January 29, 2016
Thanks to Alissa Simon, HMU Tutor, for today's post.
Let me begin by stating that organizing and leading Harrison Middleton University's Quarterly Discussions is one of the best parts of my job. The participants always challenge me as do the texts and authors. This quarter, I chose something completely foreign to me and I am absolutely ecstatic with the result. For January's discussion, we read from Euclid's definitions in The Elements as well as a selection from Alfred North Whitehead's Introduction to Mathematics. The discussion truly brought the two works to life for me, especially in comparison with each other. I will briefly describe some of our conversation below. Please enjoy this segment of our conversation, and consider joining us for our next discussion (Kafka in April). Email me at any time at: firstname.lastname@example.org .
Euclid begins The Elements with the definition of a point. It is ironic to think that it was necessary to unpack the definition of a single point, but it was. We discussed this idea as a basic, abstract unit from which structure is created. In other words, the point can be viewed as the first building block in Euclidean geometry. This sounds obvious, but consider the concept of creating a number of definitions from scratch. Euclid had the benefit of some previous mathematicians and philosophers around him, but he essentially created this mathematical process. He explained a logical structure where, previously, none existed. Euclid explained abstract definitions with simplicity and precision. Mathematics is often a field of precision, and his simple clarity of reasoning may be one of the best examples of logic to this day. That this ancient Greek text continues to survive at all may be a testament to its worth. At first, the definition of a point did not appear to be innovative, however, as I began to understand the scope of Euclid's theories and applications, I wondered at his ability to locate and define the point at all.
Euclid's Elements stem from his studies of the earth. (Geo is Greek for earth, and metron is Greek for measure.) Therefore, Euclid had the foresight to realize that his observations and measurements of the earth had larger implications, but he also realized that his observations were limited by a definiteness. In order to open them to wider applications, he needed to find a way to abstract these measurements into definitions. Therefore, Euclid began to synthesize, simplify and apply his data. While Euclid discovered geometry in a very concrete way, he catalogued it in abstract language in order to increase its applicability.
Alfred North Whitehead calls mathematics an 'abstract science'. Whitehead reinforces the need for simple, direct mathematical language when he says, “The reason for this failure of the science to live up to its reputation is that its fundamental ideas are not explained to the student disentangled from the technical procedure which has been invented to facilitate their exact presentation in particular instances. Accordingly, the unfortunate learner finds himself struggling to acquire a knowledge of a mass of details which are not illuminated by any general conception.” His statement reinforces the same instinct that drove Euclid to begin The Elements with basic definitions. Geometry exists as a bridge between the natural world and human understanding. Yet it is written in abstract language which allows for a wider application. In other words, grappling with mathematical definitions allows all of us an opportunity to solve problems for ourselves, even in unrelated fields. Geometry, discovered through concrete means, has abstract applicability, while understanding the abstract may lead us back to a specific resolution.
Therefore, when Whitehead states that our abstract and inaccurate laws may be 'good enough', he actually means to say that our approach is vital, and our precision may be good enough. The definition and application of infinity supplies one example of this idea of 'good enough'. After we attempted to unpack the idea of 'infinity', the group concluded that: if infinity is the negation of the finite, then neither of those terms are representable. Instead, they are abstract definitions meant to guide us in some way. However, the adjective 'infinite' grants us an ability to talk about infinity without addressing specific quantities. Things can be infinitely large or infinitely small...both of which signify something endless or boundless. Furthermore, while infinity cannot be grasped with certitude, indefiniteness is merely an unknown.
These beginning conversations could spark endless conversation in my mind. I find applications in science, poetry, nature, social sciences, etc. Euclid's graceful simplicity combined with Whitehead's invitation to study our world led to an exemplary conversation. Whitehead states, “The vital point in the application of mathematical formula is to have clear ideas and a correct estimate of their relevance to the phenomena under observation”. Understand the idea, clarify your parameters and explore a new phenomena. The world is as big (or small) as we make it.
Thanks to all of our participants!
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