## Towards a Soft Mathematics

Scientific progress of the New Time has placed mathematics on a pedestal of higher truth and thus has begotten an illusion of achieving perfection. On the boundary of the 20th century D.Hilbert designed “a final elucidation of the essence of infinity” considering it essential for the honour of the human mind as such ([1], p.341). Collapse of illusions started in 1931 with the appearance of the Godel’s theorem on incompleteness of formal systems, and a final solution of the continuum problem given by P.Cohen in 1963 undermined two-valued logic since the answer to the question put as “either-or” was “neither-nor”. Thus Hilbert’s programme failed and “Cantor’s paradise” turned into a hell. Saving the face of mathematics multi-valued logics, fuzzy sets, and the probability theory missed in fact the essence of the problem because being in front of ontological inexactness they tended to describe it still exactly.

Assumption for uncertainty occurred first in physics and then spread over to the rest of the sciences as a complementarity principle. The habitual determinism happened however to be deep-rooted since its shelters are still persistent and a hope for hidden parameters continues to nourish the adherents of the classic paradigm. The mathematical way of thinking remains, with Kant’s blessing, the measure of scientific value. “The genotype may be compared to a system of axioms” as was declared by C.H.Waddington in 1966 [2], despite the fact that mobile genes had been already discovered 15 years ago ([3], p.88). While mathematicians were taking pains to comprehend all the illusionary of perfect completeness and absolute exactness, physicists and biologists went on producing closed models. And the humanities, following the natural sciences, commenced to construct, with a great deal of enthusiasm, artificial classifications, artificial languages, artificial intellects. Striving for unambiguous certainty, unconditional objectivity, limiting completeness of description the traditional science digressed from the reality of life with its flexibility, openness, free will. It turned out bankrupt in the face of the global crisis being unable to predict or solve urgent problems. The crisis makes one to accept that a different methodology and a new paradigm are needed for studying viable, organic, developing objects.

Lagging behind life the science still tries to perceive current changes. Philosophers who comprehend the urgency of revising the notion of rationality [4] understand now that they will be forced to deal with “a new type of complexity related to human intuition and human emotions” ([5], p.70). The ideal of completeness yields to that of wholeness [6]. Hence the requirements that can be placed on mathematical methods are to retain a sufficient exactness without destroying the wholeness of an object under investigation.

A demand for *humanizing* mathematics arises as a *heresy*. When A.Grothendieck realized that passion for mathematics took one away from reality and alienated from an enigma of the human soul ([7], p.99), he left the Bourbaki’s group. R.Penrose having established the non-computability of consciousness asserted that a new physics was badly needed [8]. R.Hersh insisted on including mathematics into the human culture [9].

In reply to Cohen’s ironic words that “life might have been much more *pleasant* if the Hilbert’s programme were not shocked by Godel’s discoveries”, A.Parshin, on reflection, declared that “if it were not the Godel’s theorem, life all the same would have not been more pleasant – it simply could not have been in existence” ([10], p.94). He, furthermore, added that “Godel’s theorem should exist in biology as well thus showing the impossibility of a complete description of organisms in purely genetic terminology” (p.109).

The idea of *soft mathematics* gains more and more its attractiveness. Humanitarization of mathematics is being discussed as a tendency in development of modern science [11]. Association of mathematics with *soft* *sciences* looks like a promising prospect behind the Pillars of Hercules of rigid canons [12]. *Soft calculus* is considered to be a marker of a new paradigm [13]. And even strict V.Arnold ventures to speak about hard and soft mathematical models [14].

As it often happens, the new sought is found in the well-forgotten old.. One must only look at it with a renewed glance. Patiently, like Cinderella, asymptotic methods have been toiling hard for the last age in the kitchen of classical mathematics humiliated by their inferiority complex. Although justice was done to them as a kind of art [15, 16] they were never allowed to enter the true science – intrinsic inexactness was an obstacle. And, suddenly, in the end of the 20th century this *lame duck* grows invincibly into a *fine swan* of the new paradigm [17, 18]. It possesses all what has been sought: softness, flexibility, openness. And evaluation of exactness to be controlled. It is true, the exactness is always limited in the finite domain. But this is an inevitable retribution for the retention of wholeness embodied in the balance of exactness, locality and simplicity.

Substantially, asymptotic mathematics fits well into synergetics [19] which appears to be a most responsible mediator of the new paradigm in a very wide sense of the word. They are linked by the dynamism of methods directed to life: from limiting to approximating, from being to becoming, from completeness to wholeness. The Greek term *asymptotos* means *noncoincident* what stresses out that asymptotic approximation does not convert into coincidence. Similarly, wholeness does not convert into completeness.

Classical mathematics still remains an ideal identified with the *Absolute* the way to which is “through comprehension of the world harmony expressed as the harmony of numbers” [11]. But man is finite and any attempt to cognize, restrain, conquer the infinity results in paradoxes [20]. Refined models beget monsters of formalism no less dangerous than chimeras of mysticism ([21], p.199). On his endless path to God’s truth man needs a support of the human truth [22]. It is not unambiguous as well as, for instance, the notion of exactness ([23], p.7). Furthermore, it is restricted by scales of the human world. Thorough boundaries occur in the conception of continuous medium, in fractal geometry, in cosmology [24]. The limiting extrapolation assumed in classical mathematics takes away from the life natural world into an artificial space of abstractions. Asymptotic mathematics is free from this obligation. It has the Planck’s restricted border.

The system triad *exactness-locality-simplicity* that defines asymptotic mathematics proves to be not accidental. Its semantic archetype *ratio-emotio-intuitio* is present in all integral finds of the humanity and lies at the basis of modern trinitarian philosophy [25].

## References

1. Hilbert D.Foundations of Geometry. M. 1948 (Russian translation from German).

2. Towards a Theoretical Biology. I. Prolegomena. Ed. by C.H.Waddington. Birmingham: Aldine Publ.Co. 1968.

3. Golubovsky M.D. The Century of Genetics: Evolution of Ideas and Concepts. SPb: Borey Art. 2000.

4. Rationality at the Cross-roads. M. 1999 (In Russian).

5. Mainzer K. Thinking in Complexity: The Complex Dynamics of Matter, Mind, and Mankind. N.Y.: Springer. 1996.

6. Barantsev R.G. From Completeness – to Wholeness // Problems of Civilization. St.-Petersburg University. 1992. Wholeness versus Completeness // Russian Philosophy and Modern World. SPb. 1995 (In Russian).

7. Grothendieck A. Recoltes et Semailles. Fatuite et Renonvellement. 1984.

8. Penrose R. Shadows of the Mind: A Search for the Missing Science of Consciousness. Oxford. 1995.

9. Hersh R. What is Mathematics, Really? Oxford. 1997.

10. Parshin A.N. Reflections over the Godel’s Theorem // Voprosy Filosofii, 2000, No.6, p.92-109 (In Russian).

11. Panov M.I. Humanitarization of Mathematics is a Tendency of Development of Science in the 20th Century: (Is it Possible to Consider Mathematics as an Alloy of Culture, Philosophy, Religion?). A Review // RJ, Ser.3. Philosophy, 1999, No.6, p.21-30 (In Russian).

12. Devlin K. Goodbye, Descartes: The End of Logic and the Search for a New Cosmology of the Mind. N.Y. 1997. Davis Ph. Beyond the Pillars of Hercules: Soft Mathematics // SIAM Rew. 1998, No.6.

13. Kondratiev V.G., Solodukhina M.A. Soft Calculus as a New Paradigm. A Review // RJ, Ser.3. Philosophy, 2000, No.3, p.34-41 (In Russian).

14. ArnoldV.I. “Hard” and “Soft” Mathematical Models // Priroda. 1998. No.4, p.3-14 (In Russian).

15. Kruskal M. Asymptotology // Math. Models in Phys. Sci. N.J. 1963, p.17-48.

16. Babich V.M., Buldyrev V.S. An Art of Asymptotics // Vestnik Leningr. Univ. 1977, No.13, p.5-12 (In Russian).

17. Barantsev R.G. On Asymptotology // Vestnik Leningr. Univ. 1976, No.1, p.69-77 (In Russian). Asymptotic versus Classical Mathematics // Topics in Math. Analysis. Singapore e.a. 1989, p.49-64. Prospects of Asymptotic Mathematics // Math. Analysis. Problems of Theory, History and Methodics. L. 1990, p.108-120. Inevitability of Asymptotic Mathematics // Mathematics. Computer. Education. 2000. V.7. Part 1, p.27-33 (In Russian).

18. Andrianov I.V., Manevich L.I. Asymptotology: Ideas, Methods, Results. M.: Aslan. 1994 (In Russian).

19. Barantsev R.G. Synergetics and Asymptotics // Polignosis. 2000. No.4, p.135-137 (In Russian).

20. Infinity in Mathematics: Philosophical and Historic Aspects. M. 1997. (In Russian).

21. Swassjan K.A. Becoming of European Science. Erevan. 1990 (In Russian).

22. Zakharov V.D. Truth in Natural Sciences. I. Mathematics and Truth // Vestnik RUDN. Ser. Philos. 2000. No.1, p.92-105 (In Russian).

23. Dialectics of Exact and Inexact in the Modern Scientific Knowledge // Voprosy Filosofii. 1988. No.12, p.3-42 (In Russian).

24. Barantsev R.G. Conceptions of contemporary science: experience of integral approach. SPb. 2001, 80 p.

25. Barantsev R.G. Definition of Asymptotics and System Triads // Asymptotic Methods in the System Theory. Irkutsk. 1980, p.70-81. Uncertainty Principle in Asymptotic Mathematics // Perturbation Methods in Mechanics. Irkutsk. 1984, p.107-113. System Triad is a Structural Cell of Synthesis // System Studies. M. 1989, p.193-209. Becoming of a Trinitarian Paradigm // 1st Russian Philos. Congress. SPb. 1997. V.3, p.291-294. On Trinitarian Methodology // Between Physics and Metaphysics: Science and Philosophy. SPb. 1998, p.51-61 (In Russian).

#
R. G. Barantsev

Saint-PetersburgUniversity