19 August 2014

On the common belief that 1 + 1 = 2

School children are taught at an early age that 1 apple plus 1 apple equals 2 apples. They go on to learn that 1 orange plus 1 orange equals 2 oranges. Sometimes they painfully discover on their own that 1 caning plus 1 caning equals 2 canings. Eventually they conclude that 1 + 1 = 2, irrespective of what is placed immediately after the 1s and 2 or even if there is nothing after the 1s and 2. The belief that 1 + 1 = 2 is a universal truth is widespread, even among mathematicians. But is it true?

In primary school students are posed with problems of the following kind. If two apples cost Rs 20, how much would four apples cost? And they are taught that it can be solved in this fashion.

First you arrange the data as below:
2      20
4      ?

Then, if you cross multiply you find that ? (or what is in later years called the unknown variable) is equal to 40.

In primary school this seems to be a universal truth. But in secondary school the student comes in for a rude shock. He finds that there are problems that cannot be solved in this manner and in fact yield the wrong answer if he attempts to solve them this way.

Thus, if 2 workers take 20 hours to build a wall, then it is incorrect to conclude that 4 workers will take 40 hours to build the same wall. Before one can solve such a problem, one needs to decide in advance whether it is one of direct or inverse proportion.

A similar argument applies to the equation 1 + 1 = 2. Imagine that I eat one apple between 8 am and 9 am and one apple between 9 am and 10 am. How many apples do I eat between 8 am and 10 am. An operational way of arriving at an answer would be to place 1 on the number line for the first apple and then move 1 unit to the right for the second apple to find that between 8 am and 10 am I have eaten 2 apples.

Imagine, now, that I travel at a speed of 20 km/hr between 8 am and 9 am and at a speed of 40 km/hr between 9 am and 10 am. I cannot use the same operational method to conclude that between 8 am and 10 am I have travelled at a speed of 60 km/hr. Distance objects are amenable to the rule that 1 + 1 = 2 but velocity objects are not. We instinctively apply the 1 + 1 = 2 rule to apples and distances but not to velocities and think no more of it.

At the start of the 20th century A.N. Whitehead and Bertrand Russell set out to prove, among other things, that 1 = 1 = 2, from even more fundamental notions. The result was Principia Mathematica which was published in three volumes in 1910, 1912 and 1913. Ludwig Wittgenstein raised an immediate objection that the book used a circular argument in that by using sets it was using objects that were known to obey the 1 + 1 = 2 rule. But the prestige of Whitehead and Russell was such that the book was still acclaimed widely as having achieved a breakthrough.

Some years earlier Gottlob Frege had travelled a similar path. Just when his magnum opus The Fundamental Laws of Arithmetic was dusted and ready for publication he received a letter from Russell asking whether the set of all sets which are not members of themselves was a member of itself. Frege was gracious enough to conclude his second volume with this acknowledgement: "A scientist can hardly encounter anything more undesirable than to have the foundation collapse just as the work is finished. I was put in this position by a letter from Mr Bertrand Russell when the work was almost through the press."

Russell was to suffer a similar fate himself when Kurt Godel in 1931 published a paper titled "On Formally Undecidable Propositions in Principia Mathematica and Related Systems". Wikipedia paraphrases the paper thus: "Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory." More simply, you cannot prove that 1 + 1 = 2.

But formal proofs apart, there is another reason for doubting the universal truth of 1 + 1 = 2. Primary school mathematics is divided into two compartments: arithmetic and geometry. For centuries it was believed that the only geometry possible was Euclidean geometry in which parallel lines meet at infinity and the sum of the measures of the three angles of a triangle is 180 degrees. At the beginnng of the 19th century Gauss discovered the principles of non-Euclidean geometry but the idea was so outlandish that he did not publish. It was left to the Hungarian mathematician Janos Bolyai and the Russian mathematician Nikolai Ivanovich Lobachevsky to lay out the basic principles around 1830.

Even so it was not thought to be anything more than an academic exercise. Only after Einstein published the General Theory of Relativity in 1916 was it realised that the world was basically non-Euclidean and that space was not flat but curved and that the curvature varied from point to point.

If school geometry was only one of many possibilities then from symmetry considerations one should conclude that the school arithmetic in which 1 + 1 = 2 is also only one of many possible arithmetics. In fairness to Whitehead and Russell it must be pointed out that their work was published before Einstein's General Theory.

Thus one is led to the conclusion that if one cannot prove that 1 + 1 = 2 it is because 1 + 1 is not equal to 2. In deciding to apply the rule to distances but not velocities we demonstrate that we know this. But because so many of the objects of our everyday experience do follow the 1 + 1 = 2 rule we conclude that it is universally true. But this is no more than a reflection of the fact that as human beings we tend to reason inductively, accept those facts which fit our beliefs and thus reinforce them, and reject those facts which do not fit our beliefs.

What we can say at best therefore is that 1 + 1 is sometimes equal to 2 though perhaps we can even go so far as to say that 1 + 1 is often equal to 2.

Category: Mathematics

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