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After a number of years dealing with mathematics in your primary, secondary school, there still may be a question what the number is. Moreover, unless you are a professional mathematician, with PhD in your resume, I can safely assume that your frustration and fear of mathematics is still present.
What is a number? Seemingly popular approach I am using here does not reduce the strength and significant clarity of the definition. Bear with me, and listen carefully :-) You may find out many interesting things!
Here is the clearest approach to defining number.
Number is a count.
I will repeat again, number is a count. The purity and significance of this definition can not be emphasized more. While it is simple, it conveys many more important messages than other definitions and approaches you may have read about before. One of the most important message, in my view, of this definition is that it implicitly specifies what you can do with counts. Knowing what you can do with counts, you actually filter out all non mathematical concepts that may be mixed during "bad" mathematical lectures over the years. Also, thinking of numbers as counts, you define what pure mathematics is about! And that can help you answering the questions how math can be applied (about what "applied" means we will see later) in so many different fields, and what differentiate pure and applied math.
What you can do with counts is what mathematics is all about! So, what can you do with counts? You can add them, subtract them, divide, multiply. You can, then, do any number of these operations in any sequence you want. No apples, pears needed to do that! Count 5 is a universal count. It can come from counting apples, pears, cars, atoms, money, steps, seconds. That number 5, count 5 is a universal thing for all of them. While you can eat 5 apples, drive 5 cars, wait 5 seconds, with count 5 you can not do that. But you can add another number 5 to it! or deduct count of 3 from it. Or do any other "counting" operation! Note very important thing -> how you call these counts, i.e. are they integers, positive, negative, odd, even, rational, are just labels we attach and associate to the concept of a count! There is only count we are dealing it all the time.
Looking at count 5 only, i.e. number 5 only, you can not tell whether it came from apples, pears, or counting seconds. So, how you will differentiate count 5 of apples and count 5 of seconds if count, number is actually so universal concept? There is no other way than to keep track by yourself what you have counted. Technically, you will write a small letter beside the number, beside the count to remind you what it is a count of! Or you can remember that in your mind. Whatever works for you.
So, whenever you read about those exotic mathematical concepts, like matrices, determinants, integrals, equations, algebra, arithmetic, you will know one thing - it is all and only about pure counts we have just talked about. There is nothing else there. For instance, a matrix is a set of counts arranged in rectangular fashion on page. But, you do not need even that rectangle. You can just imagine in your mind the same set of counts and differentiate between them in any way you want. It just happened that it was convenient to write those numbers in a rectangular grid on paper! It was just convenience.
The logic, the reasoning in the world of counted objects is separate from the logic that deals only with counts. You can investigate properties of counts only, completely independent from the real world objects they might represent count of. This is the topic of mathematics. To find what is true about counts i.e. numbers. Hence the proof. But note here, we are really interested in counts' characteristics, no matter which objects have been counted! What these characteristics can be? We can have odd numbers, or even! We can have prime counts. Some counts can be divided by others, while some not. But, the numbers are numbers, it is us who give them names to keep track of some of their properties or just we want to deal with some numbers while leaving other numbers alone!. Naming numbers is not a mathematical operation. It just help us describe, label numbers, counts, we want to deal with.
We can compare the way we can deal with numbers to the way a sculptor deals with clay. It is only the clay that he works with and nothing else. Clay! But, what clay represents when it is shaped, what sculpture represents is not about clay! The motivation how the sculptor will twist, press, mold, shape clay is outside clay's world. Same in math! The reason why we add, subtract, divide, or even select numbers to deal with, frequently are outside mathematics! The motivation can come from us buying CDs or from an economist measuring supply and demand, or police measuring speed of the car.
Now, back to sculpture again. The sculpture can represent anything. The similar thing is with numbers. In mathematics we are dealing with numbers only, the same way sculptor deals only with clay! But, if we want to interpret math results and use math in some other fields then we will have to keep track what we have counted, measured, keep track which objects are numbers, counts associated with! That would be called applied math. And, again, the numbers can represent count of many, many different things.
In developing and understanding a subject, axioms come late. Then in the formal presentations, they come early. - Rueben Hersh.
The view that mathematics is in essence derivations from axioms is backward. In fact, it's wrong. - Rueben Hersh
[ number concept, concept of a number, number, count, numbers, counts, integers, rational numbers, concept, math, mathematics ]
