Truth Criteria : Consider the following statements:
i) Peahens (i.e., female peacocks) lay eggs around September.
ii) Water boils at 100°C.
iii) 5 divides 15 without leaving any remainder.
iv) If you add two odd numbers, the result is always an even number.
If we want to see whether these statements are valid or not, how would we go about checking them? 'Will the same method work for checking all the statements?
Consider the first hypothesis. It can only be tested empirically. That is, we would have to observe peahens and find out when they lay eggs. If we observe a large number of peahens, and find that nearly all of them lay eggs around September, then we can say that peahens lay eggs around September. Note that this empirical rule would be accepted, even if there are rare instances that do not follow the rule.
By actual experimentation, you can check that the second statement is true, only under certain conditions. By heating water under pressure, it can be shown to boil at a temperature much above 100°C. (It is this principle that is used in the pressure cooker.)
The third statement is a mathematical statement. Can you prove that it is true by observation? You may take several examples of sets of 15 objects and divide them into 5 equal parts. But you need to prove it for all sets of 15 objects. So, how would you prove it? If you understand what 5, 15, division and remainder mean, you can prove it mathematically.
What I am saying is that in mathematics, truth is only a matter of consistency and logic. The proof of a mathematical statement consists of a series of logical arguments, applied according to certain accepted rules, definitions and assumptions.
Let's go further now. Consider Statement (iv). Can you prove it empirically? If so, how would you exhaust all the possible pairs of odd numbers? Until you have' checked all the possible odd number pairs, your hypothesis would not be accepted as a mathematical rule. You may verify it for a large number of cases, but then someone may ask if it is true for 35678947321987 and 10000420001293? So, unless the result is proved formally for a general pair of odd numbers, the result is not mathematically acceptable.
One way of proving the result is:
Any odd number can be written as 2n+1, where n is some whole number. So, take two odd numbers, 2n1+1 and 2n2+1, where nl and n2 are whole numbers. The sum of these numbers is (2n1+1) + (2n2+1)-= 2(n1+n2+1) = 2m, say, where m = n1+n2+1 is a whole number.
So, (2n1+1) + (2n2+1) = 2m, which is divisible by 2, and hence is an even number. Thus, the sum of two odd numbers is an even number. Q.E.D.!
Here we have made use of definitions (odd, even, whole numbers), previously drawn results (the sum of whole numbers is always a whole number) and logic. No observations. If someone does not know these definitions and previously drawn results, it is not possible for her or him to understand this proof.
This kind of logic, which uses known results, definitions and rules of inference to prove something, is called deductive logic. This kind of logic starts from a general statement or definition, which is accepted beyond doubt, and deduces the next step in the proof. For example, suppose we accept the statements
(A) "All men are mortal", and
(B) "Raghav is a man." Then, from (A) and (B) we are forced to deduce that "Raghav is mortal".
Another kind of logic used in mathematics is inductive logic. To try and understand what it means, let us look at a non-mathematical example first.
I see a dog and find that it has a tail. I see a second dog and find that this one also has a tail. This is true for the third and fourth dogs too, and so on, for a large number of dogs. So, I conclude that all dogs have a tail. This is like the proof for the statement about peahens laying eggs in September. This kind of logic only proves that the statement has a very high probability of being true. But it does not prove that we shall never find a dog without a tail, or we shall never find a peahen which might lay eggs in March.
But mathematicians need to use a form of logic that will prove a statement for all the cases. Therefore, they use a particular form of inductive logic. It is known as mathematical induction. Let us consider an example to see how it works.
Suppose we want to prove that the sum of the first n natural numbers is where n is any natural number, that is,n(n+1)/2, where n is any natural number, that is,
(i) First we prove that the statement is true for n = 1.
(ii) Then we prove that if the statement is assumed to be true for m, then it must be true for (m+1), where m is any natural number. Notice that here we are not saying that it is true for m or that it is true for (m+l). We are saying that if it is true for m, then it is true for (m+l) also.
Now, if both (i) and (ii) are satisfied, then we can say that it is true for 1, and whenever it's true for in, it is also true for m+1.
Now, since it is true for m = 1, it is true for m+l = 1 + 1 = 2.
Since it is true for 2, it is true for 2+1 = 3, and so on.
Since there is no limit on the value of m (as long as it is a natural number); we have proved the statement for all natural numbers.
This is the structure of mathematical induction.
Let us now apply it to prove that the sum of the first n natural numbers is
(i) For n=1, the statement
1= 1(1+1)/2 = 1×2/2, which is true.
Therefore, for n=1, the statement is true.
(ii) We now assume it to be true for n=m, where m is a natural number. That is, we assume that
1+2+3+...+m+(m+1)= m(m+1)/2+ (m+1)
= (m+1) (m+1)/2
Thus, 1+2+3+... +(m+1) = (m+1) (m+1)/2
That is, the statement is true for (m+1) as well.
So, we have shown that if it is true for m, it is true for (m+1) too.
Therefore, by mathematical induction, we conclude that
1+2+3+...+n= n(n+1)/2 for all natural numbers n.
This is how inductive logic is used to prove mathematical results.
E) Which kind of logic is used for proving Pythagoras's theorem?
E) From your experience of mathematics, give at least one example each of the use of inductive and deductive logic to prove mathematical statements.
Now consider the statement:
"The square of a prime number has at least 4 factors".
How would you prove or disprove it?
Since the statement is supposed to be true for all prime numbers, it must be true for any particular one also. Let us see if it is true for 7. 7 x 7 is 49, and the factors of 49 are 1, 7, and 49. Hence, the statement is false for 7. So, 7 is a counter-example, and shows that the statement is false.
Notice the difference between our attitude to proving and disproving statements by examples. When we were proving a statement, we wanted it shown to be true for all cases. But to disprove a statement, just one example is enough.
Why don't you try an exercise now?
E) Prove or disprove the statements that
i) the sum of the interior angles of a quadrilateral is 360°,
ii) the sum of the interior angles of a pentagon is 450°.
Let us now see what helps in making mathematical statements brief and clear.