Note: “math and science” is used in this essay in the simplest rendering possible.
Kurt Godel’s incompleteness theorem is “A proof that Hilbert’s tenth problem cannot be solved. For any set of rules of inference, there are valid proofs not designated as valid by those rules.”1 In turn, Hilbert’s tenth problem was “to ‘establish once and for all the certitude of mathematical methods’ by finding a set of rules of inference sufficient for all valid proofs, and then proving those rules consistent by their own standards.” 2
Mathematician Kurt Godel was an interesting man. He was seemingly uninterested in the worship of mathematics and science and, contrary to the arrogance (Bad Philosophy) and clerical-thinking that runs amok in that world and is the embodiment of science with an “S”, Godel saw math and science as non-static things. Godel’s incompleteness theorem was an undertaking that could only have been done by someone who saw math as non-static; one who views math as static, or complete, would be incapable of the foresight, originality and skepticism involved in such a quest. Godel’s thinking was thoughtful and original, and, as thoughtful, original thinking usually goes, led to a thoughtful and original insight. Godel’s incompleteness theorem was simply that - an insight; one may do with an insight what they wish.
Bad Philosophy: The worship of math and science forbids all insight into them, as does the mindless worship of anything. (Worship itself is the admission that a thing is absolutely flawless, or, that it is profoundly flawed.) The insights provided by original and sometimes unpopular thinking by original and unpopular characters into math and science have been enlightening; one may notice that the concept of “falsifiability” is parroted endlessly, usually with a curious lack of credit to its original thinker. Math and science do not - can not - have an “expert” class; they can only have a “curious” and “non-curious” class, all other details being secondary, but not unimportant. The term “expert” is antithetical to the ability to undertake the skepticism necessary to achieve Godel’s incompleteness theorem - what a hole for your expertise when its foundations are questionable (meaning that they can be questioned). “Science,” as Feynman once put it, “is the belief in the ignorance of experts.” Who more than Godel showcased this belief when undertaking the incompleteness theorem?
I know that there are those who will get upset that a metaphor from “something like” math is used. That is fine. They will say that “borrowing metaphors from the technical sciences could be a dangerous practice.”3 This prevailing antiwisdom-pseudoprofundity (Bad Philosophy) is the product of years of Completeness as philosophy - absolute and utter rightness as comes to the foundations of math and science, or the assumption that the foundation is good because it “works”. Those same “thinkers” sneeze “cocktail party philosopher” at every question, highlighting the inability of math and science to question or even look inward and downward; seemingly admitting that the most insightful, original, and deep questions are only discussed at cocktail parties? - let them fit and snort about whatever they desire. Yes, I am sure that it “could be a dangerous practice.” Thank you for this profound revelation bestowed upon you in what is sure to be a very serious place.
The skeptics of Godel say “we cannot eschew all agreed-upon foundations because of Godel”, meaning that, in the memory of Roger Scruton, there are certain questions that just shouldn’t be asked. Those who say such a thing or see it in that way - as an attempt at saying math is “no-good”- are wrong. That is not the insight provided by the incompleteness theorem or Godel. The insight gained is that of acknowledgment of math as a thing, a non-static, ever changing thing with foundations - foundations which could be flawed, not wrong. If a wooden-beam in a beautiful cathedral has a splinter, that does not mean that the cathedral doesn’t exist or isn’t beautiful, instead, it simply means that the cathedral was built.
There is a certain genuine respect for math and science necessary to think like Kurt Godel. Much to the contrary of the current dogma of “believing” (choose your word) science and math, Kurt Godel knew that anything worth the title of math and science could, and should, hold its own. Kurt Godel was in fact treating math with the respect it deserved - seemingly to the contrary of many others. There is a good philosophy in letting the pursuit of math and science be free; as we have seen in philosophies echoed in Feynman’s proclamation about physics being “boring” if it just keeps getting deeper, the “worshipers” don’t want any good philosophy:
“I think there will certainly not be novelty, say for a thousand years. This thing cannot keep going on so that we are always going to discover more and more new laws. If we do, it will become boring that there are so many levels one underneath the other… We are very lucky to live in an age in which we are still making discoveries. It is like the discovery of America - you only discover it once.”
-Richard Feynman 4
The above Feynman quote is in fact the opposite of good philosophy and one with that philosophy would be incapable of seeing or achieving the incompleteness theorem. Those who oppose (or misunderstand) Godel’s finding embody this; Pascal’s Wager as relates to their approach to math and science. I will refer to this as “Feynman’s Wager” - math and science should only go so deep, and their foundations, even if flawed, work. Math and science are, or are not. Let us weigh the gain of math and science being, and let us weigh the loss of math science not being (perfect or complete). If you gain, you gain all of math and science., if you lose, you lose nothing (but philosophical errors). Wager, then, that math and science are. There is an infinity to gain (through Bad Philosophy) and nothing (but small philosophical matters) to lose.
Kurt Godel embodied a good philosophy - his curiosity about the afterlife, the human soul, the U.S. Constitution (see Godel’s Loophole) and his incompleteness theorem all being echoes of that good philosophy. Why are they echoes? They show a desire to understand things very deeply - not just as deep as numbers or experiment seek to understand. Godel embodied “good philosophy” - an interest in the deepest understanding possible.
David Deutsch’s The Fabric of Reality, pg. 256
David Deutsch’s The Fabric of Reality, pg. 256
James Gleick’s Genius, pg. 430
The Character of Physical Law (1965) via David Deutsch’s The Beginning of Infinity, pg.444