Just a small correction: When you say
the Method of Least Squares was a better fit than the Bell Curve
it should be noted that the method of least squares is the bell curve-that is,
the Gaussian
distribution. What CSP is distinguishing the method of least squares from is
rather the notion
that there should be a definite threshold ("limen") between perceived and not
perceived, or
between same and different. For CSP (and for us psychophysicists today) it's
all probabilistic
and a matter of more or less, and not of a definite threshold. The most recent
incarnation of
CSP's view is Signal Detection Theory, q.v.
Jay Dowling
From: Dennis Leri [mailto:[email protected]]
Sent: Tuesday, September 16, 2014 5:52 PM
To: Peirce List
Cc: [email protected]
Subject: Re: [PEIRCE-L] Re: [biosemiotics:6834] Re: Natural Propositions,
Chapter 2
Lists,
On page 29, note 22 Frederick cites Gustav Fechner's Psychophysik. In his
experiments with Jastrow, Peirce contributed three things central to the
nascent science of Psycho-Physics after Fechner: 1) a better mathematical tool;
2) a recognition of sub-threshold awareness and hence bringing into question
the usefulness of the idea of a threshold, a just noticeable difference (jnd);
and 3) a new line of research, the application of mathematics to not just
sensory differentiation but also to reasoning under uncertainty.
Peirce followed a line of reasoning different from Fechner:
... the errors of our judgments in comparing our sensations seem sufficiently
accounted for by the slow and doubtless complicated process by which the
impression is conveyed from the periphery to the brain; for this must be liable
to more or less accidental derangement at every step of its progress.
According to Peirce, the Method of Least Squares was a better fit than the Bell
Curve. Peirce:
Accordingly we find that the frequencies of errors of different magnitudes
follow the probability curve, which is the law of an effect brought about by
the sum of an infinite number of infinitesimal causes. This theory, however,
does not admit of an Unterschiedsschwelle [difference limen or threshold-DL].
On the contrary, it leads to the method of least squares, according to which
the multiplication of observations will indefinitely reduce the error of their
mean, so that if, of two excitations, one were ever so little the more intense,
in the long run it would be judged to be the more intense the majority of
times...
But what about the just noticeable difference? For Peirce the math should not
only conform to experience but be predictive as well:
If there be a least perceptible difference, then when two excitations differing
by less than this are presented to us, and we are asked to judge which is the
greater, we ought to answer wrong as often as right in the long run. Whereas,
if the theory of least squares is correct, we not only ought to answer right
oftener than wrong, but we ought to do so in a predictable ratio of cases.
(Quotes from: Peirce, Charles Sanders & Jastrow, Joseph. "On Small Differences
in Sensation" [1885], in Memoirs of the National Academy of Sciences 3: 73-83)
And Peirce's research showed that under uncertain conditions, when differences
seemed to have disappeared or alternately to have not appeared, subjects did
indeed choose correctly within a band or range of differences close to the jnd
but definitely below it. What Peirce is saying is that we can and do as human
beings make accurate judgments that are based upon differences that are not
conscious, that are below threshold. And, not only do we do that to a degree of
accuracy greater than chance but that a methodology predicated upon the Method
of Least Squares says it must be the case. Differences that make a difference
need not be conscious.
Dennis
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