Imagination, patterns, and visual proofs

What does it mean to understand that something is a pattern? In a puzzle like this:

A fairly reliable test is to ask someone to draw the shape that should replace the question mark. This implies that understanding a pattern has something to do with being able to generate more instances of the pattern in question. This in turn suggests a link between pattern recognition and imagination (in this case visual pattern recognition and visual imagination).

Some patterns are not linear sequences, and in especially complex cases understanding that there is a pattern there might be something you can do even if you can't reliably pass a "draw the rest of the image" test:

I don't think it is surprising that imagination could play this kind of role in cognition, but what does seem to be more controversial is the idea that imagination plays a role in reasoning about what is possible and what is necessary. Some evidence for the latter case is the relationship between visual imagination and mathematical proof. In general these sorts of relationships are also why I am interested in empirical research on visual imagery.

One way to study this is by looking at proof techniques in mathematics which rely on diagrams or pictures. This webpage has some nice examples of visual proofs. Here is one that the sum of the first n odd natural numbers is n²

And here is a proof that the infinite sum 1/4 + 1/16 + 1/64 + ... = 1/3

Visual proofs can also be misleading, as in this famous example of a nonproof that pi = 4. A good account of visual proofs should explain why the above proofs succeed while the below proof fails. ¹

One thing these proofs have in common is that understanding the proof presented in the image requires us to spot some kind of pattern, and then extrapolate from that pattern that certain facts must hold for all cases the pattern generates/recognizes. If this kind of pattern matching in images involves visual imagination, then an account of when visual proofs of mathematical facts are valid might help get us some way to an account of when visual imagination is a reliable tool for drawing general conclusions about all possible shapes.

Doing this for visual imagination might serve as a special case for the role that imagination more broadly plays in cognition².

In my post yesterday I wrote a series of questions about Kant's concept of schema. One interpretation for what schemata are for Kant is that they are the cognitive entities responsible for this kind of reasoning about patterns.

I will likely take a break form writing about Kant for a few days, but my next step is to take a look at parts of his writing on schematism which do not involve mathematical examples and try to see if his use of the concept of schema in those cases is consistent with the interpretation above.