This notion, that a proposition may be established as the conclusion of an explicit logical proof, goes back to the ancient Greeks, who discovered what is known as the "axiomatic method" and used it to develop geometry in a systematic fashion. The axiomatic method consists in accepting without proof certain propositionsas axioms or postulates (e.g.,the axiom that through two points just one straight line can be drawn), and then deriving from the axiom all other propositions of the system as theorems. The axioms constitute the "foundations"of the system; the theorems are the "superstructure," and are obtained from the axioms withthe exclusive help of principles of logic.(p.2)引自 I Introduction
In fact, it came to be acknowledged that the validity of a mathematical inference in no sense depends upon any special meaning that may be associated with the terms or expressions contained in the postulates. Mathematics was thus recognized to be much more abstract and formal than had been traditionally
supposed: more abstract, because mathematical statements can be construed in principle to be about anything whatsoever rather than about some inherently circumscribed set of objects or traits of objects; and more formal, because the validity of mathematical demonstrations is grounded in the structure of statements, rather than in the nature of a particular subject matter. The postulates of any branch of demonstrative mathematics are not inherently about space, quantity, apples, angles, or budgets; and any special meaning that may be associated with the terms (or"descriptive predicates") in the postulates plays no essential role in the process of deriving theorems. We repeat that the sole question confronting the pure mathematician (as distinct from the scientist who employs mathematics in investigating a special subject matter) is not whether the postulates he assumes or the conclusions he deduces from them are true, but whether the alleged conclusions are in fact the necessary logical consequences of the initial assumptions. (p.8)引自 II The Problem of Consistency
A general method for solving it was devised. The underlying idea is to find a “model” (or“interpretation") for the abstract postulates of a system, so that each postulate is converted into a true statement about the model. (p.11)引自 II The Problem of Consistency
Introducing the idea of meta-mathematics is insightful. In my field of study (sociology and educational methodology), methodology is described as pertaining to meta-methodology (borrowing Nagel and Newman’s words on p.21. it is meaningful statements about doing research and their arrangements and relations and p.24 the assertions and discipline in which the subject matter) as well as methodology (let’s say, a system of meaningless signs — but this doesnt work well, and the subject matter as a system of signs).
“The description, discussion, and theorizing about the system belong in the file marked “meta-mathematics” (p.24). But is it really meta as if the system exists before its meta -level arrangement? Or the idea of system, and everything literally, depend on what is called meta?
'If either Mt. Rainier is 20,000 feet high or Mt. Rainier is20,000 feet high, then Mt. Rainier is 20,000 feet high'. The reader will have no difficulty in recognizing this long statement to be true, even if he should not happen to know whether the constituent statement 'Mt. Rainier is 20,000 feet high' is true. Obviously, then, the first axiom is a tautology-"true in all possible worlds." It can easily be shown that each of the other axioms is also a tautology. (p. 40)引自 V An Example of a Successful Absolute Proof of Consistency
