Mathematical Infalibility

What a Naughty Boy

Is there anything more beautiful than a well-crafted mathematical proof? If you answer yes to this question, you are indeed a very sad and troubled soul. 

On occasion, I have posted with regard to mathematical topics and concepts. My own level of comprehension of the subject is limited, although I do have an honest interest. My peak of mathematical knowledge occurred during my first year of University when I undertook and passed a foundation course in calculus. Recently, I was reorganising my study and came across my old notes on the course. A wave of nostalgia overcame me, and I decided to flick through the pages full of calculus notation. I confess none of what I had written was intelligible to me. I had understood the mathematics at the time, but after the passage of 45 years, the symbols and the concepts were complete gibberish. I digress.....  

At the beginning of the 20th century, two eminent intellectuals came together to formulate a logical footing for mathematics: Bertrand Russell (for it was he) was primarily a logician, while his associate, Alfred North Whitehead, was an established academic and mathematician.

There was a time, sometime in the 19th century when haughty mathematicians proposed that mathematics was applicable to all human endeavour. But not only to endeavour but also to the inner workings of the human psyche- a bold thesis indeed. It was honestly conceived that mathematical theory had a role in the understanding of music (beyond harmony), theology, philosophy, ethics, and, god forbid, meteorology. Now we know better, apart from meteorology.    

After 4,500 years of development, mathematics evolved into a myriad of systems. Here are just a few, in no particular order: arithmetic; geometry; calculus; set theory; non-Euclidean geometry and algebra. These represent the most important branches of mathematics, but there are many others. Indeed, today, founding a new mathematical system remains the province of math. PhD students. These systems, once acknowledged by doctoral review professors, are promptly forgotten. It needs to be admitted that most of what we call mathematics has no discernible practical value but nevertheless remains an intellectual monument formulated by very clever people and only understood by a very select group of very clever people.    

Our Heroes (c1900) were perplexed that the 'so-called edifice' of mathematics was but a ramshackle affair built up over centuries upon shifting sands. Each 'advance' had been erected upon systems where the premises were taken as unequivocally true. For how long could this continue before it was discovered that a core tenet of a pivotal system was found to be false. In such a scenario, the whole of mathematics subsequent to the error would come tumbling down. At a stroke, the life's work of sages, current and past, would lie crushed to gather dust and scorn of the ages.

My reader may look askance at the previous paragraph and gape with wonder: surely we have learned that of all the sciences, mathematics is the only subject, along with logical deduction, that provides the means for the generation of true and absolute knowledge. Certainly, this is the case if all the premises in the line of mathematical deduction remain solid and true, that said, if an error is introduced into the chain of reasoning, then what follows is mere ferret shit on a stick. They decided that the best approach was to utilise the principles embodied in formal logic. From a given set of sparse logical axioms that are irrefutable, it should be possible to establish all mathematics on sound principles. If the preceding premises are true, and by maintaining logical rigour, what follows should also be true and irrefutable. A noble course/cause, no doubt. 

It was with this problem in mind that Russell and the other fella decided to undertake the daunting intellectual process of providing a firm base from which to build all of mathematics. Twas a bold endeavour, nay adventure. They were about to embark on an intellectual journey full of hardship and drama, both personal and intellectual. Initially, it was thought that the project would take a year; however, the task would take a decade to attain fruition. And even then, they had doubts about whether the work was actually complete. What followed became the 'Principia Mathematica' eventually published in three volumes (1910, 1912 and 1913). I direct my readers to look up said tomes and inwardly digest. And therein lies the problem. Even folk well-versed in logical nomenclature will struggle to follow the reasoning of these two great men. When it came to publishing this seminal work, Russell et al. quickly found out that no publisher was forthcoming. In fairness, the books were never going to be best sellers. Thus, Russell and Whitehead had no recourse but to self-publish. It has been hypothesised that at the time of release, only six individuals read the three volumes from beginning to end; it was never going to be an easy read. So now, after such immense intellectual attainment, the authors could bathe in the self-satisfaction that can only come from pure cerebral achievement- and there was also the adulation from fellow savants. Let's not forget that within the sacrosanct pages of the first volume of  Principia, it had been proved absolutely, nay conclusively, that 1 + 1 = 2; it only took 362 pages. And then, along came the German mathematician Kurt Godel.

In 1930, Godel published his 'Incompleteness Theorem'. In essence, Godel completely undermined the logical reasoning used by Russell and North Whitehead, and Russell was plunged into a deep depression. It seemed his years of concentrated work had been in vain. Godel's theorem is highly technical, complex and specialised and completely outside my understanding. However, in layman's terms, it appears that there can be no axiomatic form of arithmetic that achieves both consistency and completeness. Even the so-far infallible logic inherent in the axiomatic-deductive method proved to be flawed. Did 1 + 1 really = 2? The genius, Ludwig Wittgenstein, suggested a return to commonsense reality.  But most mathematicians thought that this was going too far.

O, the calamity/humanity! Mathematics has never really recovered from this 'basic truth' - go ask the man in the street.