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:Published (almost the same text) in J. of Indian Council of Philosophical Research, 34(2), 343-363, DOI 10.1007/s40961-016-0089
A PLURALIST FOUNDATION OF THE MATHEMATICS OF THE FIRST HALF OF THE 20TH CENTURY
Antonino Drago
Formerly at University Federico II of Naples drago@unina.it
Abstract. A new hypothesis on the basic features characterising the Foundations of Mathematics (FoM) is suggested. By means of it the several proposals, launched around the year 1900, for discovering the FoM are characterised. It is well-known that the historical evolution of these proposals was marked by some notorious failures and conflicts. Particular attention is given to Cantors program and its improvements. Its merits and insufficiencies are characterized in the light of the new conception of the FoM. After the failures of Freges and Cantors programs owing to the discoveries of respectively an antinomy and internal contradictions, the two remaining, more radical programs, i.e. Hilberts and Brouwers, generated a great debate; the explanation given here here is their mutual incommensurability, defined by means of the differences in their foundational features. The ignorance of this phenomenon explains the inconclusiveness of a century long debate between the advocates of these two proposals. Which however have been so greatly improved as to closely approach or even recognize some basic features of the FoM. Yet, no one proposal has recognized the alternative basic feature to Hilberts main one, the deductive (axiomatic) organization of a theory, although already half century before the births of all the programs this alternative was substantially instantiated by Lobachevsky's theory on parallel lines. Some conclusive considerations of a historical and philosophical nature are offered. In particular, the conclusive birth of a pluralism in the FoM is stressed.
Key-words: Foundations of Mathematics, Foundational features, Potential and actual Infinity, Theory organization, Incommensurability, Set theory, Lobachevsky theory.
1. The naive attribution of the nature of FOM to a single notion or a single theory
For many centuries the foundations of Mathematics (FoM) have been naively considered as constituted by one mathematical notion, e.g. number, point, infinitesimal, limit, etc., each one combining in itself both mathematical and philosophical meanings. Yet, already in ancient Greece a conflict among these presumed basic notions arose; numbers, potentially generating irrationals, proved to be incompatible with the basic geometrical notions obtained by means of the ruler and compass. This conflict was solved by relegating numbers to playing a non-foundational role. In modern times a new conflict arose between the basic geometrical notions and the notion of an infinitesimal. However, after the birth around the French revolution of new mathematical theories (Monges descriptive geometry, L. Carnots geometry, Lagranges calculus, Cauchys calculus) which relied on new mathematical notions, it was apparent that in Mathematics the basic notions were all mutually incompatible. As a result, a traditional notion was no longer considered capable of covering the great number and variety of the already existing mathematical theories.
However, mathematicians explored one more possibility of conceiving the FoM, again from within their science. They considered one mathematical theory as constituting the foundations of all others. In ancient times geometry played this role. After the long domination of geometry, in modern times infinitesimal analysis competed with it, leading to Lagrange to try to include the former in the latter (Lagrange 1773). But he was unsuccessful.
However, after the recognition of the relevance of non-Euclidean geometries, slowly but conclusively mathematicians realized that not only geometry, but every mathematical theory may be formulated in some distinct ways. Hence, past theories proved to be irreducible to a single theory; hence, the problem of recognizing the FoM cannot be solved by making one traditional theory basic to all others.
2. Freges and Cantors proposals of new FoM
In the second half of the 19th Century, since the dominant philosophies (mainly Kants) failed in suggesting both appropriate FoM and a clear logic (blurred by Hegels suggestion of a new, ill-defined logic), mathematicians had to interpret the basic features of Mathematics on the basis of their science only. In order to made clear these features, they, on one hand, purged from any philosophical import the basic notions of previous theories, e.g. the notions of point and line in Euclidean geometry, infinitesimals and limit in calculus, etc.. On the other hand, they had to refer to a higher level than that explored in the past, i.e. that of a single, already known mathematical theory.
However, two more attempts to identify the FoM with one theory were initiated, but by inventing an essentially new theory. Since the nature of Logic is to support all mathematical theories, this theory was suggested by Frege as basic to Arithmetic and thereafter all mathematical theories, who formulated Logic in full mathematical terms. (Notice that at that time classical logic did not have a mathematical rival and mathematicians were not aware that they were exploring only one of the two possible logics. But, some years after, the discovery of Russells antinomy obstructed the relationship between mathematics and logic. Since Freges relationship was irreformable, this antinomy constituted an insurmountable obstacle to Freges proposal, which eventually failed.
At first Cantor improved the first traditional conception of the FoM by exploring the potentialities of one notion which is present in all mathematical theories, but had never been elucidated in its potentialities, i.e. the infinity. To this end he invented a new mathematical notion, set, which he considered basic for the Mathematics as a whole. By disentangling the notion of infinity actually, actual infinity - according to a fascinating hierarchy of (infinite) degrees of infinity he then built a new mathematical theory, Set Theory (ST).
The price he paid was to overcome the borderline between Mathematics and the philosophical realm; indeed, he was unable to define a set in mathematical terms; indeed, he did not accurately define within the mathematical realm his two first notions (set and belong to) and in addition he suggested a particular methodology for dealing with the degrees of infinity, which moreover was actual infinity, without attention to potential infinity. In such a way he obtained much more than a technique (as previously was calculus, generated by the instrumental use in Mathematics of the philosophical notion of an actual infinitesimal); he included a new philosophical basis, which remained hidden to working mathematicians.
By representing in his mathematical theory the notion of infinity which each mathematical theory includes, Cantors theory was potentially capable of including all relevant mathematical theories. Hence, he could claim that this essentially new mathematical theory constitutes the basis of Mathematics as a whole.
ST was very successful for four reasons. First, Cantor produced marvellous results within an unprecedentedly explored field of fascinating research. Second, ST, being based on the revolutionary notion of infinity, appeared to be the best program for re-constructing and improving Mathematics as a whole. Third, STs declaring as primitive some basic notions allowed mathematicians to assume without question the connections of Mathematics with philosophical issues, in the hope that these meanings would be clarified by subsequent formal achievements. Fourth, ST has allowed mathematicians to persist in the traditional attitude which considers a single theory as basic.
Also after the launch of other programs those of Hilbert and Brouwer for most mathematicians, mainly those wanting mathematical knowledge to progress considered Cantors program to be the best program for re-constructing Mathematics..
However, after the discovery of Russells antinomy, concerning in a specific way ST, the original theory could no longer be defended. It was taken up again as a mere approximation to a correct theory, by calling it the nave ST. In addition, some Cantorians renewed the original proposal by adding new axioms, although these additions brought the new theory (ZFC ST) closer to contradiction (as is shown by the location of ZFCs axioms in the hierarchy of the axioms identified by reverse mathematics (Simpson 2009, p. 42)).
3. Hilberts and Brouwers programs for re-founding Mathematics
Two more mathematicians, Hilbert and Brouwer, claimed that the basis of Mathematics is neither one notion, nor one (new) theory, but one methodological issue.
In actual fact, Brouwer began from the notion of two-ness, by which, however, he menat an interior process of dividing time into past/future. Moreover, like Frege and Cantor, Brouwer claimed that the basis of Mathematics is an essentially new theory; indeed, his Arithmetic was new with respect to the previous Arithmetic to the extent that it starts not from an ideal being but from the above-mentioned process and then constructed operatively. Therefore he referred to a general method, i.e. he required that in arithmetic and all consequent theories each mathematical object has to be constructed by means of finite algorithms (later, he also declared his distrust in classical logic).
In his program for re-founding Mathematics Hilbert declared that all the basic notions of traditional theories were inadequate; rather, he gave relevance to gave relevance non mi sembra corretto (in italiano?) ha dato importanza (esclusiva) a only the specific axioms understood formally - of a theory. Moreover, initially his program was linked to an old theory, Euclidean geometry, but only in order to extract from it a general method for axiomatizing all scientific theories. (Twenty five years later, Brouwers criticisms led Hilbert to declare an additional tenet, i.e. classical logic is indispensable to a mathematician).
In sum, both worked at a higher philosophical level than Cantors level of a philosophical method embodied by a single new theory; their level was a philosophical approach constituted by the following tenets: i) previous mathematicians had built theories without accurately defining the FoM; ii) a new way to conceive the FoM is necessary(respectively, an intuitive way or a formal way); iii) each of these ways conceived Mathematics according to a monist view, excluding the other one; iv) each way is specified by a method admittedly including some philosophy; Brouwers method included a new basic process (two-ness), obliged the operations to be constructive and wanted to discover a new kind of logic; Hilberts method attributed a safety role to translate theories into their axiomatic formulations; where the axioms have to enjoy the three following properties: completeness, independence and consistency (all three properties to be proved by specific mathematical proofs according to classical logic); v) the success of this kind of re-construction of all possible mathematical theories will eventually prove the validity of the proposed foundations.
Brouwer suggested one part only of his program; e.g. the specific kind of logic to be used was defined later by one of his followers, Heyting. Instead Hilbert has the merit of having made clear (after a long meditation of twenty five years) the main mathematical and logical aspects of his program.
The entire history of Mathematics being full of conflicts among the supporters first of the different basic notions and subsequently the different basic theories, it was not surprising that in the 20th Century a radical conflict between the above two programs also arose; a heated debate concerned their manifest exclusiveness. This debate was so deep and complex that no one founder could claim to have won; even a century after it is inconclusive (Martin-Loef 2007).
It was not so much the debate but rather the development of each programme that led to significant problems. One of Hilberts followers, Goedel, by trying to positively conclude Hilberts program, instead ironically produced two theorems which barred it. On the other hand, Brouwers program not only was unsuccessful in convincing the mathematical community to dismiss the formalist versions of the mathematical theories, but also developed inconsistency in its program, since its re-construction of mathematical theories included some notions - e.g. spreads, some results obtained by the fixed-point theorem all relying on actual infinity.
4. The discover of two basic dichotomies
However, after the middle of the 20th Century, Markovs and Bishops works (Markov 1971; Bishop 1967) independently made it possible to conceive a formal dichotomy: either the mathematical constructive tools only (or, in philosophical terms, the tools making use of no more than the notion of potential infinity: PI), or the tools of classical mathematics (which make free use of actual infinity, AI, provided that the results are contradiction-free). In my opinion, this dichotomy regarding all mathematical tools surely pertains to the FoM.
At the same time Beth concluded a broad investigation into the basic mathematical and logical features of Mathematics by pointing out that the current development of Mathematics was biased by organizing a mathematical theory according to only one model, i.e. the model of deductive (possibly, axiomatic) organization (AO). Furthermore, Weyl, Beth himself, and then van Heijenoort, Kreisel and Hintikka interpreted Goedels theorems as a suggestion for finding out an alternative model of a theory organization.
My historical analyses brought to light that in the past important scientists (e.g. L. Carnot, Lavoisier, S. Carnot, Lobachevsky, Galois, Boole, Klein, Brouwer, Kolmogorov, Markov) had already founded important scientific theories outside the deductive model. A comparative analysis of the original texts of these theories suggests the following characteristic features of their common model of organization (Drago 2012, pp. 175-189), which I call a problem-based organization (PO), since the theory starts from a universal problem; it then looks for a new scientific method by reasoning through doubly negated propositions, each one of them being not equivalent to its affirmative version; thus, here the double negation law fails and such a proposition belongs to non-classical logic (e.g., intuitionist logic). In an original text of the previously mentioned theories the doubly negated propositions are grouped in some cycles of reasoning; each of them refers to a sub-problem which is solved by means of an ad absurdum proof, which concludes at the last doubly negated proposition, which is not translated into the corresponding affirmative proposition, as occurs in classical logic. A final ad absurdum proof concerning all cases of the main problem concludes a doubly negated predicate, U. At this point, the author, in the belief that his argument was sufficiently supported, converts the universal conclusion to the corresponding affirmative predicate U; from which he then derives all the consequences according to classical logic. This change of both the predicate U and the whole logic leads to a subsequent deductive development.
The above leads us to consider the FoM as constituted by two dichotomies, concerning both Philosophy and Mathematics; 1) either the use of PI only or the use of AI; which in formal terms correspond to either constructive or classical mathematics; 2) either an AO or a PO; which in formal terms correspond to either classical or non-classical logic.
5. A characterization of the previous programs. The occurrence of incommensurabilities
According to these dichotomies the attempt of the above mathematicians to discover the FoM may be characterized as follows.
Frege explored AO both within the organization of Logic and in postulating it as the premise of Mathematics; moreover, he preferred PI, but according to a vague Kantian notion of it. Cantor decisively chose AI; he translated it into the notion of the infinite number of elements belonging to a set, the act of comprehension of which may be considered a non-operational organization of them, which is suitable only for an AO of the theory. These choices of Cantors are the most powerful and promising choices for developing the mathematical work and were later supported also by Hilberts program.
Brouwer, on the other hand, chose PI. Moreover he declared his distrust of the axioms of classical logic, in particular the LEM, in view of a new kind of mathematical logic. Hence, implicitly he looked for a PO.
Notice that each of the four above scholars overtly made only one choice; Cantor, who chose AI, added choice AO implicitly; Brouwer, who chose PI, merely alluded to PO; Hilbert, who chose AO, explicitly added the choice AI after twenty five years.
However, the above mathematicians, when considered together, closely approached a description of all the four choices of the FoM. Owing to this substantial advance in knowledge of the FoM, after these proposals it was no longer possible to go back to a 19th Century conception of the FoM; alternatively, one had to renounce the search for the FoM (as in point of fact Bourbaki did).
Notice that the two alternative choices of each dichotomy are exclusive in nature. It is not possible to include in a PI mathematics all the axioms (e.g. Zermelos axiom) and results obtained by an AI mathematics; conversely, the latter does not take in account the undecidability results obtained by the former mathematics. By making use of non-classical logic, a PO theory cannot be reduced to an AO formulation of the same theory, because the law of double negation irreducibly separates them.
Hilbert and Brouwers programs being different in both their basic choices, it is no surprise if they presented radical variations in the meanings of several common notions (e.g., number one, infinity, LEM, etc.); or even some notions were declared by the opposing scholar to be non-existent (e.g. intuition, formalization, logical alternative, etc.). Not surprisingly these radical variations in meaning were reductively interpreted as either contradictions (e.g., those generated by the two kinds of logic) or impossibilities (e.g. Brouwers claim to include relevant mathematical theories in PI new formulations of them, or Hilberts claim to prove the consistency of all axiomatic theories). Hence, no common language was possible; only an accurate and attentive use of language could offer a partial translation between the two programs. In other words the comparison of the two programs generated an incommensurability phenomenon; which explains why for a century the mutual comparison of these two surviving programs generated irreducible debates and conflicts, whose final result was a no-contest situation.
6. The subsequent search for the FoM performed by the mathematicians ignoring incommensurabilities
The incommensurabilities also explain why excellent mathematicians met great difficulties in developing their own programs. Initially Brouwer rejected anything not corresponding to his basic tenets; hence, he could not achieve an understanding of the alternative choices. Yet, in order to offer ever more counter-examples of non-classical notions and techniques, Brouwer then introduced some extensions of his program - i.e. choice sequences and use of the fixed-point theorem which manifestly pertain to AI. After him, Heyting added one more enlargement; he organized intuitionist logic according to a Hilbertian axiomatic, AO, although he tried to temper his axioms system by adding a verbal proviso about the insufficiency of this formal system to reliably grasp a whole intuitionist theory. In such a way both Brouwer and Heyting extended the scope of the original program so that it included aspects of the alternative choices, i.e. AI and AO.
Owing to these enlargements, it is no surprise if the Intuitionists did not achieve - as later Bishop reproached them (Bishop 1967, pp. 1-10) - a lucid presentation of their Arithmetic as a PI theory; and ever less, I add, a comprehensive view of the FoM.
On the other hand, in order to react to both Poincars and Brouwers criticisms of his program, Hilbert introduced a Metamathematics. However, he never achieved a satisfactory definition of it, maybe because he considered it a mere scaffold to be removed after obtaining the proofs of the three required properties of an axiomatic. According to him, this new theory makes use of finitist mathematics (a vague notion which closely approaches PI) and a kind of logic on which in Hilberts opinion - Brouwer could agree because it has to apply contentual inference; (Mancuso 1999, pp. 10, 212) hence, it is the typical logic of a PO theory, i.e. intuitionist logic. I conclude that this program was the more advanced one since it was eventually capable of taking in account all the four basic choices, although the last two in an uncertain way.
However, Hilberts introduction of metamathematics counter-reacted on his original program taking in account AO and AI only. The internal comparison between the mathematical theory under scrutiny, Arithmetic considered from the perspective of AI and AO, while Poincar considered it from the perspective of PI and PO the Formalists according to the choices AI&AO - and metamathematics - whose choices, as we recognised in the above by partially interpreting Hilberts obscure declarations, are the opposite ones PI&PO -, generated an incommensurability phenomenon within the program itself.
In conclusion, Hilbert was doomed to fail to achieve knowledge of FoM for two reasons. Initially his program was confined to a partial view of FoM, since it recognized as choices AO and AI only. When he later introduced metamathematics, his program involved all the four choices, but he was unable, owing to his vague definitions of the characteristic features of Metamathematics, to recognize the mutual incommensurabilities of the two theories, as well as the radical variations in meaning of their basic notions.
In fact, Formalist mathematicians worked in a way which was naive with respect to the philosophical issues involved by the FoM. They interpreted the comparison of Arithmetic with its metamathematics in merely formal terms of deductive steps; moreover, they disregarded all the philosophical questions involved by the radical variations in meaning of the basic notions. Their minds were dominated by Leibniz' ideal of "Calculemus!"; they considered it to be the only successful strategy for comparing two mathematical theories and also their program with Brouewrs program.
Indeed, it is well-known that such a great mathematician as Hilbert, although mastering the most sophisticated mathematical tools and techniques, worked inappropriately and eventually unsuccessfully to advance his program. Even the great Ackerman and von Neumann were wrong in believing they had solved by means of a specific theorem the above comparison between arithmetic and its metamathematics.
More in general, owing to the phenomena of incommensurability generated by the different pairs of choices, no mathematician was successful in inferring inductively from the basic mathematical notions which were meant subjectively and hence presented radical variations in meanings -, a single dichotomy, and even less the structure of the two dichotomies constituting the FoM.
7. The surprising negative result and the subsequent achievement of a surprising agreement among the mathematicians
In fact, Hilberts program failed owing to a stroke of genius. Goedel was able to prove the mutual incompatibility of the consistency and the completeness of an axiomatic theory. He compared the axiomatic formalization of Arithmetic with its metamathematics through a mixing of a strict formalization (e.g. the goedelization of all the propositions of both theories), a partially intuitive (contentual) notion (the -consistency) and doubly-negated propositions not equivalent to their corresponding positive propositions (the negation of the not provability of a proposition in its goedelized version). Thus he obtained a contradiction between deduced and provable, i.e. between a positive proposition in a formalized theory and its corresponding modal proposition, which is equivalent, via S4 model, to a proposition of intuitionist logic.
It is remarkable that, when translating into integer numbers the infinite sequence of theorems (goedelization), Goedel - in order to characterize Hilberts finitism - introduced a first definition of a PI mathematics, i.e. the theory of the primitive recursive functions. In other words, he was successful in clarifying the choice PI, inaccurately defined by Hilbert in Metamathematics and incompletely presented by the Intuitionists. Moreover, after presenting the celebrated theorems, Goedel established unequivocally the independence of intuitionist from classical logic (Goedel 1933f), i.e. the full validity of the former, and hence - I add the independence of the organisation of a theory governed by this logic, PO. In sum, these two papers of Goedels offered accurate definitions of the four basic choices, which Hilberts program had introduced only partially. Yet, he failed to conclude that the FoMs are constituted by the these four choices.
In general, Goedel's results played several roles. It is not surprising that subsequently it was very difficult to interpret the meaning of Goedels theorems within the history of Mathematics, to the extent that were even perceived as a stumbling block to further research on the FoM.
Eventually, most mathematicians, instead of appreciating the result that implicitly was obtained by both programs - to have presented in some way all the four basic choices reached an odd agreement on a more tolerant view of the original motivations of both programs. The agreement recognized these programs as representing two different potentialities for a working mathematician; who can develop not only formal mathematical theories, but also intuitionist mathematical theories within a naive formalist framework. From the viewpoint of the two basic dichotomies we recognise that this agreement represents a regression to a deliberately undefined perspective on the FoM. Owing to the above mentioned internal incommensurabilities, this perspectivee can neither advance knowledge of the FoM as was the case for eighty years -, nor obtain an insight into the causes of the humiliating failure of such a long search for the FoM. No surprise if some years ago an authoritative scholar wrote the following discouraged words:
There is currently a general malaise about the logical approach to the foundations of mathematics. One main reason is that foundational thought in this century has been dominated by a few global views about the nature of mathematics logicism, formalism, platonism and constructivism each of which has proved to be defective in substantial ways, while nothing else has come to take their place. (Feferman 1998, p. 105)
As a reaction to the above compromise agreement between intuitionist and formalist mathematicians, most mathematicians espoused Bourbakis point of view, which, by completely disregarding the problem of the FoM, offers a backward, but comforting, conception of the history of Mathematics as a unilinear, cumulative progress of incessantly new objective results.
8. Re-visiting Set Theory
Cantors ST became popular for several reasons: 1) It produced surprising and marvellous results; for this reason set-theorety mathematicians forgot that a long debate on the notion whether potential or actual divided the mathematical community; moreover they attributed a backward attitude to Kronecker, who opposed ST in name of the use of arithmetic only (in fact, PI). 2) It required only a slight change mind by each follower, since ST appeared as a mere improvement of the traditional way of conceiving the FoM by means of one notion or at most one theory. ST represented a great advance for the working mathematician, because he was allowed to work comfortably in apparently the entire body of mathematical theories through an intuitive notion, or at most a key-theory. 3) Since the notion of infinity pervades the whole of Mathematics; by means of it (as incorporated by the notion of a set) Cantor pursued the goal of a systematic inclusion - by deductive means; hence he chose AO ; the set-theoretical translation of the great number of old and new mathematical theories played a unique role in both re-formulating previous theories and suggesting inter-theoretical links among all theories. 4) ST allowed mathematicians to preserve the traditional two-thousand-year-old attitude of not explicitly appealing to other disciplines; in particular, ST allowed mathematicians to confine all the connections of Mathematics with philosophical issues to the meaning of this notion only, the set, which however was declared a primitive notion. 5) It played the role of the first program capable of re-constructing and improving Mathematics as a whole.
From the viewpoint of the two dichotomies we see that both Cantor and his immediate followers deliberately chose one kind of infinity, i.e. actual (AI). Indeed, the notion of a set represents, through a mental act, an infinite number of elements as grouped in a completed totality; no constructive process (PI) obtains such a result. In addition, in 1883 Cantor wanted to organize ST deductively by starting from few principles-axioms (AO) according to classical logic, since at that time no other mathematical logic was considered possible. In sum, Cantor wanted to (informally) found ST on AI and AO.
Despite the ambiguities in its basic notions, Cantor s ST represented a great advance in the long search to identify the FoM. Whereas the previous introduction of AI (through the infinitesimal and then the e-d technique) were all dubious, in 1874 Cantor formally distinguished the set of real numbers from the denumerable, hence he clearly distinguished AI from PI, and thus began a systematic study of an entirely new realm, infinities, i.e. the basic two choices PI and AI, out of the four choices. These results represented a great advance in the history of Mathematics; they introduced what may be rightly called modern Mathematics, a Mathematics formally including the study of a dichotomy.
Nevertheless, ST also represented a regression because it reiterated the traditional way of confining the search for the FoM to exploration of a single theory. It is apparent that one theory cannot include both choices AI and PI (although most mathematicians believe that PI mathematics is a mere restriction of AI mathematics; on the contrary, eg. the undecidability results cannot be represented by the AI mathematics). Moreover, one theory cannot be built according to both kinds of organization, or equivalently both kinds of logic. By wanting to include in itself all the four basic choices ST inevitably produced antinomies.
In point of fact, Russell discovered the celebrated antinomy, concerning in particular ST. It led mathematicians to declare nave the original ST. This antinomy is caused by ambiguities in the definition of a set. Grelling recognized an ambiguity in the first definition (a set X not including itself as an element); this definition may indicate a process of construction which is never completed; hence, this set belongs to a PO theory which has to test whether this hypothesis is consistent or not. Instead of proceeding according to the PO choice the paradox asks, as a next step, a typical question of an AO theory, i.e. what is the completed result of previous process?. In a PO theory this is an inappropriate question. Goodstein recognized an ambiguity in the two possible meanings of the total quantifier, i.e. either without exception or a completed totality; the first meaning is correctly expressed by a double negation, hence it belongs to intuitionist logic (of a PO theory); the latter meaning belongs to classical logic (of an AO theory). In sum, the ignorance of both PO and intuitionist logic leads to introducing ambiguities (or rather, radical variations in meaning) in the notion of set.
Commonly, this paradox is understood as being caused by the use of the comprehension axiom, which therefore could no longer be applied without restrictions. It is not a case that set theorists claim that the notion of a set has to be considered a primitive notion, about which no question is allowed, just as one deals with an axiom within a formalist system. In such a way present mathematical practice of nave ST completely ignores the previous ambiguities and the implicit choices.
In order to preserve this omnicomprehensive program, set theorists introduced, as the only possible move, some (partial) remedies. In 1908 Zermelo has suggested an axiomatic theory of ST. He avoided the definition of a set, really an impracticable task. In order to avoid inconsistencies he rejected some kinds of set through the addition of new axioms; yet, they lead ST to more closely approach contradictions, as is shown by the hierarchy of the axioms given by the Reverse Mathematics.
However, this axiomatic of ST represented a great advance in the search of the FoM, because it was the first new mathematical theory founded on a declared pair of formal choices, AI and AO. Subsequently this theory was improved in the ZFC formulation of ST, by reiterating Zermelos choices.
These choices are the most powerful and promising ones in both mathematical and philosophical aspects. Furthermore, they are the same on which in theoretical physics Newton founded his mechanics, which in the previous two centuries had gained a so much importance as to constitute a long-standing paradigm, which influenced even Mathematics. No surprise if ST played a paradigmatic role in the minds of most mathematicians, to the extent that Hilbert proclaimed: No one will drive us from the paradise which Cantor created for us. (Hilbert 1967, p. 376)
Since according to the two dichotomies the notion of a revolution is defined as any change in the basic choices, ST, which inaugurated the formal definition of a pair of basic choices, constituted a revolution. This appraisal agrees with most mathematicians opinion on its birth.
However, other aspects of ST reduce its merits.
First of all, the birth of ST also created a false consciousness of the historical situation. Both the impetuous progress of the developments of ST and the very strong opposition by Kronecker and Brouwer led each mathematician to exclude their opponents views from a respectable Mathematics. However, the two dichotomies suggest an entirely different view, which encompass all viewpoints that are on a par; i.e. a pluralist view.
Let us recall that in order to alternatively obtain the continuum according to PI, Brouwer suggested a new way of conceiving ST. He invented the new notion of choice sequence, which frees the infinite from the notion of law; notice that here law is conceived as a negative notion (unnatural and unmanageable); hence these words, i.e. without law, constitute a double negation belonging to intuitionist logic; the same for the other way to conceive it as the possibility of un limited continuation of the sequence. Moreover, Brouwer stressed that his theory does not derive from either axioms or laws, because it his is aimed at solving a problem; i.e. how to build a continuum considered as the set of all real-number generators. Hence his theory is a PO theory and his logic is correctly intuitionist.
In a previous paper I showed that Koyrs successful categories in interpreting the birth of modern science actually translate the four choices into physical and historical notions: Dissolution [no longer PO] of the finite Cosmos [no longer PI] and spaces [AO] geometrization [AI]. (Koyr 1957; Drago 1995 a), In order to characterize the birth of axiomatic ST one may state in a parallel way: Dissolution [no longer PO] of finite mathematics [no longer PI] and set axiomatization [AO] of the actual infinite [AI]. I also discovered those categories which characterise the alternative physical theories to Newtons, i.e. chemistry, thermodynamics, etc.: Evanescence [no longer AI] of the force-cause [no longer AO] and discretization [PI] of matter [PO]. Brouwers alternative to Cantors theory may be characterized by: Evanescence [no longer AI] of the abstract comprehension axiom [no longer AO] and discretization [PI] of a set [PO]. The thus obtained new categories manifest a conflict in the foundations of ST. For this reason ST cannot claim to include the entire body of Mathematics.
One further reason, of a technical nature, comes from Goedel celebrated theorems which definitely forbid considering any theory as basic to the FoM; If ST is non-contradictory, it cannot claim a complete AO, because it includes non-decidable propositions (surely, the set of all sets represents one of them); and on the other hand, if ST is AO complete theory, it cannot be considered non-contradictory (although several mathematicians tried to avoid non-contradictoriness by introducing new axioms). (Partial consistency proofs are of limited interest). Moreover, the Loewenheim-Skolem theorem also excluded the categoricity of ST.
One more reason for the incompleteness of ST came in 1963 from Cohens proof of the independence of ZFC theory from both the continuum hypothesis and the choice axiom. (Cohen (1965), The most accredited philosophical interpretation of the first result is a bifurcation of ST, i.e. ST may be considered in two ways, with or without this hypothesis. This result implies that between the denumerable and the continuum there exists only a partial translation. That amounts to what in philosophical terms is called a mutual incommensurability. Indeed, this bifurcation of ST is a formal manifestation of the ambiguous choice at the same time of AI and PI (formally introduced in the same years within Mathematics by the birth of constructive mathematics (Markov 1962; Bishop 1967)). The same holds true for Cohens proof of the independence of the axiom of choice (the multiplicative axiom), which characterizes the theoretical dynamics of ST; indeed, it may be related to the two distinct ways of proceeding of a theory, i.e. AO and PO.
When ST is considered in pragmatic terms, one recognizes that it has obtained a spectacular array of successful analyses of conceptual apparatuses of current Mathematics; however the independence phenomena from ZFC (e.g. functions as rules, etc.) eventually showed that ST cannot, through its relying on two choices only, AI and AO, include all mathematical notions, hypotheses and theories because it produces artefacts when dealing with constructive theory and computer science, which essentially refer to PI (think of undecidabilities). Moreover, it can say nothing about the other dichotomy AO/PO, in particular about non-classical logic.
As a consequence of all in the above, ST theory has to be considered an explanatory rather than a foundational theory. In other words, it is a mathematical language which is very suitable for describing innumerable, but not all, theories. If the present working mathematicians do not object to qualifying a set as a primitive notion it is because their choice of the notion of a set as a basic notion is largely a pragmatical one.
Since set-theorists have staunchly pursued a mythical target of both achieving unprecedented results on infinity and re-formulating the whole of Mathematics (as Bourbaki claimed to have obtained), (Bourbaki 1947,, p. 7) they persisted in ignoring the alternative choices, PI and PO, as representative choices of a backward attitude. This fact constituted a considerable hindrance to a more adequate definition of the FoM by the mathematical community. Eventually set theorists entered a labyrinth of thought, as Ferreirs put it (Ferreirs 1999). A scholar was forced to recognize that the notion of a set is much more complicated than was originally thought.
Retrospectively, in a light of the philosophy of Mathematics, ST represented a first step towards including philosophical considerations in the FoM, but concealing? it within a mathematical notion (initially a set, then the notion of membership and the axiomatic). In such a way set theorists considered the entire body of mathematics in merey technical terms. In more general terms, ST represented a last attempt to preserve a monist view of Mathematics, i.e. a view which ignored the conflicts between different philosophical perspectives, as were those between Brouwer and Hilbert; in other words, through ST mathematicians wanted to preserve Mathematics as a peaceful activity.
9. Historical considerations
Retrospectively, let us consider what steps mathematicians must take in order to recognize the four choices precisely:
1) About the aim to clarify the choice AO, it is Hilberts merit to have been the first to recognize the FoM by re-constructing Euclidean geometry according to a formal axiomatization of AO (1899) and then to have suggested this model of organization as basic to all theories. Hilbert is in no doubt about classical logic:classical logic is for a mathematician as the fists are for a boxer.\(Hilbert 1967, p. 376)
2) About the choice AI, it is again a Hilberts merit to have clarified his previous ambiguities on this point, although only after after twenty five years of reflections stimulated by Brouwers criticisms.
3) In order to clarify the choice PI in more precise terms than Kroneckers (The integer numbers are made by the good God, the rest is made by the men), one had to re-formulate, by means of constructive tools only, at least a substantial part of the whole body of Mathematics. It is Brouwers merit to have launched this program and moreover to have accumulated a large number of counterparts of classical results; but not before several decades, Goedel has defined a theory of the primitive recursive functions, and after thirty more years both Markov and Bishop independently re-formulated a large part of the body of Mathematics.
4) In order to define intuitionist logic the logic governing the choice PO -, In the first years of the 20th Century Brouwer began to dismantle the faith in AO (no LEM, no axioms); later Heyting performed the task, although within a formalist framework. In 1931, when founding the semantic of intuitionist logic, Kolmogorov albeit unwarily - approached the goal [Drago 2005]. More difficult was the search aimed at defining the choice PO according to an opposite model of mathematical theory. In recent years my historical studies on several past theories conclusively defined this model.[Drago & Perno 2004; Drago 2012].
Retrospectively, we recognise that the major difficulty met by the mathematicians in trying to define the choices of FoM was to identify an alternative model of organization to AO, and hence a dichotomy regarding the kind of the organization of a scientific theory. Since Goedels results undermined Hilberts program, primarily based on AO, they implicitly suggested a strong motivation for investigating the possibility of a non-deductive organization And since the purpose of Hilberts program was to provide a solution to the crisis generated by the birth of non-Euclidean geometries the starting event of the modern problem of FoM -; mathematicians could conclude that Goedels negative results suggested first of all returning to an investigation of the early problem, i.e. to study anew the birth of non-Euclidean geometry.
As early as the 19th Century mathematicians had disregarded all the non-Euclidean geometries for the forty years following their births. For several more decades few scholars devoted historical studies to the original works about these geometries. Recent historical analysis of Lobachevskys original works showed that he never presented the first non-Euclidean geometry in axiomatically; rather, he organised his theory in a very similar way to the alternative model of PO (Lobachevsky 1840; Drago 1995 b; Cicenia&Drago 1996, Drago, Perno 2004; Drago 2007); as a matter of fact, his geometrical theory instantiated the ideal model of a PO theory in a similar intuitive terms to those in which Euclidean geometry instantiated the ideal model of an AO theory.
In addition, Lobachevsky wanted to found his theory upon PI (Lobachevsky 1835-38, Introduction). In sum, Lobachevsky's theory wanted to re-formulate the oldest mathematical theory, Geometry, according to just the two basic choices PO&PI which were the alternative ones to the dominant ones in Mathematics, AO and AI, with which he was very familiar.
In conclusion, Lobachevsky recognised the four choices and the related pluralist philosophy more closely than all subsequent mathematicians (Drago 2011). Moreover, a further historical inspection of the history of Mathematics of his times shows that around the period of the French revolution several authors (Lavoisier, L. Carnot, S. Carnot) havd already suggested this novelty by founding their theories according to the alternative choices, in particular the non-deductive organization of a theory (Drago 2012).
Hence, the most important result achieved, albeit implicitly, by Hilberts program on FoM - i.e. to introduce mathematicians to all the four basic choices - had alreay been obtained substantially a century before him by the theories of some mathematicians - mainly Lobachevsky.
In addition Lobachevsky introduced his new theory according to a pluralistic view, i.e. without invoking the suppression of different attitudes on the FoM from his own. Unfortunately for a century and half subsequent mathematicians ignored this pluralistic attitude of Lobachevsky's.
10.. Philosophical considerations
As a conclusion, I paraphrase in specific terms for the mathematicians what Burtt wrote in 1924 for the scientists in general:
Metaphysics [the mathematicians] tended more and more to avoid [Mathematics], so far as they could avoid it; so far as not, it became [in each of the above program] an instrument for [its] further conquest of the [mathematical] world (Burtt 1924, p. 303).
Retrospectively, we recognize that the correct strategy was instead to translate into accurate mathematical terms i.e. into more accurate terms than Cantors within a single theory -, only two basic notions of philosophy (already by Galileo presented as problematic), i.e. infinity and the organization of a theory, according to a pluralist attitude that already Leibniz had hinted at when he recognised two labyrinths in human reason, one regarding infinity - either actual or potential -, and another regarding the subjective way of having experience of an organization, i.e. law or freedom (of methodological research).
This correct strategy was approached closely by Beths program for establishing a new link between Mathematics and philosophy in order to overcome Kants inadequate philosophy of knowledge. Indeed, he advocated an intelligent improvement of the intuitionist program (Beth 1959, ch. 2) and expressed a wish for an alternative to AO and hence in the FoM.
As a general consideration, the main event unforeseen by mathematicians searching for the FoM was the emergence of some essential differences, i.e. incommensurabilities which are irreducible in terms of the usual working moves, i.e. deductions and calculations. We well know that in the past, owing to an incommensurability phenomenon, in that time defined as an incommensurability in the comparison between two magnitudes, the ancient Greek mathematicians bounded their work in Mathematics to the finite calculations only. Also Western mathematicians, in their research for knowing the FoM, have been bounded by incommensurability phenomena, which born in a wider context of a comparison of two theories. However, whereas the Greek mathematicians have bounded their mathematical knowledge in a deliberate way, Western mathematicians, - by confidently following Cantors and Hilberts tenets, in particular Hilberts celebrated dictum: In Mathematics does not exists Ignorebimus! - have been bounded by their ignorance of incommensurability phenomena. Unfortunately the advice of the 18th Century diplomat-philosopher-mathematician Leibniz i.e. our mind faces the two above-mentioned labyrinths - has been ignored by all philosophers and scientists too.
As a general conclusion, the major historical novelty of this long attempt to define the FoM and their incommensurabilities was its final result. This novelty is not the attribution of the victory of a program over all others, so as ultimately to establish the unity of Mathematics, but a pluralist conception of the FoM. As a matter of fact, present Mathematics includes all four alternative choices regarding the two dichotomies, i.e. both classical and constructivist mathematics, both classical and intuitionist logic. Since ancient times there had been a pluralism of basic notions and for a further time there was a pluralism of basic theories; in the last century a pluralism of different basic programs for developing the whole of Mathematics began.
According to this pluralist view, once the specific philosophical import of each couple of choices regarding the two dichotomies is chosen, then once the couple of choices regarding the two dichotomies is chosen and hence the specific philosophical import of them is assumed, then a theory develops according to the corresponding formal requirements as a purely mathematical task. That does not mean that the choices are indifferent to a working mathematician, because each mathematical theory, if conceived as a whole, includes these choices which cannot be justified in mathematical terms. Indeed, a mathematical theory includes also some philosophy, i.e. the choices regarding two dichotomies.
Acknowledgments.
I acknowledge Prof. David Braithwhaite for the numerous corrections on my English text.
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About this conclusion one may reiterate what Lanza del Vasto wrote on the foundations of science: People looked for an even deeper and securer terrain in order to lay the foundations, while the only thing firm is, as the word firm says, the firmament [that is, the philosophical realm]. (Lanza del Vasto 1943, p. 69).
Notice that it still includes actual infinity since it is claimed that it achieves the final, single point, although approached by intervals defined by a definitory series scusa cambio cos; va bene? Although it is only approached by no more than extended intervals, those of the definitory series come sarebbe in inglese? Forse dici in italiano: bench esso sia solo approssimato da intervalli estesi(Kogbetlianz 1968, App. 2). The widespread ignorance of this point proves the general inaccuracy in sharply separating the actual infinity from the potential infinity of the approximations.
For more details on the following summary of the historical events see (van Heijenoort 1967b; Sieg 1984; Mangione, Bozzi 1993; Mancosu 1998). The excellent review by Sieg unfortunately ignores the ideal element included by both the notion of Dedekinds cut - which is equivalent to the assumption of an existential quantifier -, and the idealist Cauchy-Weierstrass notion of limit (see previous fn.).
For a short, but detailed, summary of the criticisms of contemporary ST see Feferman (1998, p. 288).
(Beth 1959, ch. 1. 2). Notice the house of edition, North-Holland,, not Harper, which strangely enough published a different book with the same title in the same year.
(Weyl 1946; Beth 1950, p. 102; van Heijenoort 1967, p. 356; Kreisel 1989; Hintikka 1989). Yet two centuries before, already several scholars (Leibniz, D'Alembert and Lazare Carnot) had stressed that there exist two contrasting kinds of organization of a theory, the "empirical" one and the "rational" one. See the detailed illustration by (L. Carnot 1783, pp. 101-103). In last century a similar dichotomy, between principle theories and constructive theories, was suggested in theoretical physics by both (Poincar 1902, ch. "Optique et Electricit"), and Einstein (Miller 1981, pp. 123-142).
I follow a verificationist theory of meaning in the sense of Prawitz (1987, p. 155), as referred to this kind of proposition. It may be considered as a reduction of the demarcation suggested by Popper for a theory, to a single doubly negated proposition.
Many scholars (e.g. Gardis 1991) consider this kind of proof as reducible to a direct proof; yet, they implicitly apply to its conclusion the classical law of the double negation.
This notion was introduced in intuitive terms by both Feyerabend and Kuhn (see Ali Khalidi 2001). I accurately defined it as generated by a difference in the two pairs of choices on the two dichotomies (Drago 1987).
(Heyting 1960 p. 102). See (Franchella 1994) for a historical reconstruction of this turning point in the history of intuitionism.
For both a first introduction to, and a first appraisal on it, see both Hilbert's papers in (van Heijenoort 1967) and van Heijenoort's introductions to them. About the lack of clarity of his papers, see (Mancosu 1998, p. 161).
Indeed, the choice AI is implied by the assumption of a strong version (i.e. with quantifiers) of the induction principle; whereas the AO choice is implied by the Formalist axiomatic way to see a mathematical theory.
For instance, let us recall Hilberts merely verbal transformation of a quantifier into an operator (-calculus), claimed by him to be capable of freeing a theory from problems given by the quantifiers. See Freudenthal s severe appraisal of Hilbert s philosophy of science (Freudenthal 1972, p. 393, col. I).
(Mancosu 1998, pp. 176); see also Hilbert s positive evaluations of these wrong results (ibidem, pp. 229, 270-1).
I attempted a new interpretation of both the historical meaning and the thesis of Goedels incompleteness theorem in (Drago 1993).
The complex nature of this proof was remarked by (Davis 1965, p. 108):: the proof (actually the very statement) of the Goedel completeness theorem is non-constructive
(Kolmogoroff 1931) had implicitly proved the same result.
It was called peaceful by (Meschkowsky 1965, fn. ch. 10). Let us recall also (Smorinsky 1982, p. 459): Both sides were right; both were wrong; there was not that much actual disagreement anyway; and nowadays only an occasional eccentric still pursues an anachronistic battle. Today we recognize not two competing views on mathematics, but two types of mathematics constructive and nonconstructive with certain relations obtaining between them. Constructive mathematics is the legacy of Brouwers philosophy and the relation between constructive and nonconstructive mathematics is the legacy of Hilberts program. In contrast to Smorynskys opinion the present paper stresses that the two types of mathematics differ not in some lateral features, but in foundational features; whereas under the technical viewpoint they are merely different, under the philosophical or foundational viewpoint they are at odd.
He was unsuccessful because his definition of a set was circular.(Jan 1995, sect.s 2.1, 3.1, 5.1-3) and (Jan 2010).
This definition leaves as indefinite how the series achieves the last point. See (Kogbetlianz 1968 App. 2). See also (Heyting 1960, p. 30-31 )
(Goodstein 1965, p. 91). The underlinings are added to make manifest the two negations within a doubly negated proposition.
(Weyl 1946, p. 9) wrote: "... classical logic was abstracted from the mathematics of finite sets and their subsets... Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of set theory, for which it is justly punished by the antinomies...". Also present mathematicians avoid this choice on the kind of logic: [the] generally accepted picture [in studying foundational problems, i.e. the logic is unproblematic] is wrong. It rests on woefully inadequate analysis of the entire problem situation in the foundations of logic and mathematics. It is based upon the unnecessarily restrictive idea of what our basic logic is like (Hintikka 1997, p. 158). See also point (ii) of (Feferman 1998, p. 288)..
Actually the new basic notion, membership, also gives rise to problems, since it requires, provided that a set is defined by the axioms, the non constructivist principle of omniscience. (Bishop 1967, pp. 1-10).
(Drake 1989, p. 15). For further current criticisms to contemporary ST, see (Feferman 1998, p. 287-288).
See e.g. Fraenkel who declared ST no less revolutionary than the Copernican system, or than the theory of relativity, or even quantum theory and nuclear physics. ((Fraenkel 1967; p. 240, last words of the book):
As an example Lebesgue stated that no discussion between the two parts was possible [about AC] because they had no common logic. (quoted in (Fraenkel 1967, p. 81).:
(Heyting 1960 p. 32). The preface of the book makes use of several doubly negated propositions (7 in the first period). Yet, he confesses that in the following text he will speak in a somewhat absolute way, in contrast to intuitionist philosophy.
For a more detailed discussion, see (Fraenkel 1967, p. 325).
(Mostowski 1967, pp. 83, 102); the discussant A. Robinson, states the same (p. 101) Suppes share this opinion, yet he exorcises the birth of alternatives within Mathematics (p. 115).
Weyl represented the research for the FoM in his times by means of a graph similar to a parallelogram (see Ferreirs, op. cit., p.361) in two dimensions, which correspond to the two dichotomies. According to Weyl; the first dimension (Brouwer-Hilbert) represents the opposition between intuition (and empiricism) and. the formalist axiomatic; this dimension corresponds to the dichotomy PO-AO; the second dimension (Brouwer-Weyl) represents the dichotomy PI-AI, where AI in this case represents, according to Weyl, the smallest degree of actual infinity, i.e. the introduction of a quantifier only.
(Mostowski 1967, pp. 82-83). (Foreman and Kanamori 2010, p. vii) want to find out new axioms for founding ST which involve a simple primitive notion that is easy to understand and can be used to build or develop all standard mathematical objects; however, even a reduced set of such axioms concerning a set, has eventually led ST to find itself at the confluence of the FoM, internal mathematical motivations and philosophical speculation. (Ib., p. x) Not surprisingly S. Feferman states: I am convinced that the Continuum Hypothesis is an inherently vague problem that no one axiom will settle in a convincingly definite way. (Feferman (1999, p. 109).
Let us notice that in a more general framework of the historical development of both Mathematics and Physics, the scientists of 20th Century had the opportunity to recognize the same above-illustrated options through an analysis of the failures, rather than the successes, of the dominant programs of research in each of these two branches of science. In Mathematics, the failure of Hilbert's program w
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Also in Physics this dichotomy regarding organisation was (partially) recognized, when the Newtonian paradigm which was based on AI&AO choices (Drago 1988) - failed. According to Einstein, the theoretical organization of special relativity is the alternative one to traditional, deductive organization (Klein 1967; Miller 1989, pp. 123). Yet, neither Einstein nor other physicists were able to define PO in accurate terms (Frisch 2006).
(Lobachevsky 1835-38, Introduction; Drago 1995; Cicenia, Drago 1996).
Actually, this pluralist framework has been already suggested by several mathematical theorists of the time around the French revolution, in particular by L. Carnot (1813).
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