Related papers: Constructive mathematics
A note on "Bayesian nonparametric estimators derived from conditional Gibbs structures" by Antonio Lijoi, Igor Pr\"{u}nster, Stephen G. Walker [arXiv:0808.2863].
This is an expanded version of my review of Nina Engelhardt's book "Modernism, Fiction and Mathematics", Edinburgh University Press 2018. A considerably shortened version will appear in the Notices of the AMS.
Various 'optimistic' attempts have been made to reasonably explain the undeniable effectiveness of mathematics in its application to physics. They range over retrospective, historical accounts of mathematical applicability based on…
In their 1991 paper "Algebraic Reconstruction of Types and Effects," Pierre Jouvelot and David Gifford presented a type-and-effect reconstruction algorithm based on an algebraic structure of effects. Their work is considered a milestone in…
In this note some philosophical thoughts and observations about mathematics are expressed, arranged as challenges to some common claims.
These are notes to accompany four lectures that I gave at the School on Additive Combinatorics, held in Montreal, Quebec between March 30th and April 5th 2006. My aim is to introduce ``quadratic fourier analysis'' in so far as we understand…
Gentzen's 1936 proof of the consistency of Peano Arithmetic was a significant result in the foundations of mathematics. We provide here a modified version of the proof, based on G\"{o}del's reformulation, and including additional details…
In two previous papers, we exposed a combinatorial approach to the program of Geometry of Interaction, a program initiated by Jean-Yves Girard. The strength of our approach lies in the fact that we interpret proofs by simpler structures -…
We survey the logical structure of constructive set theories and point towards directions for future research. Moreover, we analyse the consequences of being extensible for the logical structure of a given constructive set theory. We…
The paper the title refers to is that in {\em Proceedings of the Edinburgh Mathematical Society}, {\bf 40} (1997), 367-374. Taking it as an excuse we intend to realize a twofold purpose: to atomize that important result showing by the way…
The constructive approach to mathematics has the advantage that witnesses can be extracted from statements of existence and theorems can be unwound to give algorithms. Even better, constructive theorems can be interpreted in any topos,…
We apply some tools developed in categorical logic to give an abstract description of constructions used to formalize constructive mathematics in foundations based on intensional type theory. The key concept we employ is that of a Lawvere…
Liouville's 1853 paper, in which he derived in closed form the general local solution of equation $u_{z\bar z}=\exp(u)$, is one of the few papers from the 19th century that 21st century mathematicians routinely quote as motivation for their…
The work presents two examples of simple mathematical formulas which are natural nonlinear modifications (one being a generalization) of Gielis' formula. These formulas involve a comparable number of parameters and provide non-Platonic…
AI constructivism as inspired by Jean Piaget, described and surveyed by Frank Guerin, and representatively implemented by Gary Drescher seeks to create algorithms and knowledge structures that enable agents to acquire, maintain, and apply a…
In this paper a construction of affine exterior algebra of Grassmann, with a special attention to the revisitation of this subject operated by Peano and his School, is examined from a historical viewpoint. Even if the exterior algebra over…
This chapter aims to provide a clear and understandable picture of constructive semigroups with apartness in Bishop's style of constructive mathematics, BISH. Our theory is partly inspired by the classical case, but it is distinguished from…
Logical bilateralism challenges traditional concepts of logic by treating assertion and denial as independent yet opposed acts. While initially devised to justify classical logic, its constructive variants show that both acts admit…
Starting from Greg Moore's description about Physical Mathematics, a framework is proposed in order to understand it, based on Gilles Ch\^atelet's philosophy. It will be argued that Ch\^atelet's ideas of inverting, splitting, augmenting and…
This text is a survey of the general theory of stochastic processes, with a view towards random times and enlargements of filtrations. The first five chapters present standard materials, which were developed by the French probability school…