The oldest and most famous axiomatic system is that of Euclid, in which he systematized all of the knowledge of geometry (and more) available to him over two thousand years ago. Based on five axioms, Euclid was able to rigorously prove a very large number of known and new theorems (called "propositions" in his Elements). His axioms were supposed to be intuitively true. The first four axioms dealt with line segments, lines, circles and angles (see this Wikipedia entry) and have been viewed as self-evident. In contrast, the fifth axiom, which was equivalent to the following statement: "Through a point outside a given line, only one parallel to the line can be drawn" (page 9), was not intuitively true (apparently because the two lines involved extend to infinity in two directions, similarly to asymptotes).
Since this proposed axiom was not obviously true, many mathematicians tried to prove that it logically follows from the first four axioms. Only in the nineteenth century was it demonstrated that it is NOT possible to prove the parallel axiom from the first four axioms.
This proof that it is impossible to prove a given statement is a great precursor of Gödel's incompleteness theorems.
Since the parallel "axiom" is logically independent of the first four axioms, it could be assumed to be either true or false. Euclidean geometry, which assumes that it is true, appears to be a good match for (or model of) our every day experience of 3-dimensional space. However, several mathematicians also developed alternate geometries that take the parallel "axiom" to be false (e.g., there is no parallel or there is more than one parallel line). These non-Euclidean geometries are not intuitively "true" since they do not seem to model our every day experience of 3-dimensional space. Yet, they appear to make complete sense, mathematically.
But how can a mathematical system "make complete sense" if it is not obviously true? Well, what matters is that it could be true. In other words, the system must not be self-contradictory; it may not produce theorems that are mutually incompatible, such as one statement and its negation. If an axiomatic system meets this requirement, it is said to be consistent.
The development of non-Euclidean geometries led to a major shift in the perception, if not the role, of axioms. Axioms were and remain the essential, logical foundation of all knowledge (i.e., theorems) in an axiomatic system. However, axioms did not have to be obviously true of the natural world. They just had to be mutually consistent.
First, this shift gave much greater freedom to mathematicians, who, for example, had never before considered the possibility of non-Euclidean geometries.
Second, this shift had a much greater impact on the discipline of mathematics as a whole. Mathematics became more abstract. Internally consistent systems became as interesting and important as actual models of the real world.
Hence Bertrand Russell's famous quote:
"Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."But if internal consistency is enough, and the truth of the axioms (and therefore of the theorems as well) is only secondary, what is left? The short answer: the rigorous logical reasoning that leads from axioms to theorems.
For even if the literal truth of the whole edifice is not required any longer, it must still stand firmly and securely on its foundations. In other words, each and every theorem of an axiomatic system must necessarily follow from the axioms.
To establish the validity of logical inferences requires eliminating all sources of vagueness and making all steps in the reasoning fully explicit. But natural languages, presuppositions, intuition, etc., get in the way of precision and rigor. Therefore, mathematics did not just become more abstract, it became more formal.
Pure mathematics became a symbols game. Contrast Euclidean geometry (with, for example, its axioms stated in natural language, in this case ancient Greek) to our formal system FS, where each axiom, each premise and each conclusion of each inference rule is a meaningless sequence of symbols. Reasoning in a formal system depends only on the "form" or structure of the symbolic expression. And so "axiomatic systems" became "formal axiomatic systems."
The price to pay to guarantee that theorems logically and necessarily follow from axioms and other premises is to reduce logical inference to mechanical symbol manipulation, and most importantly, to give up meaning.
What do the symbols E , 0, and 1 in FS actually mean? It is not relevant. In fact, even if they do have a meaning (not within the system, of course, but for the human being who is studying the system from the outside), it is best to completely forget about it.
The only guaranteed way to make logically valid inferences, that is, 100% error-free symbol manipulations, is to empty one's mind of any interpretation of the symbols that could lead to implicit presuppositions, bias, or misguided intuition, in other words, to act like a mechanical automaton or a digital electronic computer.
Now, remember, we gave up the need for truth and meaning. But for the whole edifice to stand up to logical scrutiny, the formal system must still be consistent.
In the next few posts, we will look at several ways of demonstrating the consistency of an axiomatic system.
No comments:
Post a Comment