If one holds that mathematics is about the world, the two fundamental questions for elementary mathematics are:
1. How does geometry relate to the world?
2. How do numbers (especially irrational numbers) relate to the world?
Chapter 1 answers the first and Chapter 4, on the basis of Chapter 2 answers the second.
As presented in Euclid’s Elements, straight lines are infinitely thin, continuous and infinitely straight. Lines on earth, at the microscopic level, are none of these. So how can Euclid’s propositions apply to shapes and relationships in the world and how can his arguments reflect and capture relationships in the world? And if everything that we manufacture is shaped and measured according to our understanding of Euclid’s geometry is that to be expected or is it a happy accident?
Answering these questions is the essential burden of Chapter 1. Some highlights:
• A closed figure with three straight edges is a triangle when, considered as a shape there are three relevant sides with no relevant bending or discontinuities.
• Euclid’s postulates are all primitive measurements, either of distance or direction
• A Euclidean argument is a series of abstract measurements. It is a recipe for establishing the asserted indirect measurement by a series of more direct measurements, reducing ultimately to Euclid’s postulates.
Chapter 1 is the first instance of the broader principle that a) we need mathematics to establish quantitative relationships to support indirect measurement b) we establish quantitative relationships by mathematical arguments that embody a series of abstract measurements. In sum, measurement is both the purpose and the method of mathematics.
One thought on “Comments on Chapter 1 “Euclid’s Method””
If I never get beyond chapter 1 (“Euclid’s Method”) of Dr. Knapp’s book “Mathematics is About the World”, I will come away from it this invaluable insight:
“The mathematical issue arises from the fact that all measurement of continuous quantities is approximate.”
Approximate? I understand Ayn Rand’s idea of perfection pertaining to moral perfection as not only possible, but certain – if one’s pursuit of happiness is grounded in the proper moral code.
What’s interesting to me (a non-scientist; non-mathematician) is his idea of perfection as “approximate and finite”. This is an eye-opener, if I do understand it. (I do not grasp what a ‘continuous quantity’ is…). I had thought that “mathematics doesn’t lie”, meaning, to me, that mathematics is a science of perfection. And in my own field, as an artist, I know what he means when he says: “…one never achieves infinite precision…All precision is finite.” (p.25) My light and dark shades in a painting are never so sharply defined, so that some color from the surrounding areas must be blended into whatever is “defined” as the more important colors, shades and shadows of the main subject etc., however such precision is relevant to the whole. He says, on page 33: “One analyzes the complex in relation to the simpler.” This means that the main subject in my painting will be (usually) the more complex. And yet, the surrounding areas are just as important as the main subject. It’s truly a complex blending of what I try to make “precise” into the “approximate”.
Who would have thought that, in the world of art, mathematics, whether one knows or understands it, does somehow enter into the artist’s choices? All the way down to making every dab of color and shade, composition, color harmony etc. on the canvas matter. Well I’ve always known this in art, but to see it said in the context of mathematics is really a new insight.
How far off I may be from understanding the mathematics of what I do, I see that the artist’s choices in blending, color harmony, design and the tiniest details matter enormously: “…there will always be a limit to the precision that is actually available and the specific precision requirement in any particular case will always be finite” (p. 34) . This shows me that I’ve been right all along: I have not been gifted from afar with my ability to master my craft.
No matter how well, or little, I understand what I’ve read so far, Dr. Knapp’s idea has got me thinking. And isn’t that the great thing?