Some Varieties of Superparadox. The implications of dynamic contradiction, the characteristic form of breakdown of breakdown of sense to which self-reference is prone

The Problem of the Paradoxes came to the fore in philosophy and mathematics with the discovery of Russell's Paradox in 1901. It is the "forgotten" intellectual-scientific problem of the Twentieth Century, because for more than sixty years a pretence was maintained, by a consensus of logicians, that the problem had been "solved"

Proof-irrelevance out of excluded-middle and choice in the calculus of constructions

We present a short and direct syntactic proof of the fact that adding the axiom of choice and the principle of excluded-middle to Coquand-Huet's Calculus of Constructions gives proof-irrelevanc

Modal logic NL for common language

Despite initial appearance, paradoxes in classical logic, when comprehension is unrestricted, do not go away even if the law of excluded middle is dropped, unless the law of noncontradiction is eliminated as well, which makes logic much less powerful. Is there an alternative way to preserve unrestricted comprehension of common language, while retaining power of classical logic? The answer is yes, when provability modal logic is utilized. Modal logic NL is constructed for this purpose. Unless a paradox is provable, usual rules of classical logic follow. The main point for modal logic NL is to tune the law of excluded middle so that we allow for a sentence and its negation to be both false in case a paradox provably arises. Curry's paradox is resolved differently from other paradoxes but is also resolved in modal logic NL. The changes allow for unrestricted comprehension and naive set theory, and allow us to justify use of common language in formal sense

Freedom, Anarchy and Conformism in Academic Research

In this paper I attempt to make a case for promoting the courage of rebels within the citadels of orthodoxy in academic research environments. Wicksell in Macroeconomics, Brouwer in the Foundations of Mathematics, Turing in Computability Theory, Sraffa in the Theories of Value and Distribution are, in my own fields of research, paradigmatic examples of rebels, adventurers and non-conformists of the highest caliber in scientific research within University environments. In what sense, and how, can such rebels, adventurers and non-conformists be fostered in the current University research environment dominated by the cult of 'picking winners'? This is the motivational question lying behind the historical outlines of the work of Brouwer, Hilbert, Bishop, Veronese, Gödel, Turing and Sraffa that I describe in this paper. The debate between freedom in research and teaching, and the naked imposition of 'correct' thinking, on potential dissenters of the mind, is of serious concern in this age of austerity of material facilities. It is a debate that has occupied some of the finest minds working at the deepest levels of foundational issues in mathematics, metamathematics and economic theory. By making some of the issues explicit, I hope it is possible to encourage dissenters to remain courageous in the face of current dogmasNon-conformist research, economic theory, mathematical economics, 'Hilbert's Dogma', Hilbert's Program, computability theory

Poincaré's philosophy of mathematics

The primary concern of this thesis is to investigate the explicit philosophy of mathematics in the work of Henri Poincare. In particular, I argue that there is a well-founded doctrine which grounds both Poincare's negative thesis, which is based on constructivist sentiments, and his positive thesis, via which he retains a classical conception of the mathematical continuum. The doctrine which does so is one which is founded on the Kantian theory of synthetic a priori intuition. I begin, therefore, by outlining Kant's theory of the synthetic a priori, especially as it applies to mathematics. Then, in the main body of the thesis, I explain how the various central aspects of Poincare's philosophy of mathematics - e.g. his theory of induction; his theory of the continuum; his views on impredicativiti his theory of meaning - must, in general, be seen as an adaptation of Kant's position. My conclusion is that not only is there a well-founded philosophical core to Poincare's philosophy, but also that such a core provides a viable alternative in contemporary debates in the philosophy of mathematics. That is, Poincare's theory, which is secured by his doctrine of a priori intuitions, and which describes a position in between the two extremes of an "anti-realist" strict constructivism and a "realist" axiomatic set theory, may indeed be true

Study of logical paradoxes

By a paradox we understand a seemingly true statement or set of statements which lead by valid deduction to contradictory statements. Logical paradoxes - paradoxes which involve logical concepts - are in fact as old as the history of logic. The Liar paradox, for instance, goes back to Epimenides (6th century B.C.?). In the late 19th century a new impetus v/as given to the investigation of logical paradoxes by the discovery of new logico-mathematical paradoxes such as those of Russell and Burali- Porti. This came about in the course of attempts to give mathematics a rigorous axiomatic foundation. Sometimes a distinction is maintained between a paradox and an antinomy. In a paradox, it is said, semantical notions are involved and a certain "oddity", "strangeness", or what may be called "paradoxical situation", resides in its construction. The resolution of a paradox is therefore not simply a matter of removing contradiction, but also requires clarifying and removing the "oddity". On the other hand, an antinomy is said to consist in the derivation of a contradiction in an axiomatic system and its resolution lies in revising the system so as to avoid the contradiction. In discussing paradoxes and antinomies, we shall not be strictly bound by this usage of these terms: we use "paradox" and "antinomy" interchangeably. Indeed, from our point of view, even antinomies in an axiomatic system ultimately need semantic clarification and thus removal of paradoxical situations

Follow the Flow: sets, relations, and categories as special cases of functions with no domain

We introduce, develop, and apply a new approach for dealing with the intuitive notion of function, called Flow Theory. Within our framework all functions are monadic and none of them has any domain. Sets, proper classes, categories, functors, and even relations are special cases of functions. In this sense, functions in Flow are not equivalent to functions in ZFC. Nevertheless, we prove both ZFC and Category Theory are naturally immersed within Flow. Besides, our framework provides major advantages as a language for axiomatization of standard mathematical and physical theories. Russell's paradox is avoided without any equivalent to the Separation Scheme. Hierarchies of sets are obtained without any equivalent to the Power Set Axiom. And a clear principle of duality emerges from Flow, in a way which was not anticipated neither by Category Theory nor by standard set theories. Besides, there seems to be within Flow an identification not only with the common practice of doing mathematics (which is usually quite different from the ways proposed by logicians), but even with the common practice of teaching this formal science

Follow the Flow: sets, relations, and categories as special cases of functions with no domain

We introduce, develop, and apply a new approach for dealing with the intuitive notion of function, called Flow Theory. Within our framework functions have no domain at all. Sets and even relations are special cases of functions. In this sense, functions in Flow are not equivalent to functions in ZFC. Nevertheless, we prove both ZFC and Category Theory are naturally immersed within Flow. Besides, our framework provides major advantages as a language for axiomatization of standard mathematical and physical theories. Russell's paradox is avoided without any equivalent to the Separation Scheme. Hierarchies of sets are obtained without any equivalent to the Power Set Axiom. And a clear principle of duality emerges from Flow, in a way which was not anticipated neither by Category Theory nor by standard set theories