The Applicability of Mathematics as a Philosophical Problem: Mathematization as Exploration

This paper discerns two types of mathematization, a foundational and an explorative one. The foundational perspective is well-established, but we argue that the explorative type is essential when approaching the problem of applicability and how it influences our conception of mathematics. The first part of the paper argues that a philosophical transformation made explorative mathematization possible. This transformation took place in early modernity when sense acquired partial independence from reference. The second part of the paper discusses a series of examples from the history of mathematics that highlight the complementary nature of the foundational and exploratory types of mathematization.     

Least Bell's Vireo breeding records in the Central Valley following decades of extirpation

The Least Bell’s Vireo ( Vireo bellii pusillus ) was listed as state endangered in 1980 and federally endangered in 1986 in response to a sharp population decline and range reduction. This vireo commonly bred in riparian forests throughout the Central Valley of California, but prior to 2005, no nesting pairs had been confirmed in the region in over 50 years. On 29 June 2005, a Least Bell’s Vireo nest was located in a 3-year-old riparian restoration site at the San Joaquin River National Wildlife Refuge in Stanislaus County, California. In 2006, a Least Bell’s Vireo pair returned to the refuge to successfully breed, followed by an unsuccessful attempt in 2007 by an unpaired female. These records are approximately 350 km from the nearest known breeding population and appear to be part of a growing number of sightings outside of the species’ current southern California breeding range. These nesting attempts lend credence to the idea that extirpated species can recolonize restored habitat by long-distance dispersal.     

Protein kinase C mediates neural induction in Xenopus laevis

Inductive cell interactions are essential in early embryonic development, but virtually nothing is known about the molecular mechanisms involved. Recently factors resembling fibroblast growth factor and transforming growth factor-beta were shown to be involved in mesoderm induction in Xenopus laevis, suggesting that membrane receptor-mediated signal transduction is important in induction processes. Here we report direct measurements of protein kinase C (PKC) activity in uninduced ectoderm, and in neuroectoderm shortly after induction by the involuting mesoderm, in Xenopus laevis embryos. Membrane-bound PKC activity increased three to fourfold in the induced neuroectoderm while the cytosolic PKC activity was decreasing, indicating that PKC activity was translocated during neural induction. A similar time- and dose-dependent translocation of activity was seen after incubation with the PKC activator 12-O-tetradecanoyl phorbol-13-acetate, which also induced neural tissue in competent ectoderm, suggesting that PKC is involved in the response to the endogenous inducing signal during neural induction.     

Rat pancreatic islet cells in primary culture: occurrence of giant cells amenable to patch clamping

Summary Neonatal and adult rat pancreatic islet cells were maintained in dissociated cell culture for up to three weeks. The unexpected occurrence of giant (40–50 m) cells was noted, some of which reacted positively to an insulin antiserum, indicating the presence of insulin. The giant cells were amenable to study using the extracellular patch clamp technique, which was used to demonstrate a population of membrane channels gating outwardly directed current in these cells.     

Proof-analysis and Continuity

During the first phase of Greek mathematics a proof consisted in showing or making visible the truth of a statement. This was the epagogic method. This first phase was followed by an apagogic or deductive phase. During this phase visual evidence was rejected and Greek mathematics became a deductive system. Now epagoge and apagoge, apart from being distinguished, roughly according to the modern distinction between inductive and deductive procedures, were also identified on account of the conception of generality as continuity. Epistemology of mathematics today only remembers the distinction, forgetting where they agreed, in this manner not only destroying the unity of the perceptual and conceptual but also forgetting what could be gained from Aristotelian demonstrative science.