A hypothesis was true if it could be proved with ‘logic’ and false if, it could not be.

Pythagoras of Samos (circa 580-500 BCE) began to introduce the notion of mathematical proof; the first firm foundation stone of mathematical understanding – and therefore of science itself – was laid. Thales of Miletus (flourished 6^th century BCE) may have been the first to introduce this notion of proof, but it seems to have been the Pythagoreans who first made important use of it to establish things that were not otherwise obvious.

However, what is mathematical proof? A proof, in mathematics, is an impeccable argument, using only the methods of pure logical reasoning, which enables one to infer the validity of a given mathematical assertion from the pre-established validity of other mathematical assertions, or from some particular primitive assertions – the axioms – whose validity is taken to be self –evident. Once such a mathematical assertion has been established in this way, it is referred to as a theorem.

Until about 80 years ago, ‘truth’ to mathematicians had been synonymous with logical proof. A hypothesis was true if it could be proved with ‘logic’ and false if, it could not be. For this reason, mathematicians had operated in a fantasy world, one in which nothing was left to ‘faith’ because everything could be proved to be either true or false within a closed logical system.

By contrast, the world familiar to the rest of us is one in which faith assume a major role in deciding truth. In particular, controversial scientific hypotheses (such as the neo-Darwinian and teleological theories about the origin of species) are widely accepted as true , even though they have not been proved and never will be.

Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. Most of mathematics is based upon a well-understood structure of rules and is considered highly logical. It is always necessary to state, or otherwise have it understood, what rules, and presuppositions are being used before any logic can be applied.

In 1931, the Austrian-born, US mathematician/logician Kurt Gödel (1906-1978) demonstrated that within any given branch of mathematics, there would always be some propositions that could not be proven either true or false using the rules and axioms of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so; you will only create a larger system with its own unprovable statements. The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.

For example, Euclidean geometry is only one of several geometries, each type of geometry being based on a different set of axioms.

Euclidean geometry is a specific mathematical structure, with its own specific axioms (including some less assured assertions referred to as postulates), which provided an excellent approximation to a particular aspect of the physical world. This was the aspect of reality, well familiar to the ancient Greeks, which referred to the laws governing the geometry of rigid bodies and their relations to other rigid objects, as they moved around in 3-dimentional space. Certain of these properties were so familiar and self-consistent that they tended to become regarded as ‘self-evident’ mathematical truths and were taken as axioms (or postulates).

Einstein’s (1879-1955) general relativity – and even the Minkowskian (Hermann Minkowski, 1864-1909) space-time of special relativity – provides geometries for the physical universe that are different from, and yet more accurate than, the geometry of Euclid, despite the fact that the Euclidean geometry of the ancients was already extraordinary accurate. Thus, we must be careful, when considering geometrical assertions, whether to trust the ‘axioms’ as being, in any sense, actually true .

Gödel’s Theorem has been used to argue that a computer can never be as smart as a human being, because the extent of its knowledge is limited by a fixed set of axioms, whereas humans can discover unexpected truths. This plays a part in modern linguistic theories, which emphasise the power of language to come up with new ways to express ideas. In addition, it has been taken to imply that you will never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself.

Gödel showed that within a rigidly logical system such as that of English logician and philosopher Bertrand Russell (1872-1970) and English mathematician and philosopher Alfred North Whitehead (1861-1947) had developed for arithmetic, propositions could be formulated that are undecidable or indemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved nor disproved. Hence, one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions. It appears to foredoom hope of mathematical certitude through use of the obvious methods. Perhaps doomed also, as a result, is the ideal of science – to devise a set of axioms from which all phenomena of the external world can be deduced.

If anything at all, mathematics needs to be rational and logical. In fact, nothing is more rational than mathematics! Nevertheless, mathematical logic, or any other ‘lineal logical system’, has very severe limitations!

Logic is a poor model of cause and effect. We use the same words to talk about ‘logical sequences’ and about ‘sequences of cause and effect’. We say, ‘If Euclid’s (he lived around 300 BCE) definitions and postulates are accepted, then two triangles having three sides of the one triangle equal to three sides of another triangle; the triangles are congruent to each other.’ And we say, ‘If the temperature falls below zero degrees Centigrade water begins to ice.’

However, the ‘if … then’ of logic (‘lineal logic’) in the syllogism is very different from the ‘if … then’ of cause and effect (‘recursive logic’).

In a computer, which works by ‘cause and effect’, with one transistor triggering another, the sequences of cause and effect are used to ‘simulate’ logic.

Some 50 years ago, people used to ask, ‘Can a computer simulate all the processes of logic?’ The answer was yes, but the question was surely wrong. They should have asked, ‘Can ‘logic’ simulate all sequences of cause and effect?’ Then the answer would have been no.

A digital computer, or computer-controlled robot, has the ability to perform tasks commonly associated with intelligent human beings. The term is frequently applied to the project of developing systems endowed with the intellectual processes characteristic of humans, such as the ability to reason, discover meaning, generalize, or learn from experience. Since the development of the digital computer in the 1940s, it has been demonstrated that computers can be programmed to carry out very complex tasks – as, for example, discovering proofs for mathematical theorems or playing chess – with great proficiency. Still, despite continuing advances in computer processing speed and memory capacity; there are yet no programs that can match human flexibility over wider domains, or in tasks requiring everyday knowledge. On the other hand, some programs have attained the performance levels of human experts and professionals in performing certain specific tasks, so that artificial intelligence in this limited sense is found in applications as diverse as medical diagnosis, computer search engines, and voice or handwriting recognition.

When the sequences of ‘cause and effect’ become circular (or more complex than circular – recursive) then the description or ‘mapping’ of those sequences onto ‘timeless’ logic becomes self-contradictory. Paradoxes are generated that pure logic cannot tolerate.

The internal operation of an ordinary buzzer (doorbell) serves as an example, a single instance of the apparent paradoxes generated in a million cases of homeostasis throughout Biology. Here the ‘if … then’ junctures are causal. The ‘if … then’ of causality contains ‘time’, but the ‘if … then’ of logic is ‘timeless’ (it does not contain ‘time’). It follows that logic is an incomplete model of causality.

Logic can often be reversed, but then the effect precedes the cause. This generalisation has been an obstacle for the psychological and biological sciences since the time of Plato and Aristotle.

The Greeks were inclined to believe in what were later called ‘final causes’. They believe that the pattern generated at the end of the sequence of events could be regarded as in some way causal of the pathway followed by that sequence. This led to the whole notion of teleology, as it was called (‘telos‘, meaning the end, or purpose of a sequence). Teleology is anathema in modern natural scientific thinking – scientific theories apparently do not need the concept of foresight and planning! 
 

Any mental system, be it science, philosophy, religion, or mysticism, must be based on impeccable arguments, using only the methods of pure logical reasoning, which enables one to infer the validity of a given assertion from the pre-established validity of other assertions. If this is not the case, you should discard the mental system.

Physical science is by its very nature a closed, mental system because it only considers the material universe. Religion is also a closed, mental system because it rigorously adheres to certain dogmatic assertions. However, mysticism (because of its very nature) must be an open, mental system, which grows in understanding all the time.