Freethought & Rationalism Archive

KeithHarwood · 06-03-2002, 06:27 PM

Quote:

Originally posted by NialScorva:
You can even use the square root of i to add another dimension.

No you can't. Square root of i is the complex number (1+i)/sqrt(2).

Ion · 06-03-2002, 07:09 PM

Quote:

Originally posted by liquid:

...
Method 2: No number is real. Our use of 'real' numbers is just as arbitrary as 'imaginary' ones.

Basically, imaginary numbers are just as valid as real numbers - neither set exists any more or less in the actual universe.

I like this clear explanation.
Indeed 'real' and 'imaginary' numbers are human conventions used in describing the nature.

06-03-2002, 07:10 PM

Wow - I guess I shouldn't have expected any easy answers.. I was kind of hoping that that KHG (?) fellow would show up and just lay it all out... in any case, I guess that thread has died - the central point has been put to rest by other means.

Thanks for your help

NialScorva · 06-03-2002, 10:48 PM

Quote:

Originally posted by KeithHarwood:


No you can't. Square root of i is the complex number (1+i)/sqrt(2).

You're right, I was thinking about <a href="http://www.cs.berkeley.edu/~laura/cs184/quat/quaternion.html" target="_blank">quaternions</a>, which are all square roots of -1. Came across them for doing some nifty tricks in a scene graph.

NeilUnreal · 06-04-2002, 08:17 AM

You could use the square root of "i" to generate a new class of numbers, but the complex representation is a conceptual order of magnitude more parsimonious since it can be expressed within the system. Generating a new square root for "i" would be like adding "i" in the first place: it would require an extension to the system.

-Neil

p.s. If you're ever stuck for something to do, try examining a class of numbers which can be used to describe division by zero (the way "i" can be used to descibe sqrt(-1)).

Bluebird · 06-04-2002, 07:36 PM

Quote:

You could use the square root of "i" to generate a new class of numbers ... Generating a new square root for "i" would be like adding "i" in the first place: it would require an extension to the system.

Ummm...I don't understand this at all. It seems to me that when everyone has been talking about adding i to a number system, they are really talking about the complex numbers (C) which are all things of the form a+bi where a and b are real numbers. Now the complex numbers are what is called an algebraically closed field. In this case, that means that every polynomial that has complex coefficients (ie values in front of the powers of x are complex) has roots (ie solutions for x) in the complex numbers. Hence x^2-i=0 which is a polynomial with complex coefficients has a solution in C, the complex numbers. But the solution to this equation is the square root of i and thus already in C.

Or, as KeithHarwood said, (1+i)/sqrt(2) is in C already so adding it will not make the numbers you are studying any larger. Does that make sense?

Bluebird

Bluebird · 06-04-2002, 08:51 PM

And as a mathematician-in-training, here's my two cents on this:

Quote:

Saying you will not consider the possibility of supernatural causes in science sounds to me like saying you refuse to deal with imaginary numbers in math simply because they don't exist.

Plenty of mathematicians spend their whole lives thinking about things that seemingly have no connection to or description in our "reality", by this I mean the natural world around us. Math gets FAR more abstract than the square root of negative one.

I see math as this other world I spend time in where I make assumptions (axioms) and definitions and then see what I can prove from those, assuming some level of consistency. I don't make claims about reality from these proofs (although some math can be used to give us good ideas about what is going on around us) and I do not assume anything until I can prove it. (well...sometimes mathematicians will say "If A is true B holds" when they don't really know if A holds.)

Now, for the supernatural world. Many people work under the assumption that ghosts exist, or there are guardian angels, etc. These are axioms people have set up for their own little world (like my math world) but they try to force their axioms on reality. Also, most people don't go out of their way to prove assertions about the supernatural, or experiment or collect data and so forth.

So, even though math may seem to have a supernatural aura about it (and most people are probably more terrified of math than ghosts) the analogy ends there, at least as far as I am concerned. Thank you for reading my two cents!

Bluebird

06-05-2002, 06:07 PM

Quote:

(and most people are probably more terrified of math than ghosts)

If you get past simple differential calculus, I am among those more terrified of math than ghosts. Horrified even. Mortified!

seebs · 06-06-2002, 08:09 PM

Quote:

Originally posted by NialScorva:

First off, numbers are a description of the world, not an existing thing. A simple '5' doesn't exist either, it's a label we use to describe a certain count of objects that just exist. If you'll note that in your example, you say we can't measure something that's i inches long. It's the "inches" part that important, that's what ties the number to reality. The fact that complex numbers cannot be tied to reality in strictly the same way says nothing about them. It's like saying green doesn't exist because the sky's color can be measured in greeness.

I disagree on both counts. Real numbers correlate directly to observable properties of the real world. A square N inches across has an area of N^2 square inches... As long as N is real.

I also don't see any grounds for saying that 5 doesn't "exist". It exists as much as the referents of most personal pronouns do; careful study of bodies has never revealed a "person" anywhere in them, but we all believe in people.

I see numbers as being just as meaningful as the reality they describe; they're just not physical objects.

Quote:

As I said, imaginary numbers can be used to measure all sorts of things. Household electricity has an imaginary component to it (which if you feed a non-zero imaginary current back onto the grid, you can make your meter run backwards and transformers explode). They're one of several ways of describing a 2-dimensional value in a single piece, and length is not a 2 dimensional value.

Is it really imaginary, or is it just "a second dimension"? Not all second dimensions have the weird qualities of imaginary numbers.

If you have a plane with X and Y coordinates, it is not *always* the case that squaring a y value gets you back onto the x axis, but squaring an x value just keeps you in x's.

Quote:

The supernatural is a completely different situation. Instead of descriptions, we're now talking about existence. The problem is that science is all about giving descriptions to existing things, and the supernatural is practically undescribable by definition. You can't describe a pattern in a unique event, or rather you can describe any number of patterns that are equally (in)valid. Even assuming that the supernatural does exist, it's never going to be useful because it can't be predicted. If it can be described and predicted, then it's natural. Not much else to it, in my view.

I tend to agree that, if it can be described and predicted, it's natural... The interesting question is whether or not there may be natural things we don't currently understand which meet the descriptions given for "supernatural" things. I don't see this as any weirder than claiming that, some day in the distant future, people will have special techniques which allow them to cause objects to float in space without any kind of physical support. Magnetism is *normal* to us now... But once it was close to magic.

NeilUnreal · 06-07-2002, 08:50 AM

Two novices and a master were watching a flag moving in the wind.

"It is the flag that moves." says the first novice.

"It is the wind that moves." says the second novice.

"It is neither, it is the mind that moves." says the master.

Two novices and a master were watching a man counting cattle using an abacus.

"There are five cows." says the first novice.

(etc.)

-Novice Neil

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06-03-2002, 07:10 PM	#13
Jerry Smith Guest Posts: n/a	Wow - I guess I shouldn't have expected any easy answers.. I was kind of hoping that that KHG (?) fellow would show up and just lay it all out... in any case, I guess that thread has died - the central point has been put to rest by other means. Thanks for your help

06-04-2002, 08:17 AM	#15
NeilUnreal Junior Member Join Date: Nov 2001 Location: USA Posts: 77	You could use the square root of "i" to generate a new class of numbers, but the complex representation is a conceptual order of magnitude more parsimonious since it can be expressed within the system. Generating a new square root for "i" would be like adding "i" in the first place: it would require an extension to the system. -Neil p.s. If you're ever stuck for something to do, try examining a class of numbers which can be used to describe division by zero (the way "i" can be used to descibe sqrt(-1)).

06-07-2002, 08:50 AM	#20
NeilUnreal Junior Member Join Date: Nov 2001 Location: USA Posts: 77	Two novices and a master were watching a flag moving in the wind. "It is the flag that moves." says the first novice. "It is the wind that moves." says the second novice. "It is neither, it is the mind that moves." says the master. Two novices and a master were watching a man counting cattle using an abacus. "There are five cows." says the first novice. (etc.) -Novice Neil

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