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Squinting December 15, 2006

Posted by amahabal in Uncategorized.

Here and I explain what I mean by squinting, and put forth a case for why this is a fruitful exploration. Sequences can often be seen in distinct and valid ways. For example the sequence “1 2 3 2 1 2 3” can be seen as an oscillating between 1 and 3, but it can also be seen as the four terms [1 2 3 2] repeating endlessly. Though both ways of understanding agree on every single term, they are psychologically nothing like each other. They have a different feel and the set of “other similar sequences” is quite dissimilar in the two cases. In the first case, a sequence that lies nearby in idea space is “1 2 3 4 3 2 1 2 3 4 3 2 1”, while in the second case, a nearby sequence is “3 4 7 5 3 4 7 5 3 4 7 5” (which is just the four terms [3 4 7 5] repeating endlessly).


Richard Feynman argued in “the character of physical law” that two theories that fit the data equally well can nonetheless be of unequal utility according to what other theories they evoke. The two different theories of what the sequence above consists of fit the data equally well. The second theory, however, misses important bits of relationships between the pieces. It ignores, for instance, that the second repeating bit (2) is one more than the first repeating bit (1). In the examples that I shall present, ignoring squinting leads to theories that ignore important aspects of the situation.


Consider the sequence “1 2 0 4 5 1 2 3 0 5 6 1 2 3 4 0 6 7”. One way of understanding the sequence is to consider each block (for example, 1 2 0 4 5) as consisting of three pieces: a pre-zero piece, a zero and a post-zero piece. In this way, the sequence can be looked at as three interlaced sequences. The first of the three interlaced is “[1 2] [1 2 3] [1 2 3 4]”, the second “0 0 0 0”, and the third “[4 5] [5 6] [6 7]”. Each of these three is a very simple sequence to understand. The only way that this theory is wanting is in that it treats its three constituents as independent of each other.


The other way of looking at this sequence is to say that it is just “1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 7” except that some terms are hidden behind a zero. Or, put differently, the zeros stand for a 3, a 4 and a 5 respectively.


Which of these is the right way? That should be rephrased as which of the two people prefer once they see both. That is an empirical question, but I put my money on the solution of the zeros standing for something else. Being able to see something as something else is a rampant cognitive phenomenon. The remainder of this post has example after example of scholarly writing pointing in that direction.


Let me point out a continuum of “seeing as.” At the one end is seeing something as an instance of a category. Seeing a creature and recognizing it as a dog is of this type. At the other end is seeing something as one particular object (instead of a category). Examples at this end include seeing somebody as being Einstein: all the following people have been so seen: Charles Hartshorne (Einstein of religious thought), Dr. Magnus Hirschfeld (Einstein of sex), and Eric Drexler (Einstein of Nanotechnology). These two ends, while not the same, clearly form a spectrum if we consider Einstein to be a category, albeit small. From here on, I do not consider this distinction.

  • Language is permeated with metonyms and metaphors. Authors like George Lakoff (metaphors we live by; women, fire and dangerous things; where mathematics comes from) believe that much of cognition is metaphorical. A simple Internet search on Google with the query “can be seen as” returns several matches where people are seeing things and situations as problems, as opportunities and as insults.



  • Science and mathematics have big formidable edifices built out of repeated seeing as. All of modeling is of course seeing something as something else. It is also easy to come up with very specific examples. Integration can be seen as summation. Though the official definition involves limits, numerical computation is in practice done via summation.


Many combinatorial identities can be reasoned about by seeing both sides of the identity as ways of choosing team members from a pool of candidates. Such solutions are usually orders of magnitude higher in elegance. You can find some pretty examples in this Wikipedia entry.


Proofs of Wilson’s theorem involve seeing the number 2 through p-2 as a group, thereby allowing the use of results from group theory. Entire branches of mathematics are based on seeing a certain type of objects as something else: Algebraic Topology (which sees sets of topologies as groups) being an example.


  • Ervin Goffman gives several acts of pretending, game playing, conning (causing somebody else to see things different from reality) and so forth in his classic Frame Analysis.



Convinced that perception is the key to intelligence, Ulam was trying to explain the subtlety of human perception by showing how subjective it is, how influenced by context. He said to Rota, “When you perceive intelligently, you always perceive a function, never an object in the physical sense. Cameras always register objects, but human perception is always the perception of functional roles. The two processes could not be more different…. your friends in AI are now beginning to trumpet the role of contexts, but they are not practicing their lesson. They still want to build machines that see by imitating cameras, perhaps with some feedback thrown in. Such an approach is bound to fail…”

Rota, clearly much more sympathetic than Ulam to the old-fashioned view of AI, interjected, “but if what you say is right, what becomes of objectivity, and idea formalized by mathematical logic and the theory of sets?”

Ulam parried, “what makes you so sure that mathematical logic corresponds to the way we think? Logic formalizes only a very few of the processes by which we actually think. The time has come to enrich formal logic by adding to it some other fundamental notions. What is it that you see when you see? You see an object as a key, a man in a car as a passenger, some sheets of paper as a book. It is the word as that must be mathematically formalized…. until you do that, you will not get very far with your AI problem.”

To Rota’s expression of fear that the challenge of formalizing the process of seeing a given thing as another thing was impossibly difficult, Ulam said, “do not lose your faith— a mighty Fortress is our mathematics,” a droll but ingenious reply in which Ulam practices what he is preaching by seeing mathematics itself as a fortress.




1. Alex Linhares - January 2, 2007

Great post, just made a link to it. I could not find a backtrack here, though. Best, — Alex

2. Alex Linhares - January 2, 2007
3. Neeraj - January 22, 2007

So is there any pun intended in the program’s name ?? 🙂

4. TṽIXXʔ̏ŌN꒼I - March 2, 2007



5. kerrjac - August 26, 2007

Another consideration is that a large difference between how 123-123 vs. 2341 2341 registers psychologically is in musical rhythm, which is more intrinsic to our perception of the information than the information itself. I find that if a lot of information is broken down into too many categories, or if a little information is categorized into too many groups, then it’s too much for me to handle. The magic numbers from research in working memory is 7-9 bits of information. I suspect that there’s something going on with music and working memory here.

6. Michael Tim - February 28, 2009

I love your site!

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