Concrete to abstract-how mathematical ideas grow, Mathematics

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.

Posted Date: 4/24/2013 1:55:50 AM | Location : United States

Related Discussions:- Concrete to abstract-how mathematical ideas grow, Assignment Help, Ask Question on Concrete to abstract-how mathematical ideas grow, Get Answer, Expert's Help, Concrete to abstract-how mathematical ideas grow Discussions

Write discussion on Concrete to abstract-how mathematical ideas grow
Your posts are moderated
Related Questions
Tom is cutting a piece of wood to form a shelf. He cut the wood to 3.5 feet, but it is too long to fit in the bookshelf he is forming. He decides to cut 0.25 feet off the board. Ho

joey asked 30 randomly selected students if they drank milk, juice, or bottled water with their lunch. He found that 9 drank milk, 16 drank juice, and 5 drank bottled water. If the

Which of the subsequent numbers will yield a number larger than 23.4 while it is multiplied by 23.4? When multiplying through a number less than 1, you get a product in which i

(a) Determine the matrix that first rotates a two-dimensional vector 180° anticlockwise, and then per- forms a horizontal compression of the resulting vector by a factor 1/2 (leavi

Find out some solutions to y′′ - 9 y = 0 Solution  We can find some solutions here simply through inspection. We require functions whose second derivative is 9 times the

The first definition which we must cover is that of differential equation. A differential equation is any equation that comprises derivatives, either partial derivatives or ordinar

Calculate the introduction to Probability? Probability refers to the chance that an event will happen. Probability is presented as the ratio of the number of ways an event can

Consider the following interpolation problem: Find a quadratic polynomial p(x) such that p(x0) = y0 p’(x1) = y’1 , p(x2) = y2 where x0 is different from x2 and y0, y’1 , y2 a

Estimate the area between f ( x ) =x 3 - 5x 2 + 6 x + 5 and the x-axis by using n = 5 subintervals & all three cases above for the heights of each of the rectangle. Solution