Monday, February 28, 2011
Known as ‘the father of geometry’, Euclid is thought to have lived somewhere between 330 BCE and 260BCE. He taught mathematics in Alexandria and wrote what has been called the most enduring mathematical work of all time, the ‘Stoicheia’ or ‘Elements’. This thirteen volume work was a comprehensive compilation of geometrical knowledge, based on the works of Thales, Pythagoras, Plato, Eudoxus, Aristotle, Menaechmus and others.
In the dialogue that follows, Euclid and one of his students, Heractus, discuss the study of the laws of the natural world, the study of Nature itself, and research at university.
Euclid: All of my work is based upon Nature – the world God created. The straight line is the basis of my work. However, this is a theoretical concept, since the straight line is merely all that man sees of the World at his feet. Straight lines are found to be nothing of the sort if one steps back from Earth, since what we, men, call a straight line is that portion of the horizon of our Earth – it seems straight, but in reality, is the curvature of the planet.
To produce a line that was straight, one would have to draw a tangent to the Earth’s globe that would stretch to infinity without bending – that would be a true straight line. What we have to work with is only the semblance of that true straight line.
Heractus: How does that affect your work here on Earth?
E: It makes me realize that my geometry is merely an attempt – a scholarly attempt – to fathom out God’s Creation.
H: So what you are actually doing when unfolding your laws is unraveling God’s plan of His Creation. What is your motivation for doing that? Are you trying to equal God?
E: Not at all. No scholar presumes so much. In fact, the opposite is the case; that in the attempted unraveling, one comes to see the complexity of Nature – of God’s Creation, and this gives a scholar the inspiration to go on – not to approach God in His Genius, but to be inspired by it to achieve more than he would otherwise achieve.
God does but show only so much of His Creation to draw scholars in to a world of discovery that may take a lifetime – many such lifetimes to discover, and even then may leave much left to be found.
That is the beauty of the study and contemplation of Nature – of the Natural world and the laws that hold it all together – that try as you may, you can never come even close to discovering all there is to discover.
H: But isn’t that frustrating, the thought that one can never find what there is to be found?
E: You should think a minute more – several minutes more – and then ask your question again, coming to realize the full import of what you are saying.
If the totality of the laws and intricacies of Nature could be determined, what would we be left with? You recall the story of the Tower of Babel, when men built a tower so high as to be able to reach God.
H: Yes, I do.
E: Then you recall that before men committed their folly, God confounded them in their work by giving them the different tongues of the world so that they could not understand each other and so could not go on.
H: What then?
E: If you take the Elements, as I have called my work, you may also see the resemblance to that confounded tower of Babel.
Instead of confounding me with surrounding me with people who spoke different languages, I was given a complexity to unravel, which was ultimately impossible, come to many approximations as I eventually did.
Don’t you see that in my unraveling of the laws that hold our universe together, I was, in my own way, trying to reach out to God.
I could not actually reach God, no one can, but that shouldn’t stop us from applying what He endowed us with – the mind – to move into the abstractions that are nothing more or less than God’s plans of Nature.
H: So you could say that all study is the same; attempting to get nearer to God.
E: Yes, exactly. It matters little that what one person – let us say an undergraduate student – strives to find what others have already found. It is that student’s point of discovery at that point in time – it is that student discovering something first, just as it was when I first worked on geometrical laws all those years ago. The student discovering something for himself is exactly akin to that first discovery of mine. That student finds the joy of discovery in the same way that I did. A discovery for one is no less a discovery than any others.
H: What of research at University? Can we say that a student conducting original research is moving in uncharted waters?
E: Exactly. If that student is working at the forefront of knowledge, as he or she must be to claim that what is being researched is original, then that student is placed in the position I was when I was discovering for myself. Research is climbing that Tower of Babel, safe in the realization that it is only one step in an infinite number of steps – that any research, however humble it may appear to those who are more learned, is that finding for oneself some aspect of God’s Creation – in that way, research is a righteous occupation worthy of any one of us mortals.
Deductive reasoning is that type of reasoning which constructs or evaluates deductive arguments and then uses them to show that a conclusion necessarily follows on from a set of premises or hypotheses. If the argument reaches the conclusion predicted in the logic of the argument, then that argument is valid, provided that its premises were true. Deductive reasoning is a method of increasing knowledge.
H: Your mathematical observations were based upon deductive reasoning, were they not
E: They were, and deductive reasoning remains the modus operandi of all scientific investigation.
H: Why is that so?
E: Because for us to be confident in saying that something in the natural world is this way or that, we must be able to call upon proof, and proof stems from using deductive logic, with premises that must be true if the subsequent argument can be said to be valid. Notice here that I did not say that the argument is true, but valid.
H: What does that mean?
E: That it can be held up to scrutiny and found to be repeatable – is the basis of research not twofold – rigour being one, repeatability the other?
That is the nature of our ways of determining the laws of the natural world; by using a logic that is both impersonal and objective. Men have been persecuted for using deductive reasoning; it goes against any authority other than those of its own premises.
In other words, at one time in the history of societies and civilizations, something said that clearly was at odds with either popular belief (or we might say, popular ignorance or prejudice) or the higher authority of autocratic rulers was either ignored as being erroneous, or condemned as the heresy of the day.
H: So, it could be said, could it not, that this form of logic was instrumental in bringing forth a more rational mode of thought?
E: Quite so. I would go further and say that once such logic took hold in the minds of people, the assumed correctness of autocratic authority began to be toppled. In this way, the reasoning I used in the postulates, axioms and notions which formed my groundbreaking work on geometry, is akin to philosophy; it is the basis of a positive philosophy that has been responsible for changing the world of superstition and belief; to the modern world we all inhabit – a world in which logic and reason supersede authority and localized power, which led to people being kept in ignorance of how what we now call the physical sciences operate.
H: Not merely those, important as they are to our understanding of the planet and our place in it, but also in our place in civilized society, not merely as quartered slaves – serfs, if you will, but free men in our own right and standing as human beings. It is also responsible for fuelling the quest for scientific knowledge ever since, has it not?
E: So you say. With the benefit of hindsight, which future generations will be blessed with pertaining to the value of my work on geometry, and the ‘repercussions’ that stemmed from it, we may well say that the world – our portion of it – changed dramatically from that day in the field of mathematics, and the other sciences, as well as the philosophical underpinning to how we ought to be governed, rationally and impartially without recourse to prejudice masquerading as divine providence.
H: Mankind has truly benefited from your findings, possibly in ways that you could not have foreseen at the time.
E: That is undoubtedly so, but I may also add that had I had the time, inclination and energy to follow the full implications of my work, that I would undoubtedly have reached the conclusion that such a world as you find yourself in a thousand years from now would have come to pass; that is the nature and I might say the beauty and full value of deductive reasoning.
Robert L. Fielding
Posted by Justice at 10:46 PM