Well, tough. This is how art works: a consensus emerges among practitioners that some things are better than other things, even though nobody can articulate in words just what makes the good things so good, or the bad things bad. The discipline of aesthetics is kind of irrelevant to art and architecture: it doesn't have much useful content, and it's not helpful in judging work. Most artists/architects don't use it in any way. It's just windy philosophizing.
The taste of food is similar. If I'm a restaurant critic and I tell you that a certain restaurant is good, you need to have a way of assessing that opinion which doesn't involve requiring it to be written in some kind of positivistic, quantified, apodictic form.
I trained as an architectural historian, and let me tell you, architectural quality can't be put into words. Architects and arch. historians have intuitive opinions. There is no science of architectural criticism. Alexander was the kind of person who thought such a science was possible.
The most important clues to the mediocrity of his buildings are the facts that a) they aren't widely imitated and b) they don't usually appear in fine books about architecture, trad or otherwise.
Alexander was more of a mathematician than an architect. It's a long time since any one person did both of those things well.
Christopher Alexander thinks he is the messiah of building. The buildings he has built are not good, though, and not all of the other architects in the world are as stupid as you might want to believe. "A Pattern Language" just isn't the great gift to humanity that some people, eager for prescriptions, want it to be.
I don't know where you got the idea that there is a first guess that comes from a table. The table lookup just brings x into a suitable range, given A.
Shades of "Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?"— a Taylor series can only do so much!
If you search for "perpetual motion machine" you can find a ton of cheap knockoffs of this device. You can get a badly made wooden knockoff for $50 or quite a bit less?
One interesting thing is that category theory provides a way to precisely describe a Most General Unifier. It is a example of a coequalizer.
This isn't a really basic and accessible result, but IMO it gives a good flavour of what category theory is suitable for: formally describing constructions we didn't previously have the tools to express precisely.
The taste of food is similar. If I'm a restaurant critic and I tell you that a certain restaurant is good, you need to have a way of assessing that opinion which doesn't involve requiring it to be written in some kind of positivistic, quantified, apodictic form.