How to study reality

Zeitgeist 2 Minutes

SMBC on how to get access to the secrets of reality.

A teacher asking kids: who wants to learn something that will reveal the secrets of nature? They ask what. It's math! One kid says he'll only be interested if the examples use cookies.
Click through for source and red button caption: https://www.smbc-comics.com/comic/secrets-2

I guess this is philosophical comic sharing week for me.

Somewhat related, I’ve been slowly working my way through Sean Carroll’s The Biggest Ideas in the Universe: space, time, and motion. I know a lot of you got this info watching his video series during the pandemic. I did watch a few of those, but never had the time back then to just sit through all of it. So now I’m reading about it instead, which is fine since it’s a mode I learn better with anyway.

Carroll’s main endeavor in this book, and for the second one coming in May on quanta and fields, is to provide a relatively gentle introduction to the math involved in physics. The idea is to get you familiar with the mathematical structure of the theories, without having to learn to actually solve anything with them, that is, without having to become a practitioner. He’s not the first to take this approach, but the first I’ve seen be comprehensive about it.

Like many books in this genre, it does start off pretty gentle. Carroll doesn’t assume you know calculus or trigonometry, but he does assume a comfort level with algebra. And like many others, at a certain point, the book accelerates and feels far less gentle. Often steps get skipped, with the assumption that they’re obvious. (One benefit of reading rather than watching is being able to linger on these cases.) Despite my historical challenges with math, I’ve been able to follow along, at least until the chapters getting into spacetime geometry, which are turning into a struggle.

Among the hints about life I wish I could transmit to my younger self, one is the importance of repetition for learning anything, particularly something like math. I always found math homework agonizing, and so as a boy did the minimum possible. Unfortunately my foundational math classes happened during a period when math teachers weren’t keen on making students turn in their homework. I did eventually learn my lesson, you have to get your repetition in by doing the homework, but not before the damage was done.

Which isn’t to say I’m completely helpless. My education went through introductory calculus, trigonometry, statistics, and other basics. When my back is to the wall, I can usually get by. A career in computer programming helped somewhat. One of my issues with plain math is how abstract and seemingly unconnected it is with anything in the world. Learning to read equations as a sort of mini-program that reality executes helps me deal with them in scientific papers, or in books like Carroll’s.

This seems to be in line with a lot of discussions I’ve read over the years about finding a better way to teach math. Many of my teachers seemed to delight in the abstract nature that I found so unappealing, and so taught from that perspective. They saw a beauty in those abstractions, and seemed unable to grasp that most people simply don’t. Math education that starts and stays grounded in practical problem solving is probably a better avenue for most students.

So yes, the keys to understanding the nature of reality is math, and the keys to understanding math is doing the homework. As adults, those of us who didn’t get a great launch with it have to get by as best we can.

What do you think? Any tips and techniques you’ve found useful for dealing with math? Or for learning to appreciate its beauty in the way so many math teachers do?

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48 thoughts on “How to study reality”

  1. I am in the other camp. 🙂 For me, the whole point of maths is its “emptiness”, which can be filled in many ways. From talking to others, who were taught it lagrely via practical applications, I often found that they missed the larger picture — that of the structure of maths itself. That makes it difficult to progress e.g. from school algebra to abstract algebra with its fields, rings, algebras… Without a firm grasp of that “emptiness”, how can one truly appreciate such marvels as the connection between simple exponentiation (a short-hand for repeated multiplication) and trigonometry, out of which the Euler’s formula drops out so miraculously?

    Mind you, maths as an abstraction can be also taught badly (as I was, alas) as unconnected abstract silos. Took me a while to get over that. So, I suppose you need some balance between the two extremes , but I suspect there aren’t many educators capable of that.

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    1. I suppose if someone was a good enough teacher they could inspire that appreciation for the emptiness. But I think the better way to have sold me on math was as a superpower, for actually accomplishing things, as the comic makes clear. The closest it got in my education was having word problems at the end of the problem set. But all of the instruction had been abstract, so we dreaded the word problems. Really, the instruction should have started with them. They could save the abstractions for the end.

      That said, STEM instruction in the public schools where I live was always a serious compromise. I remember the science textbooks being the most bland things imaginable, not revealing any of the drama involved in discovering the facts. It was only years later that I came to understand that a lot of that was to avoid offending social conservatives on school boards. So it shouldn’t be surprising that the math education was in the same vein.

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  2. I loved this post and the cartoon that led it off–they definitely speak to my condition, as the Quakers say. I know I feel better when I spend a few minutes in the day working a math problem or at least watching someone else work a problem. Having said that, time gets away from me, and I often drop the math for weeks or months at a time. But, much like an indicator species in a healthy forest, it’s a sign of a healthy life when I have enough time in the day to think mathematically for a minute or two. I remember now that I even wrote about this late-in-life discovery a few years back: https://wordpress.com/post/thesubwaytest.com/2745

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    1. The link didn’t work for me, but I think I catch your meaning. And it has occurred to me that I could probably buy a math workbook or something and do a problem or two a day. Within a few years, I’d probably be a lot sharper at this stuff. Definitely something to consider. Like you, I suspect it’d fall by the wayside on may busy days, but keeping it as a todo might be a good idea.

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  3. @selfawarepatterns.com
    Good mathematicians don't always make good teachers. They didn't have to struggle with it, and so rely on a lot of "intuition". Those of us who struggle understand the smaller steps, building a far more conscious competence.
    Nothing is left as obvious, and the struggle is affirmed, not denigrated.

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  4. @selfawarepatterns.com Being mediocre at maths, but loving the conceptual power it gives has always put me in a funny position. I can see a solution, but can't always do the algebra. It's partly what got me into computers. I like Carroll's approach, because not everyone has to be able to solve the stuff. Also, that allows you to get into topics beyond real and complex analysis, as taught at school. Graph theory, number theory, etc.
    Teaching A-level computer science has opened up a lot of areas.

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    1. @grant_h @selfawarepatterns.com

      I think you're right about those for whom the math came too easily. I definitely got that feeling from my college math teachers.

      The algebra is my weak link too. I was just noting to someone that I wonder if buying an algebra workbook and doing a problem or two a day for a few months / years wouldn't help.

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      1. @selfawarepatterns @selfawarepatterns.com buying a book most certainly would. Having to stand up and teach topics has forced me to be a lot better at them than when I started. The laws of Boolean Algebra are slowly permeating my brain, but it's taken a few years.
        I am tempted, but daunted, at picking up high school algebra. I'm just a lot more interested in other branches of maths now. So group theory, and category theory. But what foundations are needed, since "everyone" starts with analysis?

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        1. @grant_h @selfawarepatterns.com

          I know what you mean about teaching. I've had to teach internal classes for employees, and I always come away with a much better knowledge of whatever the subject was. In one case, I was literally just a night or two ahead of the class in figuring it out.

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  5. I took a course in college, “Mathematical modes of thought”. The textbook was “Mathematics in daily life: making decisions and solving problems.” Which was a complement to my coursework in Behavioral Statistics (for Psychology). All of which led eventually to this – https://www.xlibris.com/en/bookstore/bookdetails/782364-election-2016 which solves the problem of hate in the world. For <$30. you can "get it". It's up to you. Please, if you choose to read it, consider writing a review and dropping it on Amazon, B&N, Xlibris, Goodreads, and WordPress. Thanks.

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  6. I did well in math until high-school integral calculus. At about this time, the social environment of the early 1970s became more interesting to me, and I more or less took Timothy Leary’s advice. I didn’t drop out exactly, but I certainly tuned in and turned on.

    By then I had glimpsed the astonishing interconnectedness of mathematics. A few years later, in an introductory logic course, the professor disparaged Plato’s Forms, and I came to their defense by pointing out that with math, there is often the sense of stumbling upon the same truth again and again from different directions, implying an independent reality that we can only discover.. He stroked his heavily bearded chin for a moment and then told me to “Hang on to that.”

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    1. Sounds like you had your act together much better than I did in high school. I meandered around, taking basic math courses, before finding out I really needed algebra if I was going to college. I then struggled in those algebra courses. I didn’t get to calculus until college (late 1980s). I also didn’t get any exposure to Plato until then, and it wasn’t much, just some limited reading in a western civ history course.

      All in all, there was a lot of stuff I wished I’d managed to get exposed to in school, that I only read about much later, and in a much more fragmented fashion.

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  7. Isn’t reality in the eye of the beholder? Or, is that something else?

    I don’t think the key is solely math, although math is useful. There seems to me to be two kind of things in the world: things that can be described with the math of physical laws; and things are too complex to be explained by laws where science has to use statistics to force the math on them. Both have equal weight in explanation. The world would be very boring if only math alone could explain it and it would be too chaotic and unstable if math explained none of it.

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    1. There are many things people take to be a natural kind that I think turn out to be in the eye of the beholder, at least to some degree (life, religion, consciousness, etc).

      I’d say the math is useful for a lot more than people like to admit. But it’s not all fundamental physics. My BS is actually in financial accounting, which is often called upon to figure out a way to quantize the value of things that people insist is incalculable. That’s why the comic includes “get rich” in the list of math’s superpowers.

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  8. I also loved math until I got to Integral Calculus in university and then I also followed Timothy Leary which, incidentally, I don’t regret. My loves were wide, spanning math, logic, philosophy, psychology, fine arts, music, and women. What I loved about math was its cleanliness or purity, what you called its ’emptiness’. I believe there is a connection between logic (including math) and aesthetics (including all forms of art and creativity). I really liked Sabine Hossenfelder’s book, “Existential Physics” for explaining reality from the physicist’s point of view, and it was very light on the math behind it, summarizing it in plain English. Highly recommended.

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    1. Just looked up Timothy Leary. (The name was familiar but couldn’t recall the details.) Interesting. And mostly before my time. (My teenage years were in the 80s.)

      I haven’t read Hossenfelder, except for her old blog posts. Carroll’s shot at explaining the math seems far from perfect, but I’m finding it a pretty good step up from the typical science book for popular audiences. At least I was before getting to the ramp up toward general relativity.

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  10. I’m sure my math skills are far worse than yours. For me I’m not sure it’s a matter of making the math practical. I took a class in college called “Applications of Modern Math” which I was told was supposed to be better for “English” people like me since it involved word problems and real life situations, but that class just about killed me. I didn’t learn a thing either. I focused on passing that class to the point that I blew off many of my other classes. I ended up with a paradoxical A in math and a C in creative writing. I wish I had blown off the math class instead.

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    1. I know the feeling. Although reading Carroll’s book, I discovered that I retained more from my old math classes than I thought. If you made an A, you might be surprised by how much you could recall if you needed to, or how fast you could relearn the basics. Of course, if you’re like me, you probably also have a mild PTSD that’s triggered when try to follow a mathematical explanation, much less try to solve anything.

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      1. Massive PTSD. Especially over word problems. I spent several hours each day with a math tutor and still had no clue what was going on. The reason I did well was because the guy graded on the curve, and I don’t think many bothered to get tutoring (even though it was a free resource).

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        1. You just reminded me of a class (I think it was chemistry) where the I once made a 39/100 on a test which turned out to be B on the curve. Lazy teachers making impossible tests and then adjusting to get the right stats. It was more hazing than teaching, and was one of the reasons I changed out of STEM into business.

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          1. Ouch. I’m sure I would’ve done much worse, if it makes you feel better. In my case, whatever I learned in math class I promptly dumped out of my mind the second the test was over. There was no way in hell I’d ever major in anything with even the tiniest speck of math in it. In your case, it’s a damned shame, since you would’ve been great in STEM.

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  11. I’m the kind who loves it abstract. My degree was in maths. I actually found all the more practical examples at school annoying and silly, because I couldn’t see why the people involved really needed to know, or why they couldn’t eg just measure the distance they needed to know. But just understanding the different patterns and the order that arises from them, is an almost mystical experience for me.

    Perhaps a better way to “sell” kids on maths is by explaining it as not just the solution to this particular problem, but a way of solving *any* analogous problem. I spoke with my grandad about this a little, and he suggested maths is a tool, and so it’s not so surprising it seems useless before you understand how it’s used, and how the tool can be used in so many ways. The difficulty of the sale is that maths really is so unlimited in its applications.

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    1. That’s what I hear from many math proteges.

      Stephen Wolfram, several years ago, put out something arguing that the best way to get kids interested in math was to have them use tools to solve problems upfront, and only later learn the details. (It’s similar to how most of us get drawn into software development.) Of course that was self serving, since he was advocating the use of math software to do it. But I do think he had a point. People don’t need to know everything they can solve with it, just an idea of the kinds of problems they could use it for.

      I’m pretty sure if anyone had told me that trig can be used to calculate the distances to the planets and stars, or that calc is used in rocket equations, I would have been a lot more interested. Who knows, I might have eventually developed an appreciation for those abstractions. But starting with it made math a very steep climb.

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  12. Interesting historical fact. As most folks know Gregor Mendel’s experiments in genetics eventually provided the key to explain and justify Darwinian evolutionary theory. Before the rediscovery of Mendel’s work biologists assumed “continuous variation” in species to explain evolution. But that theory kept running into problems. Mendel even sent a copy of his work to Darwin thinking it might help his research. Darwin ignored it. Maybe because Darwin was not good with math and Mendel’s work was mathematical. Mendel’s work eventually provided an explanation for discontinuous variations in species which explains Darwins theory.

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    1. Most of the stuff I’ve read says Darwin was never aware of Mendel’s work. But Darwin’s work created a stir, and he became a famous figure pretty quickly. He probably received an enormous amount of material from other researchers. Which meant he likely never processed Mendel’s, or if he did, didn’t grasp its significance. Unfortunately from what I’ve read, most of the scientific community didn’t either, until it was effectively rediscovered decades later. Sometimes a scientist never knows the effects of their work.

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  13. Yes, rain on my parade! I’ve read it both ways. Either way Mendel’s work was published and well known. So, Darwin should have been aware of it. But I’ve also read that Darwin was not good with mathematics which may explain why he didn’t read it or if he did easily make the connection to evolution. Also Mendel worked with hybrids and it was not clearly obvious that Mendel’s mechanism applied to mutations as well. But I do like the math aspect to the story and I’m sticking to it!

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    1. Sorry, didn’t mean to rain. And, not having done any deep dive on this, I’m certainly in no position to say that isn’t what happened.

      No worries on the protocol. It is good to do what you did and make sure the other person sees it. With WordPress’ new commenting system (which is mostly an improvement), anyone other than the blog owner won’t see replies that aren’t directly to them.

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      1. I do like all the personal stories about math. For me I was blessed with gifted mathematics teachers before entering university. Having a good background in math was a bonus for me. I was not intending to study any of the so-called STEM subjects. Nevertheless to graduate I was required to take one math and one laboratory science. I wanted to avoid spending any time in a lab so I asked if I could take 2 advanced math classes instead. My wish was granted and I saved all those (sorry STEM folks) boring hours in a lab. I know, I know, it was good for my education. But I want more time to read in my favorite subjects.

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        1. I don’t blame you. The one university lab class I took (chemistry) was awful. I hated everything about it, from the utter lack of camaraderie the rules enforced, to the stupidly uncomfortable goggles they made us wear, to the fact that it required six hours of time a week for only one hour of credit. I’d forgotten about that as one of the factors in deciding to change majors.

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  14. I’d like to add a clarification to potentially help straighten out a misleading idea that’s commonly implied, and certainly in that comic strip. The clarification is that actually math in itself tells us nothing about reality. This is quite simple to demonstrate. Imagine a fake world that’s run by physical laws just as our is, though things work differently in this fake world since causality functions differently. Observe that mathematics would remain just as true in that world as it is here, thus demonstrating that math tells us nothing about reality in itself. This should be the case for anything that’s true by definition, or “a priori”. To potentially learn about reality we must also rely upon our senses to assess empirical observations.

    In general I think it’s effective to consider math as a language, though an invented language that’s technically quite simple when compared against the highly advanced languages that we evolved to speak. Some might question this. Math a simple language? Observe that there’s nothing that can be done in math that can’t also be done in the far more advanced language of English, though massively more can be done in English than can ever be done in math. The difference in perception I think is that we evolved to speak languages like English, though not invented languages like math.

    That said, I’m very much on the “practical” rather than “abstract” side of things. I was maybe sufficient in high school math. In college I was disgusted by the arbitrary nature of the mental/behavioral sciences that most interested me, though fortunately physics inspired me by having exactly what soft sciences did not, or effective descriptions of how things work. So I squeaked by in several calculus courses while loving my lower division physics courses. Once I qualified for upper division physics however it became clear that my mind functions too slowly to keep up. So I completed a gratuitous degree in economics, since I could at least respect the understandings achieved in it, though have made my living doing construction work. Blogging seems to fill the academic void, and at least sometimes gives me hope that I might help improve our still pathetic mental and behavioral sciences.

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    1. Definitely math by itself isn’t sufficient. It has to be useful for making accurate predictions (or retrodictions) about empirical data, or at least predictions more accurate than random chance. But I don’t think the comic implies that only math is needed, just that it’s an important component. And it’s math that allows us to take the observed physics in labs and use it to extrapolate what happens inside the core of the earth, stars, in the early universe, or other contexts.

      On calling math a “simple” language, I guess it depends on how you’re measuring complexity. From what I’ve read, the ancient Greeks used regular language to discuss their mathematics. That worked, but it probably limited how complex things could get. Algebra and beyond likely required improvements in notation to be useful.

      I think what we can say is that math is much more precise. Regular language is often maddeningly ambiguous, ambiguities we regularly run into in our discussions here. Certainly the variables in math can have ambiguous meanings, but the structures and relations they’re in seem harder to “weasel” with than in straight language.

      An interesting hybrid are computer programming languages. They’re definitely a language, but because they’re instructions, you can’t really do ambiguity. (Although it’s easy to mean one thing while coding something else.) I’ve always had an easier time with programming than straight math, even though they’re often seen as related disciplines. As I noted in the post, I think it’s because a program is always about actual processes.

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      1. Sounds like we’re pretty square on this Mike. And actually what I meant about anything possible in math is also possible in English, merely references how every element in mathematics has an associated term in English that’s defined to function the same way. Of course actually doing math in the form of English gets ridiculous fast. You’d virtually always need a computer to crunch such math. But the point is that in an objective sense, math is a tiny language that isn’t able to describe much of what we commonly need to describe, while English is amazingly more broad and has endless potential to grow. Sometimes this can be difficult to appreciate since we evolved to speak natural languages but didn’t evolve to speak invented languages.

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  15. I know that it is only a comic and therefore in a sense only a joke. And yet to me it conveys a sense of infinite sadness. The Church of Technocracy truly believes it and Sam Altman seeks $7trl in his attempt to cure (rule?) the world.

    So no, the keys to understanding the nature of reality is not math. Or at least certainly not maths alone.

    Maths fails to answer almost all of the really important questions embodied in the single word “why”.

    Maths is a tool essential to science but has little use in fields of kindness or decency. Little use in happiness or sense of purpose.

    Of course, this little comic strip was not looking to become philosophical in any sense.

    But maths alone will not make for a “better” world and will only help us to “understand” its bare mechanics.

    The reality of life needs other tools. And even a physicalist would have to admit that the combination of atoms into entities of extreme complexity means that maths and physics need to be supplemented by many other fields to make “sense”.

    Many talk pure “science”. Some realise that such a limited outlook is a tragedy.

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    1. I segued from the comic into a discussion of the mathematics of physics, mainly because I happen to be reading Carroll’s book. But the sense the comic is talking about it is much broader. So it includes things like accounting, finance, economics, statistics used in social sciences, computational neuroscience, software engineering, and much more.

      In those senses, math is a tool, at least for most of us. But like all tools, it can be used for good or ill. The same quantitative techniques can coordinate the logistics of a massive relief effort after a natural disaster, the efforts involved in a military invasion, or the horrors of state sponsored genocide.

      I do agree that it’s not all math. But math seems crucial. And, as part of science, it does help us understand the world’s mechanics. It seems like understanding those mechanics has done more to help people live longer and better than any of the other approaches throughout history. Granted, it’s also provided more efficient ways to kill people than any other method, but knowledge is always a double edged sword.

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