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Philosophy of Mathematics

Reflexive Substantion of an One-Way Ascendancy of Mathematics over Ethics

Krassimir D. Tarkalanov
Plovdiv University, Plovdiv, Bulgaria

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ABSTRACT: Russell and Popper are concordant with Plato with respect to the independence of mathematics upon the sensations. Beth shares the opinion of the complete independence between the world of science and mathematics and that of psychology. Essenin-Vol'pin's opinion is of an ascendance of ethics and jurisprudence over mathematics. For the first time, the position of Plato, Russell, and Popper are substantiated in this paper through Hegel's reflexive natural scientific method. The external activation of numbers into interaction through arithmetical operations, adopted by him, has been taken as a basis of this substantion. This is the reason why mathematical rules of reasoning are exact-they represent a pure product of the 'third world.' The rules of ethics and the related humanities are their reflective approximate reverberations. Ascendancy of the rules of such types of science over mathematics is impossible due to the irreversibility of the reflexion.

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The problem of the interaction between the psychical and the thinking worlds as reverberations of the material one has been treated much earlier by ancient philosophy. Plato excludes any dependence of mathematics, it being the most brilliant representative of the mental world, of the sensations. Russell [1] (I. pp. 237-238) is concordant with the above. He considers that the mathematical truth is "applicable solely to the symbols," the symbols being "words," that "do not signify anything in the real world." Thus, the correct opinion, pointed out, remains unsubstantiated, since nowhere is it related to the philosophical categories.

In the substantion, offered by this paper, we proceed from the assumption that the variety of the mathematical symbols, at any rate, is reduced to and ensues from the aim: namely-to study the quantitative characteristics of "the qualities" from "the being." That connects the mathematical symbols with "the real world," i.e.-it reveals the possibility of a substantiating, since those characteristics interact. Following the construction of the foundations of mathematics, we should agree that the interaction among its concepts (i.e. the rules of the mathematical reasoning) is reduced to the interaction among the natural numbers. Hegel defines them reflexively [2], [3] ensuing from "the qualities" of "the beig" which (conversely) indicates that the mathematical truth denotes something "in the real world."

Russell has pointed out that "Hegel's philosophy is very difficult-he is ...the most difficult to grasp of all great philosophers" [1] (III., p. 337), thus associating him with the philosophers "willing to spread confusion in mathematics" [1] (III.,p. 460). Houwever, we shall show here that his reflexive definitions, pointed out, do, indeed, connect the mathematical truth with the real word and serve as a substantiation of an one-way ascendancy of mathematics over ethics and the related humanities. Moreover, we shall use for that purpose the clarity introduced by those definitions of Hegel in the way of activating the numbers into interaction. He adopts the concept that "the essence is ... a first negation of the being" [2] (p. 10) and that the reflexion is a natural scientific method of increasingly deep penetration into the essence of the phenomena studied. Reflexion is an infinite series of consecutive negations starting from the quality studied. The following formulation of Hegel has given us justification for such a brief descriptive definition of it, namely: "The determinativeness of the formation of the being has for its justification the being; and it is in correlation with the other. ... Therefore, the other here is not being with a negation or with a limit, but a negation with a negation ..." [2] (p. 18). The natural scientific character of the reflexion method for studying "the real world" has been substantiated by means of applying to the primary philosophical category "quality" of the primary "operator" (dialectical) "negation." The infinity of the reflexive process has been substantiated by the law of negation of the negation and simultaneously the irreversibility of the consecutive (every two) steps of the reflexion too. In addition to the above Hegel adduces many other arguments, not quoted here, in order to support and substantiate those ones as well as other significant traits of its characteristics.

Thus, reflexion is a method of forming the system of the concepts in their "irrepressible pure course which does not take in anything from the outside" [3] (p. 49). Ensuing from a given "quality" and following precisely the same "pure course" (since he uses only the primary "operators" for "negation" and "taking a unity") Hegel has consecutively defined the concepts of limit, pure quality, quantum, number [3] (p.145, p. 225, pp. 49-50, pp. 250-252). The quantum corresponds to a concrete number of concrete objects. Due to the unusually wide variety of such objects as well as to the sufficiently appreciable deviation from their concrete qualities in the course of reflexion (a repeate application of "negation" and "taking a unity") the quantum is a rather vast abstraction used for studying initial qualities. When successively applying the primary "operators" Hegel has defined the even more abstract concept of number.

We shall here make a slight deviation: the definition of a number from a quantum finds its respective dialectical reverberation in the application by Frege of the abstraction principle in the settheory. As Russell especially remarks in "Philosophy of the Logical Analysis" [1] (III., p. 462) that means the transition of all sets with an equal number of elements to the number of these elements. Thus we denote the dialectical substantion (given by Hegel about 70-80 years earlier) of the absraction principle formulated in an exact way in mathematics much later.

We go back to Hegel's definition of a number, already treated, as an object of mathematics as well as a means of studying the qualities. Immediately after that we point (this time with arguments and motives well grounded and adduced) that this definition represents a concept which is quite distant from the initial "qualities." Therefore, we are justified to consider the number from the objective "third world" [4] as an objective content of the thinking and the knowledge. Hegel has given a well-founded and substantiated answer to the question how should we study this object as well, i.e. (in essence) what is the object of mathematics and what are its rules of reasoning. Hegel writes:"Arithmetic studies the number and its figure, or, more precisely, arithmetic does not study them-it operates with them. Since the number represent an inert precisation, an indifferent one, it should be put in correlation one to the other from the exterior. The ways of such a transition are, in fact, the types of calculus." [3] (p. 254). In other words, the rules of reasoning in mathematics should be reduced to forms of studying the rules of the arithmetical operations with the numbers. Due to their "indifference" and their external reduction into correlation one to the other, the study of the rules of such correlation represents a pure product of the objective thinking process, i.e.-of "the third world."

In every science the rules of reasoning have been developed and formed as objective ones. The difference in their application ensues from the degree of an objectivity of the science's concepts. As we pointed out, being the most abstract, the mathematical concepts are, thus, the most objective. This abstract remoteness of theirs (with objective retained intact, however) excludes the possibility of the concrete manifestations of the qualities (which can affect the sensations and emotions) influencing the rules of reasoning in mathematics. Conversely, those rules are a product of the pure objective thinking from "the third world." Its objectivity means synonimity and, therefore, the rules of reasoning in ethics and in the humanities represent an aproximate reverberation of mathematical ones. In this way the ascendancy of mathematics over those sciences has been reflexively substantiated, as is indicated both in title of the present paper and in its summary immediately following.

We have one task left, namely-to substantiate in the same reflexive way its unidirectionality (its one-way character). It is necessary that we should do that due to the emergence of different opinions about that. Beth [5] (p. 55, where some of his works are quoted including one written with Piagét as co-author) opposes "all kinds of variants of psychological traits in logic and mathematics as well as of logical ones-in psychology." Essenin-Vol'pin [6] (p. 104) writes: "I cleam that the reverse attempt is much more feasible, namely: to base mathematics if not strictly and solely on ethics, at least on those logical principles which in those ethical (as well as juridical) sciences aim at elaborating a system of rules...." In [6] again he points out that "the searches for truth has always been connected with the problems of justice" Our reflexive substantiation contradicts a part of Beth's position: concerning the repudation of logical traits in psychology. This substantiation confirms his position, namely "against all variants of psychological traits in logic and in mathematics" and rejects completely Essenin-Vol'pin's opinion of mathematics being based on the logical principles of ethics. Actually, we shall add two reflexive arguments to the substantiated elaboration of the rules of reasoning in the humanities as an approximation to the mathematical ones as already pointed out above. One of the arguments is the irreversibility of the consecutive (every two) steps in the natural science method of reflexion, already pointed out, while the second one (which, to a certain extent, represents a corollary to the first) is the indicated elimination of the possibility of the concrete manifestations of the qualities, which can affect the sensations and emotions, to affect over the rules of reasoning in mathematics. In other words, the logical principles in mathematics cannot be based on those belonging to the sphere of the humanities, since the principles in the latter are being influenced by their concepts which, however, cannot be fully separated from the sensations and emotions. Their rules are approximations of the mathematical ones, obtained due to the influence of their concepts which do not belong to "the third world." That is quote another question if those concepts could ever remain entirely objective since it seemed impossible for the subject to disregard and not take into consideration the emotions when forming emotional concepts. Therefore, it is not justified always to connect "the search for the truth" with "the problems of justice." We have already mentioned Russell's opinion stating that mathematical truth is "applicable solely to the symbols" (as well as the content of those symbols). The justice, however, in the humanities is not symbolical.

Thus, our object has been reached and our task completed: a reflexive one-way ascendancy of mathematics over ethics has been substantiated, completed to the opinions quoted for which no substantiation through the principal philosophical categories has been achieved. We have simultaneously substantiated the inexpedience to construct completely axiomatically ethics and their related sciences, as Spinoza and Christian Wolf had done. In that connection Hegel writes somewhere else that "The so-called geometrical method of pholosophizing leads to extreme dogmatism, to suppressing dialectical thinking... That barbaric pedantry or rather pedantic barbarism... would inevitably lead to a point at which the geometrical method would be denied all confidence at all." Of course, the reflexive method itself does not exclude, but rather demands, an improvement and perfection of the concepts and rules of reasoning in the humanities. Hegel, however, defends an inexpedience of any striving after complete axiomatisation.

Following the clause of the philosophical essence of the papers submited we do not treat some more special questions of philosophical content, demanding a thorough inquiry into mathematics and physics however (a part of such questions might be treated elsewhere by us). We should think, however, that we cannot but mention by way of an information some of them, mainly the ones connected with the content considered as well as the works quoted. One of those questions is that of the ultraintuitionistic programme [6] concerning the foundations of mathematics and natural science thinking, connected with the logicism of Russell [1] as well as the intuitionism in mathematics [7]. We would not aim at discussing such a programme. Our point would be, however, rather to stress the fact that Hegel had got ahead of his time by a whole century as far as the intuitionistic trend in mathematics is concerned. Hegel in [2] after a whole series of reflexive definitions had come to the point and had substantiated the opposites or, more particularly, the opposite numbers. The above statement had been included in his Remark 2. concerning the Law of the Excluded Third [2] (p. 63-64). We can trace there a dialectical substantiation for not accepting by Hegel the Law of the Excluded Third a whole century before the intutionistic trend in mathematics would repudiate it through the irreversibility of the consecutive (every two) steps of the reflexion. Two consecutive dialectical negations do not lead to the initial quality. Some internal motives and stimuli for not adopting that Law had emerged quite independently in quantum mechanics (Max Born, My Life and My Views, N.Y., 1968). It can be said as well, that his reflexive method of knowledge from [2] had "anticipated" the independent existence of opposite geometries, which Lobachevsky, Bolyai, Gauss etc. had realized (driven and stimulated by internal mathematical motives). On his part Lobachevsky had applied the complementation principle (of the different theories-geometries) again a whole century before Niels Bohr offered that principle to be applied in optics (rf. to Max Born's book).

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[1] B. Russell, History of Western Philosophy and its Connection with Political and Social Circumstances from the Earliest Times to the Present Day, George Allon&Unwin (Publishers), Ltd. London, 1945.)Bulgarian translation:

I. V.1. Ancient Philosophy, Chr. Botev Publishing House, Sofia, 1994.

II. V.2. Medieval Catholic Philosophy, Chr. Botev Publishing House, Sofia, 1995.

III. V.3. Modern Philosophy, Chr. Botev Publishing House, Sofia, 1996.)

[2] G.W.F. Hegel, Science of Logic, v.2, Moscow, 1971. B. second "Doctrine for Substance" (in Russian).

[3] G.W.F. Hegel, Science of Logic (Bulgarian translation: First part, Objective Logic..., Sofia, 1965.).

[4] Popper K.R., Epistemology without a Knowing Subject // Van B. Rootselear and J.F. Staal (eds.), Logic, Methodology and Philosophy of Science, III. Proceedings of the Third International Congress for Logic, Methodology and Philosophy of Science, Amsterdam, 1967. North-Holland, 1968.

[5] V.P. Philatov, Malahov V.S. (compilers), Philosophical Dictionary, Modern Philosophy, 19-20 c. (Second edition), Publisher "Gal-iko," Sofia, 1996 (ISBN 954-8010-54-2, in Bulgarian).

[6] Essenin-Vol'pin A.S., On the Anti-traditional (Ultraintuitionistic) Programme of the Foundation of Mathematics and Natural-Science Thinking, Vopros'i philosophii, Nr 8(1996), 100-136 (in Russian).

[7] A. Heiting, Ituitionism (An Introduction), Amsterdam, 1956 (Russian translation: "Mir " Press, Moscow, 1965.).

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