A little kid, who does not know what 'color' is, comes across a bag of colorful billiard balls. what would be his reaction? His childish curiosity drives him towards the bag. He is attracted by the appearance of the balls...I mean, the colors, though he doesn't know what it is....
Most children stop there. But, imagine, an extraordinary( hypothetical, if you feel so) child, who can proceed further. The next thing he would do is, look for similar balls (balls which look like each other) The child has a way to tell whether two balls look like each other or not, by visual inspection. (looking like each other in our language means same color).
Next, the child can divide the bag of balls in to groups of like balls. Every pair of balls within a group would look alike. Hence, each group can be represented by a single ball. If the child is given new balls, he can easily put them in to respective groups. Or, if it doesn't look like any of the group representatives, it makes a new group.
Now, he is close to defining color. The representatives of each group are not balls, they are colors. He can name each group at his will. This is precisely what man did, over generations. The names he gave were red blue green et al.
Now, a formal look at the procedure adopted by the child. An important comment to be made at this point is about the way the child decides if two balls are alike. If balls 1 and 2 are alike , and balls 2 and 3 are alike, inevitably, balls 1 and 3 will fall in the same group. Hence, they should look alike. Formally speaking, his definition of alike should be trasitive. By using 'two balls look alike', and not 'one looks like the other', I have already meant that the like is symmetric. A little more thought will convince you that the like needs to be reflexive as well, for successful classification. Thus, it gives a reason to the mathematical definition of equivalence relation. Formally, those groups arising out of the equivalence relation are called classes. Once the classes are made, a mathematician may do various things with his classes....order them etc.
This is how most of the seemingly undefinable terms like color, size, mass, charge, cardinality, etc are defined(?).
The purpose of this blog is to express my thought flow, which at the moment says, mathematics is a way of thinking, formalized.