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 <jonzing...@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/ >
- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/