The Vehicles of AI — Analog and Digital Computers
Before we embark on our exploration into the world of Artificial Intelligence, let us start off with the vehicles, the wheels of the ride — The Computer, Analog or Digital. The fundamental difference in these two classes of computing is in the way information is represented.
Analog, as the word signifies, stands for representation of natural phenomena, as they are. For example, the sine wave.
The sine wave represents continuum. There are no discontinuities, the graph extends from -∞ to +∞ (‘∞’ stands for infinity), without a break in between. Analog computers operate on data that are continuous using the principle of differential amplification that is realized in hardware, commonly using op-amps.
Digital computers, on the other hand, manipulates on digitized analog data, or the continuous data made discrete.
The image above depicts the digitization of data, at regular intervals, from the analog counterpart. The data thus digitized is manipulated upon, by the rules of a formal system like Boolean Logic. The fundamental building block of digital computers is the switch (for example, switching states between 0 and 1 in binary), which is implemented using transistors embedded inside Integrated Circuits (ICs). We will discuss about the history of implementing digital logic in our later posts.
The deluge of the digital computer completely has reduced its analog counterpart to nothingness, over the past few decades. This is mostly because shopworn appliances came out cheap and high performing with the digital technology. Nevertheless, let us delve into the fundamentals of computing for a while.
In analog or digital computing, Mathematics is central. In fact, it is the very star around which the entire heavenly system of computing and communications revolve. This makes it imperative for us to comprehend Mathematics, and what it represents. The difference between analog and digital computing traces its origins to pure Mathematics, and Formal Mathematical Systems like Boolean Logic.
Pure Mathematics and Formal Mathematical Systems
Just stumbled upon the following, from over the web:
‘Formal mathematics is like spelling and grammar — a matter of the correct application of local rules. Mathematics is like literature — it brings a story to life before your eyes and involves you in it, intellectually and emotionally.’ Henceforth, whenever we mention Mathematics, we mean Pure Mathematics.
Mathematics is the ‘Science of Patterns in the Universe’. Just much the same way that we have languages to communicate our ideas, Mathematics captures the language of the Universe- right form speaking the curve, that Halley’s comet traverses in its lifetime, to how much value I owe towards my credit bill. Yes — Mathematics has no standing on its own. It is just the transcript of the story that is being enacted. Without the story, the transcript has no significance. Let us look on Mathematics, from the ground up.
The Journey of Mathematics
Natural numbers represented whole quantities: three candies, two blocks. The number system was devised for the purpose of counting, thus starting with the Natural Number System. The system of counting furthered down, with the addition of zero and negative numbers, motivated by the transactions of money. For example, I have nothing in hand, and I owe him three apples. This represents a deficit of three in my account, or symbolically, my account stands at -3 (him being the only person I deal with). This gave rise to the Integer Number System. The idea of fractions, also known as rational numbers, was conceived when the Egyptians sought a way to split food among people. Finite Math encompassed the operation of division, if the denominator was non-zero. This gave way to the Rational Number System.
So far, so good. Till here, we may notice the correspondence of the world of numbers to worldly day-to-day transactions. From here, Mathematics takes a wholly different identity, transfiguring into the abstract. The need for the inclusion of infinite operations (operations not limited to divisions) as with the square root of a non-perfect square, like √2 , and the newer expansions discovered for π (Pi) in the late 1600s, led to a whole new facade of Mathematics — Abstract Math. A couple of early definitions for the infinite series expansion of π is as follows.
The ‘realness’ of Mathematics takes a new turn, from just whole objects to unwholly ones like π (Pi). The new class of the irrationals takes form. The incorporation of irrationals like √2, infinite expansions like that of π, extended the number system to Real Numbers. The new number system is designated as ‘R’. We are left agape at the beauty of Nature and capturing its signature through Abstract Mathematics. An equally important irrational, the exponential or ‘e’ follows the almost same story.
Importantly for our study, it is from here that the number system sheds its ‘discrete’ persona. Mathematics is not anymore just for counting. It literally is really to stand for anything in Nature. Mathematics couples itself with Nature tightly with the Real Number System.
In 1539, Gerolamo Cardano, an Italian physician, had in the process of solving a general cubic equation, encountered strangely a term depicting the square root of a negative number. The solution though, itself does not have the square root of a negative number. This later led to the inclusion of the imaginary number ‘i’ to the number system, thus extending the number system as The Complex Number System. Mathematics stood for the first time, ‘for the idealization of infinite refinement, for which there is no clear a priori justification from Nature,’ as mathematician Roger Penrose puts it. Mathematics had transcended from being an invention, to be a discovery.
Now that we have a general overview of the number systems and how they came about, let us look a bit into formal Mathematical systems next week.
Next week: Formal Mathematical Systems.
Previous week: Can Machines Think?
