[FRIAM] How is a vector space like an evolutionary function?

[Jon Zingale]

At first glance, the commonality is one of contingency. *Vector spaces*
are contingent on underlying *fields* like *evolutionary functions* are
contingent on *underlying goals*. Before jumping to the conclusion that I
believe that evolutionary functions *are* vector spaces, let me mention
that in place of vector spaces I could have said monoid, algebra, module,
or an entire host of other higher-order structures. What is important
here is not the particular category, but the way that these higher-order
structures are *freely* constructed and the way that they relate to their
associated underlying structures[⁛].

While some mathematicians will argue that these structures *apriori* exist,
one can just as easily interpret the goal of such a construction to be
the design of new structures. In a sense, a vector space is designed for
the needs of a mathematician and founded upon the existence of a field.

Consider the field of integers modulo 5, here named 𝔽5. This object can
be thought of as a machine that can take an expression (3x7 + 2/3),
give an interpretation (3⊗2 ⊕ 2⊗2), and evaluate the expression
(3⊗2 ⊕ 2⊗4 ≡ 4) relative to the interpretation. Now 𝔽5, is an *algebraic*
object and so doesn't really have a notion of distance much less richer
*geometric* notions like origin or dimension[ℽ]. This object can do little
more than act as a calculator that consumes expressions and returns values.
However, through the magic of a *free* construction, we can consider the
elements {0,1,2,3,4} of 𝔽5 as tokenized values, free from their context
to one another. Where previously they could be compared to one another:
added, multiplied, etc... now they are simply *names*, *independent* and
*incomparable* to one another. For clarity here, I will write them
differently as {⓪,⓵,⓶,⓷,⓸} to distinguish them from the non-tokenized
field values. "What does this buy us", you may ask? Now, when we consider
mixed expressions like 5*⓵ + 7*⓶ + 12*⓷ + 2*⓶, we can agree to sort
like things (5*⓵ + 9*⓶ + 12*⓷) and otherwise let this expression remain
*irreducible*. The *irreducibility* here buys us a notion of dimension[↑],
and we quickly find that many of the nice properties we would like of a
space are suddenly available to us. Crucially, these properties were no-
where to be found in the original underlying field. This is to say, that
these properties arise as a kind of *epiphenomena* wrt the underlying field.

The properties now granted to us via the *inclusion of tokenized values*
*as generators* is one half of the story. Dual to the inclusion is another
structural map named evaluation. This map, like a gen-phen map, *founds*
all of the higher-order operations by giving them a direct interpretation
below in the underlying field. Taken together, the inclusion map and the
evaluation map do a bit more. They assure a surprising correspondence
between the number of ways one can linearly transform spaces and the
number of ways one can map tokenized values into another. This fact is
often stated as "a linear transformation is determined by its action on
a basis".

Structures arising from constructions like the one above are ubiquitous
in mathematics and demonstrate a way that epiphenomena (vector, inner-
product, tensor, distance, origin, dimension, theorems about basis) can
arise from the design of higher-order structures while relating to the lower
-order structures they are founded upon. My hope is that drawing this
analogy will be found useful and produce a spark for those that know
evolutionary theory better than I[†].

Jon

[⁛] See the description of the free vector space construction from the
introduction to chapter 4 of 'Categories for the Working Mathematician'.

[ℽ] Some here probably wish to exclaim, "but wait, I can define a metric
on 𝔽5!" I wish to deflect this by asserting that the idea of a metric is
a geometric notion and that philosophically it may be cleaner to consider
the metric as being defined not on 𝔽5, but on 𝔽5 *construed* as a space,
Met(𝔽5) say.

[↑] The tokens ⓵, ⓶, and ⓷ in the expression above play the role of
independent vectors. An expression like 4*⓶ + 2*⓷ can now be interpreted
as moving 4 steps in the ⓶ direction, followed by moving 2 steps in the
⓷ direction.

[†] Just about everyone.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://redfish.com/pipermail/friam_redfish.com/attachments/20200726/4fd2d606/attachment.html>