At their extremes, the modernization of ancient mathematical texts (absolute presentism) leaves nothing of the source and the refusal to modernize (absolute antiquarism) changes nothing. The extremes exist only as tendencies. This paper attempts to justify the admissibility of broad modernization of mathematical sources (presentism) in the context of a socio-cultural (non-fundamentalist) philosophy of mathematics.

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense (...) of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences. (shrink)

Mark Balaguer ha elaborado una peculiar variante del platonismo matemático –denominada ‘full-blooded platonism’ o ‘FBP’– para solucionar el problema de Benacerraf sobre la inaccesibilidad de las entidades abstractas. Según FBP, todos los objetos matemáticos consistentemente caracterizables existen, aunque de modo contingente. En este trabajo quisiera mostrar que la plenitud ontológica y la contingencia modal no pueden converger en una teoría de objetos matemáticos filosóficamente respetable. Para esto argumento que FBP no cubre algunos factores elementales de confiabilidad epistémica y que envuelve (...) un criterio de plenitud tanto matemática como metafísicamente implausible. (shrink)

In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including $-\omega, \,\omega/2, \,1/\omega, \sqrt{\omega}$ and $\omega-\pi$ to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG, it may be said to contain (...) “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system ℝ of real numbers bears to Archimedean ordered fields. In Part I of the present paper, we suggest that whereas $\mathbb{R}$should merely be regarded as constituting an arithmetic continuum, No may be regarded as a sort of absolute arithmetic continuum, and in Part II we draw attention to the unifying framework No provides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis. In addition to its inclusive structure as an ordered field, the system No of surreal numbers has a rich algebraico-tree-theoretic structure—a simplicity hierarchical structure—that emerges from the recursive clauses in terms of which it is defined. In the development of No outlined in the present paper, in which the surreals emerge vis-à-vis a generalization of the von Neumann ordinal construction, the simplicity hierarchical features of No are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations of No as an absolute continuum. (shrink)

Suppose we were to ask some students of philosophy to imagine a typical book of classical German philosophy and describe its general style and character, how might they reply? I suspect that they would answer somewhat as follows. The book would be long and heavy, it would be written in a complicated style which employed only very abstract terms, and it would be extremely difficult to understand. At all events a description of this kind does indeed fit many famous works (...) of German philosophy. Let us take for example Hegel's Phenomenology of Spirit of 1807. The 1977 English translation published by Oxford has 591 pages, and as for style here is a typical passage: Self-consciousness found the Thing to be like itself, and itself to be like a Thing; i.e., it is aware that it is in itself the objectively real world. It is no longer the immediate certainty of being all reality, but a certainty for which the immediate in general has the form of something superseded, so that the objectivity of the immediate still has only the value of something superficial, its inner being and essence being self-consciousness itself. The object, to which it is positively related, is therefore a self-consciousness. It is in the form of thinghood, i.e. it is independent ; but it is certain that this independent object is for it not something alien, and thus it knows that it is in principle recognized by the object. It is Spirit which, in the duplication of its self-consciousness and in the independence of both, has the certainty of its unity with itself. This certainty has now to be raised to the level of truth; what holds good for it in principle , and in its inner certainty, has to enter into its consciousness and become explicit fork. (shrink)

Mathematicians use theological metaphors when they talk in the kitchen of mathematics. How essential is this talk? Have theological considerations and religious concepts influenced mathematics? Can mathematical models illuminate theology? Some authors have given positive answers to these questions, but they do not seem final. It is unclear how religious views influenced the work of those mathematicians who were also theologians. Religious background of some mathematical concepts could have been inessential. Mathematical models in theology have no predictive value. It is, (...) however, important to continue the recently initiated search for the mutual influences of mathematics and theology., 2016.). (shrink)