Arithmetic, Set Theory, Reduction and Explanation

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

Accounting for primitive terms in mathematics

The philosophical problem of unity and diversity entails a challenge to the rationalist aim to define everything. Definitions of this kind surface in various academic disciplines in formulations like uniqueness, irreducibility, and what has acquired the designation “primitive terms”. Not even the most “exact” disciplines, such as mathematics, can avoid the implications entailed in giving an account of such primitive terms. A mere look at the historical development of mathematics highlights the fact that alternative perspectives prevailed – from the arithmeticism of Pythagoreanism, the eventual geometrisation of mathematics after the discovery of incommensurability up to the revival of arithmeticism in the mathematics of Cauchy, Weierstrass, Dedekind and Cantor (with the later orientation of Frege, who completed the circle by returning to the view that mathematics essentially is geometry). An assessment of logicism and axiomatic formalism is followed by looking at the primitive meaning of wholeness (and the whole-parts relation). With reference to the views of Hilbert, Weyl and Bernays the article concludes by suggesting that in opposition to arithmeticism and geometricism an alternative option ought to be pursued – one in which both the uniqueness and mutual coherence between the aspects of number and space are acknowledged

Pragmatic Holism

The reductionist/holist debate seems an impoverished one, with many participants appearing to adopt a position first and constructing rationalisations second. Here I propose an intermediate position of pragmatic holism, that irrespective of whether all natural systems are theoretically reducible, for many systems it is completely impractical to attempt such a reduction, also that regardless if whether irreducible `wholes' exist, it is vain to try and prove this in absolute terms. This position thus illuminates the debate along new pragmatic lines, and refocusses attention on the underlying heuristics of learning about the natural world

After Hermeneutics?

Recently Alain Badiou and Quentin Meillassoux have attacked the core of the phenomenological hermeneutic tradition: its commitment to the finitude of human understanding. If accurate, this critique threatens to render the whole tradition a topic of merely historical interest. Given the depth of the criticism, this essay aims to establish a provisional defense of hermeneutics. After briefly reviewing each critique, it is argued that Badiou and Meillassoux themselves face rather intractable difficulties. These difficulties, then, open the space for a hermeneutic response, which is accomplished largely by drawing on the work of Paul Ricoeur. We close with a suggested program for hermeneutic thought

Wholeness as a Conceptual Foundation of Physical Theories

A description of physical reality in which wholeness is the foundation is discussed along with the motivation for such an attempt. As a possible mathematical framework within which a physical theory based on wholeness may be expressed, elementary embeddings along with the Wholeness Axiom are suggested. It is shown how features of wholeness such as wholeness being indescribable, more than the sum of parts, locally accessible and giving rise to a self-similar, or holographic, type of order are reflected in the mathematics. It is also shown how all the sets in the mathematical universe may be expressed as emerging from the dynamics of wholeness. Moreover, it is indicated how the mathematics may be further developed so as to connect up with a physical interpretation.Comment: 12 pages, 1 eps figure, submitted to "Physics Essays

Zermelo in the mirror of the Baer correspondence, 1930–1931

AbstractAround 1931 Zermelo had an extended correspondence with the young Reinhold Baer concerning the edition of Cantor's collected works. Some of the letters also deal with Skolem's paradox and Gödel's first incompleteness theorem. Whereas Zermelo's letters are lost, most of Baer's letters are contained in the Zermelo Nachlass. Besides giving insight into Zermelo's reaction to Skolem's and Gödel's results, the letters also demonstrate Baer's clear understanding of the behavior of models of set theory and of the relevance of Gödel's first incompleteness theorem

Irreversibility and typicality: A simple analytical result for the Ehrenfest model

With the aid of simple analytical computations for the Ehrenfest model, we clarify some basic features of macroscopic irreversibility. The stochastic character of the model allows us to give a non-ambiguous interpretation of the general idea that irreversibility is a typical property: for the vast majority of the realizations of the stochastic process, a single trajectory of a macroscopic observable behaves irreversibly, remaining "very close" to the deterministic evolution of its ensemble average, which can be computed using probability theory. The validity of the above scenario is checked through simple numerical simulations and a rigorous proof of the typicality is provided in the thermodynamic limit