Discrete Categories and Continuous Change

 

The gap between the continuous and the discrete has been the object of discussion for a long time.

How do discrete categories arise in this world of continuous change?
What’s the relationship between the potentially infinite world  and our discrete, finite descriptions?
What are category boundaries?
What’s this horseness, the distinguishing quality of a horse?
How do we distinguish cats and dogs?
Or, the ultimate question of Artificial Intelligence: What is an a?
a
These are rather tame examples.

hofstadter8a1 The more fancyful letters to the left are seldom used for bulk text.
Unfamiliar fonts slow down reading, at least initially, but even these letters are easily recognized.

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. Of course, taxonomies and decision trees are useful. Even more useful are their probabilistic counterparts. Medical expert systems using the maximum entropy method will give quite impressive results, often more reliable than the average human expert.

D.R. Hofstadter discusses at great length how difficult it is to parameterize the set of all A. The Metafont program by D.E. Knuth will give you a small submanifold, parameterized by a few numbers. The parameterization must be constructed by a human, and this task is a difficult one. He even suggests an analogy to the gap between true statements and provable statements in an axiomatic system. If this were true, the set of all A would be a „platonic set“, existing somewhere up in the sky, accessible only to the powers of the mind, forever unreachable by mere machines.

I do not believe this. I like my fellow animals.

The chasm between the continuous and the discrete is so wide and deep that we cannot hope to cross it in one giant leap. Let’s try several small ones!

 Date Posted: 07 Mrz 2009 @ 01 35 PM
Last Modified: 22 Mai 2010 @ 09 45 AM
Posted By: Hardy
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