Mathematical Reasoning

Let’s relate math with literature and start our journey towards showing the existence of mathematics in all aspects of life. ~Vansh Jatin Mehta

Mathematically acceptable Statements.A Statement is a sentence which is not an order or an exclamatory sentence and of course a statement is not a question.

For example.What is your name? Get out of the room.Hurrah!we won.These are not statements.In the context of mathematics statement is only accepted as a statement if it is either true or false in short if it is conclusive.There is no space for ambiguous statements to be accepted as statements in context of mathematics.

Now what do you mean by ambiguous statements?The word ‘ambiguous’ needs some explanation. There are two situations which make a statement ambiguous. The first situation is when we cannot decide if the statement is always true or always false.

For example, “Tomorrow is Sunday” is ambiguous, since enough of a context is not given to us to decide if the statement is true or false.

The second situation leading to ambiguity is when the statement is subjective, that is, it is true for some people and not true for others. For example, “Cats are intelligent” is ambiguous because some people believe this is true and others do not.Now since we are aware of ambiguous statements there must be unambiguous statements (obviously the mathematically acceptable Statements).

Now the next question arises is what are the tools to establish conclusion in unambiguous statements.The main logical tool used in establishing the conclusion of an unambiguous statement is deductive reasoning. To understand what detective reasoning is all about let us begin with the Puzzle for you to solve.Suppose you are given 4 books.

And your told that these books follow the rule. Front cover has name of subjects and back cover has the colour.” If the back cover is red it implies that the book is math”.What is the smallest number of books you need to turn over to check if the rule holds true.Of course you have the option of turning over all the books and checking.

But can you manage with turning over a few number of books?

Notice that the statement mentions that if the back cover of the book is red then the book is math. It does not mention that every math book should have its back cover red. It may or may not be true. The Rule also does not state that Non- math book must have a colour other than red only.It may or may not be true.So out of the four books given to us how many should be turn?

So do we need to turn ‘math’?No, whether there is red or other colour in the back it doesn’t matter as the rule will still hold true.

Also what about ‘green’? No , as it won’t matter if it is math or other subject on the other side as the rule will still hold true.

But you need to turn ‘red’ and ‘science’.If ‘red’ has other subject in the front cover the the rule has been broken .

Also, if ‘science’ has red in its back cover then the rule has been broken.

The kind of reasoning we have used to solve this puzzle is called deductive reasoning. It is called ‘deductive’ because we arrive at (i.e., deduce or infer) a result or a statement from a previously established statement using logic. For example, in the puzzle above, by a series of logical arguments we deduced that we need to turn over only ‘science’ and ‘red’.Deductive reasoning also helps us to conclude that a particular statement is false, because it is a special case of a more general statement that is known to be false.

For example, once we prove that the product of two even numbers is always even, we can immediately conclude (without computation) that 674839394 × 42637482 is even simply because 674839394 and 42637482 are even.

Deductive reasoning has been a part of human thinking since Adam and Eve.For example “iceberg in Antarctica will melt only if if the temperature touches 104 degree fahrenheit”. And “iceberg started melting in antarctica from today” This implies that today the temperature has touched 104 degrees fahrenheit.

And since we are human beings we always have misconceptions in our daily life. For example:If your Crush smiles at you you conclude she likes you.While it may be true that “the girl smiles at you implies she likes you” on the other hand, it may also be true that ” if she smiles at you, you look like a joker”.

Why don’t you examine some conclusions that you have arrived at in your day-to-day existence, and see if they are based on valid or faulty reasoning?