Fractally Analogical Thinking

Fractally Analogical Thinking


January 18th, 2021 [WIP]


"Good mathematicians see analogies.
Great mathematicians see analogies between analogies."

- Stefan Banach

An interesting question popped into my head today: what if we nest analogies? In other words, what if we treat analogies as recursive operators?

So we're on the same page, when I say "analogy" I'm referring to what you're probably thinking of*:* if you were thinking of a comparison between two objects/ideas, 1) ok, fair 2) I'll address that shortly. a word analogy, or word comparison of the form \(a : b :: c : d \) .

You'll normally hear such analogies read aloud as "\(a\) is to \(b\) as \(c\) is to \(d\)." What this really means is that "the simplest relationship which maps \(a\) to \(b\) also maps \(c\) to \(d\)." For example, the relationship (mapping/function) in the simple analogy \(\textrm{sky : blue :: grass : green}\) can be summarized in one word as "color" because \(color(\textrm{sky}) = \textrm{blue}\) just as \(color(\textrm{grass}) = \textrm{green}\).

A meta-analogy, or analogy between analogies, is one in which each element (\(a\), \(b\), \(c\), and \(d\)) is itself an analogy of the form \(a : b :: c : d\).


Here's a simple example:


\(\textrm{sky : blue :: grass : green}\)

\(:\)

\(\textrm{dog : bark :: cat : meow} \)

\(::\)

\(\textrm{sugar : sweet :: lemon : sour}\)

\(:\)

\(\textrm{ice : cold :: fire : hot}\)


Each analogy (line) in this meta-analogy is straight-forward; their mappings can be respectively summarized as color, sound, taste, and temperature. However, an interesting relationship emerges between these analogies when they are arranged together into one meta-analogy. Taking a step back, the broader meta-analogy could be summarized as "sight is to sound as taste is to touch."


The additional transformation which makes the four analogies cohere into one meta-analogy could be thought of in one word as "sense." \(sense(\textrm{color}) = \textrm{sight}\) just as \(sense(\textrm{temperature}) = \textrm{touch}\) (and since 'sound'You could argue that \(sense(\textrm{sound}) = \textrm{hearing}\) but 'sight is to sound' sounds better than 'sight is to hearing' so I'm sticking with it. and 'taste' are already senses, applying the "sense" function to them has no effect).

The mapping in the meta-analogy as a whole is a little harder to pinpoint. It clearly relates the different human senses, but the relationship is not obvious. What is the simplest relationship which not only maps sight to sound, but also taste to touch? I have no idea. Maybe a synesthete would be able to intuit this analogy in a way us normies could never fathom.

Before I continue to explore this interesting idea, I want to pause to entertain another: what if analogies of the form \(a : b :: c : d\) are already meta-analogies themselves? Some of you might have paused when I earlier defined an analogy as a four-word semantic comparison, because there is another (likely more popular) definition of analogy: "a comparison between two objects, or systems of objects, that highlights respects in which they are thought to be similar." Not four objects, but two.

Wikipedia similarly defines analogy as a cognitive process of information transfer from a source subject to a target subject.


So what's going on here? Is an analogy a comparison between two objects (2-analogy) or four (4-analogy)? One hypothesis is that all 2-analogies are 4-analogies in disguise. The domains \(a\) and \(c\) are explicitly stated, but the ranges \(b\) and \(d\) are left to inference. Let's see if this pans out:



Analogical Thinking



Analogical thinking (analogical reasoning, or reasoning by analogy) is the way humans transfer learning from one domain to another. It is what allows us to cross-apply problem-solving techniques. In this way, it can be thought of roughly as a human analog to machine transfer learning (as if this wasn't getting meta enough):

\(\textrm{machine : transfer learning :: human : analogical thinking}\)

Reasoning by analogy is fundamental to human thought, but imperfect in many ways. Let's first see a few examples of reasoning by analogy, so we can get a clear sense of how it can be helpful, while also seeing firsthand some of its drawbacks.


Everything before that line ^ was written before I discovered category theory lol. Turns out 2-analogies are morphisms and 4-analogies are functors.

TODO



A tool for discovery



Applying some of the interesting properties of analogical thinking has tremendous potential to help us discover new knowledge. Kepler famously reasoned his way to a causal theory of planetary motion by analogy. In anatomy, analogous structures are those which server similar functions despite being evolutionarily unrelated.

In fact, analogical reasoning is central to and formalized by mathematics in category theory: see isomorphisms*. * e.g. an invertible matrix is a linear isomorphism between vector spaces.

of words and language



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of Truths

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