The Isomorphism of Ideas

An analogy works by realizing that two ideas, or two parts of those ideas, are the same thing.

Learning about derivatives in calculus, you may get the sneaking feeling that it reminds you of an odometer and speedometer on a car. That’s not a coincidence, the speedometer actually is the (absolute) first derivative of the amount of distance you travel.

If you learn about physics and realize that voltage in electricity is analogous with height in gravity, that’s also not a coincidence. They’re both ways of indicating potential energies.

Even if the two things being compared are totally different, an analogy seems to show that at least part of the pattern representing the ideas is the same. If I make a comparison between Viking and Greek mythology and see that Odin and Zeus are two patriarchal heads of a pantheon of other deities. They might have many other dissimilarities, but at least part of the structure of the two ideas is the same.

The Mathematics of Analogy

There is actually a type of math which deals with these seeming-differences but internal-sameness: isomorphism.

Essentially, isomorphism is asking whether two completely different descriptions of objects and their relationships, can be redescribed as being the same as something which looks completely different. Or in our case, is there some subset of the objects and relationships for two ideas which can be rearranged to actually look the same.

Unfortunately, isomorphism is a hard problem. It’s not known if there’s a general way to quickly figure out whether two sets of objects and their relationships can be renamed to match another pattern.

However, despite this difficulty, most of the time we want to create an analogy, we don’t need a perfect match, all the time, nor do we need to work with hugely complex bundles of relationships. To make an analogy, you only need several parts of each idea to match each other.

Analogies and Learning

Analogies work so well for explaining things because your brain seems equipped to exploit this isomorphism between ideas.

Consider learning a new idea. The idea may have a lot of details to get right. Those details also have relationships with the other details. Learning the idea involves practicing those details and their connections until you can recall the idea easily.

Now, if I told you that this new idea is actually isomorphic with another idea you’ve already practice the job becomes a lot easier. You can simply pair the features of the idea with each other and then you can exploit the fact that you already understand the connections and relationships in the first idea.

Most of the time this happens, I think, it’s so banal as to not even merit attention. When I’m learning capital cities of countries, and someone tells me that Paris is the capital of France, I’m exploiting the fact that I already have this mastered knowledge of other countries having capital cities, and that the capital is usually the seat of the government, it is often, but not always, the largest city in the country, etc.

The only time this mental feat is even noteworthy is when the isomorphism is rather clever. Because it may not seem obvious that complex numbers and two dimensional vectors are actually the same in many ways, learning of this isomorphism can be quite useful in helping to demystify the notoriously hard concept of an imaginary number.

I think people sometimes make the mistake of thinking analogy works, as a process, only when it’s of this rather clever type. Most analogous reasoning is rather obvious, but it is still an essential component of learning new ideas.

Exploiting Isomorphism

My original ideas on holistic learning, put a lot of emphasis on finding these isomorphisms between ideas. If you can find an analogy, you can learn a complicated idea much faster because you exploit the fact that you’ve already mastered the pattern. Given that many ideas are at least partially isomorphic to one another, this creates a powerful principle for learning quickly.

I’ve since grown a bit more skeptical of hunting for analogies as being the primary way to learn something. Mostly because finding isomorphisms is a hard problem that depends, in no small part, to already somewhat understanding the idea. Finding these isomorphisms becomes a task for the master of a subject, not the complete beginner.

Instead, I feel the foundation for learning has to be based on active practice. Sometimes raw practice will lead to this epiphanies that two ideas are actually related. Sometimes that insight will come much later, once you’ve already practiced the ideas somewhat.

However, I still believe that analogous reasoning is an important part of learning, even if it’s not always an easily-controllable process. Finding isomorphisms between sets of knowledge is a potent tool in understanding.

Which Subjects Have the Most Analogies?

Perfect analogies occur when the two ideas in question share an identical deep structure. In other words, they are completely isomorphic to one another.

This rarely happens by coincidence. In fact, the more complicated an idea is, the less likely you’d have a perfect isomorphism between ideas if the two ideas were really unrelated. The speedometer/odometer relationship and first order derivatives aren’t coincidentally the same, they’re actually the same.

Math, is very often the link that connects the two sets of ideas. Because math is pure abstraction—stripping away the details to have only the unlabelled relationships themselves exposed—it is very often the analogy that connects the two sets of ideas.

I might even go so far as to say that all true analogies (meaning complex sets of ideas which are completely isomorphic) are mathematical. Math, then, might be simply the description of what things in the world really have the same logical relationships to one another.

Math can be about calculation. But it can also be an inventory of all the basic types of patterns and logical relationships which can exist in the world for there to be analogies between them.

Things Which Look Different Being the Same

The fact that isomorphism is probably a hard, theoretical problem, means that we may never get to a point where all the possible analogies of different ideas are known. But that also makes it special when you do discover that two things you were looking at which seemed different, are actually the same.

I’ve spoken about examples from math and physics, but I think this equally applies to all aspects of life. Many of the blog articles I’ve written which have superficially different titles, concepts and definitions, are about the same ideas. In many cases, even I haven’t realized that certain seemingly different ideas are expressing the same point.

A lot of learning is pegged as the act of seeking out more facts and data. But I think it’s equally a task of seeing how things which look different are actually the same. That connection between ideas mattering as much as the ideas themselves.

  • Luis Garcia de la Fuente

    Very interesting post. I guess Classical Mythology was conceived as an isomorphism of human life, so people could easily make sense of their desires, lives and eventually deaths.

  • Luis Garcia

    Very interesting post. I guess Classical Mythology was conceived as an isomorphism of human life, so people could easily make sense of their desires, lives and eventually deaths.

  • I’ve been thinking that there is great value in “comparative” studies that offers nothing “new” but show connections, correspondences between two or more seemingly different things. Earlier this year, I gave a tech talk that solved the same problem with different programming languages. It was not only well-attended, but well-received. “This is really the same thing as that” really helped a lot of people who were familiar with only one of the languages, for example. So I’ve decided to leverage this comparative approach more in the future.

  • Franklin Chen

    I’ve been thinking that there is great value in “comparative” studies that offers nothing “new” but show connections, correspondences between two or more seemingly different things. Earlier this year, I gave a tech talk that solved the same problem with different programming languages. It was not only well-attended, but well-received. “This is really the same thing as that” really helped a lot of people who were familiar with only one of the languages, for example. So I’ve decided to leverage this comparative approach more in the future.

  • Augusto Pereira

    I believe that the concept that you’re trying to talk about here is that of a functor between categories. An isomorphism only relates two objects of the same category. For example, while the complex numbers form a two-dimensional *real* vector space (and any two vector spaces of the same finite dimensions are isomorphic), they also have other structures, such as that of a field, which is arguably what makes them important (it is a theorem that there can be no other fields of higher dimension). A functor, on the other hand, relates two different structures. A simple example is that of the fundamental group of a topological space: by using this functor, one can deduce topological properties of some space by purely algebraic means, which I believe corresponds more accurately to the generally agreed upon definition of “analogy”.

  • Augusto Pereira

    I believe that the concept that you’re trying to talk about here is that of a functor between categories. An isomorphism only relates two objects of the same category. For example, while the complex numbers form a two-dimensional *real* vector space (and any two vector spaces of the same finite dimensions are isomorphic), they also have other structures, such as that of a field, which is arguably what makes them important (it is a theorem that there can be no other fields of higher dimension). A functor, on the other hand, relates two different structures. A simple example is that of the fundamental group of a topological space: by using this functor, one can deduce topological properties of some space by purely algebraic means, which I believe corresponds more accurately to the generally agreed upon definition of “analogy”.

  • sami

    I really wonder what you are thinking about “first principle thinking” which was mentioned by Elon Musk. I definitely did not want to undermine the message of the post contrary to this I would like to harmony to evaluate the different way of thinking. For further reading: http://www.businessinsider.com/elon-musk-first-principles-2015-1

  • sami

    I really wonder what you are thinking about “first principle thinking” which was mentioned by Elon Musk. I definitely did not want to undermine the message of the post contrary to this I would like to harmony to evaluate the different way of thinking. For further reading: http://www.businessinsider.com

  • Kenneth Bruskiewicz

    Another prospective source of metaphors might be population ecology or evolutionary ecology.

    Biology seems very domain specific, until one views it as economics by any other name. The Social Darwinists of course believe this, but irregardless of how palatable that point of view is politically, there are ways where the metaphor is effective.

    Genes and organisms optimize over scarce bundles of resources, but instead of money it’s energy.
    Businesses metabolize or catabolize the output of other businesses, either through partnerships or mergers.
    Niche-filling organisms and niche markets filled by small business aren’t that different an idea.

    From there you can talk about more or less the same kind of pressures pushing themselves down on different embodiments of rules. Business principles follow the rule of consumers and government regulation, while genetic principles follow the rule of the jungle. But that’s a technical point when they’re quite similar from this alien point of view.

  • Kenneth Bruskiewicz

    Another prospective source of metaphors might be population ecology or evolutionary ecology.

    Biology seems very domain specific, until one views it as economics by any other name. The Social Darwinists of course believe this, but irregardless of how palatable that point of view is politically, there are ways where the metaphor is effective.

    Genes and organisms optimize over scarce bundles of resources, but instead of money it’s energy.
    Businesses metabolize or catabolize the output of other businesses, either through partnerships or mergers.
    Niche-filling organisms and niche markets filled by small business aren’t that different an idea.

    From there you can talk about more or less the same kind of pressures pushing themselves down on different embodiments of rules. Business principles follow the rule of consumers and government regulation, while genetic principles follow the rule of the jungle. But that’s a technical point when they’re quite similar from this alien point of view.

  • Kenneth Bruskiewicz

    Elon Musk also separately claimed in his Reddit AMA that knowledge can only be acquired by hanging it on past knowledge, i.e. a “semantic web”. I don’t know what Scott thinks, but I figure that the only reason that metaphors fail is that they weren’t particularly deep enough in the first place, for you to get away with using them novelly without running into their limitations.

    Elon’s background is in physics/chemistry, where the term “first principles” gets tossed around a lot. To appreciate what he means, it’s worth taking a couple hard physics courses. Essentially it’s working with assumptions in a way such that you’re able to fully describe the behavior of the objects in the problem, but with the scarcest set of assumptions possible. If you make too many assumptions, it’s likely that they’ll come together in a way where you’ll believe in a result that is wrong.

    Metaphors have a role to play in that kind of problem solving, especially inspiring you to see a certain system as being part of a larger class of systems. But they should only frame you exploration of solutions, and not control it.

  • Kenneth Bruskiewicz

    Elon Musk also separately claimed in his Reddit AMA that knowledge can only be acquired by hanging it on past knowledge, i.e. a “semantic web”. I don’t know what Scott thinks, but I figure that the only reason that metaphors fail is that they weren’t particularly deep enough in the first place, for you to get away with using them novelly without running into their limitations.

    Elon’s background is in physics/chemistry, where the term “first principles” gets tossed around a lot. To appreciate what he means, it’s worth taking a couple hard physics courses. Essentially it’s working with assumptions in a way such that you’re able to fully describe the behavior of the objects in the problem, but with the scarcest set of assumptions possible. If you make too many assumptions, it’s likely that they’ll come together in a way where you’ll believe in a result that is wrong.

    Metaphors have a role to play in that kind of problem solving, especially inspiring you to see a certain system as being part of a larger class of systems. But they should only frame you exploration of solutions, and not control it.

  • Scott Young

    Not sure what this has to do with first principles, can someone explain the connection I’m missing?

  • Scott Young

    Not sure what this has to do with first principles, can someone explain the connection I’m missing?

  • Scott Young

    Good point. Perhaps I’m misusing terminology.

    The implicit model I’m working on is one where knowledge roughly corresponds to some kind of graph of semantic knowledge. The isomorphism is then when two different graphs are isomorphic to each other. So two sets of knowledge having subsets which are isomorphic (or partially isomorphic) could make sense if knowledge is representable as a graph. I don’t think that’s far-fetched–neurons themselves relate much like graphs–so any knowledge represented in a neural substrate would therefore be related to graphs.

    That point was rather speculative, so I didn’t include it, but the idea of isomorphism is more specifically talking about graph isomorphism. Perhaps as a more general concept of analogies outside of human cognition (or if my relating neuronal structures to graphs is incorrect) functors might be a better term.

  • Scott Young

    Good point. Perhaps I’m misusing terminology.

    The implicit model I’m working on is one where knowledge roughly corresponds to some kind of graph of semantic knowledge. The isomorphism is then when two different graphs are isomorphic to each other. So two sets of knowledge having subsets which are isomorphic (or partially isomorphic) could make sense if knowledge is representable as a graph. I don’t think that’s far-fetched–neurons themselves relate much like graphs–so any knowledge represented in a neural substrate would therefore be related to graphs.

    That point was rather speculative, so I didn’t include it, but the idea of isomorphism is more specifically talking about graph isomorphism. Perhaps as a more general concept of analogies outside of human cognition (or if my relating neuronal structures to graphs is incorrect) functors might be a better term.

  • Scott Young

    Interesting. I think seeing the same thing from multiple perspectives helps us understand it better. I’d be interested in your talk.

  • Scott Young

    Interesting. I think seeing the same thing from multiple perspectives helps us understand it better. I’d be interested in your talk.

  • Manuel Arango

    Interesting. It reminds me of linear transformations on linear algebra. A set can turn into another and yet both of them may have some “basic” principles.

  • Manuel Arango

    Interesting. It reminds me of linear transformations on linear algebra. A set can turn into another and yet both of them may have some “basic” principles.

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