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

Behavioural Economics: Classical and Modern

In this paper, the origins and development of behavioural economics, beginning with the pioneering works of Herbert Simon (1953) and Ward Edwards (1954), is traced, described and (critically) discussed, in some detail. Two kinds of behavioural economics – classical and modern – are attributed, respectively, to the two pioneers. The mathematical foundations of classical behavioural economics is identified, largely, to be in the theory of computation and computational complexity; the corresponding mathematical basis for modern behavioural economics is, on the other hand, claimed to be a notion of subjective probability (at least at its origins in the works of Ward Edwards). The economic theories of behavior, challenging various aspects of 'orthodox' theory, were decisively influenced by these two mathematical underpinnings of the two theoriesClassical Behavioural Economics, Modern Behavioural Economics, Subjective Probability, Model of Computation, Computational Complexity. Subjective Expected Utility

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

To the Beat of Different Drumer....Freedom, Anarchy and Conformism in 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 calibre in scientific research within University environments. In what sense, and how, can such rebels, adventurers and nonconformists 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 Wicksell, 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 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 dogmas.Non-conformist research, macroeconomics, foundations of mathematics, intuitionism, constructivism, formalism, Hilbertís Dogma, Hilbertís Program, computability theory

Semantic and Mathematical Foundations for Intuitionism

Thesis (Ph.D.) - Indiana University, Philosophy, 2013My dissertation concerns the proper foundation for the intuitionistic mathematics whose development began with L.E.J. Brouwer's work in the first half of the 20th Century. It is taken for granted by most philosophers, logicians, and mathematicians interested in foundational questions that intuitionistic mathematics presupposes a special, proof-conditional theory of meaning for mathematical statements. I challenge this commonplace. Classical mathematics is very successful as a coherent body of theories and a tool for practical application. Given this success, a view like Dummett's that attributes a systematic unintelligibility to the statements of classical mathematicians fails to save the relevant phenomena. Furthermore, Dummett's program assumes that his proposed semantics for mathematical language validates all and only the logical truths of intuitionistic logic. In fact, it validates some intuitionistically invalid principles, and given the lack of intuitionistic completeness proofs, there is little reason to think that every intuitionistic logical truth is valid according to his semantics. In light of the failure of Dummett's foundation for intuitionism, I propose and carry out a reexamination of Brouwer's own writings. Brouwer is frequently interpreted as a proto-Dummettian about his own mathematics. This is due to excessive emphasis on some of his more polemical writings and idiosyncratic philosophical views at the expense of his distinctively mathematical work. These polemical writings do not concern mathematical language, and their principal targets are Russell and Hilbert's foundational programs, not the semantic principle of bivalence. The failures of these foundational programs has diminished the importance of Brouwer's philosophical writings, but his work on reconstructing mathematics itself from intuitionistic principles continues to be worth studying. When one studies this work relieved of its philosophical burden, it becomes clear that an intuitionistic mathematician can make sense of her mathematical work and activity without relying on special philosophical or linguistic doctrines. Core intuitionistic results, especially the invalidity of the logical principle tertium non datur, can be demonstrated from basic mathematical principles; these principles, in turn, can be defended in ways akin to the basic axioms of other mathematical theories. I discuss three such principles: Brouwer's Continuity Principle, the Principle of Uniformity, and Constructive Church's Thesis

Sraffa's Mathematical Economics - A Constructive Interpretation

The claim in this paper is that Sraffa employed a rigorous logic of mathematical reasoning in his book, Production of Commodities by Means of Commodities (PCC), in such a way that the existence proofs were constructive. This is the kind of mathematics that was prevalent at the beginning of the 19th century, which was dominated by the concrete, the constructive and the algorithmic. It is, therefore, completely consistent with the economics of the 19th century, which was the fulcrum around which the economics of PCC was conceived.Existence Proofs, Constructive Mathematics, Algorithmic Mathematics, Mathematical Economics, Standard System.

Introducing choice sequences into mathematical ontology

Tese de mestrado, Filosofia (Epistemologia e Metafísica), Universidade de Lisboa, Faculdade de Letras, 2012A ideia de objectos matemáticos que estão em permanente desenvolvimento no tempo foi pela primeira vez avançada por L.E.J. Brouwer. Na matemática intuicionista estes objectos são concebidos como sequência infinitas de números naturais que em qualquer estágio do seu crescimento têm apenas um número finito de valores, além disso, tais valores podem ser livremente escolhidos, no sentido em que a sua produção não necessita de ser determinada por nenhuma regra matemática definida. Tais objectos são denominados de sequências de escolha. O presente trabalho tem como objectivo fornecer uma resposta à sequinte questão: são as sequências de escolha legítimos objectos matemáticos? A resposta que iremos propor e à qual iremos argumentar favoravelmente é a seguinte: tais objectos não podem ser considerados objectos matemáticos legítimos. Com esta tese em vista, iremos discutir as propriedades intrínsecas das sequências de escolha relativamente à maneira como são incorporadas no contexto matemático e as suas implicações. Seguindo esta metodologia pretendemos atingir um cabal entendimento filosófico das consequências em que incorremos ao aceitarmos sequências de escolha como objectos da ontologia matemática e das razões que temos para não as aceitarmos como tal.Abstract: The idea of mathematical objects which are in a permanent state of growth in time was by the first time defended by L.E.J. Brouwer. In intuitionistic mathematics these objects are conceived as infinite sequences of natural numbers that at any stage of growth have only finitely many values defined. Additionally, these values may be freely chosen, in the sense that their generation has not to follow any determinate mathematical rule. These objects are called choice sequences. The present work aims at providing the answer to the following question: are choice sequences legitimate mathematical objects? The answer that we will propose and argue for is a negative one: that they cannot be considered legitimate mathematical objects. In order to do this we will discuss the intrinsic features of choice sequences concerning the way they are incorporated into a mathematical framework and their implications. Following this methodology we expect to achieve a good philosophical understanding of the consequences of accepting choice sequences into our mathematical ontology and of the reasons that we have not to accept them as such