Lect 13

Continuing the discussion of the discrete. Keep in mind that the background is to rethink the concept of computation in algorithm vis a vis Turing's paper among other things. We've looked at analogies with architecture and other fields and now we're looking at his thought experiment discretely, that is in terms of one aspect of what his thought experiment requires which is the discrete terms of what is the output, 0 and 1. The blog entries were terrific because they approach that style of thinking.
We want that grammatical grasp of our concepts and the flexibility of thought to see it redefine our architectural ones. On algorithm and geometry our points suggested no internal connection save in a few instances. Algorithm expanded into any notion of operation means geometry is constructed from operations but not defined by them but by geometrical axioms. Geometry as an organization specific to matter is a language by which to characterize it but doesn't describe the process of formation. Topology, on the other hand might be more internally connected. There is always an order to the execution of the steps and they are related to each other in specific fundamental ways.

We were asking what is discrete. Today the elements, objects, entities, etc which remain constant (a variable can be one of these I suppose) upon which the rule set acts. Q: is the space in which a sequence run an "element"? We possibly reduced this with the fruit shopping analogy to an if/then condition. And this might seem on the whole more to do with toplogy. It certainly brings us back to the problem of logic.
On a few definitions:
What is a rule space as opposed to a rule set?
Can there be an infinite rule set?
Can the terms of an algorithm magically change? (Can 0 just become 1)? In contrast to: Can the rules or rule ste change?

Are we looking at anything like Descartes' invention of analytic geometry? (I mean the axiomatization of geometry? The application of algebraic procedures to geometrical terms?)

What is discrete?
1) Are Rule Sets Discrete
2) Are Objects, elements, terms and entities discrete? (That is, those things upon which the rule set acts)
3) Is a step in the carrying out of the rule set discrete?
4) Is the if/then conditional discrete?

One thing to keep in mind is that computation has changed the relation between geometry and toplogy - but this is perhaps a feature of analytical computing.

Another thing to consider is whether algorithm and toplogy are possible more connected than algorithm and geometry.

In other words, algorithm might act on geometrical elements but in order for it to act geometry isn't essential to its function. The if/then and the question of loops and so forth seem to suggest an internal connection with topology.

In terms of continuous, we mean possibly divisible. Do we mean infinitely divisible?
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