November 2, 2009
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Orthogonal versus independent: The linear algebra of truth
Philosophers are quite promiscuous in
their usage of the word “orthogonal”.For instance: Daniel
Dennett writes in Freedom Evolves that free will and
determinism are orthogonal. The Stanford Encyclopedia of Philosophy
article on “Moral anti-realism” has a section arguing that
subjectivism and relativism are orthogonal. Accommodationists toward
religion will often claim that science and religion are orthogonal.What these philosophers seem to mean
is that the concepts are independent,
that when one changes it is not logically necessary that the other
change as well. This is no doubt correct; but orthogonal is
a far stronger notion than this, one which has not at all been
demonstrated.Two
vectors are (linearly) independent if
they are not parallel or antiparallel; similarly, two concepts are
independent if they are not perfectly correlated or anticorrelated.
Determinism and free will, subjectivism and relativism, science and
religion, are surely independent.But
two vectors are only orthogonal if
they are perpendicular, if they share no linear components
whatsoever. Similarly, two concepts should only be considered
orthogonal if they are completely uncorrelated.
This is difficult to find in nature, though it does occur—typically
in completely unrelated domains. Belief in free will is basically
orthogonal to theoretical claims about dark matter.Clearly
determinism and free will are not completely
uncorrelated—in theory and in
practice beliefs about determinism are closely connected with beliefs
about free will. Similarly, subjectivism and relativism are closely
correlated, and belief in science and belief in religion are strongly
anticorrelated (hence, 95% of the National Academy of Sciences is
atheist or agnostic).It’s
not a big deal, I suppose; but I think more consistent usage might
alleviate some confusion. When one person says “orthogonal”
and another objects that the two concepts aren’t completely
unrelated, and the first
responds that this isn’t what he meant… basically the two are
talking past each other until both can agree about what “orthogonal”
really means.
