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So when does Mathematics begin? Consider these: there are three apples, three papers, three bowls, three shoes, three (model) trees, three pens, three books, or three magnets, on a table. Notice that, we all agree that in the apple case and the book case, the apples are as many as those books. Generally speaking, no matter what these objects are, we all use the term "three" to say about the number of these object. This indicates, "three" is the critical word for describing the "quantity" about these cases. By this inspiration, we can only discuss the number "three", and ignore the objects on the table. That is, we can abstract the concrete objects in the real world, and come into the abstract world of pure numbers.
Quantity is composed of a number and an object, or a number and a unit. The quantity "Three balls", "Seventy centimeter" are examples of each case. By the word "centimeter" we mean the length about the standard ruler from the scale 0 to the scale 1, and then the predicate "seventy centimeter" presents seventy times of the length of a centimeter, and is about the length of our hands.
In mathematical world, we feel interested only in the part of numbers, while in physical world, a person might take more care about the choice of unit, for convenience of describing physical phenomenons.
From now on, our exploration is from quantity, concerning the real world, to numbers, by which I mean, intuitively, the beginning of the mathematical world.