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

[uǝlƃ ↙↙↙]

Hm. I'm not sure epiphenomena is the right word. When you say "these [e.g. sorted token] properties were nowhere to be found in the underlying field", I'm not sure that's quite true. There's a sense in which each token is grouped by addition already 4+1=4+1+4+1 ... All you (seem to) have done is remove the reduction rule. E.g. 4+1↛0. I.e. the structure of the groupings seems, at least somewhat, causally related to the underlying field.

Hopping, then, up to the premature registration of and reliance upon some structure like a vector space or seed spreading, I think it might be important to relax your claim that the higher order structures are *epi*phenomenal. I.e. allow that there *might* be some causal relation to the underlying mechanism(s) and small-scoped goals to the function, but the project is to find out *if* that's the case and if so, what is that causal relationship.

To try to be a little clearer, it may be important to start out with the falsifiable claim that they're purely epi, then try to constructively demonstrate particulars of the forward map (from generator structure to phenomenal structure).

On 7/26/20 9:08 PM, Jon Zingale 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[†].


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