How is a vector space like an evolutionary function?
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How is a vector space like an evolutionary function?
 
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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/ |
Re: How is a vector space like an evolutionary function?
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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 <[hidden email]> wrote:
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In reply to this post by jon zingale
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[†]. -- ↙↙↙ uǝlƃ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/
uǝʃƃ ⊥ glen
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Re: How is a vector space like an evolutionary function?
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In reply to this post by Frank Wimberly-2
I probably should have added an abstract :) What I am getting at here is that
*free* object constructions build higher-order structures relative to those they are built from and relative to the target of their associated right-adjoint (often the category of sets via a forgetful functor). That these higher-order structures then support notions that may not exist in a direct way relative to the structures they are built from, one can view these newly supported notions as a kind of *epiphenomena* relative to the underlying structure. While it is not meaningful to speak about the length of a set, we can support a higher-order monoidal structure where length is a meaningful notion. For a monoid relevant example[𝜆], consider the following recursive definition of the length of a list: len :: [a] -> Int len [] = 0 len [t] = 1 len (t:ts) = len [t] + len ts What I seek to show is that this definition follows directly from an adjunction that constructs *free* monoids from sets. A functor F :: Set -> Mon builds from the elements of a set X, the set of all possible words on X and equips this set with a multiplication (++) and an identity element ([]). This functor has a right 'forgetful' functor which forgets the monoidal structure and returns the set of all possible words on X, X*. We can then look at morphisms from this object (X*, ++, []) into a natural number monoid (ℕ, +, 0), taking concatenation to addition and the empty word to the additive identity (the first and third rule in the definition above). Now, the second rule (len [t] = 1) is a little arbitrary and finds its meaning in the interaction between the underlying category of sets and the higher-order monoidal category. There in Set, we find a composition that ultimately gives len its character. The composition of Functors (G∘F) gives a natural map, η, which includes X into X* as an inclusion of generators. Looking at η: X -> X* and the map const1 :: X -> ℕ (Where ℕ is given by the same forgetful functor G and all elements of X are mapped to the number 1), we find that the unique function G(len) :: X* -> ℕ that makes the triangle commute and respects the functorial conditions gives via the monoidal structure the notion of length we seek. The structural functors (G and F) can be seen as *founding* a category of monoids upon a category of sets, and dually *structuring* the category of sets by the category of monoids. [𝜆] This example is from Benjamin Pierce's 'Category Theory for Computer Scientists'. -- Sent from: http://friam.471366.n2.nabble.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/ |
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Now I'm even more worried that epiphenomena is not the right concept, even in it's (I think) less common Pyrrhonian form. To the extent that the phenomenal layer can be treated as (at least somewhat) independent of its generative layer or, further, the extent to which the outer layer might "structure" the inner layer, I think they've graduated to primary phenomena.
On 7/27/20 9:56 AM, jon zingale wrote: > That > these higher-order structures then support notions that may not exist in a > direct way relative to the structures they are built from, one can view > these newly supported notions as a kind of *epiphenomena* relative to the > underlying structure. > > [...] The structural > functors (G and F) can be seen as *founding* a category of monoids upon a > category of sets, and dually *structuring* the category of sets by the > category of monoids. -- ↙↙↙ uǝlƃ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/
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Re: How is a vector space like an evolutionary function?
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To the extent that I understand Nick's idea, a satisfactory theory of
evolutionary function must admit a means for describing epiphenomena and crucially this epiphenomena must be non-mysterious. The theory may be considered useful if it distinguishes goals from designs, and presents testable relations between the two[⇅]. Further, Nick offers a model, by way of Elliot Sober, of a child's toy that demonstrates a special case of epiphenomena called a spandrel[Ϡ]. I argue here that Sober's model can be characterized within a class of algorithms whose analysis yields a way to reason about epiphenomena and whose structural relations (given by a free construction) are foundational to the class. In Sober's model[†], we are given a bucket with circular, triangular, and square-shaped holes in the lid. Additionally, there are matching circular, triangular, and square-shaped blocks. Each block type has an associated color: red, yellow, and blue. By some form of magic, as the blocks enter the bucket through their associated hole in the lid, the blocks become sorted in the bucket. Circular blocks find their way to the bottom, triangular blocks to the middle, and the square ones to the top. Epiphenomenologically, the blocks are found to be sorted by color. The key shuffle is a shuffling algorithm that produces a shuffled list from a given one. Given an ordered list of things, we write on each thing a random number. Next, we sort the list of things by the random numbers, placing a smaller number to the left of any larger. Lastly, we erase the numbers from the things and we are left with our things in a shuffled order. Epiphenomenologically, the things are found to be shuffled. Common to both is exploitation of structural relations between a list of things and a list of pairs of things. In the key shuffle case, we imagine the shuffle to 'live' in a category of lists: keyShuffle :: [a] -> [a], but the algorithm itself relies on a category of lists of pairs that is not apparent from the type signature alone. intermediary :: [(r, a)] -> [(r, a)] Through this additional structure, the shuffle manifests via sorting, the sorting of the random numbers causes a shuffling of the given things. In the case of Sober's algorithm, the pairing of color and shape is decidedly more direct. The sorting of shapes gives rise to a sorting of colors, but again, it is effected through a structural map (ideally a monomorphism) identifying shape with color. The parallel can be drawn more explicitly: Sober: Maps from colors to shapes are monomorphisms, sorting on shapes gives distinctly sorted colors. Shuffle: Maps from place-values to random numbers are monomorphisms, sorting on random numbers gives distinctly shuffled place-values. Of course, the monomorphism condition is something of a red herring. To mix things up a bit, consider what happens when instead of color we assign to each shape-type a prime number, and write on each block a number that is divided by the prime associated with each shape-type. Now again, say, all the circles end up at the bottom and the numbers on these blocks are all divisible by 2, the triangles with numbers divisible by 3, and the squares with numbers divisible by 5. Now, things are more confusing because we could have written the number 10 on a square block and found that though it was divisible by 2, the block found its way to the top. Composites: Maps from numbers to shapes are non-monomorphisms[⋔], sorting on shapes gives non-distinctly sorted numbers. More interestingly, and closer to the real-life examples discussed by Eric and Nick in vFriam, for a large number of blocks and randomly assigned numbers to shapes (respecting the divisibility constraint), we find that all of the circles carry numbers divisible by two, and fewer squares carry the same. Still, there will be a bunch of mixture. On the one hand, mixture, on the other, meaningful and quantifiable distinctions. To my mind, it is the structural relationship between a category of lists of things and a category of lists of pairs of things that tie these examples together[⋌]. Here, the *design* of an algorithm that shuffles things and the *design* of a child's toy that *epiphenomenologically* 'sorts for color' are possible via the exploitation of intermediary types, and the structural relations between them. I wish that I could have written about this more definitively, but I feared not writing anything at all. My apologies, quite a bit more could and should have been said. [Ϡ] From Wikipedia: https://en.wikipedia.org/wiki/Spandrel_(biology) [⇅] Computer science is flush with examples of algorithms where the instruction set given to a computer gives rise to surprising behavior exploited by an algorithmancer to satisfy a particular task. The creative act of designing algorithms 'lives' between two worlds, one of pointers and operations and another of pleasing effects. See the reference, in the above post, to the styrofoam herding robot and the algorithms in this post for examples of *design*. That these algorithms exploit epiphenomena to perform tasks and that they are amenable to analysis, makes them exemplary for the design and construction of a general theory. [†] Taken on faith here as I have never read Sober's actual account. Instead, the model is recounted from my conversations with Nick. [⋔] Not only non-monomorphism, but there are many possible choices of representative. That a given composite could be mapped to one or another shape is what gives this example its peculiarity. [⋌] Hiding in the background are additional structural maps, 𝛿 :: X -> X∏X and 𝜀 :: (A∏B, A∏B) -> (A,B), which serve to found the free construction relied on here and similar to the map in the earlier posts. Again, note that while the adjoints *found* the relationship between the higher-order structure and the lower, determinations between the related categories can often be fairly loose. Consider the functorial relation between a vector space and its dual, the relationship exists whether or not we specify a basis. For a different and novel example, see Example 2.2.1 on page 55 of Emily Riehl's book: http://www.math.jhu.edu/~eriehl/context.pdf -- Sent from: http://friam.471366.n2.nabble.com/ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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Re: How is a vector space like an evolutionary function?
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For those algorithmically inclined readers, I coded up the examples in
Haskell. While the soberSort, keyShuffle, and compositeSort could each be written in a few lines, I took the time to build out a KeySortable type class and wrote the examples relative to it. There are 4 files involved: Spandel.hs, MixedSpandrel.hs, Shape.hs, and Sortable.hs The first two simply implement the examples. Shape.hs defines a notion of ordered shapes and provides a generator for producing random lists of shapes. Sortable.hs has the class implementation details. There you will find that I have defined a _Pair_ datatype and then extend the Ord class to it (ordering based on the first in the pair, the 'primary phenomena') and to the Bifunctor class. Lastly, I define a KeySortable class, contingent on Bifunctor, and extend it to include _Pair_. Aside from the class constraint, creating an instance of KeySortable needs the usual product maps (η, 𝜀), but then gets _sort_ and _shuffle_ for free. For interested readers, the code can be found here: https://github.com/jonzingale/Haskell/tree/master/epiphenomena -- Sent from: http://friam.471366.n2.nabble.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/ |
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