Over-Generalization and its Paradox

How do we define a point? Because I’ve never seen one, have you? It’s certainly not that drop of ink you put on a paper with a point-ed pen. If you look closer, it’s huge! A swarm of bacteria might be throwing a party on it. In fact, if you can see it, then it’s not a point.

A point is an abstraction. It doesn’t exist in reality; and yet, it’s a surprisingly simple concept to imagine. In fact, you need to infinitely zoom in to see a point; and to do that in time, it will literally take you forever. How can we speak of points and infinities without ever being able to point them out? It must be something so fundamental in how our mind works that we do it so effortlessly.

Infinity can be defined as simply counting forever: 1, 2, 3, 4, … , infinity? How about infinity+1? Nope. That’s still the same ‘countable’ infinity. Infinity is not a number. It’s an imagined limit of a repetitive process. It’s a combination of repetition in time (1, 2, 3…) and timelessness (forever). To create infinity, we use time to count and then transcend time with a hypothetical end.

The same thing happens with a point; but in space. To imagine a point, we infinitely (and repetitively) zoom into space until we finally reach the ‘bottom’ where we can finally see a single point. A point transcends space just like infinity transcends time. And they both do it by combining our knowledge of an exact recursive process (1+1+1…) with an assumed conclusion: that if this process is done forever until the end of time, we can reach a point or an infinity.

This process of taking limits of repetitions is very useful for creating abstract objects that are beyond our physical reality. In fact, with recursions, we imagine numbers, with which we imagine equations, with which we construct models and theories that transform an otherwise random world into one that follows universally repetitive patterns (e.g. scientific laws). Is repetition purely a construct of the mind or is it a fundamental building block of reality?

Our experience is basically a compilation of repetitive impulses of waves: sound waves vibrate our ears, light waves impact our retina, and touch sends repetitive neural firings to our brain. The brain takes those waves and uses its own repetitive neural firings and vibrations in a complex network where it builds more and more abstract concepts with simpler cycles. It basically compresses the infinite sources of unsynchronized oscillations into simple repetitive patterns; losing information, but gaining understanding.

Intelligence is the ability to distill the seemingly random set of observations to simple patterns that can be used to recreate the reality we experience. These patterns are then used to predict things we haven’t seen yet. The distillation part is deductive and is accompanied by a loss of information. The recreation part is inductive and is usually done by repetition to build the patterns. How much can we compress the reality we observe while still honoring it? And how much of the reality can we reconstruct with the patterns we deduce?

The only way we can answer these questions is by trial and error. The ratio of compressed patterns to reconstructed observations is somehow related to how much we understand our reality. We want our patterns to be as simple as possible but we also want them to apply to as many observations as possible.

Generalization is a measure of how much of our reality we can reconstruct with those patterns (or algorithms). The more a pattern can generalize, the more we are confident of its veracity as a source of our reality. Ideally, a unifying pattern of everything will generalize over all possible observations of reality. This is where excessive hope turns into inevitable fallacy. Reality is not made of points or infinities.

While the ability to generalize might be the defining factor of intelligence, it is also the source of ignorance. Our excessive hope that the patterns we know can be used to repetitively recreate a reality beyond our experience have always taken us to places where the exception is the rule. Reality might be made of repetitive patterns, but the mind is very limited in its ability to reconstruct those patterns from a countable number of abstracted concepts.

In part, this is due to the fact that the information relevant for some parts of reality is irrelevant for other parts of it. That is, the information we lose when deducing patterns of one part of reality might be essential for the patterns of other parts of it. For example, scientific laws deduced at a certain scale of physical quantities tend to eventually fail at other scales. It turns out, physical objects behave in very different patterns at different scales, requiring very different forms of deduction and induction. In this sense, we are “scale myopic” both in theory and in experience.

Scientists have repetitively been proven short-sighted by generalizing theories that apply on a given scale to apply on all, cosmic and atomic, scales. In the 18th century, at the beginning of the industrial revolution, the physical world was interpreted as a big machine; following the laws of mechanics. In 1814, Laplace expressed it like this: if an intellect (sometimes called a Demon) knows the precise location and momentum of every atom in the universe, their past and future values for any given time are entailed. That is, the universe is deterministic, like a machine with gears. Newton’s laws of mechanics could be used to reconstruct both the motion of surrounding planets and that of an apple falling from a tree; it must be true for everything else in the universe (like atoms and galaxies).

A century later, scientists discovered that smaller things don’t behave mechanically at all; their properties somehow combine those of waves and particles: a mind boggling contradiction. In fact, the only confusing thing about those tiny things is that they are unlike anything we’ve ever experienced before. Their behavior is hard to imagine because we can’t generalize our experience with big objects to that with very tiny ones.

Around the same time, Laplace’s Demon had to go to hell. Scientists discovered that uncertainty is a fundamental property of nature. It’s impossible to know the precise location and momentum of every atom in the universe; no matter what. The world can’t be a machine. To add another blow, the mechanical view of time, which could be measured by mechanical clocks, was also proven incomplete. Every object has its own watch! How can such an intuitive concept as time be also wrong? The rest of the 20th centuries is full of those disappointing realizations forcing scientists to switch from a certain philosophy to its complete opposite. Even the perspective that the seemingly random events in the universe are governed by fundamentally deterministic laws has been flipped on its head. Nowadays, some physicists believe that simple physical laws ’emerge’ from the statistics of random underlying events. If nothing else, the scientific discoveries of the past century teach us a very good lesson: to never trust our intuitive generalizations.

False or excessive generalization don’t only happen across physical experience. They also happen at the heart of logic. A couple thousand years ago, Pythagoras led a sort of religion that had a set of beliefs, some with foundations in math. The universe, he believed, is all made of ratios of natural numbers: rational numbers (like 2/3 and 5/4). Accordingly, understanding the world is just a matter of finding the right ratios that explain its building blocks. The idea makes sense, and it easily generalizes. It beautifully describes harmonics in music and characterizes geometrical shapes. But at some point, one of his followers, Hippasus, the length of a diagonal of a square cannot be a ratio. It had to be something more complicated: an irrational number. Naturally, this undermined a big part of Pythagoras’s religion.

Something similar happened in the 1930’s. Hilbert, Russell, Whitehead, Frege and a bunch of famous mathematicians were all about creating a system that can ‘prove’ any statement in math from basic logical axioms. It makes sense that math is all ‘logical’ and that, with enough symbol manipulations, any true mathematical statement should be provable. But then, almost out of nowhere, a German mathematician named Kurt Godel, proved that an axiomatic mathematical system will always have true statements that cannot be proven. The intuitive generalization that all true logical statements are provable failed.

These examples are both terrifying and exciting. They are terrifying because they expose how little we know about the world we live in. We can’t rely purely on our intuition as a source of understanding. It is always likely that our assumptions are wrong. And this is exactly why we should be excited about the possibilities. Life is an adventure with infinite gems waiting to be discovered. No one knows how we will see the world in a hundred years.

Failed generalizations in social domains are countless. All forms of superstition, sexism and racism stem from oversimplified generalizations and a refusal to question one’s assumptions. Essentially, we build patterns about things and people, and we’re always wrong; especially when the thing we’re trying to explain or predict is more complex than we’ve assumed. But we have to build these patterns to survive. As the statistician George Box expressed it succinctly: “all models are wrong, but some are useful”.

The key is to realize that all generalizations have boundaries. What applies at normal speeds, doesn’t apply at high speeds. What works on the scale of large objects, fails at small scales. We often have too much confidence in our intelligence, secretly hoping that our patterns can explain EVERYTHING; from the beginning of the universe to the tiniest building block of matter. Most religions are culpable of perpetuating unverified universal beliefs, because they’re in the business of making people feel better by bringing certainty to their hearts; but governments and scientific institutions are often as guilty of excessive generalizations.

So generalizations are always bounded. Is this a good conclusion? Hardly. Our generalizations always fail at one point or another but we can’t claim that this is always going to be the case. Logically, this claim can’t be true. Because it is a generalization itself! If it’s true, then it’s not true: a contradiction. Does this mean that the assumption “all generalizations are bounded” is false? Is there, after all, at least one unbounded generalization? It probably only means that hope is always justifiable.

This paradox is similar to theLiar’s paradoxin its self-referential nature. Abstracting the problem bit, we can think of generalizing patterns as properties. For example, we can ask whether a set of observations can be described by a pattern P or not: if it can, it has property P. If an observation can be described and predicted using Newton’s second law, it has the property “Newton’s second law”. Accordingly, the hypothesis “the generalizing pattern X doesn’t apply to at least one observation” translates to the statement “the set of observations that have property X is bounded”. This statement can be assigned a property B; that is, all sets of observations where this statement is true have the property B. Now, we make the claim “all generalizations are bounded” or “all sets of observations have property B”. Does this statement (expressing a property) have property B or not? If it does, then it doesn’t. The property B of non-absolute-generality applies to all other properties, including itself. But unlike the liar’s paradox, the property in question is boundedness not truth. Bertrand Russell discovered the liar’s paradox when mathematicians where trying to formulate all of math in terms of logic (and failed). The lesson is clear. You can read more about it here.

It’s not hard to accept the fact that we can’t make absolute generalizations in theory; who cares if quantum mechanics is incomplete. But it’s not easy to do it in practice. After all, how can we accept that everything we believe is false at some point or another? What’s the point in believing anything then? We all have things we believe are absolutely true, regardless of the situation, and we rarely question our deep-rooted assumptions.

Learning how to embrace our ignorance and accept fundamental paradoxes of life is not a new art. Some ancient cultures might have emphasized it more than ours. For example, Jainism, an ancient Indian religion, is founded on three principles: non-violence, non-attachment, and non-absolutism. This last one, also called* **anekāntavāda*, is the recognition that truth, as expressed by language, can never be absolute. This Jain doctrine states that reality is too complex for us to understand, always expressing itself in unpredictable ways. This idea naturally encourages learning, tolerance and openness to other people’s perception of reality. Similarly, Zen Buddhists contemplate seemingly paradoxical statements/poems, called Koans, to learn how to understand and embrace fundamental paradoxes of life. Greek philosophy was founded on a deep awareness of ignorance that Plato attributed to Socrates: “the only thing I know is that I know nothing”.

We have to live with the desire to make sense of the world but also with the awareness that we can’t make general statements that are always right. We have to embrace the uncomfortable realization that our generalizations are both right and wrong, depending on the context and the assumptions. We have to be wise as much as we’re intelligent. That is, we have to realize that our facts and beliefs, no matter how powerful, are as bounded as we are.