What is a number, really?

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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 ]

Concept of a Set and of a Number

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For instance, let's take a look at the cars on a highway, apples on a table, coffee cups in a coffee shop, pears in a basket. Without our initiative, our thought action, will, our specific direction of thinking, objects will sit on table or in their space undisturbed and unanalyzed. They are apples, cars, coffee cups, pears. But then, on the other hand, we can think of them in any way we wish. We can think how we feel about them, are they edible, we can think about theory of color, their social value, utility value, psychological impressions they make. We can think of them in any way we want or find interesting or useful, or we can think of them for amusement too. They are objects in the way they are and they need not to be members of any set, i.e. we don’t need to count them.

Now, imagine that our discourse of thought is to start thinking of them in terms of groups or collections, what whatever reason. Remember, it's just came to our mind that we can think of objects in that way. The fact that the apples are on the table and it looks like they are in a group is just a coincidence. We want to form a collection of objects in our mind. Hence, apples on a table are not in a group, in a set yet. They are just spatially close to each other. Objects are still objects, with infinite number of conceptual contexts we can put them in.
Again, one of the ways to think about them is to put them in a group, for whatever reason we find! We do not need to collect into group only similar objects, like, only apples or only cars. Set membership is not always dictated by common properties of objects. Set membership is defined in the way we want to define it! For example, we can form set of all objects that has no common property! We can form a group of any kind of objects, if our criterion says so. We can even be just amused to group objects together in our mind. Hence, the set can be specified as “all objects we are amused to put together”. Like, one group of a few apples, a car, and several coffee cups. Or, a collection of apples only. Or, another collection of cars and coffee cups only. All in our mind, because, from many directions of thinking we have chosen the one in which we put objects together into a collection.

Without our initiative, our thought action, objects will float around by themselves, classified or not, and without being member of any set! Objects are only objects. It is us who grouped them into sets, in our minds. In reality, they are still objects, sitting on the table, driven around on highways, doing other function that are intrinsic to them or they are designed for, or they are analyzed in any other way or within another scientific field.

Since, as we have seen, we invented, discovered a direction of thinking which did not exist just a minute ago, to think of objects in a group, we may want to proceed further with our analysis.
Roughly speaking, with the group, collection of objects we have introduced a concept of a set. Note how arbitrary we even gave name to our new thought that resulted in grouping objects into collections. We had to label it somehow. Let's use the word set!

Now, if we give a bit more thought into set, we can see that set can have properties even independent of objects that make it. Of course, for us, in real world scenarios, and set applications, it is of high importance whether we counted apples or cars. We have to keep tracks what we have counted. However, there are properties of sets that can be used for any kind of counted objects. Number of elements in a set is such one property. If we play more with counts and number of elements in a set we can discover quite interesting things. Three objects plus six objects is always nine objects, no matter what we have counted!  The result 3 + 6 = 9 we can use in any set of objects imaginable, and it will always be true. Now, we can see that we can deal with numbers only, discover rules about them, in this case related to addition that can be used for any objects we may count.

Every real world example for mathematics can generate mathematical concepts, mainly sets, numbers, sets of numbers, pair of numbers. Once obtained, all these pure math concepts can be, and are, analyzed independently from real world and situations. They can be analyzed in their own world, without referencing any real world object or scenario they have been motivated with or that might have generate them, or any real world example they are abstracted from. How, then, conception of the math problems come into realization, if the real world scenarios are eliminated, filtered out? Roughly speaking, you will use word “IF” to construct starting points. Note that this word “IF” replaces real world scenarios by stipulating what count or math concept is “given” as the starting point.
But, it is to expect. Since a number 5 is an abstracted count that represents a number of any objects as long as there are 5 of them, we can not, by looking at number 5, tell which objects they represent. And we do not need to that since we investigate properties of sets and numbers between themselves, like their divisibility, which number is bigger, etc. All these pure number properties are valid for any objects we count and obtain that number! Quite amazing!

Moreover, even while you read a book in pure math like "Topology Fundamentals" or "Real Variable Analysis" or "Linear Algebra" you can be sure that every set, every number, every set of numbers mentioned in their axioms and theorems can represent abstracted quantity, common count, and abstracted number of millions different objects that can be counted, measured, quantified, and that have the same count denoted by the number you are dealing with. Hence you can learn math in the way of thinking only of pure numbers or sets, as a separate concepts from real world objects, knowing they are abstraction of so many different real world, countable objects or quantifiable processes (with the same, common count), or, you can use, reference, some real world examples as helper framework, so to speak, to illustrate some of pure mathematical relationships, numbers, and sets, while you will still be dealing, really, with pure numbers and sets.

There may be, also, a question, why it is important to discover properties of complements, unions, intersections, of sets, at all? These concepts look so simple, so obvious, how such a simple concepts can be applied to so many complex fields?

Let’s find out! Looking at sets, there is really only a few things you can do with them. You can create their unions, intersections, complements, and then find out their cardinalities, i.e. sizes of sets, how many elements are there in a set. There is nothing else there. Note how, in math, it is sufficient to declare sets that are different from each other, separate from each other. You don’t have to elaborate what are the sets of, in mathematics. You do not even need to use labels for sets, A, B, C,… It’s sufficient to imagine two (or more) different sets. In mathematics, there are no apples, meters, pears, cars, seconds, kilograms, etc. So, if we remove all the properties of these objects, what properties are left to work with sets then? Now, note one essential thing here! By working with sets only, by creating unions, complements, intersections of sets, you obtain their different cardinalities. And, in most cases, we are after these cardinalities in set theory, as one of the major properties of sets, and hence in mathematics. Roughly speaking, cardinality is the size of a set, but also, after some definition polishing, it represents a definition of a number too. Hence, if we get a good hold on union, complement, intersection constructions and identity when working with sets, we have a good hold on their cardinalities and hence counts and numbers. And, again, that's what we are after, in general, in mathematics!

As for real world examples, you may ask, how distant is set theory or pure mathematical, number theory from real world applications? Not distant at all. Remember the fact how we obtained a number? A number is an abstraction of all counted objects with the same count, of all sets of objects with the same number of elements (apples, cars, rockets, tables, coffee cups, etc). Hence, the result we have obtained by dealing with each pure, abstracted number can be immediately applied to real world by deciding what that count represents or what objects we will count that many times. Or, the other way is, even if we dealt with pure math, pure numbers all the time, we would've kept track what is counted, with which objects we have started with. There is only one number 5 in mathematics, but in real world applications we can assign number 5 to as many objects as we want. Hence, 5 apples, 5 cars, 5 rockets, 5 thoughts, 5 pencils, 5 engines. In real world math applications scenarios it matter what you have counted. But that fact and information, what you have counted (cars, rockets, engines, ..) is not part of math, as we have just seen. Math needs to know only about a specific number obtained. Number 5 obtain as a number of cars is the same as number 5 obtained from counting apples, from the mathematical point of view. But, it can and does represent sizes of two sets, cars and apples. For math, it is sufficient to write 5, 5 to tell there are two counts, but for us, it is practical to drag a description from the real world, cars, apples, to keep track what number 5 represents.

[ concept of a set, math, math tutoring, mathematics, number concept, number definition, numbers, set, set concept, sets and numbers, tutoring ]