Particular to General : When I say 'tail', what do you think of? Do you think of the tail of a horse, or of a monkey? Or do you think of the tail of your pet dog?
The tail of a particular animal has many features that are not part of the concept of 'tail'. For instance, my horse has a dark tail, two feet long. I could describe the thickness of its hair, its colour, at what angle it is inclined to the body, and so on.
But would this description fit the tail of any horse? Wouldn't some of these features need to be changed from horse to horse? So, if I want to apply this concept to all horses, I need to form an image of a tail which is not bound by the particular properties of my horse's tail.
Now, I notice that cows and dogs also have similar things attached to their bodies. So, I further generalise my concept of tail to include the tails of all animals.
So, while generalising from a particular case to cover more and more cases, we leave out some features of the specific example, and pick out what is common to the various examples. We abstract their common aspects and form a general concept.
Isn't this the same way we form the concept of a quadrilateral? This concept is the result of examining squares, rectangles, trapezia, etc., and picking out their common properties, namely, that they are all closed figures with four sides. So, we form a general concept of a closed four-sided figure, and call any such figure a quadrilateral.
You must have noticed by now that as we generalise ideas we are moving towards more and more abstraction.
You may like to try these exercises now.
E1) Write down an example each, related to natural numbers and fractions, to, show the movement from particular to general.
E2)) Is 'moving from particular to general, the same as 'moving from concrete to abstract'? Why?
And now let us consider another important aspect of mathematics.