Unstated Assumptions

We have covered a lot of concepts & skills in math. It is time to remind ourselves, that behind many mathematical statements, there are unstated assumptions. Otherwise, our understanding would not be correct. In fact, math knowledge grew by questioning some of these assumptions, leading to new ways of understanding the concepts. Let us see a few examples.

5 + 3 = 8. This seems to be a very simple truth. But is it true in the following cases?

1. We have a set of 5 apples & another set of 8 green fruits. When we put them together would we always get a total of 8 fruits? What if some of the 8 green fruits were apples which were classified both as apples & green fruits? Here is a case where both sets had common elements. The statement 5 + 3 = 8 is true only when both the sets are disjoint or have no common members. This issue also led many mathematicians to believe that arithmetic should start with Set Theory.

   1. There is another problem of the same kind which happens in daily life. If we mix 5 litres of milk and 3 litres of sugar, will we get 8 litres of sweetened milk? No because some of the sugar will dissolve in milk and occupy the molecular spaces between the milk molecules. Here again the sets are not disjoint. But physics tells you that in terms of weight 5 kgs of milk and 3 kgs of sugar will always give you 8 kgs of sweetened milk!

2. If we travel 5 kms by bus and 3 kms by cycle, then are we 8 kms from the starting point? It again depends on the direction of travel. The topic of integers and vectors arose from this issue.

   1. But what is clear is that the total distance traveled would be 8 kms.

   2. But the question “how far from the starting point?” has infinite answers. The least answer is 2 kms, The biggest answer is 8 kms. The actual answer can be anything between 2 & 8, depending on the direction of travel.

In school level mathematics we assume that the entities added are discrete from each other and do not overlap. In the travel problem we assume that the direction of travel could only be in a particular direction and its reverse.