Truth and Proof
28/04/10
Philip Veasey (Chair of WLHS and onetime Professor of Mathematics) gave a talk which he hoped would clarify some
misunderstandings that he felt were apparent in the group's discussions on Science earlier this year.
He described the meaning of truth and proof in a spectrum of increasingly powerful systems moving through:
 Propositional Calculus
 Predicate, Modal and Deontic Logic
 Pure and Applied Mathematics
 Science
 Engineering and Technology
In formal systems such as logic and mathematics, statements are defined as things that have the property
of being either true or false. The truth or falsity of a statement is ultimately determined by whether it
can be derived from the axioms by using the rule(s) for deduction. Such proofs make no reference to the
world. Philip defined the properties of:
 Consistency  a contradiction cannot be deduced from the axioms
 Completeness  all true statements, expressible in the system, can be deduced from the axioms in a
finite number of steps
He explained how Gödel's Theorem had shown that only the simplest system, Propositional Calculus, was
consistent and complete. This is why, for instance, when we are unable to prove something in Mathematics,
it is always possible that it is true but can never be proved. Philip also mentioned some of the classic
logical paradoxes, such as "This statement is false", to support his contention that the foundations of the
rationality of which humanists are so proud are less solid than most people assume.
Although formal systems are abstractions that are independent of the world, logic attempts to demonstrate
valid ways of reasoning in the world. Mathematics seeks to extend this defining abstractions of what can
be seen in the world, e.g. numbers and space. Applied mathematics is the use of such abstractions to solve
real world problems. Pure mathematics looks at these abstractions of the world and finds patterns in them
leading to higher level abstractions such as the idea of a Group, in abstract algebra.
He continued by emphasising the change as we move through to Science. Now, instead of truth being a
property of statements made in a formal system without reference to the "real world", truth, or at least
our willingness to go along with something as being the closest we have got to the truth, is determined
by empirical evidence. The next step, into Engineering and technology, takes us into a world where the objective
is not the truth but a solution to a practical problem. Truth, if anything, is the success of the solution.
Philip said that, although there may be limits to the rationality, he always tested theists by asking them
whether they were prepared to accept a contradiction if their religion demanded it. This rapidly identifies
those with whom further discussion is pointless.
He finished by urging people to learn about the new science of Complexity which gives theoretical explanations
of evolutionary processes. The mathematics proves that under the right conditions in a complex adaptive system
such as the Economy, the Weather or Life on Earth, order will inevitably emerge from chaos. A good understanding
of this can change the way we view many things. He recommended the book "The Web of Life" by Fritjof Capra.
P.V.  15 May 2010
