Friday, January 20, 2012

On the composition of natural numbers

Hi readers! 

I've finally stopped doubting myself for calling you that. I mean you gotta have some serious readers if your hits reach 7900, right? Anyways, since you do exist, I wanna thank you for reading my blog! I hope I've kept you entertained.

For the past two weeks, I've been in India, my homeland. I've been through many hours of driving... more precisely being driven. These long car rides served as great opportunities for me to think about things.

I have been pouring over how natural numbers are composed. In that, I mean I have been wondering what natural numbers like 27 or 98 really mean and how these numbers are constructed from basic logical principles. A recent car ride to New Delhi gave me great insights.

But first of all, what are natural numbers? David Berlinski wrote in his book "One, Two, Three. Absolutely Elementary Mathematics" that natural numbers function as tools, allowing us to distinguish things from a thing, and one set of things from another set of things. According to him, natural numbers create things altogether! I think natural numbers are the numbers we naturally know. They are the numbers that we use to get about daily life. They help us in counting. They are basically the numbers 1,2,3... and so on towards infinity. They are also known as the non-negative whole numbers.

To get how natural numbers are composed, one must first understand the idea of the number zero. Zero means something, and that something is nothing. Before we arrive at anything, we must first have nothing. The notion of nothing, or zero, is defined by the symbol "0".

The next pillar of this number system is a consistent unit, something we define as "1". This unit is left as a symbol, and is not given an extra descriptor such as 1 cow or 1 dog, to maintain generality and applicability to various scenarios.

With the fundamental ideas of "0" and "1", other unique symbols can be constructed. I'm going to show you the logical basis for the construction of the symbols "2", "3", "4", "5", "6", "7", "8' and "9". All of this may seem trivial to you right now, but if you think about it, none of this has to be. Keep your mind open, and you'll see some amazing implications later on.

(A) Firstly, this condition is "0":




(B) This condition can be understood as "1":



(C) This next condition looks very different from (B) and is greater than (B) by one unit, and thus needs to be given another symbol "2":



(D) This next condition looks different from (C) and is greater than (C) by 1 unit, and is thus given another symbol "3":



Using the same ideas, the following symbols are inherited by the following conditions:










Well, what comes next? Our educated nature would force us to say 10. But remember, 10 has yet to be invented. We are still in the midst of logical construction.

So here we are... We need to invent another symbol for the condition which has 1 more unit then 9.  Well, that symbol could very well be this:



But that's not what are forefathers chose. They decided that the condition which has 1 more unit than 9 should be recognized as another pillar of the natural number system, and that it deserves a unitary recognition, like 1 has. But of course, the condition which is 1 unit greater than 9 is greater than 1, so it cannot look like 1. It must be unique in its looks. They gave it the symbol "10". "10" means, reading from left to right, 1 complete unit of 1 unit greater than 9 and no other units of 1 (hence the zero):



I would like to call 10 the "grouping unit", since it groups 1 unit more than 9 units into 1 group. I would also like to call what the grouping unit represents the "grouping unit value". In our system, the grouping unit 10 has a grouping unit value of 1 unit greater than 9. Also, notice that 10 is a construction of the symbols defined earlier on, and not a new symbol altogether. Human economy is at play here, for construction of a new symbol to denote 10's unitary property would complicate life.

You may ask at this point, why was this grouping unit (10) invented? Well for one, if it wasn't, we would have to keep conjuring new symbols and would eventually run out of options. You may also ask, why is the chosen grouping unit value 1 unit more than 9? Why can't the grouping unit value be 8 or 9 instead? A possible answer- our forefathers realized we all have 10 fingers and considered that number sacrosanct. But it should be noted here that the grouping unit 10 could have other possible grouping unit values like 8 or 9. If the grouping unit 10 had a grouping unit value of 9, 10 would then mean 1 unit of 9 and no other units of 1. But the fact is we chose a grouping unit value of 1 unit greater than 9 for the grouping unit 10, and that's that.

This breakthrough allowed us to represent larger numbers conveniently. For example, 27 would represent 2 units of 10 and 7 additional units of 1. 98 would represent 9 units of 10 and 8 additional units of 1. This convenience would last up till 99. One more unit would create 10 units of 10. How can we represent this? Well, a new grouping unit can be invented- "100". "100" reads, from left to right, 1 unit of 10 units of 10, no units of 10 and no additional units of 1 (hence the two 0s).

In a similar fashion, adding a zero to the right of the nth grouping unit creates a new (n+1)th grouping unit, each (n+1)th grouping unit containing 10 occurrences of the nth grouping unit. An example- converting 100 to 1000 by adding an additional zero. 1000 would consequently mean 10 groups of 100s and no other groups of 100, 10 or 1. Adding zeros to such grouping units also constructs and defines the various powers of 10.

With the help of 10 and its powers, we are able to compose natural numbers in an efficient and sustainable manner; using only the symbols of 0 to 9 in a certain order.

QEF

I hope this has been enlightening to you, because it certainly has been for me. Now, for something else. What would happen if we human beings were born instead as funny-looking alien creatures with 6 fingers, instead of 10, assuming we had the same brains and intelligence? By that I mean what would happen to our natural number system? Something really weird!

The ideas of 0 and 1 would be the same to us, but our grouping unit of 10 would no longer have a grouping unit value of 1 unit greater than 9. Because we would have 6 fingers, our grouping unit of 10 would now have a grouping unit value of 6. So now numbers would proceed like this:



The numbers would continue like this, jumping up by 5 units on what we recognize as multiples of 5:



What would the last term be? Would it jump up to 60? Just like 10 units of 10 is grouped as 100 in our system... now, 6 units of 6 would qualify as 100. Thus, ? = 100.

The grouping units of 10, 100, 1000... in this system would now represent what we know as 6, 36, 216... in our system. 5 then becomes the last figure that appears on any placement, instead of what we know as 9. For example, 5 + 1 = 10, 15 + 1 = 20, 55 + 1 = 100 and 555 + 1 = 1000.

This is all really mind-blowing! It is also difficult to comprehend because we are so accustomed to our own number system. Imagine the pains involved in switching to another natural number system with a different grouping unit value!

If you liked this demonstration, you can also try to construct numbers for intelligent aliens with different number of fingers, like from 2 till 9 fingers. It'll be an interesting exercise. After that, try doing the same thing for aliens with 1 finger and 10 fingers (us). The last two experiments will be especially eye-opening. You'll get to see how arbitrary our number system is, and how little reason there is for us to prefer one number system (and grouping unit value) over another.

Picture from: http://www.clker.com/clipart-27994.html
LaTeX math codes: http://www.codecogs.com/latex/eqneditor.php

Monday, January 2, 2012

My problem of induction

I wrote this essay for an application to a special university program. Because I had to keep within a tight word limit, I omitted lengthy explanations of concepts. For that matter, I'll probably write another post. But anyways, here's the essay:

Back in August 2011, I was taking an evening walk and started thinking about Newton’s Law of Gravitation. The law relates the gravitational force between two point masses to their individual masses and their separating distance. This law is said to hold for all masses at all separating distances.

I began wondering how Newton derived his law. I deduced that he must have carried out experiments on masses which were separated by certain distances. But then something occurred to me. Newton could only have tested a finite number of masses at a finite number of separating distances to verify his formula. This is because it is physically impossible to test all combinations of mass and distance, them being infinite.

But how then can Newton’s observation be called a universal law? Couldn’t there be untested masses and distances over which Newton’s Law doesn’t hold? I was perplexed by this thought.

I read extensively and chanced upon the problem of induction by David Hume. According to the problem of induction, inductive statements, which are generalizations of past experiences, are unjustified because past experiences used for their justification may not be similar to present or future experiences. This principle implies that Newton’s Law is flawed because of its inductive nature.

I used to believe that laws of science were absolute and unquestionable. But the problem of induction made me realize the fragility of scientific theories. This destroyed my love for science.

The problem of induction also convinced me that my everyday actions were unjustified. I’ll explain... Let’s say as a child I touched a pot of boiling water and discovered it was hot. If I were to see another pot of boiling water today, I will remember my past experience and will not touch the pot I see, assuming it to be hot. But philosophically, this decision is unjustified. The pot today may be of a different nature than it was back then. It might just be cold. By extending this idea, I began to realize my everyday actions were unjustified. I became nihilistic. I even began to doubt my own existence, because according to the problem, it is perfectly possible for me to stop existing in the next moment.  

I was devastated. But then, I began reading up on solutions to this problem. Regarding the validity of science, I found Karl Popper’s philosophy particularly relevant. According to him, science is not a perfect way to probe the Universe, but it is the best way we have. Understanding that science was the best way to explore the Universe brought back my love for it, and I accepted its inherent imperfection. I began to find the imperfection beautiful, because it creates mystery and fuels curiosity.

Regarding the philosophical validity of our everyday actions, I read and realized that though our everyday actions may be imperfect, they are the most rational options available to us. If we don’t follow these rational approaches, we will be reduced to a state of perpetual contemplation and will become unproductive in the process. 

Ultimately, pragmatism is the solution to this problem of induction, or at least it provides the greatest consolation for us imperfect human beings. I learnt the importance of being pragmatic through this experience. I also understood the provisional nature of our existence and of science. I now perceive the world as a huge puzzle, one that I may or may not be able to solve. But I’m going to act morally and rationally, to solve this puzzle in the way I feel is right!

Picture source: http://namestormers.com/company-names-blog/naming-philosophies-from-the-naru-naming-guru/