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

[jon zingale]

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



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