This discussion, complex world vs simple rules, is very old and goes back to Plato and Aristotle. Plato explained our ability to recognize each and every A: All concrete examples partake,in varying degrees, in the ideal A. As these ideals do not exist in the world of the senses, he postulated some kind of hyperreality, the world of ideas, where they exist in timeless perfection. Our souls come from there to this world, and all recognition is re-cognition. Of course. this stuff is hard to swallow for a programmer trying to build some damned machine. A good prototype is better than nothing, but the ideal A has so far eluded any constructivist attempt.
His critic Aristotle did not believe in the world of ideas. As a taxonomist, he described the camel by its attributes. If the distinguishing attributes are present, it’s a camel. Else it’s not a camel.
Characterizing an ‘A’ by its attributes has proved harder than it seems. What is an attribute? Which attributes are useful? Is this line a short line, or is it already long? Is this a round bow or an edge? Not every A looks like a pointy hat! Does a very characteristic feature compensate for the lack of three others? Even if we have good features, there may be no simple rules. There is this well-known “Rule”: No Rule without Exception: Even folk wisdom discourages any attempt to catch A-ness in a simple net of if and else.
We should not expect that the concept of A-ness can be expressed by such simple means. The set of all “A” ( it exists as a platonic set somewhere, or doesn’t it? ) may be pulled back into R^n as a set of grayscale images, but do we really know about its geometric structure? For large n, it is a thin subset, a complicated geometric object. The metric of R^n will preserve its local structure, but that’s all. It does not tell us about the concept of A-ness. We should expect that large amounts of memory are necessary.
FYI: (shameless plug) I’ve tried to illustrate my ideas about a connection between the topology of finite spaces, continuous maps, product spaces and quotient spaces and the factorization of classifying maps on my website, learning-by-glueing.com. It’s not finished, any comments are welcome.
This discussion, complex world vs simple rules, is very old and goes back to Plato and Aristotle. Plato explained our ability to recognize each and every A: All concrete examples partake,in varying degrees, in the ideal A. As these ideals do not exist in the world of the senses, he postulated some kind of hyperreality, the world of ideas, where they exist in timeless perfection. Our souls come from there to this world, and all recognition is re-cognition. Of course. this stuff is hard to swallow for a programmer trying to build some damned machine. A good prototype is better than nothing, but the ideal A has so far eluded any constructivist attempt.
His critic Aristotle did not believe in the world of ideas. As a taxonomist, he described the camel by its attributes. If the distinguishing attributes are present, it’s a camel. Else it’s not a camel. Characterizing an ‘A’ by its attributes has proved harder than it seems. What is an attribute? Which attributes are useful? Is this line a short line, or is it already long? Is this a round bow or an edge? Not every A looks like a pointy hat! Does a very characteristic feature compensate for the lack of three others? Even if we have good features, there may be no simple rules. There is this well-known “Rule”: No Rule without Exception: Even folk wisdom discourages any attempt to catch A-ness in a simple net of if and else.
We should not expect that the concept of A-ness can be expressed by such simple means. The set of all “A” ( it exists as a platonic set somewhere, or doesn’t it? ) may be pulled back into R^n as a set of grayscale images, but do we really know about its geometric structure? For large n, it is a thin subset, a complicated geometric object. The metric of R^n will preserve its local structure, but that’s all. It does not tell us about the concept of A-ness. We should expect that large amounts of memory are necessary.
FYI: (shameless plug) I’ve tried to illustrate my ideas about a connection between the topology of finite spaces, continuous maps, product spaces and quotient spaces and the factorization of classifying maps on my website, learning-by-glueing.com. It’s not finished, any comments are welcome.