On Constructive Axiomatic Method

In this last version of the paper one may find a critical overview of some recent philosophical literature on Axiomatic Method and Genetic Method.Comment: 25 pages, no figure

Doing and Showing

The persisting gap between the formal and the informal mathematics is due to an inadequate notion of mathematical theory behind the current formalization techniques. I mean the (informal) notion of axiomatic theory according to which a mathematical theory consists of a set of axioms and further theorems deduced from these axioms according to certain rules of logical inference. Thus the usual notion of axiomatic method is inadequate and needs a replacement.Comment: 54 pages, 2 figure

The Uses of Argument in Mathematics

Stephen Toulmin once observed that `it has never been customary for philosophers to pay much attention to the rhetoric of mathematical debate'. Might the application of Toulmin's layout of arguments to mathematics remedy this oversight? Toulmin's critics fault the layout as requiring so much abstraction as to permit incompatible reconstructions. Mathematical proofs may indeed be represented by fundamentally distinct layouts. However, cases of genuine conflict characteristically reflect an underlying disagreement about the nature of the proof in question.Comment: 10 pages, 5 figures. To be presented at the Ontario Society for the Study of Argumentation Conference, McMaster University, May 2005 and LOGICA 2005, Hejnice, Czech Republic, June 200

From Euclidean Geometry to Knots and Nets

This document is the Accepted Manuscript of an article accepted for publication in Synthese. Under embargo until 19 September 2018. The final publication is available at Springer via https://doi.org/10.1007/s11229-017-1558-x.This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if (a) it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, (b) the information thus displayed is not metrical and (c) it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions.Peer reviewe

Individualized vs. Generalized Assessments: Why RLUIPA Should Not Apply to Every Land-Use Request

Courts and advocates alike have struggled to articulate a coherent rule regarding when the Religious Land Use and Institutionalized Persons Act (RLUIPA) should apply to local governments\u27 land-use decisions. When it is applied too broadly, RLUIPA runs roughshod over the ability of state and local governments to control their own land-use patterns, and it is inconsistent with the Supreme Court\u27s First Amendment and federalism precedents. When applied too narrowly, RLUIPA fails to provide a remedy for victims of religious discrimination. This Note explains the legally cognizable—but previously unrecognized—differences between the types of land-use decisions that local governments make, and it argues that RLUIPA should apply to individualized assessments, such as use permits and variances, but that RLUIPA should not apply to generalized assessments, such as requests for zoning-ordinance amendments. This Note uses two recent Ninth Circuit cases—one of which would have been decided differently if the court had used the proposed distinction—to illustrate how an analysis of individualized and generalized assessments would work in practice

Why 'scaffolding' is the wrong metaphor : the cognitive usefulness of mathematical representations.

The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least (and plausibly for others), scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary algebra indicates that Menary is on the right track with his ‘enculturation’ view of mathematical cognition. Moreover, these examples allow us to elaborate his remarks on the uniqueness of mathematical representations and their role in the emergence of new thoughts.Peer reviewe

A deterministic version of Pollard's p-1 algorithm

In this article we present applications of smooth numbers to the unconditional derandomization of some well-known integer factoring algorithms. We begin with Pollard's $p-1$ algorithm, which finds in random polynomial time the prime divisors $p$ of an integer $n$ such that $p-1$ is smooth. We show that these prime factors can be recovered in deterministic polynomial time. We further generalize this result to give a partial derandomization of the $k$-th cyclotomic method of factoring ($k\ge 2$) devised by Bach and Shallit. We also investigate reductions of factoring to computing Euler's totient function $\phi$. We point out some explicit sets of integers $n$ that are completely factorable in deterministic polynomial time given $\phi(n)$. These sets consist, roughly speaking, of products of primes $p$ satisfying, with the exception of at most two, certain conditions somewhat weaker than the smoothness of $p-1$. Finally, we prove that $O(\ln n)$ oracle queries for values of $\phi$ are sufficient to completely factor any integer $n$ in less than $\exp\Bigl((1+o(1))(\ln n)^{{1/3}} (\ln\ln n)^{{2/3}}\Bigr)$ deterministic time.Comment: Expanded and heavily revised version, to appear in Mathematics of Computation, 21 page

The twofold role of diagrams in Euclid's plane geometry

International audienceProposition I.1 is, by far, the most popular example used to justify the thesis that many of Euclid's geometric arguments are diagram-based. Many scholars have recently articulated this thesis in different ways and argued for it. My purpose is to reformulate it in a quite general way, by describing what I take to be the twofold role that diagrams play in Euclid's plane geometry (EPG). Euclid's arguments are objectdependent. They are about geometric objects. Hence, they cannot be diagram-based unless diagrams are supposed to have an appropriate relation with these objects. I take this relation to be a quite peculiar sort of representation. Its peculiarity depends on the two following claims that I shall argue for: (i) The identity conditions of EPG objects are provided by the identity conditions of the diagrams that represent them; (ii) EPG objects inherit some properties and relations from these diagrams