On Deduction (1975-1999)

"The advantage of semantic networks over standard logic is that some selected set of the possible inferences can be made in a specialized and efficient way. If these correspond to the inferences that people make naturally, then the system will be able to do a more natural sort of reasoning than can be easily achieved using formal logical deduction." (Avron Barr, Natural Language Understanding, AI Magazine Vol. 1 (1), 1980)

"It is actually impossible in theory to determine exactly what the hidden mechanism is without opening the box, since there are always many different mechanisms with identical behavior. Quite apart from this, analysis is more difficult than invention in the sense in which, generally, induction takes more time to perform than deduction: in induction one has to search for the way, whereas in deduction one follows a straightforward path." (Valentino Braitenberg, "Vehicles: Experiments in Synthetic Psychology", 1984)

"Deduction is typically distinguished from induction by the fact that only for the former is the truth of an inference guaranteed by the truth of the premises on which it is based. The fact that an inference is a valid deduction, however, is no guarantee that it is of the slightest interest." (John H Holland et al, "Induction: Processes Of Inference, Learning, And Discovery", 1986)

"It is difficult to distinguish deduction from what in other circumstances is called problem-solving. And concept learning, inference, and reasoning by analogy are all instances of inductive reasoning. (Detectives typically induce, rather than deduce.) None of these things can be done separately from each other, or from anything else. They are pseudo-categories." (Frank Smith, "To Think: In Language, Learning and Education", 1990)

"This absolutist view of mathematical knowledge is based on two types of assumptions: those of mathematics, concerning the assumption of axioms and definitions, and those of logic concerning the assumption of axioms, rules of inference and the formal language and its syntax. These are local or micro-assumptions. There is also the possibility of global or macro-assumptions, such as whether logical deduction suffices to establish all mathematical truths." (Paul Ernest, "The Philosophy of Mathematics Education", 1991)

"A mathematical proof is a chain of logical deductions, all stemming from a small number of initial assumptions ('axioms') and subject to the strict rules of mathematical logic. Only such a chain of deductions can establish the validity of a mathematical law, a theorem. And unless this process has been satisfactorily carried out, no relation - regardless of how often it may have been confirmed by observation - is allowed to become a law. It may be given the status of a hypothesis or a conjecture, and all kinds of tentative results may be drawn from it, but no mathematician would ever base definitive conclusions on it. (Eli Maor, "e: The Story of a Number", 1994)

"Mathematicians apparently don’t generally rely on the formal rules of deduction as they are thinking. Rather, they hold a fair bit of logical structure of a proof in their heads, breaking proofs into intermediate results so that they don’t have to hold too much logic at once. In fact, it is common for excellent mathematicians not even to know the standard formal usage of quantifiers (for all and there exists), yet all mathematicians certainly perform the reasoning that they encode." (William P Thurston, "On Proof and Progress in Mathematics", 1994)

"Model building is the art of selecting those aspects of a process that are relevant to the question being asked. As with any art, this selection is guided by taste, elegance, and metaphor; it is a matter of induction, rather than deduction. High science depends on this art." (John H Holland," Hidden Order: How Adaptation Builds Complexity", 1995)

"The methods of science include controlled experiments, classification, pattern recognition, analysis, and deduction. In the humanities we apply analogy, metaphor, criticism, and (e)valuation. In design we devise alternatives, form patterns, synthesize, use conjecture, and model solutions." (Béla H Bánáthy, "Designing Social Systems in a Changing World", 1996)

"For the scientific materialist the materialism comes first; the science comes thereafter. We might therefore more accurately term them 'materialists employing science'. And if materialism is true, then some materialistic theory of evolution has to be true simply as a matter of logical deduction, regardless of the evidence." (Philip E Johnson, "The Unraveling of Scientific Materialism", 1997)

"In everyday language, an axiom is often held to be a 'self-evident truth'. That phrase betrays some awfully sloppy thinking. A truth can be evident, but self-evident? Evidence is what convinces people that something is true. So what on earth is 'self-evident' supposed to mean? A truth that convinces itself? [...] To those who were seeking foundations for mathematics, axioms were much more prosaic. They weren't truths at all, let alone evident ones, and certainly not self-evident ones - assuming that such a slogan means anything at all. Axioms were assumptions. A place to start. A collection of statements that mathematicians agreed to accept. You are free to challenge them if you wish, but even if you do, you won't change mathematics (though you might create some more, branching off in a new direction). From the axiomatic viewpoint, mathematics consists of the deductions that are made, once the axioms are accepted as a starting point. It's like a game. If you want to play football, then you follow the rules of football. Of course you are free to change the rules - but then you're playing a different game." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"When a book is being written, it is a maze of possibilities, most of which are never realised. Reading the resulting book, once all decisions have been taken, is like tracing one particular path through that maze. The writer's job is to choose that path, define it clearly, and make it as smooth as possible for those who follow. Mathematics is much the same. Mathematical ideas form a network. The interconnections between ideas are logical deductions. If we assume this, then that must follow - a logical path from this to that. When mathematicians try to understand a problem, they have to thread a maze of logic. The body of knowledge that we call mathematics is a catalogue of interesting excursions through the logical maze."(Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)