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 […]
Tag: abstraction
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 […]
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 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 […]
Beauty, Form, and Euclid’s Elements Part 3
In the previous posts to this series, https://thinkingbeautifully.org/form-beauty-and-euclids-elements-part-2/ I set out to articulate a perspective on the opening of Euclid’s Elements as arising by abstraction of forms that arise from sensible perceptions of things experienced in the real world. I aimed to make the case that certain ones of his Definitions and the forms his Postulates […]
Form, Beauty, and Euclid’s Elements (part 2)
In the previous installment https://thinkingbeautifully.org/form-beauty-and-euclids-elements-part-1/ (upon which this part depends), I gave what I feel is the correct beginning in how to arrive at and understand basic notions in mathematics. In doing so, I attempted to re-appropriate medieval concepts of matter, form, and abstraction in order to understand what constitutes the objects and subject-matter of […]
Form, Beauty, and Euclid’s Elements (part 1)
I want to begin exploring, or perhaps recovering, a way of understanding the discipline of mathematics as one that integrally involves a perspective of beauty, not a “beauty cold and austere, like that of a sculpture,” (Bertrand Russell) but a beauty that irradiates from the form of anything that can be understood as having being. […]