What we learn vs. what we know
(last edited April 17, 2021)
The runaway train of machine (and deep) learning has, in my opinion, reached an alarming level — alarming in the most negative sense. It is one thing to be excited about a useful technology (say, one that is really good at finding useful patterns and associations in massive amounts of data), but to extrapolate from this unsound, and in many cases utterly false narratives, is dangerous to progress in science. Dangerous because we live in a culture that tech giants and celebrity-like figures now have more impact on the general discourse of science than Aristotle, Russell, and Copernicus ever had. It is now Elon Musk and the ‘God(Fathers)’ of deep learning that we must refer to if we want a rational understanding of the world we live in. In this post I hope to dispel a fallacy — that I never dreamed it would ever be need to be dispelled — namely that learning from experience (training) is how we come to ‘know’ or how our cognitive capacities are built up — in fact, I will argue the opposite, namely that a very small part of our knowledge is learned and that the vast majority of our cognitive apparatus is an incremental ‘process of either discovering a fact/truth, or being told of the discovery’.
Skills vs. Knowledge
Perhaps ML = AI is such an attractive hypothesis because the average person has the common (folk) belief that we know by learning. But in fact this is not scientifically true — at least not when learning is learning by seeing examples and trial and error every time optimizing by minimizing some error. There is a very ambiguous use of the phrase ‘I know’. It is a confusion between ‘skills’ and ‘knowledge’ and it is rooted in our everyday discourse where we say ‘I know x’ in both cases: in case x was a skill or x was a piece of knowledge. So some kid might say ‘I know how to ride a bicycle’ or ‘I know how to play acoustic guitar’, but here they are not referring to any piece of knowledge, but to a skill. It is certainly different from knowing that the circumference of a circle is 2πr. How are the two different? Well, ‘John’s knowing of how to ride a bicycle’ could be imperfect, basic, and certainly different from ‘Mary’s knowing of how to ride a bicycle’. But ‘John’s knowing that the circumference of a circle is 2πr’ cannot have degrees of knowledge, either John knows that fact or he doesn’t. Moreover, ‘John’s knowing that the circumference of a circle is 2πr’ cannot be different from ‘Mary’s knowing that the circumference of a circle is 2πr’. And yet another and very crucial difference between a skill and a piece of knowledge is this: John might, or might not, know how to ride a bicycle, and if he doesn’t then KnowHow(john, ride(bicycle)) would be false. But ‘the circumference of a circle is 2πr’ is true, whether John knows it, or not — in fact, whether we all know it or not (it could be a truth that we did not discover yet) and it is true also whether humanity existed or not!
The moral of this short introduction? What we learn we can learn differently. Conversely, what is true in-spite of us, and thus what we cannot know differently, cannot be (and is not!) learned, but we simply come to know it (either by instruction, or by investigation/deduction/research, etc.). The graphic below shows the difference between the two.
Thus, if x-LargerThan-y and y-LargerThan-z then x-LargerThan-z whether I experience it or not, and whether I know it or not. The relation LargerThan is transitive, end of story. You can learn how to play guitar differently from Sally, or drive a car differently from how Steve drives a car, but all of you either know, or don’t know, that ‘water boils at 100 degrees Celsius’. If you don’t drive a car ‘properly’ like Johnny does, then you are not wrong, just different. But if you don’t believe that ‘water boils at 100 degrees Celsius’ then you simply don’t know that fact. And if you believe that ‘water boils at 65 degrees Celsius’ well, let’s just say that you’re in the same category of those who still believe the earth is flat (Incidentally, I discovered recently that there is in fact a flat-earth society!)
If we learn skills, is all knowledge innate?
In a very important paper that I read a few years ago (entitled “Is Logic Innate”) the authors conclude based on empirical evidence and quite a bit of experimentation that, like language, much of our logical and reasoning apparatus is innate and thus not learned:
Evidence is presented showing that young children adhere to universal semantic principles that characterize adult linguistic competence across languages. Several a priori arguments are also offered in favour of logical nativism. These arguments show that logic, like Socratic virtue and like certain aspects of language, is not learned and cannot be taught — thus supporting a strong form of innateness. Crain, S. & Khlentzos,D. (2008) Biolinguistics 2.1: 024–056, 2008. Available here
Like in language, our reasoning faculty is not subject to discussion and thus not subject to being picked-up by us — by our experience in the world. Below I will simply prove that logical implication is one such example. It is valid in spite of us, and precisely because it confirms with the physical reality of the world we live in. Below is the truth table of logical implication (‘→’) (or material implication, or material consequence, or material conditional, etc.), which is a relation or a connective that is used of two statements p and q as such: p → q meaning p implies q (or, if p then q, or q is conditional on p).
According to the truth table, p → q is false only in one case, when p is true (T) and q is false (F), and is true in all the other cases. Intuitively, the semantics of implication says the following: you cannot infer falsity from truth, while all other situations are possible. At a first glance it seems like a contrived rule, one that some esoteric logician made up. How could S
S = if the moon is made of cheese then there are winged pink elephants that fly
be a true statement? Well, we should think of the implication as saying nothing we know so far (or there are no established facts that we know of) that ‘precludes the conditional’ S from being true. And that, is a sound argument. But let’s see how in fact this is a valid rule. Not only is it valid, it is not up to us to change it, and it is in fact something we discovered as a valid fact in our physical reality! We will do that using set theory. The diagram on the right-hand side of the figure above is the set-theoretic interpretation of material implication (here U is the ‘universe of discourse’ that contains all sets). Note that p → q in logic corresponds to saying the set P corresponding to p is a subset of the set Q corresponding to q (so the set of dogs is a subset of the set of animals, is equivalent to saying dog → animal, or if some ‘x is a dog’ then ‘x is an animal’, etc.)
Now let us look at the set-theoretic (actually physical!) interpretation of logical implication. The red circle is the situation where p and q are both true — in set terminology, we are talking about an object that is a member of the set P and and an object that is also a member of the set Q. Now this situation can in fact happen, as shown on the right hand side (we can have some x that is in P and in Q at the same time). The situation of the green circle can also happen: we could be in Q (q is true), but not in P (p is false). The blue circle is also a situation that can physically be materialized: we could be in a situation where we are not in P (p is false) and we are also not in Q (q is false). But what about the orange triangle? Can we have a place in the landscape on the right where we are in P but not in Q? That situation, cannot in fact happen, and that’s why this is the only situation where the material implication is false! Thus, what might at a first glance sound like an esoteric logical rule concocted by some logician, this in fact is a sound rule that corresponds to reality and it is not up to us to learn it or unlearn it. It is a fact of our (meta)physical reality — and a fact that some came to discover at some point — and the rest of us were then told/instructed to know this fact (unless there’s someone out there that came to know this fact by ‘experiencing logical implication’ !?).
What am I saying thus far? Most of our knowledge is not learned, it is in fact there. We come to know of it by instruction or by deduction or discovery. What we learn are skills and they are in fact a small part of our reality — skills that do not change any fact in life. Now there are many skills that we like to automate and that’s why it is certainly very useful to have a technology that can acquire certain skills by trial and error (aka, training) (falling off the bicycle several times and sounding terribly bad on the guitar for some period of time — or, forever in my case). But this addiction to thinking that we can have knowledge by trial and error is not only misleading — it is actually in conflict with everything that has been established in science since before antiquity. So calm down deep learners, and be humble. A statistical correlation machine cannot learn that
if Location(x, L) and PartOf(a, x) then Location(a, L)
A robot needs to know this basic rule: if it moves an object from location L1 to location L2, it has in fact moved every part contained in that object to location L2. This cannot be learned, because it is not a skill. This is a basic fact in the physical reality we live in, regardless of my experience and yours. And it is not our sensory devices that made us all sense that ‘the circumference of a circle is 2πr’ — that is a true fact that was arrived at (deductively) and then we were taught/told so.
The ML community (deep or otherwise) should tone down their rhetoric and be humble as to the limits of ML. The AI = ML fallacy should be dispelled because it hampers progress in AI. Discovering patterns and correlations in data is a useful technology, but it has a very small impact on AI and cognition. AI is a lot more than ‘data’ science and this is too important of a subject to remain in the hands of hackers.
