Dear Shalender,

All we can do as human beings is to use our senses to communicate our perceptions of the "inner" level of the universe as truly as possible in order to learn and gather as objectively as possible any information about the "outer" universe.

As physicists we learn from Kant that space and time are the essential forms of our human sensibilities. Therefore any "serious" formulation of nature must start with movements of perceived entities in space time.

Fortunately we have today excellent theories portrayed in rigorous mathematics that describe energy and momentum as related to their conjugate variables (operators) time and space that in principle (except the unification problem has not been completed) can be applied to most relevant questions in our universe.

Dear Shalender,

All we can do as human beings is to use our senses to communicate our perceptions of the "inner" level of the universe as truly as possible in order to learn and gather as objectively as possible any information about the "outer" universe.

As physicists we learn from Kant that space and time are the essential forms of our human sensibilities. Therefore any "serious" formulation of nature must start with movements of perceived entities in space time.

Fortunately we have today excellent theories portrayed in rigorous mathematics that describe energy and momentum as related to their conjugate variables (operators) time and space that in principle (except the unification problem has not been completed) can be applied to most relevant questions in our universe.

I see that almost all answers are essentially based on a Cartesian duality. I 'd rather follow D. Makbeth, Realising Reason, where we can find:

"According to the theory of evolution by natural selection, evolution occurs if there is, first, variation, an ongoing source of novel features, second, a mechanism of their replication or inheritance, and third, differential fitness among those replicable variations. Over time, millions and millions of years, biological evolution realizes beings that are properly described as alive, as having biological natures and natural habitats. Living things are not, in other words, merely things that can do things, as, for instance, mere mechanisms are; they are instead constitutively instances of kinds or species. Living things have natures, and thereby a distinctive mode of being (relative to the being of a mere mechanism, a merely causal structure). They have characteristic forms of life."

In this sense we are already structured! Of course we may also speak about "gestalts" which are perceived as non-analytic or holistic structured wholes. Perceiving then defining is to much Cartesian!     

Yes Drostas,

Living entities communicate in the most general sense from molecular assemblies in cells to civilizations. The concept of perception does not portray the whole "story" of evolution, but it is nevertheless a necessary requisite.

Dear Erkki,

I like you synthesis of my name! Drostas=Dros(sos)(C0)stas.

A living entity makes mistakes and sometimes constructive ones.

> General Relativity assumes that there already exists a space-time non-trivial topology.

GR assumes more than topology: it assumes that the Universe is a pseudo-Riemannian manifold (so, geometry, beyond topology). The topology of the Universe is usually (often) "trivial" -- what's non-trivial about the topology of classical open or closed Universes?

String theory also usually assumes topology and geometry of the target space as given, so does not differ much in assumptions from classical GR. Even though this might seem not logical -- as gravity is a derivative entity in string theory (graviton = certain excited state of fundamental string, same as are photons and other bosons; and fermions in superstring theories), so the geometry of the target space should be also a derived concept, from certain coherent state of ST gravitons. But such "inconsistency" is not something unusual, the same superficially flawed consideration is used in QFT when one considers classical background fields (say, classical macroscopic EM field, in background of which QED phenomena are studied -- not bothering that such a classical EM background must perhaps have a representation as certain coherent state of photons). 

So, back to your questions: yes, ST also assumes certain topology and even geometry of the target space; superficially in the same way as GR does. There are however deeper differences with GR, due to

1) different way in which strings (vs pointlike particles = probes of GR) see geometry of the background (target) space -- one manifestation of this is that there exist different (from the point of view of classical geometry => different for GR) target spaces that are the same from ST point of view (T-duality);

2) rich zoo of ST objects that dictate certain non-trivial geometry for the target space (say, heavy beasts like NS5 branes or black strings).

The target space can be also not a manifold but generalizations thereof (orbifold, conifold).

> I propose that we should start from a different philosophical basis of "Relative Existence", which will not need presumption of absolute topology. I believe that this paradigm will give us deeper insights in physics.

I am not sure from the whole discussion if you mean by "topology" what is usually meant by it (cf. whatever set of axioms/definition in general topology e.g.). Topology is very useful instrument, so it's natural scientists want to see it in their theories. Geometry on the other hand is more often considered not necessarily existing a priori (say, near Big Bang in certain theories) -- but if one refuses existence of classical geometry then some alternatives must be proposed for describing observables. Certain branches of string theory do this (see Matrix theory e.g.) -- geometry is then to appear as a derived concept in certain conditions (say, at late times after the Big Bang; in such conditions something with a group of symmetries appears that has a subgroup isomorphic to the group of diffeomorphisms of a (pseudo-Riemannian) manifold; which is then interpreted as emergence of geometry in this framework)

All in all all this is quite exciting stuff.

Hi Andrey> In the context of General Topology mathematics, a trivial topology is the one, which has no structure, which means null set of set of all sets: http://en.wikipedia.org/wiki/General_topology. 

The euclidean topology is not a trivial topology and euclidean topology is defined by a flat metric space. Even pseudo-Riemannian is a well defined topology but it is not a flat metric space. Topology is generic concept of nearness and distance defined in the set theoretical form. In a trivial topology all elements are at the same level with each other so there is no ordering and thus no concept of distance.

General Topology is based on the set theoretical concept, which is based on the concept of "absolute existence". The concept of absolute existence means if something exists, it has to exist somewhere. That is how the natural numbers are constructed starting from a null set.

"Relative existence" however means if something exists, it exists relative it's neighbors. It need not exist inside anything, it just exists in relation to something else.

Well, ok. I guess we've agreed so far that the Universe on scales above ~Planckian is a smooth manifold (or a generalized structure, which locally looks like a manifold almost everywhere), and that in part defines its topological properties. Such a picture agrees with observations at currrently accessible energies, and so is reflected in all theories striving to describe our real Universe.

Any set can be equiped with trivial topology (in which only open sets are the whole set or null set); the topological structure of our world is richer, and it is assumed richer from the start in any theory of this world. 

Shalender,

There are some subtle points involved here. Your question, first of all has a metaphysical flavour as it concerns the language (mathematics) as utilized to obtain knowledge about the universe.

In this vein many theories are brought up, in particular string related "models".The scientists behind this development do not distinguish between, see the quote below, the language (mathematics) and the physical universe and hence would make your question meaningless.

The following sentences are quoted from an upcoming publication, soon to be put on my RG page:

Many physicists today are undeniably epistemologists, however, brane and string

scientist seem to have a different ontology in that they consider the vibrating string to be the particle itself rather than merely dictating the properties of its host particle. This view may not be commensurate with a quantum theoretic hierarchy of concepts, i.e. representing nature via accumulation of knowledge through unbiased and systematic research. Indeed the absolute being of material bodies, obtains self-referential consequences, provoking further discussions in relation to Gödel’s incompleteness theorem(s) [4]. In short Gödel’s self-referential account will be the guide for dynamical processes, already emerging on the molecular level, introducing viable communication protocols via Poisson distributed channels, for details see more below. In contrast modern p-branes, while being spatially extended strings, originate in mathematical concepts that appear to conflate matter with its mathematical description, i.e. do not exhibit the proper fabric for a self-referential examination. We will here advance an ontology that entangles the physical- and the mental world, while embedding evolution as a template for self-organization and associated teleodynamical processes.