The seven step series

[Bruno Marchal]

On 28 Jul 2009, at 17:36, m.a. wrote:

Bruno,

             I have searched my notes for an exposition of BIJECTION and found only one mention in an early email which promises to define it in a later lesson. Do you have a reference to that lesson or perhaps an instant explanation of it? Thanks,

                                                                              Chief Ignoramus

I hope you are not stuck by that, given that the cartesian product does not rely on the understanding of "bijection". 

An instant explanation of bijection is this:

Suppose you have two sets A and B, and you would like to know if they have the same number of elements. For example:

A = {a,b,c,d,e,f,g}

and 

B = {1,  2,  3,  4,  5,  6,  7}

Suppose that you cannot count. You forget the lessons for counting!

But you have ficelles, I mean ropes, cords, or strings.

So you can line up the two sets, and try to attach to each elements of A a piece of rope, and joint them to one element of the set B.

IF you succeed doing that, and respecting the one-one or 1-1link, and getting all the elements of B (the "onto" condition), THEN you have shown the existence of a bijection between the two sets.

Let us see if that work on the example.  The "----------" represent the pieces of rope. OK?

a  ----------  1

b  ----------  2

c  ----------  3

d  ----------  4

e  ----------  5

f  ----------  6

g  ----------  7

So there is a bijection between A and B. 

The bijection *is* that association, as we will defined much later(*).

Other bijections can exist between A and B, like

a  ----------  7

b  ----------  2

c  ----------  3

d  ----------  4

e  ----------  5

f  ----------  6

g  ----------  1

It is enough that one exist, to conclude the sets have the same "number of elements", or same cardinal.

Convince you that if two sets have different number of elements, there is no bijection in between. It has to be 1-1, and "onto" (no missing element).

Exercise (but no hurry): 

Verify if you can see some bijections existing between the powerset of a set with 2 elements, and B_2. The same for the powerset of a set with 3 elements, and B_3.

Bruno

!*) Oh! I can give you the particular mathematical bijection, existing between A and B, given that I have already define the notion of couple.

It is the set of couples representing naturally those rope association:

{(a,1), (b,2) (c,3) (d,4), (e,5), (f,6), (g,7)}.    Take it easy, and be sure you have read what I say about the couples before.

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

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[Bruno Marchal]

Ronald,

On 28 Jul 2009, at 12:51, ronaldheld wrote:

>
> Bruno:
> I meant the mathematical formalism you are teaching us. When we
> eventually get to the UDA steps, I wil be better able to do that
> assessment.
>

OK.
Note that the first 6 steps have already be done recently, with Kim,  
and even before. But there is no problem to come back on this, later.  
The key point there consists in explaining the first person  
indeterminacy, and its invariance for set of transformations (adding  
delays in the computation, going from "real" to "virtual", etc.).
You may prepare yourself by reading the relevant portion of the sane04  
paper.  Eventually the seventh step itself somehow recapitulates the 6  
preceding steps, so it is OK.

Best,

Bruno


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




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[Bruno Marchal]

SOLUTIONS

OK. I give the solution of the exercises of the last session, on the cartesian product of sets.

I recall the definition of the product A X B. 

A X B    =   {(x,y) such that x belongs to A and y belongs to B}

 I gave A = {0, 1}, and B = {a, b}.


In this case, A X B = {(0,a), (0, b), (1, a), (1, b)}

The  cartesian drawing is, for AXB :


a     (0, a)   (1, a)

b     (0, b)  (1, b)

        0          1


Exercise: do the cartesian drawing for BXA.

Solution:

1     (a, 1)   (b, 1)

0     (a, 0)  (b, 0)

        a          b

You see that B X A = {(a,0), (a,1), (b,0), (b, 1)}

You should see that, not only A X B is different from B X A, but AXB and BXA have an empty intersection. They have no elements in common at all. But they do have the same cardinal 2x2 = 4.

1)
Compute
{a, b, c} X {d, e} =

I show you a method (to minimize inattention errors):

I wrote first {(a, _),  (b, _), (c, _), (a, _),  (b, _), (c, _)}  two times because I have seen that {d, e} has two elements.

Then I add the second elements of the couples, which comes from {d, e}:

{(a, d),  (b, d), (c, d), (a, e),  (b, e), (c, e)}

OK?

{d, e} X {a, b, c} = {(d, a), (d, b), (d, c), (e, a), (e, b), (e, c)}

{a, b} X {a, b} = {(a, a), (a, b), (b, a), (b, b)}

{a, b} X { } = { }.

OK?

2)
Convince yourself that the cardinal of AXB is the product of the  
cardinal of A and the cardinal of B.
A and B are finite sets here. Hint: meditate on their cartesian drawing.

Question? This should be obvious. No?

3) Draw a piece of NXN.    (with, as usual, N = {0, 1, 2, 3, ...}):

.        .          .         .         .          .         .           .

.        .          .         .         .          .         .        .

.        .          .         .         .          .         .     .

5    (0,5)  (1,5)  (2,5)  (3,5)  (4,5)  (5,5)  ...

4    (0,4)  (1,4)  (2,4)  (3,4)  (4,4)  (5,4)  ...

3    (0,3)  (1,3)  (2,3)  (3,3)  (4,3)  (5,3)  ...

2    (0,2)  (1,2)  (2,2)  (3,2)  (4,2)  (5,2)  ...

1    (0,1)  (1,1)  (2,1)  (3,1)  (4,1)  (5,1)  ...

0    (0,0)  (1,0)  (2,0)  (3,0)  (4,0)  (5,0)  ...

          0       1         2       3         4       5  ...

OK? 

N is infinite, so N X N is infinite too.

 Look at the diagonal: (0,0) (1,1) (2,2) (3,3) (4,4) (5,5) ...

definition: the diagonal of AXA, a product of a set with itself,  is the set of couples (x,y) with x = y.

All right? No question? Such diagonal will have a quite important role in the sequel.

Next: I will say one or two words on the notion of relation, and then we will define the most important notion ever discovered by the humans: the notion of function. Then, the definition of the exponentiation of sets, A^B, is very simple: it is the set of functions from B to A.

What is important will be to grasp the notion of function. Indeed, we will soon be interested in the notion of computable functions, which are mainly what computers, that is universal machine, compute. But even in physics, the notion of function is present everywhere. That notion capture the notion of dependency between (measurable) quantities. To say that the temperature of a body depends on the pressure on that body, is very well described by saying that the temperature of a body is a function of the pressure.

Most phenomena are described by relation, through equations, and most solution of those equation are functions. Functions are everywhere, somehow.

I have some hesitation, though. Functions can be described as particular case of relations, and relations can be described as special case of functions. This happens many times in math, and can lead to bad pedagogical decisions, so I have to make a few thinking, before leading you to unnecessary complications.

Please ask questions if *any*thing is unclear. I suggest the "beginners" in math take some time to invent exercises, and to solve them. Invent simple little sets, and compute their union, intersection, cartesian product, powerset.

You can compose exercises: for example: compute the cartesian product of the powerset of {0, 1} with the set {a}. It is not particularly funny, but it is like music. If you want to be able to play some music instrument, sometimes you have to "faire ses gammes",we say in french; you know, playing repetitively annoying musical patterns, if only to teach your lips or fingers to do the right movement without thinking. Math needs also such a kind of practice, especially in the beginning.

Of course, as Kim said, passive understanding of music (listening) does not need such exercises. Passive understanding of math needs, alas, many "simple" exercises. Active understanding of math, needs difficult exercises up to open problems, but this is not the goal here.

Bruno

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

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[John Mikes]

Hi, Bruno,

let me skip the technical part and jump on the following text.

F u n c t i o n  as I believe is - for you - the y = f(x) form. For me: the activity - shown when plotting on a coordinate system the f(x) values of the Y-s to the values on the x-axle resulting in a relation (curve). And here is my problem: who does the plotting? (Do not say: YOU are, or Iam, that would add to the function concept the homunculus to make it from a written format into a F U N C T I O N ).

 

John M

 

On Wed, Jul 29, 2009 at 11:59 AM, Bruno Marchal <marchal@...> wrote:

SOLUTIONS

OK. I give the solution of the exercises of the last session, on the cartesian product of sets.

I recall the definition of the product A X B. 

A X B    =   {(x,y) such that x belongs to A and y belongs to B}

 I gave A = {0, 1}, and B = {a, b}.


In this case, A X B = {(0,a), (0, b), (1, a), (1, b)}

The  cartesian drawing is, for AXB :


a     (0, a)   (1, a)

b     (0, b)  (1, b)

        0          1


Exercise: do the cartesian drawing for BXA.

Solution:

1     (a, 1)   (b, 1)

0     (a, 0)  (b, 0)

        a          b

You see that B X A = {(a,0), (a,1), (b,0), (b, 1)}

You should see that, not only A X B is different from B X A, but AXB and BXA have an empty intersection. They have no elements in common at all. But they do have the same cardinal 2x2 = 4.

1)
Compute
{a, b, c} X {d, e} =

I show you a method (to minimize inattention errors):

I wrote first {(a, _),  (b, _), (c, _), (a, _),  (b, _), (c, _)}  two times because I have seen that {d, e} has two elements.

Then I add the second elements of the couples, which comes from {d, e}:

{(a, d),  (b, d), (c, d), (a, e),  (b, e), (c, e)}

OK?

{d, e} X {a, b, c} = {(d, a), (d, b), (d, c), (e, a), (e, b), (e, c)}

{a, b} X {a, b} = {(a, a), (a, b), (b, a), (b, b)}

{a, b} X { } = { }.

OK?

2)
Convince yourself that the cardinal of AXB is the product of the  
cardinal of A and the cardinal of B.
A and B are finite sets here. Hint: meditate on their cartesian drawing.

Question? This should be obvious. No?

3) Draw a piece of NXN.    (with, as usual, N = {0, 1, 2, 3, ...}):

.        .          .         .         .          .         .           .

.        .          .         .         .          .         .        .

.        .          .         .         .          .         .     .

5    (0,5)  (1,5)  (2,5)  (3,5)  (4,5)  (5,5)  ...

4    (0,4)  (1,4)  (2,4)  (3,4)  (4,4)  (5,4)  ...

3    (0,3)  (1,3)  (2,3)  (3,3)  (4,3)  (5,3)  ...

2    (0,2)  (1,2)  (2,2)  (3,2)  (4,2)  (5,2)  ...

1    (0,1)  (1,1)  (2,1)  (3,1)  (4,1)  (5,1)  ...

0    (0,0)  (1,0)  (2,0)  (3,0)  (4,0)  (5,0)  ...

          0       1         2       3         4       5  ...

OK? 

N is infinite, so N X N is infinite too.

 Look at the diagonal: (0,0) (1,1) (2,2) (3,3) (4,4) (5,5) ...

definition: the diagonal of AXA, a product of a set with itself,  is the set of couples (x,y) with x = y.

All right? No question? Such diagonal will have a quite important role in the sequel.

Next: I will say one or two words on the notion of relation, and then we will define the most important notion ever discovered by the humans: the notion of function. Then, the definition of the exponentiation of sets, A^B, is very simple: it is the set of functions from B to A.

What is important will be to grasp the notion of function. Indeed, we will soon be interested in the notion of computable functions, which are mainly what computers, that is universal machine, compute. But even in physics, the notion of function is present everywhere. That notion capture the notion of dependency between (measurable) quantities. To say that the temperature of a body depends on the pressure on that body, is very well described by saying that the temperature of a body is a function of the pressure.

Most phenomena are described by relation, through equations, and most solution of those equation are functions. Functions are everywhere, somehow.

I have some hesitation, though. Functions can be described as particular case of relations, and relations can be described as special case of functions. This happens many times in math, and can lead to bad pedagogical decisions, so I have to make a few thinking, before leading you to unnecessary complications.

Please ask questions if *any*thing is unclear. I suggest the "beginners" in math take some time to invent exercises, and to solve them. Invent simple little sets, and compute their union, intersection, cartesian product, powerset.

You can compose exercises: for example: compute the cartesian product of the powerset of {0, 1} with the set {a}. It is not particularly funny, but it is like music. If you want to be able to play some music instrument, sometimes you have to "faire ses gammes",we say in french; you know, playing repetitively annoying musical patterns, if only to teach your lips or fingers to do the right movement without thinking. Math needs also such a kind of practice, especially in the beginning.

Of course, as Kim said, passive understanding of music (listening) does not need such exercises. Passive understanding of math needs, alas, many "simple" exercises. Active understanding of math, needs difficult exercises up to open problems, but this is not the goal here.

Bruno

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

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[Bruno Marchal]

Hi John, and the other.

John motivates me to explain what is a function, "for a mathematician".

On 30 Jul 2009, at 17:53, John Mikes wrote:

Hi, Bruno,

let me skip the technical part

OK. But I remind you this current thread *is* technical.

and jump on the following text.

F u n c t i o n  as I believe is - for you - the y = f(x) form. For me: the activity - shown when plotting on a coordinate system the f(x) values of the Y-s to the values on the x-axle resulting in a relation (curve). And here is my problem: who does the plotting? (Do not say: YOU are, or Iam, that would add to the function concept the homunculus to make it from a written format into a F U N C T I O N ).

But I have not yet say what is a function. I just mentioned that "they are everywhere" to open the appetite of the audience.

F u n c t i o n  as I believe is - for you - the y = f(x) form.

You take a risk believing things - for me -. 

Actually the "y = f(x)"  form will come later, with the goal of distinguishing clearly the key difference between a function and the many possible forms of a function.

Ah! but you force me to define what is a function right on (for a mathematician of course). Take it easy. You can skip to the sum-up line below.

OK, ready? I mean the others among those who pursue this mathematical shortcut toward the seventh step (the UD step, actually).

We have already seen functions. If you remember the bijection between A = {a,b,c,d,e,f,g} and B = {1,  2,  3,  4,  5,  6,  7}.

a  ----------  7

b  ----------  2

c  ----------  3

d  ----------  4

e  ----------  5

f  ----------  6

g  ----------  1

I said that the following set of couples 

{(a,1), (b,2) (c,3) (d,4), (e,5), (f,6), (g,7)}

was a nice "set theoretical" representation of the bijection, and that the bijection is an example of function. We can give it a name, F, for example.

F = {(a,1), (b,2) (c,3) (d,4), (e,5), (f,6), (g,7)}.

A function is a mathematical object, actually a set, which embodies an association between the elements of two sets. Here the two sets involved are A and B. 

A is said to be the domain of F. B is said to be the range of F. And the function itself, F,  get a nice set theoretical  "form" of a set. The set of all the couples which determine or define the association. Here it is the set {(a,1), (b,2) (c,3) (d,4), (e,5), (f,6), (g,7)}.

Arbitrary set of couples will appear as very good way to describe relation, in general. 

But for function a key condition, the functional condition, has to be applied:

      - If (a, b) belongs to F then if (a, c) belongs to F we have that b = c.   (the functional condition).

This means, that if F is a function from the set A to the set B, you cannot associate to one object of A, many objects of B. 

For example the temperature in a place can be a function of time, because at each moment of time you will not associate two temperatures.

It is the key point for seeing that a function from A to B, describe a very general notion of dependency.

We will be interested in functions from N to N. (With N = {0, 1, 2, ...}. Where examples abound.

Take the function which associates to each natural number its successor.

The function is (or is represented "fully") by the infinite set of couples

{(0, 1), (1,2), (2,3), (3,4), (4,5), (5, 6), (6, 7), ...}

We will be interested in function having two arguments. Those will be the function from NXN to N. Example: take addition. This is a function, because when you add any numbers, 3 and 6, for example, 3+6, you don't expect two results. So the functional condition is respected. OK? So the function addition can be defined or represented by the set

{((0, 0), 0), ((0, 1) 1) ...  ((4, 8) 12) ... }

With the numbers, all the operations are functions. The same with the sets. 

To sum up: a function is a set of couples, most of the time infinite, respecting the functional condition.

A good training consists in searching all functions between little sets:

Exercise: 

1) how many functions and what are they, from the set {0, 1} to himself. What are the functions from {0, 1) to {0, 1}?

Solution:

{(0,0), (1,0)}   the constant function which associates zero to any value of its argument.

{(0,1), (1,1)}   the constant function which associates one to any value of its argument.

{(0,0), (1,1)}  the identity function, which output its argument as value.

{(0,1), (1,0)}, the NOT function, which associate 0 to 1, and 1 to 0. 

There is four functions from {0, 1} to {0, 1}.

2) how many functions, and what are they, from the set cartesian product {0, 1} X {0, 1} to {0, 1}

Among them many are celebrities, you know. The AND, the OR, and many (how many?) others.

For a beginner in math, this is not at all an easy exercise. The real useful exercise is to try to understand the enunciation of the question. We will take the time needed.

3) A bit tricky perhaps: how many functions exist from { } to { } ?

----------------------------------------- SUM UP LINE ----------------------

So functions, once mathematical objects, are just set of couples, verifying a condition. We will be interested in the functions from N to N. Each such function is an infinite set of couples.

Some function have some form, or related expression. Not all though (as we will see), and we will have to study the relation between form and function. Many functions will lack a form, and this will not prevent them to play some role in the life of those who have a form.

John, we will see who plot which functions and why. I promise.  ;)

Bruno

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

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[Miroslav Dobsicek]

>> I am in a good mood and a bit picky :-) Do you know how many entries
>> google gave me upon entering
>> Theaetetical -marchal -bruno
>
>
> Well 144?
>
> Good way to find my papers on that. The pages refer quickly to this  
> list or the FOR list.

I am sorry for the delay, I've just got back from my vacation.

Hmm. The above written search should not return any references to your
papers/letters as the minus sign in front of your name asks for an
exclusion.

Given that it works as supposed google then gives only 1 hit in my
location (Sweden). That hit is a translation of the word "Theaetetical"
into some eastern characters. Thus, I end up with zero meaningful hits
and a feeling that you might be the only one using this word.

That makes me insists a little bit more (in a very polite way) that,
occasionally, your work is
 "difficult to read unless one is willing to undertake long
  discussions, clarifications and position adjustments."

I am writing this in a reference to your complains that sometimes you
have troubles to get enough relevant feedback to your work.

> I let those interested to meditate on two questions (N is {0, 1, 2, 3,  
> 4, ...}):
>
> 1) What is common between the set of all subsets of a set with n  
> elements, and the set of all finite sequences of "0" and "1" of length  
> n.
> 2) What is common between the set of all subsets of N, and the set of  
> all infinite sequences of "0" and "1".
>
> Just some (finite and infinite) bread for surviving the day :)

I am going to catch up with the thread ...

Cheers,
 mirek

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[Bruno Marchal]

On 02 Aug 2009, at 23:20, Mirek Dobsicek wrote:

I am in a good mood and a bit picky :-) Do you know how many entries

google gave me upon entering

Theaetetical -marchal -bruno

Well 144?

Good way to find my papers on that. The pages refer quickly to this  

list or the FOR list.

I am sorry for the delay, I've just got back from my vacation.

Hmm. The above written search should not return any references to your
papers/letters as the minus sign in front of your name asks for an
exclusion.

Given that it works as supposed google then gives only 1 hit in my
location (Sweden). That hit is a translation of the word "Theaetetical"
into some eastern characters. Thus, I end up with zero meaningful hits
and a feeling that you might be the only one using this word.

That makes me insists a little bit more (in a very polite way) that,
occasionally, your work is
"difficult to read unless one is willing to undertake long
 discussions, clarifications and position adjustments."

I am writing this in a reference to your complains that sometimes you
have troubles to get enough relevant feedback to your work.

Come on Mirek: "Theaetetical" is an adjective I have forged from "Theatetus".

"Theatetus" gives 195.000 results on Google.

"Theatetus" wiki 4310.

By "theatetical notion of knowledge", I mean the "well known" attempts to define "knowledge" by Theaetetus in Plato's Theaetetus. The most known definition is "truye justified belief", that Bill taylor just mentionned on the FOR list recently as:

"This old crock should have been given a decent burial long ago."

I guess I will have to make a comment ...

My work is, without doubt, very difficult to read because it crosses three or four fields: "mathematical logic", "philosophy of mind" and "computer science";  + quantum mechanics to evaluate the plausibility of the derived computationalist physics. This does not help in an epoch of hyper-specialization.

I am also using a deductive approach in the philosophy of mind. I am apparently the first to *postulate* "mechanism".  Most philosophers of mind accept mechanism as the only rational theory, or reject it with some passion. Few, if any, use it as an hypothesis, in a deductive strategy. Then mathematical logic is virtually unknown, except by mathematical logicians, who, for historical reasons, do not want to come back to the earlier philosophical motivations: they want to be accepted as pure mathematicians. Except the philosophical logicians, who in majority criticized classical logic, and see philosphy as a mean to criticize classical philosophy. Mathematicians are so used to classical philosophy, that they consider it as science, and hate to be remind that this is still a philosophical. 

I have no feedback for purely contingent reason related to facts which have nothing to do with the startling feature of the conclusion of the reasoning. Up to now, I heard continuously about critics on an imaginary work I have never done. The price of the best PhD thesis that I got in France has eventually only spread those rumor from Brussels to elsewhere.

All real scientist who have studied my work and have accepted to meet me, or to write a real report on it, have understood it. True, some took a rather long time to understand, but that is normal: the subject matter is very complex, and still taboo, especially for the atheists, and other religious-based thinkers. But when they study it, they quickly discover that I use the scientific method, that is I am just asking a question, what is wrong with the following reasoning? ... The reasoning is decomposed in "easy" steps, so people accepting (for personal belief or for the sake of the argument) the hypotheses and wanting to reject the conclusion have a way to put their fingers on some problems.

UDA has been judged to obvious and simple in Brussels, and that is why I have augmented the thesis with the AUDA, which unfortunately is considered as ... too much simple for logicians, and too much difficult for non logicians. But AUDA is not needed at all to understand the simple and clear result: if we are digitalisable machine, the laws of physics emerge from a statistics on computations, in a verifiable way (quantitatively and qualitatively). The result is very simple and clear: the reasoning which leads to that result is much more subtle and difficult.

I am not at all pretending that reasoning is correct. Science progress when people do errors, but we have to find them, and sometimes, if we don't find them, we have to accept momentarily the conclusion, perhaps with the hope an error will be find later. But the attitude of a (tiny but influencing) part of the community consists in hiding the reasoning, or deforming it completely. This can't help. 

Some people, even here recently (see 1Z's post) and recently on the FOR list, attributes me a curious theory, where they confuse the conclusion with the postulate (which deprives the work of *any* meaning). But the theory I am studying is the old "mechanist theory", in its modern digital version, and nothing else. So, if they have a genuine interest in the subject, we would begin to learn something if they can criticize some point in the reasoning, instead of ignoring it, or attributing it statements without ever referring to a relevant piece of text. Of course they can't point on such text, given that such information exists only in their mind. They repeat rumors, and have clearly not take time to read the papers.

The fact that the result would contradict the current paradigm does not help, of course, but is not, yet, the source of the problem.

I let those interested to meditate on two questions (N is {0, 1, 2, 3,  

4, ...}):

1) What is common between the set of all subsets of a set with n  

elements, and the set of all finite sequences of "0" and "1" of length  

n.

2) What is common between the set of all subsets of N, and the set of  

all infinite sequences of "0" and "1".

Just some (finite and infinite) bread for surviving the day :)

I am going to catch up with the thread ...

Welcome back Mirek. Feel free to ask for any clarification, position adjustments, question, at any level ...Do you understand what is the comp hypothesis? 

Bruno

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

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[John Mikes]

Bruno and Mirek,

 concerning Theateticus vs. Theaeteticus:

 in my strange linguistic background I make a difference betwee ai and ae - the spelling in Greek and Latin of the name. As far as I know, nobody knows for sure how did the 'ancient' Greeks pronounce their ai - maybe as the flat 'e' like in German "lehr" while the 'e' pronounciation might have been clsoer to (between) 'make' and 'peck' - the reason why the Romans transcribed it by their ONE letter "ae", (lehr) and not as English would read: 'a'+'ee'. The spelling you gave points to this latter. The Latin 'ae' is not TWO separate letters (a+e), it is a twin, as marked in the Wiki article

..."Theætetus"... and not Theaetetus

which looked strange to me from the beginning  .

(I wonder if the e-mail reproduces the (ae) one sign? look up in Wiki's Theaetetus Dialogue (in the title with the wrong spelling) the 1st line brings the merged-together double 'æ'.)

*

English spelling always does a job on classical words, the Greek 'oi' has been transcribed into Latin sometimes as 'oe' and pronounced as in "girl" (oeuvre) while many think it was a sound like what the pigs say: as "oy". then comes America, with it's Phoenix (pron: feenix)....

I don't think the Romans were much better off, centuries after and a world apart from the ancient (classical for them) Greeks.

And who knows today if the great orator was Tzitzero or Kikero to turn later into Tchitchero?

*

"The Old Man" did quite a job on us at the tower of Babel.

*

[[ - I am enjoying your 'other' post where you spelled out my own vocabulary as indeed thinking functions as relations, lately not as a static description, but also the interchanging factor - ]]

 

John

 

 

On Mon, Aug 3, 2009 at 4:55 AM, Bruno Marchal <marchal@...> wrote:

On 02 Aug 2009, at 23:20, Mirek Dobsicek wrote:

I am in a good mood and a bit picky :-) Do you know how many entries

google gave me upon entering

Theaetetical -marchal -bruno

Well 144?

Good way to find my papers on that. The pages refer quickly to this  

list or the FOR list.

I am sorry for the delay, I've just got back from my vacation.

Hmm. The above written search should not return any references to your
papers/letters as the minus sign in front of your name asks for an
exclusion.

Given that it works as supposed google then gives only 1 hit in my
location (Sweden). That hit is a translation of the word "Theaetetical"
into some eastern characters. Thus, I end up with zero meaningful hits
and a feeling that you might be the only one using this word.

That makes me insists a little bit more (in a very polite way) that,
occasionally, your work is
"difficult to read unless one is willing to undertake long
 discussions, clarifications and position adjustments."

I am writing this in a reference to your complains that sometimes you
have troubles to get enough relevant feedback to your work.

Come on Mirek: "Theaetetical" is an adjective I have forged from "Theatetus".

"Theatetus" gives 195.000 results on Google.

"Theatetus" wiki 4310.

By "theatetical notion of knowledge", I mean the "well known" attempts to define "knowledge" by Theaetetus in Plato's Theaetetus. The most known definition is "truye justified belief", that Bill taylor just mentionned on the FOR list recently as:

"This old crock should have been given a decent burial long ago."

I guess I will have to make a comment ...

My work is, without doubt, very difficult to read because it crosses three or four fields: "mathematical logic", "philosophy of mind" and "computer science";  + quantum mechanics to evaluate the plausibility of the derived computationalist physics. This does not help in an epoch of hyper-specialization.

I am also using a deductive approach in the philosophy of mind. I am apparently the first to *postulate* "mechanism".  Most philosophers of mind accept mechanism as the only rational theory, or reject it with some passion. Few, if any, use it as an hypothesis, in a deductive strategy. Then mathematical logic is virtually unknown, except by mathematical logicians, who, for historical reasons, do not want to come back to the earlier philosophical motivations: they want to be accepted as pure mathematicians. Except the philosophical logicians, who in majority criticized classical logic, and see philosphy as a mean to criticize classical philosophy. Mathematicians are so used to classical philosophy, that they consider it as science, and hate to be remind that this is still a philosophical. 

I have no feedback for purely contingent reason related to facts which have nothing to do with the startling feature of the conclusion of the reasoning. Up to now, I heard continuously about critics on an imaginary work I have never done. The price of the best PhD thesis that I got in France has eventually only spread those rumor from Brussels to elsewhere.

All real scientist who have studied my work and have accepted to meet me, or to write a real report on it, have understood it. True, some took a rather long time to understand, but that is normal: the subject matter is very complex, and still taboo, especially for the atheists, and other religious-based thinkers. But when they study it, they quickly discover that I use the scientific method, that is I am just asking a question, what is wrong with the following reasoning? ... The reasoning is decomposed in "easy" steps, so people accepting (for personal belief or for the sake of the argument) the hypotheses and wanting to reject the conclusion have a way to put their fingers on some problems.

UDA has been judged to obvious and simple in Brussels, and that is why I have augmented the thesis with the AUDA, which unfortunately is considered as ... too much simple for logicians, and too much difficult for non logicians. But AUDA is not needed at all to understand the simple and clear result: if we are digitalisable machine, the laws of physics emerge from a statistics on computations, in a verifiable way (quantitatively and qualitatively). The result is very simple and clear: the reasoning which leads to that result is much more subtle and difficult.

I am not at all pretending that reasoning is correct. Science progress when people do errors, but we have to find them, and sometimes, if we don't find them, we have to accept momentarily the conclusion, perhaps with the hope an error will be find later. But the attitude of a (tiny but influencing) part of the community consists in hiding the reasoning, or deforming it completely. This can't help. 

Some people, even here recently (see 1Z's post) and recently on the FOR list, attributes me a curious theory, where they confuse the conclusion with the postulate (which deprives the work of *any* meaning). But the theory I am studying is the old "mechanist theory", in its modern digital version, and nothing else. So, if they have a genuine interest in the subject, we would begin to learn something if they can criticize some point in the reasoning, instead of ignoring it, or attributing it statements without ever referring to a relevant piece of text. Of course they can't point on such text, given that such information exists only in their mind. They repeat rumors, and have clearly not take time to read the papers.

The fact that the result would contradict the current paradigm does not help, of course, but is not, yet, the source of the problem.

I let those interested to meditate on two questions (N is {0, 1, 2, 3,  

4, ...}):

1) What is common between the set of all subsets of a set with n  

elements, and the set of all finite sequences of "0" and "1" of length  

n.

2) What is common between the set of all subsets of N, and the set of  

all infinite sequences of "0" and "1".

Just some (finite and infinite) bread for surviving the day :)

I am going to catch up with the thread ...

Welcome back Mirek. Feel free to ask for any clarification, position adjustments, question, at any level ...Do you understand what is the comp hypothesis? 

Bruno

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

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[Miroslav Dobsicek]

> Come on Mirek: "Theaetetical" is an adjective I have forged from
> "Theatetus".
> "Theatetus" gives 195.000 results on Google.
> "Theatetus" wiki 4310.

Of course, after all you reference the dialogue Theaetetus in your
papers thus one can easily match the word Theaetetical agains it.
Let me quickly summarize the experience I had with "theatetical notion
of knowledge" while reading one of your papers for the first time.

Maybe I am an ignorant, then shame on me, but I have not read the
Theaetetus. So I took a look at the Wikipedia and read

 "In this dialogue, Socrates and Theaetetus discuss three definitions of
knowledge: knowledge as nothing but perception, knowledge as true
judgment, and, finally, knowledge as a true judgment with an account.
Each of these definitions are shown to be unsatisfactory."

Hmm that really helps .., I told to myself and continued with reading.
With an uneasy feeling of stepping into the water I eventually settled
down to conclusion that you likely mean something as "true justified
belief".
I really wished you wrote it more straightforwardly without turning your
readers quite unnecessarily down to the Theaetetus and inventing new
words such as "Theaetetical".

Anyway, I'd like to stop discussing this issue :-) since my only point
was to give you a hint why I said that it is not easy to read your
papers/letters.

> Feel free to ask for any clarification, position
> adjustments, question, at any level ...Do you understand what is the
> comp hypothesis?

Let us see if I get it right. Your comp hypothesis is
1) I'm a machine,
2) Each possible computation is Turing-computable,
3) Natural numbers and their relations do exist.

This should not be confused with other quite common comp hypothesis that
the universe is a big computer. This hypothesis entails the existence of
a physical computer.


Ad 1) I take the position that "I" is only a convenient temporary
pointer to a part of universe. The pointer "Socrates' thoughts" is of
the same quality.

Ad 2) Breath taking. While 1) and 3) are assumptions of the kind "OK,
let's think for a while that ...", 2) has the status of a thesis. I
don't have any firm position on what could an objective reality be (and
without a justification I tend to think it is inaccessible to us), but
if there is any objective reality, 2) could be a statement about it.

Ad 3) If natural numbers and their relations are the only entities which
do exist then me, you, everything is a recipe of a Turing-computable number.

OK, that is it. This is how I understand to your starting assumptions.

Mirek





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[Bruno Marchal]

John,

Thanks for those informations. I thought that the "æ" was just a french, if not an old french, usage.

Note that when I wrote "Theatetus", it is just a mispelling. I tend to forget that second "e", but your remark will help me to remind it. Note that Miles Burnyeat, in his book " The Theaetetus of Plato, and Levett in his traduction wrote simply "Theaetetus". But in french too, more and more people forget to attach the "o" and "e" in words like oeuvre, or soeur (sister).

Bruno

On 04 Aug 2009, at 15:05, John Mikes wrote:

Bruno and Mirek,

 concerning Theateticus vs. Theaeteticus:

 in my strange linguistic background I make a difference betwee ai and ae - the spelling in Greek and Latin of the name. As far as I know, nobody knows for sure how did the 'ancient' Greeks pronounce their ai - maybe as the flat 'e' like in German "lehr" while the 'e' pronounciation might have been clsoer to (between) 'make' and 'peck' - the reason why the Romans transcribed it by their ONE letter "ae", (lehr) and not as English would read: 'a'+'ee'. The spelling you gave points to this latter. The Latin 'ae' is not TWO separate letters (a+e), it is a twin, as marked in the Wiki article

..."Theætetus"... and not Theaetetus

which looked strange to me from the beginning  .

(I wonder if the e-mail reproduces the (ae) one sign? look up in Wiki's Theaetetus Dialogue (in the title with the wrong spelling) the 1st line brings the merged-together double 'æ'.)

*

English spelling always does a job on classical words, the Greek 'oi' has been transcribed into Latin sometimes as 'oe' and pronounced as in "girl" (oeuvre) while many think it was a sound like what the pigs say: as "oy". then comes America, with it's Phoenix (pron: feenix)....

I don't think the Romans were much better off, centuries after and a world apart from the ancient (classical for them) Greeks.

And who knows today if the great orator was Tzitzero or Kikero to turn later into Tchitchero?

*

"The Old Man" did quite a job on us at the tower of Babel.

*

[[ - I am enjoying your 'other' post where you spelled out my own vocabulary as indeed thinking functions as relations, lately not as a static description, but also the interchanging factor - ]]

 

John

 

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

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[Bruno Marchal]

Hi Mirek,

Long and perhaps key post.

On 04 Aug 2009, at 15:32, Mirek Dobsicek wrote:

Come on Mirek: "Theaetetical" is an adjective I have forged from

"Theatetus".

"Theatetus" gives 195.000 results on Google.

"Theatetus" wiki 4310.

Of course, after all you reference the dialogue Theaetetus in your
papers thus one can easily match the word Theaetetical agains it.
Let me quickly summarize the experience I had with "theatetical notion
of knowledge" while reading one of your papers for the first time.

Maybe I am an ignorant, then shame on me, but I have not read the
Theaetetus.

There is no shame in being ignorant. Only in staying ignorant :)

I feel a bit sorry with my last post. I hate to look like patronizing, but it is a professional deformation. Apology.

Note that all the Theaetetus' stuff is really needed just to motivate the "arithmetical" definition of the knower, alias the first person, alias the "universal soul", and this concerns AUDA (the "arithmetical UDA), which should be done normally after getting straight the UDA's point, ... unless you are mathematical logician, who are the only one who can find AUDA more "easy" than UDA.

So I took a look at the Wikipedia and read

"In this dialogue, Socrates and Theaetetus discuss three definitions of
knowledge: knowledge as nothing but perception, knowledge as true
judgment, and, finally, knowledge as a true judgment with an account.
Each of these definitions are shown to be unsatisfactory."

Socrates asks Theaetetus to define "knowledge". That is a very difficult question.

Then Socrates shows that all attempts made by Theaetetus lead to difficulties. He literally concludes that the problem is open, and this is debated in the philosophical literature since then.

The remarkable thing is that if you accept to modelize "account" by "sound machine provability", which can be done for the not too complex machine, like Peano Arithmetic provers, or Zermelo Fraenkel Set Theory prover, the definitions of Theaetetus make sense, and can be use to show, at least, that many philosopher are deductively invalid in  their critics. Actually, even the critics by Socrates have to be weakened. 

All the arithmetical hypostases (in the Plotinus paper) are variant of Theaetetus definition.

The main one (corresponding to Plotinus "primary hypostases) are the following one:

 

p (the truth of p)

Bp (the provability of p, the account of p)

Bp & p (the provability of p when p is true)

The amazing thing is that the incompleteness theorem can be used to show that, about sound machine, we have

Bp <-> Bp & p.

But this equivalence is true but not provable by the machine, making the ideal knower already obeying a different logic than the ideal prover. This introduces a non trivial notion of first person for the machines.

If you remember G and G*, the equivalence between proof and knowledge belongs to G* minus G. The corona of the true but unprovable (by the machine) statements. Yet they prove (know) the same arithmetical "p", yet, from those points of view, it appears very different.

Hmm that really helps .., I told to myself and continued with reading.
With an uneasy feeling of stepping into the water I eventually settled
down to conclusion that you likely mean something as "true justified
belief".

I have found dozen of different translations, in french and in english, of the greek expressions. 

I really wished you wrote it more straightforwardly without turning your
readers quite unnecessarily down to the Theaetetus and inventing new
words such as "Theaetetical".

In french students are burned alive if they dare to create new adjective, and I thought that in English we have more freedom, but I may be wrong. Sorry.

Anyway, I'd like to stop discussing this issue :-) since my only point
was to give you a hint why I said that it is not easy to read your
papers/letters.

There are other reasons, if only the difficulty of the subject. 

Feel free to ask for any clarification, position

adjustments, question, at any level ...Do you understand what is the

comp hypothesis?

Let us see if I get it right. Your comp hypothesis is
1) I'm a machine,

OK. This of course could be interpreted in many ways, and that is why I have introduce the quasi-operational "yes doctor". It makes clear hat the "I" is the conscious first person I, not the third person body.

2) Each possible computation is Turing-computable,

OK. That is Church thesis. Very few people doubt it, but it is a refutable statement. If a human find a well defined function with an account of how human can compute it, but no machine can, then CT will be refuted. Only Kalmar did pretend to have such a function, but eventually his "function" was not well defined.

3) Natural numbers and their relations do exist.

This is arithmetical realism. Just a way to prevent infinite discussion about intutionism and ultrafinitism.  It is no more than the belief of all mathematician that the excuded middle principle is freely used on arithmetical proposition. Except for some philsopher, or mathematicians, but only at the pause cafe, or during the week-end, this is a widespread belief. But given that the result could be considered as a bit weird, I usually prefer to make it explicit. It can be retrieved from the (classical) Church thesis. Church thesis does not make sense without arithmetical realism. This is something I intend to show precisely in the UDA-7 thread.

This should not be confused with other quite common comp hypothesis that
the universe is a big computer.

Yes. It is different in principle. It is different unless I am the physical universe, which I doubt. Comp is really indexical mechanism: "I am a machine", or "I see no change after the functional substitution at the right level", or "I survive classical teleportation", etc.

This hypothesis entails the existence of
a physical computer.

Not necessarily. It could depend by what you mean by "physical".

Ad 1) I take the position that "I" is only a convenient temporary
pointer to a part of universe. The pointer "Socrates' thoughts" is of
the same quality.

This I do not understand. "I", used by Mirek, is a person, subject of consciousness. I think you confuse that first person "I", and its (current) body.

At that stage it is better to be reasonably agnostic about a universe. "I" is the person who survives teleportation, with a new body. With mechanism, "I" can be said the owner of the body.

Ad 2) Breath taking. While 1) and 3) are assumptions of the kind "OK,
let's think for a while that ...",

Hmm... Most scientist believes implicitly or explicitly in "1)" and "3)". Only Penrose happens to be explicitly arguing that "1)" is false, but note that he defends "3)" eloquently. John Searles, a philosopher, pretends that he believes that "1)" is false, in case of digital machine, but then he reasons in most place like if it was implicitly believing it, or being very fuzzy about what is non mechanical in biology. Only "fairy tales" kind of religious person have (fairy tales) theory of the soul, in which case they would say "no to the doctor" (even in principle: I add this because I would, in most cases,  say "no" to the doctor for technical reasons probably).

2) has the status of a thesis. I
don't have any firm position on what could an objective reality be (and
without a justification I tend to think it is inaccessible to us), but
if there is any objective reality, 2) could be a statement about it.

You don't need to have a firm position on what could an objective reality be.

You need only to believe that you can be objective on a part of that reality. In particular you need to be objective with sentences like:

x divides y if and only if it exists a number z such that y = x*z.

if 4 divides 8 then 2 divides 8

For any natural number x, if 4 divides x then 2 divides x.

Goldbach conjecture is false or true.

Etc.

I think it is reasonable to bet on the objectivity of those statements. And besides, they are used in the practice of all other exact sciences all the times. You don't have to believe more than that for being an arithmetical realist.

Ad 3) If natural numbers and their relations are the only entities which
do exist then me, you, everything is a recipe of a Turing-computable number.

No. Not at all. Sorry. Gosh, you will be very surprised if you follow the UDA-7. On the contrary. Arithmetical truth VASTLY extends the computable domain. Most relations between numbers are not Turing emulable.

And then, why do you introduce suddenly the idea that numbers and their relations would be the only entities existing for me?

This is not part of the assumption. The assumption are not more than;

A - There is a level of substitution such that I can survive a digital functional substitution made at that level.

B - Church thesis  (classical Church thesis, not the intuitionist one, thus I use classical realism, classical logic, mathematical logic)

What you say is in the conclusion of the UDA reasoning (although in a rough and simplified form(*)).

If only number exists, then it would be like I am proposing a new theory. I could have done that, but this is not what I have done. What I give is a constructive proof that if I am machine, and CT is correct, then the laws of physics have to be reduced and derived from the laws of number (or any recursively isomorphic structure). The movie graph argument is what makes this obligatory. The notion of fundamental matter loose its meaning. Matter becomes a Moiré effect lived by the number when they infer relations from their point of views. You can still believe in matter if you want too, but you cannot use it to explain even the physical observation, which have to emerge from special number's points of view.

That is what UDA shows.

Now AUDA is only used to illustrate that incompleteness in computer science and logic provide consistency to such a view. But as I said, Theaetetus all what the multiple explanations of what could be knowledge are directly translatable in arithmetic thanks again to incompleteness. For this you need Solovay theorem, and you need to read Boolos or Smorynski, or Smullyan's book in logic, or study sane04. I am surfing on the shoulds of giants here: like Post, Gödel, Kleene, Löb, and Solovay.

OK, that is it. This is how I understand to your starting assumptions.

You may have to revised or I may have ben unclear, or the result is too much counter-intuitive, and things take time. People are not accustomed to see a proof in philosophy or theology, but I show, or try to show, that the very common comp belief is much strangest, and far less reductionist than what most materialists believe.

The intended result, in a nutshell, is the incompatibility of very weak forms of materialism with digital mechanism.

By a weak form of materialism I mean any doctrine which posits a primitive or substantial matter and reduces the mind to it (computationally or not computationally). Materialism and mechanism, which are always thought as ally are really just epistemologically incompatible.

OK? I mean do you see that such a result is far from trivial? UDA1-7 conveys already the basic understanding of what happens. The Movie graph captures the mind-body problem in the comp frame, and is responsible for the inadequacy of not just physics, but of any particular universal machine.  What you see is not the product of a universal machine (be it physical or mathematical), but of an infinity of them. At first sight the mystery is now: why does the physical world looks so computable. That is the unavoidable white rabbit problem.  Who knows, it could one day lead to the refutation of comp. But AUDA, computer science and self-reference logics suggest that comp could provide a coherent picture after all, capable of explaining where the laws of physics come from, without eliminating the person.  AUDA lead to identify physics with one of the Theaetetus-like definition of a form of knowledge, and this is enough to extract a logic of what is observable, and this makes comp testable. I give a tool capable of measuring the degree of computability of nature. But alas, it does not really distinguish comp and many weakening of comp. Most of the "gods" still pay taxes ("gods" are defined by NON Turing emulable self-referential entities ...). Up to now the comp logic is not contradicted by quantum logic.

I have to go. This is an important post. Other people, like Jones recently, get me wrong on the starting assumption. It is really just "I am a machine" made precise. Then computer science makes it possible to reason and prove things with that assumption. I have no original theory. I propose just a proof, a reasoning or an argument.

Bruno

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

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[Miroslav Dobsicek]

Hi Bruno,

Bruno Marchal wrote:
> Hi Mirek,
>
> Long and perhaps key post.

Thank you a lot for a prompt and long reply. I am digesting it :-)

Just some quick comments.

> There is no shame in being ignorant. Only in staying ignorant :)

I've ordered the dialogue from a second-hand book shop :-) The Stanford
encyclopedia says
 "Arguably, it is his (Plato) greatest work on anything."
So I'll give it a try :-)


>> judgment, and, finally, knowledge as a true judgment with an account.

> The remarkable thing is that if you accept to modelize "account" by
> "sound machine provability",

This is probably the key problem for me. I know next to nothing about
provability, the logic of provability, PA/ZF provers.

I know that quite often you reference Boolos 1993 - The Logic of
Provability. I took a look at it at Google Books preview but ... there
is something missing in my education. From the beginning I am puzzled
with "Why?, what?". What a headache :-)

> In french students are burned alive if they dare to create new
> adjective, and I thought that in English we have more freedom, but I may
> be wrong. Sorry.

I'd grant this freedom to rational native speakers only :-)


> x divides y if and only if it exists a number z such that y = x*z.

I don't dare to correct your english but "there is/exists a number ..."
is what I would write.

>> Ad 3) If natural numbers and their relations are the only entities which
>> do exist then me, you, everything is a recipe of a Turing-computable
>> number.
>
> No. Not at all. Sorry. Gosh, you will be very surprised if you follow
> the UDA-7. On the contrary. Arithmetical truth VASTLY extends the
> computable domain. Most relations between numbers are not Turing emulable.

Aha! Then I really have a wrong mental picture of your work. I
understood to arithmetical realism along the lines of this quotation
from the Stanford article on realism:

"According to a platonist about arithmetic, the truth of the sentence '7
is prime' entails the existence of an abstract object, the number 7.
This object is abstract because it has no spatial or temporal location,
and is causally inert. A platonic realist about arithmetic will say that
the number 7 exists and instantiates the property of being prime
independently of anyone's beliefs, linguistic practices, conceptual
schemes, and so on."

So I thought that you essentially take
 a) Numbers and their properties and relations exists.
 b) Now, since you don't assume existence of anything else => your body,
your bike and coffee must emerge as patterns in the world of numbers.
 c) Taking the Church-Turing thesis, these patterns are Turing-computable.
 d) Definitely, the vast majority of all patterns is not Turing-computable.

This is how I have thought about your working framework. Notice, that I
don't talk about what you try to show, argue for, want to end up with etc.

Cheers,
 Mirek

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[Bruno Marchal]

Hi Mirek,


On 05 Aug 2009, at 00:52, Mirek Dobsicek wrote:

> I've ordered the dialogue from a second-hand book shop :-) The  
> Stanford
> encyclopedia says
> "Arguably, it is his (Plato) greatest work on anything."
> So I'll give it a try :-)


I love that book, and it is also my favorite piece of Plato.
To be sure, I don't think it is needed to understand neither UDA nor  
AUDA, but it can help.


> This is probably the key problem for me. I know next to nothing about
> provability, the logic of provability, PA/ZF provers.
>
> I know that quite often you reference Boolos 1993 - The Logic of
> Provability. I took a look at it at Google Books preview but ... there
> is something missing in my education. From the beginning I am puzzled
> with "Why?, what?". What a headache :-)

You miss an introductory course on mathematical logic.
Have you herad about Gödel's incompletness theorem. Boolos book  
explains the sequel.

I thought that, after Hofstadter best selling book on Gödel's theorem  
(Gödel, Escher, Bach), it would be possible to talk on mathematical  
logic to the layman, like we can talk on physics to the layman. But I  
was wrong. Gödel's theorem is not yet part of the common knowledge,  
and when it is used by non mathematician, in general it is abused.



>> x divides y if and only if it exists a number z such that y = x*z.
>
> I don't dare to correct your english but "there is/exists a  
> number ..."
> is what I would write.

Thanks.

>
>
>>> Ad 3) If natural numbers and their relations are the only entities  
>>> which
>>> do exist then me, you, everything is a recipe of a Turing-computable
>>> number.
>>
>> No. Not at all. Sorry. Gosh, you will be very surprised if you follow
>> the UDA-7. On the contrary. Arithmetical truth VASTLY extends the
>> computable domain. Most relations between numbers are not Turing  
>> emulable.
>
> Aha! Then I really have a wrong mental picture of your work. I
> understood to arithmetical realism along the lines of this quotation
> from the Stanford article on realism:
>
> "According to a platonist about arithmetic, the truth of the  
> sentence '7
> is prime' entails the existence of an abstract object, the number 7.
> This object is abstract because it has no spatial or temporal  
> location,
> and is causally inert. A platonic realist about arithmetic will say  
> that
> the number 7 exists and instantiates the property of being prime
> independently of anyone's beliefs, linguistic practices, conceptual
> schemes, and so on."

That is quite correct. All mathematicians are realist about  
arithmetic, and most are realist about sets. But set realism is a much  
more stronger belief than arithmetical realism.
Comp necessitates arithmetical realism if only to be able to state  
Church thesis. Theoretical computer scientist are realist, because  
they belief that all machine either stop or not stop.


>
>
> So I thought that you essentially take
> a) Numbers and their properties and relations exists.

Yes, but some people put to much sense in "exists". It is the  
mathematical usual sense, like when you derive "there exists a prime  
number" from the statement "17 is a prime number". No need to invoke  
Plato Heaven, in the assumption.



>
> b) Now, since you don't assume existence of anything else => your  
> body,
> your bike and coffee must emerge as patterns in the world of numbers.

I am agnostic. I assume neither that something else exists nor that it  
does not exist, and then I prove from the assumption that we are  
turing emulable, that physics is no more the fundamental science. I  
prove that if we are machine then matter has to be an emerging  
epistemological concept, and physics is a branch of machine biology/
psychology/theology, or mathematical computer science.
"b)" is obviously non valid. The fact that bike an coffee must emerge  
from numbers is really the conclusion of the whole UD reasoning. It is  
not because I don't assume them, it is because their independent  
existence is shown contradictory.
I show that mechanism makes physicalism epistemologically  
inconsistent. Even if matter really exists, it cannot be used to  
justify our belief in matter. A slight application of Occam razor  
eliminates matter, at that stage.

>
> c) Taking the Church-Turing thesis, these patterns are Turing-
> computable.

Not at all. The world of number is provably not Turing-computable.  
Only a very tiny part of the world of number is computable. There is a  
whole branch of mathematical logic devoted to the study of the degree  
of non computability of the relations existing among the numbers.
Church thesis asserts only that the *computable* patterns are Turing  
computable. It is just the assertion that Turing computability can be  
used to define computability.

>
> d) Definitely, the vast majority of all patterns is not Turing-
> computable.

I don't understand.


>
>
> This is how I have thought about your working framework. Notice,  
> that I
> don't talk about what you try to show, argue for, want to end up  
> with etc.

My framework, comp, is just the hypothesis that I can survive with an  
artificial digital brain (even material, if you want). That's all.

The negation of comp is "my soul/person/consciousness" is not Turing  
emulable. Or I say "no" to all doctors. Or "I don't survive classical  
teleportation done at any level".

I use the term "computationalism"  in its standard usual traditional  
sense. If you assume explicitly that computationalism needs the brain  
to be a material object, then the UDA can be seen as a reductio ad  
absurdo.
The conclusion of UDA is that Materialism is incompatible with  
Computationalism. It is not obvious, but given that the numerous  
attempts by materialist to solve the mind body problem have failed, it  
is not so astonishing that a solution of the mind body problem needs  
some "scientific revolution". The "revolution" is the reversal between  
physics and the "theology of numbers" (the study of what numbers can  
believe in, can know, can bet on, etc.).

Bruno


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




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[John Mikes]

Bruno, just to take off some mal-deserved feathers:

I think Theaetetus has two different 'e' sounds one after the other (anybody can pronounce him better?) and in Hungarian we have them (' e ' like in 'have' and e' like in 'take') with a 3rd variation where the accent is not applied: a closed and an open ' e ' sound (instrumental in dialects). So I have no problem to pronounce the discussing gentleman as The'-etetus. Maybe he called himself (?) Te-aythetos? Ask Plato you are close to him.

 

(And I always proudly thought that Hungarian - vs. English - has a simple vowel-code in an unchanging uniform pronunciation...).

German proverb: "Fremdworter sind glucksache" (= foreign words are a matter of luck). A friend added: you can NEVER know what they mean.

John

On Tue, Aug 4, 2009 at 11:06 AM, Bruno Marchal <marchal@...> wrote:

John,

Thanks for those informations. I thought that the "æ" was just a french, if not an old french, usage.

Note that when I wrote "Theatetus", it is just a mispelling. I tend to forget that second "e", but your remark will help me to remind it. Note that Miles Burnyeat, in his book " The Theaetetus of Plato, and Levett in his traduction wrote simply "Theaetetus". But in french too, more and more people forget to attach the "o" and "e" in words like oeuvre, or soeur (sister).

Bruno

On 04 Aug 2009, at 15:05, John Mikes wrote:

Bruno and Mirek,

 concerning Theateticus vs. Theaeteticus:

 in my strange linguistic background I make a difference betwee ai and ae - the spelling in Greek and Latin of the name. As far as I know, nobody knows for sure how did the 'ancient' Greeks pronounce their ai - maybe as the flat 'e' like in German "lehr" while the 'e' pronounciation might have been clsoer to (between) 'make' and 'peck' - the reason why the Romans transcribed it by their ONE letter "ae", (lehr) and not as English would read: 'a'+'ee'. The spelling you gave points to this latter. The Latin 'ae' is not TWO separate letters (a+e), it is a twin, as marked in the Wiki article

..."Theætetus"... and not Theaetetus

which looked strange to me from the beginning  .

(I wonder if the e-mail reproduces the (ae) one sign? look up in Wiki's Theaetetus Dialogue (in the title with the wrong spelling) the 1st line brings the merged-together double 'æ'.)

*

English spelling always does a job on classical words, the Greek 'oi' has been transcribed into Latin sometimes as 'oe' and pronounced as in "girl" (oeuvre) while many think it was a sound like what the pigs say: as "oy". then comes America, with it's Phoenix (pron: feenix)....

I don't think the Romans were much better off, centuries after and a world apart from the ancient (classical for them) Greeks.

And who knows today if the great orator was Tzitzero or Kikero to turn later into Tchitchero?

*

"The Old Man" did quite a job on us at the tower of Babel.

*

[[ - I am enjoying your 'other' post where you spelled out my own vocabulary as indeed thinking functions as relations, lately not as a static description, but also the interchanging factor - ]]

 

John

 

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

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[Bruno Marchal]

Well, given that nobody dare to ask question, I will play the role of the idiot myself.

On 30 Jul 2009, at 21:22, Bruno Marchal wrote:

Exercise: 

1) how many functions and what are they, from the set {0, 1} to himself. What are the functions from {0, 1) to {0, 1}?

Solution:

{(0,0), (1,0)}   the constant function which associates zero to any value of its argument.

{(0,1), (1,1)}   the constant function which associates one to any value of its argument.

{(0,0), (1,1)}  the identity function, which output its argument as value.

{(0,1), (1,0)}, the NOT function, which associate 0 to 1, and 1 to 0. 

There is four functions from {0, 1} to {0, 1}.

OK. But a function from A to B, (with A and B being sets) is simply a set of couples (x, y) with x in A, and y in B. No?

So it seems you are forgetting the functions  {(0, 0), (0, 1)}, no? 

ANSWER: no!   Not all set of couples are function. In particular {(0, 0), (0, 1)} is not a function because the "input" 0 has two outputs: 0 and 1. The functional condition is not met, and a function is a set of couples provided it obeys to the functional condition.

Let me introduce vocabulary, to ease the talk. If the couple (a, b) belongs to the function F, I will say that a is an argument of F, and that b is a value of F. I will write F(a) = b, for (a, b) belongs to F. We also say that F send a on b.

Later, when we will arrive at the notion of computable function: argument will be called input, and value will be called output. The idea is that a function F is a sort of generalized machine: you give argument and it gives back a value. When F *is* mechanical, you give an input and the machine gives back an output.

With that vocabulary, the functional condition can be stated by saying than a function F cannot gives back two values for one argument. A function cannot send an elements on two elements.

To help your mind, think about typical "physical" function. For example the temperature in a room in function of time. For example: we look at the temperature a 1 o'clock, then 2, 3, 4, etc.

T = {(1, 24), (2, 25) (3, 24), (4, 24), (5, 23), (6, 23), (7, 23), (8, 23), (9, 22), (10, 20), (11, 19) ...}

Meaning, at time 1 there is 24 degrees celsius. At time 2, there is 24 degrees celsius, etc.

The functional condition is respected: you cannot have two values of the temperature at the same time.

Does this help?

Another example of a typical physical function. Movement of a mobile in space SPACE in function of TIME. This can be described by a function M from the set of moment in TIME in the set of position in SPACE:

M is a function from TIME to SPACE. 

M = {(t1, p1), (t2, p2), (t3, p3) ... }

where t1, t2, t3, are time coordinate, and p1, p2, p3 are space coordinate.

Again, this will be a function because the mobile cannot be in two places at the same moment.

OK?

2) how many functions, and what are they, from the set cartesian product {0, 1} X {0, 1} to {0, 1}

Among them many are celebrities, you know. The AND, the OR, and many (how many?) others.

For a beginner in math, this is not at all an easy exercise. The real useful exercise is to try to understand the enunciation of the question. We will take the time needed.

First let us compute {0, 1} X {0, 1}. This is simple, as it is the "rectangular" set of all couples (x, y) with x and y each in {0, 1}. Thus it gives 

1   (0, 1)  (1, 1)

0   (0,0)   (1, 0)

      0        1

that is {0, 1} X {0, 1} = {(0,0), (1, 0), (0, 1), (1, 1)}.

Let us build one function F1 from {(0, 0), (1, 0), (0, 1), (1, 1)}  to  {0, 1}.

Here the arguments are (0, 0), (1, 0), (0, 1), (1, 1), and we must decide on which, among 0 and 1, those couples will be sent.

I am lazy, so I will send them all on 0. I will get the constant function 0.

F1((0, 0)) = 0

F1((1, 0)) = 0

F1((0, 1)) = 0

F1((1, 1)) = 0

OK?

Thus F1 is the set of couples {((0, 0), 0), ((1, 0), 0), ((0, 1), 0), ((1, 1), 0)}. Note that here the arguments are themselves couples.

Another one: the constant F2 which sends all couples on 1:

F2((0, 0)) = 1

F2((1, 0)) = 1

F2((0, 1)) = 1

F2((1, 1)) = 1

Thus F2 is the set of couples {((0, 0), 0), ((1, 0), 0), ((0, 1), 0), ((1, 1), 0)}.

If you remember how many binary strings of length 4 can exist, you can guess that we have 16 (2x2x2x2) such functions.

Exercise: find them all. Beginners have to train themselves, if only to develop (slowly but surely) the familiarization with the definitions and notations.

3) A bit tricky perhaps: how many functions exist from { } to { } ?

Solution: A function F from { } to { } has to be a SET of couples (x, y) with x in { }, and y in { }. But { } is empty, so there are no such couples, so F is an empty set of couples, so F is the empty set. F = { }. So there is ONE function from { } to { }. That function is the empty set itself, and is sometimes called the empty function.

OK?

Now, a few new material. It is just vocabulary.  

-------------------------------------------------------------------------------------

SET EXPONENTIATION

Definition. If A and B are sets, we define A to the power of B = A^B, by the set of all functions from A to B.

Thus the exercise above could have been written:

1) compute {0, 1} ^ {0, 1}

2) compute {0, 1} ^ ({0, 1} X {0, 1}) and card({0, 1} ^ ({0, 1} X {0, 1}))   where  "card" = "cardinal of" = "number of elements of"

3) compute { } ^ { } and card({ } ^ { })

Little subject research. If card(A) = n, and card(B) = m. What is card(A^B)? In english: how many functions exist from a set of n elements in a set of m elements? Hint: ask yourself how many choices you have at each step of the construction of a function from A to B.

For the sequel, i suggest you reread everything I have said about "bijection". All right?

Any trouble?

Courage. We are no so far from introducing the computable functions, which is an obvious prerequisite to get the mathematical notion of computation, which is needed to understand the notion of computational supervenience, and get both UDA-7 and UDA-8.

Bruno

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

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[Miroslav Dobsicek]

> 3) compute { } ^ { } and card({ } ^ { })

> If card(A) = n, and card(B) = m. What is
> card(A^B)?

I find it neat to write | {} ^ {} | = | { {} } | = 1 :-)
It's almost like ASCII art. Just wanted to signal that I'm following.

mirek

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[Bruno Marchal]

On 11 Aug 2009, at 15:32, Mirek Dobsicek wrote:

>
>
>> 3) compute { } ^ { } and card({ } ^ { })
>
>> If card(A) = n, and card(B) = m. What is
>> card(A^B)?
>
> I find it neat to write | {} ^ {} | = | { {} } | = 1 :-)

You will make panic those who are not familiar with symbols!


>
> It's almost like ASCII art. Just wanted to signal that I'm following.

Thank for telling me.


OK, people, good time to solve the problems. Please don't read this  
post, unless you find it is the good time for you to do some math. If  
not, postpone until a good time. Technical posts have to be studied,  
not read. Take the time needed. Tell me if I am too quick.

The solution of "3)" has been given.

Let us look at:


>> If card(A) = n, and card(B) = m. What is
>> card(A^B)?


card(A) = n
This means A is a finite set with n elements.

card(B) = m
This means B is a finite set with m elements.

Let us simplify by supposing that m = 3, and n = 2. Hoping that the  
reasoning done for finding the solution on the particular case will  
inspire the reasoning for finding the solution in the general case.

Let us imagine that A is the set {a, b, c}, with its three elements,  
and that B is the set {1, 2}, with its two elements.

And let us try now to remember what is the question.

The question is: what is card(A^B)?

Well, card(A^B) is the number of elements of A^B.   By definition of  
the cardinal.

What is A^B?

Well, A^B is the set of functions from B to A. By definition of set  
exponentiation.

Well, if the question was just "what is card(A^B)?", this would  
provide the best solution, or the best note if you want. But the  
teacher provided the information that A has n elements, and that B has  
m elements, and intuitively we can bet that the number of functions  
from a set to another can depend on the number of elements of each  
sets involved, so that "what is card( A^B)?" meant probably how to  
compute card(A^B) in function of card(A) and card(B).

Ok, we decided to look on the particular case with A = {a, b, c}, and  
B = {1, 2}.

A^B = the set of all functions from B to A.

That is the set of functions from {1, 2} to {a, b, c}.

Well, let us try to find, or to build, one function from B to A.

But , here a moment of panic can occur, (empirical observation). For  
the unnameable sake, what *is* a function? What is a function from B  
to A.
Well, if it is an open manual home work, such panic can be eased by  
looking in the math notes. You may remember the motivation or the  
informal sense of what a function represents, which is a relation of  
dependency, and this is in the most general sense, so that all  
possible dependency are tolerated. For a function from B to A, it  
means the element of A depends in function of the elements of B. Such  
a dependency is well described by a couple (x, y) with x in B and y in  
A.

we have (x,y) belongs-to F representing the meaning that y depends "in  
the function F" of x.

Think about x as time and y as temperature.

So, a function from B to A is just a set of couples (x, y) with x in B  
and y in A, with the functional restriction that x is not send to two  
different values y. At each time x, you can have only one temperature y.

That is, here: a set of couples (x, y) with x in {1, 2} and y in {a,  
b, c}, and such that if (1, x) belongs-to F, no other (1, y) belongs  
to F.

Let us build one function from {1, 2} to {a, b, c}.

OK, 1, from B,  can determine what in A ? Well, we have three  
possibilities a, b and c. OK, i will use my free will to decide that  
for this function I want now, 1 will determine a. So I put the couple  
(1, a) in the function.

At this stage, the "function" looks like {(1, a)}.

Finished?

No, a function from a set to another one gives a values, outcomes,  
outputs for all elements of its domain. I have to say what is  
determine by 2, in B. OK, I will use my free will again, and decide to  
add the couples (2, a).

At this stage, the function looks like {(1, a) (2, a)}.

Finished?

Yes.

We do have a function from B to A. The set {(1, a) (2, a)} describes  
completely a function from B to A, a so-called "constant function".  
think of 1 and 2 as moment of times, and think of a, b, c, as possible  
temperature. The function  {(1, a) (2, a)} describe a case here the  
temperature is constant and equal to a.

Finished? No, we have to find all functions from B to A. All functions  
from {1, 2} to {a, b, c}.

Well actually, we need to find only the number of such functions. For  
1 I have three choices, then for 2, I have still three choices, and  
the choices are independent, so that for each choice the remaining  
three choice will lead to distinct functions, this make 3 X 3  
functions = 9 functions:

{(1, a) (2, a)}
{(1, a) (2, b)}
{(1, a) (2, c)}

{(1, b) (2, a)}
{(1, b) (2, b)}
{(1, b) (2, c)}

{(1, c) (2, a)}
{(1, c) (2, b)}
{(1, c) (2, c)}

so A^B = {{(1, a) (2, a)}, {(1, a) (2, b)}, {(1, a) (2, c)}, ... ,  
{(1, c) (2, c)}}, and card(A^B) = 9. In this case. This give all the  
way the a, b, c can depend on 1 and 2.

I stop here. I let you train on the following question:

How many functions from {a, b, c} to {1, 2}?

How many functions from {1, 2, 3, 4, 5} to {a, b, c, d, e}?

What is the general solution, in term of cardinal n and m of the sets  
involved ?   (the original question).

Take your time, and ask any question. This is the type of stuff rather  
easy for exact scientists, and rather new for those who buried math in  
their unconscious in high school, so take each your own time.

I hope I am not too long. We will see many many examples of functions.

Bruno








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




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[Miroslav Dobsicek]

> Well, A^B is the set of functions from B to A. By definition of set  
> exponentiation.

I'd just like to point out that Bruno in his previous post in the seven
step serii made a small typo

 "A^B - the set of all functions from A to B."

It should have been from B to A. The latest post is correct in this respect.

mirek


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[Bruno Marchal]

On 11 Aug 2009, at 22:24, Mirek Dobsicek wrote:

>
>
>> Well, A^B is the set of functions from B to A. By definition of set
>> exponentiation.
>
> I'd just like to point out that Bruno in his previous post in the  
> seven
> step serii made a small typo
>
> "A^B - the set of all functions from A to B."

I wrote that? I was wrong. Thanks for saying.


>
>
> It should have been from B to A.


Yes!




> The latest post is correct in this respect.

Thank God!

Apologies for typos, mispelling, and believe me, I can do even bigger  
mistakes. I will. Be vigilant.

Bruno



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




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[Bruno Marchal]

On 11 Aug 2009, at 22:24, Mirek Dobsicek wrote:

Well, A^B is the set of functions from B to A. By definition of set  

exponentiation.

I'd just like to point out that Bruno in his previous post in the seven
step serii made a small typo

"A^B - the set of all functions from A to B."

It should have been from B to A. The latest post is correct in this respect.

And now Simplicius is coming back and asks: " but why do you define the exponentiation of sets, A^B, by the set of functions from B to A?".

The answer of the sadistic teacher: this is a DEFINITION, and is part of the program. If you have complains about the program, write a letter to the minister of education.

Hmm...

A better answer is given by the solution of the preceding exercise:

If card(A) = n, and card(B) = m. What is

card(A^B)?

It happens that if A^B is defined as the set of functions from B to A, then card(A^B) is given by card(A)^card(B)

How many functions exist from a set with m elements in a set with n elements? n^m.

Hope you see that n^m is NOT equal to m^n (when n and m are different). 3^4 = 3x3x3x3 = 81, and 4^3 = 4x4x4 = 64.

2^7 = 128, 7^2 = 49.

In that way, A^B generalizes for set what n^m is for numbers.

And why card(A^B) = card(A)^card(B) ?

You can see this in the following way: let card(A) = m, and card(B) = n. We must understand why card(A^B) = n^m.

For example a function from {a, b, c, d, e, f, g} in {0, 1, 2, 3, 4}. To fix the idea. So m = 7, and n = 5. OK?

Let us build an "arbitrary" function F. Well,we begin with "F = {(a, ...", and we have to say where "a" is sent. We have five (n) choices, and then we have to choose where b is sent, and we have again n choices, and for each first choice any second choice is acceptable so we have 5 (n) choices multiplied by 5 (n) choices, itself multiplied by 5 (n) choices, as many times there are elements in the starting set, that is 7 (m). This gives 5 x 5 x 5 x 5 x 5 x 5 x 5, that is 5^7. or more generally n x n x n x n x ... x n, m times.

OK?

We will be interested in N^N. That is, the set of functions from N to N. 

The set of computable functions will be an important subset of that set.

Let me give a precise definition of bijection, as I promise.

I need two rather useful definitions. 

 - I will say that a function from A to B is ONTO, if all elements of B appears in the couples of the function. Note that card(B) has to be less or equal to card(A) to make that possible, by the functional condition.

 - I will say a function is ONE-ONE, if two different elements of A are sent to two different elements of B. Note that card(A) has to be less or equal to card(B) to make that possible. 

The condition one-one is the reverse of the functional condition. The functional conditions says that an element cannot be sent on two different elements (a time cannot give two temperature!), and the one-one condition says that two different elements cannot be sent on one element.

Exercises: build many examples with little finite sets. You may search examples for infinite sets.

OK. The definition of bijection. I will say that a function is a bijection between A and B if it is both a function ONTO from A to B, and a function ONE-ONE from A to B. we say more quicky that f is a bijection if f is both onto and one-one.

Exercises:   for "2)"  below, the real needed exercise is:  "do you understand the question?" Unless you like to count things, but such skills are not needed for the sequel. 

1) Convince yourself that if A and B are finite sets, then there exists a bijection between A and B if and only if card(A) = card(B).

2) If A has n elements (card(A) = n), how many bijections exists from A to B? 

     Again start with simple examples, and try to generalize.

     For example, how many bijections from {a, b, c} to {1, 2}. How many bijections from (a, b, c} to {a, b, c}?

3) can you find, or define a bijection between the infinite set N, and the infinite set E = {0, 2, 4, 6, 8, ...} (E for even).

4) Key questions for the sequel, on which you can meditate:

- is there a bijection between N and NxN?      (NxN = the cartesian product of N with N)

- is there a bijection between N and N^N?

Bruno

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

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