Perspectives on Truth and Certitude

February 3, 2013 at 7:39pm

In a somewhat hilarious(ly awesome) physics class moment last week, my teacher dropped the Socratic Method on us, which culminated in a statement suggesting that we knew a particular fact only because someone told us and we had believed them. This assessment was, unfortunately, probably true. So what do we know about what we believe to be truths? Let's peruse my collection of saved quotations. (Cue new css blockquote style!)

This is a koala.

Falsifiability

"In the human brain, there is gullibility. How gullible are you? Is your gullibility located in some "gullibility center" in your brain? Could a neurosurgeon reach in and perform some delicate operation to lower your gullibility, otherwise leaving you alone? If you believe this, you are pretty gullible, and should perhaps consider such an operation."

- Douglas Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid

GEB is one of my absolute favorite works, and I learn more from it every time I attempt to read it. It means a lot more to me now that I've become more experienced in computer science and relevant symbolic/cognitive/logical/mathematical thinking skills, than it did in freshman year when I first read it. Anyway, how do we verify that the things other people tell us are true? Interestingly enough, the theory of empiricism dictates that knowledge can only be acquired through the senses. John Locke put forth the theory of the tabula rasa, or blank slate, on which human experiences and direct sensory impressions inscribe and shape the understanding. In other words, seeing (and touching, hearing, tasting, etc.) is believing. Of course, I've never personally read Locke's An Essay Concerning Human Understanding, so how do I know what he thinks? Because the internet told me? Perhaps one of the internet's most important functions besides delivering cat videos is the unprecedented rate of information exchange. Even if I never go to Australia in my life, I can see pictures of koalas on the internet and therefore know they exist. But if I subscribe to a strict necessity of proof, should I question the existence of koalas? Nevertheless, I'm at least reasonably sure that koalas exist. Maybe I should read some Locke and visit Australia, you know, just in case.

Confirmation Bias

"The human understanding is no dry light, but receives an infusion from the will and affections; whence proceed sciences which may be called "sciences as one would." For what a man had rather were true he more readily believes. Therefore he rejects difficult things from impatience of research; sober things, because they narrow hope; the deeper things of nature, from superstition; the light of experience, from arrogance and pride, lest his mind should seem to be occupied with things mean and transitory; things not commonly believed, out of deference to the opinion of the vulgar. Numberless, in short, are the ways, and sometimes imperceptible, in which the affections color and infect the understanding."

- Sir Francis Bacon, Novum Organum

Inherent cognitive biases exist in the human understanding. Confirmation bias is the notion that once we believe something is true, we will go out of our way to reinforce our belief, even if we ignore other evidence to the contrary. I experience this sometimes in physics class when we do experiments. A preconceived notion of how physics works, even if it's not necessarily wrong, will lead to biasing lab results toward a "correct" outcome, even though that's bad science. Dr. House would probably counter confirmation bias with Occam's Razor, or the belief that the simplest explanation that assumes the least is most likely the true explanation. I feel like that's a pretty good heuristic, of course not an absolute rule. By eliminating things assumed for the purpose of aligning data with one belief or another, we arrive at more objective decisions. In early 2012, a team of scientists firing neutrinos from CERN measured a faster-than-light neutrino and the scientific community exploded with debate. The idea was entertained, but treated with skepticism as further experimentation and the discovery of a faulty optic cable debunked the experimental FTL results. Of course we shouldn't dismiss new theories that don't comply with established ones, even those as sacrosanct as relativity, but we should also adhere to Bacon's belief in good experimentation, a healthy dose of questioning, and the scientific method to verify our results.

Certainty in Logic

"Admittedly, the present state of affairs where we run up against the paradoxes is intolerable. Just think, the definitions and deductive methods which everyone learns, teaches, and uses in mathematics, the paragon of truth and certitude, lead to absurdities! If mathematical thinking is defective, where are we to find truth and certitude?"

- David Hilbert, On the Infinite

David Hilbert wrote the words above in response to the development of a theory of infinite cardinality by Georg Cantor from his work developing set theory. In essence, Cantor introduced distinctions between countably infinite sets (such as integers, \(\{..., -1, 0, 1, 2, ...\}\)) and uncountably infinite sets (all real numbers, \(\{...,0.000...1, 0.000...2, ..., 0.99999, 0.9999999...,...\}\). There are infinite integers; there are also infinite real numbers. However, there are a lot more real numbers than integers. (What's the smallest number you can think of greater than 0?) This recalls Zeno's Dichotomy paradox where to take a step, one must traverse an infinitely uncountable distance. Before you can go half way, you have to go a fourth of the way, and before that an eight of the way, and so forth. (The infinite cardinality notion features heavily in John Green's beautiful novel The Fault in Our Stars, which I highly recommend.) Cantor's work was heavily criticized, but from our modern perspective, set theory is pretty super awesome. (In the same article, a couple of sentences later, Hilbert also famously said, "No one shall drive us out of the paradise which Cantor has created for us.") So where do we find certitude in mathematics? GEB introduced me to Gödel's Incompleteness Theorems, which prove that mathematics cannot prove itself consistent, nor can any formal system. In essence, we know that nothing has disproved the consistency of arithmetic so far, and we must take on faith that nothing ever will, because we have to assume that things in arithmetic are true but cannot be proven as such. The consequences of an inconsistency discovery on the life of a young mathematician is explored in the amazing short story "Division By Zero" by Ted Chiang. As if the Incompleteness Theorems weren't enough, Tarski's Undefinability Theorem proves that arithmetic truth cannot be proven by arithmetic. Anyway, similarly to the Incompleteness Theorems, Alan Turing left his mark as a father of computing when he introduced the Halting Problem, demonstrating that it is undecidable whether or not a program will halt or not. (I previously wrote about his work on undecidability.) Even in a discipline like mathematics that strives to represent truth through symbolism, there is so much that is unknowable.

Can we really know for sure that anything is true? Maybe, maybe not. However, we can get close enough for knowledge still to be fulfilling. Perhaps by believing in the truth value of certain things, we imbue them with truth.