For a fantastic explanation of isomorphism, I recommend
the book
Gödel,
Escher,
Bach by
Douglas Hofstadter.
The book
_Godel,_Escher,_Bach_ is a
Pulitzer Prize
winning novel written in the 1970s that looks at the
fundamentals of how
humans think and what constitutes
thought.
The subtitle on the book is "A single golden strand in
the spirit of Lewis Carroll."
An isomorphism is a map between one situation and another. It is similar to a mathematical function. For each input or set of inputs there is a unique output or set of outputs. The difference is that an isomorphism is always reversible -- an output or set of outputs always yields the original input or set of inputs. It is a one-to-one mapping between two sets.
Traditionally, examples of isomorphisms are the writing or symbology we use to emulate some aspect of reality. The numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are isomorphic to the ideas of one-ness, two-ness, three-ness, etc. When we see two circles atop one another, there is nothing in the structure that implies eight-ness (the concept of the value eight). But in our minds, we have built up an isomorphism, that is a mapping, between one idea (the visual image) and another (the mathematical idea of eight).
V + V = X is true if your isomorphism is for Roman Numerals.
But a different isomorphism, that of string concatenation,
yeilds V + V = VV
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