Concrete to Abstract : Mathematics, like all human knowledge, grows out of our concrete experiences. Let us take the example of three-dimensional shapes. Think about how you came to understand the concept of "roundness" and of a sphere. Was your mental process something like the following?
We see all sorts of objects around us. While dealing with them, we notice that some of these things, like a ball, an orange, a water melon, a 'laddu', have the same kind of regularity, namely, a roundness. And so, the notion of 'roundness' gradually develops in our mind. We can separate the objects that are round from those that aren't. We also realise that the property of roundness, common to all the round objects, has nothing to do with the other specific attributes of these objects, like the substance they are made of, their size, or their colour. We gradually separate the idea of 'roundness' from the many concrete things it is abstracted from. On the basis of the essential property of 'roundness', we develop the concept of a sphere. Once we have formed this concept, we don't need to think of a particular round object when we're talking of a sphere. We have successfully abstracted the concept from our concrete experiences.
In a similar way, we learn to abstract the concept of 'redness', say. But there's a major difference between this concept, and mathematical concepts. Firstly, every mathematical concept gives rise to more mathematical concepts. For example, related to the concept of a sphere we generate the concepts of radius, centre, surface area and volume of a sphere.
Secondly, we can think of various purely abstract and formal relationships between the related concepts. For instance, examine the relationship between a sphere and its volume. Irrespective of the size of a sphere or the material it is made of, the relationship is the same. The volume of a sphere depends on its radius in a certain way, regardless of how big or small the sphere is.
Thus, not only can we abstract a mathematical idea from concrete instances, we can also generate more related abstract ideas and study relationships between them in an abstract manner. These abstract mathematical ideas exist in our minds, independent of our concrete experiences that they grew out of. They can generate many more related abstract concepts and relationships amongst themselves. The edifice of ideas and relationships keeps growing, making our world of abstractions larger and larger.
You may like to think of another example of this aspect of the nature of mathematics.
E8) Would you say that the number system developed in this way? If so, how? Let us now consider another way in which mathematics grows.
This is closely related to what we have just been discussing.