Feferman, S. and G. Jäger, Systems of explicit mathematics with non-constructive μ-operator. Part I, Annals of Pure and Applied Logic 65 243-263. This paper is mainly concerned with the proof-theoretic analysis of systems of explicit mathematics with a non-constructive minimum operator. We start off from a basic theory BON of operators and numbers and add some principles of set and formula induction on the natural numbers as well as axioms for μ. The principal results then state: (...) BON plus set induction is proof-theoretically equivalent to Peano arithmetic PA; BON plus formula induction is proof-theoretically equivalent to the system <0 of second-order arithmetic. (shrink)

The aim of this paper is the analysis of uniformity in Feferman's explicit mathematics. The proof-strength of those systems for constructive mathematics is determined by reductions to subsystems of second-order arithmetic: If uniformity is absent, the method of standard structures yields that the strength of the join axiom collapses. Systems with uniformity and join are treated via cut elimination and asymmetrical interpretations in standard structures.

This is a critical analysis of the first part of Go¨del’s 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Go¨del’s discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of the truths of mathematics in an absolute sense, and the latter consists of all humanly demonstrable truths. The question is whether these coincide; if they do, no formal axiomatic system (or Turing machine) can (...) comprehend the mathematizing potentialities of human thought, and, if not, there are absolutely unsolvable mathematical problems of diophantine form. (shrink)

The background to the development of proof theory since 1960 is contained in the article (MATHEMATICS, FOUNDATIONS OF), Vol. 5, pp. 208- 209. Brie y, Hilbert's program (H.P.), inaugurated in the 1920s, aimed to secure the foundations of mathematics by giving nitary consistency proofs of formal systems such as for number theory, analysis and set theory, in which informal mathematics can be represented directly. These systems are based on classical logic and implicitly or explicitly depend on the (...) assumption of \completed in nite" totalities. Consistency of a system S (containing a modicum of elementary number theory) is su cient to ensure that any nitary meaningful statement about the natural numbers which is provable in S is correct under the intended interpretation. Thus, in Hilbert's view, consistency of S would serve to eliminate the \completed in nite" in favor of the \potential in nite" and thus secure the body of mathematics represented in S. Hilbert established the subject of proof theory as a technical part of mathematical logic by means of which his program was to be carried out its methods will be described below. In 1931, Godel's second incompleteness theorem raised a prima facieobstacle to H.P. for the system Z of elementary number theory (also called Peano Arithmetic, and denoted below by PA) since all previously recognized forms of nitary reasoning could be formalized within it. In any case, Hilbert's program could not possibly succeed for any system such as set theory in which all nitary notions and reasoning could unquestionably be formalized. These obstacles led workers in proof theory to modify H.P. in two kinds of ways. The rst was to seek reductions of various formal systems S to more constructive systems S 0. The second was to shift the aims from foundational ones to more mathematical ones. Examples of 1 the former move are the reductions of PA to intuitionistic arithmetic HA, and Gentzen's consistency proof of PA by nitary reasoning coupled with quanti er-free trans nite induction up to the ordinal , TI( 0), both obtained in the 1930s (cf.. (shrink)

In his book Shadows of the Mind: A search for the missing science of con- sciousness [SM below], Roger Penrose has turned in another bravura perfor- mance, the kind we have come to expect ever since The Emperor’s New Mind [ENM ] appeared. In the service of advancing his deep convictions and daring conjectures about the nature of human thought and consciousness, Penrose has once more drawn a wide swath through such topics as logic, computa- tion, artificial intelligence, quantum physics (...) and the neuro-physiology of the brain, and has produced along the way many gems of exposition of difficult mathematical and scientific ideas, without condescension, yet which should be broadly appealing. 1 While the aims and a number of the topics in SM are the same as in ENM, the focus now is much more on the two axes that Pen- rose grinds in earnest. Namely, in the first part of SM he argues anew and at great length against computational models of the mind and more specifi- cally against any account of mathematical thought in computational terms. Then in the second part, he argues that there must be a scientific account of consciousness but that will require a (still to be found) non-computational extension or modification of present-day quantum physics. (shrink)

This paper continues investigations of the monotone fixed point principle in the context of Feferman's explicit mathematics begun in [14]. Explicit mathematics is a versatile formal framework for representing Bishop-style constructive mathematics and generalized recursion theory. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications (Feferman's notion of set) possesses a least fixed point. To (...) be more precise, the new axiom not merely postulates the existence of a least solution, but, by adjoining a new constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. Let T 0 + UMID denote this extension of explicit mathematics. [14] gave lower bounds for the strength of two subtheories of T 0 + UMID in relating them to fragments of second order arithmetic based on Π 1 2 comprehension. [14] showed that T $_0 \upharpoonright$ + UMID and T $_0 \upharpoonright$ + IND N + UMID have at least the strength of (Π 1 2 - CA) $\upharpoonright$ and (Π 1 2 - CA), respectively. Here we are concerned with the exact reversals. Let UMID N be the monotone fixed-point principle for subclassifications of the natural numbers. Among other results, it is shown that T $_0 \upharpoonright$ + UMID N and T $_0 \upharpoonright$ + IND N + UMID N have the same strength as (Π 1 2 - CA) $\upharpoonright$ and (Π 1 2 - CA), respectively. The results are achieved by constructing set-theoretic models for the aforementioned systems of explicit mathematics in certain extensions of Kripke-Platek set theory and subsequently relating these set theories to subsystems of second arithmetic. (shrink)

This paper is mainly concerned with proof-theoretic analysis of some second-order systems of explicit mathematics with a non-constructive minimum operator. By introducing axioms for variable types we extend our first-order theory BON to the elementary explicit type theory EET and add several forms of induction as well as axioms for μ. The principal results then state: EET plus set induction is proof-theoretically equivalent to Peano arithmetic PA <0).

This paper continues investigations of the monotone fixed point principle in the context of Feferman's explicit mathematics begun in [14]. Explicit mathematics is a versatile formal framework for representing Bishop-style constructive mathematics and generalized recursion theory. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications possesses a least fixed point. To be more precise, the new (...) axiom not merely postulates the existence of a least solution, but, by adjoining a new constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. Let T$_0$ + UMID denote this extension of explicit mathematics. [14] gave lower bounds for the strength of two subtheories of T$_0$ + UMID in relating them to fragments of second order arithmetic based on $\Pi^1_2$ comprehension. [14] showed that T$_0 \upharpoonright$ + UMID and T$_0 \upharpoonright$ + IND$_\mathbb{N}$ + UMID have at least the strength of $\upharpoonright$ and, respectively. Here we are concerned with the exact reversals. Let UMID$_\mathbb{N}$ be the monotone fixed-point principle for subclassifications of the natural numbers. Among other results, it is shown that T$_0 \upharpoonright$ + UMID$_\mathbb{N}$ and T$_0 \upharpoonright$ + $IND_\mathbb{N}$ + UMID$_\mathbb{N}$ have the same strength as $ $\upharpoonright$ and, respectively. The results are achieved by constructing set-theoretic models for the aforementioned systems of explicit mathematics in certain extensions of Kripke-Platek set theory and subsequently relating these set theories to subsystems of second arithmetic. (shrink)

The context for this paper is Feferman's theory of explicit mathematics, a formal framework serving many purposes. It is suitable for representing Bishop-style constructive mathematics as well as generalized recursion, including direct expression of structural concepts which admit self-application. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications possesses a least fixed point. To be more precise, the (...) new axiom not merely postulates the existence of a least solution, but, by adjoining a new functional constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. The upshot of the paper is that the latter extension of explicit mathematics embodies considerable proof-theoretic strength. It is shown that it has at least the strength of the subsystem of second order arithmetic based on $\Pi^1_2$ comprehension. (shrink)

The context for this paper is Feferman's theory of explicit mathematics, a formal framework serving many purposes. It is suitable for representing Bishop-style constructive mathematics as well as generalized recursion, including direct expression of structural concepts which admit self-application. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications (Feferman's notion of set) possesses a least fixed point. (...) To be more precise, the new axiom not merely postulates the existence of a least solution, but, by adjoining a new functional constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. The upshot of the paper is that the latter extension of explicit mathematics (when based on classical logic) embodies considerable proof-theoretic strength. It is shown that it has at least the strength of the subsystem of second order arithmetic based on Π 1 2 comprehension. (shrink)

The determinism-free will debate is perhaps as old as philosophy itself and has been engaged in from a great variety of points of view including those of scientific, theological, and logical character. This chapter focuses on two arguments from logic. First, there is an argument in support of determinism that dates back to Aristotle, if not farther. It rests on acceptance of the Law of Excluded Middle, according to which every proposition is either true or false, no matter whether the (...) proposition is about the past, present or future. In particular, the argument goes, whatever one does or does not do in the future is determined in the present by the truth or falsity of the corresponding proposition. The second argument coming from logic is much more modern and appeals to Gödel's incompleteness theorems to make the case against determinism and in favour of free will, insofar as that applies to the mathematical potentialities of human beings. The claim more precisely is that as a consequence of the incompleteness theorems, those potentialities cannot be exactly circumscribed by the output of any computing machine even allowing unlimited time and space for its work. The chapter concludes with some new considerations that may be in favour of a partial mechanist account of the mathematical mind. (shrink)

This paper is concerned with the determination of the proof-strength of the power set axiom relative to axiom systems for Feferman's explicit mathematics. As conjectured by Feferman, we obtain that the presence of the power set axiom does not increase proof-strength. Results are achieved by reducing the systems including the power set axiom to subsystems of classical analysis. In those cases where only the induction axiom is available, we make use of the technique of asymmetrical interpretations.

This paper is a direct successor to 12. Its aim is to introduce a new realisability interpretation for weak systems of explicit mathematics and use it in order to analyze extensions of the theory PET in 12 by the so-called join axiom of explicit mathematics.

From 1931 until late in his life (at least 1970) Godel called for the pursuit of new axioms for mathematics to settle both undecided number-theoretical propositions (of the form obtained in his incompleteness results) and undecided set-theoretical propositions (in particular CH). As to the nature of these, Godel made a variety of suggestions, but most frequently he emphasized the route of introducing ever higher axioms of in nity. In particular, he speculated (in his 1946 Princeton remarks) that there might (...) be a uniform (though non-decidable) rationale for the choice of the latter. Despite the intense exploration of the \higher in nite" in the last 30-odd years, no single rationale of that character has emerged. Moreover, CH still remains undecided by such axioms, though they have been demonstrated to have many other interesting set-theoretical consequences. (shrink)

The paper starts with an examination and critique of Tarski’s wellknown proposed explication of the notion of logical operation in the type structure over a given domain of individuals as one which is invariant with respect to arbitrary permutations of the domain. The class of such operations has been characterized by McGee as exactly those definable in the language L∞,∞. Also characterized similarly is a natural generalization of Tarski’s thesis, due to Sher, in terms of bijections between domains. My main (...) objections are that on the one hand, the Tarski-Sher thesis thus assimilates logic to mathematics, and on the other hand fails to explain the notion of same logical operation across domains of different sizes. A new notion of homomorphism invariant operation over functional type structures (with domains M0 of individuals and {T, F} at their base) is introduced to accomplish the latter. The main result is that an operation is definable from.. (shrink)

“Small” large cardinal notions in the language of ZFC are those large cardinal notions that are consistent with V = L. Besides their original formulation in classical set theory, we have a variety of analogue notions in systems of admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics and recursive ordinal notations (as used in proof theory). On the face of it, it is surprising that such distinctively set-theoretical notions have analogues in such (...) disaparate and relatively constructive contexts. There must be an underlying reason why that is possible (and, incidentally, why “large” large cardinal notions have not led to comparable analogues). My long term aim is to develop a common language in which such notions can be expressed and can be interpreted both in their original classical form and in their analogue form in each of these special constructive and semi-constructive cases. This is a program in progress. What is done here, to begin with, is to show how that can be done to a considerable extent in the settings of classical and admissible set theory (and thence, admissible recursion theory). The approach taken here is to expand the language of set theory to allow us to talk about (possibly partial) operations applicable both to sets and to operations and to formulate the large cardinal notions in question in terms of operational closure conditions; at the same time only minimal existence axioms are posited for sets. The resulting system, called Operational Set Theory, is a partial adaptation to the set-theoretical framework of the explicit mathematics framework Feferman (1975). The.. (shrink)

Kurt Gödel (1906-1978) did groundbreaking work that transformed logic and other important aspects of our understanding of mathematics, especially his proof of the incompleteness of formalized arithmetic. This book on different aspects of his work and on subjects in which his ideas have contemporary resonance includes papers from a May 2006 symposium celebrating Gödel's centennial as well as papers from a 2004 symposium. Proof theory, set theory, philosophy of mathematics, and the editing of Gödel's writings are among the (...) topics covered. Several chapters discuss his intellectual development and his relation to predecessors and contemporaries such as Hilbert, Carnap, and Herbrand. Others consider his views on justification in set theory in light of more recent work and contemporary echoes of his incompleteness theorems and the concept of constructible set"--Provided by publisher. (shrink)

The context for this paper is Feferman's theory of explicit mathematics, T 0 . We address a problem that was posed in [6]. Let MID be the principle stating that any monotone operation on classifications has a least fixed point. The main objective of this paper is to show that T 0 + MID, when based on classical logic, also proves the existence of non-monotone inductive definitions that arise from arbitrary extensional operations on classifications. From the latter (...) we deduce that MID, when adjoined to classical T 0 , leads to a much stronger theory than T 0. (shrink)

This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are proof-theoretically equivalent to Feferman's.

The aim of this article is to give the proof-theoretic analysis of various subsystems of Feferman's theory T1 for explicit mathematics which contain the non-constructive μ-operator and join. We make use of standard proof-theoretic techniques such as cut-elimination of appropriate semiformal systems and asymmetrical interpretations in standard structures for explicit mathematics.

Ambiguity is a property of syntactic expressions which is ubiquitous in all informal languages–natural, scientific and mathematical; the efficient use of language depends to an exceptional extent on this feature. Disambiguation is the process of separating out the possible meanings of ambiguous expressions. Ambiguity is typical if the process of disambiguation can be carried out in some systematic way. Russell made use of typical ambiguity in the theory of types in order to combine the assurance of its (apparent) consistency (“having (...) the cake”) with the freedom of the informal untyped theory of classes and relations (“eating it too”). The paper begins with a brief tour of Russell’s uses of typical ambiguity, including his treatment of the statement Cls ∈ Cls. This is generalized to a treatment in simple type theory of statements of the form A ∈ B where A and B are class expressions for which A is prima facie of the same or higher type than B. In order to treat mathematically more interesting statements of self membership we then formulate a version of typical ambiguity for such statements in an extension of Zermelo-Fraenkel set theory. Specific attention is given to how the“naive” theory of categories can thereby be accounted for. (shrink)

When I was a teenager growing up in Los Angeles in the early 1940s, my dream was to become a mathematical physicist: I was fascinated by the ideas of relativity theory and quantum mechanics, and I read popular expositions which, in those days, besides Einstein’s The Meaning of Relativity, was limited to books by the likes of Arthur S. Eddington and James Jeans. I breezed through the high-school mathematics courses (calculus was not then on offer, and my teachers barely (...) understood it), but did less well in physics, which I should have taken as a reality check. On the philosophical side I read a mixed bag of Bertrand Russell, John Dewey and Alfred Korzybski (the missionary for “General Semantics” in Science and Sanity, a mish-mash of the theory of types, non-. (shrink)

1. One theory or many? In 2004 a very interesting and readable article by Lenore Blum, entitled “Computing over the reals: Where Turing meets Newton,” appeared in the Notices of the American Mathematical Society. It explained a basic model of computation over the reals due to Blum, Michael Shub and Steve Smale (1989), subsequently exposited at length in their influential book, Complexity and Real Computation (1997), coauthored with Felipe Cucker. The ‘Turing’ in the title of Blum’s article refers of course (...) to Alan Turing, famous for his explication of the notion of mechanical computation on discrete data such as the integers. The ‘Newton’ there has to do to with the well known numerical method due to Isaac Newton for approximating the zeros of continuous functions under suitable conditions that is taken to be a paradigm of scientific computing. Finally, the meaning of “Turing meets Newton” in the title of Blum’s article has another, more particular aspect: in connection with the problem of controlling errors in the numerical solution of linear equations and inversion of matrices,Turing (1948) defined a notion of condition for the relation of changes in the output of a computation due to small changes in the input that is analogous to Newton’s definition of the derivative. The thrust of Blum’s 2004 article was that the BSS model of computation on the reals is the appropriate foundation for scientific computing in general. By way of response, two years later another very interesting and equally readable article appeared in the Notices, this time by Mark Braverman and Stephen Cook (2006) entitled “Computing over the reals: Foundations for scientific computing,” in which the authors argued that the requisite foundation is provided by a quite different “bit computation” model, that is in fact prima facie incompatible with the BSS model. The bit computation model goes back to ideas due to Stefan Banach and Stanislaw Mazur in the latter part of the 1930s, but the first publication was not made until Mazur (1963).. (shrink)

The point of departure for this panel is a somewhat controversial paper that I published in the American Mathematical Monthly under the title “Does mathematics need new axioms?” [4]. The paper itself was based on a lecture that I gave in 1997 to a joint session of the American Mathematical Society and the Mathematical Association of America, and it was thus written for a general mathematical audience. Basically, it was intended as an assessment of Gödel’s program for new axioms (...) that he had advanced most prominently in his 1947 paper for the Monthly, entitled “What is Cantor’s continuum problem?” [7]. My paper aimed to be an assessment of that program in the light of research in mathematical logic in the intervening years, beginning in the 1960s, but especially in more recent years. (shrink)

In addition to this being the centenary of Kurt Gödel’s birth, January marked 75 years since the publication (1931) of his stunning incompleteness theorems. Though widely known in one form or another by practicing mathematicians, and generally thought to say something fundamental about the limits and potentialities of mathematical knowledge, the actual importance of these results for mathematics is little understood. Nor is this an isolated example among famous results. For example, not long ago, Philip Davis wrote me about (...) what he calls The Paradox of Irrelevance: “There are many math problems that have achieved the cachet of tremendous significance, e.g. Fermat, 4 color, Kepler’s packing, Gödel, etc. Of Fermat, I have read: ‘the most famous math problem of all time.’ Of Gödel, I have read: ‘the most mathematically significant achievement of the 20th century.’ … Yet, these problems have engaged the attention of relatively few research mathematicians—even in pure math.” What accounts for this disconnect between fame and relevance? Before going into the question for Gödel’s theorems, it should be distinguished in one respect from the other examples mentioned, which in any case form quite a mixed bag. Namely, each of the Fermat, 4 color, and Kepler’s packing problems posed a stand-out challenge following extended efforts to settle them; meeting the challenge in each case required new ideas or approaches and intense work, obviously of different degrees. By contrast, Gödel’s theorems were simply unexpected, and their proofs, though requiring novel techniques, were not difficult on the scale of things. Setting that aside, my view of Gödel’s incompleteness theorems is that their relevance to mathematical logic (and its offspring in the theory of computation) is paramount; further, their philosophical relevance is significant, but in just what way is far from settled; and finally, their mathematical relevance outside of logic is very much unsubstantiated but is the object of ongoing, tantalizing efforts.. (shrink)

What Gödel accomplished in the decade of the 1930s before joining the Institute changed the face of mathematical logic and continues to influence its development. As you gather from my title, I’ll be talking about the most famous of his results in that period, but first I want to indulge in some personal reminiscences. In many ways this is a sentimental journey for me. I was a member of the Institute in 1959-60, a couple of years after receiving my PhD (...) at the University of California in Berkeley, where I had worked with Alfred Tarski, another great logician. The subject of my dissertation was directly concerned with the method of arithmetization that Gödel had used to prove his theorems, and my main concern after that was to study systematic ways of overcoming incompleteness. Mathematical logic was going through a period of prodigious development in the 1950s and 1960s, and Berkeley and Princeton were two meccas for researchers in that field. For me, the prospect of meeting with Gödel and drawing on him for guidance and inspiration was particularly exciting. I didn’t know at the time what it took to get invited. Hassler Whitney commented for an obituary notice in 1978 that “it was hard to appoint a new member in logic at the Institute because Gödel could not prove to himself that a number of candidates shouldn’t be members, with the evidence at hand.” That makes it sound like the problem for Gödel was deciding who not to invite. Anyhow, I ended up being one of the lucky few. (shrink)

This is a critical reexamination of several pieces in van Heijenoort’s Selected Essays that are directly or indirectly concerned with the philosophy of logic or the relation of logic to natural language. Among the topics discussed are absolutism and relativism in logic, mass terms, the idea of a rational dictionary, and sense and identity of sense in Frege.

Four requirements are suggested for an axiomatic system S to provide the foundations of category theory: (R1) S should allow us to construct the category of all structures of a given kind (without restriction), such as the category of all groups and the category of all categories; (R2) It should also allow us to construct the category of all functors between any two given categories including the ones constructed under (R1); (R3) In addition, S should allow us to establish the (...) existence of the usual basic mathematical structures and carry out the usual set-theoretical operations; and (R4) S should be shown to be consistent relative to currently accepted systems of set theory. This paper explains how all but parts of (R3) can be met using a system S extending NFU enriched by a stratified pairing operation; to meet more of (R3) a stronger system S∗ is introduced, but there are still some real obstacles to meeting this requirement in full. For (R4) it is sketched how both S and S∗ are shown to be consistent. (shrink)

In his 1918 monograph \Das Kontinuum", Hermann Weyl initiated a program for the arithmetical foundations of mathematics. In the years following, this was overshadowed by the foundational schemes of Hilbert's nitary consistency program and Brouwer's intuitionistic redevelopment of mathematics. In fact, not long after his own venture, Weyl became a convert to Brouwerian intuitionism and criticized his old teacher's program. Over the years, though, he became more and more pessimistic about the practical possibilities of reworking mathematics along (...) intuitionistic lines, and pointed to the value of his own early foundational e orts. Weyl 's work in Das Kontinuum has come to be recognized for its importance as the opening chapter in the actual development of predicative mathematics, whose extent has been plumbed both mathematically and logically since the 1960s. (shrink)

In the year 1900, the German mathematician David Hilbert gave a dramatic address in Paris, at the meeting of the 2nd International Congress of Mathematicians—an address which was to have lasting fame and importance. Hilbert was at that point a rapidly rising star, if not superstar, in mathematics, and before long he was to be ranked with Henri Poincar´.

The final two volumes, numbers IV and V, of the Oxford University Press edition of the Collected Works of Kurt Gödel [3]-[7] appeared in 2003, thus completing a project that started over twenty years earlier. What I mainly want to do here is trace, from the vantage point of my personal involvement, the at some times halting and at other times intense development of the Gödel editorial project from the first initiatives following Gödel’s death in 1978 to its completion last (...) year. It may be useful to scholars mounting similar editorial projects for other significant figures in our field to learn how and why various decisions were made and how the work was carried out, though of course much is particular to who and what we were dealing with. My hope here is also to give the reader who is not already familiar with the Gödel Works a sense of what has been gained in the process, and to encourage dipping in according to interest. Given the absolute importance of Gödel for mathematical logic, students should also be pointed to these important source materials to experience first hand the exercise of his genius and the varied ways of his thought and to see how scholarly and critical studies help to expand their significance. Though indeed much has been gained in our work there is still much that can and should be done; besides some indications below, for that the reader is referred to [2]. (shrink)

In the first part of this article, Feferman outlines his ‘conceptual structuralism’ and emphasizes broad similarities between Parsons’s and his own structuralist perspective on mathematics. However, Feferman also notices differences and makes two critical claims about any structuralism that focuses on the “ur-structures” of natural and real numbers: it does not account for the manifold use of other important structures in modern mathematics and, correspondingly, it does not explain the ubiquity of “individual [natural or real] numbers” in that (...) use. In the second part, Feferman presents a summary of his reasons for the skepticism he has towards contemporary metamathematical investigations of set theory. That skepticism led him to reject the Continuum Problem as a definite mathematical one. He contrasts that attitude sharply to Parsons’s “great sympathy for the current explorations of higher set theory.”. (shrink)

Like Heisenberg’s uncertainty principle, Gödel’s incompleteness theorem has captured the public imagination, supposedly demonstrating that there are absolute limits to what can be known. More specifically, it is thought to tell us that there are mathematical truths which can never be proved. These are among the many misconceptions and misuses of Gödel’s theorem and its consequences. Incompleteness has been held to show, for example, that there cannot be a Theory of Everything, the so-called holy grail of modern physics. Some philosophers (...) and mathematicians say it proves that minds can’t be modelled by machines, while others argue that they can be modelled but that Gödel’s theorem shows we can’t know it. Postmodernists have claimed to find support in it for the view that objective truth is chimerical. And in the Bibliography of Christianity and Mathematics (yes, there is such a publication) it is asserted that ‘theologians can be comforted in their failure to systematize revealed truth because mathematicians cannot grasp all mathematical truths in their systems either.’ Not only that, the incompleteness theorem is held to imply the existence of God, since only He can decide all truths. Even Rebecca Goldstein’s book, whose laudable aim is to provide non-technical expositions of the incompleteness theorems (there are two) for a general audience and place them in their historical and biographical context, makes extravagant claims and distorts their significance. As Goldstein sees it, Gödel’s theorems are ‘the most prolix theorems in the history of mathematics’ and address themselves ‘to the central question of the humanities – ‘what is it to be human?’ – since they involve ‘such vast and messy areas as the nature of truth and knowledge and certainty’. Unfortunately, these weighty claims disintegrate under closer examination, while the book as a whole is marred by a number of disturbing conceptual and historical errors. On the face of it, Goldstein would appear to have been an ideal choice to present Gödel’s work: she received a PhD in Philosophy from Princeton University in 1977 and since then has taught philosophy of science and philosophy of mind at several.... (shrink)

In the summer of 1957 at Cornell University the first of a cavalcade of large-scale meetings partially or completely devoted to logic took place--the five-week long Summer Institute for Symbolic Logic. That meeting turned out to be a watershed event in the development of logic: it was unique in bringing together for such an extended period researchers at every level in all parts of the subject, and the synergetic connections established there would thenceforth change the face of mathematical logic both (...) qualitatively and quantitatively. Prior to the Cornell meeting there had been nothing remotely like it for logicians. Previously, with the growing importance in the twentieth century of their subject both in mathematics and philosophy, it had been natural for many of the broadly representative meetings of mathematicians and of philosophers to include lectures by logicians or even have special sections devoted to logic. Only with the establishment of the Association for Symbolic Logic in 1936 did logicians begin to meet regularly by themselves, but until the 1950s these occasions were usually relatively short in duration, never more than a day or two. Alfred Tarski was one of the principal organizers of the Cornell institute and of some of the major meetings to follow on its heels. Before the outbreak of World War II, outside of Poland Tarski had primarily been involved in several Unity of Science Congresses, including the first, in Paris in 1935, and the fifth, at Harvard in September, 1939. (It was the latter which brought him to the United States and fortuitously left him stranded there following the Nazi invasion of Poland.) Much attention had been given to logic at these congresses and to Tarski’s own work, in particular, through the deep interest in it of Carnap, Quine and others. Following the end of the war, Tarski forged new alliances, especially in the United States logical and mathematical communities. To begin with, as part of the year-long celebration of the two-hundredth anniversary of the founding of Princeton University, a high-level conference on the Problems of Mathematics was held there in December 1946.. (shrink)

We show that, if non-uniform impredicative stratified comprehension is assumed, Feferman's theories of explicit mathematics are consistent with a strong power type axiom. This result answers a problem, raised by Jäger. The proof relies upon an interpretation into Quine's set theory NF with urelements.

In this paper we discuss extensions of Feferman's theory T 0 for explicit mathematics by the so-called limit and Mahlo axioms and present a novel approach to constructing natural recursion-theoretic models for systems of explicit mathematics which is based on nonmonotone inductive definitions.

This paper studies systems of explicit mathematics as introduced by Feferman [9, 11]. In particular, we propose weak explicit type systems with a restricted form of elementary comprehension whose provably terminating operations coincide with the functions on binary words that are computable in polynomial time. The systems considered are natural extensions of the first-order applicative theories introduced in Strahm [19, 20].

This article does not give the answer to the title question, but is only limited to studying the possibility of giving it. In particular, the author defends that it is legitimate to pose the fundamental question of the philosophy of mathematics and offers several criteria for such a question. As a first approach we propose the question which is incorrect and requires rectification, but is understandable: ‘What is Mathematics?‘. We consider three groups of strategies of responding to it: (...) 1) the question is naive and does not require an answer; 2) the answer is contained in the mathematical knowledge itself; 3) only philosophy can give the answer. Further rectifications to the basic question are questions about the essence and existence of mathematics. The fundamentalist and cultural-historical philosophy of mathematics gave their not fully satisfactory answers to this question in the past century. The dualism problem of philosophy of mathematics is fundamentally resolved through a certain kind of dialectics, which requires the philosophy of mathematics to submit to metaphysics. The unity of existence and essence is considered in the unity of the subject of dialectic - the mathematician who generates the knowledge as self-realization in a series of existential choices, as a means of saving transcendence - the existence ‘toward being‘ rather than ‘toward nothing‘. Positive view of the mathematics turns out to be the explicit expression of negative unity i.e. the subject. (shrink)

The historian of philosophy often encounters arguments that are enthymematic: they have conclusions that follow from their explicit premises only by the addition of "tacit" or "suppressed" premises. It is a standard practice of interpretation to supply these missing premises, even where the enthymeme is "real," that is, where there is no other context in which the philosopher in question asserts the missing premises. To do so is to follow a principle of charity: other things being equal, one interpretation (...) is better than another just to the extent that the one produces a better argument than the other. We show that this principle leads to paradoxical conclusions, including the following: there is no objectively correct interpretation of any real enthymeme found in the text of a major philosopher; an interpreter will not regard a real enthymeme of a major philosopher as adequately interpreted until he has found a way of reading it that makes it into a good argument; every classical philosopher is infallible and omniscient; major philosophers never disagree. These conclusions are preposterous, but there are indications that they are in fact being reached, as we show by means of a case study of recent scholarship on Plato's Third Man Argument. To avoid the overinterpretation and anachronism that result from the unrestrained use of the principle of charity, one must employ a counterbalancing principle of parsimony: to seek the simplest explanation for the text under discussion. We study the role of the principle of parsimony by means of a mathematical case study, involving the suppressed premises in Euclid's Elements. Here the principle of parsimony plays a larger role than it does in the interpretation of philosophical texts, leading to a sharper distinction between Euclid's geometry and Euclidean geometry than we find between Plato and Platonism. We conclude by comparing two models of interpretation, which we call prospective and retrospective. Although the prospective model of interpretation leads to Platonism rather than to Plato, we argue that it still has a place in Platonic scholarship. (shrink)