Making concrete analogies and big picture

When a politician says that voting for independence “would be like leaving an ocean liner for a lifeboat, without paddles, on a stormy sea”, a researcher in the crowd will yell at the top of their lungs:

Citation needed

Their skeptics is good and necessary, because with just a little carelessness the analogy will become a fallacy. Yet, analogical reasoning is one of the first cognition tools for scientists to formulate theory, design experiments, and explain to others. One notable example is the Energy Systems Language, proposed by Howard Odum, the father of system ecology, which describes ecological systems as… electrical circuits. Without analogy our train of thoughts will not be concise and intuitive. Yet, in my opinion it isn’t used to its maximum potential.

Too much I see weak analogies; all of which are serious for the development of science. Every time I see a research comparing features by features, I just feel pity for the time and energy the authors had spent. With all my respect, I think even when a weak analogy is extended to a whole book, it’s still going the path leading nowhere. On another hand, a strong analogy only needs to be one sentence, but intuition can come like a flood.

Having an analogy is an important step, but that’s not enough. All we need to do is to make it more concrete.

Making a concrete analogy

At the beginning of chapter 10 in Norwegian Forest, Murakami writes:

Thinking back on the year 1969, all that comes to mind for me is a swamp – a deep, sticky bog that feels as if it’s going to suck off my shoe each time I take a step. I walk through the mud, exhausted. In front of me, behind me, I can see nothing but the endless darkness of a swamp.

The author describes the helplessness of the character by comparing it with walking around in swamp. But only who actually have experienced that feeling, either in the mud or in their real life, can be touched. For those who are inexperienced on the issue, the feeling is not vividly clear, hence the analogy is weak. But if the prose stops comparing feature to features and concentrates solely at the physical phenomenon, then the psychological issue will suddenly be more straightforward.

Here is my take on turbulent flow, using as the closing words for an article about a personality disorder:

When a smoke begins to smolder, it first maintains its stability. But with just a little turbulence, the smoke becomes an uncontrollable chaos. Swirling currents will be generated to radiate heat outwardly, rolling together and causing more and more energy to be lost. And after the energy is completely depleted, it will dissolve into the surroundings and leave not even a single mark behind.

Creating concrete analogy is not just about reading Wikipedia pages about the phenomenon, it’s about shifting the mindset from explaining to contemplating. Superficially, that prose is about a physical phenomenon only, with physics jargons like stability, turbulent, chaos, current, heat, energy. But this is an analogy for non-physicists; the readers don’t need to understand what they mean. So in their perspective, what does this look like?

When a ______ begins to ______ , it first maintains its ______.

These gaps must be filled to make sense of the world, so unconsciously the readers will projected themselves into it; what is filled depend on their background. For example:

When my mom begins her anger, she first maintains her calm.

In other words, the readers see themselves in there.

For that child, the words to fill in are totally understood, but they cannot connect those materials in a logical sequence to make sense of the reality. They can sense when their mom is going to explode, but her calm at the same time refutes that, and they cannot understand why. To a child, anger is anger and calm is calm; their co-existence is simply unthinkable. The fluctuation of the phenomenon is clear, but the mismatch between what-should-be and what-it-is effectively blocks the forming of the key question that makes the reality make sense again: “why can mom be angry and calm at the same time?”

I think, if we want to boost interdisciplinary research, we need to help forming the questions we have no idea what, of people we have no idea who. The questions are easy to answered once they are formed, but the unusual connections have to be formed first. Once the analogy has done its job for making sense of the data, it will be abandoned, so there is no need to worry about unintended fallacy.

Researchers in philosophy, social sciences and humanities earnestly need concrete analogies, but since superficially they are physical phenomena, only physicists can write it, because they understand the math underlie it. Sure, collaborations can be formed, but mostly as computational modellings, not attacks on the jargon barrier once and for all. To actually reset one’s perspective to have the fresh eyes again but still preserve the wisdom they have spent years to have, there must be a way to make the big picture from rigorousness.

Constructing the big picture

Note: superficially this section seems to focus only about math, just because math is the best example for it.

There are two forces that split every author in half. On one hand, they are encouraged to present the big picture instead of details, that transferring knowledge is an equally important task as creating knowledge. On the other hand, generations have to learn to live with the fact that in order to write a concise, precise, accurate, rigorous article, using jargons is necessary and unavoidably. Sure, the target audience of research papers are researchers, but are jargons and big picture really mutually exclusive?

Perhaps popular science books will provide the big picture? No. I have never seen any article explaining the Peter-Weyl theorem to a kid. They just talk about historic stories or philosophical lessons, using local jokes and local embellishments to locally make the terms attractive. It’s good to know about blackholes, multiverses or antiparticles, but why can’t Fourier transform be presented in a such magnificent manner when it is so fundamental in every field of science?

If the feeling of finally understand a theorem is too good, then why can’t others see this beauty? Unfortunately this happens most in math-related fields, because math jargon is most alienating to daily usage. Although an untrained person can both be confusing when being asked what political representation or mathematical representation is, it’s easier to understand

the activity of making citizens’ voices, opinions, and perspectives “present” in public policy making processes

rather than

the homomorphism from an abstract algebraic object on a vector space.

To an outsider, this feels like all Greek.

What is the most important thing for a child to learn language? Context. They don’t say “OK, I have a will power to learn my native language because I really want to effectively communicate with my parents, and today I gonna learn the word Ostentatious. I will open up the dictionary and learn its definition by heart. I’ll highlight the keywords, take margin notes and watch some lectures in YouTube until I see its importance in the adult community.” No, they learn it because it straightforwardly connects to their personal life.

To learn a new word its connections to the context must be seen. To understand the context the meaning of individual words must be understood. For a word to transition from the exotic state to the meaningful state, this catch-22 must be handled at once. Dictionaries and grammar exercises can accelerate the process, but the learners need to see how their lives would change before spending time learning it. No-one learns English to become a native English speaker. They learn to become a new person.

It’s true that it takes blood, sweat and tears to explore new knowledge, but if math is really the language of the universe, then why don’t we simply view it as a foreign language and apply knowledge of language acquisition on it? That way, however complexity the topic is, it can always be presented in a playful style that even a kid can understand, and goes straight to their hearts.

So, from a formal definition, how to see the big picture from it?

Trick 1: Placing the term next to its most important keyword


Formal definition: [Every object has a potential energy.] Potential energy is the energy possessed by an object because of its position relative to other objects. (Wikipedia)

→ Even when an apple suddenly appear in the middle of the space, it immediately has a potential, a relative distance between it and the Earth.

Explain: The two terms are not the same, but (1) “relative distance” is crucial to understand “potential”, and (2) the readers are smart enough to know that they are not the same. The most important thing here is that “potential” is introduced lightly, as if it is just a synonym of “relative distance”. Later usage of “potential” in different contexts will refine the understanding of the readers unconsciously. (To be fair, this is translated from my Vietnamese version, in which “potential/propensity” does have a sense of “relative distance”.)

By moving the word to be defined from the beginning of the sentence into the middle, you can reach the mindset of not defining it when you are defining it. When the verb “to be” is presented, there is no prioritized keyword to keep the readers from overwhelming. Worse, having to define something when the topic is not about it is like making a distracting footnote there. On the contrary, with this mindset, not only the term can be understood right where it’s introduced, but also it is connected to the context immediately (“as an intrinsic property”, “of the apple”, “in the relationship”, “with the Earth”), which the definition can never do. It’s the context that actually does all the hard work for us, and it happens in the unconsciousness. In other words, the definition is sprout out throughout the text.

Only use this trick when the term needs to be used repeatedly. A weaker version of this trick is to simply give examples of the term, which is presented somewhere in the next trick.

Trick 2: Replace intermediate terms with their definitions


Formal definition: A representation U(G) on V is irreducible if there is no non-trivial invariant subspace V with respect to U(G). (Wu-Ki Tung, Definition 3.5)

→ When a representation on a space is reduced to the point that only that space and {0} are its only two subspaces that can hold their vectors from being pulled out, then we have an irreducible representation.

Explain: In the author’s perspective, the formal definition must be short and succinct; in their vision the students’ worlds must rock after this. But for the students, when the definitions of non-trivial invariant subspace is right in Definition 3.4, then they probably haven’t digested it completely. And when they refer to Definition 3.4, they probably haven’t completely digested Definition 3.3, which is half page long and completely unrelated to the 3.4…

By nesting definitions within definitions and rewriting it, the true intuition of the term “irreducible representation” before it is conceptualized is restored. Notice how 4 times negation is used in the formal definition (ir-, no, non-, in-), flipping the meaning back and forth. Logically, they can be canceled out, but perhaps linguistically the 4 negation version is more rigorous than the none one? If the first trick is to see how a term naturally emerges from the context, then this trick is to realize how a chain of logic is just dynamics from terms to terms; the more flips the meaning has the more rigorous it is.

I think, underneath all logical connectives (and, or, not, if, iff) and quantifiers (∀,∃) are just the way to transform the familiar to the unexpected, and with creative we can translate it into human language: impossible, necessary, unavoidable, indispensable, can’t exist without, together with, there is no way, never again, etc. Doing this won’t lead you into the wrong track, because the formal definitions are always there to be the lighthouses. The non-jargon words in the turbulent flow that everyone can understand are actually the crystallization of innumerable experiments, and of course falsifiable.

Only after understand this dynamic can the students appreciate the single-valued of a well-formed concept, and then start using it. Unfortunately this also means that they will forget how they get to it at the first place, so when they are asked out of the blue, they just screw up.

Trick 3: Contradict yourself

Solving contradictions is the reason why the ideas survive and are worth the attention, but unfortunately it’s invisible in the definitions. Logical unexpectations create confusion and curiosity without the need to sugarcoat or cherrypick anything, therefore evoking emotions without being bias. Press Ctrl + F and search for “but” in this article, and you will see many of them. And that’s just the tip of the iceberg.

Presenting contradictions in every move is what every filmmaker and game designer know very well. One by one, the readers experience a chain of tiny exhaustions and relieves, making them feel what the confusions and excitements the author feels during the research. Sentence structures will be longer to allow rooms for complicate ideas, but they are also more straightforward.

The exemplar of all time in making contradiction should belong to this Mark Twain’s quote:

The difference between the almost right word and the right word is really a large matter—it’s the difference between the lightning bug and the lightning.

There are at least 3 comparisons:

  • the lightning bug and the lightning
  • the almost right word and the right word
  • the almost right word and the right word and the lightning bug and the lightning

The word “almost” usually means almost 100%, but the last comparison plummets it to 0%, effectively making it a negation! Each comparison is actually a contradiction too.

Another exemplar is the couplet of Poincaré and an unknown poet:

Mathematics is the art of giving the same name to different things
Poetry is the art of giving different names to the same thing

Reading any list of proverbs and quotes, and you will see the same pattern. Between the comparison there is a link connects the two, but they must be kept implicitly; otherwise there won’t be a leap to excitation.

Some tips to make that leap:


The keyword and the term in trick 1 must be italicized to slow down the reading flow of the reader. The way a lecturer pauses or uses gesture during the talk will reveal the nature of the emphasis:

  • Bold: used when the emphasis shouldn’t be fleeting and need to refer later, when the emotion is pushed to extreme, or when there is an invisible link between two or more emphases
  • Italic: used when the words naturally emerge and dissipate into the flow

Other kinds:

  • Underline: sentence breakdown; a segment with multiple words is different to multiple one-word segments
  • Quote: wording choice, which can be a new coined term, or a sarcasm
  • Semicolon: mood switcher, switching from explaining mood to summary mood and vice versa

Logic splitting

For example if we have this logic:

A = B = C = D

Then A and C can be considered as “synonyms”, so are B and D, and the logic can be split into separate sentences (the equal signs are just examples, and the wording really depends on the context):

→ When A exists there is B. This forces C to become D.

The jump between two statements actually makes them amplify each other. By allowing each of them is in a different “world” of their own, the readers can be brought from one perspective to another. This is actually a development of point (1) and (2) in trick 1.

If instead we have a chain of logic:

A ⇒ B ⇒ C ⇒ D

then each statement can be used in different paragraphs, effectively become the backbone of the article.

* * *

What do all of these tricks even mean? They mean to shift the focus from isolated words to the relationships between them, hence from local details global insight will emerge. By accepting that rigor cannot be acquired in one day, and by pushing the idea further by purposely blurring and twisting the rigorousness, the meaning flowing throughout the words is unveiled. Simplifying the jargons and stimulating the ideas are just a mild way to distort the facts, not about putting them into context.

Imagine the whole article is a heatmap, then the terms and complex ideas are the heat sources, as the readers need to spend more energy on them to understand. An ideal heatmap shouldn’t be too hot (too condensed) or too cold (too uninformative), and your job is to control the heat flows so that . Each punctuation, word, phrase, sentence, paragraph should not be self-contained ideas, but are answers for previous twists and are new twists by their own. Without contradiction, the idea is weak.

The takeaway message is in the last sentence. I repeat: do contradict things. The way to find the balance point between so many contradict things is to act contradictorily. Crazy ideas can be thought at any time, but only by weighting the opposite side at the same time that bias can be avoided. It’s not about stimulating the ideas, but about bringing them to the readers’ hearts.

A good writer will follow writing guidelines closely, break the problems step-by-step, explain their arguments point-by-point, imagine they have only one chance to persuade a busy colleague. But a skillful writer will play with it, cumulatively build up tensions and drop them, envision strong analogies and right words so that the jargons intertwine with the surroundings naturally, yet doing so effortlessly and whimsically, like the whole thing is just a football game, which is entertaining and tensing at the same time.

Should I prove that math is really a language before continuing the analogy about Greek? Is that a fallacy, or am I just stating a thinking that perhaps everybody believes in, but never seriously think that it can be true, and hence just denying a reality that they want to believe in? Why reluctant to accept that? Am I missing something?

To be clear, I want to emphasize some disadvantages of this:

  • Take a lot of time and energy to prepare
  • Very easy to confuse readers instead of making it vividly
  • Not really useful when actionable-ness or rigorousness is important.

So while it’s not best to use in face-to-face communications or research papers, it’s perfect for textbooks’ introductions. Instead of putting others’ quotes at the beginning of a chapter, why don’t we make them ourselves?

I think, if we are serious to find the theory of everything, then we should also be serious to improve our writing skills. I also have a deeper theory behind this, explaining how to find the fresh perspective again when you feel lost, and proposing a theory of perspective. Have fun.


From pure logic and imagination, we construct a sphere upon the real world. From the top, a bright spot shines through the surface, and shine on the real world the sphere’s image.

On the real world, a straight line will extend to infinity, and two lines after passing each other will never meet again. But when looking upwards the sphere, they are just images of circles going through the top, repeatedly passing and meeting each other. We see that they not just reunite at infinity, but also reunite their past in the future.

Whenever the sphere spins, everything around us changes. Distant things will move close unexpectedly, and familiar ones will leave us softly. This may seem absurd, but also evident at the same time. Evident, but cannot be grasped, for it comes from a place out of our sight. Cannot explain the unreasonable, nor can explain the obvious, that ambiguity would be frustrating.

Let’s gather all the ambiguities altogether, and name it as x. With just a simple question, a puzzle piece is flipped. And by perseverance, the symmetry within will emerge. Turns out it’s the symmetry. They will run along a circle, imitate the symmetry of the sphere, creating periodic movements, the simplest of which is the pendulum.

Untold pendulums are imprinted in every single thing, swinging perennially. All things are just combinations of them. Each pendulum has its own rhythm, it will move slowly at two opposite ends, but faster during the middle of the swing. This is why the link between two obvious points is so faded. Sometimes it is so faded that no-one can possibly consider that the two extremes are just the same thing.

It will resonate when acted by its own rhythm. The resonance, even transient, is enough to evince it. The pendulums can also join together to form waves. The waves might be invisible, but can spread out throughout space, recurrent over time, ready to resonate to anything share its rhythm.

* * *

Notice the first four sentences of the 4th paragraph:

  1. Let’s gather all the ambiguities altogether, and name it as x.
  2. With just a simple question, a puzzle piece is flipped.
  3. And by perseverance, the symmetry within will emerge.
  4. Turns out it’s the symmetry.

Each of them has double meanings. They blend history of math with cognitive psychology:

Sentence History of math Cognitive psychology
1 The naming of the unknown Concept formation and the role of labeling
2 The use of equations The jump in cognition
3 The discovery of groups The attitude change
4 The use of groups in physics Everyone has distortion, but no one can ever think that they have

So later on, each seeding point can be developed to a paragraph. When talking about the naming of the unknowns, the concept formation can be primed in there. So in the next paragraph, the reader will experience a sudden switch that is so logical.

  1. They will run along a circle, imitate the symmetry of the sphere, creating periodic movements, the simplest of which is the pendulum.

The 5th sentence describes the Peter – Weyl theorem, constructed by repeatedly using trick 2 on the actual theorem. (At the time of writing it I thought that there exists a projective group whose characters form the basis for the Lebesgue space, like ℤ/nℤ, but it’s not.)

This example is meant to be a preamble of a whole book, meaning that it doesn’t attempt to explain anything. Instead, it primes the readers, so that when the concepts are officially explained, they will have a feeling of coming up themselves

Dunning, Mike, Pendulum
Gentner, D., K.J. Holyoak, K.J. Holyoak, and B.N. Kokinov, The Analogical Mind: Perspectives from Cognitive Science, Bradford Books (MIT Press, 2001)
Jullien, F., The Propensity of Things: Toward a History of Efficacy in China, trans. by J. Lloyd, The Propensity of Things (Zone Books, 1999)
Segerman, Henry, Octahedron (Stereographic Projection)
Settles, Gary, Laminar-Turbulent Transition, 10 March 2009
Stillwell, J., Mathematics and Its History, Undergraduate Texts in Mathematics (Springer New York, 2004)
Tung, W.K., Group Theory in Physics (World Scientific, 1985)






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