Recursive Synthesis and the Foundations of Mathematics

``Theoretical mathematics'': Toward a cultural synthesis of mathematics and theoretical physics

Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and destructive consequences. Serious caution is required, and the issue should be considered before, rather than after, obvious damage occurs. With the hazards carefully in mind, we propose a framework that should allow a healthy and positive role for speculation.

Responses to ``Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics'', by A. Jaffe and F. Quinn

This article is a collection of letters solicited by the editors of the Bulletin in response to a previous article by Jaffe and Quinn [math.HO/9307227]. The authors discuss the role of rigor in mathematics and the relation between mathematics and theoretical physics.

Recursion and Combinatorial Mathematics in Chandashaastra

Contribution of Indian Mathematics since Vedic Period has been recognised by the Historians. Pingala (200 BC) in his book on 'Chandashaastra', a text related to the description and analysis of meters in poetic work, describes algorithms which deal with the Combinatorial Mathematics. These algorithms essentially deal with the conversion of Binary numbers to Decimal numbers and vice versa, finding the value of 'n choose r', evaluating 2^n, etc. All these algorithms are recursive in nature. There are also evidences of making use of stack variables to stack the intermediate results for later use. …

A Simple Approach to the Tiling Problem Using Recursive Sequence

The tiling problem has been a famous problem that has appeared in many Mathematics problems. Many of its solutions are rooted in high-level Mathematics. Thus we hope to tackle this problem using more elementary Mathematics concepts. In this report, we start with the simplest cases, with the smaller numbers: the number of ways to tile a $2 \times n$, $3 \times n$, $4 \times n$ rectangular board using $2 \times 1$ domino tiles, where the number of rows is fixed and we present a recursive formula based on $m$ and the earlier terms. This allows us to deduce the non-recursive formula for each case …

Revisiting the mathematical synthesis of the laws of Kepler and Galileo leading to Newton's law of universal gravitation

Newton's deduction of the inverse square law from Kepler's ellipse and area laws together with his "superb theorem" on the gravitation attraction of spherically symmetric bodies, are the major steps leading to the discovery of the law of universal gravitation (Principia, 1687). The goal of this article is to revisit some "well-known" events in the history of science, and moreover, to provide elementary and clean-cut proofs, still in the spirit of Newton, of these major advances. Being accessible to first year university students, the educational aspect of such a coherent presentation should no…

开放准备、人类解释与教师合成：面向AI丰富高等教育的人尺度方法论

Open Preparation, Human Explanation, and Instructor Synthesis: A Human-Scale Methodology for AI-Rich Higher Education

本文提出一种人尺度方法论，通过信息学案例，强调将证据信任从工具禁用转向实时解释与观察，构建了操作周期、现实框架及口头证据模型，回应AI与主动学习的教育需求。