Saturday, September 3, 2011

What is a number, really?

In this post I would like to introduce a concept of a number. I want to show a method how we can conceptualize a number, to actually understand what is it. This introduction will provide multiple benefits for anyone interested in deeper, fundamental understanding of mathematics. It can provide answers about what exactly is the subject of research in mathematics. The explanation can be a useful and effective starting point for all those creative minds who ask “why mathematics?” “what for we have to calculate all that?”, and “what is the number actually?”.

Let’s say, to start with, that we can conceptually, visually or in our minds, differentiate objects among themselves, and that we can, and then we want, to count them. When we, for whatever reason, group objects in some collections, we can be in situation to determine which collection has more objects, if we want to. Don’t forget, we don’t have any numbers, or names for counts, defined yet.

How we can determine which set, which collection has more elements, objects?

We can compare collections of objects by matching, pairing, elements from one collection with objects, elements from another collection. Pairing has to be, obviously, one to one.  Let’s say, we have a set of apples and set of pencils, as shown in the Picture 1. We want to find out a number of elements in each set.

We can start first by determining which, if any, set has more elements. In order to do that,  we only need to match, to pair, one object from one set with an object  from another set. If no objects are left unmatched then two sets have the same number of elements. As we can see in the Picture 1, by pairing apples and pencils, no objects are left unmatched, hence these two sets have the same number of elements. We do not have the name yet for that count, for that number, but, the good thing is we know what we are talking about! We are talking about certain number of elements, defined by the exact match of two collections. That property is what we are after, that “numberness” is what we are after.

Now, note one very interesting thing. If we replace apples with pears, and pair, match every pear with every pencil, we can see that the match is again achieved! The pairs are again complete and no pears or pencils are left unmatched. Hence, these two collections, two sets, have the same number of elements. Now, let’s introduce another collection of objects, say watches. Pair the watches with pears. As we can see it’s the same count, complete match, hence the same number of elements. Note very, very interesting observation here! No matter what objects we are matching, as long as the match is one to one, and as long as the pairing is complete, we have the same number of elements in two collections.

It is this property, this “numberness”, common in pairing two sets, that we call a number. You see, no matter what kind of objects are in two sets, if they match, that property, is the actual number. That is the concept of a number! It is, at this point, completely arbitrary how we are going to label this numerical concept we have just discovered and defined. Word, symbol, reference for it is completely arbitrary. In English, it is called number five, and the numeric symbol is 5. Label for our new concept is really of very minor significance at this point. The concept we obtained is way more important than the tag we will use to reference it in our speech. Of course, we could start with three object and obtain number three, or seven objects, and obtain number seven, etc. Notice how we, now, have this set property to work with, set property related to its number of elements, the quantity of objects, that we have abstracted from any two sets of objects, that we can call a number, or a count! That’s the actual concept of a number. Notice, also, how “number” in its essence, is not even a noun, but more like an adjective, that describes “quantitative” aspect of two sets, the number of paired elements of two collections.




Picture 1. One example of number conceptualization.


As long as we know that this labels, 1,2,3,…represents that property of one to one pairing between the elements of two sets (with the goal to determine if they have the same number of elements) we are on a good path to work with numbers and quantification.

You may ask at this point “well, I don’t always see two sets when I count objects of one set, I just count them without any pairing with the elements of another set”. Good question! What you actually do, by, say, counting CDs, in your collection, is matching them with the set of natural numbers in your mind, which is completely ok. But, note, you have natural numbers at your disposal to use them for counting other objects, while in our previous explanation we are actually just defining the numbers! We qwere after very definition of a number. Once the numbers are defined, as we did for number five, you can use that number five and others in counting any kind of objects!

Look at that number five, universal count of five, for any kind of objects. Here is one more interesting conclusion. You see, how at this point, we can deal with counts only! We can deal with a count 3 and a count 5, which we call a number 3 and a number 5, regardless which objects they represent count of! If we want to add them, it will always be 3 + 5 = 8, no matter what we have counted! This is exactly what pure math is about! We, now, can use our generalized knowledge that 3 + 5 = 8, and utilize it any real world situation, for instance, if we have 3 cars and we buy 5 more cars we will have 8 cars.

Note that “purity” of math is just related to the fact that we do not care what we have just counted. We were only interested in adding, subtracting, dividing, pure counts, pure numbers we have abstracted from real world objects counting.

If we want to mix pure math and real world scenarios and objects we are counting, it’s easy! We can just put a small letter beside the number, to keep track what we have counted. Hence, in physics we have 3m + 5m = 8m, for distance, length in meters. Then, also we can put 3 apples + 5 apples = 8 apples in agriculture studies. We essentially do two steps here, during the additions of real objects. When we want to add 3m + 5m, we actually separate pure numbers from the meters counted, we enter with these two numbers the world of pure math, where we do calculation of numbers only, 2 + 3 = 8, and then we go back to real world of meters (because we have those small letters to remind us what we have counted) with the result 8, and associate the name of the object, in this case it’s a physical unit of length, or distance, which is meter, (m), to the number 8. And, voila, we have just used pure math in the real world application!

[ applied math, applied mathematics, concept of a number, concept of a set, natural numbers, number, set and number, set theory ]