The Old Guys First
Euclid:Point, Line, proofs that rely on reductio ad absurdum.
Kepler:tedious processing
Newton:conversion of tedious processing into systems (calculus, algebra, etc.)...finding equations that give the right numbers instead of searching for the numbers the way that Kepler did.
Boole:Boolean Logic
Frege:Syllogism

Formal Systems
While some of the words used in relation to formal systems have alternate meanings in other fields, the concept of the system itself remains quite clear. Like most systems, there exist both components and rules that govern how these components operate within the system. Using the example of the "MU" system, the authors Godel, Escher and Bach present terms that are used to describe the elements of a formal system. To begin with, we (the readers) are given an axiom. An axiom is an existing theorem which is already known. A theorem is a string of symbols that forms the basis of the system at hand. Within the system, theorems are created, starting with the given axiom, according to rules of production. These rules determine how axioms may be manipulated in order to generate new theorems. This process is referred to as the derivation of a theorem. When we look at formal systems, we observe two realms of thinking. Work done within the system is called the Mechanical mode because of its rote way of dealing with the information. When we step back to think about the processes involved and attempt an intellectual analysis that will ameliorate understanding of the system, we are thinking in the Intelligent mode. One requirement of a formal system is that one can perform a test that will definitively show that all axioms within a system are determinable. Such a test is called a decision procedure. While at times decision procedures can also be done for theorems, it is not a requirement that any finite test exist for a theorem.

Logic
Actually, I'm not sure why my notes say that Frege was responsible for syllogism. Aristotle definitely had some part in that. Anyway, a basic syllogism reads something like, It is true that A gives B, and that B gives C; therefore, it is also true that A gives C. The example we used in class was, "All horses are mammals. All mammals are warmblooded. Therefore, all horses are warmblooded."

The other things that we have done with logic is to use base 10 numbers in logic tables. Once we've done this, we can relate it back to the creation of circuits
Basically, the logic tables help us to perform functions on our base 10 numbers, which are designated as being either "P" or "Q". Here's an example:

P	Q	P "or" Q	P "and" Q
1	1	1	1
1	0	1	0
0	1	1	0
0	0	0	0

This gets a little trickier as the function that you are performing changes. Not only that, but this can all be translated to circuits.

Scary Cyborgs?
So what comes next? There are lots of issues surrounding the future of the internet, computer, and humanity. Watch Bladerunner and think about The Matrix and Frakenstein. Do we know what we're creating? Are we changing the way our memories work and how we process information? Interesting stuff....For more about CYBORGS follow the link...