Monday, April 21, 2014
'R' is fer Realz, yo!
'R' is for Let's Get Real (the Real Number line)
... because this reality you're living in?
It isn't. It's entirely manufactured for you and by you, and you have no clue that you're living in this unreality that you think isn't.
It's as simple, and as pervasive, as this:
The 'Real Numbers'?
Let's take a step back.
First, as you know, there was counting, and that started, naturally, from the number one.
But even that statement is so obviously flawed and ridiculous on the face of it to a modern-day mathematician. Why are you starting with the number one? Why aren't you starting with the number zero?
This isn't an argument over semantics, by the way, this has real (heh!), fundamental impact, for the number one, in counting, is not the identity. You cannot add one to a number and get that number back, and if you can't do that, you don't have a category (a 'Ring'), and if you don't have that, you have nothing, because your number system has no basis, no foundation, and anything goes because nothing is sure.
But getting to the number zero, admitting that it exists, even though it represents the zero quantity, so technically, it refers to things that don't exist (and therefore a fundamental problem with the number zero in ancient societies) ... well, there was a philosopher who posited that the number zero existed.
He was summarily executed by Plato and his 'platonic' buddies because he had spouted heresy.
If number zero exists, then you had to be able to divide by it, and when you did, you got infinity. And, obviously, infinity cannot be allowed to exist, so no number zero for you.
We went on for a long, long time without the number zero. Even unto today. You study the German rule, then you learn your multiplication tables starting from which number? Not zero: one.
"One times one is one. One times two is two. One times three is three."
Where is the recitation for the "Zero times ..."?
And I mean 'we' as Western society, as shaped by Western philosophy. The Easterners, lead by the Indians, had no problem admitting the number zero, they even had a symbol for it, 0, and then playing in the infinite playground it opened up, and therefore Eastern philosophy thrived, flourished, while Western philosophy, and society, stagnated, ...
... for one thousand years, ...
Just because ... now get this, ... just because people refused to open their eyes to a new way of seeing the world, ...
... through the number zero.
That's what happened when finally West met East, through exchange of ideas through trade (spices) and the Crusades (coffee served with croissants), and philosophers started talking, and the number zero was raised as a possibility.
Mathematics, mathematical ideas, and ideas, themselves, exploded onto the world and into though. Now that there was zero, there was infinity, now that there was infinity, and it was tenable, people now had the freedom to explore spaces that didn't exist anymore. People could go to the New World now, both figuratively and literally.
For growth in Mathematics comes from opening up your mind the possibilities you wouldn't ('couldn't') consider before, and growth in Mathematics leads to opening the mind further.
Take, for example, the expansion of the counting numbers, from not admitting zero to, now, admitting it, yes, but then the fractional numbers. You could count fractionally now.
From zero to infinity there were an infinite number of numbers, but, with fractions, we now found that from zero to one, there were an equal number of infinite number of fractions.
The really neat discovery was that if you put all the fractions in one set, and you put all the counting numbers into another, there was a one-to-one correspondence between the two.
An infinity of counting numbers was the same size as an infinity of counting-by-fraction numbers.
So, infinity was the biggest number, fer realz, then, eh?
No, not really.
Because then came what we call the 'Real Numbers' (which aren't, not by a long shot), and then we found an infinite number of numbers between one-half and one-third.
But the thing with these numbers?
The were rationals (fractional) in there, to be sure, but they were also irrationals.
There were numbers like π and τ and e and other weird and wonderful numbers, and the problem with these numbers was that there was no correspondence between them and the rational numbers. There was no way you could combine rational numbers in any form and point directly to τ, for example. These numbers were transcendent.
What's more: they were more. There were infinitely more transcendent numbers, irrational numbers, than there were rationals.
And not even countably infinite more, they were uncountably more infinite.
There was an infinite that was bigger than infinity, and this we call the Continuum.
Why? Because between zero and one and then between one and two there's this measured, discrete, gap, and this we use for counting. There's a measured, even, step between the counting numbers, and even between the fractional numbers: you can count by them, because between them there is this discreteness.
Between the Reals there's no measurable gap. You can't count by them, and you can't add 'just this much' (every time) to go from τ to π ...
(Heh, actually, you can: π = τ + τ, but then what? What's the 'next' irrational number? There's no such thing as the 'next' irrational number, because no matter what 'next' number you choose, there will always be an uncountably infinite number of numbers between that 'next' number and the number you started from, so your 'next' number isn't the next number at all, and never will be.)
So, wow, the Reals. Lots of them. They cover everything, then, right?
Not even close.
There are numbers that are not numbers.
For example, what is the number of all the functions that yield the number zero?
There are, in fact, an infinite number of those.
How about all the functions that give you ('eventually') π?
... Ooh! There are several different ones to find π, aren't they?
Yes. Yes, there are. In fact, there are an uncountably infinite number of functions that compute π.
Now, wait. You're saying, geophf, that there are uncountably infinite number of functions to find each and every Real number and that the Real numbers are uncountable as well, so that means...
Yeah, the Continuum reigned supreme for just a while as the biggest-big number, but it was soon toppled by this Powerset infinity (my term for it, it's actually called something else).
Now, I don't know the relation between the functions that yield numbers, and the functions that construct functions that do that.
But do you see where we're going with this?
As big as you can stretch yourself, there's new vistas to see in mathematics (and meta-mathematics, let's not neglect that, now, shall we?).
But we still haven't scratched the surface.
Is the world quantized, like the rational numbers? Or is it a continuum like the Reals?
Or is something else, even something more?
Direct current comes to you in a straight, steady line.
The thing about DC ('direct current')? It sucks.
(Just ask Marvel.)
If you want clean, pure, ... powerful, ... well: power, over any kind of distance, you have to ask ... not our friend Tommy (Edison), but our wild-child Tesla.
He proposed to Edison that we should use AC ('alternating current') to provide electricity, and Edison threw him out of his lab, that idiot, telling him never to show his face there again.
Guess how your electricity is delivered to your home today?
The thing about alternating current? It's a wave-form, and not only that, it's a triple wave-form. How do real numbers model electricity? Well, with DC, you've got one number: "That there is 5 volts or 5 amps or 1.21 gigiwatts."
Boy, that last one came out of the blue: like a bolt of lightning!
But if it's alternating current, then you need the sine and cosine functions to describe your power. Functions? Wouldn't it be nice if it were just a number?
Yes, it would be nice, and there is a number to describe wave-form functions, like your alternating current.
They are called 'imaginary numbers,' because, if you look hard enough on the number line, with good enough eyes, eventually you'll see the number π or τ or e or 1, or 2, or 3, or even zero.
But no matter how hard you strain your eyes, you will never see a number with an imaginary component, because why? Because most imaginary numbers, being on the curve of the wave-form are either above or below the number line. They 'aren't' numbers, then, because they're not on the numberline.
I mean, come on! The square-root of negative one? Why would anybody do this? Unless they were daft or a bit batty.
The thing is, without imaginary numbers, we wouldn't have the forms to get our heads around alternating current.
Most of the world, except those very lucky few who lived within a mile or two of a power plant, would be in darkness.
And the computer? Pfft! Don't get me started. Hie thee to the nunnery, because we are now back in the Dark Ages.
Or at most in the Age of 'Enlightenment' where you had to run for cover when the landlady bellowed "'ware slops!" ... unless you wanted raw sewage mixed with rotted fruit on your head.
But now, here we are, because we have both Real and imaginary numbers, together giving us the complex number set (which, it turns out, is not bigger than the Reals, as there is a one-to-one correspondence between each real number and each complex number. Fancy that! An infinity 'more' number of complex number above and below the Real number line gives the same number of complex numbers as Reals).
Not even close.
Last I checked, I don't live in Flatland, and, last I checked, nor do you.
Complex go 'above' and 'below' the Real number line, but ... what about the third dimension? Is there numbers to model us in three dimension?
What would such numbers look like?
And here's a stunner. If I were on Mars, or the Moon, and you were here, reading this blog post, how would I know where to look to see you?
The Moon, and Mars, too, has their own three-dimensional frames of reference, and the Earth has its own, too (it's called geocentric). So, to draw a line from a satellite (such as the Moon, known as 'Earth's satellite') so that it can look down at a spot on the Earth, you actually have to use a four-dimensional number to connect the two three-dimensional frames of reference so that one can look at the other. This four-dimensional number is called the Quaternion.
It's simple, really, it's just rocket science.
And it's really ... mind-bending, at first, wrapping your head around the math and drawing pictures, or using both your hands, three fingers out indicating both sets of axes, and you use your nose to draw the line connecting the two, and then you scream, 'but how do I measure the angles?'
Not that I've worked on satellite projects or anything. cough-EarthWatch-cough.
But nowadays, you can't get into making a realistic game without having quaternions under your belt. Monster sees you, monster charges you, monster bonks you on the head. Game over, thank you for playing, please insert $1.50 to
die ... that is to say, to try again.
The thing is: how does the monster 'see' you? The monster has it's own frame of reference, just as you do. The monster exists in its own three-dimensional coordinate system, just as you do. If you were standing on a little hillock, would you expect the monster not to see you because you're slightly elevated?
Of course not! The monster sees you, the monster bonks you. All of this happens through transformation of disparate coordinate systems via quaternions.
Now that's something to impress people with at cocktail parties.
'Yeah, I spent all day translating coordinate systems using quaternions. Busy day, busy day."
Just don't say that to a mathematician, because he'll (in general, 'he') will pause, scratch his head then ask: "So you were checking out babes checking you out?"
Then you'll have to admit that, no, not that, you were instead avoiding your boss trying to catch your eye so he could hand you a whole stack of TPS reports to work on over the weekend.
Like I ... didn't. Ooh. Ouch! Guess who was working through Easter?
Okay, though. Four dimensions. We've got it all, now, right?
Not if you're a bee.
Okay, where did that come from?
Bees see the world differently from you and me.
Please reflect on the syntax of that sentence, writers.
Bees see the world differently (not: 'different') from you and me (not: 'from you and I').
Gosh! Where is the (American) English language going?
(But I digress.)
If you look at how they communicate through their dance, you see an orientation, but you also see some other activity, they 'waggle' (so it's called the 'waggle-dance') and the vigor at which they do their waggle communicates a more interesting cache of nectar. There are other factors, too.
The fact of the matter: three dimensions are not enough for the bee's dance to communicate what it needs to say to the other drones about the location, distance, and quantity of nectar to be found.
So, it has its waggle-dance to communicate this information. Everybody knows this.
Until, one little girl, working on her Ph.D. in mathematics, stopped by her daddy's apiary, and, at his invitation, watched what he was doing.
"Daddy," she said, "those bees are dancing in six dimensions."
Guess who changed the topic of her Ph.D., right then and there?
Combining distance, angle from the sun, quantity of interest, ... all the factors, the bees came up with a dance.
They had only three dimensions to communicate six things.
The thing is, nobody told the bees they had only three dimensions to work with. So they do their dance in six dimension.
If you map what they are doing up to the sixth dimension, it gives a simple (six-dimensional) vector, which conveys all the information in one number.
Bees live in six dimensions, and they live pretty darn well in it, too.
Or, put this way: 80% of the world's food supply would disappear if bees didn't do what they did.
You are living in six dimensions, or, more correctly, you are alive now, thanks to a little six-dimensional dance.
Okay, but what if you're Buckaroo Banzai?
Pfft! Eight dimensions? Light weight.
In String Theory, we need at least ten dimensions for super strings, and twenty-six dimensions for some types of strings.
So, 'R' is for real numbers.
The neat thing about numbers, is ... they can get as big as you can think them.
And they're good for a Ph.D. thesis ... or two.http://logicaltypes.blogspot.com/2014/04/f-is-for-function.html