answer ...

This could clear things up for you.

http://www.friesian.com/imagine.htm

Somehow I doubt that it actually will, but I feel as though it should.

[joshs]

But i to the i'th power is around .21

[luke warm]

helene_t, on Jun 16 2006, 09:06 AM, said:

Let's assume that there is a solution to the equation x^2 = -1

why?

[barmar]

luke warm, on Jun 16 2006, 06:49 AM, said:

my interests lean more toward philosophy and not mathmatics, but the statement "the square root of -1=the square root of -1" is tautologous... i see a definite internal contradiction when a person says that i represents a number for which no value exists other than the -1 one supposedly gets from squaring it

i don't dispute that imaginary numbers are used, i've just never read a satisfactory explanation as to why, or seen *any* example with an answer ...

What you apparently don't realize is that ALL of mathematics is just an abstract system devised by people to help them calculate things. Whole numbers happen to have a direct correspondence to countable objects, which we find intuitive, but as more and more complicated number systems are devised (fractions, negatives, real numbers, imaginary numbers, quaternions, matrices, etc.) they become more abstract. But they're all just defined by the way they're used.

As an example, the number -1 only exists as the solution to the equation x+1=0. It's not like you can hold -1 apples in your hand. Is there really a big difference between defining negative numbers this way and defining imaginary numbers as the solution to quadratic equations?

The statement "the square root of -1 is i" is not a tautology, it's a definition -- it gives a convenient name to a quantity that is described by the equation (so we don't have to write sqrt(-1) all the time). It's no different from "the ratio of the circumference of a circle to its diameter is pi".

[luke warm]

Quote

What you apparently don't realize is that ALL of mathematics is just an abstract system devised by people to help them calculate things.

thanks, it's a lot clearer now

all i said in my first post was that hawking uses imaginary spade/time "... to get around even the possibility of a creator, hawking et al have to rely on imaginary numbers..."

in his article he says, "However, many people were unhappy with the idea that the universe had a beginning, because it seemed to imply the existence of a supernatural being who created the universe." (a concept einstein had no trouble grasping)

hawking and others, to get around the need of a creator, proposed imaginary time and space, as shown by this quote from the article, "But one wouldn't have to appeal to something outside the universe, to determine how the universe began."

he also had to have some way to suspend the 2nd law of thermodynamics, so he says that the big bang, while being determined by the laws of physics, possessed no such laws (they had broken down at the singularity)...

he's quite obviously a genius... but others were just as obviously so, and others could exercise their genius without presupposing anything at all concerning the existence of God

[david_c]

Hawking might describe the motivation of the theory differently if he was writing for an audience of theoretical physicists. In his essays for a general audience I think he rather overplays the philosophical aspects. Surely the main reason he likes the theory is because it is sound theoretical physics.

[luke warm]

probably he would... even so, i think it's fairly obvious that the no creator presupposition plays a large role... some of the philosophical arguments are too hard to handle, as he has repeatedly stated... so the philosophy of the matter imo does play a large role in his thinking

[whereagles]

I have a friend working at the European Space Agency. He says putting stuff in orbit is far more difficult that what Hawking seems to believe. Space travel is definitely going to be impossible unless we come up with some major conceptual breakthrough. And I really mean major.

Now, if the brane world scenarios proved to be true, that would be a whole different ball game...

[Al_U_Card]

To quote a reasonably current popular alternative rock song....."Insane in the membrane"

[barmar]

luke warm, on Jun 17 2006, 09:48 AM, said:

all i said in my first post was that hawking uses imaginary spade/time "... to get around even the possibility of a creator, hawking et al have to rely on imaginary numbers..."

There are many different theories that try to address this, I don't think they all rely on imaginary space/time. For instance, there are theories involving a succession of universes -- after expanding for a few billion years, gravity eventually pulls everything in to a Big Crunch, and this crunch results in an explosion that is the Big Bang for the next universe. And string theory includes some theories where a Big Bang occurs when a pair of branes intersect as they're floating through the multiverse -- the universe is essentially a bubble created from that interaction.

I'm sure the mathematics for all of these things make use of complex numbers, but that's hardly surprising. Complex numbers are used for much more mundane tasks, like electrical engineering, and they come up all the time in most advanced physics.