Wednesday, August 25, 2010
Subscribe to:
Post Comments (Atom)
LINKS
Frequent Commenters Here
Blogs I Visit Every Day or Two
My novel, Temping.
"...like tincting with absinthe..."
LUTHERAN SURREALISM
Photo of Blogger
Blog Archive
- March (2)
- February (8)
- January (11)
- December (9)
- November (11)
- October (9)
- September (12)
- August (11)
- July (9)
- June (14)
- May (9)
- April (8)
- March (12)
- February (9)
- January (13)
- December (14)
- November (16)
- October (20)
- September (15)
- August (15)
- July (14)
- June (19)
- May (20)
- April (17)
- March (19)
- February (14)
- January (15)
- December (14)
- November (23)
- October (26)
- September (16)
- August (18)
- July (21)
- June (18)
- May (18)
- April (16)
- March (16)
- February (14)
- January (21)
- December (11)
- November (20)
- October (16)
- September (15)
- August (13)
- July (18)
- June (19)
- May (16)
- April (14)
- March (15)
- February (15)
- January (19)
- December (13)
- November (11)
- October (11)
- September (9)
- August (10)
- July (12)
- June (12)
- May (15)
- April (17)
- March (14)
- February (14)
- January (21)
- December (15)
- November (8)
- October (13)
- September (11)
- August (10)
- July (13)
- June (13)
- May (11)
- April (16)
- March (17)
- February (16)
- January (21)
- December (18)
- November (25)
- October (25)
- September (19)
- August (24)
- July (23)
- June (11)
- May (15)
- April (12)
- March (12)
- February (15)
- January (17)
- December (15)
- November (14)
- October (11)
- September (10)
- August (13)
- July (23)
- June (19)
- May (13)
- April (20)
About Me
- Kirby Olson
- Upstate, NY
- PLEASE READ THE INTRO AND ANNE WALDMAN'S EXAMPLE BEFORE CONTRIBUTING! Send to Kirbyolson2@gmail.com.
10 comments:
Ah. You've decided to restart your mathematical studies. Here's your next lesson:
2.
You're getting way ahead of me. I haven't mastered this lesson yet!
Kirby,
You're getting way ahead of me. I haven't mastered this lesson yet!
Ah, yes. Mastering one is harder than most people appreciate. Just remember your preliminaries...
0
1 = S 0
I.e., there are two kinds of natural numbers. 0, which is the atomic basis of the natural number system, and natural numbers are successors of other natural numbers.
In this, 1 is the first non-atomic natural number, and so is different structurally from 0. So the starting point of your studies really should be 0. Have you mastered that lesson?
How exactly do you get from zero to 1? I think I should go back to zero.
But since zero is a circle, I find it hard to leave that number. It's so inviting, so familiar. why do the other numbers exist?
Kirby,
How exactly do you get from zero to 1? I think I should go back to zero.
Ah. There are two axioms for the successor function that are relevant here.
1. for all natural numbers n, n ≠ S n
2. for all natural numbers n, 0 ≠ S n
The first says that no natural number has the property you predicate for zero, i.e., that it is its own successor. The second goes further, and says that zero isn't the successor of any natural number (let alone itself).
It's so inviting, so familiar. why do the other numbers exist?
A cynical answer might be, "keeping bankers and mathematicians employed." A better answer is Kronecker's, "God made the natural numbers, all else is the work of man." Of course, Kronecker said it in German.
From a mathematician's point of view, the interesting thing about the natural numbers, 0, 1, 2, etc., is that they represent the simplest possible infinite set. Indeed, the classical definitions of infinite (e.g., the existence of a function to the set that is 1-1 but not onto, per Dedekind), show that there are natural 1-1 functions from the natural numbers into any infinite set. Thus, in a mathematically precise way, the natural numbers form a minimal infinite set. Therefore, understanding the natural numbers is the gateway to understanding infinity.
In the beginning there was only 0.
And then there was 1.
And then there was 2.
...
I am sure someone has pointed out this book to you.
http://www.amazon.com/One-Two-Three-Infinity-Speculations/dp/0486256642
I think no one begins with zero. They start with 1, and then 2 and so on. Eventually someone asks where they came from before they were at 1, and this is what zero is, but I'm not sure it exists the same way as the other numbers. It seems made up.
Zero is the number you use when you don't have something.
How many apples do you have?
In the olden days, if you didn't have an apple, you'd open your mouth wide, speechless, and it would form a circle, which explains the shape of the number 0.
I think you guys are thinking a bit too sequentially...
It's not about what comes before 1.
It's about what you have when you throw 1 away.
And then you have how many left?
We'll designate that open-mouth 0.
Or it's more about what you have when you don't have anything.
Zero.
William,
I am sure someone has pointed out this book to you.
http://www.amazon.com/One-Two-Three-Infinity-Speculations/dp/0486256642
My Ph.D. is in Math, specifically Math Logic, more specifically the computability theoretic properties of random sets. Realistically, Gamow is beyond Kirby, and I'm beyond Gamow. So it goes ...
I think no one begins with zero. They start with 1, and then 2 and so on.
Well, I think Brett nailed this one. Natural numbers are essentially the canonical elements of the finite sets under the bijection relation, i.e., the finite cardinals. You need to account for the empty set, hence 0.
Let me tell you a little story. When I was a pup, in the Ph.D. program at Illinois, one of my profs was Gaisi Takeuti. Takeuti told a story about how he'd once been asked to give a seminar to the Philosophers. Anyway, he started with Peano Arithmetic, let the set of natural numbers be {0,1,...}. This was literally his starting point. At this point, the philosophers got entangled in an argument as to whether the natural numbers began with 0 or 1, and that was pretty much the end of the seminar.
Anyway, Gaisi had the typical mathematician's point of view on this. He'd prefer to start with zero (because of the motivation via finite cardinals), but it really didn't matter -- the resulting theories essentially equivalent, so you it really doesn't matter which you pick, just so long as you do and get on with the job.
And I'll note that most CS types start with zero, if only because of the array indexing conventions of C.
Post a Comment