With illumination, abstraction, the rendering of propositions elaborated, the elements and development of mathematics may now be described. First and foremost, a definition may be made by identifying with a term what is sensibly perceived and rendered according to its correct usage. For example, numbers, such as 1, 2, 3, etc., may be defined as […]

# Author: Jim Turner

## The Mathematician’s Mind Redux, II

Within Hadamard’s cycle, there is a movement of incubation to illumination in which ideas in a mathematician’s mind are surveyed in search of resolving a specified problem. In this process, ideas are evaluated based upon their relevance to the content of the problem and connections are sought between these ideas based upon their relevance to […]

## The Mathematician’s Mind Redux, I

After a long hiatus, I have decided to re-introduce and elaborate upon what I take to be an account of what it means to think beautifully about mathematics. In an effort to do so, I will be following a path that will engage, at least in its initial stages, the work of the mathematician Jacques […]

## Creational Thinking – Part III: Abstraction and Order

We continue to sketch an account of creational thinking as a way of understanding mathematics and its relationship to reality. In our narrative so far, we had begun to develop a perspective of objects of reality as creative by adopting a hylomorphic view of them as the sensible and intelligible united in matter. As objects […]

## Creational Thinking – Part II: Things, Abstractions, and Formations

We begin our journey in developing a notion of creational thinking by giving an account of physical objects in hylomorphic terms and describe how by abstraction human understanding distinguishes such an object in terms of the sensible and the intelligible. In the course of laying out such an account, we will be expanding upon our […]

## Creational Thinking – Part I: Introduction

In a previous series of posts (beginning here https://thinkingbeautifully.org/mathematical-understanding-as-seen-within-a-framework-of-beauty-part-1/), I described a particular perspective on thinking beautifully in mathematics. In that description, I aimed to channel the thinking of Medieval thinkers such as Thomas Aquinas and Bonaventure, but also by incorporating the thinking of such twentieth century philosophers as Bernard Lonergan and Michael Polanyi. Unfortunately, […]

## Mathematical Understanding as Seen Within a Framework of Beauty (part 4)

Following upon my previous posts, where I began to explore the possibility of framing mathematical understanding within the Thomistic modes of beauty proportio and claritas, I now consider the third mode of beauty, that of integritas. This mode brings beauty in mathematical understanding to its fulfillment and serves to complete and unite the other two […]

## Mathematical Understanding as Seen Within a Framework of Beauty (part 3)

In the first two parts of this series, I began outlining how mathematics and mathematical understanding can be framed within the Thomistic modes of beauty: proportio, claritas, and integritas. In particular, I defined mathematics as the science whose subject-matter is measurable orders: objects understood as parts united into whole, having a distinction of same or […]

## Mathematical Understanding as Seen Within a Framework of Beauty (part 2)

In part 1 of this series, I began to explore how mathematical reasoning can be understood within a Thomistic framework of beauty as expressed in the modes proportio, claritas, and integritas. In that post, I arrived at an initial description of the subject-matter of mathematics as pertaining to quantitative being understood as parts ordered into […]

## Mathematical Understanding as Seen Within a Framework of Beauty (part 1)

In a previous series of posts (part 1 being here: https://thinkingbeautifully.org/form-beauty-and-euclids-elements-part-1/) I began to describe how mathematics could be understood as an endeavor of human discovery and invention by showing how form and the pursuit of beauty underlies successes in such efforts. In particular, I focused on the opening of Book I of Euclid’s Elements […]