Mathematics is the most ancient of sciences. However, in the beginning was the word. We must remember that the olden “logos” lives in logics and logistics rather than grammar. The order of mind and the order of store are the precious gifts of our ancestors.

The intellectual field resides beyond the grips of the law of diminishing returns. The more we know, the huger become the frontiers with the unbeknown, the oftener we meet the mysterious. The twentieth century enriched our geometrical views with the concepts of space-time and fractality. Each instance of knowledge is an event, a point in the Minkowski 4-space. The realm of our knowledge comprises a clearly bounded set of these instances. The frontiers of science produce the boundary between the known and the unknown which is undoubtedly fractal and we have no grounds to assume it rectifiable or measurable. It is worth noting in parentheses that rather smooth are the routes to the frontiers of science which are charted by teachers, professors, and all other kinds of educationalists. Pedagogics dislikes saltations and sharp changes of the prevailing paradigm. Possibly, these topological obstructions reflect some objective difficulties in modernizing education.

The proofs are uncountable of the fractality of the boundary between the known and the unbeknown. Among them we see such negative trends as the unleashed growth of pseudoscience, mysticism, and other forms of obscurantism which creep into all lacunas of the unbeknown. As revelations of fractality appear the most unexpected, beautiful, and stunning interrelations between seemingly distant areas and directions of science.

The revolutionary changes in mathematics in the vicinity of the turn of the twentieth century are connected not only with the new calculus of the infinite which was created by Cantor. Of similar import was the rise and development of mathematical logic which applied rigor and analysis to the very process of mathematical demonstration. The decidable and the undecidable, the provable and the improvable, the consistent and the inconsistent have entered the research lexicon of the perfect mathematician. Mathematics became a reflexive science that is engaged not only in search of truths but also in study of its own methods for attaining these truths.

Aristotle's logic, the paradoxes of Zeno, the razor of William of Occam, the donkey of Buridan, the Lebnizian Calculemus, and Boolean algebras are the outstanding achievements of mankind which cast light on the road to the new stages of logical studies. Frege immortalized his name by inventing the calculus of predicates which underlies the modern logic.

The twentieth century is marked with deep penetration of the ideas of mathematical logic into many sections of science and technology. Logic is a tool that not only organizes and orders our ways of thinking but also liberates us from dogmatism in choosing the objects and methods of research. Logic of today is a major instrument and institution of mathematical freedom.

S. Kutateladze

October 2, 2006

Footnote. A fragment of the article “What Is Boolean Valued Analysis?” Siberian Advances in Mathematics, 2007, V. 17, No. 2, 91–111.

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