20 Years of Big Ideas, Compressed
I recently turned 30, and as part of my celebration, I’m going to do a whirlwind tour through some of the ideas that have fascinated me and enriched my life in the last 20 years (Yes, I was a fairly contemplative 10-year-old;-).). I recommend setting aside more time than usual to consume this post or to divide reading between a couple of sessions.
First, a little bit of ideas-meta about the personal context in which the ideas have been embedded. Foremost, I am tremendously grateful for the force of curiosity in my life. In my now decades-long engagement with depression, curiosity has helped keep me going and doing the activities that I know are healthy but that sometimes require great cognitive effort to do. On the days when I am in a lot of pain or apathy, the simple question of, “But won’t it be interesting to see what comes of all this?” helps me to stay in the game as an active player and not merely as a spectator of my own dysfunction.
I should also say that skepticism and doubt about the value and results of my curiosity are frequent internal visitors. As someone with some serious academic training and sensibilities, I know that I’m an academic nobody, and there’s no sign of that changing any time soon. I have no publications in prestigious journals. I only have a master’s, not a PhD. If the only justification for spending large amounts of time thinking about reality and doing mathematics and physics that I find compelling is to somehow be a peer of great academics from outside the academy, then realistically I should quail inwardly. It’s unlikely that my work will ever be taken seriously by folks who control what work gets attention and traction among researchers. If I start out convinced that I’m going to revolutionize the academic establishment’s understanding of a subject from outside of it, then statistically, I can expect to be disappointed.
What I’ve decided is that none of that is the point. The curiosity and enjoyment of investigation can be enough if I let go the idea of a grand purpose justifying this application of myself. If it influences how other people think in constructive ways, then that’s wonderful, regardless of who those folks turn out to be. If not, then it’s still worthwhile to do this for bringing about only my own satisfaction.
IDEA 1: ENERGY
My memory is hazy, but when I was 8, I started thinking about energy in relation to bodies and food. For example, I knew that my body was some sort of energy-driven machine, because I could lose weight by expending more energy than I consumed through food (Weirdly, one of the helpful insights gained through being a child with anorexia.), where energy was measured in calories. Then, a couple of years later, when my grandfather died, I thought about what happened to his body after we buried it and what it meant that bodies decompose. That his body could change into a form that plants or animals could metabolize, and that humans too consumed plants and animals, and plants made their own food through non-living resources, implied to me that living and non-living things must share a common underlying material, which to me was this energy thing that people said one got from food. From there, I concluded that bodies are a sort of condensation of energy, and so must be everything else.
Now, my understanding was in a sense very primitive. I don’t claim to have deduced anything like Einstein’s equation relating matter and energy or achieved the sophistication of modern physics, which recognizes and quantifies various types of energy and their relationships to other physical quantities. What’s important is that this was the start of ongoing thoughts about energy in many different contexts: human and environmental health, physics, productivity, psychology, and music. Learning to manage my energy as much as my time is just one example of how keeping this concept close has improved my life.
IDEA 2: SCALE AND MEASUREMENT
Why don’t ants have brains like ours? Why don’t we have brains like ants? This was one of my springboards into thinking about scale at around age 12. At first, I only considered spatial scale: for example, ants and humans could have brains that are exactly scaled-larger/-smaller versions of each other, but because ants and humans are different sizes we’ll never have strictly identical brains. When I was learning about insect anatomy, however, it was clear that ants and humans don’t even have structurally identical brains. How much of that is due directly to the very different sizes of our bodies and not to the fact that ants and humans perform different activities?
I’ve continued thinking about scale in the contexts of time (E.g. statistical properties of canary song differ depending on whether you’re considering phenomena that happen across multiple seconds or within a single second.), physical laws, the behavior of populations of people vs. of individuals and small groups, analysis (a field of mathematics), and processes/algorithms.
A nearby concept is measurement. I’ve thought a lot about what measurement is and mechanisms for measuring across math, physics, and engineering. From a mathematical perspective, measurement is largely about rules of assigning numbers to different sets of objects, which is a natural generalization of our notions of mass and distance across time and space. From a physical and engineering perspective, measurement is a lot about figuring out how to arrange a situation so that you can measure what you intend to measure with whatever levels of accuracy and precision are needed for your application. In engineering, it’s clear from first-person experience that the observer is part of and influential in the physical system that determines what measurement outcomes will be observed.
This way of thinking demystifies some of the quantum-mechanical story of measurement, which treats measurement as this special operation that might derive its magical nature from consciousness being directed at a physical system, etc. Some demystifying comes from a more refined understanding of what quantum states actually are, but some of it comes from not viewing the observer as physically separable from the system under measurement, even if it is a distinct component of it. As an intuition pump for what I’m trying to get at here, sight is the result of a physical interaction that is as intrinsic to our bodies as it is to photons, even though photons certainly don’t need our bodies in order to zoom around. When a photon collides with our retina, the properties of the photon and our retina affect the final phenomenon of sight. In the same way, an observer taking measurements isn’t a necessary component for a system’s basic existence. But when the observer wants to measure the system and so interact with it, the observer will cause certain physical changes to the system that will be a part of what the observer measures.
IDEA 3: CONTINUITY
If you look at space at smaller and smaller scales, do you ever run out of space? Can you break space? If so, what would disconnect the components? If space can be endlessly subdivided, does it make the most sense to consider it as being comprised of a collection of definite/discrete points? What could be an alternative? I spent a large amount of time in high school math classes thinking about this. When we learned about the fundamental concepts of Euclidean geometry, we were given points in space as primitives. I didn’t find their existence nearly so obvious, even though I accepted that the concept was extremely useful. I went to grad school to study geometry, in particular, largely because I wanted to get to a place with these questions about points where I was satisfied. I now know more about points and spaces (mostly that have little apparent connection to physical reality), but in some sense I’m no closer to answers now than I was 15 years ago.
IDEA 4: RELATIVENESS
A powerful idea in mathematics is the Yoneda Lemma, which roughly says that you can learn everything you need to know about a mathematical object by studying the ways to get to/from it from other objects. This is compelling in-and-of-itself, but it takes on a whole other level of significance for me when I consider that the generality of this statement is such that it holds for all objects at all times we might consider them. This means that the objects of mathematics are as much the web of connections between them as they are discrete entities in their own right. Now think about social structures, or scientific knowledge, or highway systems. A person, discovery, or road doesn’t mean much if it isn’t connected to or interacting with any other people, discoveries, or roads. It’s equally the whole system that defines the individual components. So whenever we make a statement describing a mathematical object—or person, or fact, etc.—it can be very illuminating to follow up with the question: Compared to what? What is the reference class of the object we’re considering, and how are we making the comparison from the object we’re focusing our attention on to that reference class?
IDEA 5: COMPUTATION
What’s the best way to think about what computation really is? If we start by thinking about how PCs work, there’s a notion that we can represent any quantity we might want to use as a sequence of 0s and 1s. How many place-values we have available to us depends on the physical properties of the hardware that we’re using. So, in principle, the difference between any two values stored in the same set of hardware can be measured by the minimal number and locations of the bit-flips that need to happen in order to change one sequence of binary digits into the other. Then computation is the activity of hardware’s physical substrate changing in correspondence with 0s turning into 1s and vice versa. So what are the differences between this type of computation and what brains do? Heck, what about between the silicon chips on a PC and what atoms in the wider universe are doing all the time, changing between various possible states? Do we require that computation have intention behind it? If so, we now have to scope the problem of figuring out what intentionality is into the problem of figuring out what computation is. Not necessarily a negative, but something to consider.
IDEA 6: APPROXIMATION
Newton/Leibniz calculus formalized the concept of a limit, a quantity to which a sequence of values can get arbitrarily close, even if no value in the sequence is ever equal to the limit. We might say, then, that if a sequence has a limit, then each value in the sequence is an approximation of said limit. We can then go on to say things about how “good” an approximation each value is by measuring the difference between it and the limit according to various metrics. In engineering, we often use a truth model of physical parameters where we consider the value of physical parameter to be best modeled by a real number (This term has a precise mathematical meaning.). Because real numbers can require an infinite number of digits/decimal places to represent (and therefore compute with), in practice we never actually handle real numbers as engineers, but a subset of the rational numbers, those rational numbers which we can represent with a finite (and usually fixed, as per hardware on measurement and computing devices) number of digits. How many digits we need, i.e. how close to our truth model we need to get, is inferred from the goals and constraints of the tasks we set ourselves.
IDEA 9: OSCILLATION
This might seem like a weird one, but oscillation is fundamental in our present models of energy and time. Electromagnetic waves, e.g. visible light, for example, are modeled quite gainfully as oscillations. And when one considers how to define time, it’s hard to avoid a parametric/operational definition as the count of variations—i.e. oscillations—of a signal. This is the classic tick/tock of a clock or the conducting pattern of a baton. In embedded systems and music composition, this notion of a pulse to arrange activity across different parts of the system at hand is fundamental. What’s more, the mathematical properties of the trigonometric oscillating functions are rich and aesthetically elegant.
IDEA 10: HIERARCHY
Hierarchy is a structure that’s very prevalent in human modeling of computation/mathematics, social organization, and physical phenomena. I don’t claim that hierarchy isn’t useful, but I do contend that there are situations where it is a default that folks could benefit from questioning. To what extent, for example, is a given company’s best output achieved by implementing a hierarchical organization of labour, and in what respect(s)? In pay distribution? Power distribution? Task distribution? A combination of the three? When thinking about physical phenomena, too, there’s a tendency to organize things hierarchically according to physical scale; at the bottom of the hierarchy, for example, we might put (anti)quarks and what other building blocks we know to be indivisible into smaller components. At the top of the hierarchy we might put the whole of existence. We then preference physical size as governing the relationships between the items in the hierarchy. What other structures, such as webs, radial clusters, clusters with loops, etc. could be useful models to inspire or reflect ways in which we interact with each other or with thinking about how to understand the world? For example, it occurred to me 6 years ago that mathematical truths could be modeled as nodes in a web (in math-speak, a graph) connected by implications. Under that formulation, a mathematical proof is a valid path between nodes in a directed graph where we allow cycles (loops).
IDEA 11: FINITENESS
As an entry point to this topic, recall what I was saying earlier about how, in practice, we can only compute with numbers that have a finite decimal representation. This leads me to consider whether the underlying truth model of certain physical parameters (such as speed, time, spatial extent, mass) being in some fundamental way real-valued quantities is a “good” model of reality, philosophically. Practically speaking, using this systemic “rational-as-real” approximation is extremely useful, and I’m not suggesting engineers and scientists dispense with it. But is it the best we can do for practical purposes? And what would it do to our understanding of physics to not allow ourselves any real- or complex-valued physical quantities? How far could that take us to making accurate predictions of physical systems or toward understanding the mechanisms of physical processes, even if their outcomes are generated by processes too complex to fully observe, analyze, or predict?
IDEA 12: LOCALITY & SPACE
Where is an object? To me, this is a question that quickly leads to many subtleties. For one thing, where is defined relative to a model of space. Is space an infinite collection of points? Are there any “gaps” between them? What is the space’s shape? How do we, as agents, learn what the shape of the space is? It took humanity making some key geometric observations to figure out that we live on a sphere (more accurately, an oblate spheroid). We’re still making complicated observations to figure out what the larger shape of the universe is. Assuming we’re in any given space, how are we going to agree to describe locations in this space? For example, by coordinates? If so, then which coordinate system? And if we want to ask, “How far”?, that invites lots of questions about what we mean by distance in our space and what are our units of measurement.
IDEA 13: ABSTRACTION
One model of what abstraction is is the process of “forgetting” information. For example, think about how children draw. A stick figure is a fantastic abstraction of the shape of a human body—so much so that we use it in graphic design produced by adults all the time! There’s a correspondence between the stick figure and human bodies that leaves out enormous amounts of information, but just enough is preserved that we can recognize the symbolic relationship between the symbol and its referent. Abstraction is one of the central processes of doing mathematics, and it allows us creative freedom to explore what’s conceivable.
An interesting set of problems that I encounter relating to abstraction have to do with when it’s useful and when it isn’t. In software design and mathematics, the impulse and direct movement toward abstraction have had profound and powerful consequences for our knowledge in those fields developing. Conversely, in poetry and other creative work, I find that transparent abstraction often weakens what I produce. I hypothesize that one of the reasons humans consume other people’s creative work is for vicarious experience of all the rich, concrete details of life that there isn’t enough time or other resources for us to experience in full. To put it poetically, the poem should be at most the shadow of abstraction, rather than the object itself.
A more relatable problem, perhaps, is when it’s helpful to abstract about individual human beings and when it isn’t—which implicitly raises a lot of interesting questions about what our purposes are in making these decisions. For example, consider the “I don’t see color/race.” claim. The intent is positive, but as many others have pointed out, there are many situations where awareness of and sensitivity to an individual’s skin tone is more constructive than ignoring/forgetting about it.
IDEA 14: BOUNDEDNESS
I really enjoy the concept of boundedness as an intermediary between finiteness and infinity. What I’m getting at here: consider the region of the usual number line between 0 and 1, inclusive of the endpoints. Let’s represent that using interval notation as [0,1]. How many numbers are in that interval (i.e. a particular type of set of numbers). Well, that’s actually a harder question to answer fully than one might think at first. First you have to specify which type of number you mean—integers? Rational numbers? I won’t go into details, but beyond there being infinitely many real numbers in that interval, there is a whole “order” of infinity more than them than there are rational numbers in that same interval. It’s really an overwhelming quantity compared to any we deal with concretely in our day-to-day activities. So it’s amazing in a way that we can “locate” any given number in that interval pretty precisely, because that interval is bounded; if I give you the number pi/4 and tell you to place it on the number line, for example, you can ignore most of it and put it fairly precisely between 0 and 1.
This idea is rich enough when we think about it only in a mathematical setting and what it means for various sets of mathematical objects, like sequences or series or manifolds. But I borrow the underlying mathematical models of “solution spaces” all the time and import them into solving engineering or other problems in my life. Here, the added value of thinking about orders of infinity is close to zero. The really useful bit is considering the difference between the set of things one could do in solving a problem to be finite or infinite. If it’s infinite, or if the set of outcomes of changes you might make is infinite, what you really want to do next is see whether you can find bounds for those sets. Sometimes you can’t, and then you need to be especially crafty. But often you can find a good-enough model of a problem that allows you to stay in the land of the finite, which can make it much easier to produce useful work.
IDEA 15: NUMBER
It’s pretty amazing to me that human children ever grok what number is, even if they (and most adults) never learn the formal mathematical definition of it. Appealing to the earlier notion of locality that I raised, numbers aren’t things that have a place as part of their set of defining features. You can have a chair with four legs located in a particular place, but knowing the location of that chair tells you nothing about the number four. Instead, the situation is more like that four is one of the possible sizes we can get if we try to put finite sets into one-to-one correspondence with each other (roughly, to pair up all their elements). This requires space in that we need to have a setting in which distinct objects to exist, but where objects are in that space is irrelevant for counting them up, in principle. If I have a finite set of apples and a finite set of oranges, and for each apple I can pair it with exactly one orange, and vice versa for the set of oranges, then those sets are in one-to-one correspondence. So they must have the same size. What is that size? Well, let’s take it to be the amount of pairs we were able to make. ‘Four’ is a name we have for one of those pair-quantity options. The way I’m talking about this is extremely imprecise relative to actual set theory—and doesn’t even touch how to approach defining anything but the natural numbers (i.e. 0 (optional), 1, 2,…) but hopefully it conveys the basic idea.
IDEA 16: SYMBOLIC REPRESENTATION & SYSTEMIC CONSISTENCY
Is force a vector? The knee-jerk answer to this is, “Yes!”. I say that this answer is an approximation of the truth that for certain purposes isn’t close enough. Within the constructed “fiction” that is the realm of symbols where vectors are real, yes, we can say that a vector represents a force, and we usually shortcut this cognitive step by saying that the vector is a force, or vice versa. Now, I’m not suggesting that physicists go through the semantic rigmarole of qualifying every statement with something like, “relative to the constructed model that is our understanding of the physical world I define X force to be Y vector.” I’m in pain just thinking about it. But from an armchair on a leisurely evening, where does this notion of force come from within our sensed experience, and isn’t it amazing that we have shorthanded all of that experience over generations into coherent symbolic frameworks that correspond in a meaningful way to physical phenomena beyond what we’ve experienced directly? We are a part of the structure of the universe, we are interacting with other parts of that structure, and the structure that we are can become aware of many aspects of the structure within and outside of ourselves. Moreover, we can formalize our experience of structure into language (usually, the language of mathematics). The usefulness of that language depends greatly on its consistency with measurements and observations. In this sense, considering human beings as integrally part of our universe’s structure, we are harnessing the internal consistency of that larger structure.
IDEA 17: PROOF & EPISTEMOLOGY
How do we know what’s true? What is our definition of truth? How do we mentally navigate between truths—for example, start at a true statement and get to a different statement that’s also true? I find that the answers to these questions differ between types of human activity. In math, the answers to these questions are extremely well defined for practicing professional mathematics. If you really want to examine the logical foundations of mathematics (which is really an application of philosophy, as I see it), one can start to question these things and consider the consequences for mathematical knowledge. But one doesn’t need to do this in order to arrive at true mathematical statements given definitions of truth that basically everyone doing math at an advanced level agrees on.
In poetry, by contrast, there is the concept of the “truth” of poetic language that is distinct from the truth-value of declarative statements that might be contained in a poem, but it’s a notion that depends on a given individual’s experience of a poem. A poem is something that can feel true; truth is sort of a personalized quantity measured by the visceral response it evokes. It’s pretty cool that groups of people do at least seem to experience certain poems in similar ways—at least enough for certain poems to emerge over time as great examples of the craft (though it’s important to be mindful that group judgements of artistic quality depend not just on individuals’ visceral reactions or reflections, but also sociopolitical context for the work, etc.).
In other fields, such as statistics or engineering, there is a notion of “realms” of truth; there’s the truth as we measure it and the truth of what is, independent of our measurement of it. And when we measure, we can only achieve approximate images of the “underlying” truth, which is usually some mathematical model of physical phenomena we have enough empirical evidence to accept as our ultimate truth model against which to judge our measurements.
IDEA 18: PERCEPTION AS ACTION
We are at least partially the architects of our internal realities. So, to the extent that we can control our own architect-ing, we can change our internal reality. But what are the mechanisms for doing this? One is to modulate the moment where we formulate what a stimulus is and react to it. Let’s consider the phenomenon of stage fright, as an example. For someone who has stage fright, there’s some set of stimuli that initiates a fear-based reaction and that has to do with being on stage or being in front of an audience, or both. For instance, perhaps the sight of a spotlight sets a person’s state toward fear due to their inner model of what an audience is: a judgmental, threatening, dangerous entity that is looking to humiliate them based on their performance. That pattern of response is as legitimate in some sense as any other, but it’s not helpful if you have to say, deliver a speech without stammering for your job. An active process that one can engage in before having to get up on stage might be to reformulate internally what an audience is. To me, for example, if I’m lucky enough to receive an audience, I formulate the audience to be an entity that has come to receive something from me. If I do my job right, I can help service their need! That’s a very positive and calming motivation for me. Once we recognize their existence, we can question our negative perceptions whenever we want to alter our experiences, actions, and outcomes.
WHAT OF ALL THIS?
I could easily spend an entire lifetime investigating any one of the above topics. It’s tempting to invent a false dichotomy where one thinks, “Well, I’m not a professional in field X, therefore I can’t contribute anything useful to X” (In my case, X = mathematics, physics, or philosophy most of the time). But I don’t think that’s sound reasoning. And at the very least, if we follow the claim’s suggestion, it functions as a cop-out for trying to think more interesting thoughts and to engage more deeply with certain aspects of the world. In other words, it’s an encouragement to back away from some of my favorite parts of life. To state my claim positively, I think it’s possible to contribute useful ideas to an area of human knowledge without necessarily having 40 hours every week to focus on that area. That doesn’t mean it’s going to be easy or even possible to break into the academic establishment or get published, and those factors have an enormous amount of influence over whether one’s work will actually be used and not just be useful in principle. There are things we do simply because we love and enjoy and feel called to them. I call myself an engineer these days, and I consider that true. Simultaneously, the more abiding truth seems to be that my calling is to seek reality. Whatever happens in my career, that’s happiness enough.