In the previous paragraph a simple example, which shows that the real num-
bers are necessary for assigning truth values, is given by colour name strategy.
Similar problems arise with many other correspondences between languages
and the world, for example consider the statement: ’J ones is six feet tall’.
This is unverifiable, the statement for which truth-values can be assigned is:
’J ones is six ± δ feet tall’ where δ is some error, usually of a statistical na-
ture. It is normally taken that physical measurements are real valued quantities
with real valued errors, and therefore the real numbers are needed to describe
the physical world. There is a chance that there is a theory of everything (TOE)
and this has been reviewed by Taubes (1995) [46]. It could be argued that TOE
would require that objects have only discrete properties, say by having length
a multiple of the Planck length etc., but present quantum mechanics requires
Hilbert spaces which in turn require properties of the real numbers, so that a
requisite TOE would have to change this; in any case it seems unreasonable
that philosophy of language and mathematics should legislate the nature of fun-
damental physics. A better reason for discarding the real numbers would be to
assume that humans have only a finite number of integer valued brain states,
so that the assignment of a finite number of truth values should suffice for the
assignment of meaning as perceived by the brain. The indications are that real
valued quantities are needed for describing cognition and hence brain states.
Three examples of this are:
1. Wynn (1992) [52] p.323 describes the accumulator theory of how peole
learn to count, this requires that people can perceive temporal duration,
a real quantity, and then when sufficient amount of this has accumulated
set an integer valued quantity one higher,
2. a clock measuring real valued temporal duration is needed for many skills
involving timing Shaffer (1982) [41],
3. and most perceptual models involve real valued quantities, see for example
Massaro and Friedman (1990) [29].
These three examples suggest that real valued quantities should be used for
describing brain states; however Hopfield (1984) [19] has shown that continuous
neurons, which is what real neurons are, can often be described by two-state
McCullock-Pitts neurons, suggesting that for psychological measurements biva-
lent quantities might suffice. The problem is in fact a worse than just requiring
real numbers for physical and psychological measurements. Real numbers oc-
cur in pure mathematics and a general account of meaning should account for
meaning in such a-priori languages, because apart from any other reason math-
ematics is good at describing the physical world, Wigner (1960) [50]. It appears
that the segmentation problem does not have a solution involving only integers.
The pure mathematical reason for this is that accounting for the meaning of the
real numbers cannot be done by assigning an integer number of truth values,
there is the requirement of a new axiom when constructing Ж. To put this an-
other way having discrete truth values is not sufficient, even with brittle re-write
21