When I was in high school (back in 2009) I used to think a lot in terms of “right” and “wrong”, “real” and “fake”, and “true” and “false”. Being 18 years old, I was certain I had discovered many truths about the world. Like many other people, young and old, I used to think science was “right” and religion was “wrong”. Fortunately, my education included a course in Epistemology, the branch of philosophy concerned with knowledge, i.e. how we acquire it, what it is and how to justify it. Despite the fact that it was not given by an expert on the topic, the course tackled my beliefs just enough for me to challenge all of them. I questioned the way that I had acquired all my “knowledge” and suddenly I realized that most of it was uncertain (and I say “most of it” because I quickly “concluded” that my mathematical knowledge was certain: I could redo the “proofs”, but more on that later). Almost all that I knew had been said to me by another person, i.e. a teacher, a parent, a friend or an acquaintance. Even if I accepted that they meant to tell me the truth, they didn’t know most of those things by experience. Even worse, throughout the years I observed that many people learn something by experience and generalize it fallaciously to other situations. Also, many of the things people “knew” were also taught to them by others. The uncertainty was so big that I decided to “start from zero” and ask myself “what can I know with 100% certainty?” Suddenly, I found myself questioning everything! Even my own reality! How do I know that I am not in a computer simulation? How do I know that I am not dreaming? How do I know that the sun will rise tomorrow if a flip of a coin does not always get the same result? i.e. if I see a series of six heads, that doesn’t mean that the seventh toss will be heads again. How do I know that the sun doesn’t behave like that; but just with a probability different than 1/2? Why should I trust more on people who say it will rise than the people that say “the end is coming”? I realized that I didn’t know how to approach the problem of answering those questions confidently, so I decided to do some research about it: to see if somebody else had come up with some kind of “mathematical proof” that I could follow so I could be certain of what I “knew”.
It turned out that philosophers had already asked those very same questions. Renè Descartes made the dreaming question I did above. David Hume found the “problem of induction” (the coin and sun question I’ve done), and there had been modern versions of the “computer simulation” scenario like the “brain in a vat” idea or even the movie The Matrix. Most of the information I found on the subject, treated those ideas under the label of “skepticism” and the reactions to it were varied but never fully satisfactory for me.
As I understood it, people in philosophy tried to defeat the “problem of induction” and other “skeptic problems” using arguments. An example of this is the “here is a hand” argument. However, (at least for me) none of the arguments were strong enough to prove the falsehood of the skeptical ideas. Other people tried to change the definition of knowledge. You can see some of those approaches in these video series by Wireless Philosophy. Nonetheless, the more I did research on the subject, the more I found that nobody had done a good proof against most skepticism. The only group of sentences that I found compelling was the beginning of Renè Descartes’ famous argument “Cogito ergo sum” which I read directly from his “Discourse on the Method“:
“I decided to pretend that everything that had ever entered my mind was no truer than the illusions of my dreams, […]. But no sooner had I embarked on this project than I noticed that while I was trying in this way to think everything to be false it had to be the case that I, who was thinking this, was something. And observing that this truth ‘I am thinking, therefore I exist’ was so firm and sure that not even the most extravagant suppositions of the sceptics(sic) could shake it, I decided that I could accept it without scruple as the first principle of the philosophy I was seeking.” – Renè Descartes
I liked the fact that the approach I took was similar to that of great minds like Renè Descartes (and David Hume). However, I did not like the parts that followed after his discovery of “this truth”. You see, Descartes made some arguments from his “cogito ergo sum“ that I could not accept as easily (check the source above for details). For me, his arguments felt like a mind rushing desperately to find certainty and God, instead of a mind rigorously trying to find the next “certainty”. Moreover, some philosophers have argued that Descartes’ “I am thinking, therefore I exist” presupposes a distinction between the self (“I”) and other things. Seeing that philosophers had done all of this, I considered studying philosophy. However, I soon got tired of those endless debates that objected everything but were never fully convincing. That took me to the next thing I felt to be certain: mathematics!
Albeit there were some mathematical things that I had not proven in my life, I had “proved” many, and those proofs were convincing for me. There was truth in mathematics! I was certain (back then) of it! So, based on this, and apart from other reasons (like the fact that I had always enjoyed math), I decided to study Mathematics as my major.
While studying mathematics I quickly felt that mathematicians had done a lot more than philosophers in the matters of truth and knowledge. I learned their methods and appreciated their power: the algebraic tricks, the visualization tricks, the standard tricks, some modeling tricks, the probability tricks, the trick of seeing something as another thing more manipulable to solve your current problem, the categorification of concepts, the codification of concepts, the formalization of concepts, and many more methods.
It didn’t pass long before I was introduced to Mathematical Logic and Gödel’s “Incompleteness Theorems”. I quickly fell in love with the topic. It turned out Mathematicians had already studied (with their super cool and powerful methods) the very thing that I was interested in learning!: whether Mathematics itself was a source of truth (not exactly what the theorems say, but mathematical logic was just the right area to study the topic). The subject was so well studied that the period of time when it was researched (and the consequences it brought to the philosophy of mathematics) had a somber name!: “the foundational crisis of mathematics“.
Without getting too technical, I delved more and more into mathematical logic and the foundations of mathematics, always searching for what the “experts” had to say about truth (and proofs) in mathematics. I even subscribed to the Foundations of Mathematics mailing list; I joined the HoTT Café Google group and the Homotopy Type Theory Google group; I participated in various workshops and wormshops; I read the n-Category Café blog; I studied propositional logic, 1st order predicate logic, forcing, category theory, formal systems, the Zermelo-Fraenkel-Choice (ZFC) system, the Martin-Löf Type Theory (MLTT) System, the Homotopy-Type Theory (HoTT) System, the Elementary Theory of the Category of Sets (ETCS) System, I even did my (170 pages) Undergraduate Thesis describing some systems that serve as foundations of mathematics… But in the end, the concept of “truth” (and proof) in mathematics always seemed to end in a philosophical debate: it depends on whether if you are a formalist, platonist, empiricist, logicist, conventionalist or whichever philosophical school you choose.
Moreover, the concept of (formal) proof of a theorem depended on the selection of an (axiom) system, and the choice of that system is done by the working mathematician! The systems are agreed by the community! Moreover, in practice, many mathematicians just solve problems based on a finite selection of “well-known facts” (alleged “theorems”) and “(fair) hypothesis”. As a consequence of this, not even mathematics is certain! It changes with time! It changes depending on the community using it! It appears to be just a human construct!
Ok, what do we have so far? We have seen the persistence of philosophical skepticism. It is so strong that it even messes with mathematical certainty and its objectivity. But those are just philosophical ideas. In practice, mathematics is useful to predict reality, right? In practice, the general theory of relativity and other scientific theories surely are the pinnacle of human advancements, right? In practice, the evolution of technology is evidence that we actually know something about the world, right?
If you study just a little how society works, you’ll see that the assertions in the last paragraph, even in practice, do not hold that strongly. For the past year, I have been following closely how research is done. Sure! Mathematics is a fantastic way to communicate our ideas, but the success of science and technology seems to be more about plain and simple “trial and error” and many “coincidences”. The most current example that comes to my mind is the development of Deep Learning: people wanted computers to “learn”, people had many educated guesses about how to do it, people used the tools at their disposal and finally, it worked! A deep neural network was beating every other approach to machine learning. Nevertheless, the experts on the topic usually assert that they do not know how (large) “neural networks” work exactly (see first lines of this research institute).
The behavior of science is very well explained in Thomas S. Kuhn’s “The Structure of Scientific Revolutions“. Nowadays people are applying the methodology of science to almost everything! They are constraining systems, proposing models, conducting experiments, refining models, refuting hypothesis, opposing theories, creating new things, and it keeps working!
The purpose of this first blog post was two-fold. The first was to show my readers that uncertainty arises everywhere. The second was to uplift that first conclusion by stating that human “knowledge” does not need to be 100% certain in order for us to attain great things. The key to the advancement of science and technology is collaboration and constant educated-guesses…