# Number and function for non-mathematician

Hi Norman,

Le 20-juin-06, à 04:04, Norman Samish a écrit :

I've endured this thread long enough!  Let's get back to something I can understand!

This means that at some point you have stop to understand. It is easier to help if you say so at the moment when you stop to understand. I mean this both for the Universal Dovetailer Argument (UDA) and the "mathematical UDA" (diagonalization stuff). Although UDA does not belongs to math, it is a rigorous reasoning showing that IF someone assume the COMP hypothesis, THEN the physical world appearances MUST emerge from the platonic relations that occur among numbers. Then the mathematical UDA, alias the interview of the universal (lobian) machine, is only a more precise version of UDA so that a (universal) machine can understand it and present it in a refined testable way..
The diagonalizations I have done are needed if only just for explaining what is a Universal Machine, how they could exist, etc.

Not that there's anything wrong with that, but we must acknowledge that Bruno speaks a language that very few of us can understand

Mmmhh... I was hoping talking a language *all* can understand. The language of numbers and functions (elementary math). By "all" I mean all *universal entity* having a minimum introspective ability (and later even this sentence will shown to be "understandable" by universal *machine*, but this, of course, will need some amount of work). Of course I am sure you are a universal machine (at least), but this will be shown in due course.

Norman, I would bet you have only a trouble with the math notations, not with the math itself. As a math teacher I know that about 99.99999.... of those who thinks they have problems with math have only problem with notations or with motivation. I would like to help you to transcend that notation problem. I guess you have a problem with the notation for functions, and perhaps you need no more than a reminding of the basic definitions.

I'll reply, "Because your audience is shrinking!  I've plotted the Audience vs. Topic, and find that, in 12.63 months, there is a 91% probability that, if the topic doesn't become understandable to one with an IQ of 120, your audience will be zero, and the only expositor will be Bruno.

Fromthis I can infer you have some knowledge on numbers. At least of the numbers 12.63, 91, 100, 120, 0 and 1. :-)

Well, I am sure you have some knowledge about the natural numbers:

N = the collection of all numbers like 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...

I guess you remember that the expression "378" denotes usually the number given by three times 100 added to seven times 10 added with eight times 1. All right? So a number like 36 denotes really the number of strokes in the following diagram:

| | | | | | | | | |
| | | | | | | | | |
| | | | | | | | | |
| | | | | |

Three boxes of ten strokes, and a box of six strokes. All right?
In those arrangements the number ten plays a conventional role. Why ten? Probably because we have ten fingers(*).

One of the key notion in math is the notion of function. In our setting, we can begin to limit ourself to function from N to N. But first, what is a function?
A function from A to B is just anything which for each object in A associates one object in B. Here A = B = N, and thus, a function from N to N is an association of one number for each number.
If "f" is a name for some function, "f(n)" represents the value f associates to n. See examples below.
Now there is an infinity of numbers (cf N= {0, 1, 2, 3, 4, 5, ...}, so a function can be represented by an infinite table of values:

Exemple: Hal Finney mentionned the factorial function: it associates to each number n, the product of all non nul numbers least or equal to n, except if n is null in which case the factorial of n is one, thus we have:

Factorial(0) = 1
Factorial(1) = 1
Factorial(2) = 2
Factorial(3) = 3 times 2 = 6
Factorial(4) = 4 times 3 times 2 = 4 times 6 = 24
Factorial(5) = 5 times 4 times 3 times 2 = 5 times 24 = 120
Etc.

A function can be represented by an infinite set, which is just the set of all the associations provided by the function. Example:

Factorial = {(0, 1) (1, 1) (2, 2), (3, 6) (4, 24) (5 120) (6, 720) ...}

But function from N to N can also be represented just by their value on 0, 1, 2, ...

Factorial = 1 1 2 6 24 120 720 ...

Other examples: the function Double which sends n on 2 times n:

Double = 2, 4, 6, 8, 10, 12, 14, 16, ...

Norman, are you ok with such a talk.

Concerning the motivation, just remember that old name for a computer: a number crunching machine. Indeed, even if do some text processing with a computer, whatever you type on the keyboard can be coded or interpreted as a number, the working of the computer can be interpreted in term of a function from numbers to numbers, the output again is a number which can be (re)interpreted as a text.

Norman, and all non mathematician, please tell me if you understand this post, before trying to read the "diagonalization (solution)" post.

I'll reply, "Damn!  I was hoping to learn something!"

Tell me if this post helps, and in case it doesn't, tell me where you have a problem. If your problem bears on the philosophical motivation, please tell us (that is interesting too).

We need just to define what is a bijection, and you will be ready for the diag posts.

Bruno

(*)Not (currently) important remark; But it helps a lot for the learning of how to separate accidental truth relying on the choice of the notation, and truth on numbers (or functions) which does not depend on the choice of notation. To be a "round" number, like 10, 20, ... is notation dependent. To be an odd number or to be a prime number is not dependent on the choice of notation. This can help to figure out the non conventional part of math.

Could you find what "378" would represent for an extraterrestrial having only eight fingers? (assuming they allows one symbol by digit, and using like us 0 for the number of "a" in the word "bbb", 1, for the number of "a" in the word "babb", by 2 the number of "a" in the word "bababb", etc. By 7 they mean of course the number of "a" in the word "bbbbababababaaabbb". All right?
But they have only 8 digits, (four at each hand, for example). So they can use only (by our assumption that numbers are written with as many elementary symbols than we have digits) with the eight symbols: "0", "1", "2", "3", "4", "5", "6", "7".
Like us they simplify the notation for numbers by packages. Only they use package of eight instead of ten. So for them, "36" will represent (3 times 8) + 6 = the number *we* are used to wrote "30".
How then will they write our number 36, where the "3" means three packages of ten, and "6" means
six unities? well 36 = 32 + 4 = (4 times 8) + 4 = 44.
Can you count like an extraterrestrial who has only two fingers (one at each hand)? The answer is:

0,
1,
10 = (1 times 2) + 0) = 2 (in base ten)
11 = (1 times 2) + 1) = 3
100 = (1 times 4) + 0 times 2 + 0 times 1) = 4
101 = 5
110 = 6
111 = 7
1000 = 8
1001 = 9
1010 = 10
1011 = 11
etc. ... OK?

...must go now,

Bruno

http://iridia.ulb.ac.be/~marchal/

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