On Axioms (2000-2009)

"Mathematics is not placid, static and eternal. […] Most mathematicians are happy to make use of those axioms in their proofs, although others do not, exploring instead so-called intuitionist logic or constructivist mathematics. Mathematics is not a single monolithic structure of absolute truth!" (Gregory J Chaitin, "A century of controversy over the foundations of mathematics", 2000)

"We start from vague pictures or ideas […] which we encapsulate by rules, and then we discover that those rules persuade us to modify our mental images. We engage in a dialog between our mental images and our ability to justify them via equations. As we understand what we are investigating more clearly, the pictures become sharper and the equations more elaborate. Only at the end of the process does anything like a formal set of axioms followed by logical proofs" (E Brian Davies, "Science in the Looking Glass", 2003)

"A recurring concern has been whether set theory, which speaks of infinite sets, refers to an existing reality, and if so how does one ‘know’ which axioms to accept. It is here that the greatest disparity of opinion exists (and the greatest possibility of using different consistent axiom systems)." (Paul Cohen, "Skolem and pessimism about proof in mathematics". Philosophical Transactions of the Royal Society A 363 (1835), 2005)

"An axiomatic theory starts out of some primitive (undefined) concepts and out of a set of primitive propositions, the theory’s axioms or postulates. Other concepts are obtained by definition from the primitive concepts and from defined concepts; theorems of the theory are derived by proof mechanisms out of the axioms." (Cristian S Calude, "Randomness & Complexity, from Leibniz to Chaitin", 2007)

"Human language is a vehicle of truth but also of error, deception, and nonsense. Its use, as in the present discussion, thus requires great prudence. One can improve the precision of language by explicit definition of the terms used. But this approach has its limitations: the definition of one term involves other terms, which should in turn be defined, and so on. Mathematics has found a way out of this infinite regression: it bypasses the use of definitions by postulating some logical relations (called axioms) between otherwise undefined mathematical terms. Using the mathematical terms introduced with the axioms, one can then define new terms and proceed to build mathematical theories. Mathematics need, not, in principle rely on a human language. It can use, instead, a formal presentation in which the validity of a deduction can be checked mechanically and without risk of error or deception." (David Ruelle, "The Mathematician's Brain", 2007)

"If you have the rules of deduction and some initial choice of statements as sumed to be true (called axioms), then you are ready to derive many more true statements (called theorems). The rules of deduc tion constitute the logical machinery of mathematics, and the axioms comprise the basic properties of the objects you are interested in (in geometry these may be points, line segments, angles, etc.). There is some flexibility in selecting the rules of deduction, and many choices of axioms are possible. Once these have been decided you have all you need to do mathematics." (David Ruelle, "The Mathematician's Brain", 2007)

"In mathematics, the first principles are called axioms, and the rules are referred to as deduction/inference rules. A proof is a series of steps based on the (adopted) axioms and deduction rules which reaches a desired conclusion. Every step in a proof can be checked for correctness by examining it to ensure that it is logically sound." (Cristian S Calude et al, "Proving and Programming", 2007)

"Mathematics as done by mathematicians is not just heaping up statements logically deduced from the axioms. Most such statements are rubbish, even if perfectly correct. A good mathematician will look for interesting results. These interesting results, or theorems, organize themselves into meaningful and natural structures, and one may say that the object of mathematics is to find and study these structures." (David Ruelle, "The Mathematician's Brain", 2007)

"Reducing theorems to a small number of axioms turns out to be deeply reminiscent of what scientists do. The mark of a good scientific theory, after all, is that it describes a large number of observations of the world while making only a small number of assumptions." (Marcus Chown, "God’s Number: Where Can We Find the Secret of the Universe? In a Single Number!", 2007)

"The fact that we have an efficient conceptualization of mathematics shows that this reflects a certain mathematical reality, even if this reality is quite invisible in the formal listing of the axioms of set theory." (David Ruelle, "The Mathematician's Brain", 2007)

"We axiomatize a theory not only to better understand its inner workings but also in order to obtain metatheorems about that theory. We will therefore be interested in, say, proving that a given axiomatic treatment for some physical theory is incomplete (that is, the system exhibits the incompleteness phenomenon), among other things." (Cristian S Calude, "Randomness & Complexity, from Leibniz to Chaitin", 2007)

"When we introduce the concept of a group, we do this by imposing certain properties that should hold: these properties are called axioms. The axioms defining a group are, however, of a somewhat different nature from the ZFC axioms of set theory. Basically, whenever we do mathematics, we accept ZFC: a current mathematical paper systematically uses well-known consequences of ZFC (and normally does not mention ZFC). The axioms of a group by contrast are used only when appropriate." (David Ruelle, "The Mathematician's Brain", 2007)

"Why are proofs so important? Suppose our task were to construct a building. We would start with the foundations. In our case these are the axioms or definitions - everything else is built upon them. Each theorem or proposition represents a new level of knowledge and must be firmly anchored to the previous level. We attach the new level to the previous one using a proof. So the theorems and propositions are the new heights of knowledge we achieve, while the proofs are essential as they are the mortar which attaches them to the level below. Without proofs the structure would collapse." (Sidney A Morris, "Topology without Tears", 2007)