Largest number?

I hope you read the intro. If not, please start at the beginning here: Home. 

Here is the first mathematical proof I have selected for us to talk about. It is so simple my 9 year old cousin and I talked about it for an hour and loved every minute of it. But, don't skip this one just because I say it is simple. It will highlight the core principle of proofs and epistemology, meaning the nature of knowledge, and using that simple core, we can get to more interesting and challenging cases. 

What are largest numbers? Let's start with a practical question: is there a largest number that corresponds to the oldest human alive? You can guess that there is. Say, 120. That's no proof, but, we can survey the literature, news articles and confirm our guess. How about the age of the known universe? That's about 13.8 billion years. How many days or seconds is that? Two pretty large numbers considering 365 days in an average year and 24*60*60 = 86400 seconds in each day: 5.037 trillion days and 435 million trillion seconds. 

Well, of course, you'll say there is no largest number, everybody knows that! Yes, indeed, but, let's make sure we know truly. I will get tired of writing "true knowledge", "knowing truly", "knowing for sure," so on, so I will use the word "know" only when we have a solid proof, the mathematical or the philosophical kind. And, before you get disheartened, let's do it quickly (but without haste). Here is our proof:

Let's imagine that there is a number so large, it is the largest. (This is the key "assumption" step.) Let's call it L. 

Now, let's add 1 to L and call the sum L+1 = M. 

We observe that M is larger than L. But, wait, that's in direct violation of our assumption. And, because there is no other assumption and every other thing we say (or "assert") is true (but, please check it for yourself, don't believe me!), we conclude that our assumption must be false. There can't be any number that's larger than all the others. And, that completes our proof. 

Wasn't that easy? Some of you already know this and may think that this is simple and unimportant. We beg to differ. What we just did is known as the mother of all proofs, or the father of all critical thinking. It has at least two good old names you might be familiar with:

1. Proof by contradiction, or the latin term: reductio ad absurdum. 

2. Modus tollens (method of denying)

The first set of names is hopefully obvious by now. We proved by use of a contradiction. The latin term suggests that we "reduced" our confused thinking to a clear absurdity, but in the process we found out something that's true! And, that's also the only case when I will use these wonderful words we have: "true", "truth".

In other pages, we will explain  the term"modus tollens" and make use of this seemingly simple mechanism with many names to prove fascinating results from not only the foundation of thinking, i.e., logic and philosophy, but also math, specifically number theory, geometry, and so on. But, for now I hope you are having fun!