Hilbert's finitist program was not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration with Bernays during the period from 1917 to 1922. These notes reveal a dialectic progression from a critical logicism through a radical constructivism toward finitism; the progression has (...) to be seen against the background of the stunning presentation of mathematical logic in the lectures given during the winter term 1917/18. In this paper, I sketch the connection of Hilbert's considerations to issues in the foundations of mathematics during the second half of the 19th century, describe the work that laid the basis of modern mathematical logic, and analyze the first steps in the new subject of proof theory. A revision of the standard view of Hilbert's and Bernays's contributions to the foundational discussion in our century has long been overdue. It is almost scandalous that their carefully worked out notes have not been used yet to understand more accurately the evolution of modern logic in general and of Hilbert's Program in particular. One conclusion will be obvious: the dogmatic formalist Hilbert is a figment of historical (de)construction! Indeed, the study and analysis of these lectures reveal a depth of mathematical-logical achievement and of philosophical reflection that is remarkable. In the course of my presentation many questions are raised and many more can be explored; thus, I hope this paper will stimulate interest for new historical and systematic work. (shrink)

Moral relativists often defend their view as an inference to the best explanation of widespread and deep moral disagreement. Many philosophers have challenged this line of reasoning in recent years, arguing that moral objectivism provides us with ample resources to develop an equally or more plausible method of explanation. One of the most promising of these objectivist methods is what I call the self-interest explanation, the view that intractable moral diversity is due to the distorting effects of our interests. In (...) this paper I examine the self-interest explanation through the lens of the famine debate, a well-known disagreement over whether we have a moral obligation to donate most of our income to the global poor. I argue that objectivists should reduce their confidence that the persistence of the famine debate is due to the distorting influence of self-interest. If my argument is on target, then objectivists may need to supply a stronger explanation of moral disagreement to defend their view against the threat of moral relativism. (shrink)

Alasdair MacIntyre has developed a theory of the rationality of traditions that is designed to show how we can maintain both the tradition-bound nature of rationality, on the one hand, and non-relativism, on the other. However, his theory has been widely criticized. A number of recent commentators have argued that the theory is either inconsistent with his own conception of rationality or else is dependent on the standards of his particular tradition and therefore fails to defuse the threat of relativism. (...) In the present essay, I argue that this objection is mistaken. (shrink)

People disagree about whether individuals in rich countries like the United States have an obligation to aid the world’s poorest people. A tempting thought is that this disagreement comes down to a non-moral matter. I argue that we should be suspicious of this view. Drawing on psychological evidence, I show that we should be more pessimistic about our ability to attribute the disagreement to a difference in factual beliefs.

Faced with endlessly repeated opportunities to save drowning children, most people think morality intuitively permits us to indulge in at least some goods that are not nearly as important as a child’s life. Some philosophers argue that this intuition gives us an important (though defeasible) reason to think we may sometimes permissibly refuse to save a life even when we can do so at insignificant cost. I argue that recent psychological experiments should make us wary of this claim.

Disagreement has been grist to the mills of sceptics throughout the history of philosophy. Recently, though, some philosophers have argued that widespread philosophical disagreement supports a broad scepticism about philosophy itself. In this paper, I argue that the task for sceptics of philosophy is considerably more complex than commonly thought. The mere fact that philosophical methods fail to generate true majority views is not enough to support the sceptical challenge from disagreement. To avoid demanding something that human reasoning cannot supply, (...) sceptics must show that philosophers have sufficient overlap to resolve their disagreements in particular concrete cases. (shrink)

Wilfried Sieg. Calculations by Man and Machine: Conceptual Analysis.

A common response to ongoing disagreement about abortion has been to look for overlap between the prolife and prochoice sides of the debate. In recent years, however, both opposing camps in the debate have claimed to be able to establish their respective positions on the basis of the same common ground. Faced with the apparent failure of philosophers to settle their differences about abortion by means of shared values, the question naturally arises: what should we do about this? It is (...) possible to discern two main options in the scholarly literature. The first is to keep doing what we are already doing: namely, trying to create consensus on the abortion issue by appealing to analogies designed to reveal common ground. One example of this approach comes from a recent attempt by Charles Camosy to find areas of agreement with Peter Singer and other defenders of abortion.See Charles Camosy, “Common Ground on Surgical Abortion?—Engaging Peter Singer on the Moral Status of Potential Persons,”. (shrink)

Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame (...) is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas. To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at (see link below). (shrink)

The incompleteness theorems constitute the mathematical core of Gödel’s philosophical challenge. They are given in their “most satisfactory form”, as Gödel saw it, when the formality of theories to which they apply is characterized via Turing machines. These machines codify human mechanical procedures that can be carried out without appealing to higher cognitive capacities. The question naturally arises, whether the theorems justify the claim that the human mind has mathematical abilities that are not shared by any machine. Turing admits that (...) non-mechanical steps of intuition are needed to transcend particular formal theories. Thus, there is a substantive point in comparing Turing’s views with Gödel’s that is expressed by the assertion, “The human mind infinitely surpasses any finite machine”. The parallelisms and tensions between their views are taken as an inspiration for beginning to explore, computationally, the capacities of the human mathematical mind. (shrink)

Herbrand's Theorem, in the form of $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\exists } $$ -inversion lemmata for finitary and infinitary sequent calculi, is the crucial tool for the determination of the provably total function(al)s of a variety of theories. The theories are (second order extensions of) fragments of classical arithmetic; the classes of provably total functions include the elements of the Polynomial Hierarchy, the Grzegorczyk Hierarchy, and the extended Grzegorczyk Hierarchy $\mathfrak{E}^\alpha $ , α < ε0. A subsidiary aim of the paper is to show (...) that the proof theoretic methods used here are distinguished by technical elegance, conceptual clarity, and wide-ranging applicability. (shrink)

David Hilbert was one of the great mathematicians who expounded the centrality of their subject in human thought. In this collection of essays, Wilfried Sieg frames Hilbert's foundational work, from 1890 to 1939, in a comprehensive way and integrates it with modern proof theoretic investigations.

The Carnegie Mellon Proof Tutor project was motivated by pedagogical concerns: we wanted to use a "mechanical" (i.e. computerized) tutor for teaching students..

It is a remarkable fact that Hilbert's programmatic papers from the 1920s still shape, almost exclusively, the standard contemporary perspective of his views concerning (the foundations of) mathematics; even his own, quite different work on the foundations of geometry and arithmetic from the late 1890s is often understood from that vantage point. My essay pursues one main goal, namely, to contrast Hilbert's formal axiomatic method from the early 1920s with his existential axiomatic approach from the 1890s. Such a contrast illuminates (...) the circuitous beginnings of the finitist consistency program and connects the complex emergence of existential axiomatics with transformations in mathematics and philosophy during the 19th century; the sheer complexity and methodological difficulties of the latter development are partially reflected in the well known, but not well understood correspondence between Frege and Hilbert. Taking seriously the goal of formalizing mathematics in an effective logical framework leads also to contemporary tasks, not just historical and systematic insights; those are briefly described as “one direction” for fascinating work. (shrink)

The Belgian polyarchy has developed into a «particracy» in which the political parties have a preponderant impact on the political decisionmaking.As in every polyarchy, the political parties fulfill a set of vital functions in the Belgian system. But that does not explain its «particratie» particularity. The particracy arises as a consequence of four factors. The parties are fixed by the «zuilen» into an «internal-imperialistic» role. The power of the parties increased considerably, an increase of which the politization of the instruments (...) for, and the agencies of, policy is one of the most important, together with the capability of making and unmaking issues. The impact and power of the opponents of the parties in the decision-making process have been diminished to a considerable extent. This weakening involves the king, the parliament, theadministration, the electorate, and, ultimately, the cabinet because of the growing impact of the parties on policy design and policy output as well as on the appointment of ministers. Finally, the choice of the consociational pattern of politics of accomodation by the elite indicates the instrumental function of particracy in Belgian politics. (shrink)

Recent decades have seen a shift away from the traditional view that Aquinas's theory of the natural law is meant to supply us with normative guidance grounded in a substantive theory of human nature. In the present essay, I argue that this is a mistake. Expanding on the suggestions of Jean Porter and Ralph McInerny, I defend a derivationist reading of ST I-II, Q. 94, A. 2 according to which Aquinas takes our knowledge of the genuine goods of human life (...) and their proper ordering to one another to be self-evident only to the wise who are able to discern the truth about our God-given human nature. I then show that this reading provides a better account of Aquinas's view than two recent alternatives: John Finnis's brand of inclinationism and Daniel Mark Nelson's virtue-based interpretation. (shrink)

Recent years have seen a number of interpreters defend the claim that Nietzsche’s perspectivism is an epistemological doctrine. This interpretation of perspectivism leads to the worry that Nietzsche cannot offer any arguments for his view by means of which he may convince his opponents. To rescue Nietzsche from this justificatory problem, some interpreters have recently turned to the notion of “internal reasons,” or reasons that have force within multiple perspectives because they are based on shared standards. In this paper I (...) show that the task for supporters of Nietzsche is considerably more complex than it may seem at first sight. I argue that there is no way to know in advance of empirical investigation on a case-by-case basis whether perspectivism can be saved from undermining any compelling reasons that Nietzsche might provide in its defense. (shrink)

Wilfried Sieg and John Byrnes. AnModel for Parallel Computation: Gandy's Thesis.

Dedekind’s structuralism is a crucial source for the structuralism of mathematical practice—with its focus on abstract concepts like groups and fields. It plays an equally central role for the structuralism of philosophical analysis—with its focus on particular mathematical objects like natural and real numbers. Tensions between these structuralisms are palpable in Dedekind’s work, but are resolved in his essay Was sind und was sollen die Zahlen? In a radical shift, Dedekind extends his mathematical approach to “the” natural numbers. He creates (...) the abstract concept of a simply infinite system, proves the existence of a “model”, insists on the stepwise derivation of theorems, and defines structure-preserving mappings between different systems that fall under the abstract concept. Crucial parts of these considerations were added, however, only to the penultimate manuscript, for example, the very concept of a simply infinite system. The methodological consequences of this radical shift are elucidated by an analysis of Dedekind’s metamathematics. Our analysis provides a deeper understanding of the essay and, in addition, illuminates its impact on the evolution of the axiomatic method and of “semantics” before Tarski. This understanding allows us to make connections to contemporary issues in the philosophy of mathematics and science. (shrink)

We establish by elementary proof-theoretic means the conservativeness of two subsystems of analysis over primitive recursive arithmetic. The one subsystem was introduced by Friedman [6], the other is a strengthened version of a theory of Minc [14]; each has been shown to be of considerable interest for both mathematical practice and metamathematical investigations. The foundational significance of such conservation results is clear: they provide a direct finitist justification of the part of mathematical practice formalizable in these subsystems. The results are (...) generalized to relate a hierarchy of subsystems, all contained in the theory of arithmetic properties, to a corresponding hierarchy of fragments of arithmetic. The proof theoretic tools employed there are used to re-establish in a uniform, elementary way relationships between various fragments of arithmetic due to Parsons, Paris and Kirby, and Friedman. (shrink)

Alonzo Church's mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of Gödel's general recursiveness, not his own λ (...) -definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of effective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the character of my answers is reflected by an alternative title for this paper, Why Church needed Gödel's recursiveness for his Thesis. In Section 5, I contrast Church's analysis with that of Alan Turing and explore, in the very last section, an analogy with Dedekind's investigation of continuity. (shrink)

Contemporary society tends to behave very ambiguously towards tradition. Religious and moral customs that are still widely practiced are also often critically questioned or simply abandoned. Similarly, attitudes towards the role of science and technology in today’s global community are highly paradoxical. Some practices reveal a deeply rooted belief in the potentials of science and technology to increase happiness, on both the individual and the collective levels. Other practices and discourses are trying to prove the opposite, recommending to treat the (...) belief in scientific and technical progress as a myth that is mature for de-mythologization . Traditional myths should be rehabilitated instead. While the former opinion may be traced back to the spirit of modernity, the latter is called “postmodern,” hinting that the modern mentality has either come to an end, or reached a stage of profound transformation. Tradition is then no longer a taboo, but identified as what founds and guarantees a community’s value-system. Only the restoration of basic symbolic distinctions may prevent a culture from sliding into moral arbitrariness, in which only “good feelings” define moral standards and commitments are replaced by a “liberalism of neutrality” . A genuine moral and religious renewal may start from the rediscovery of the all-encompassing, mysterious dimension of life. (shrink)

Wilfried Sieg. Calculations by Man and Machine: Mathematical Presentation.

Wilfred Sieg. Mechanical Procedures and Mathematical Experience.

Religious sects and their practices have become highly topical in the recent past in Germany. The call for criminal sanctions in order to tackle actual or presumed misconduct is becoming louder and louder. The author, a criminal law professor, analyses the present discussion and puts it into a constitutional law perspective. Any civil society that aims to optimise the scope of the personal freedoms of its members must withstand the temptation of creating a special regime of criminal law rules just (...) to combat religious sects and their practices. Criminal law has tobe applied uniformly in the whole society. Therfore, criminal sanctions can only be handed out, when practices infringe intolerably with the individual' s rights protected by the law. (shrink)

The first part of this study is a reflection on the writings of Lowi and Briefs, and deals with the transition from the ideal of liberty to interest-group liberalism in the United States. In the second part I offer an analysis of Hegel’s attempt to combine the ideals of justice and liberty within a political economy. The third part deals with further issues of economic justice raised by the Bishops’ recent Pastoral Letter.

This essay offers a philosophical analysis of the role of the father-figure in the family. I argue that a Cartesian approach to this question is useless, and that Hegel, while he offers the beginning of an adequate analysis, falls short of the multiple-function model which an adequate analysis requires.

In this paper, I will first draw attention to the central fact that Freud describes in his article on “Negation”; i.e., “recognition of the unconscious on the part of the ego is expressed in a negative formula.” Technically, such a negative formula is called denial. Second, I will ask whether such denials are accidental or necessary in the life of consciousness. To answer this question I will use the philosophical system of Hegel. I will make use of his ideas at (...) two levels of generality. At the first level of generality, I will analyze one of Hegel’s most famous and influential passages: “Lordship and Bondage.” Hegel presents this passage as a necessary moment in the development of consciousness. If we find in that passage a necessary denial, then we can argue that denial is not just a fact but is, at least in any philosophical anthropology akin to Hegel’s, a necessity. At the second level of generality, we will then quote two statements by Hegel, one about the nature of the will and one about the path to truth for consciousness. Both statements will be clarified by Hegel’s analysis of “Lordship and Bondage.” Both statements will be used to demonstrate that Hegel affirmed explicitly the necessity of negativity in the life of consciousness. (shrink)

Contemporary philosophy is said to focus on particular issues, rather than on comprehensive syntheses. The following contribution intends to join this trend by offering some reflections on the “animal rights” debate, which is to be situated within the wider context of environmental philosophy. While classical Western concepts of man were anthropocentric, recent cultural developments have triggered a rediscovery of Nature, especially of nonhuman animals, while focusing on their affiliations with us, humans. Appropriate relations with those animals require a respectful attitude (...) on the human side, as if those animals had full moral and legal rights. But is this not an illusion? Can we talk about real “rights” for animals, or should we just remain aware of them having their own feelings and take care not to hurt them, unless for a “serious” cause? While taking note of the wide variety of animal species and habitats , the answer to this question may have its bearings on one’s personal choices regarding food, clothing, entertainment, etc. (shrink)

The paper discusses the problem of global distributive justice. It proposes to distinguish between principles for the domestic and for the global or intersocietal distribution of wealth. It is argued that there may be a plurality of partly diverging domestic conceptions of distributive justice, not all of which need to be liberal egalitarian conceptions. It is maintained, however, that principles regulating the intersocietal distribution of wealth have to be egalitarian principles. This claim is defended against Rawls's argument in The Law (...) of Peoples that egalitarian principles of distributive justice should not be applied globally. Moreover, it is explained in detail, why Rawls's “duty of assistance to burdened societies” cannot be an appropriate substitute for a global principle of distributive justice. (shrink)

Wilfred Sieg and Dirk Schlimm. Dedekind's Analysis of Number: Systems and Axioms.

In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of number-theoretic functions and we show the equivalence of the new approach with the classical one.

Wilfried Sieg. Formal Systems, Church Turing Thesis, and Gödel's Theorems: Three Contributions to The MIT Encyclopedias of Cognitive Science.

In his classic paper On Computable Numbers Turing analyzed what can be done by a human computor in a routine, “mechanical” way. He argued that mechanical op-erations obey locality conditions and are carried out on configurations satisfying boundedness conditions. Processes meeting these restrictive conditions can be shown to be computable by a Turing machine. Turing viewed memory limitations of computors as the ultimate reason for the restrictive conditions. In contrast, Gandy analyzed in his paper Church’s Thesis and Principles for Mechanisms (...) what can be done by “discrete deterministic mechanical devices” and appealed to physical considerations to motivate restrictive principles. These devices include, crucially, ones that carry out parallel processes, e.g., cellular automata. Following Turing’s methodological ways and guided by Conway’s “Game of Life” as a paradigm, Gandy formulated four restrictive principles and proved that devices satisfying them are computationally equivalent to Turing machines. (shrink)