Re: theory and practice: ying and yang

erk schrieb:
> Alexandr Savinov wrote:
> <SNIP>
>

>>I think that the concept of continuum was invented in order to make the
>>life of mathematicians easier. Try to transfer elementary mathematical
>>notions onto discrete case and you understand that it is impossible.
>>Having continuum solves the problem.

If you ask about some ready solution then it is does not exist. But when I was writing about such a future theory I meant that it is possible to make a list of principles such a theory should satisfy. For example, the model needs to be able to describe itself. Equivalent statements: a set has to include intself as a member; a space has to be one of its points, a dimenstion has to be considered a normal value or point; infinity has to be reachable etc. Currently any theory is based on some absolute things called axioms, particular elements, exceptions etc. (speed of light in physics; infinity, zero and one in mathematics, delta-functions, variables,..., thousands of examples, actually).

>>Real numbers do not help in solving contemporary problems and this why I
>>find this apparatus almost useless. They are used and will be used by
>>inertia.

The problem cannot be solved by simply introducing another kind of numbers - there should be more general solution. Of course, I cannot give a *concrete* example but I could give a number of hints, principles, directions, criteria for such a new theory but unfortunately it is not well organized and is not a topic of this forum. In very general terms consider a construct consisting of a number of elements. But they are not a set because having a set means that we need one more element which is somewhere outside this construct. So where is this particular element, which allows us to consider these elements as one construct? The hypothesis (assumption of the theory) is that it is one of these elements. Of course, such a hypothesis produces more questions than gives answers but as a said it is only a direction. There is also other principles and hints. If one develops them then it is possible to produce a kind of new theory (not necessarily correct, of course - just a kind of view of the world). For example, it is very desirable for any element to have an opposite and a dual element so we add this property. And so on. Periodically we need to provide meaningful interpretations. For example, some elements play the role of space with respect to other elements which are points. Eventually such a construct will represent a model. One of its properties will be closeness with respect to the scale, i.e., if we gradually decrease the scale then we return to the same point from large scale (micro world and macro world coincide). Fantastic picture is not it?

-- 
alex
http://conceptoriented.com