Mapping Realities

Introduction

When I was in college, my mom took in a cat that later gave birth to a litter of kittens. One day, watching the mother cat tenderly care for her newborns, she smiled and said, “See the love of a mother!” This made me uncomfortable. I understood love as being something distinctly human. Yet, the fact is, people with pets and who know all the subtleties of their behavior do indeed see evidence of something like love, something like guilt or shame, and even something like altruism.

Photo by Art G.

Is it problematic to take the concept of “love”, which is human, and map it to certain feelings and behaviors in animals? Animal love is not human love. Animals are earthly creatures, but St. Paul says that love is eternal.

Love never ends. As for prophecies, they will pass away; as for tongues, they will cease; as for knowledge, it will pass away. For we know in part and we prophesy in part, but when the perfect comes, the partial will pass away. (1 Cor 13:8-10)

Notice that whatever aspects of love that we find in animals, we also find in human beings, whereas the reverse is not true. There is therefore a reduction of the concept of love when we take it from human to animal. Reducing “love” in this way can make us uncomfortable, like me with my mom. I wanted to ask, “Are you saying that love is just an animal instinct?”

This is where the concept of a map (i.e. a mapping) can be helpful. Once you recognize that you are employing a map, you do not mind the reductions as much. In fact, you might even enjoy them. They bring to light all kinds of symmetries in disparate “regions” of reality. They allow the wisdom of one reality to be applied to another. And they contribute to the beauty of the (phenomenal) world. Before going further, let’s explain what we mean by a map.

Maps in General

Plain-old maps

Think about any plain-old map, the kind that no one uses anymore because of Google Maps. At some point, someone had to go out there and survey their vast, three-dimensional city and somehow capture it on a single page. By doing so, they designed a handy, miniature version of a vast reality that fits in one’s pocket.

On a map, the length of your pinky may be equivalent to one hundred miles. Think about how much information is compressed into the length of your pinky. In fact, it is not compressed; it is reduced. Information is lost. Walk down any street and there will be innumerable details to notice: trees, lamp posts, potholes, storm drains, etc. All of this information is lost in the process of making the map. If your city vanished and someone tried to recreate your street looking at a map, they would have to arbitrarily decide on all those countless details because the map lost that information in the mapping.

Even though a map loses information, it still represents reality accurately, unless it is an erroneous map. For example, if the map says you will, when driving north on such and and such street, see the church, the library, and then the McDonald’s, but you instead see the church, the McDonald’s, and then the library, you would be absolutely certain that the map was wrong. An accurate map may lose information but it cannot be wrong. Reduced though it may be, it must always remain faithful.

Mathematical maps

In mathematics, a map is usually defined from one set to another. For example, you could create a map from all integers to the powers of 2 (…,1/4, 1/2, 1, 2, 4, 8, …). You could design the map in the following way. Take whichever integer is provided, say $n$, and spew out $2^{n}$. Let’s call the map $\phi$. So, $\phi(5)=2^5=32$. Here are a few more examples of the map $\phi$ at work:

$n$	$\phi(n)$
$3$	$2^3=8$
$4$	$2^4=16$
$-2$	$2^{-2}=\frac{1}{4}$

The map from New York City to a foldable page loses lots of information. What about this map, $\phi$? The simple way we know that $\phi$ does not lose information is that you can reverse the map and get back precisely to where you started. If I tell you that $64$ is a power of 2, you could tell me exactly which number to provide to $\phi$ to get $64$: $\phi(6)=64$. The map works backwards and forwards perfectly well. This is unlike the foldable map of New York City. If you had never been to New York, seeing the map would not give you enough information to visualize New York because the map is a reduction with information loss.

When a map can be reversed (inverted) cleanly, it is called injective. With an injective map, $\phi(x)=y$ means that $\phi^{-1}(y)=x$.

There are also maps in mathematics that are not injective and so which lose information, like the city map. For example, I could define $\pi$ as a map from 3-dimensional space to 2-dimensional space by just dropping the third coordinate. So $\pi( (x,y,z) ) = (x,y)$. This would be the equivalent of a shadow. It flattens the 3d object onto a surface. Think about an elaborate 3-dimensional entity, say someone you love, and what their shadow looks like on the ground. The shadow is completely faithful to the original. The head is not upside down and there are only four limbs. And yet so much information is lost in the flattening. Someone who never met your loved one could not reconstruct their image solely from their shadow.

When a map is not injective, its inverse has many possibilities. Thus, $\pi^{-1}( (1,1) )$ is not any single point. It consists of all the points in 3d space that end up “on top of each other” at $(1,1)$ in the shadow. So there are many points in 3d space could be fed to $\pi$ to produce the same $(1,1)$ shadow coordinate.

Another example of a non-injective map is that between any number and its parity (whether it is even or odd). Say this map is called $\epsilon$. If I say $\epsilon(x)=odd$, you would not be able to determine which number I chose for $x$ because there are infinitely many odd numbers.

Simple, Profound, and Stretched Maps

Simple and automatic maps

Returning to our discussion of “animal love” for a moment, suppose you heard someone say of his cat, “She loves me!” The next day, you heard him say the same thing about his wife. For that matter, consider any similar statement like, “He’s patient” or “She’s smart.” Presumably listeners know how to interpret these statements differently depending on whether we are talking about an animal or a person. Hopefully no one expects that the cat is smart like a person is or that a man’s wife loves him in the same way that his cat does. This happens automatically. We map concepts to their respective co-domains intuitively. Even if you preferred that people use a different word than “love” for animals, you would still understand what they were trying to say. The map is automatic.

There are also conceptual metaphors, like when someone says, “I am under tremendous pressure at work.” The map from physical pressure to psychological stress is so natural that we might not even realize that pressure is a physical phenomenon. It means the system is not at equilibrium, there are strong forces at play, and perhaps there is only so much the system can bear (for so long) before it breaks down. Through a metaphor, the map ultimately unveils a metaphysical reality: pressure extends beyond the physical realm.

Profound maps

Some maps may not be immediately obvious to everyone, but there are so many interconnections that the jump from one domain to the other is quite natural. Mary Powathil recently shared this article with me, which locates the profound connections between the life of a monk and the life of an artist. Sister Teresa Jackson writes,

Both monks and artists of all types are drawn, perhaps driven, by the transcendent. Whether the force that beckons them on is named God or is a more personal vision, it is always something beyond, something that transcends the complacent here and now where many people seem content to dwell. Both prayer and creativity are expressions of a primal feeling that there is something more, something that may be hard to express or articulate but powerful and compelling. - Sister Teresa Jackson

It is hard not to read that and see the truth of the map that Sister Teresa highlights. There are innumerable maps of this kind and many wise people locate and employ them for good. A similar map is often drawn between excellence in life and greatness in sports. In fact, most games are some kind of reductive map of reality, with less ambiguous rules and more objective markers of success. This is what makes sports maps so natural. As St. John Paul II wrote:

This is the logic of sport, especially Olympic sports; it is also the logic of life: without sacrifices, important results are not obtained, or even genuine satisfaction.

• St. John Paul II¹

Pope Leo XIV too reminded us just last month “that in life, as in the game, no one is saved alone.” He prays that every sport “become a parable of life lived with you, working with joy and effort, living with humility in defeat and with gratitude in the victory you offer in your Resurrection.”²

You may notice a profound map between, say, playing the piano and being a manager. When you first started, you had no idea what you were doing, made a lot of mistakes, and had to concentrate to make it through (resulting in a kind of “stiffness”). In time, you became more of “a natural” and the activity became more enjoyable, effortless, and effective. Of course, this map takes us across many domains because it is the map that interrelates all skill development (and more broadly, growth in virtue). It is precisely because such a map unites these areas that we can make general conclusions that apply to all of them at once. For example, the four stages of competence describes the developmental journey of every skill:

Hierarchy of Competence adapted from Noel Burch, image by Igor Kokcharov

Stretched maps

We can even make maps across great metaphysical distances, though these may make us quite uncomfortable. When I was in high school, I remember feeling uneasy that software companies had a position called “Product Evangelist.” An evangelist is supposed to be someone who shares the good news of the forgiveness of our sins in Christ. Either these companies have created a supernatural product or they have stretched the meaning of evangelization. “Stretched maps” like this do not always employ the same word. Sometimes, they simply map the concept.

In the Christian sense, one aspect of faith is believing in that which is not physically available to the senses. It is “seeing” the unseen and being firmly convicted of it. We see a similar notion in the concept of vision found in both corporate and nonprofit spaces. The visionary sees something that is not visible or even in the realm of foreseeable possibilities. They see it so clearly and are so convinced by it that, through their conviction, they bring the vision to reality. Jesus says, “All things are possible for one who believes.” (Mark 9:23) We do not ignore the innumerable differences and distinctions between the two. Vision and faith exist in different planes of reality, and yet that is precisely why we can make a reductive map between them. They both effect great change in the world on the basis of belief.

To take another example, people sometimes say that we are “programmed to [X],” where X is some human behavior like “build relationships” or “seek affirmation.” By this, they are making a map between human behavior and the behavior of a computer running code. Software developers know that complex software with many components interacting in innumerable combinations results in truly sophisticated behavior. With machine learning, the possibilities of encoding complex and dynamic behavior has risen to new heights. In all these senses, DNA is indeed a type of code that ultimately produces human behavior. I once estimated that there are approximately 250 Bibles worth of code (DNA) in every one of our cells. Code which is this complex ends up producing highly sophisticated behavior. Clearly, there is some truth to the map between human behavior and computer code. Yet, for example, we cannot imagine a computer program “building relationships,” even if it was programmed to do so. The map is significantly reductive. It flattens the spiritual, interpersonal, interior, or subjective dimensions of what building human relationships is fundamentally about. The map between human behavior and computer code is a stretch, which leads to a modern fallacy:

1. A reductive map is used to (unwittingly) reduce human behavior to the co-domain of behaviorism.
2. Another map is used to locate the equivalent capability in computers.
3. People celebrate the equivalence of human behavior and computer code.

But if the map is employed correctly, i.e. if it is recognized as reductive in the first place, it can be very helpful.

How Maps Fail Us

Homomorphisms

Feel free to skip this subsection if it’s not your cup of tea.

Another helpful analogy can be drawn from mathematics. (We are making a map from mathematical maps to maps in other domains). When a map preserves all the operations of the domain in the co-domain, it is called a homomorphism. Take the map $\phi$ we outlined above. Since $5+5=10$ in the realm of integers under addition, we want to see whether this relationship is maintained in the domain of powers of 2 under multiplication. We ask, does $\phi(5)+\phi(5)=\phi(10)$? $\phi(5)+\phi(5) = 2^5 \cdot 2^5 = 2^{10} = \phi(10)$. Any such operation is preserved in this map, which means that not only do we have a map of elements from one domain to another, but the interrelationships of the elements are preserved as well. For a one-to-one map like this, this is rather straightforward. What about in a reductive map?

Going back to $\pi$, which was the map from a 3d object to its 2d shadow, let us see if it preserves relationships. We know that $(1,2,3)+(4,5,6)=(5,7,9)$. We must check if $\pi\left((1,2,3)\right)+\pi\left((4,5,6)\right)=\pi\left((5,7,9)\right)$. This map is pretty easy to test, as we only have to strip the last coordinate. $\pi\left((1,2,3)\right)+\pi\left((4,5,6)\right) = (1,2)+(4,5) = (5,7)$. Since $\pi\left((5,7,9)\right)=(5,7)$, we know that the map is a homomorphism.

Aside: Here is an example of a map that fails to be a homomorphism (and how to correct it). Let $\epsilon$ be the map from integers (addition) to $+1$ or $-1$ (under multiplication) depending on whether it is even or odd. If $\epsilon$ returns $+1$ for odd numbers, notice what happens. $\epsilon(3)=+1$, $\epsilon(5)=+1$, $\epsilon(3+5)=\epsilon(8)=-1$. But $\epsilon(3)\cdot\epsilon(5)=1\cdot1=+1$. Since $\epsilon(3)\cdot\epsilon(5) \ne \epsilon(3+5)$, $\epsilon$ is not a homomorphism. If we redefine $\epsilon$ so that odd numbers are $-1$ and even numbers are $+1$, the problem goes away. $\epsilon(3)=-1$, $\epsilon(5)=-1$, $\epsilon(3+5)=\epsilon(8)=+1$. And $\epsilon(3)\cdot\epsilon(5)=-1\cdot-1=+1$.

Broken maps wreck havoc

In recent times, people have implicitly mapped the concept of faith to “manifesting”.

Manifesting is the idea that, through the power of belief, we can effectively “think” a goal into becoming reality. It’s a form of “magical thinking,” or the need to believe that one’s hopes and desires can have an effect on how the world turns. The general concept of manifesting is centuries old but has gained new adherents in recent years through the popularity of books like The Secret; online searches related to manifesting spiked during the lockdown period of the Covid-19 pandemic and remain strong.³

The map from faith to manifesting does not preserve key relationships. For example, Christianity holds that “faith without works is dead” (James 2:26). The fact that the map cannot represent this core “duality” is a red flag. To validate some conceptualization or schema of reality, we often see if we can find a map to a well-established reality that preserves the key relationships. The article from Psychology Today goes on to do just that. It compares manifesting to well-established concepts in psychology.

On its own, manifesting is a flawed approach to achieving a goal. But if the positive, confident mindset that manifesting embraces is combined with focused, planned, and practical actions such as clearly stating and prioritizing one’s goals; gaining new expertise; and taking steps to overcome both internal and external obstacles, it can potentially contribute to success.

Manifesting is an popular concept, but it has not developed sufficient nuance to represent reality. For this reason, it can mislead people and cause harm. Compare this to vision, which we looked at above. In organizational literature, vision is always part of a broader ecosystem which includes strategy, SMART goals, etc. Like faith, it is intertwined with action.

Let us take another example.

The traditional manager at work may be considered a father figure. A good boss balances encouragement and admonitions so that employees are both motivated and challenged. He gives them exposure to the wider world and does his best to cultivate their unique gifts in a positive environment. He shows understanding when something goes wrong, but he also does not let one of them get away with shirking responsibility. And of course, he provides for their sustenance, like an archetypical father!

There are clearly aspects of truth in this map. The problem is that there is one fundamental difference between the two that makes the map impossible and highly problematic. The primary objective of a good manager is the mission of the institution, and the support of his employees is subordinated to this end. The primary objective of a good father is the upbringing of his children and external goals are subordinated to their well-being. When people lose sight of this key difference, it never ends well. For various psychological needs or cultural reasons, they see a map between their boss and a father figure. As a result, they get enraged when they experience “rejection” from their boss who must prioritize the extrinsic, expected outcomes. Likewise, managers, making the same mistake, expect their employees (especially ones that they have personally mentored) to be loyal to them in a familial sense as a kind of substitute child.

All maps are broken from some perspective

In some sense, the only map in real life that can preserve every single relationship is the map from a reality onto itself. We have shown how the map from faith to manifesting is problematic, but even the map from faith to vision can be shown to be problematic. Faith is belief in God, not in oneself, and this makes a world of difference.

So, all we can say is that, when seen from a particular angle, two realities can be considered roughly the same or else can have a relationship-preserving/reductive map between them even if this would not be true when seen from another angle. To be able to find the angle where a map can be found between two disparate realities requires mental flexibility, openness, creativity, and humility.

Recognizing a map is not the same as saying “it’s all the same,” as some people naively suggest about all religions. It requires careful thought and nuance to find the maps and perspectives that highlight commonalities. (For example, from a humanistic perspective or with regards to their civilizing/ordering function, religions are often similar.)

Conclusion

Leonardo da Vinci - RCIN 912579, Recto Studies of water, and a seated old man

The Renaissance artists were especially adept at finding harmonies across nature. Leonardo da Vinci saw the same phenomenon in eddies of flowing water as he did in curls of hair.

Observe the motion of the surface of the water, which resembles that of hair, which has two motions, one of which depends on the weight of the hair, the other on the direction of the curls; thus the water forms turning eddies, one of which follows the impetus of the main course while the other follows that of incidence and reflection.

• Leonardo da Vinci ⁴

Great scientists, like Einstein, employed a creative interplay of intuition and rigor or of imagination and logic to bring together seemingly disparate realities under a common map. He did so in a remarkable way by recognizing a deep equivalence between gravity (being pushed down to Earth) and acceleration (being pushed back in your seat as your car speeds up). Indeed, Einstein’s great dream and dying wish was to find a unifying map that brought together even more of the universe, especially gravity and electromagnetism.

Wisdom requires that we unveil the Logos hidden in plain sight and which unites all things, since all was created through him and there is nothing that was made without him. Getting out there and finding good maps is a great service to humanity for many reasons but especially because it builds authentic unity even across the farthest reaches of matter and spirit.

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1. https://www.vatican.va/content/john-paul-ii/en/homilies/2000/documents/hf_jp-ii_hom_20001029_jubilee-sport.html ↩

2. https://www.vaticannews.va/en/pope/news/2026-06/pope-s-june-prayer-intention-for-the-values-of-sports-2026.html ↩

3. https://www.psychologytoday.com/us/basics/manifesting ↩

4. Translated quote taken from https://www.ias.edu/ideas/lavin-leonardo-chaos ↩

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