The Nature of Consciousness

Piero Scaruffi

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These are excerpts and elaborations from my book "The Nature of Consciousness"

Formal Systems

David Hilbert, a German mathematician of the beginning of the 20th century, is credited with first introducing the question of whether a mechanical procedure exists for proving mathematical theorems (fully in 1928). His goal was to reduce Mathematics to a more or less blind manipulation of symbols through a more or less blind execution of formal steps. Already implicit in Hilbert's program was the idea that such a procedure could be carried out by a machine.  The discipline of formal systems was born, with the broad blueprint that a formal system should be defined by a set of axioms (facts that are known to be true) and a set of inference rules (rules on how to determine the truth or falsity of a new fact, given the axioms). By applying the rules on the axioms, one could derive all the facts that are true.

A formal system employs the language of propositions (statements that can only be true or false and can be combined by binary operators such as “not”, “and” and “or”) and predicates (statements with a variable that can be quantified existentially or universally, i.e. can be only true or false relative to “at least” one value of the variable or “for every” value of the variable). For example, the fact that Piero Scaruffi is a 51-year old writer could be expressed as: “writer (Piero) AND age (Piero, 51)”. The fact that teachers are poor can be expressed with the expression: "FOR EVERY x, teacher(x) -> poor (x)"; that translates as: every individual that satisfies the predicate "teacher" also satisfies the predicate "poor". The fact that some teachers are obnoxious can be expressed as: "FOR AT LEAST ONE x teacher(x) -> obnoxious (x)".

The language of Logic is not very expressive but it lends itself to logical reasoning, i.e. deduction.

This generation of mathematicians basically pushed logical calculus to the forefront of the tools employed to investigate the world. The apparatus of formal systems became the apparatus that one must use to have any scientific or philosophical discussion. Implicit in their program was the belief that the laws of logic “were” the laws of thought.

 


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