Mathemagic

A graphic novel on the philosophy of Mathematics? A friend pointed out the existence of this unlikely book, titled 'Logicomix', and I just had to get it.

Based on the life and work of Bertrand Russell, the book views his mathematical pursuits as arising from events in his personal life. Orphaned at an early age, Russell turned to mathematics to find answers to the reality of existence. He became obsessed with the methods of creating mathematical proofs using logic, but as his understanding grew, Russell eventually became dismayed at the realization that much of mathematics rested on unproven axioms. This began a lifelong quest for the logical foundations of mathematics that would be in harmony with reality.

The book introduces a number of other eminent mathematicians and philosophers that were contemporaries of Russell and gives a layman's perspective into their complex works. Here's a few of the star cast of geniuses:

Cantor: Invented Set Theory and spent decades creating a mathematics of infinity. He was an outcast of the then mathematical community which viewed his work as a threat to the existing mathematical establishment. Was confined to an asylum and treated for severe depression for much of his life. He described the existence of orders of infinities, where one set of infinities was larger than another.

Hilbert: A polymath who led the movement to create a complete axiomatic foundation of mathematics. The pursuit, however, was later shown by Gödel to be impossible.

Russell: The protagonist of the the book, he showed the contradictions in Set Theory through self-referential propositions (Russell's paradox). Cantor's set theory, extended by others, postulates that any definable collection is a set. In layman's language, Russell's paradox can be presented as follows: A town has a single barber, who shaves those and only those men in town who do not shave themselves. Who shaves the barber? This seemingly well-defined set leads to a contradiction that destroys naive set theory. An even simpler statement that illustrates the contradiction "This statement is False". (If it is, then it's not and if it's not, then it is). He also gave a more than 300 page proof for 1+1=2.

Wittgenstein: A philosopher and student of Russell. Claimed to have solved all the problems of philosophy. His Tractatus Logico-Philosophicus is considered a masterpiece, though I have to admit I've neither understood its thrust nor made a serious attempt to do so.

Gödel: While the rest of the mathematical world had bought into Hilbert's idea of creating a complete system of axioms which could logically test the validity of any conceivable hypothesis, Gödel showed that this was an impossibility. His two Incompleteness Theorems, published when he was 25, show that for any logically consistent set of axioms that can prove arithmetic, a proposition can be created that can be shown to be unprovable. The ability to fully appreciate his work would require decades of training and high mathematical aptitude. In its absence, it is useful to consider what people who satisfied this criteria said of him. 

"Kurt Godel's achievement in modern logic is singular and monumental - indeed it is more than a monument, it is a landmark which will remain visible far in space and time. ... The subject of logic has certainly completely changed its nature and possibilities with Godel's achievement." John von Neumann

Like many mathematical geniuses, was mentally disturbed for most of his life, and died by starving himself when there was no one to taste his food, which he feared was poisoned.

And so the book subtitled 'An Epic Search for Truth', ends with a humbling realization that mathematics, and more generally logic, cannot answer every question that can be asked. But this realization is to be used with caution, as many specialized results are abused and exploited by those who do not understand it fully, to make assertions to satisfy their own agendas and beliefs. I do not presume to understand the complexity of many of the ideas listed above and consciously resist the temptation to extrapolate them here.

At some point in the future, I intend to read 'Gödel, Escher, Bach' to explore some more of the mind-numbing complexity of the relationship between logic and reality. 

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