I just read on AskWoody.com that Martin Gardner, the longtime author of the Scientific American column “Mathematical Games”, has died at the age of 95.
I am a bit surprised at how it is affecting me.
As a kid, I loved his column. I took pride in solving the puzzles. The activity taught me focus and concentration. It taught me problem solving. It taught me to think.
But most of all, it gave me my first exposure to the notions of logical incompleteness and unequal infinities.
With the naivety of youth, I believed that the universe was black and white. Statements were either true or false, and essentially all knowledge (using a sufficiently large book) could be expressed in language.
Further, infinity was infinity. the biggest thing there is. Nothing bigger than that, despite the common schoolyard escalation of taunts:
“You have cooties.”
“Well, you have cooties plus one.”
“You have cooties plus two.”
“Oh, yeah? You have cooties times one hundred”, cleverly upping the stakes by switching from simple addition to the more powerful multiplication.
“Well, you have cooties times infinity!”
“Oh yeah? You have cooties times infinity-plus-one.”
… and so on.
Still, infinity came to connote – at least to me – a sense of maximality, the sum totality of everything, of the universe. Of The Universe.
Gödel’s Incompleteness Theorem states that no consistent system of axioms whose theorems can be listed by an “effective procedure” (essentially, a computer program) is capable of proving all facts about the natural numbers. For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.
This has profound implications for reductionist attempts – to which I was inherently attracted in my youth – to reduce all knowledge to a set of indisputable, logically consistent axioms and deduction mechanisms. Essentially, if you depend upon axiomatic logic and consistency, you are limited in what you can prove. There exist statements that are expressible within the given system, but whose truth or falsity are not provable using only the tools of the system.
This result undermines language as the sole medium of defining knowledge, since language will always produce statements that are outside the reach of provable truth or falsity. It casts doubt on the binary notion of truth, opening up the mathematical discipline of fuzzy logic.
It even strikes closer to home for me, contradicting something my father explicitly told me when I was a child, one of his core beliefs: “If you can’t describe what you know, then you do not know it.”
Gödel’s method of proof was his famous Diagonal Lemma, in which he starts with the logical system, its symbols and its constructs, and then proceeds to build an example of his elusive unprovable statement.
Using a similar diagonalization technique, Georg Cantor was able to demonstrate that the infinity that represents the natural numbers (1, 2, 3, 4…) was a “lesser” infinity than the infinity representing the real numbers (the numbers that represent measurable distances on a line), the former being a “countable” infinity and the latter an “uncountable” one. Further, he proved that there are ever-higher, unlimited levels of “greater” infinities, all of which are “uncountable”.
All this just blew me away. It fired my imagination. It opened my mind to an extraordinary universe, to universes, to multiverses. It inspired me to embrace the beauty of abstraction. It led me to study mathematics, first as an undergraduate, then as a graduate student, through to my doctorate degree.
Abstraction, modeling, language, truth, philosophy, knowledge, infinities, alternate possibilities, universes.
For me, it all started with Martin Gardner. My life would not be the same without the contribution that he made to it. I’m saddened at the news of his passing, but grateful for the door he opened for me.