Complexity of the Geological Framework and Use of Mathematics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 12

“it's difficult to say”, etc.). He has proposed methods for translating a geologist's
opinion into subjective probabilities. Grayson's ( 1960 ) topic deals primarily with
drilling decisions by oil and gas operators. A sign hanging in the office of one of the
operators interviewed by him states: “Holes that are going to be dry shouldn't be
drilled”, although some years previously this particular operator had drilled 30 con-
secutive dry holes. De Finetti ( 1972 ) used this paradox to urge the geologist to
express his predictions in a probabilistic manner rather than translating them into
the inadequate logic of absolute certainty.
The human mind allows the formulation of hypotheses which are flexible to the
extent that they may immediately incorporate all new facts before the hypotheses
could be properly tested. On the other hand, the advantage of using the logic of
mathematics is that it is indisputable and, when random variables are used, it is
possible to check the deductions against reality. P.A.M. Dirac (in Marlow 1978 ) has
advocated the use of mathematics in physics as follows: “One should keep the need
for a sound mathematical basis dominating one's search for a new theory. Any
physical or philosophical ideas that one has must be adjusted to fit the mathematics.
Not the other way around. Too many physicists are inclined to start from
preconceived physical ideas and then try to develop them and find a mathematical
scheme that incorporates them. Such a line of attack is unlikely to lead to success”.
This is good advice for all scientists.
1.3 Use of Curves
1.3.1 Trend-Lines
It can be argued that all geological processes are deterministic. However, it is
usually not possible to use purely deterministic expressions in the mathematical
equations used for representation because of uncertainties or unknown causes. In
many geological situations, the spatial variability of measurable features can be
divided into a regional systematic component (loosely called “trend” or “drift”) and
more local, unpredictable fluctuations (residuals from the trend). The trends may
have been generated by broad-scale deterministic processes. For example, average
grain size of sand particles increases towards a beach. However, larger and smaller
particles may coexist everywhere in the sampled area in different proportions.
Systematic variations called “trends” are, in part determined by the density of
sampling points. If many more measurements are performed locally, a residual
from a more regional trend or drift can become the trend for a more local survey
area. It is appropriate to use deterministic functions for “trends” when it is kept in
mind that they do not necessarily describe the result of deterministic processes. We
have a choice of using empirical functions such as polynomials or functions
corresponding to curves or surfaces that are theoretical predictions for geological
phenomena. For example, Vistelius and Janovskaya ( 1967 ) pointed out that the