The quality of your thinking is determined by the quality of the concepts upon which that thinking is based; and the quality of those concepts is determined by the precision with which they are delineated.
1. How would you construct a definition?
Let’s say you need to define a concept, how would you go about it? The short, easy and right answer is: it depends, but that’s also kind of useless. Having a few ideas on how a definition should look like can be quite helpful, even if we decide to do something completely different that time.
To know what a definition should look like, it’s useful to first ask: why would you want to have a definition in the first place? Well, it’s fundamental for understanding and communication. For instance, I would argue that most arguments are due to people discussing the same word/concept but having different definitions in their minds. In the words of semiotics (the study of meaning-making) they are using the same signifier but different signifieds, leading to the misunderstanding.
This purpose of knowledge discovery and transfer was already discussed in a previous post about Aristotle’s account of causation (the Greek word that is commonly translated to cause in this context, aitia, actually means explanation). Here we will take a different perspective which in a way encompasses that of Aristotle.
2. What is a definition?
Yes, that’s a little bit recursive.
- In order to define a definition, we need a definition of definition.
It may seem just a mind game, but this problem is more common than it may appear at first. In science, in order to study something, one first needs to define the object of study, but sometimes you don’t know what you are studying and you are making the study precisely to find that out! This is sometimes called Meno’s Paradox, following Plato’s dialogue, or Learner’s Paradox for a more descriptive name.
Anyway, somehow this is not much of an issue, the solution for these recursive problems is to start with an initial proposal (even one that is knowingly wrong) and iteratively refine it. Solvitur ambulando as the saying goes.
2.1 Taking the etymological shortcut
Here we have an alternative path though, we can actually leverage the power of etymology instead. From the Latin de (completely) + finire (to bound), to define would mean to set an enclosure. Interestingly, closure is the fundamental concept for defining mathematical structures in abstract algebra1. For instance, the numbers are defined that way.
In The unfolding of language, linguist Guy Deutscher argues that all of our abstract concepts were built by interpreting physical concepts metaphorically. How would we venture into the abstract world otherwise, if all we have at our disposal is the reality that surrounds us?
For instance, have initially meant “holding in your hand”, but then it was elevated to the abstract notion of “possession” (there’s no way you can see “possession” with your eyes) and furthermore to “obligation” (“I have to do this”, where you can think of an obligation as a task that you own, even if you don’t want to).
Anyway, I just wanted to digress a little bit with the hope that I can share my fascination for how a physical concept such as enclosure can have such abstract significance. Defining a term is saying at what point the thing stops being that thing, if you move two meters more, you are no longer in France, you are in Italy; if you have a vegetarian dish, and you add some meat, it’s no longer vegetarian. Enclosure determines the inside as much as the outside, in some way, saying what a thing is is also saying what it is not, and sometimes that’s a better path towards definition.
2.2 The limits of the limiting, impossible definitions
In this line, mathematician Paul Halmos emphasises the importance of giving counterexamples when explaining a concept. An interesting edge case is that of Hindu philosophy; the response to the question “what is Brahman (universal consciousness)?”, is ”neti, neti”, which is Sanskrit from “not this, not that”. When it comes to fundamental aspects of being such as consciousness, the best we can do is to say what it is not, basically anything that we can point to; in this sense the concept is never enclosed and remains open.
A lot of philosophy revolves around these open concepts, especially in the east. I’ve heard in Korean zen they do just fine with a single koan, the question: “what is this?“. The question is not directed to any particular object, but to the whole of experience; certainly it can take an entire life to answer. Other questions can be “what is a thought?”, “who or what are you?”, “what is music?“. Open concepts are so because they are indefinable. Note the fact that we cannot set limits on them does not mean they are unlimited, or infinite (again from the Latin ”in” meaning negation, and “-finis” meaning limit).
Something can be indefinable for two reasons:
- (epistemic): we lack the knowledge required to define it,
- (ontologic): the thing in itself is indefinable. In the second case you could say the thing is infinite, in the first case you could say the thing looks as if it were infinite to us.
2.3 The value of choosing the right limits
Closed concepts are also a relevant object of study in philosophy, because they can be enclosed in multiple ways. Sometimes one is better than other, others it’s a matter of taste. Is freedom the absence of restrictions (laws) or is freedom guaranteed by restrictions? For Descartes, two key categories in this regard were clarity and distinctness. To me clarity is a trivial pre-requisite, it’s opposed to obscurity, meaning a concept is clear if it’s accessible to the mind. Distinctness is where the money is, an idea is distinct if its boundaries are well defined, that is, there’s no doubt of whether a particular belongs to that entity or not.
Confusion2 and poor thinking are born out of concepts with blurry limits, you think you are talking about A, and you reach a sensible conclusion, but it turns out it was actually B, and everything you built on top of A does not apply. That is why philosophy places such importance in the proper definition of concepts. If there’s a discipline that cares even more about this it would be mathematics, where precision is required at every step of a demonstration. This is how mathematics and philosophy make better thinkers, by teaching them how to create a structure of concepts that are easily discernible and confusion-proof.
3. Making good definitions
So building good definitions is key, which also means it’s probably not easy, since otherwise everybody would do it and there would be no need to talk about this. Let’s review the kinds of definitions there are, along with some useful patterns for building them. Almost anything worthwhile is born out of an iterative process of refinement, definitions are no different, so having some templates to get the process started can be very useful, even when the templates themselves have shortcomings.
But first, a key pre-requisite for a definition to make sense: the definiens has to be more intelligible than the definiendum. In plain terms: the definition you give has to be more understandable than the thing you were defining. It’s kind of obvious, but it’s easily overlooked3. It also provides valuable insight, defining is also about linking concepts, particularly simpler concepts to more complex ones. The more structured your web is, the easier it is to navigate.
3.1 The infrastructure of concepts
Since we’ve been working with a spatial metaphor all along (definition as an enclosure), I figured we could continue the trend and classify how concepts relate to each other into three categories:
- Vertical upwards. You take several small elements and you give a name to the group they form. E.g.: the European Union. The key operation here is composition.
- Vertical downwards. You take an element and you break it down into subcategories. Basically the inverse of the above. The key operation here is partition.
- Horizontal. Elements can be similar, unrelated or antithetical to each other. E.g.: obscurity is the absence of light. The key operation here is comparison.
If we used the old Greek terms we would rename these as:
- Synthesis.
- Analysis.
- Analogy.
The last step in understanding these categories is seeing what role they fulfill:
- Synthesis brings clarity. When our mind is cluttered with concepts we inevitably lose sight of some of them, as if the ones upfront blocked the view of the rest. In this sense, there’s an obscurity in multiplicity.
- Analysis brings distinctness. By breaking down to smaller partitions we can be more precise in the borders that delineate a concept.
- Analogy brings both at the same time. If we identify that Y is analogous to X, we can use everything we learned for X and apply it on Y. In other words, we translate all the clarity and distinctness of X into Y, conditional on the validity of the analogy.
3.2 A blueprint of definitions
Since defining is also a form of concept relating, we can use the ways in which concepts can be related to describe the ways in which definitions can be made.
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Synthetic inverse. Sometimes we define concepts by enumeration of the elements that are part of that concept. These enumerations are often not exhaustive, for instance, you may ask “what is a dog?” and I may just point at one. These definitions excel at clarity, but lack distinctness: if that was all you knew, would you be able to tell apart a dog from a wolf?
In order to maintain distinctness the enumeration needs to be exhaustive, but that often leads us to the problem of cardinality or multiplicity. In math, the definition of a concept (e.g., a topology, a metric, a limit) rarely has more than 5 elements or properties, usually 3. The way this is done is by layering definitions; an object, in fact, may have 9 defining properties, but you can describe it in terms of 3 elements, each with defined by 3 such properties. Building intermediate hierarchies is fundamental to preserving distinctness without sacrificing clarity. -
Analytical inverse. Defining a concept by looking up (in the hierarchy) instead of down, may be less intuitive but is more fruitful. This is the basis for Aristotle’s brilliant idea of using genus + differentia. Genus is the “generic” part, the one that’s common to other elements, differentia is the specific difference that defines the definiendum as a partition of the genus. Let’s make it simpler: the emperor tamarin is a squirrel-sized monkey allegedly named for its beard’s resemblance to the German emperor Wilhelm II. “Monkey” is the genus, but there are many kinds of monkeys, what makes this one special? the differentia, the beard.
The clever thing about this approach is that, by saying “monkey”, I already told you a lot of information. For instance, that it likes to hang around trees, probably in tropical places, or that they are relatively close to us in the evolution ladder. The definiendum inherits all the attributes of the genus which allows for great distinction (precision) without sacrificing clarity. This is by the way, how classes are defined in object oriented programming.
Tamarin Emperor. Certainly deserves the title. What a wonderful name for such a wonderful creature. Credit: Kevin Barret.
- Analogical. Here etymology comes in handy once again: it turns out “oil” originally meant “olive juice”, which probably was the first form of oil the Romans, Greeks or whoever, encountered. Then we can speculate that when they discovered sunflower oil, they may have said: “it’s like olive juice, but coming out of sunflower”; hence the name “sunflower oil”, which etymologically would mean “sunflower olive juice”. This is analogous to the genus and differentia schema discussed above; by saying it’s like olive juice, you can already tell it’s liquid, does not mix with water, is probably caloric and can be used to cook. This is so useful that at some point we forgot oil was olive juice and we started saying “olive oil”; this way, we could reserve “oil” for the generic concept. The paritcular was elevated to universal, and now when we say sunflower oil it’s more of an actual genus and differentia structure.
Another example of an analogical definition would be: “ETA was the Spanish IRA”. Of course it only works if you know about the IRA, which stands for Irish Republican Army, a resistance movement turned terrorist organisation striving for the independence of Northern Ireland from the UK. Analogical definitions excel at efficiency, they are the fastest way to increase understanding by taking the most advantage of the existing pool of knowledge and deploying it in new contexts. However, it’s also prone to imprecisions, as it rests on a fundamental assumption that’s hardly ever exactly true, namely that the analogised elements are completely equivalent.
The latter is closer to the original sense of analogy, which meant a relation of proportionality. A clearer example of an analogy of proportion would be: “Afghanistan was to the Soviets what Vietnam was to the Americans”. What’s being compared is not the objects themselves, but the way they relate to each other (a/b vs c/d). It’s somewhat mathematical, given the knowledge of 3 (b, c, d) you can obtain information about the 4th (a), by first extracting the common relationship (c/d) and then applying it onto the other term (c/d · b = a). On the other hand, cases of direct comparison such as the oil one are called analogies of attribution.
In math analogies are called isomorphisms and are a fundamental tool in abstract algebra. You often read sentences like: “all groups of x kind are isomorphic to this group”, meaning that by studying a single group you can understand the structure of all of the other ones. This is so valuable that there’s a field called Representation Theory, which aims at identifying points of similarity, so that well developed tools can be deployed elsewhere4.
3.3 What about time?
We’ve discussed an ordering of concepts in space, what about if we had done it in time? Then we would have split definitions according to whether the definiens precedes, coexists or succeeds the definiendum. Which basically make up the Aristotelian account of explanation that we referenced in the introduction. To be more specific:
- Efficient cause. A definition in terms of the elements that cause the object in question.
- Material and formal causes. A definition in terms of the elements that need to exist synchronously for the object to exist5.
- Final cause. A definition in terms of the effect the object has.
For instance, you could define a human being in terms of its evolutionary history (1.), a cathedral in terms of its materials and shape (2.) and firemen in terms of the function they serve (3.).
Wrap up
To bring closure to this post let’s connect with the beginning.
The quality of your thinking is determined by the quality of the concepts upon which that thinking is based; and the quality of those concepts is determined by the precision with which they are delineated.
This is both a descriptive and a prescriptive statement; that is, it describes how good thinking works and prescribes how to think better. We’ve explored the why and the how, but the most important part, practice, is inevitably left unaddressed. To me the best advice is to study abstract disciplines such as philosophy (perhaps preferably ontology) and pure mathematics. The second best advice is to explore the concepts you use in your daily life:
- What are their bounds? when does a thing stop being that thing? where is it unclear (edge cases)?
- How do they relate to other concepts synthetically, analytically or analogically? what’s the best (the most clear and distinct) you can come up with?
- What kinds of misunderstandings can you identify where people argue due to discussing the same concept while holding slightly different definitions?
Footnotes
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In algebraic topology the fundamental concept for defining spaces is openness and in analysis it’s all about using inequalities to set bounds on the quantities of interest. ↩
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Once again etymology comes in handy. “Confound” comes from the Latin con (together) + fundere (fuse or melt), so a “confusion” is to fuse together two or more things, or in plain language: mix things up. The opposite is to set them apart, to distinguish them. ↩
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At least to me, it’s sadly not uncommon to find definitions that are more obscure than the term being defined. It’s also not easy because clarity/obscurity depends a lot on who you are talking to. For instance, for native English speakers “extravagant means weird” is a well ordered definition in terms of clarity. However, Romance language speakers, have the word “extravagant” in their vocabulary as it comes from Latin, but not “weird”, which comes from Proto-Germanic. Thus for them, “weird” can be more obscure, and strangely enough it would make sense to say “weird means extravagant”. ↩
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If you want to see this in action Michael Penn shows a very nice toy example in this video. ↩
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The main difference between form and matter is that matter is extensive but form is intensive. The same blueprints that describe a real cathedral can describe a toy cathedral, but you cannot use the same materials. In some way matter is worldly and form other-worldly, it belongs to the world of ideas. ↩