“How could anyone hope to approach the concept of beauty without deeply studying the nature of formal patterns and their organizations and relationships to Mind.” – Douglas R. Hofstadter
Euclid was wrong. He was also completely right.
It is within the details of that pair of statements that a remarkable amount of modern (last 200 years) mathematics (and physics, biology, chemestry, and the social sciences) has thrived.
From Einstein’s Relativity Theory to Mandelbrot’s Fractals to Godel’s Incompleteness to Chaos Theory to Cognitive Science, it was the correction and/or reinterpretation of Euclid that sparked the fire.
- What does ‘straight line’ mean?
- Why can’t two parallel lines cross at infinity?
- Is the earth really a sphere?
- What constitutes a valid ‘proof’?
- Is ‘logic’ a priori?
- Are geometric shapes Platonic forms that acurately describe nature?
- Does a ‘line’ have two sides?
- What the hell is a ‘point’?
- What’s the point?
Those are only a small fraction (<— see what I did there) of the questions that have occupied the LIVES of thousands of mathematicians, scientists, and philosophers for a damned long time.
And what is quite cool is that many of them have been answered… and the answers are freakin’ SWEEEEEEEEET!
Without those questions, we wouldn’t know that black holes are REAL; that gravity bends time; that nearly ALL of nature follows the rules of fractal geometry and chaos/complexity theory; and on and on and on…
Modern Sci Fi would suck. Computer animation would blow. And we wouldn’t have iphones, or hip hop, or magic. (‘Magic’ is just another name for really trippy shit that our brains have a hard time understanding – irrespective of whether science is doing a good job explaining it or not.)
What’s better is that we are only just getting started!
There is far more unexplained phenomenon than there is explained – by a LOT.
To Infinity And Beyond!
In our quest to understand how the Theories of Mind and Body apply to making us stronger people (and how we can use that to develop stronger communities and cultures) – a new type of holistic, interdiciplinary science I’m calling Strength Theory – we will be diving into multiple boundry problems that touch upon all of these scientific areas (and more).
Mathematics (and other ‘analogy languages’) will be our Linqua Franca. We will use it to clarify and make intuitive principles that are hard (if not impossible) to get across via Engllish (or any other ‘natural’ language) because of their inherent LACK of clarity.
Natural languages are obfuscating almost on purpose. Single words can mean tons of different things depending upon context – sometimes totally outside of context! Sentences when spoken can be written in multiple ways that completely change their meaning. Without visual cues from a speakers face, we often wouldn’t have a clue what they were saying – making phone conversations periodically hilarious, and the ‘art of podcasting’ significally harder than the ‘art of Youtubing’.
Despite our cultures pathological fear of anything that looks “mathy”, math is one of the worlds greatest inventions, bringing a rigor and clarity to discourse that keeps conversations far more civil on on point. Basically… the opposite of the internet forums and Facebook!
I remember distictly the drastically different tones between my graduate courses in the Mathematics department versus the courses I took in the graduate departments in the applied subject areas I was working in: biology, political science, and philosophy.
In those other areas – largely devoid of mathematics in any real sense – conversations and debates quickly would become personal battles driven by emotional stubbornness. In contrast to that, arguments in a mathematics class, while periodically heated, were quickly (and easily) dispelled. The reasons actually have less to do with the nature of the subject (the objects it discusses), and more to do with the nature of the LANGUAGE of that subject (the way it discusses those objects).
Science at large has tried to adopt as much of this language as they can in part to keep arguments civil and on-point. Progress requires many things, but among them is the ability for smart people to have debates without talking past one another.
In other words: Mathematics is the language of love.(1)
The Mind Thinks Math
The SECOND reason we are going to dive into the formalization of mathematics is that this is the foundation upon which much of our current understanding of the Mind is based. Weird, I know.
Logic, Math, & Metamathematics play a significant role in the understanding of what the connection between a Brain and a Mind is, and what we mean when we say words like self-reference, recursive, or self-similar.
You can’t talk seriously about Cognition without using those words. Yet those words are fundamentally mathematical.
What Euclid Got Wrong
Euclid was one of the greatest geniuses to ever walk the earth. His Elements was among the most important creations by any human – ever. Most of what he “discovered”, or “created” (or however you want to look at it) is still true today, and will be forever. A world without Euclid is simply not one I want to live in.
He had one horrible flaw, however: He was human!
If only he’d been a God, he wouldn’t have made any mistakes. Alas, he wasn’t a God, and he DID make a few mistakes.
The most important of which fall into THREE main categories.
- The nature of his definitions made clarity very hard
- He had major logical gaps in his proofs
- His treatchment of logic was implicit rather than explicit
However, these flaws are so subtle, so hidden, it literally took 2,000 years for anyone to notice!
In the next post in this series I will describe these 3 issues in more depth. Then we will start fixing them via what mathematicians call ‘formalizing’. That is where the language is math/love is built. And we’ll learn how this formalizing of mathematics strikes to the core of modern science.
Welcome to the new world of Chaos and Complexity. The water is boiling hot, frozen cold, and yet quite nice all at the same time.
- Mathematics = Language of Love (LoL). [↩]