A Proposed Solution to Two Puzzles in Mathematical Mapping

Abstract

Identifying different things as the same thing, mathematical equality is puzzling,
same thing (A=A) not being different things (A=B), and different things (A=B) not
being same thing (A=A). Resolution occurs considering a translation function
determining a relational field. Hereby inductive generation of an element in a domain
terminates when fused into an indistinguishable whole with an element in a converse
domain. A lacuna occurring, precedent and subsequent of the lacuna cannot be known to
be related. Generation is recursive or iterative. Recursive generation identifies
aspect, occurring when within a fixed set of elements, only one sequence is traceable
in marginal order from most like archetype to most like autotype and back in
reciprocating order. Iterative generation identifies instance, occurring when within
a fixed set of elements, more than one sequence is traceable in order from most like
archetype to most like autotype and back in varying order. A=A identifies a
corecursive hyperset, reciprocally generating the same transmutative sequence in
inverse order from alternate limits. Iterative sequence is recursive archetype
identifying reverse autotype. A=B identifies a communicative ring with unity,
reciprocally generating the same transmutative sequence in variable order from
alter-nate limits. Iterative sequence is discursive archetype not identifying reverse
autotype.