One plus two equals three. That’s a true statement. As true as they come actually. You cannot argue with that. Of course, if you wish to sound smart, you may say what if we are using a binary basis? (The system used by computers to count). But we ignore the heuristics here. The numbers are defined in a basis they make sense in. Arithmetic thus makes sense. But does it? Let’s look around some of the most fundamental pillars of logic and see if they hold up to scrutiny.
In fact, why does 2 follow 1? Why not 1.3 or 1.999? What does it mean when we define addition as we have defined above? Well, it’s easy to define any natural number (apart from 1) actually. You can define any successive (natural) number by saying that you increment it by 1. That’s neat. But it shifts the burden of proof to defining what “sum” is and more importantly what “1” is.
Dedekind and Peano spent some portion of their lifetimes on this. In mild terms, they started by taking axioms. Mathematicians use this lingo, whenever, there is a true statement which cannot be proven within that system. It’s basically an assumption. But don’t let that dishearten you yet. Because, lo, “addition” or “sum” is defined. This means that we can say no more than . You can’t say anything more. You can’t explain that. It’s a fact which you have to swallow. We don’t say why. It’s not important. But mainly because we can’t.
Numbers are one of the first real abstractions performed by the human mind. You can have one apple or two apples. Or three or a stomach ache if you want more! But, when we count, we are performing one of the most fundamental operations in Mathematics, addition. You are basically abstracting certain density distribution repetitions as discrete objects which should be counted. Of course, the density distribution might remain uniform extendedly like in space for example. (I don’t mean outer space here, but any space actually. Sure, outer space would also work!) In such a case, drawing a curve or line through them, we see we cannot subdivide them into discrete points, or separated density distribution repetitions. Perhaps, if we become tiny enough, we might be able to see whether the pencil lead which drew the line fell at discrete steps or not. But for now, we make the assumption, that space is continuous, backed heavily by experiments and the need for physical laws we have formulated to hold forte. So how do we count these infinite set of points. (Infinity, by the way, is a number which is larger than the largest number you can think of.) Turns out we don’t need to. Instead, we make use of another approximation. We know that we can only count discrete repetitions. And we can count things which can be counted. (This (apparently) rules out trying to count the number of sand grains in the Sahara desert or the number of water droplets in the Pacific. Possibly, one could approximate the total volume and use elementary chemistry: Avogadro’s no. etc arrive at about a limiting value. And since this would be a finite a number, it would be countable. The number of points in a line though is ‘proven’ to be uncountable and infinite. Of course, infinite things, if they can be properly labeled can be counted as well: the set of Rational Numbers is an example.) Of course, the first two are far easier to count than the last one. Because a point is smaller than the smallest thing you can imagine. And water drops and sand grains are pretty large in comparison to any “stuff” that we may imagine!
As we notice that the problem always arises with infinities. Here we get a problem because we can’t count an infinite set of points. Instead, we take a certain distance and define it to be our 1. Then we proceed like normal counting. Each time we subdivide our interval, we redefine our smallest unit of measure. It’s also known as the Least Count. Thus, depending on how deep we wish to go, we can count in intervals of 1, 0.1, 0.00000…1 etc. That’s entirely up to us. But as we know that there are always infinite sets of numbers between any 2 of such smallest units.
That ends part 1.0. Next, we’ll start with part 2.0. Nah, 1.1. Or perhaps 1.01. But it’s just my notation anyway. And also the rest of the world’s.