What is obvious?
What does it mean for something to be obvious? Lots of things seem to be obvious to lots of people but are they really? I think in many cases this is an illusion or the result of a sort of amnesia. This is important because one often believes that the things they consider ‘obvious’ are thought that way by many others. This kind of thinking can lead to difficulties in communication, act as a hinderance to people truly understanding ideas and blind us to the progress we have made as a civilisation to reach the point we currently occupy.
Very few things in life are ‘obvious’ and this is true in maths as much as anything else. All of the things you see and use in maths were once not there. Integers were not dug up in tar pits, pi did not roll down a mountain, the equals sign did not sprout up from the ground. Every proof, every concept, every symbol that you come across in maths was once not there, was once pushing out the frontier of mathematical thinking.
There is a strange dichotomy at play here. Many of the things we are quick to call ‘obvious’ today were invented or discovered by some of the minds we consider the greatest in history. I know at some point I’ve considered the concept of expected value ‘obvious’ but do I really have any right to do that? After all, this theorem was only the result of collaboration between two of the greatest mathematicians of all time, Fermat and Pascal.
If it took geniuses like Fermat and Pascal to come up with expected value why should I get to ever think that these concepts are ‘obvious’? Even the concepts of zero or negative numbers had to be invented. Many of the greatest minds of their ages didn’t come up with these ideas that we would almost consider too basic to even discuss. John Salvatier talks in his essay ‘Reality has a surprising amount of detail‘ about how “after you see them [important details] they quickly become so integrated into your intuitive models of the world that they become essentially transparent” and I think this is a wonderful view point on this. Very few ideas are obvious, we are just all too quick to integrate them and then forget about what advances they really represent.
In fact expected value is such an unusual idea that the ancient Greeks played games that flew squarely in the face of its logic. They used to use bones as dice where the faces were different sizes, if one looked at base rates it would show that the larger faces would appear on top more often. The amazing thing is that based on the scoring system if you used this fact you could have gained a positive expected value from playing with this in mind and yet there is little evidence that players actually followed this strategy.
Many people, I am sure, will remember being in the first maths lesson of a year and hearing the teacher tell them that there are no dumb questions. I also imagine that most people nodded in a sort of ritualistic agreement while thinking to themselves that there clearly were dumb and obvious questions that someone could ask.
To make my point about how few things are truly obvious I want to suggest some questions that one might have thought weren’t worth asking at some point or another so that I can show what concepts actually asking these ‘obvious’ questions can lead to.
What if the angles in a triangle didn’t add to 180˚? Well then you get introduced to the world of non-Euclidian geometry which is the basis for how we navigate around the world. Also it is worth pausing to see if you can actually answer why they add up to 180˚.
What if 1=12? What a preposterous idea that is. Unless one decides to think about the fact that what this actually describes is modular arithmetic modulo 11. Without this idea we would lose large amounts of number theory preventing us from having basically any cryptography. Oh and this is actually the exact set up with have with analog clocks.
What if you could square root a negative? Well then you wouldn’t really get MATHERROR but instead find a whole set of imaginary numbers that are used extensively in real world physics and engineering.
What about if probabilities were not just between 0 and 1? Well if you go the negative route then Richard Feynam has an entire paper for you on how these are useful in quantum mechanics. He points out as well that on some level these are no stranger than negative numbers, after all you can’t ever hold -5 apples.
There are so many versions of this in Maths and I’m sure in many other places you look. I challenge you to consider almost any of the most basic concepts you have about maths or any other topic and see if you really understand it and what it would mean if it wasn’t true. In many cases I believe you’ll find that if it wasn’t true instead of breaking something it leads to a vast new world of possibilities.
Why are some things obvious?
It seems surprising the amount of things one might be tempted to call obvious. If we stick with maths and abstract reasoning there doesn’t seem to be any reason for us to be any good at them. The Wason selection task is a particularly good example of this. It exposes participants to two logic puzzles, one with abstract concepts, letters and numbers, and one with very concrete characteristics, people drinking at a bar. The statements people have to verify and the logic as to how to verify them are identical and yet people do markedly better in the concrete rather than the abstract example. We simply don’t seem to be built for abstract reasoning. If one wants any more convincing how strange it is that we can reason, I would only ask them to show me any animal struggling to survive because it can’t differentiate.
Our ability to reason abstractly is a kind of neurological alchemy, we can take in some thoughts, think about our thoughts and then end up with better thoughts without any external stimulus. Despite the marvel that is reasoning I want to look at two ways that things are rendered as being more ‘obvious’ than they might otherwise.
The first point I want to touch on is that many things are labelled ‘obvious’ because of a sense of inbuilt amnesia we all suffer from. Eliezer Yudkowsky introduced me to the concept of inferential steps. These mark areas where one needs to know one concept to be able to understand another. For example to understand multiplication you need to understand what numbers are. He talks about how at the beginning of civilisation almost nothing was beyond one inferential step away. You might have private information as to where an oasis was but everyone understood the concept of an oasis. This is not true today.
As society gets more specialised we have to travel down roads with more inferential steps, one thing we are not so good at doing is remembering just how difficult each of those steps was to take when we first had to make it. The main point I want to make here is that some things may seem ‘obvious’ but it would be a drastic mistake to think they are objectively obvious, often it is just you looking down path and assuming everyone is much closer on it to you than they actually are.
Alfred North Whitehead wrote that “Civilisation advances by extending the number of important operations which we can perform without thinking of them” and this takes me to my second reason why we can view things as obvious. Over time civilisation has built remarkable scaffolding for the mind, like language, writing, Arabic numerals, that allows us to compress and manipulate information in ways far beyond what we otherwise could achieve.
If one wonders how powerful these tools are at making the difficult clearer the case of Greek and Roman numerals provides a good example. These two cultures, especially the Greeks, had many of the minds we consider some of the finest at Mathematics in history and yet they were hampered. The Roman numeral system allowed additional and subtraction but was much more difficult to use for any more complex tasks. It also suffered from a lack of compressibility, take the number 1748, for us this has only 4 bits of information we need to recall, one for each order of magnitude, for a Roman there are 10 bits of information to remember, MDCCXLVIII. We all seem to have a numbness to numbers over a certain length and so this method of recording integers would almost certainly make them more difficult to mentally manipulate.
Our current system does much of the heavy lifting for us. It compresses and makes numbers malleable to a far greater extent then the Roman system, allowing small children to accomplish ‘obvious’ tasks that under the previous counting system would have been anything but simple.
One interesting and humbling corollary from this is that there will almost certainly be topics and concepts that people in the future, perhaps even children, can learn, think about and manipulate that to them will seem obvious but which to someone living today may be beyond their ability to even comprehend.
A third reason some things can seem obvious comes down to unknown ignorance. Many ideas really are quite complex but can often be communicated at some level in fairly straight forward terms. The example I frequently think about from this point of view is evolution. Almost everyone if you asked would say that they get evolution, weaker things die and stronger things live, but in fact this only touches a small part of what it is. Some characteristics of an animal don’t make it more likely to survive but do make it more likely to find a mate, other characteristics go the other way. Not all beneficial mutations will be carried forward in the population. Organisms can change over time for entirely non evolutionary/natural selection reasons. There are so many intricacies to the idea of evolution that unless you are motivated to look for more it is very easy to feel like one has a reasonable grasp on how organisms come to be how they are.
This is related to the idea of the Dunning Kruger effect where when one is initially learning about a topic they are rapidly growing their knowledge of it but are not growing their awareness of how large the topic actually is at the same rate. In many situations it is all too easy to overestimate ones knowledge and underestimate the complexity of a topic making one feel like it is much simpler than it is.
These three areas can lead us into thinking either that some topic is inherently easy to grasp, or that others grasp some topic as well as we do when this might not be the case. Both can lead to the dismissal of considering what gap there might be either between your knowledge of some area and true knowledge of it or between your knowledge of an area and someone else.
Why spend time with obvious things?
I want you to think about Meditations by Marcus Aurelius. Meditations is a strange book, it repeats itself a lot, it is fragmented, there doesn’t seem to be much, if any, structure to it past the first chapter. When one considers why it was written though a lot of this makes sense. It was written to help Marcus, the then emperor of Rome, keep certain ideas in mind at all times. Many of the subjects are things like, don’t get annoyed when dumb people do dumb things because you can’t control them. This seems like a pretty obvious bit of advice so why is it written so much? Whilst these statements seem obvious when you read them it is much more difficult to have these ideas so internalised that when someone is being dumb in real life instead of getting annoyed at them and then realising later that you shouldn’t have you are able, in the moment, to calm down about it.
This idea of repetition also relates to the idea of multiple proofs. Why do some concepts in maths have over 100 different ways of proving them? One likely answer is that mathematicians like to do maths for the sake of it but there is another answer worth considering. Some of these proofs may just be others dressed differently, other proofs will attack a problem from a completely different angle revealing something new about the underlying concept. By spending time repeatedly attacking some obvious belief from different angles one often gains far more knowledge and intuition about the idea than you would expect when you set out.
Another reason to spend time considering obvious things is that there is a big difference between obvious to verify and obvious to discover. This is such an important distinction to make that one of the Millennium prizes in maths is for a solution to the P vs NP problem looking at whether every problem whose solution can be verified quickly can be solved quickly. One more intuitive problem to consider when thinking about this is something like a sudoku puzzle. If the answer given is correct then it is obviously correct but that does not mean at all that it was obvious to solve. I think it is all too easy to conflate obvious to verify with obvious to come up with when in reality they can be very very far apart.
There is one further idea I want to touch on which is the conflation of ‘obvious’ with ‘necessary’. By this I mean looking at something that couldn’t be any other way and labelling it as ‘obvious’ because no other outcome was possible. Considering maths again there are many examples of things that can’t be any other way given a set of axioms and definitions and are yet incredibly non obvious to discover or prove. Consider Euler’s equation e^iπ +1=0, based on all the definitions this equation must always have been true and yet few would say that because it must be true it is in any way obvious. Spending time looking at these ‘necessary’ results can, similar to the above, help lead you to discover far deeper insights about the topics you are dealing with.
The next time you come across some advice or some bit of maths that seems obvious to you it is worth pausing to consider which bits are obvious. Is it obvious in the sense that it is simple to verify? Is it obvious in the sense that you could have come up with the reasoning yourself? Very few things are ‘obvious’ and it can often act as a kind of semantic stop sign past which you don’t think to dig much deeper. This is always always a mistake and there is almost always more to find.