Mathematical Monsters

Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways between these two monstrous habitats. I start from Jeffrey Jerome Cohen’s influential account of monster culture and explore how well mathematical monsters fit each of his seven theses. The mathematical monsters I discuss are drawn primarily from three distinct but overlapping domains. Firstly, late nineteenth-century mathematicians made numerous unsettling discoveries that threatened their understanding of their own discipline and challenged their intuitions. The great French mathematician Henri Poincaré characterised these anomalies as ‘monsters’, a name that stuck. Secondly, the twentieth-century philosopher Imre Lakatos composed a seminal work on the nature of mathematical proof, in which monsters play a conspicuous role. Lakatos coined such terms as ‘monster-barring’ and ‘monster-adjusting’ to describe strategies for dealing with entities whose properties seem to falsify a conjecture. Thirdly, and most recently, mathematicians dubbed the largest of the sporadic groups ‘the Monster’, because of its vast size and uncanny properties, and because its existence was suspected long before it could be confirmed

Poincare and the analytic tradition

There is no substantive agreement about the philosophical views of analytic philosophers. Nevertheless, for much of them logical analysis is widely recognized to be important. It is true that Poincare used no logical analysis but refused nevertheless the old metaphysics. Indeed, the analytic tradition of philosophy of science is perhaps better characterized by several overlapping similarities, which are the clincher for my main thesis: Analytic philosophy of science has one of its origins in the philosophical network in France around 1900 and, especially, in Poincare

Poincare and the analytic tradition

There is no substantive agreement about the philosophical views of analytic philosophers. Nevertheless, for much of them logical analysis is widely recognized to be important. It is true that Poincare used no logical analysis but refused nevertheless the old metaphysics. Indeed, the analytic tradition of philosophy of science is perhaps better characterized by several overlapping similarities, which are the clincher for my main thesis: Analytic philosophy of science has one of its origins in the philosophical network in France around 1900 and, especially, in Poincare

Between Scylla and Charybdis - on the place of economic methods and concepts within ecological economics

Ecological Economics inherently faces a challenge akin to sailing between Scylla and Charybdis. In Greek mythology these are two monsters located on opposite sides of a narrow strait, and falling victim to one or other of them is unavoidable. In the recurring process of establishing and refining its conceptual foundations, Ecological Economics runs the risk of, on the one hand, losing important insights by trying to be radically different from mainstream economics and, on the other hand, becoming a redundant appendix to mainstream environmental economics by routinely applying its concepts and methods. We argue that avoiding both fallacies is possible by using Ecological Economics’ orientation towards sustainability as a guiding principle. The scientist’s power of judgment supports her decision concerning which methods are suitable for tackling a given sustainability problem. The intersubjective quality of judgment prevents the resulting methodological pluralism from drifting toward arbitrariness

How Future Depends on the Past and on Rare Events in Systems of Life

A paraîtreInternational audienceThe dependence on history of both present and future dynamics of life is a common intuition in biology and in humanities. Historicity will be understood in terms of changes of the space of possibilities (or of " phase space ") as well as by the role of diversity in life's structural stability and of rare events in history formation. We hint to a rigorous analysis of " path dependence " in terms of invariants and invariance preserving transformations, as it may be found also in physics, while departing from the physico-mathematical analyses. The idea is that the (relative or historicized) invariant traces of past organismal or ecosystemic transformations contribute to the understanding (or the " theoretical determination ") of present and future states of affairs. This yields a peculiar form of unpredictability (or randomness) in biology, at the core of novelty formation: the changes of observables and pertinent parameters may depend also on past events. In particular, in relation to the properties of synchronic measurement in physics, the relevance of diachronic measurement in biology is highlighted. This analysis may a fortiori apply to cognitive and historical human dynamics, while allowing to investigate some general properties of historicity in biology

A Fractal Epistemology for Transpersonal Psychology

The role of science has been controversial within the nascent field of transpersonal psychology. Traditional linear and reductionist models are insufficient to address rare and unreproducible states of mind, fringe rather than normative experiences, and highly personal or culturally specific aspects of awareness. Through a fractal epistemology this paper introduces novel metaphors, models, and methods within a more holistic, organic, and synthetic branch of science. Principles of the epistemology illuminate observer dependence, fuzzy boundaries, recursive patterns, and higher dimensional phenomena that emerge within the infinite expanses between ordinary, finite (Euclidean) dimensions