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Spatiotemporal Geometry 59

POSTULATE 2.9: Complementarity considers a multiple parallel pro-
cessing of field realizations which are diverse yet necessary for a com-
plete understanding of the phenomenon of interest.
In light of complementarity, uncertainty manifests itself as an ensemble of
possible field realizations that are all in agreement with what is known about
the phenomenon of interest. These realizations are different but complementary
facets of the description of the actual phenomenon. In other words, the different
realizations are not contradictions, rather, they are complementary aspects of
a seamless unity.
The concept of complementarity includes the actual (but usually unknown
to us) realization of the natural field as well as several non-actual but possible
field realizations. While the actual realization is located in the physical world,
the possible realizations are located in a conceptual (or logical) world. A
"possible field realization" is not the same as a "conceivable field realization."
Just as there exist possible realizations that are not conceivable (on the basis of
our current knowledge, technological abilities, etc.), there also exist conceivable
realizations that are not possible because they violate rules of logic, natural
laws, etc. In science, we consider only physically possible realizations, i.e.,
realizations which are characterized by the same physical laws, data, etc. as
the actual realization of the natural field.

COMMENT 2.9 : Complementarity ma y offer a n interesting interpretation

of th e cause-effect concept: I n mathematical logic, the use o f material condi-

tionals fi.e. , propositions of the form "ifxi> thenxz" o r "xi ~ > Xi," where.

Xi is the antecedent and X2 is the consequent) requires us to admit various

possible realizations besides the actual (but unknown) one. A conditional

then may be true not in terms of how things are, but of how they would

be in an appropriate field realization. A conditional xi—> X2 is true in a

realization if X2 is true in the same realization in which xi is also true. In

other words, an event xi fnay be considered as causing an event X2 if both

Xi and X2 occur in the observed realization, but in the vast majority of the
other realizations in which xidoes not occur, X2 does not occur either. We

revisit material conditionals in Chapter 4-
Putting Things Together: The Spatiotemporal
Random Field Concept
Spatiotemporal random field (S/TRF) modeling of natural phenomena has led
to considerable successes over the last few decades. Conceptually, the S/TRF
model is a combination of the three fundamental ideas (the Spatiotemporal
continuum, field, and complementarity) of the preceding sections. Here we will
use large and small Roman characters to denote random fields and random
variables, respectively; Greek characters will be used to denote realizations
(data values, etc.). The following S/TRF definition is often used in modern
Spatiotemporal geostatistics.