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

[Frank Wimberly]

Jon,

I'll think about that more.  An initial reaction is that I'm surprised that
you call monoids, rings, etc "higher structures".  They have less structure
than a vector space, don't they?  Is it because they're more general?

Frank

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Sun, Jul 26, 2020, 10:09 PM Jon Zingale <jonzingale at gmail.com> wrote:

> 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.
> - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
> FRIAM Applied Complexity Group listserv
> Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
> un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
> archives: http://friam.471366.n2.nabble.com/
> FRIAM-COMIC http://friam-comic.blogspot.com/
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://redfish.com/pipermail/friam_redfish.com/attachments/20200726/6f8e0b94/attachment.html>