We Hold These Truths to be Self-Evident
So starts the American Declaration of Independence. It is where the American ideal starts. It is the foundation of American political philosophy. It was written by Thomas Jefferson and ratified by congress on July 4th 1776. The complete sentence from which it is taken is:
We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.
It is one of the best-known sentences on Earth.
Euclid – the Father of Geometry
And it owes at least part of its success to Euclid. Euclid is best known for his Geometry. But the content of his book, The Elements, is not original. Euclid appears to have collected his theorems and propositions from others in his area. They are not simply a collection of results. Euclid was guided by the principle:
Prove all things. Hold fast to that which is true.
Euclid wanted a treatise which could be believed. He was not trying to impose his will on the student. He allowed anyone to follow his thoughts and to accept or reject the logic of his arguments. So, any statement made in proving a theorem, had to have a reason attached to it. The reasons had to come from four different areas:
- The Definitions
- The Postulates
- The Axioms
- Previous theorems
The Definitions – well they tell us what Euclid means when he says "rhombus". This is just as important today as it was when he wrote them. Language changes over time – the original meaning of "rhombus" is "bull-roarer" – but, depending on where you were born and raised, the same word may have different meanings (do you know what a bull-roarer is?). Hence the need for Definitions.
The Postulates are a list of the things you are allowed to do. Are you allowed to draw a line between two points? And so on.
Now the Axioms. These are the things that Euclid believes to be so obvious that they need to be said but they don't need to be proved. So, the first is: Things equal to the same thing are equal to each other. If I'm as tall as you and you're as tall as her, then I'm as tall as her.
The Theorems are the properties of geometrical figures and the relationships between them that, together, lead to a desired conclusion. The result of any theorem, "The base angles of isosceles triangles are equal" for example, may be quoted in subsequent theorems as a reason – every statement in Euclidean Geometry must have a reason.
So:
- We define our terms
- We state what we are allowed to do
- We tell the reader what we consider self-evident
- We can use previous results
And every statement must be proved. Nothing is to be taken on faith.
Euclid in action
Euclid's method has been used for 2400 years. It has been held in such high regard that, for centuries, scholars have used his method whenever they wished to communicate their own ideas and discoveries.
If you watch courtroom drama on television, you will see Euclid's method in action. The British legal system was deliberately cast in Euclid's mould. The doctrine of precedent, the appeal to the Constitution and the necessity of proof, are all based on Euclid.
Newton's original works on Physics and Mathematics are written as propositions and theorems. And the American Declaration of Independence owes its structure to Euclid: "We hold these truths to be self-evident" is an Axiom. The Constitutions of governments around the world are part Definition, part Postulate and part Axiom.
State the Axiom!
And on a personal level, we have all been made to look foolish in a classroom because of many things. But one of those things is the teacher assuming that what he or she considers to be self-evident, is therefore self-evident to everyone in the classroom. And it isn't!
Much misery could be avoided if teachers stated their axioms before launching into a new topic. That would give the poor student a chance to question the axiom and either accept it or reject it. How many students who have said: "I don't understand Algebra, Calculus, …" would have been helped if they knew what the teacher thought they knew – before the lack of this knowledge convinced them they couldn't understand due to some character or intellectual flaw?
Some students are blessed with the ability to seek out the axioms required for any new work they do. Some teachers do so as well. But many students, and teachers, would not think of doing so, simply because they don't know the power and necessity of axioms, those things we consider self-evident, that may not be self-evident to anyone else.
Not paying attention to axioms can cause problems for centuries. Zeno of Elea proposed several paradoxes which have survived for over 2000 years. In the 1880's, Weierstrass provided the axiom necessary to refute them. It's called the epsilon-delta definition of limits and (now) is a foundation idea in Calculus. Its unlikely Zeno knew of it and, in making up his paradoxes, probably deluded himself as well as the majority of people who discussed them for the next two millennia.
In contrast, Newton starts his book Principia Mathematica with the Laws of Motion. For Newton these were Axioms in the sense that they were necessary for all the things that Newton was able to deduce in Physics and they were incapable of proof.
Subsequently, other physicists challenged these laws/axioms as experiments seemed to disprove them. And other geometers have challenged Euclid's axioms. These challenges have resulted in new physics and new geometries. If you're a fan of Star Trek you've seen what Hollywood has been able to do with both of these. Of course, one way to be famous is to challenge an axiom, to prove that it is false or that it is not unique.
Aristotle, a contemporary of Euclid, stated that human beings are subject to three Laws of Thought:
- A thing is itself.
- Everything is either A or not A, where A is some quality like Red.
- Nothing is both A and not A.
They also are axioms:
- An apple is an apple.
- Everything is either an apple or not an apple.
- Nothing is both an apple and not an apple.
The quandary of Quantum Mechanics
And we believed them up till the start of the 20th century. Then a couple of physicists came up with an idea called Quantum Mechanics. It was based on evidence that could not be explained by current theories. In a nutshell, Quantum Mechanics says that Aristotle's Three Laws of Thought may be OK for apples. But for subatomic particles, Aristotle's laws needed an extra couple of words. Here is what Quantum Mechanics says:
- A thing is not itself.
- It is not the case that everything is either A or not A.
- Some things are both A and not A.
Many people cannot accept Quantum Mechanics. Many of them know that there is something terribly puzzling about what Quantum Mechanics says. But putting the problem into axiomatic form allows us to see what the problem is. It's not a problem with us, it's a problem with what Quantum Mechanics claims is axiomatic. We can then accept Quantum Mechanics axioms and see where they lead or we can reject the new axioms and return to Newtonian physics.
You may be comforted to know that NASA uses Newtonian physics for all space voyages ï¿½ no Einsteinian or Quantum physics at all.
But when we have the axioms in front of us, we know where any problems lie. What we do about it is up to us. We have life, liberty and can pursue happiness. And that sort of liberty allows us to choose to reject axioms – or accept them.