Mathematical Logic and the Ideal of Apodictic Science · 2009-12-20 · Metalogicon (2000) XIII, 1...

Metalogicon (2000) XIII, 1

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Mathematical Logic and the Ideal of Apodictic Science

Antonino Drago

1. Aristotle's ideal for a scientific theory.

In present scientific education the deductive scheme is considered the best presentation of a scientific theory. Although any scientific researcher knows that his activity follows unforeseable paths, the deductive way of presenting the results constitutes almost an obligation which rare persons avoid, at the cost of unfavourable feelings in the audience.

The origin of this habit is recognized in the greatest scientific works that followed deductive scheme; it is enough to recall Newton's Philosophiae Naturalis Principia Mathematica (1687) and, before him, Euclides' Principles. Actually, the latter book is contemporary to Aristotle's suggestion in Second Analytic for an ideal of a scientific theory. Beth translated this ideal in modern terms.1 Since Galilei this ideal left space to experimental evidence too, without changing its inner structure. Then Hilbert improved the deductive method and suggested a program for considering it as the final method of mathematics.

Mathematical logic is born in opposition to some philosophers - e.g., Hegel – that emphasized by means of common language a radical alternative to the deductive way of reasoning.

It began by formalizing the logic of natural language as a mathematical language. After a starting period, Peano, Frege, Hilbert, Russell eventually suggested formally deductive systems

1 E.W. Beth: Foundations of Mathematics, Harper, New York, 1959, ch. I, 2; in particular, p. 47.


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of mathematical logic; in their hope, any ambiguity of common language was avoided and moreover the appliucation of deductive scheme ascribed to these systems a full dignity as autonomous scientific theories. By following this trend, Frege planned to found the whole mathematical theory on the set of axioms of mathematical logic. Then, Russell's antinomies stopped Frege's program but did not shake Hilbert's confidence in axiomatic; whose program in its turn was considered as the firm basis for assuring certainty to the mathematical thought. Not even the shock of Gödel's theorem moderated this confidence in the large majority of subsequent mathematicians - as present situation shows. 2. The long search for an alternative to Aristotle's ideal

As a fact, some authors attempted to find out an alternative to Aristotle's ideal. The most impressive writing is Lukasiewicz' solemn speech about his inner motivation for producing a new logic, i.e. many-valued logic. The Polish logician “confesses” to be fighting “a spiritual war”, that is an “ideal struggle for the liberation of the human spirit” from the “logical coercion given by Aristotle's science as a system of principles and theorems connected by logical relationships. The concept came from Greece and has reigned supreme... The creative mind revolts against this concept of science... A system of three-valued logic... destroys the former concept of science, based on necessity... Logic is a free product of man, like a work of art”.2

2 J. Lukasiewicz: Selected Works, North-Holland, Amsterdam, 1970, p. 84-85.


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To my knowledge, the subsequent literature does not mention Lukasiewicz' attitude. Rather, it was Gödel's theorem that seems to have elicited a new search; indeed, at least van Heijenoort stated that owing to Gödel's theorem the deductive scheme is unsatisfactory for representing a mathematical theory.3

Beth also stressed several times a similar view in more moderate but more accurate terms. Without quoting Lukasiewicz, he maintained that Aristotle's ideal constitutes a very bias to present-day, theoretical activity in science, since “there are two types of science..., rational science” and “empirical science”, as well as there are “two fundamentally divergent types of scientific method”.4 He scrutinized past theories for achieving an example of an alternative structure but he was able to recognize little more than a philosophical divergence, without sketching an alternative scheme.

After two decades a Symposium puts the same problem of an alternative to Aristotle's ideal; a solution was looked for by investigating the theories suggested by several scientists of both ancient and modern times.5 However, no conclusive result was achieved. Recently, more accurate analyses on Aristotle's conception of a scientific theory stressed that the great philosopher actually suggested more than apodictic science; when he wrote his Physics, the scientific theory was organized through no axioms, but several problems to which he offered solutions.6

3 J. van Heijenoort: “Gödel's Theorem”, in Encyclopedia of Philosophy, Mac Millan, 1967, p. 480. 4 E.W. Beth: Natuurphilosophie, Goerinchen, 1948; “Towards an up-to-date philosophy of natural sciences”, Methods, 1 (1949) 178-185; Les fondements logiques des Mathématiques, Gauthier-Villars, Paris, 1958; Foundations of Mathematics, op. cit.. 5 J. Hintikka, D. Grunder, E. Agazzi (eds.): Pisa Conference, vol. II, Reidel, Boston, 1981. 6 G. Cambiano: “Il metodo ipotetico e le origini della sistemazione euclidea della geometria”, Riv. Fil., 58, (1967) 115-149; V. Sainati: “La teoria aristotelica dell'apodissi”, Teoria, 10, (1990) 3-47. A similar attitude for founding a theory on problems was followed by Archimedes.


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Among contemporary logicians, Lorenzen distrusts the deductive ideal; he quotes Pascal as supporting a science founded upon common language only.7

A further contribution comes from a historical analysis on the history of logic. Lewis first in 1918 and then Nidditch, Wang and Thiel stressed that there exist two different traditions in logic;8 Nidditch qualified them as the “mathematics of logic” vs. “the logic of mathematics”, while Thiel stressed “the dichotomy of a Boole-Peirce Schroder...vs. a Peano-Whitehead-Russell tradition”. More detailed was K. Lorenz's paper, whose titled suggests an alternative in a sharp way: “Rules vs. Theorems”.9 Although these words refer merely to statements or rules, the arguments of the paper actually concern the kind of organization of a logical theory; either as a deductive scheme or as aimed to offer a method for solving problems. In particular, he recalls that some historians of logic remarked that in Aristotle's writings the nature of syllogism results to be ambiguous, inasmuch as it may represent at the same time rules and theorems. Moreover, in Topics(100a18) Aristotle's definition of a logical theory is at variance with his celebrated ideal: “...to find a method by which... we shall avoid saying anything self-contradictory” (here and in all the following quotations, emphasis is added for reasons presented at p. 23).

As a conclusion, there is evidence for a fundational problem about the kind of organization of a logical theory. This problem is

7 P. Lorenzen: Constructive Philosophy, U. Massachussets P., 1987, sect. 14. 8 C.I. Lewis, C.H. Langford: Symbolic logic, Dover, 1959. The references to Nidditch and Wang are in C. Thiel: “Scrutinizing an alleged dichotomy in the history of mathematical logic”, VIII LMPS, Moscow, 1989, vol. 3, 254-255. 9 K. Lorenz: “Rules vs. Theorems”, J. Phil. Logic, 2 (1973) 352-369. A similar, more vague analysis is given by J. van Heijenoort: “Logic as Calculus and Logic as Language”, in R.S. Cohen, M.W. Wartowsky (eds.): Boston St. Phil. Sci., 3, 1967, Reidel, 440-446. According to him, there was an opposition between a Frege-Russell's axiomatic approach and a Löwenheim (-Boole-Schröder) approach”; yet “the opposition is... dissolved... during the twenties” (p. 444).


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more deep and more comprehensive than the problem of finding out a suitable - either intuitive or mathematical - notion to be put as the one basic notion of a logical theory, or even the problem of making clear the meaning of some basic connectives. On the other hand, this problem, being a general one, is not a merely philosophical problem because it pertains properly to mathematical logic, inasmuch as any logician has to decide it when he is formulating a logical theory as a system. 3. A further historical analysis: Some proposals for non-apodictic logical theories One can improve the above analysis on the history of logic in order to discover more suggestions for an alternative scheme to the apodictic one. Various facts can be collected; I bound myself to emphasize those that constitute in my opinion the most relevant ones. I will list them in a chronological order.

An unfortunate fact for our effort for understanding the foundations of science was Leibniz's misfortune since the 18th Century. Really, he was the last philosopher who was able to constructively work in the foundations of modern science. In particular, his analysis on the basic ideas of human knowledge suggested to him two labyrinths of human reason, i.e. the notion of infinity and the opposition between law and free will.10 This opposition represents - according to the above-mentioned Lukasiewicz' words - what constitutes in subjective and moral terms a conflict between two attitudes, i.e. to follow an apodictic ideal for a scientific theory - as a theory is taught to pupils - and

10 G.W. Leibniz: Nouvelles lettres et opuscules inédits, Paris, 1857, 178-185.


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to search for a new scientific method capable to solve a crucial problem - as it is pursued by a working scientist -. In correspondence to this opposition Leibniz enounced two basic logic-philosophical principles, i.e. the principle of non- contradiction and the principle of sufficient reason.

The above conflict was then qualified by D'Alembert in theoretical science. He stressed that it is hopeless to complete a theory by improving a deductive theory, since such a scheme “leaves always some holes”.11 Rather, he suggested, one has to start from empirical facts and then to build the theory according to them. Although his criticism was very wise (Gödel theorem is preconized by it), unfortunately his constructive suggestion was not enough for sketching an alternative organization of a scientific theory. However, D'Alembert and his followers reduced the status of a principle from a self-evident statement to a merely empirical statements. Unfortunately, this hint for an alternative was unfeasible by logicians.

A strong suggestion for an alternative just in logical theory came then from Condillac. His Logique

12 claimed to suggest an entirely new way of arguing (the book was offered as a gift by the French Directory to any pupil of École Polytécnique in 1796). As a fact, no principles (and not even definitions) are presented by this theory, which rather is organized on a problem, in particular on the problem of qualifying the notion of equality. Moreover, Condillac supported the view that through a suitable reform natural language will become a well-made language, so that its accuracy will be the same than that a formal language enjoys.13

Unfortunately, Condillac emphasis on natural language rather than on formalisms alienates him from the dominant tradition in mathematical logic. The subsequent historical trend in

11 J. Le Ronde D'Alembert: “Élémens”, in J. Le Ronde D'Alembert, D. Diderot (eds.): Éncyclopédie Française, Paris, 1744, vol. 17. 12 E. Bonnot de Condillac: Logique, Paris, 1784. 13 W.R. Albury: “The order of ideas” in J.A. Schuster, R.R. Yeo (eds.): The Politics and Rethoric of Scientific Method, Reidel, 1986, 203-226.


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logic was exactly the opposite one to his suggestions; i.e. logic was formalized as a mathematical language, notwithstanding some serious foundational problems. To my knowledge, Lewis and Longford only reported a paradox which lies at the bases of formal logic. They recognized14 that “... in the case of logic, it would appear that this procedure [the deductive method] involves a puzzle: unless logic is taken as granted, we can make no deduction; but it is logic which is to be deduced”(p. 115); on the other hand “to assume logical principles as principles of one's deduction, in order to deduce a body of logical laws from a few which have been postulated, is obviously a circular process...”(p. 116). “Thus... any attempt to develop exact logic by the usual methods of mathematical deduction in general, must involve some circularity”(p. 117).

Then, the authors think to solve the above paradox by following the “logistic method”; the rules in accordance with which proofs are given “are rules of more or less mechanical operations upon the symbols...”(p. 118). Apart the suspicious vagueness of “more or less”, it is clear that authors did not have in mind Gödel's results, which do not allow to assume the consistency of such a formal system of mechanical rules; thus, the paradox is still unresolved. 4. Instances of non-apodictical logical theories: From Boole to intuitionism

An accurate historical analysis shows more than mere proposals. Indeed, many historical facts contrasted the winning 14 C.I. Lewis, C.H. Langford: op. cit., pp. 116-117. See also C.I. Lewis: A survey of Symbolic Logic, U. California P., Berkeley, 1918, p. 355.


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trend established by mainly Frege's and Russell's works; even the starting theory of mathematical logic, i.e. Boole's one, and several new logical theories born by essentially lacking of formal principle-axioms. In the following, I will offer a list of all such theories, at the extent I was able to recognize them.

At variance with the present-day image of mathematical logic, Boole introduced his theory without any axiom-principle, rather by “principles that are to guide us....”.15 He puts a crucial problem, which in his times was considered as a metaphysical problem: “to investigate the fundamental laws of [the] operations of the mind by which the reasoning is performed” (p. 1). In particular, “Syllogism, conversion, etc., are not the ultimate process of Logic...” (p. 10). Rather, Boole followed the analogy, which he thinked to hold true inasmuch “as not to conflict with an established... result of observation” (p. 21). “Ultimate laws of Logic... forbid, as it would seem, the perfect manifestation of science in other form...”. All that led him to the statement “Nothing in the nature of Language prevents us [in order “to represent an idea”] from using a mere letter in the same sense” as algebra does (p. 26).

As a next instance of a non-apodictical theory is intuitionistic logic. Brouwer did not accept current foundations of both mathematics and logic. He wrote about the “unreliability of the logical principles”.16 In particular, the principle that at his times constituted the very distinctive mark of mathematical logic from current philosophical logic, i.e. the principle of the excluded middle, was rejected. Brouwer's logic presents no formalization, no organization of the theory, no assured principles. However, Brouwer characterized the nature of its principles. According to him, they are neither Platonist notions, nor first axioms of a deductive system; “logical principles are not [even] directories, but regularities” (p. 108). His notion of a logical theory is 15 G. Boole: The Laws of Thought, (1854), Dover, New York, 1950. About analogical reasoning, see also p. 402ff.. 16 L. E. J. Brouwer: “The unreliability of the logical principles”, (1908) in Collected Works, North-Holland, 1975, 107-111.


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empirical in nature – this is just the opposite one to the Aristotelian notion of a theory -, because “logic is an applied mathematics”,17 not an abstract theory; it “belongs to ethnography...”.18 By illustrating the intuitionism and in particular intuitionistic logic, he presents them through some fundamental “acts”, just for suggesting a method of research rather than an abstract theory.

Unfortunately, this empirical content is not evidentiated by Brouwer in a satisfying way; for ex., when he refers to logical sentences he says that “the meaning of a logical proposition is the “purpose” or “intention” of a mathematical construction satisfying “given conditions”,19 where nobody - if not someone already persuaded, e.g. an intuitionist -, agrees to attribute an empirical content to “purpose” or “intention”.

Who formalized first intuitionistic logic, A. Heyting, perceived a basic choice on the kind of organization of the theory. He advised that “It is to be remembered that no formal system can be proved to represent adequately an intuitionistic theory. There always remains a residue of ambiguity...”.20 Moreover, he emphasized a distinction between a "descriptive" function of the "axiomatic method" - according to which one can deduce hypothetically too - and a "creative" function of the same method - according to which one follows Platonist ideas - which is rejected by the intuitionism.21

Although unable to recognize an alternative, present-day illustrations of intuitionistic logic emphasize that “for

17 A. Heyting: L'intuitionnisme, Gauthier-Villars, 1955, Paris, p. 19. 18 L. E. J. Bouwer: op. cit., p.74. 19 A. Heyting: L'intuitionnisme, op. cit., p. 17. 20 A. Heyting: Intuitionism, North-Holland, 1966, p. 102. 21 A. Heyting: “Axiomatic method and intuitionism”, in Y. Bar-Hillel et al. (eds.): Essays on the Foundations of Mathematics, North Holland, 1961, 237-249, p. 239. His unability to be more accurate is in pag. 14 of Intuitionism, op. cit., where he states the induction principle by means of a “theorem” drawn from some basic properties of number theory.


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intuitionistic mathematics [and hence, logic too], in the tradition of Brouwer, informal rigor is the principal, perhaps the only source of the mathematical knowledge”.22 They appeal to an almost general use of "informal rigor" in order to build the theory, also because “it seems arbitrary to restrict mathematics, in a narrow formalist sense, to his subjective distillate...” (p. 835); but even more because: “Our experience in translations of one system in another ... shows that we have a good deal of freedom also in our interpretation of intuitionist logic” (p. 839). As a conclusion of some decades of research, the task is recognized as an unaccomplished one: “The proof-interpretation [of logic] provided at least an informal insight in the mysteries of intuitionistic truth, but it lacked the formal clarity of the notion of the truth in classical logic with its completeness property”.23

On the contrary, a clear-cut interpretation of intuitionistic logic has been suggested by Kolmogoroff in 1932, as an activity of solving problems.24 This interpretation disregards all philosophical premises as well as rigorous definitions of the basic notions (e.g. a problem); it is an objectivistic interpretation, inasmuch as it may be considered as representing e.g. the activity of a teacher in a classroom. According to it, any logical statement is related to a problem - i.e. to find out the means for solving a given problem. Moreover, any principle represents no more a source of theorems that may be achieved by deductively investigating the content of it, but rather a methodological principle by which one searches the solution of the problem at issue.

22 A. S. Troelstra, V. van Dalen: Constructivism - An Introduction, North-Holland, Amsterdam 1988, p. 834. 23 D. van Dalen: “Intuitionistic logic”, in D. Gabbey, F. Guenthner (eds.): Handbook of Philosophical Logic, Reidel, 1986, ch. III. 4, 225-339, p. 228-229. 24 A. Kolmogoroff: “Zur Deutung der Intuitionistischen Logik”, Math. Zeitfr., 35, (1932) 58-65.


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5. Instances of non-apodictic, logical theories: After the birth of intuitionistic logic

When he launched the above-mentioned “spiritual war”, Lukasiewicz introduced an entirely new mathematical logic in which the number of values is different from those of classical logic. The three-valued logic was suggested by referring to matrices only - a method he and Tarski exploited25 in the attempt of escaping from the apodicticity of laws and principles. In fact, the matrix method adequately specifies the logical connectives, yet it says nothing about the way of connecting them in a systematic theory, but by listing again the set of axioms - just the classical way.

Rather, a very interesting theory has been offered by Skolem in 1922.26 It was the birth of theory of recursive functions, whose relevance for the study of the logical foundation of mathematics cannot be underestimated. Although his exposition is qualified as "naive" by the editor of Skolem's collected works, today it has to be considered as a completely developed theory. Skolem was led by a methodological principle, i.e. "it is impossible to give a complete formalization of recursive arithmetic exclusively", because any more general formalization may be proved to be incomplete by adding a new principle of recursion. Moreover, since the title of his paper Skolem specified in a clear-cut way his main problem; i.e., to provide “a logical foundation for arithmetic without the use of apparent [infinite] logical variables” and then he stated it again in the middle of the paper (p. 304). In sum, the theory as a whole represents a method for accomplishing the 25 J. Lukasiewicz: Selected Works, op. cit., p. 131-152. See, in particular, footnotes no.s 4 and 6. 26 T. Skolem: “The foundation of elementary arithmetic established by means of the recursive mode of thought without the use of apparent vairiables ranging over the infinite domains” (1923) in J. van Heijenoort: From Frege to Gödel, Harvard, 1967, 302-333.


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solution of such a problem, instead to offer a set of theorems, all deduced from some axioms.

The complete abolition of the axioms-principles was accomplished by Jaskowsky27 and Gentzen28 when they introduced natural deduction. However, what is suggested by these authors is merely a local attitude about the single argument of the theory, not a holistic approach to the logical system.

A step in this last direction was performed by Lorenzen.29 His purpose was to reconstruct the whole system of a logical theory. However, he puts the problem not in his general abstract terms, but by means of a social situation; a dialogue - or conflict, or game - between two parties, possibly two persons. Put in this way, any logical statement, as causing a conflict between the two parties is a problem.

One more instance is to be remembered. Grzegorczyk puts the problem30 of “which connectives can be called implications, negations, conjunctions and disjunctions with respect to given rules R [i.e. relations between propositions] and assumptions X [i.e., some subsets of language L]”, where the last ones have to be considered as merely “the starting points for some argumentation”. By this approach the logical system starts from language and then directly reaches the connectives qualifying a particular logical theory - intuititionism, in this case.

For brevity’ sake, I leave out computability theory by Turing and Church, although it matters to our subject.

27 S. Jaskowsky: “Recherche sur le système de la logique intuitionniste”, Act. Congr. Int. Phil. Scient., Paris, 1935, VI, 58-61. 28 S. Gentzen: “Untersuchungen über das Logische Schliessen”, Math. Zeitfr., 39, (1935), 176-210; 405-431. 29 P. Lorenzen: op. cit., sect. 6. 30 A. Grzegorczyk: “An approach to logical calculus”, Studia Logica, 30, (1972) 33-43.


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6. A characterization of the alternative organization

All the above show that in mathematical logic too the kind of organization of the theory matters; moreover, it is relevant even beyond an early stage of the theory or lateral developments.

My previous studies31 on both mathematical and physical theories recognized an option on the kind of organization of the theory as a constitutive element of the very foundations of both mathematics and physics.

CHARACTERISTIC FEATURE

ARISTOTELIAN ORGANIZATION

PROBLEM-BASED ORGANIZATION

Philosophical origins Ideal truth (Plato) Objectivity (Aristotle

Principle of sufficient reason (Leibniz)

Philosophical tension From empirical data to self-evident principles (Euclid, Descartes, Newton)

From a metaphysical problem to a new scientific method (Galilei, Lavoisier, L. Carnot)

Metaphysics Self-evident principles, basic notions (Euclid, Descartes, Newton)

The given universal problem (Galilei, Lavoisier, L. Carnot)

Truth Principles, plus deductive method (Aristotle), plus expe-rimental evidence (Newton)

Common knowledge, plus inductive method (L. Carnot)

Evidence By self-evident principles (Aristotle)

By cumulated experience (Galilei, Lavoisier)

Beginnings of the theory Principles - axioms (Aristotle, Euclid, Descartes)

Methodological principles (Galilei, Lavoisier, L. Carnot)

Definitions Rigorous (Descartes) Irrelevant (Condillac) Characteristic step Double fault in the principles:

idealism and then a correc-ting realism (Berkeley)

Main problem through a double negated sentence (Lavoisier, L. Carnot)

Logic Classical : A or not-A (Aristotle)

Non-classical: not not-A ≠ A (Brouwer, Prawitz)

Method of development of the theory

Fixed, the deductive one (Aristotle, Euclid,

To be discovered, inductive (Galilei, Lavoisier,

31 A. Drago: “La chimica classica come esempio di teoria organizzata su un problema centrale”, in F. Calascibetta, E. Torracca (eds.): Atti II Congr. Naz. St. e Fond. Chim., Acc. Naz. Sci. XL, V, 12, Roma 1987, 315-326.


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Descartes) L. Carnot) Relationship mathematics-physics

Stated by self-evident principles (Descartes, New-ton)

Suggested by operative procedures (L. Carnot, S. Carnot)

Kind of mathematics Idealized (AI) (Newton) Constructive (PI) (L. Carnot, Dalton)

Mathematical objects Beings of reason (Newton, Leibniz)

Auxiliary variables (L. Carnot)

Mathematical technique Differential equations (Newton, Euler, Lagrange)

Symmetry (L. Carnot, Barut)

Consistency Absolute? (Hilbert) Relative only (Lobachevsky) Completeness Certain? (Newton, Hilbert) Possible (L. Carnot, S. Carnot,

Lobachevsky) Reality Applications of the theory More problems, machines Self-image A pyramid of

statements; truth-injection from the top (Lakatos)

Concentric problems, truth-expanding

Caricature Philosophical knowledge; "Operation was successfull, yet patient died"

Engineer's theories; immature stage of development; obscure content

In particular, anyone among Lazare Carnot's theories - geometry, calculus and mechanics - appears to follow a non-Aristotelian scheme that was qualified as “problematic” by the major historian of Carnot.32 A study on L. Carnot's works as well as on the works by S. Carnot on thermodynamics, by Lobachevsky on non-Euclidean geometry, by chemists and by several subsequent authors led me to recognize some specific features which sketch through a definite scheme the wanted alternative to that Aristotelian organization which was then improved by Hilbert in the formal axiomatic and currently is meant as the deductive method.33 A table summarizes these features.

One can give reason of a greater difficulty than in mathematics and physics for recognizing in mathematical logic an

32 C. C. Gillispie: Lazare Carnot Savant, Princeton U.P., Princeton, 1971, p. 87. 33 A. Drago: “In Gödel's theorem a consequence of the two kinds of organization of a scientific theory?”, in Z.W. Wolkowski (ed.): First International Symposium on Gödel's Theorems, World Scientific, Singapore, 1993, 107-135.


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alternative to the Aristotelian ideal. Here, a crucial role was played by Lewis and Langford's paradox. The strong Aristotelian tradition monopolizing the minds of logicians felt that either the paradox forbids for the eternity any formalization of logic, or logic has to be formalizable in some way and then Aristotelian organization may be confirmed. Since mathematical logicians decided that the paradox had to be overcome by an authoritative move - as logistic school did - no more they could be aware of an alternative to their a priori scheme, but only of mere variations of it. 7. A program for refounding mathematical logic

From the above elucidation of the alternative organization to the Aristotelian ideal, one can venture to suggest a reformulation of mathematical logic by following for the first time in a rigorous way the above recognized, alternative organization. One can hope that this reformulation will structure mathematical logic in a consequent way as much as the deductive organization does; moreover, it may reveal unrecognized features of this field of research, till now obscured by the pervasive role played by deductive method. Owing to the lack of space, I bound myself to suggest some hints for such a reformulation.

As first, let us remark that our discovery of an option at this high level of generality enlightens Hilbert's axiomatic. This one surely represents a decisive step for a better understanding of the foundations of science; its great merit was to start a philosophy of mathematics in more general terms than some basic notions (e.g. set, intuition, etc.) or even some theoretical structures (Bourbaki). Yet, its axiomatics is only one of two choices. His arrogance consisted in its claim to be able to cover adequately the whole body of mathematical and logical theories. The failure of Hilbert's axiomatic which occurred relatively late was caused just by its


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partiality w.r. to the real content of a scientific theory; that is the very meaning of Gödel's theorem - as van Heijenoort stated.

It was scarcely noticed what I call - by following Berkeley's criticism to Newton's calculus - the double fault of Aristotelian organization. The first fault consists in idealizing a principle at a so high level of generality to reach metaphysics (in Berkeley's criticism, the idealization of real numbers till metaphysical entities - as dx is). The second fault is then necessary in order to come back to real world (in Berkeley's criticism, the suppression of dx

2 as if it was null).34 One can see that Lewis and Langford's paradox on foundations of logic - as quoted in § 3 - is an instance of this double fault; a fault comes in when one idealizes logic in a set of a priori axioms and then a second fault occurs when, in order to come back to reality, one refers the control of arguments to a little set of mechanical rules.

By improving Feyerabend's and Kuhn's vague definitions, I define incommensurable two theories that differ in their kind of organization. Indeed, in a deductive theory there are plain problems only to be solved in a deductive way, never there is a universal problem; on the other hand, when the scientific method is not given a priori - as the received deductive method is -, but has to be found out as a new achievement, the role of an axiom is devoided of relevance.

As a consequence, radical variations in meaning of basic notions occur. No surprise if the debate on the meaning of logical connectives was so long and so inconclusive w.r. to the original aim of looking for the very foundations of logic. Even to bound logic to propositional calculus only - and so to avoid the “labyrinth” of the kind of infinity - was not enough to this philosophical approach for obtaining a common agreement on the foundations of mathematical logic. No surprise if the mere

34 A. Drago: “A characterization of Newtonian Paradigm” in G. Debroeck, G.B. Scheurer (eds.): Newton's Philosophical and Scientific Legacy, Kluwer Acad. P., 1988, p. 239-252.


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propositional calculus applied to a physical theory - quantum mechanics - led this field of research in a modern “labyrinth”.35

The crucial variation in meaning concerns first of all principles. In an Aristotelian organization they are axioms-principles; in the alternative organization they are heuristic, or better, methodological principles. All that agree with Leibniz suggestion of two basic principles of our reason, i.e. the principle of non-contradiction - the typical principle of a deductive scheme - and the principle of sufficient reason - the typical principle of a heuristic inquiry.36

Moreover, in all books originating the above-mentioned mathematical and physical theories of two centuries ago, the main problem of any theory is stated by means of a double negated statement which cannot be translated in a positive statement for the lack of scientific evidence. As an instance, in L. Carnot's mechanics inertia principle is the following one: “When a body is at rest it cannot change its state by itself (=if not by other bodies)...”.37 On the contrary, Newton's previous version appears to be a positive statement lacking experimental evidence inasmuch as its content is generalized to all bodies and to metaphysical wills: “Any body perseveres in its state of rest...”.38 35 B. von Frassen: “The Labyrinth of Quantum Logic”, in R.S. Cohen, M. W. Wartowsky (eds.): Logical and Epistemological Studies in Contemporary Physics, Reidel, Boston, 1974, 176-209. 36 W. G. Leibniz: La monadologie, Delagrave, Paris, 1970, 75-76. By taking in account the option on the kind of organiztion, one can even fulfill Leibniz' program for a “Scientia Generalis”: “Leibniz' <Scientia Generalis> reinterpreted and accomplished by means of modern scientific theories”, in C. Cellucci et al. (ed.): Logica e Filosofia della Scienza. Problemi e Prospettive, ETS, Pisa, 1994, 35-54; “The modern fulfilment of Leibniz’ program for a Scientia Generalis”, in H. Breger (ed.): VI Int. Leibniz Kongress. Leibniz und Europa, Hannover, 1994, 185-195. 37 L. Carnot: Principes fondamentaux de l'équilibre et du mouvement, Deterville, Paris, 1803, p. 49: “Un corps, une fois mis en repos ne sauroit en sortir par soi-même…”. 38 I. Newton: Philosophiae Naturalis Principia Mathematica, London, 1678, p.12.


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The same difference can be recognized in the well-known debate between Poincaré and Hilbert about the mathematical principle of induction. The former one maintained that there are two versions of it, a formal one and an intuitive one, being the latter one irreducible to the former version.

Actually, at least one time Poincaré stated the above principle in the following way:

(NC(o)& NC(n)! NC(n + 1)) !~ "n ~ NC(n)

and not !n NC(n) ; where NC(n) is the predicate “no contradiction”.39 Brouwer repeated the same statement.40

It is well known that Kolmogoroff,41 Glivenko42 and Gödel43 independently suggested just an operation of double negation for translating a classical proposition to an intuitionistic proposition.

39 H. Poincaré: Science et Méthode, Flammarion, Paris, 1912, p. 187. 40 L. E. J. Brouwer: “Intuitionism and Formalism”, in Collected Works, op. cit., p. 88. 41 A. N. Kolmogoroff: “On the principle of “tertium non datur””, Math. Sbornik, 32 (1924/25) 646-667 (Engl. transl. in J. van Heijenoort (ed.): From Frege... op. cit., 416-437). 42 Glivenko: “Sur la logique de M. Brouwer”, Bull. Acad. Sci. Belg., 15 (1929), 225-228. 43 K. Gödel: “Eine Interpretation der Intuitionistische Aussagenkalkül”, Ergeb. Math., 4 (1932-33) 34-38, and also in J. Hintikka (ed.): The Philosophy of Mathematics, Oxford, 1969, 128-129. See also D. Prawitz, P.-E. Malmnäs: “A Survey of some connection between classical, intuitionistic and minimal logic”, in A. Schmidt, K. Schütte (eds.): Contributions to Mathematical Logic, Nort-Holland, Amsterdam, 1968, 215-228. In last paper the rule for the quantifier ∀ is to pass unchanged. I contest it since this rule was added subsequently to the content of the papers of the three above authors; and moreover since I think that an intuitionist logician has to deal cautiously with totalities.


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By looking back, one can give an explication why the problematic organization of a logical theory was discovered only indirectly and in some particular cases. Really, it was very intricate to recognize the failure of the law of double negation, since in almost all languages this law is blurred by a rhetoric use of sentences whose meanings are merely affirmative; moreover, sometimes two negations mean an emphasized negation also (e.g. in Italian, “Non c’è nessuno”).

All that converges to the following conclusion. Classical logic and intuitionistic logic do not represent together two different theories of informal logic but they represent two alternative kinds of organizations of mathematical logic as a system. In particular, the polemics between Hilbert and Brouwer was overemphasized as the shock of two absolutists theories; instead, they represented two modalities of the presentation of same subject.44 What Brouwer and Heyting discovered was actually the alternative organization of the same, ancient logic, but considered as a heuristic way of reasoning; or in Kolmogoroff's terms, as a logic for problem solving.

To state that intuitionistic logic is the same as the organization of logic when it is based upon a crucial problem, obliges me to specify which may be this main problem. By means of an historical analysis of the past debate about foundations on logic I can list a lot of candidates.

Previously, I noted that at its very beginnings mathematical logic was organized just upon a problem whose sentence is double negated: “Nothing in the nature of Language prevents us from using a mere letter in the same sense”. From this problem started the historical process of formalizing logic. Later on, the polemics Hilbert-Brouwer suggested a new version of the main problem. One can sum up the main Hilbert's thesis by the following sentences: “Intuitive logical reasoning is formalizable”. To which Brouwer opposed: “Intuitive logical reasoning cannot be formalizable” i.e. the mere negation of the previous sentence. 44 S. Koerner: The Philosophy of Mathematics, Harper, New York, 1960, 146-7.


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Rather, one can wisely state that “It is not true that intuitive logical reasoning cannot be formalized”, i.e. a double negated statement.45 That is just what Heyting did at variance with Brouwer. Let us note that through three sentences we grasped the core of a long and acute polemics about which is the main problem in mathematical logic.

However, both Boole's sentences and the last sentence are not satisfying us, since they include the words “intuitive reasoning”, which belong to the domain of pre-mathematical logic. Then, we can quote the words which a protagonist, i. e. Heyting, stressed as the great problem: “One cannot give a provably unsolvable problem”. This sentence is closer to the wanted problem since its words all pertain to the field of mathematical logic. It can be put as the methodological principle of Kolmogoroff's interpretation of the intuitionistic logic. Seemingly, to put such a methodological principle as the starting point of his logic does not imply changes in the development of this system. In this way we obtained a self-reliant foundation of logic no more representing a mere interpretation of intuitionistic logic.

But there exists a double negated sentence representing a specific problem of the mathematical way to be followed for formalizing logic? In my opinion, a suitable candidate is the following problem: “It is not true that by negating a true proposition one obtains a false proposition”. In fact, this problem (“What means a negation?”) pertains properly to mathematical logic. It could be put as the methodological principle of

45 Similar sentences are the following ones: “… absolute certainty [=lack of error] for human thought is impossible and even makes no sense” by A. Heyting: Intuitionism, op. cit. p. 7. “It is impossible to reduce the possibilities of thougth to a finite [=not infinite] number of rules that thought can previously lay down” by R. Epstein: The semantic foundation of logic, vol. I, Nijoff, Kluwer, 1990, 195-212, p. 198. See also J. Barwise: “The situation in Logic” in B. Marcus et al. (eds.): LMPS VII, North-Holland, 1986, 183-203, p. 184. In 1925 paper, Kolmogoroff too recognized a main problem in logic; “to explain why illegitimate use of the excluded middle principle goes unnoticed and does not lead to a contradiction”; see V.A. Uspensky: “Kolmogoroff and mathematical logic”, J. Symb. Logic, 57, (1992) 385-412, p. 389.


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intuitionistic logic.46 Classical logic denies this problem by the law of double negation. But if one puts the sentence as a problem, then classical logic becomes incommensurable and a new way of arguing has to be explored.

One more problem appears to be appropriate to the same task: “It is not true that equivalence is not identity”. This problem is suggested by Troelstra47 when illustrating at best the foundations of intuitionism. This kind of problem pertains properly to logic as a scientific theory. It is very relevant that already Condillac and, more early, Leibniz, both put the same problem as the one, key problem of the whole science.

However, according to the historical evidence I cumulated in mathematics and physics, the foundations of a scientific theory include more than the option on the kind of organization: they include one option more, on the kind of infinity, whether the potential infinity only or the actual infinity.48 In mathematical logic this option concerns the way of formalizing predicate calculus, i.e. just that calculus that always represented the major problem of the formalization of logic.

46 A similar conclusion is reached by also D. Prawitz: "Meaning and Proofs", Theoria, 43 (1977) 6-39. 47 A.S. Troelstra, D. van Dalen: op. cit., p. 839-840 and A.S. Troelstra: “Intuitionism and Philosophy of Mathematics”, in G. Corsi, G. Sambin (eds.): Nuovi problemi della logica e della filosofia della scienza, vol. II, CLUEB, Bologna, 1991, p. 226. Let us note that in the first paper the author adds that “The problematic, this is the justification of the axiom. A matematician is usually not interested in axioms if he feels that there is no interpretation (model) for them, that is if he does not have an intuition... ” (p. 215). Then, she suggests something like to a different organization of a theory, i.e. a “concept analysis”. Author’s attitude appears in agreement to intuitionistic viewpoint, i.e. to start from a subjective, intuitive notion, as the notion of “concept” is; the best example seems Turing's theory of computability. Yet, in my opinion the structure of the organization of this theory remains unclear. 48 A. Drago: “How the mathematical notion of infinity matters to theoretical physics?”, in A. Diaz, J. Echeverria, A. Ibarra (eds.): Structures in Mathematical Theories, S. Sebastian, 1990, 141-146; “I quattro modelli della realtà fisica”, Epistemologia, 13 (1990) 303-324.


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There is no space to deal with this question. However, it can be said that the very alternative to the formalistic logic - which added the choice for actual infinity to the choice for a deductive method - is a theory of logic which is founded rigorously on both potential infinity only and which is organized as a system putting a crucial problem as its main problem. Singe scientific theories presenting the latter choices share as mathematical technique group theory49 - or somewhat similar -, I guess that a new foundation of logic as a full alternative to Formalist mathematical logic is a theory whose basic mathematics is group theory.

Acknowledgement I am grateful to Prof. G. Criscuolo since I learned much from

dialogues with him.

49 A. Drago: “Una caratterizzazione del contrasto tra simmetrie ed equazioni differenziali”, in A. Rossi (ed.): Atti XIV e XV Congr. Naz. St. Fisica, Conte, Lecce, 1996, 15-25. A. Drago, A. Pirolo: "A new formulation of quantum mechanics by means of symmetries" in A. Rossi (ed.): The Foundations of Quantum Mechanics, Kluwer Acad. P., 1995, 229-237.

Mathematical Logic and the Ideal of Apodictic Science · 2009-12-20 · Metalogicon (2000) XIII, 1...

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Transcript of Mathematical Logic and the Ideal of Apodictic Science · 2009-12-20 · Metalogicon (2000) XIII, 1...