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Axiomatic Design 273

Synergy is gained when Axiomatic Design is integrated with other
design methodologies as needed such as design optimization methods,
TRIZ to identify appropriate design parameters to resolve coupling,
and probabilistic design analysis.

Appendix 8A*: Automatic Design Theorems
and Corollaries

The study of most famous examples (such as Euclidean geometry,
Newton laws, thermodynamics, and the axiomatic branch of modern
mathematics†) of the axiomatic disciplines reveals several common
threads. For example, Euclid’s axiomatic geometry opens with a list of
definitions, postulates, then axioms, before proving propositions. The
aim is to present geometrical knowledge as an ordered list of proven
facts, a historical paradigm of disciplines with axiomatic origin.
Newton’s laws were deliberately set up to emulate the Euclidean style.
The laws open with a list of definitions and axioms, before proving
propositions. While the axioms are justified empirically, consequences
of the axioms are meant to be drawn deductively. Modern mathemat-
ics and empirical knowledge are two streams that can be observed in
disciplines that emerge from an axiomatic origin.
In axiomatic design, the goal is to systematize our design knowledge
regarding a particular subject matter by showing how particular
propositions (derived theories and corollaries) follow the axioms, the
basic propositions. To prove a particular proposition, we need to appeal
to other propositions that justify it. But our proof is not done if those
other propositions themselves need justification. Ultimately, to avoid
infinite regress, we will have to start our proofs with propositions that
do not themselves need justification. What sorts of propositions are not
in need of justification? Answer: the axioms. Therefore, differentiation
of axioms from other postulates is needed. The label “axiom” is used to
name these propositions that are not in need of justification.
Nevertheless, historically, various distinctions have been made
between axioms and postulates. We will encounter two ways of draw-
ing the distinction, one based on logical status, and the other based on
status relative to the subject matter of the theory. Axioms are self-
evident truths. For example, the Independence Axiom and the

*El-Haik (2005).
†Axiomatic theories in modern mathematics include modern axiomatic geometry
(Euclidean and non-Euclidean geometries), Peano’s axioms for natural numbers, axioms
for set theory, axioms for group theory, order axioms: linear ordering; partial ordering,
axioms for equivalence relations. Not the sort of axiomatic theory we’ll be considering in
this book.