I have come across one of the most ridiculous comments regarding science today and the comment comes from none other then conservative activist John Smithson. Smithson tries to take science down a peg or two by claiming that science, with its foundations in mathematics, is based on faith.
Postulates, Smithson argues, are nothing but truths without evidence, and because postulates simply exist without the need to be proven, one must have faith in their validity in order for everything based on them to function. Smithson makes this argument by trying to say that God is a truth that requires no evidence of existence, much like a postulate, but Smithson is mistaken.
The basic answer to your question is that we have to start somewhere.
The essence of mathematics (in the sense the Greeks introduced to the world) is to take a small set of fundamental "facts," called postulates or axioms, and build up from them a full understanding of the objects you are dealing with (whether numbers, shapes, or something else entirely) using only logical reasoning such that if anyone accepts the postulates, then they must agree with you on the rest.
Postulates require logical reasoning - religion does not.
In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.
In mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Unlike theorems, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow (otherwise they would be classified as theorems).
Logical axioms are usually statements that are taken to be universally true (e.g., A and B implies A), while non-logical axioms (e.g., a + b = b + a) are actually defining properties for the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom," "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.
Essentially, Smithson questions truths such as the basic 4 + 1 axioms of Euclidean geometry, which state the following:
- To draw a straight line from any point to any point.
- To produce [extend] a finite straight line continuously in a straight line.
- To describe a circle with any center and distance [radius].
- That all right angles are equal to one another.
- The parallel postulate: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
I never realized that all right angles being equal to one another was based solely on faith! Smithson writes that by my arguing his advocating of his faith, I should feel "quite stupid," but I'm pretty confident that I am not the intellectually inferior party here. Smithson believes the "godless world is built around science." If that is the case, then he must be living in sin, because last time I checked, Smithson's blog didn't miraculously appear on the internet because of his faith - it is due to the wonderful world of science, brought to you by postulates, of course...
Kevin, you did not present my argument accurately, but that is how you operate. A postulate is not an illogical truth. While one might argue for a postulate's "logic," a postulate remains an unprovable axiom that is used because it works, not because there is a mathematical equation proving it. THAT, my friend, is a fact. And where did I get my information? In an Algebra II class taught at a community college.ReplyDelete
Maybe this will help -- a definition found in Wikipedia or is that too conservative for your unbelieving taste?
"in traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths."
God is every bit the axiomatic truth that is a scientific postulate. Life simply does not work without God. I am wondering, do all Catholics take your position ?
You seem to have quoted the same Wikipedia entry as me but you fail to understand the difference between postulates and God.ReplyDelete
Mathematics, as you had written, requires a "system of postulates" and is "ultimately based upon FAITH."
One does not need faith to see postulates, such as the axioms of Euclidean geometry, because such truths are self evident.
Mathematics differs from religion, which is solely based upon faith. People can not see God, heaven, or hell, and so they must rely on their faith to prove that such things exist.
God may be a truth but it still requires faith, while a postulate does not. That is the difference I was illustrating...