[From Bill Powers (2010.06.23.0639 MDT)
Martin Taylor 2010.06.22.16.07
BP: I’ve managed to spread abroad some wrong thinking that I did in 1957,
or incomplete thinking that I thought was complete, with just as much
self-confidence as I had in that dim distant past. Thank goodness I
didn’t decide to make an argument with the editors of Science. My
brain has evidently been reworking this problem all night, because when I
awoke this morning I saw my error, and of course wondered what ELSE I
have been so confidently wrong about. Confidence is a fragile
flower.
This all began, back then, with a valid realization about perception: we
have no way of detecting a transformation that is common to ALL
perceptions. This is why we can’t know what reality really is. So when
Stevens published his Power Law I thought I saw his error (other than the
sin of taking my name in vain as well as misspelling it). What I didn’t
yet see in 1957, and forgot in replaying my thinking from then, was that
perception of the magnitude of a number might follow the same law as
perception of the magnitude of anything else. When you try to describe
the magnitude of a sensation by comparing it with the magnitude of a
number, you’re matching two transformations, not the quantities
themselves.
Martin, you evidently saw the same thing when you went through all those
cross-comparisons of one perception with another, back in that same era.
Of course you also found, as I did, how poor the approximation to a law
was, but that didn’t keep either of us from going on to play with various
mathematical approximations.
My mistake, in yesterday’s post and in 1957, was to forget that log(S)
versus log(I) is not the same thing as S versus log(I). I was vaguely
thinking, then and yesterday, that S meant the numeric measure of the
sensation, not the experience of it or the magnitude of the neural signal
representing it. The experience of it might be a perfectly linear
function of its physically measured (ruler-measured, nice idea Martin)
magnitude, but if the perception of number magnitude is logarithmic, we
will seem to have a power law, or logarithmic relationship, when
the whole of the nonlinearity is due to the way we perceive numbers.
I’m inclined to believe in linearity for one reason: the tracking model.
That linear model fits actual behavior within the tolerances we learn, in
physics 101, to expect in a student laboratory – around 3% of the range
of the data. Maybe a nonlinear model might improve the fit a bit, but on
a linear scale we’re already 97% of the way to perfection, so how much
work is it worth to eliminate that last three per cent error? On a log
scale, of course , a three per cent error looks the same under any
magnification we care to apply so we can never eliminated it – we just
move it to lower orders of magnitude. I don’t know if I said that
comprehensibly.
The problem is that an unknown portion of the apparent nonlinearity might
be due to the way we perceive the magnitudes of numbers with the rest
being due to the way perception works – and we have no way at present to
know what the proportions are. To use Martin’s distinction, we need to
measure both the perceptual response and the stimulus intensity with the
same ruler, or at least with rulers having known relative lengths. That
means that real psychophysics simply has to await the day when
neuroscience can identify a perceptual signal and measure its variations
on a linear scale while the corresponding stimulus is varied and measured
at the same time on a linear scale. I don’t think we’re very close to
knowing how to do that.
Best,
Bill P.