by Jasper Gilley
“Mathematics is, to a large extent, the invention of better notations.” – Richard Feynman
Elon Musk recently started his most ambitious company yet. Considering he is currently the CEO of a company that seeks to make humanity a multi-planetary species, a company that wants to halt anthropogenic climate change, and a company that wants to eliminate urban traffic congestion, such a statement should (at the very least) elicit incredulity.
Elon’s new company, Neuralink, seeks to develop a brain-machine interface (BMI) – a device that would allow the human brain to communicate directly with computers (and vice versa.) I’ll let Wait But Why explain:
Everyone working on BMIs is grappling with either one or both of these two questions:
1) How do I get the right information out of the brain?
2) How do I send the right information into the brain?
The first is about capturing the brain’s output—it’s about recording what neurons are saying.
The second is about inputting information into the brain’s natural flow or altering that natural flow in some other way—it’s about stimulating neurons.
These two things are happening naturally in your brain all the time. Right now, your eyes are making a specific set of horizontal movements that allow you to read this sentence. That’s the brain’s neurons outputting information to a machine (your eyes) and the machine receiving the command and responding. And as your eyes move in just the right way, the photons from the screen are entering your retinas and stimulating neurons in the occipital lobe of your cortex in a way that allows the image of the words to enter your mind’s eye. That image then stimulates neurons in another part of your brain that allows you to process the information embedded in the image and absorb the sentence’s meaning.
Inputting and outputting information is what the brain’s neurons do. All the BMI industry wants to do is get in on the action.
The potential implications of the mass commercialization of BMI technology are incredibly significant. We could, for instance, alter our brains at will, consume experiences-on-demand (à la the Holodeck or the Matrix), and merge our consciousness with AI. These possibilities, however, are really too enormous for us to fully comprehend, especially as so much of the underlying technology has yet to be developed. I would actually argue that, because of the speculative nature of such possibilities, reading speculation on them gives you a better insight into the emotional predispositions of the speculator than the actual content itself. Thus, for a non-emotion-based discussion, it may be more helpful to consider a better-defined consequence of the advent of BMIs.
Notations
If you found a 10-year-old and asked him to evaluate the following mathematical expression, he’d probably be stumped:
However, most 10-year-olds probably have an intuitive understanding of the concept of area, and he’d probably be able to find the area of a triangle (which is essentially the same thing as evaluating the above integral.) Over the next eight years of his life (at least), this 10-year-old will dedicate a substantial amount of time to learning ways to express intuitive facts about the structure of the universe (which he already knows) in terms of abstract notations like the above integral symbol. If he’s on a standard math track, he’ll learn how to evaluate the above integral during his senior year of high school. But it’s important to note that he won’t have learned anything new about the structure of the universe − he’ll simply have learned a new way of doing an old trick.
Consider a slightly more concrete example. If I asked you to give a parameterization of a circle with radius 1, you’d probably do one of three things:
- You’d not know what a parameterization is
- You’d know what a parameterization is, but you wouldn’t know the parameterization of a circle with radius 1 off the top of your head
- You’d tell me that a circle with radius 1 can be parameterized as p(t)=<cos(t),sin(t)>.
If either of cases 1 or 2 applied to you, the equation p(t)=<cos(t),sin(t)> would mean very little. After reading case 3, you now know that it is a parameterization of a circle with radius 1, but it is by no means obvious why that is so. More importantly, it’d be an abstract fact residing in your brain, not an intuitive reality, and you’d probably forget it in 30 seconds.
Now consider the following GIF:
The elegance of this GIF lies in the way it translates abstract notation (such as cos(t)) into intuitive reality (that is, concept.) Obviously, students don’t learn math by GIFs − but doing lots of problems effectively achieves the same goal of giving one an intuitive understanding of a physical reality.
The same could be said of writing. Memorizing verb conjugations isn’t fun or interesting or intellectually stimulating by itself, but when you stop having to think about verb conjugations (because they’re intuitively obvious), you can begin using them to communicate (that is, transfer neuron firing patterns from your brain to someone else’s.) Essentially, mathematical notation is to physical reality as language is to ideas.
This is why math and writing are so reviled by some students. It’s not that they “don’t like school” or any other explanation they or society might believe. It’s simply that they still see notations as something to be mastered for their own sake, and are responding rationally to their worldview (the human brain is built not to memorize mathematical notations or language conventions, both of which are arbitrary, but to have ideas and observe the universe, both of which are very non-arbitrary.)
The Death of Notations
At its core, a brain-machine interface will be a device that implants neuron firing patterns from one brain into another. Fundamentally, ideas and intuitions about the universe are nothing more than neuron firing patterns − when both you and I understand the parameterization of a circle, identical (or at least similar) patterns of neurons fire in our brains. Therefore, a BMI connection would allow a math professor to give a farmer a perfect understanding of the parameterization of a circle, instantly, without any arduous math classes (which the farmer would probably be rather loth to undergo.) Likewise, an economist could give said farmer an understanding of inflation. More weirdly still, a musician could give the farmer the experience of listening to Tchaikovsky’s Sixth Symphony. This would probably be the most mind-bending for our poor farmer, because he previously did not realize what classical music is. He might have previously considered it a method of entertainment for the bored urban elite, but he ended up experiencing the suicide note of a homosexual man in late 19th-century Russia. I know from personal experience that one simply does not look at reality the same way again after listening to the entirety of Tchaikovsky’s Sixth Symphony.
If you can learn without studying, why would you have it any other way? That is, if you could intuitively know that the integral from 0 to 10 of the function 2x is equal to 100, and (of course) the significance of such a computation, why would you memorize the (ultimately mechanical) algorithm that derives that result or the (ultimately arbitrary) notation used to symbolize it?
I think that there is no good reason for you to do so. While all but the most diehard math fans will rejoice at this pronunciation, you quickly run into thorny problems when you start considering the implications of BMIs for your discipline of choice. Would there be no reason for literature fans to read Crime and Punishment, or music fans to listen to Tchaikovsky’s Sixth Symphony, or art fans to see Guernica?
Ultimately, it depends on the underlying difference between Guernica and the Fundamental Theorem of Calculus. The point of something subjective, like art, literature, or music, is that everyone experiences it in a different way. Two people looking at Guernica very well may draw very different nontrivial conclusions about it − and that is precisely why it is a great painting. For the most part, however, you either understand the Fundamental Theorem of Calculus or you don’t. So, in a BMI-equipped world, you might use BMIs to learn calculus, but still see Guernica in person, because any experience you get of Guernica via a BMI will ultimately be someone else’s, not your own.
That being said, perhaps there is a nontrivially subjective element to math/science/social science, etc., especially once you start attempting to produce new math/science/social science (that is, do research.) Perhaps, then, the mathematicians of the future will learn the Fundamental Theorem of Calculus using a BMI, but learn to prove Fermat’s Last Theorem more traditionally (whatever “traditionally” means in the context of math.) Or, better yet, perhaps they’ll be forced to prove Fermat’s Last Theorem from scratch (remind me not to be a future mathematician if that’s the case.)
One way or another, the advent of the brain-machine interface will bring about the biggest changes to human education, communication, and consumption of art since the printing press first appeared nearly 600 years ago.
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Featured image is the painting Guernica by Pablo Picasso.