Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

Theoretical determinism, as it is usually ascribed to Laplace, is neither verifiable nor falsifiable and has therefore no real content. It is not the same as predictability of actually observable phenomena. On the other hand, predictability is not an abstract principle; rather it is true to a certain degree, depending on the phenomena considered. It can be discussed only by examining the scientific state of affairs. This is done in some detail for classical statistical mechanics. Much of a recently published (...) debate on determinism (Amsterdamski et al. 1990) is thereby obviated. (shrink)

Poincaré was a persistent critic of logicism. Unlike most critics of logicism, however, he did not focus his attention on the basic laws of the logicists or the question of their genuinely logical status. Instead, he directed his remarks against the place accorded to logical inference in the logicist's conception of mathematical proof. Following Leibniz, traditional logicist dogma (and this is explicit in Frege) has held that reasoning or inference is everywhere the same — that there are no principles of (...) inference specific to a given local topic. Poincaré, a Kantian, disagreed with this. Indeed, he believed that the use of non-logical reasoning was essential to genuinely mathematical reasoning (proof). In this essay, I try to isolate and clarify this idea and to describe the mathematical epistemology which underlies it. Central to this epistemology (which is basically Kantian in orientation, and closely similar to that advocated by Brouwer) is a principle of epistemic conservation which says that knowledge of a given type cannot be extended by means of an inference unless that inference itself constitutes knowledge belonging to the given type. (shrink)

All the attempts to find the justification of the privileged evolution of phenomena exclusively in the external world need to refer to the inescapable fact that we are living in such an asymmetric universe. This leads us to look for the origin of the “arrow of time” in the relationship between the subject and the world. The anthropic argument shows that the arrow of time is the condition of the possibility of emergence and maintenance of life in the universe. Moreover, (...) according to Bohr’s, Poincaré’s and Watanabe’s analysis, this agreement between the earlier-later direction of entropy increase and the past-future direction of life is the very condition of the possibility for meaningful action, representation and creation. Beyond this relationship of logical necessity between the meaning process and the arrow of time the question of their possible physical connection is explored. To answer affirmatively to this question, the meaning process is modelled as an evolving tree-like structure, called “Semantic Time”, where thermodynamic irreversibility can be shown. (shrink)

Let me start with a well-known story. Kant held that logic and conceptual analysis alone cannot account for our knowledge of arithmetic: “however we might turn and twist our concepts, we could never, by the mere analysis of them, and without the aid of intuition, discover what is the sum [7+5]” (KrV, B16). Frege took himself to have shown that Kant was wrong about this. According to Frege’s logicist thesis, every arithmetical concept can be defined in purely logical terms, and (...) every theorem of arithmetic can be proved using only the basic laws of logic. Hence, Kant was wrong to think that our grasp of arithmetical concepts and our knowledge of arithmetical truth depend on an extralogical source—the pure intuition of time (Frege 1884, §89, §109). Arithmetic, properly understood, is just a part of logic. (shrink)

The paper investigates possible readings of the later Heisenberg's remarks on the nature of quantum states. It discusses, in particular, whether Heisenberg should be seen as a proponent of the epistemic conception of states – the view that quantum states are not descriptions of quantum systems but rather reflect the state assigning observers' epistemic relations to these systems. On the one hand, it seems plausible that Heisenberg subscribes to that view, given how he defends the notorious "collapse of the wave (...) function" by relating it to a sudden change in the epistemic situation of the observer registering a measured result. On the other hand, his remarks on quantum probabilities as "potentia" or "objective tendencies" are difficult to reconcile with such a reading. The accounts that are attributed to Heisenberg by the different possible readings considered are subjected to closer scrutiny; at the same time, their respective virtues and problems are discussed. _German_ Diese Arbeit untersucht mögliche Lesarten von Heisenbergs späten Bemerkungen über die Natur von Quantenzuständen. Insbesondere wird die Frage diskutiert, ob Heisenberg als Vertreter der epistemischen Auffassung von Quantenzuständen gelten kann – der Idee, dass Quantenzustände nicht Beschreibungen der objektiven Eigenschaften von Quantensystemen sind sondern die epistemischen Beziehungen von Beobachtern zu diesen Systemen widerspiegeln. Einerseits erscheint es plausibel, dass Heisenberg dieser Sichtweise zustimmt, wenn man sich ansieht, wie er den berüchtigten,,Kollaps der Wellenfunktion" verteidigt, indem er ihn mit einer plötzlichen Änderung in der Kenntnis des ein Messergebnis registrierenden Beobachters in Zusammenhang bringt. Andererseits sind seine Bemerkungen über quantenmechanische Wahrscheinlichkeiten als,,Potentia" oder,,objektive Tendenzen" mit einer solchen Lesart nur schwer in Einklang zu bringen. Die Positionen, die Heisenberg den verschiedenen möglichen Lesarten zufolge vertritt, werden eingehenden Untersuchungen unterzogen; gleichzeitig werden ihre jeweiligen Vorzüge und Probleme diskutiert. (shrink)

Artificial intelligence (AI) technologies are used in various domains of human activities, and one of these domains is scientific research. Now, researchers in many scientific areas try to apply AI technologies to their research and automate it. These researchers claim that the ‘automation of science’ will liberate people from non-creative tasks in scientific research, and radically change the overall state of science and technology so that large-scale innovation results. As I see it, the automation of science is remarkable in another (...) respect: since science is one of the most distinctive human activities, the tendency of automating it prompts us to reconsider the aspects of our humanity itself. One of these aspects is concerned with human creativity, on which this article focuses. In this article, I address two questions concerning the automation of science: first, ‘Can AI makes creative discovery?’; second, ‘What implications may the automation of science have on science and society?’. Scientific discovery is said to be one of the most creative phases of scientific research. I show that, though there are no reasons in principle why AI could not make creative discovery, we do not at present have enough knowledge how to realize it. If the automation of science nevertheless proceeds, the cultural values science as a creative activity has might be reduced, and it might alter the state of scientific community and its relationship with society in some undesirable way. Therefore, I conclude, we need to specify desirable ways of introducing AI technologies into science and devise measures against demerits of automating science. (shrink)

This volume presents research that integrates decision making and creativity within the social contexts in which these processes occur. The volume is an essential addition to and expansion of recent approaches to decision making. Such approaches attempt to incorporate more of the psychological and socio-cultural context in which human decision making takes place. The authors come from different disciplines and also belong to a broad spectrum of research traditions. They present innovative chapters dealing with both theoretical and empirical aspects of (...) decision making in different personal and organizational contexts. All chapters are written from the perspective that human decision making is inherently social and more or less creative. The volume addresses fundamental questions about the nature of human decision making as it occurs in different social contexts. Thereby, it becomes essential reading for researchers in decision making and for advanced students in psychology, management science, informatics, and related disciplines. (shrink)

This paper aims to shed light on the broader significance of Frege’s logicism against the background of discussing and comparing Wittgenstein’s ‘showing/saying’-distinction with Brandom’s idiom of logic as the enterprise of making the implicit rules of our linguistic practices explicit. The main thesis of this paper is that the problem of Frege’s logicism lies deeper than in its inconsistency : it lies in the basic idea that in arithmetic one can, and should, express everything that is implicitly presupposed so that (...) nothing is left unsaid. This, in fact, is the target of Wittgenstein’s critique. Rather than the Tractatus, with its claim that logicism attempts to say something that can only be shown, it is the Philosophical Investigations, with its argument by regress against the thesis that every rule which one can follow must be of an explicit nature, that is of real significance here. (shrink)

Theoretical determinism, as it is usually ascribed to Laplace, is neither verifiable nor falsifiable and has therefore no real content. It is not the same as predictability of actually observable phenomena. On the other hand, predictability is not an abstract principle; rather it is true to a certain degree, depending on the phenomena considered. It can be discussed only by examining the scientific state of affairs. This is done in some detail for classical statistical mechanics. Much of a recently published (...) debate on determinism is thereby obviated. (shrink)

The article attempts to reconsider the relationship between Leibniz’s and Kant’s philosophy of geometry on the one hand and the nineteenth century debate on the foundation of geometry on the other. The author argues that the examples used by Leibniz and Kant to explain the peculiarity of the geometrical way of thinking are actually special cases of what the Jewish-German mathematician Felix Hausdorff called “transformation principle”, the very same principle that thinkers such as Helmholtz or Poincaré applied in a more (...) general form in their celebrated philosophical writings about geometry. The first two parts of the article try to show that Leibniz’s and Kant’s philosophies of geometry, despite their differences, appear to be preoccupied with the common problem of the impossibility to grasp conceptually the intuitive difference between two figures (such as a figure and its scaled, displaced or mirrored copy). In the third part, it is argued that from the perspective of Hausdorff’s philosophical-geometrical reflections, this very same problem seems to find a more radical application in Helmholtz’s or Poincaré’s thought experiments on the impossibility of distinguishing distorted copies of our universe from the original one. I draw the conclusion that in Hausdorff’s philosophical work, which has received scholarly attention only recently, one can find not only an original attempt to frame these classical arguments from a set-theoretical point of view, but also the possibility of considering the history of philosophy of geometry from an uncommon perspective, where especially the significance of Kant’s infamous appeal to “intuition” can be judged by more appropriate standards. (shrink)

This essay consists of two parts. In the first part, I focus my attention on the remarks that Frege makes on consistency when he sets about criticizing the method of creating new numbers through definition or abstraction. This gives me the opportunity to comment also a little on H. Hankel, J. Thomae—Frege’s main targets when he comes to criticize “formal theories of arithmetic” in Die Grundlagen der Arithmetik (1884) and the second volume of Grundgesetze der Arithmetik (1903)—G. Cantor, L. E. (...) J. Brouwer and D. Hilbert (1899). Part 2 is mainly devoted to Hilbert’s proof theory of the 1920s (1922–1931). I begin with an account of his early attempt to prove directly, and thus not by reduction or by constructing a model, the consistency of (a fragment of) arithmetic. In subsequent sections, I give a kind of overview of Hilbert’s metamathematics of the 1920s and try to shed light on a number of difficulties to which it gives rise. One serious difficulty that I discuss is the fact, widely ignored in the pertinent literature on Hilbert’s programme, that his language of finitist metamathematics fails to supply the conceptual resources for formulating a consistency statement qua unbounded quantification. Along the way, I shall comment on W. W. Tait’s objection to an interpretation of Hilbert’s finitism by Niebergall and Schirn, on G. Gentzen’s allegedly finitist consistency proof for Peano Arithmetic as well as his ideas on the provability and unprovability of initial cases of transfinite induction in pure number theory. Another topic I deal with is what has come to be known as partial realizations of Hilbert’s programme, chiefly advocated by S. G. Simpson. Towards the end of this essay, I take a critical look at Wittgenstein’s views about (in)consistency and consistency proofs in the period 1929–1933. I argue that both his insouciant attitude towards the emergence of a contradiction in a calculus and his outright repudiation of metamathematical consistency proofs are unwarranted. In particular, I argue that Wittgenstein falls short of making a convincing case against Hilbert’s programme. I conclude with some philosophical remarks on consistency proofs and soundness and raise a question concerning the consistency of analysis. (shrink)

In a definition (∀ x )(( x є r )↔D[ x ]) of the set r, the definiens D[ x ] must not depend on the definiendum r . This implies that all quantifiers in D[ x ] are independent of r and of (∀ x ). This cannot be implemented in the traditional first-order logic, but can be expressed in IF logic. Violations of such independence requirements are what created the typical paradoxes of set theory. Poincaré’s Vicious Circle Principle (...) was intended to bar such violations. Russell nevertheless misunderstood the principle; for him a set a can depend on another set b only if ( b є a ) or ( b ⊆ a ). Likewise, the truth of an ordinary first-order sentence with the Gödel number of r is undefinable in Tarki’s sense because the quantifiers of the definiens depend unavoidably on r. (shrink)

Joseph Bertrand's 1888 evidencing that assignment of a probability depends upon what one chooses to know or not and to control or not, congruent with Grad's 1961 evidencing that statistical entropy depends upon what one deems relevant or not in formalization and measurement, radically undermine common sense realism; mean values are symbols, but symbols of what? For that very reason, recent clever conceptualizations of the quantum measurement process via partial tracing do not restore realism: How could deliberate ignorance generate a (...) reality? Beyond this, Born's and Jordan's quantal wavelike probability calculus, entailing algebraic nonseparability and spacetime nonlocality, blurs “reality” still more radically. Thus information stands out as the master word, with its two reciprocal aspects: knowledge and organization. (shrink)

In economics in situations where there is uncertainty one has to attribute some attitude to handling this uncertainty to individuals. The original idea was to assume that “people do not make systematic mistakes” for which Muth coined the term “rational expectations”. This was replaced by a much more formal vision which suggested that people fully understand how the economy evolves. In this paper I will argue that the foundations of the “rational expectations” hypothesis which has underpinned most recent modern macroeconomic (...) models, which have, in turn, been used for policymaking, are unsound. From a philosophical point of view the idea that agents can not only forecast the evolution of their environment but also their own evolution seems unjustified. If agents do manage to coordinate on specific expectations these may not be rational and can be self fulfilling. From an econometric point of view, when there are changes in the evolution of the economy, it is not rational to satisfy the hypothesis. Furthermore the empirical and experimental evidence suggests that we should abandon this route. (shrink)

The aim of this paper is to present and discuss the philosophical views concerning mathematics of the founders of the so called Warsaw Mathematical School, i.e., Wacław Sierpiński, Zygmunt Janiszewski and Stefan Mazurkiewicz. Their interest in the philosophy of mathematics and their philosophical papers will be considered. We shall try to answer the question whether their philosophical views influenced their proper mathematical investigations. Their views towards set theory and its rôle in mathematics will be emphasized.