The years 1715 to 1792 are known in many countries as either “The age of enlightenment” or “The age of reason”. This is for the very simple reason that in the previous 100 years the Western civilised world had undergone a series of momentous scientific and geographical discoveries leaving the average intellectual no longer obliged to turn to the Church to say whether or not their ideas or theories were in line with Christian teaching because the myriad of Christian denominations which had sprung up since the reformation, approximately between 1511 and 1567, meant that the authority once automatically vested in the church in all matters of scientific and sociological morals, ethics and facts was considered no more likely to be right than the theories of political philosophers, mathematicians or discoverers.
So a whole century of changes in every walk of life culminated towards its closing years in the French revolution and abolition of hereditary privilege in France, financial and trading freedom in Italy, Spain and England and writers of all types in every country no longer felt obliged to seek ecclesiastical permission to be published. Also, the establishment of independence in North America and the founding of penal colonies in Australasia were the foundation stones of the English speaking world as we know it today.
This short proof concerns one example of how freedom of intellectual enquiry gave rise to perhaps the most intriguing puzzle, problem and obvious but unprovable hypothesis in the history of maths. Christian Goldbach, a keen amateur mathematician – yes in those heady days of original thought even mathematicians had original ideas, not allowed in schools nowadays – wrote to a friend of his in Switzerland, Leonhardt Euler, one of the most distinguished mathematicians in the history of the discipline. Christian suggested to his friend that all even numbers were the sum of two prime numbers. He allowed for only one odd number to be the sum of two primes, ie 3, as it was 1 + 2 . But He could find no even number that was not the sum of two primes.
The only problem was that,while it seemed obviously true, and nobody could, or at least has as yet, disproved it, he could not formulate an acceptable proof of his conjecture within the accepted rules of traditional mathematics. A large prize was deposited in a Swiss bank which was to be awarded to the first person to come up with such a proof. Now this was some 270 years ago, but still nobody has managed to formulate an acceptable proof. The prize money is still earning compound interest at 5 % and by now would make the winner one of the richest academics in the world. The most powerful computers in today’s world cannot disprove Goldbach so why can the conjecture not be proven?
Well, personally, I have always believed it is simply because it is too obvious. Can one actually prove, for instance, that 1 plus 1 equals 2. Not by any other method than saying that 1 is what we call a single unit of something, 2 is 2 such units and so on. But this is no more than giving a definition of, or naming the noun which corresponds to, a mathematical symbol or tool. Well, surely, that is all that Goldbach’s conjecture does. He simply SAID IT THE WRONG WAY ROUND. What he should have said is “an even number is any number that is the sum of two primes.”( 2 ,of course is sum of 1 +1). That is a perfectly valid and true definition of an even number. There are others, such as “an even number is any number which can be divided exactly by any other number higher than one.” But the important thing about my definition of an even number is that by its linguistic composition it obviates the necessity to take Goldbach’s conjecture any further, in order to prove it, than to state what an even number is. Doing this proves both the conjecture and satisfies the mathematical logic inherent in proving all theorems; ie using nothing more than the numerical value and meaning of a number’s name to make a mathematical point.