Why is it useful to think vertically? A presentation of procedural thinking.
This essay is the first in a series of three essays on three dimensions of thinking.
The first dimension in line is the vertical one: up and down. It represents a mode of thinking that builds upon something else. This “something else” will usually be an axiom.
Two familiar examples of this are mathematics and logic. Mathematics builds upon certain axioms and simply follows the rules. As such mathematics is really an axiomatic system, i.e., a system derived from certain basic axioms. When we write that 2+2=4 we are simply following rules that are already set by these basic axioms. We are not in the business of challenging those rules when doing maths. We simply follow them.
The other example, namely logic, also does this. One basic axiom of logic is that a thing cannot be something and not be that thing at the same time. For example: A cup cannot be red and not red at the same time. It is amazing to think that from this axiom we have made propositional logic, quantificational logic, syllogistic logic, and more.
More common names for vertical thinking include “procedural thinking” and “logical thinking”. The reason I use the vertical dimension as a metaphor is that we might understand this sort of thinking as building a house. The foundation of the house, i.e., the axiom, is not being disputed when thinking vertically. Now we concern ourselves with building whatever the foundation tells us to build. And when we build, we usually build upwards. This is why the vertical dimension of thinking is procedural thinking. We are continuing the process upwards.
At the university the bachelor’s degree usually teaches the student some kind of vertical thinking. For example: If you study physics, then you will learn to think within certain theories of physics. The usual suspects would probably be Newtonian physics, the theory of general relativity, and quantum physics. These theories present rules for thinking, and the student learns to think within these rules.
Note that these three theories are not reconcilable, and as such we might imagine them as three houses being built on different foundations. Because they are on different foundations, different rules apply to them. Thus, the houses look different because the rules, i.e., the “instructions” by which they were built are different. Thus, within one academic field alone we have already found several foundations that yield different “houses of thought”. These different houses are not a product of vertical thinking. Vertical thinking will never yield irreconcilable products. Why? Because that would mean that the product contradicts the foundation, i.e., the axiom.
However, there is one way that vertical thinking may yield contradictory results. If the axiom on which it is based is contradictory, then vertical thinking will reveal this contradiction. This is what happens when we analyse an argument using formal logic with the method known as reductio ad absurdum. This method goes looking for a contradiction in an argument by assuming the opposite of the conclusion. If this assumption can stand, then that means that the original argument is contradictory, as it can support the negation of its original conclusion.
If we were to expand our view, we might go to another field entirely. Take medicine for example. Medicine has something in common with physics, as both medicine and physics study nature in some way. Physics tries to “unveil” the universe in theory. Medicine tries to do so as well, but in a way that adheres to the goal of treating diseases.
However, they are two very distinct disciplines. Despite the physicist’s great knowledge of theory she will likely not be as able to treat a disease as a physician. Nor does the physicist follow a Hippocratic oath, and this reveals that ethics are of great concern to the physician, but of rare concern to the physicist. This reveals that the basic axioms of vertical thinking within medicine and physics are different. And it is these axioms that a student first learns.
The fact that the first degree a student takes is a degree in vertical thinking within some framework is a testament to how essential we regard this kind of thinking. It is when we think vertically, within a certain logic, that we specialise ourselves within certain fields. This dimension of thinking has been a favourite in the Occident for a long time, because it yields such impressive results. Who knew that by studying electronics we would eventually have smartphones?
Vertical thinking also comes with its downsides. Because the foundation is set, and this thinking only concerns itself with building the house by the rules of the foundation, it may feel forced and lacking creativity. Indeed, it is this kind of thinking that computers excel at. Thus, vertical thinking’s high standing might be coming to an end as artificial intelligence will do this thinking for us. This might then make room for more for creative and deep thinking in schools.
But why is vertical thinking still so important? Why is not for example lateral thinking far more important?
To answer these questions let us imagine that Mary decides to take a course in logic. Let us also say that she disagrees with starting with vertical thinking, and she starts the course by thinking laterally. This means that she will try to learn logic not by studying and following the rules that are already set, but by trying to come up with new rules for logic. Do you see the problem that she will face?
Trying to come up with rules of logic without having some understanding of the rules already existing requires a brilliant intuition. Indeed, most universities agree that lateral thinking is the most powerful form of thinking, but to produce good lateral thinking within a field it is usually necessary to master vertical thinking within said field.
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