**Mass** is a physical quantity, which is treated as one of the basic characteristics of matter and determines its inertial and gravitational properties; mass is usually designated by the letter *m*. — *This is a conventional definition of mass as used in manuals and encyclopedias*.

In physics the majority of scientists still put in the forefront the theorem of everything, i.e. they try to formulate a theory that would unify all forces observed in physics. Today only three of the six forces have been succesfully unified: the electromagnetic, the weak and the strong forces. The efforts to include the nuclear forces, the quantum mechanical forces and the gravitational forces in the unification are still ongoing. In the unification efforts, the fundamental basis of physics remains completely unclear: there is no succesful examination of the determination or derivation of initial physical notions such as mass, charge, particle, de Broglie wavelength, spin, photon, etc. General relativity, as a phenomenological abstract theory, is also unable to clarify the situation: it is based on separate notions of mass and space, or space-time, which exist independently from each other (even though mass may influence space).

On the other hand, the physical notion of **mass** directly stems from a mathematical theory of the constitution of physical space. Set theory, topology and fractal geometry allow us to construct the real physical space as a mathematical lattice of topological balls, which was called [1,2] the tessel-lattice and which possesses fractal properties. In this approach to the fundamentals, the theorem of something occupies the first place, i.e. a peculiar object becomes primary, which is typical for set theory. Then, having a definition of the primary 'something', we can study its behaviour in the tessel-lattice, i.e. space mosaically composed of primary bricks or topological balls.

In the tessel-lattice a local deformation can appear in the form of a volumetric fractal deformation of a topological ball. The creation of such local deformation in the mathematical lattice - the tessel-lattice - can be associated with the physical notion of **mass**. A surface fractal deformation of a topological ball defines the electric state of the ball (see electric charge). Thus a mass of a canonical particle has to be considered as a ratio of the initial volume $V_{\rm o}$ of the degenerate ball in the tessel-lattice to a volume $V_{\rm def}$ of a ball that has undergone a fractal volumetric deformation, i.e.

here ${\rm const }$ is a dimension constant and the ratio $V_{\rm o}/V_{\rm def} >1$.

In this way the mass of a canonical particle can be determined. This particle (its kernel) is characterised by a local deformation described by expression (1). During the motion the kernel experiences a fractal decomposition: owing to the interaction with adjacent cells of the tessel-lattice, which forms the real space, the whole deformation (1) of the kernel gets scattered such that the particle becomes surrounded by elementary excitations that carry fragments of the particle deformation, i.e. particle's mass. The dynamics of the canonical particle is the subject of submicroscopic mechanics.

A particle surrounded by these mass excitations has a reflection in orthodox quantum mechanics: this is the well-known particle's $\psi$-wave function. These elementary excitations, carriers of mass, have been referred to as inertons. As a result, the $\psi$-wave function determined previously in an abstract phase space gains a real physical meaning as the field of inertia of the particle.

**Bibliography**

[1] M. Bounias and V. Krasnoholovets, Scanning the structure of ill-known spaces: Part 2. Principles of construction of physical space, *Kybernetes: The International Journal of Systems and Cybernetics* **32**, No. 7/8, pp. 976—1004 (2003) (also http://arXiv.org/abs/physics/0212004).

[2] M. Bounias and V. Krasnoholoves, The universe from nothing: A mathematical lattice of empty sets, *International Journal of Anticipatory Computing Systems* **16**, pp. 3-24 (2004) (also http://arXiv.org/abs/physics/0309102).