Discrete vs continue.Does problem solved?

[Yuri Danoyan]

In the Book P.Fraenkel, Yehoshua Bar-Hillel, FOUNDATIONS OF SET THEORY, North-Holland, 1958 you can read following :

"The bridging of the chasm between the domains of the discrete and the continuous,or between arithmetic and geometry, is one of the most important - may, the most important - problem of the foundations of mathematics....Of course, the character of reasoning has changed, but,as always, the difficulties are due to the chasm between the discrete and the continuous - that permanent stumbling block which also plays an extremly important role in mathematics, philosophy,and even physics."

 

This is present day problem or not to comparision of 1958 ?

 

Because "discrete" and "continue" are very abstract notions, in future for concreteness we will talking about discrete and continue symmetries.Then we will see that problem solved or not.

Edited July 17, 2008 by Yuri Danoyan
multiple post merged

[Yuri Danoyan]

See my thread in future

http://www.scienceforums.net/forum/showthread.php?t=34145

[prometeu]

A beautiful example is gamma function.

$n!=\int_0^{+\infty}x^ne^{-x}dx$

This equality can be used to define the continuous factorial function with n = (-1,+inf), so for n natural we get the usual value n!.

 

A beautiful example is gamma function.

[n!=\int_0^{+\infty}x^ne^{-x}dx]

This equality can be used to define the continuous factorial function with n = (-1,+inf), so for n natural we get the usual value n!.

[Yuri Danoyan]

What do you mean?

[prometeu]

Factorial n! is a discrete function with n=0,1,2,.... Gama function is a continued one with n>-1 that has the property to be equal to n! for n=0,1,2,.... In this case discrete is a special case of continue.

The definition of contiue is I belive: between evrey two elements of a set exist an element. So continuity is defined in discrete terms.

Say something because I like symmetries and of course group theory.

[Yuri Danoyan]

In this case discrete is a special case of continue.

 

My version: continue is special case of discrete.

In any case all solutions are approximations.

My approximation continue symmetry by discrete binary very simple and euristic....

 

Foundations of Set Theory (Studies in Logic and the Foundations of Mathematics) (Hardcover)

by A.A. Fraenkel (Author), Y. Bar-Hillel (Author), A. Levy (Author) "In Abstract Set Theory 1) the elements of the theory of sets were presented in a chiefly genetic way: the fundamental concepts were defined and..." (more)

Key Phrases: quantifiers over class variables, intuitionistic attitude, relative selector, Second Axiom of Restriction

 

http://www.amazon.com/Foundations-Theory-Studies-Logic-Mathematics/dp/0720422701/ref=si3_rdr_bb_product

Edited July 25, 2008 by Yuri Danoyan
multiple post merged

[Yuri Danoyan]

heuristic

A heuristic is a method to help solve a problem, commonly informal. It is particularly used for a method that often rapidly leads to a solution that is usually reasonably close to the best possible answer. Heuristics are "rules of thumb", educated guesses, intuitive judgments or simply common sense.

 

Metasymmetry idea--heurustic idea.

Edited July 29, 2008 by Yuri Danoyan
multiple post merged

[Yuri Danoyan]

Problem discrete/continue have solution if:

1.Concrete, as discrete symmetry/continue symmetry;

2.Minimal approximation used,binary notation;

3.Metasymmetry notion introduced;

4.To bring in an income as applications to physics;

 

Azriel Levy only alive author abovementioned book.Resently i send him question "Does this quotation is valid today or not?"

 

His answer to me:

 

Dear Mr. Danoyan,

No new edition of the book is planned. It is beteer for new authors to have their say.

There is now less interest in foundations than, say, in a century or half a century ago. The advances in the technical directions isolated the purely foundational problems and they became less interesting to mathematicians. The people who still work on the foundatinal problems are the philosophers of mathematics, who may answer your question.

Best regards, Azriel Levy

 

Foundations of Set Theory (Studies in Logic and the Foundations of Mathematics) (Hardcover)

by A.A. Fraenkel (Author), Y. Bar-Hillel (Author), A. Levy (Author)

Quotation from Chapter IV,p.210-220.The abyss between discretenes and continuity.

 

 

Other letters:

 

Dear Dr Wilczek

 

What Are You thinking about next quotation ?

>> The bridging of the chasm between the domains of the discrete and the

>> continuous,or between arithmetic and geometry, is one of the most

>> important - may, the most important - problem of the foundations of

>> mathematics....Of course, the character of reasoning has changed, but,

>> as always,the difficulties are due to the chasm between the discrete and

>> the continuous - that permanent stumbling block which also plays an

>> extremly important role in mathematics, philosophy,and even

> > physics.(P.Fraenkel,Yehoshua Bar-Hillel, FOUNDATIONS OF SET

> > THEORY,North-Holland,1958

 

Hi,

 

I don't know how to rate it in importance, since I don't see any specific suggestion of a paradox or anomaly in Nature related to it, but it does bother me that the real number continuum, which is so fundamental to our present formulation of physics, is so complicated and apparently artificial when considered as a logical construction.

 

All best wishes,

Frank Wilczek

February 08 2003

 

 

From Gerhard t'Hooft

 

I don't perceive this as a chasm - or a stumbling block. Surely symmetry is playing a central role in physics, both discrete and continuous.

Indeed, both kinds of symmetries are bewing used as tools to construct theories; gauge fields are fields whose equations can only be understood in their relation to local, continuous symmetries.

In quantum mechanics, particle theory, condensed matter theory, superconductivity, symmetries are at the centre of our formalisms.

I wouldn't call such an essential aspect of our theories a

stumbling block.

 

Cordially,

G. 't Hooft

June 09 2004

Edited August 3, 2008 by Yuri Danoyan
multiple post merged

[Yuri Danoyan]

Reminding about chasm between discrete and continue....

 

Foundations of Set Theory (Studies in Logic and the Foundations of Mathematics) (Hardcover) by A.A. Fraenkel (Author), Y. Bar-Hillel (Author), A. Levy (Author)

Quotation from Chapter IV,p.211.The abyss between discretenes and continuity