Dear List Members,
I fear that there is some misunderstanding about my previous post with
Subject "Heidegger and Mathematics".
I did not stated that I proved anything, or I try to mathematicize
Heidegger's fundamental ontology. What I said that I make/made some research
and I have _questions_ and _cojectures_ which __can be interpreted__
as a _trial_ for mathematization of Heidegger's fundamental ontology
in a mathematical enviroment. The word, _interpret_ also very important here.
And _can be_ stands for to express the possibility, or opinion, feeling
etc. So, I do not regard it as a sure thing.
> I have some questions and conjectures in mathematical logic and in
> physics, which can be interpreted, as a trial to formulate Heidegger's
> fundamental ontology in a precise, mathematical way. [...]
Mathematics is mathematics. It is not physics, philosophy or something
else. Connections with other subjects goes thorough interpretation.
(At least in my opinion. Naturally, it can be a subject of an argument.)
Here is a part of my unpublished and unfinished paper. It is written
in LaTeX.
The aim of this paper to show that there exists a (highly) non-trivial
self-representable system, which have a lot of interesting property.
For example it is possible to construct/find a model of an observer.
Because all the mathematically esteblished physical theories probably
do not have this property, I think that self-representability is a
very important property, assuming second order arithmetic is OK.
In the _process_ of _testing arithmetic_ I feel, that the hidden question
is that the "mathematical Dasein" can continue the establishing bigger
and bigger parts of math, giving them a new ontological status.
It is a never ending process, because we cannot prove that second order
arithmetic does not have contradiction.
(This is the point, where _time_ is coming into the picture.)
Good reading ... .
Tibor O'dor
Ps.: Do not forget, it is only a manuscript.
-------------------------------------------------------------------------
\documentstyle{article}
\title{ Toward Information Physics: \\ \\
Mathematical Problems and
Phylosophical Notes \\ \\
(part of the manuscript)
}
\author{Tibor \'Odor}
\begin{document}
\maketitle
\newtheorem{Conjecture}{Conjecture}
\newtheorem{Question}{Question}
\newtheorem{Theorem}{Theorem}
\newtheorem{Definition}{Definition}
\newtheorem{Note}{Note}
\def\mod{{\rm mod}}
\def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt
\hbox {\vrule width.#2pt height#1pt \kern#1pt
\vrule width.#2pt}
\hrule height.#2pt}}}}
\def\square{\mathchoice\sqr56\sqr56\sqr{2.1}3\sqr{1.5}3}
\def \qed{$\square$}
\def \qued{\,\,\qed}
\def\<{<}
\def\>{>}
\section{Testing the Arithmetic:
The Effectivization of Hilbert's Program}
We, the children of the computer age, sometimes have to compute with
extremely large numbers. We use extremely big computations for
example to control atomic reactors and air traffic. The validity of
our theories several times rely on the assumption that arithmetics
works properly for extremely large numbers.
So it would be very disturbing, if there would be a contradiction in our
arithmetic.
Hilbert's program was to prove the Peano axiom system of arithmetic
does not contain contradictions. However, it proved to be too ambitious.
Hilbert's program failed because of G\"odel's result, so we can
never be sure.
It is very disappointing. However, one can argue, that in certain
situations it is not necessary to know that the Peano arithmetic does
not contain contradiction, but it is enough to
know, that in that range of magnitude in which we use the numbers,
for certain limited operations it works properly, it does not
lead to contradiction.
It is natural to assume that we ``know'' the ``small'' numbers well,
and we did not find any contradiction using them say up to $C=20$.
This is the overall experience of the mankind (ten finger on the
hands and ten on the legs---for almost everyone) from Papua New
Guinea to the Los Alamos Super Computer Center of NSF. We know that
certain manipulations, for example addition, multiplication, substraction,
permutations (generated by some simple permutations) etc with these
numbers, if we do not apply them too many times (say, not more than
20 times) does not reveal any kind of contradiction.
Let us call this number $C_0(=20)$ be our number of ``confidence''.
The boundaries of our realm of confidence (what I think we
inherited from our ancestors, and tested by the long evolutionary
history of the mankind) are too vague, so it is worth define more precisely.
Let $\Pi_0$ be a fixed set of manipulations, namely multiplication, residue
of the division $a$ by $b$, comparison, cyclic permutation, changing
two elements, substitution into formulas.
The pair $(C_0, \Pi_0)$ we call the fundamental realm of confidence.
But what about the other magnitudes? The number $20$ seems to be too
small, although for most of us \$$2^{20}$ would be enough
(at least in present rates, even in Canadian ... ).
We want to extend our realm of confidence (or, to find the
contradiction!) in arithmetic---using only the knowledge (originated
in our every day experience) that our fundamental realm of confidence
does not reveal any contradiction.
It seems to be an intriguing question whether it is possible or not.
Our question is more precisely, which we can call the ``Effectivization
of Hilbert's Program'' is as follows.
\begin{Conjecture}
There is an increasing recursive sequence
$$
(C_0, \Pi_0), (C_1, \Pi_1),\dots, (C_n, \Pi_n), \dots
$$
and a recursive proof attempt that $(C_n, \Pi_n)$ does not reveal
contradiction using only the $(C_{n-1}, \Pi_{n-1})$-tested manipulations
and the knowledge that $(C_{n-1}, \Pi_{n-1})$ does not reveal contradiction,
proving at a certain $n_0$ that $C_n\to \infty$ and
$\bigcup_{n\in {\bf N}} \Pi_n$
gives all the possible manipulations on numbers, or we find a contradiction
at some step.
\end{Conjecture}
It seems to be impossible for the first sight, because, sooner
or later---because no proof exists that the Peano axiom system does not
contain contradiction,---we have to compute by arbitrarily big numbers,
and we have to make manipulations arbitrarily many times.
But, it is imaginable, that checking $(C_n, \Pi_n)$ we can use only
very limited type of manipulations in most of our computations, which
validity we can check in $(C_{n-1}, \Pi_{n-1})$, even if
we use them far more time than $C_n$. And this is the point, this
makes the problem.
The problem seems to be original, very deep and difficult
for the author.
(Understanding some of Martin Heidegger's thoughts on fundamental
ontology in ``Time and Beeing'' on a certain way lead the author to
ask and investigate this and the next question.)
\section{ Self-Representable Physical Theories }
Given a Physical Theory ${\cal F}$, for example the Newtonian
mechanics, the special or general relativity, quantum mechanics.
We can ask the following intriguing question: Can we construct a
model of measurement ${\cal M}{\cal F}$ in ${\cal F}$ to every law of
${\cal F}$ from its objects which check the validity of the Physical
Theory ${\cal F}$, with absolute [with arbitrarily big] precision?
The problem is here that we have to construct both the phenomena
{\it and} the measurement devices, measuring the phenomena from
the objects of ${\cal F}$, and the measurement devices has show the
same result what the Physical Theory predicts for the phenomena.
It seems to be a natural assumption that a Physical Theory which
gives a complete description of the Universe has to be self-representable.
It seems to be a deep and difficult question wether there exist or not a
self-representable Physical Theory.
\begin{Conjecture}
The Newtonian mechanics, the special or general relativity, quantum
mechanics, B\"ohmian quantum mechanics are not self-representable.
\end{Conjecture}
\begin{Question}
Which is the simplest self-representable mathematical system?
Does there exist a self-representable theory/system at all?
\end{Question}
If the ``Effectization of Hilbert's Program'' is possible, then
arithmetic would be the most simple self-representable mathematical
system.
\end{document}
--- from list heidegger@xxxxxxxxxxxxxxxxxxxxxxxxxx ---
I fear that there is some misunderstanding about my previous post with
Subject "Heidegger and Mathematics".
I did not stated that I proved anything, or I try to mathematicize
Heidegger's fundamental ontology. What I said that I make/made some research
and I have _questions_ and _cojectures_ which __can be interpreted__
as a _trial_ for mathematization of Heidegger's fundamental ontology
in a mathematical enviroment. The word, _interpret_ also very important here.
And _can be_ stands for to express the possibility, or opinion, feeling
etc. So, I do not regard it as a sure thing.
> I have some questions and conjectures in mathematical logic and in
> physics, which can be interpreted, as a trial to formulate Heidegger's
> fundamental ontology in a precise, mathematical way. [...]
Mathematics is mathematics. It is not physics, philosophy or something
else. Connections with other subjects goes thorough interpretation.
(At least in my opinion. Naturally, it can be a subject of an argument.)
Here is a part of my unpublished and unfinished paper. It is written
in LaTeX.
The aim of this paper to show that there exists a (highly) non-trivial
self-representable system, which have a lot of interesting property.
For example it is possible to construct/find a model of an observer.
Because all the mathematically esteblished physical theories probably
do not have this property, I think that self-representability is a
very important property, assuming second order arithmetic is OK.
In the _process_ of _testing arithmetic_ I feel, that the hidden question
is that the "mathematical Dasein" can continue the establishing bigger
and bigger parts of math, giving them a new ontological status.
It is a never ending process, because we cannot prove that second order
arithmetic does not have contradiction.
(This is the point, where _time_ is coming into the picture.)
Good reading ... .
Tibor O'dor
Ps.: Do not forget, it is only a manuscript.
-------------------------------------------------------------------------
\documentstyle{article}
\title{ Toward Information Physics: \\ \\
Mathematical Problems and
Phylosophical Notes \\ \\
(part of the manuscript)
}
\author{Tibor \'Odor}
\begin{document}
\maketitle
\newtheorem{Conjecture}{Conjecture}
\newtheorem{Question}{Question}
\newtheorem{Theorem}{Theorem}
\newtheorem{Definition}{Definition}
\newtheorem{Note}{Note}
\def\mod{{\rm mod}}
\def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt
\hbox {\vrule width.#2pt height#1pt \kern#1pt
\vrule width.#2pt}
\hrule height.#2pt}}}}
\def\square{\mathchoice\sqr56\sqr56\sqr{2.1}3\sqr{1.5}3}
\def \qed{$\square$}
\def \qued{\,\,\qed}
\def\<{<}
\def\>{>}
\section{Testing the Arithmetic:
The Effectivization of Hilbert's Program}
We, the children of the computer age, sometimes have to compute with
extremely large numbers. We use extremely big computations for
example to control atomic reactors and air traffic. The validity of
our theories several times rely on the assumption that arithmetics
works properly for extremely large numbers.
So it would be very disturbing, if there would be a contradiction in our
arithmetic.
Hilbert's program was to prove the Peano axiom system of arithmetic
does not contain contradictions. However, it proved to be too ambitious.
Hilbert's program failed because of G\"odel's result, so we can
never be sure.
It is very disappointing. However, one can argue, that in certain
situations it is not necessary to know that the Peano arithmetic does
not contain contradiction, but it is enough to
know, that in that range of magnitude in which we use the numbers,
for certain limited operations it works properly, it does not
lead to contradiction.
It is natural to assume that we ``know'' the ``small'' numbers well,
and we did not find any contradiction using them say up to $C=20$.
This is the overall experience of the mankind (ten finger on the
hands and ten on the legs---for almost everyone) from Papua New
Guinea to the Los Alamos Super Computer Center of NSF. We know that
certain manipulations, for example addition, multiplication, substraction,
permutations (generated by some simple permutations) etc with these
numbers, if we do not apply them too many times (say, not more than
20 times) does not reveal any kind of contradiction.
Let us call this number $C_0(=20)$ be our number of ``confidence''.
The boundaries of our realm of confidence (what I think we
inherited from our ancestors, and tested by the long evolutionary
history of the mankind) are too vague, so it is worth define more precisely.
Let $\Pi_0$ be a fixed set of manipulations, namely multiplication, residue
of the division $a$ by $b$, comparison, cyclic permutation, changing
two elements, substitution into formulas.
The pair $(C_0, \Pi_0)$ we call the fundamental realm of confidence.
But what about the other magnitudes? The number $20$ seems to be too
small, although for most of us \$$2^{20}$ would be enough
(at least in present rates, even in Canadian ... ).
We want to extend our realm of confidence (or, to find the
contradiction!) in arithmetic---using only the knowledge (originated
in our every day experience) that our fundamental realm of confidence
does not reveal any contradiction.
It seems to be an intriguing question whether it is possible or not.
Our question is more precisely, which we can call the ``Effectivization
of Hilbert's Program'' is as follows.
\begin{Conjecture}
There is an increasing recursive sequence
$$
(C_0, \Pi_0), (C_1, \Pi_1),\dots, (C_n, \Pi_n), \dots
$$
and a recursive proof attempt that $(C_n, \Pi_n)$ does not reveal
contradiction using only the $(C_{n-1}, \Pi_{n-1})$-tested manipulations
and the knowledge that $(C_{n-1}, \Pi_{n-1})$ does not reveal contradiction,
proving at a certain $n_0$ that $C_n\to \infty$ and
$\bigcup_{n\in {\bf N}} \Pi_n$
gives all the possible manipulations on numbers, or we find a contradiction
at some step.
\end{Conjecture}
It seems to be impossible for the first sight, because, sooner
or later---because no proof exists that the Peano axiom system does not
contain contradiction,---we have to compute by arbitrarily big numbers,
and we have to make manipulations arbitrarily many times.
But, it is imaginable, that checking $(C_n, \Pi_n)$ we can use only
very limited type of manipulations in most of our computations, which
validity we can check in $(C_{n-1}, \Pi_{n-1})$, even if
we use them far more time than $C_n$. And this is the point, this
makes the problem.
The problem seems to be original, very deep and difficult
for the author.
(Understanding some of Martin Heidegger's thoughts on fundamental
ontology in ``Time and Beeing'' on a certain way lead the author to
ask and investigate this and the next question.)
\section{ Self-Representable Physical Theories }
Given a Physical Theory ${\cal F}$, for example the Newtonian
mechanics, the special or general relativity, quantum mechanics.
We can ask the following intriguing question: Can we construct a
model of measurement ${\cal M}{\cal F}$ in ${\cal F}$ to every law of
${\cal F}$ from its objects which check the validity of the Physical
Theory ${\cal F}$, with absolute [with arbitrarily big] precision?
The problem is here that we have to construct both the phenomena
{\it and} the measurement devices, measuring the phenomena from
the objects of ${\cal F}$, and the measurement devices has show the
same result what the Physical Theory predicts for the phenomena.
It seems to be a natural assumption that a Physical Theory which
gives a complete description of the Universe has to be self-representable.
It seems to be a deep and difficult question wether there exist or not a
self-representable Physical Theory.
\begin{Conjecture}
The Newtonian mechanics, the special or general relativity, quantum
mechanics, B\"ohmian quantum mechanics are not self-representable.
\end{Conjecture}
\begin{Question}
Which is the simplest self-representable mathematical system?
Does there exist a self-representable theory/system at all?
\end{Question}
If the ``Effectization of Hilbert's Program'' is possible, then
arithmetic would be the most simple self-representable mathematical
system.
\end{document}
--- from list heidegger@xxxxxxxxxxxxxxxxxxxxxxxxxx ---