Warning. Skip this post if you hate mathematics.
Can we learn about Google algorithm by applying Inverse Mapping techniques? In the absence of tangible evidence, Google algorithm is assumed to be a black box. The black box can either be a proprietary software package or a stand-alone code (mostly it is). An inverse mapping is a function whose kernel can be inferred from its image. An example of this is a recursive function such as Fibonacci sequence. In our case, however, the black box is not an inverse mapping. The initial condition, X=FUNCTION(N,Y=1), and the kernel, are insufficient to infer the corresponding values of N. Therefore, the knowledge of different ordinate pairs (X,Y) are insufficient to deduce N.
Okay, so what do I want to prove now?
Using Axiomatic Set Theory and Inverse Mapping I want to prove:
- Google, and any other Mainframe based search engine cannot spider and index the entire net.
- You have no chance to know how exactly they works.
- They (Google, MSN, Yahoo) don’t know themselves how exactly it works.
The theory in one leg
Since its inception, there have been some mathematicians who have objected to using set theory as a foundation for mathematics, claiming that it is just a game which includes elements of fantasy. Though the axioms of set theory are fairly straightforward, they form the basis for several branches of mathematics including Algebra, Topology, Category Theory, etc.
[Read more about Axiomatic Set Theory] ; [Read more about Inverse Functions]
However, most of the work on set theory is done in Zermelo-Fraenkel (ZF) set theory. In this picture Ernst Zermelo 1871-1953 (left) and Adolf Fraenkel 1891-1965 (right).
It’s axioms are as follows (source: Aron Wall):
The above shows a top set (red) containing blue sets, with each blue set containing green/purple sets, which are assumed to be all be different. The Axiom of Choice says you can always find a set containing exactly set from each of the blue sets, even with an infinity of blue sets or green sets. The purple sets in the above diagram form a choice set. Here are some other ways to state the axiom of Choice:
- For any two cardinalities, either one is bigger, or they are equal. For this to be violated, you need a choice situation like above with no choice set, and then the number of B’s will be incomparable with the number of x’s. One would think there would be more x’s, but that requires there to be an x for every B. You would have to be able to choose a specific one for each B– the Axiom of Choice.
- Every set is well orderable. You can prove that any infinite set can be divided into an infinite number of well orderable pieces each smaller than the infinite set. But you need the Axiom of Choice to do this “all the way down” and get a well ordered set.
Conclusion
Today’s search engines are bunch of huge programming code running on huge mainframes (OK, call it “datacenters”) with numberless patches on it. They’re using recursive functions which produce chaotic results that even their engineers cannot tame and put in relevant order. Developing a new search engine is MUST. Meanwhile if you’re running a SEO business, just stick to the white hat things and get points for the parts in the formula we do knows.
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