  This paper concentrates on relationships of formal systems with biology. The
paper is based on previous papers by the author. We have freely used texts of
those papers where the formulations are of use, and we have extended the
concepts and discussions herein considerably beyond the earlier work. We
concentrate on formal systems not only for the sake of showing how there is a
fundamental mathematical structure to biology, but also to consider and
reconsider philosophical and phenomenological points of view in relation to
natural science and mathematics. The relationship with phenomenology comes
about in the questions that arise about the nature of the observer in relation
to the observed that arise in philosophy, but also in science in the very act
of determining the context and models upon which it shall be based.We examine
the schema behind the reproduction of DNA. The DNA molecule consists of two
interwound strands, the Watson Strand (W) and the Crick Strand (C). The two
strands are bonded to each other via a backbone of base-pairings and these
bonds can be broken by certain enzymes present in the cell. In reproduction of
DNA the bonds between the two strands are broken and the two strands then
acquire the needed complementary base molecules from the cellular environment
to reconstitute each a separate copy of the DNA. At this level the situation
can be described by a symbolism like this. DNA = <W|C> -------> <W| E |C>
--------> <W| |C><W| |C> = <W|C><W|C> = DNA DNA. Here E stands for the
environment of the cell. The first arrow denotes the separation of the DNA into
the two strands. The second arrow denotes the action between the bare strands
and the environment that leads to the production of the two DNA molecules. The
paper considers and compares many formalisms for self-replication, including
aspects of quantum formalism and the Temperley-Lieb algebra.
