Mathematical Discourse vs. Mathematical Intuition
نویسنده
چکیده
One of the most uninformative statements one could possibly make about mathematics is that the axiomatic method expresses the real nature of mathematics, i.e., that mathematics consists in the deduction of conclusions from given axioms. For the same could be said about several other subjects, for example, about theology. Think of the first part of Spinoza’s Ethica ordine geometrico demonstrata or of Gödel’s proof of the existence of God, which are both fine specimens of Theologia ordine geometrico demonstrata. To the objection, ‘Surely theological entities are not mathematical objects’, one could answer: How do you know? If mathematics consists in the deduction of conclusions from given axioms, then mathematical objects are given by the axioms. So, if theological entities satisfy the axioms, why should not they be considered mathematical objects? Hilbert says: “If in speaking of my points”, lines and planes “I think of some system of things, e.g. the system: love, law, chimney sweep ... and then assume all my axioms as relations between these things, then my propositions, e.g. Pythagoras’ theorem, are also valid for these things”. Similarly he might have said: If in speaking of my points, lines and planes, I think of a suitable triad of theological entities, and assume all my axioms as relations between these things, then my propositions, e.g. Pythagoras’ theorem, are also valid for these things. Indeed, if mathematics consists in the deduction of conclusions from given axioms, then it has no specific content. So it is simply impossible to distinguish geometrical objects, such as ‘points, lines and planes’, from ‘love, law, chimney sweep’, or a suitable triad of theological entities. This is vividly ilustrated by Russel’s statement that “mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true”.
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