Objections Due To Axiomatic Mathematics
All of geometry and mathematics is based on a few simple assumptions, or unprovable axioms, from which logical consequences are drawn to form a bewilderingly complex assembly of proofs and laws. While the initial assumptions of Euclid's Elements or the Peano Axioms have been chosen to be simple and "self-evident", there can be no assurance that they do not result in some subtle error far down the chain of reasonings.
Indeed one consequence of any axiomatic basis is Godel's Theorem which demonstrates there must be mathematical laws which are both true and logically impossible to prove. In this sense, any mathematical structures we could devise must always be an incomplete theory. And as Galileo opined:
"Philosophy is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth."
None of these considerations imply that mathematics or science is wrong but rather that there is no way to be absolutely certain of their results. Simply speaking, we tend to prefer those theories that currently match all observations and are simplest.
As a case in point, a central underpinning of the clockwork universe was Newton’s law of gravitation which fostered the sense that nature was precisely predictable in the heavens as well as on earth. Newton’s putative gravitational fields were apparently pervasive and conclusive. Trajectories of cannon balls and planetary orbits were predicted with spectacular success. It was only much later that a deviation from Newtonian dynamics of roughly 43 arc seconds per century was detected in the position of the planet Mercury. This is an amount only discernible with telescopes and precision instruments and demands a leap of faith in your grandfather’s unwavering attention to astronomical detail. But the consequences were profound.
Newton’s gravitational fields are now recognized to be a complete illusion. And at no little increase in complexity, we now prefer to believe that the inertia of moving bodies carries them along straight lines through a space warped by massive objects in an unseen dimension. Whether this reflects reality any more than Newton’s inverse square law is problematical because Einstein’s General Relativity is not compatible with quantum theory and therefore known to be somehow fundamentally in error as well.
Thus as science increases our knowledge of the wonders of nature, so in like proportion does our appreciation of its limits.
Read more about this topic: Clockwork Universe Theory
Famous quotes containing the words objections, due, axiomatic and/or mathematics:
“Miss Western: Tell me, child, what objections can you have to the young gentleman?
Sophie: A very solid objection, in my opinion. I hate him.
Miss Western: Well, I have known many couples who have entirely disliked each other, lead very comfortable, genteel lives.”
—John Osborne (1929–1994)
“The Oriental philosophy approaches easily loftier themes than the modern aspires to; and no wonder if it sometimes prattle about them. It only assigns their due rank respectively to Action and Contemplation, or rather does full justice to the latter. Western philosophers have not conceived of the significance of Contemplation in their sense.”
—Henry David Thoreau (1817–1862)
“It is ... axiomatic that we should all think of ourselves as being more sensitive than other people because, when we are insensitive in our dealings with others, we cannot be aware of it at the time: conscious insensitivity is a self-contradiction.”
—W.H. (Wystan Hugh)
“It is a monstrous thing to force a child to learn Latin or Greek or mathematics on the ground that they are an indispensable gymnastic for the mental powers. It would be monstrous even if it were true.”
—George Bernard Shaw (1856–1950)