A slight detour

Today, I couldn't get much done on the electronics side, cuz I stumbled on this book about the history of maths right from the start. In the process I found some quite interesting things. Like,

1. The first person I need to blame for the continuous headache I face in physics and engineering which is called mathematics, is Mr.Pythagoras. Yes, the very same who was the first to identify the relation, a^2 + b^2 = c^2 as the relation between the sides of a triangle. What we hadn't been told before is the he was also the first guy to come with the concept of "number relations" as the math we know today. Also the first person to come up with the concept of mathematical deductive reasoning, that is treating numbers as real things rather than expressing a property of something else physical, and manipulating them to come up with .... formulas.

2. The whole mess I never seem to quite grasp, rational, irrational, real and imaginary numbers came about cuz of today's harmless square root of 2. Yes, even though Pythagoreans were able to correctly deduct using their "mathematical" methods that, the diagonal of a unit square ( having side length as 1 unit) would be the square root of 2 (taking square as placing two triangles along their hypotenuse), they could not, rather did not have the means to describe square root of two. For thousands of years, numbers were just 1,2,3 and so on. And hence, according to the Pythagorean philosophy that nature is mathematical, real. And here came along a value, that could neither be given as a whole number nor a fraction of two whole numbers. Thus shaking the foundations of the pythagorean idea of nature is maths and maths is nature, cuz there was something present in nature that he could not describe using his math. Later came along imaginary numbers (as opposed to real) thanks to renaissance thinkers, and we know what that led to. :)

3. The most important proof of Pythagorean idea that everything in nature occurs according to mathematical laws was got by his monochord experiment where he showed that the difference between music and noise was that music occurred according to mathematical patterns or in today's view musical notes followed universal numerical proportions ( which was hi tech math back then) {This is where my problem is, the view that something in nature is occurring according to mathematical rule, rather than math being a tool to express abstraction of a natural phenomena in a simple manner}

4. Another thing I realized is that what I have been given till now, or rather been able to grasp in all my math education is simply number knowledge. I just know how numbers behave and never quite been able to treat them as "real" things and manipulate them, which is what modern math really seems to be. I am nowhere close to the "deductive reasoning" that math actually is, cuz of course I find the entire symbols and formulas thing utterly baffling.

5. Speaking of formulas, the word formula in mathematical context is usually defined as "a formula is a fact, rule, or principle that is expressed in terms of mathematical symbols. Examples of formulas include equations, equalities, identities, inequalities, and asymptotic expressions." Pretty standard right? Yes. But its wrong. The origin of the word formula comes from two bits, form and a suffix -ula.

Form: which usually means a defined physical shape (or at least is used in that sense in most contexts) also has one more meaning, an idea, abstraction, or an ideal prototype and also "behavior".

-ula is actually the feminine(interesting!) form of the suffix -ule which is used to indicate small as in capsule or globule.

And so formula is actually the expression of a ideal prototype that is true for most cases or an abstraction expressed in a smaller or compact way, in this case using symbols. Sounds more "real" to me now.

But the current context of formula interestingly came from the usage as "words used in religious rituals" which quite makes sense considering the mystical leanings of Pythagoras, no doubt he would have made the holy deduction of the relation between the sides of a triangle a full fledged ritual. But here's the twist. The current meaning of formula as a "rule" or fact actually is credited to one Carlyle (Could it be? I wonder ;)) who used it in the context "rule slavishly followed without understanding". Wasn't very far from describing a math formula to most students today I must say :)

6. Apparently, Archimedus (the Eureka! guy) was one of the top techno scientists of his time, as acknowledged by none other than Galileo and Newton. He was the ideal scientist who could not just describe natural phenomena mathematically, but also expose new phenomenons hidden from simple logic. And he was a techno scientist cuz he used his theory to improve technology, for example, his discovery of the concept of center of gravity was used to design better ship hulls so they would turn better.

7. But the first mention of the concept of an engineer was by a Roman Architect(ancient engineers) named Vitruvius. He was the first person to say that theoretical "knowledge" and artisan type "know how" could and should be merged together to make better and robust machines. And essentially, the role of an engineer is of a person who is familiar and understands the theory and also the practical implications of it, the know how, and acts as bridge between the scientist and the artisan was born out of his idea of marriage of knowledge and know how. He also acknowledged that an architect(read: engineer) needed to have the know how and knowledge of multiple streams so as to be effective as the practices of multiple technologies would be tested by his judgment (read decision or design). He also distinguished between machines, crude devices that are cumbersome to operate and do simple tasks and engines that were embodiments of sophisticated ideas and were designed with such knowledge and perfect know how to be sleek and easily operatable, thus displaying greater application of "knowledge" in the design. These things we take for granted today, were revolutionary ideas back then. And now I clearly see the importance of "knowledge" and "know how" for an engineer.

8. A major mystery in my mind just got solved!!! Why is our schooling system so pathetic? This question has plagued my mind as far as I can remember. And now I know why. The last big revolution in formalized and secular education (closest to our modern idea of schools), were universities, started in the late middle ages before renaissance. And in those universities the degree granting structure echoed the back then familiar craft guild structure. Now craftsmen as we all know, cared only about the know how rather than the theory, and that's why even today most schooling doesn't tell the why of things and simply stuffs the know how in our heads. No shit! 

9. Two people I would really like to mention, even though out of sequence are Plato and

Aristotle. They both kinda dint agree with the whole "maths is world, world is maths" notion of pythagoras, for which I really like them :) They both believed maths and the rules of maths were too ideal to be able to encompass the physical world. They believed in what they saw, which is pretty cool. But by today's standards they would be called simpletons I guess.

I figure that in the current world of probablity, advanced calulus and string theory these might seem like diminutive things, but for me it was an insight into the "why" for math. I do see how this math has made life simple for us in many ways, but for math in the scientific and technical context, the amount of complexity should be justified. And they say, if you know the why, you can bear any how. :)

All that fuss about math

Though arithmetic is as real and concrete as anything, but that's where the reality of math ends for me.

To me math is not what runs the world, math is simply man's attempt to rationalize and categorize nature. It is like my attempt and essentially my motivation for reading about everything, to get rid of the feeling I have of something that is out there, just beyond the grasp of my mind. It is similar to a feeling of uncertainty. Math was basically man's attempt to get rid of uncertainty of the things that were happening around him, an answer for his need for control, which reached its climactic and ironic end with the introduction of the field of probability into math. :)

And so numbers are as real as "chair" "hair" or "space" in the English language. And math is simply a language that gets new words in its vocabulary derived out of the original set sometimes, sometimes incorporated anew to enable it to able to describe all of nature - a different way of looking at nature, in which we try to see what is unseen to the untrained eye, look a little deeper and try to find the patterns and the abstracts. And just like any other language, maths, its symbols and its grammar needs to mastered. The problem here is that there are so many rules derived out of the fundamentals, so many abstractions, you are forced to accept them as real just for simplicity's sake. Case in point would be calculus. Integral Calculus in my view is nothing but an abstraction of an extremely large addition, but the abstraction itself has so many rules associated with it, we are forced to look at it like it were in itself a fundamental mathematical action and so as we keep building on top of it, we lose track of the fact that all those symbols, are just a way to tell us something about reality rather than real things. Just like calling a four legged form with a back and a seat, a chair becomes natural for us and chair is very real, even though it consists of different things like a seat, a back and legs.

Just a thought :)

Absorbing Capacity

Take a simple circuit:

9v battery --> diode --> resistor --> led --> ground 

all connected in series.

Now in this ckt, we know that the function of  the diode is to act as a valve, the resistor limits the current that is supplied to the led so that it does not burn out. That how's the conventional current flows right?

But the actual flow of electrons in the physical world is from the negative terminal to the positive terminal.
So the electrons actually reach the led first and then the resistor and so on.

Then how can the resistor influence the flow of electrons to the led if it actually lies after the led in the ckt?

Might sound like a silly but can get you stumped at times.

My search for an answer led to the following explanation and its pretty good too :

Most of us in our beginner classes would have been familiar with the comparison of resistance with a pipe in which water flows. Now we compare voltage as the suction capacity of the motor trying to lift water ( ie electrons) from a well to the surface. Irrespective of the width of the pipe, only a specific amount of water can the pulled up by the power of the motor.
Now if we placed a thinner pipe in the middle of two thicker pipes, the thinnest pipe is still going to determine the rate of flow of water through the system no matter how wide the other pipes.

The same explanation applies to the flow of current too. And hence we can say that the absorbing capacity (electron absorbing capacity) of the resistance is high and the highest absorbing capacity determines the max current in the system.

Interesting Fact

Today, as I was reading about "signals" I constantly kept coming across the term analog signal and discrete signal and digital signal. And I couldn't connect with those terms .. As usual. So I started wondering why is an analog signal called analog? Where did the term discrete come from. And out came the dictionary!

Analog signal : The US version of "Analogue" it simply means analogous ie similar to something else. An analog signal is simply a signal ( ie a symbol) which is exactly similar to another quantity, the quantity that it is modeling. Thats why is analog.

Discrete : The word discrete comes from the latin word discerner ie dis + cernere (to separate) meaning to divide. Discrete would then describe an object that has separate parts or form. Which I guess would rightly describe a run-off-the-mill discrete signal.

Digital signal : I guess most of us would be aware, digit means toe or finger. I think the name digital signal is simply a reflection of the fact that there are finite number of levels in a digital signal, mostly two of course, like the finite number of fingers.

And the way I understand it, continuous and discrete, do not describe the range of values or the kind of values the signal can take, rather, the time axis or the number of values that are taken by the system or the way the signal is sampled basically.

Analog and digital, on the other hand describe the values of the signal. So does the term quantized. An continuous-time system can be quantized and so can a discrete-time system.

You can either measure the signal continuously for ten minutes to get a continuous signal or measure for 1 second every one minute, ten times to get a discrete signal. That'ss the difference. But when you measure, if the signal takes any value, then its analog, or if it takes only specific steps of values, its quantized, and if it only takes a specified set of values, its digital.

So this was today's interesting fact.
Till next time.

All those Theories!

All this while I have been studying all these theories. Some seem to co-exist, some seem to complement, and some contradict. Why so?

Why all these theories anyway? What is the use?

Every time I read about a theory, I seem to understand where its coming from, atleast superficially. But when I try to compare it with another theory, I come up with a blank. I don't know why... Its like there is some kind of block, that won't let me get to the depth of it.

Its like as I study, over time, science seems to go deeper and deeper into reality, and suddenly, as if someone slapped it, it takes a step back, and starts viewing things in a completely different perspective. And that slap part started with Quantum Theory. Its like it turned everything upside down. I have always had trouble visualizing it, and I am sure I am not the only one.

Even my education, I only now realize, had been like that. And perhaps that is why I found it confusing. My syllabus is college too had a big role to play in that. Every semester we were introduced to new and new subject and concepts without any background or any obvious correlation whatsoever. It was like, one day, you take the radio, pull it apart and study it. Next day you take the fan, pull it apart and study it. Third day you take your computer pull it apart and study it ... and so on. There was absolutely no structure to it.

But the most mind boggling part of all this is the question of which theory to apply where??

All learning happens in start and stops and not always in a straight line, and that is true for development of science too. But I wish I had a little more clarity.

An Observation

In going thru all the material I realized one thing: How much modern theories depend on mathematics.

Mathematics has been a base for science since time immemorial, but I see how extensively maths is being used whether it is to prove what you thought, or it is to find out new patterns and relationships in nature.

That is what is biggest advantage of math. For the ones who know how to use it, math can reveal aspects of nature that would never be revealed using normal human reasoning and conventional language. Its miraculous in that sense.

But truely knowing it and understanding it, knowing how to use it, is no easy task.

Information Theory

I have now moved on to computers and information theory.

I did try to do some research on accelerated charge in a field .. and got one piece of info that hadn't struck me till now ... All electromagnetic phenomena on earth is happening in the earth's magnetic field and every charge that accelerates on earth accelerates in the omnipresent magnetic field of earth so would always generate an EM wave.

Now the next obvious question would be would an accelerated charge in a absolute vacuum devoid any field still radiate EM wave out of interaction with its own field? Couldn't find an answer for that yet.

Anyways moving on to the current topic : Computers and Information Theory.

Today's "Computer" stands for much more than what the name implies. Today's computer is more of a "simulator" than just a calculator. That idea was first conceived by Alan Turing, british scientist who said there could be a machine that would generate the solution of any problem that can be solved provide all the required steps are pre determined ... An algorithm is predefined.

This machine could be built, today we call them computers, only because of technological advancement of semiconductor physics, and the encoding of "rules of logic" into logic circuits.
Transistors, after its invention in bell labs, underwent a major revolution in terms of application when logic circuits were created, they were now capable of "processing" a set of input signals, converting them into a predefined set of output signals, and then these signals could be interpreted and rendered in the form the user required. A graph. A table. A equation. Anything basically. So computer could now simulate the entire process, that was defined in the algorithm, and give us the results instantly. Now that is immense power if you really think about it.

Personally this is what I have liked about computers and software programming the best. The power to create anything without any rules. If you can write a code for it, you can practically make anything. :)

With the advent of computers, information became even more readily available. All sorts of information. And that became a problem in itself. But the one I'm concerned about is the Engineering problem. And as we all know, Information theory means Shannon's thoery.

Again, Shannon's theory is something I have always hated during my engineering. It just never made sense to me. If we already know something its information content is nil, and if we get an unknown signal, that has information. Its totally counter intuitive. Until I saw it from an engineers perspective. From a Receiver's perspective. If I receive a signal, which I already knew was coming, communicating it was pretty useless to me isn't it? On the other hand, if I receive a unknown signal, now my job would be to find out whether or not its a valid signal. Its like jigsaw puzzle solving. If I already know where a piece should be and what the piece is, I would put it there anyway, but for every new signal I receive, I have to first find out whether its a valid piece of the puzzle, and if it is, then I try to put it somewhere where it would help form the whole content or the picture. So the new signal is a clue, full of some information. That is the way I understand it as of now.

Another interesting thing about Shannon is that he was the one who came up with the idea of logic circuits. He first researched into how the telephone switching relays worked and said they could be designed even more efficiently incorporating Bool's "laws of human thought". And that they infact embodied that exact idea. Now if systems displayed use of logic, then vice versa logic "circuits" could be used to design systems. That is where all of digital electronics began.

Shannon also showed that the digital/binary encoded bit stream was the most efficient way of data transfer that allowed self correction.

As electronics engineer I can really appreciate the far reaching applications of his theory... And the reason it prevailed was simply because he took out the subjectivity. He said, I don't care what the content is, I don't care if you are sending a word doc, a music file, a image or a radio transmission, my theory applies to all, he generalized it and reduced it down to the most basic level so much that it could be applied practically to anything. Genius.