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RUSSELL_KANT_EUCLID

                                             September 13, 2020

Bertrand Russell, jamming around on the subject of
the connections between geometry and reality:

    "It was formerly supposed that Geometry was the study of the
    nature of the space in which we live, and accordingly it was
    urged, by those who held that what exists can only be known
    empirically, that Geometry should really be regarded as
    belonging to applied mathematics. But it has gradually
    appeared, by the increase of non-Euclidean systems, that
    Geometry throw's no more light upon the nature of space than
    Arithmetic throws upon the population of the United
    States. Geometry is a whole collection of deductive sciences
    based on a corresponding collection of sets of axioms. One set
    of axioms is Euclid’s; other equally good sets of axioms lead
    to other results. Whether Euclid’s axioms are true, is a
    question as to which the the pure mathematician is indifferent;
    and, what is more, it is a question which it is theoretically
    impossible to answer with certainty in the affirmative."

        "The Study of Mathematics" (1907)
        originally published in _New Quarterly_
        collected in "Mysticism and Logic and Other Essays" (1917)
        p. 71


            I puzzled over what Russell might be trying to
            say here because I would've thought that the
            truth of Euclidian geometry would be answered
            "with certainty", but *in the negative*--
            modern physics insists we live in a
            non-Euclidian universe of curved space.

            Then it dawned on me to think about the
            date of publication: 1907.
            Einstein's theory of General Relativity
            wasn't published until 1915.

            In retrospect, Russell looks somewhat
            prescient there-- at least it's occurred
            to him that Euclid wasn't at all firmly
            established.



    "Kant, rightly perceiving that Euclid's propositions
    could not be deduced from Euclid's axioms without the
    help of the figures, invented a theory of knowledge to
    account for this fact; and it accounted so successfully
    that, when the fact is shown to be a mere defect in
    Euclid, and not a result of the nature of geometrical
    reasoning, Kant's theory also has to be abandoned. The
    whole doctrine of a priori intuitions, by which Kant
    explained the possibility of pure mathematics, is wholly
    inapplicable to mathematics in its present form."

        "Mathematics and Metaphysicians" (1901)
        originally published in _International Monthly_
        collected in "Mysticism and Logic and Other Essays" (1917)
        p. 96

    I'm afraid this all strikes me as Russell
    assuming the conclusion he wants...
                                       
    Isn't it the case that what we have here are
    two ways of knowing, and both get to the
    same place?  Why prefer one path or the other?


It doesn't strike me as a terrible thing to
work out the rules of geometry empirically,        One of Roger Bacon's
using figure drawing to gain insight.              examples in support
                                                   of experimentalism
                                                   was drawing a triangle.



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