Technological incompetence

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The idea of this little essay is to discuss the impact of technological advancement on human competence in various fields. I will first discuss a very specific example – mathematics development – in great detail, and then reel off several other examples that have had a similar trajectory. Finally, I will generalize and discuss the implications.

In the days of the nuclear bomb project at Los Alamos, Bethe and Feynman used to compete with each other in a game of mental arithmetic. There would be some problem – a logarithm to be calculated, a trig function value to look up, the square of some large number – and Bethe would know a clever method for getting the answer. He could interpolate between values of logarithms in his head, and had memorized tables of trigonometric functions. He knew various algorithms for taking squares of large numbers. Bethe was able to beat the younger Feynman by the sheer number of tricks he’d mastered.

The tricks of arithmetic Bethe had learned over the decades are now obsolete – if you need a logarithm, get any pocket calculator for 15 dollars. Researchers were happy to move on to calculators when it became possible, since the actual process of computing logarithms was never of any interest; only the end result had some practical value.

Methods become obsolete in other fields, too. A century ago many knew how to hunt; today, it’s a rarer skill. Farming was a common profession, but how many farmers do you know now? Riding horseback used to be an imporant skill, now it’s at best a hobby. In each of these examples, technology eradicated the need for the skill: hunting is unnecessary with industrial meat production, the average farmer now produces way more food than a century ago so that not as many are needed, and horses were mostly replaced by cars.

A skill, then, becomes obsolete when it is not practiced for its own sake but only for some practical end, and that end becomes achievable by other means. Many aspects of mathematics obviously fit this bill: the guy trying to figure out the total cost of his shopping today doesn’t care which way he gets the result, he just needs to know if he can afford his groceries.

However, mathematics is not only a practical matter, but a tool of thought, useful basically on its own and not only for its practical results. In the following, I will argue for the learning of "useless" skills to obtain mastery. I will focus on mathematics in physics in particular, since that’s my forte.

Use and misuse of tools in mathematics and physics

What’s in thinking?

What is a tool? Suppose you learned calculus by simply using Mathematica (the symbolic math program) for every integral. For you, the following mathematical expression

\[\begin{aligned} I = \int _0 ^\pi \sin (x) dx\end{aligned}\]

means "type Integrate[Sin[x],{x,0,Pi}] in to Mathematica"¹. You know that you need to do this if you wanted to figure out the area enclosed by the sin function and the x-axis over the interval $x\in [0,\pi ]$.

Is this an adequate way to think about integrals? I don’t think so. Learning calculus is not merely about knowing how to practically compute the integral of some set of functions.

The concept behind an integral is to break up a function in to smaller pieces along with the integration interval. By treating each piece separately, summing them and then going to the limit of the length of the pieces approaching zero, you can solve an integral.

This insight contains much information that simply knowing the value of this or that particular integral doesn’t. For example, it becomes quite obvious that integrals could be used to compute the length of any arc, since you can just break it in to smaller pieces and add them up. With a little imagination, you can see that areas of two-dimensional surfaces or three-dimensional volumes might be computed in the same way.

What does that have to do with learning useless skills, in this case, computing integrals by hand? It’s almost impossible to get a handle on the conceptual foundations of integrals without actually doing them by hand.

Suppose you wanted to apply these concepts for computing the integral of some function over a complicated path. To do this, you have to know about parametric functions and how to change integration variables; Mathematica won’t do that for you. You have to first set up the problem in such a way that solving it becomes automatic. You won’t know how to do that without having actually computed a bunch of integrals by hand; only then will you gain a strong intuition for when an integral is ready to be done by computer.

Let’s consider another example, derivatives. High school students know a bunch of formulae for derivatives of simple functions – polynomials, trig, and so on. They are considerably easier to calculate with a computer than integrals are, so should you know how to find formulae for derivatives?

Once again, the concepts are intimately tied to the computation. One popular way to get intuition for derivatives is to find the $\frac{d }{dx }x^2 = 2x$ formula by the following process:

\[\begin{aligned} y=x^2 \implies y + dy = (x+dx)^2 = x^2 + 2xdx + dx^2 \\ \implies dy/dx = 2x\end{aligned}\]

where the last implication follows from $dx^2$ being vanishingly small since $dx$ is assumed to be very small already. This gives you intuition for why the derivative of $x^2$ is $2x$ by explicit computation, much like parametric functions give intuition for integrals. Only by actually finding a bunch of derivatives and then applying them to problems will you master the concept of derivative.

In pure mathematics and math as applied to physics you are often dealing not with this or that particular integral, but rather a class of integrals, and you have to manipulate them without specifying exactly the integral you’re doing. In that case, knowing general rules of computation for integrals is indispensable. Likewise having a strong grasp on derivatives is crucial when dealing with functions simply satisfying some set of constraints.

These examples hopefully convince the reader that knowing how to calculate something and understanding it conceptually are linked. Algorithmic facility leads to conceptual understanding. This is unsurprising; is not a cowboy much more likely to have an in-depth understanding of horse riding than I am? Yet there seems to be nothing conceptually difficult about sitting atop a horse!

Familiarity and practice lead to understanding. Having done a bunch of practical computations gives you a nose for problems involving the relevant concepts. I will next seek to demonstrate this idea with historical examples where algorithmic² development lead to conceptual development in both mathematics and physics.

Plan C

The famous early 20th century mathematician Felix Klein divided³ the development of mathematics in to three plans:

• Plan A: dividing mathematics in to distinct subfields and working on each separately, relying little on concepts from other fields;

• Plan B: seeking connections between different fields, trying to create unifying frameworks and applying techniques from one field to another;

• Plan C: algorithmic development, trying to figure out how to compute things efficiently.

Is plan C the least of these, something to be cast aside as a mere "formal development" of great practical value but of no fundamental importance? No!

In the 1600s, Mercator converted the fraction $(1+x)^{-1}$ in to a series, and found thereby a series for $\log (1+x)$ via the following integral:

\[\begin{aligned} \log (1+x) =\int _0^x \frac{1}{1+x}dx \end{aligned}\]

and Newton later applied a similar algorithm to the general binomial series, obtaining a host of other similar identities. Newton did not seek to base these ideas on some rigorous foundation nor did he use tools from other disciplines; the algorithm Mercator had used suggested it could be used over and over again in more complicated cases, and so Newton did.

Leibniz’s invention of the symbols $d/dx, \int dx$ was clever extremely suggestive; the use of infinitesimals immediately suggests a connection between derivatives and integrals and allows one to find derivatives for various functions. This, too, is purely an algorithmic development: of course we might denote derivatives, second derivatives and so on by any number of less intuitive symbols.

A similar story is found in the adoption of the Hindu-Arabic numeral system in Europe. Al-Khwarizmi wrote a very influential book, "Algoritmi de Numero Indorum"⁴, which contained algorithms for working with our modern numeral system. They were adopted because they turned out to be superior for practical calculations, but it would not be difficult to argue that this algorithmic improvement was also a conceptual one.

We need not go all the way to medieval times to find examples. Gröbner bases are a computationally expedient tool used in commutative algebra and algebraic geometry. The development of this method was purely algorithmic: it casts old concepts in to a more computationally convenient form. Yet it has also improved our conceptual ability in its fields of application; they are especially helpful with understanding the properties of sets of non-linear equations.

Let’s talk about physics next. Feynman diagrams are the greatest example of algorithmic-conceptual connection in the recent history of physics. Feynman did not derive the rules for drawing the diagrams by any rigorous method. He employed a completely algorithmic and intuitive approach, coming up with the rules as he needed them to get to the correct result. The diagrams are now ubiquitous in particle physics: they visually capture what the process "looks like", and make it easy to perform any scattering calculation entirely algorithmically. Indeed, the process is consistent enough to be implemented on a computer. The underlying math – path integrals – also provide the basis for some philosophical interpretations of quantum mechanics.

Dirac’s work, too, had heavy influence from the algorithmic way of thinking. He came up with the "bra-ket" notation now universal in quantum mechanics to facilitate easier calculations. This notation allows its user to see expressions that may look quite different as being manifestations of the same underlying quantum states, a great conceptual improvement over the previously messy and disparate methods of doing quantum calculations. A related contribution is the "delta function", which Dirac pulled out of thin air because it was useful for computation, ignoring rigor; this "function" is in fact not a function in the mathematical sense, but has found fruitful applications in the theory of distibutions.⁵

Another example from physics can be found in the development of tensor networks, which are a computational method used in many fields of quantum mechanics. This development began as purely algorithmic: there was a need to find a way to solve large quantum systems. Yet they have also facilitated theoretical studies of quantum information and allowed the derivation of strict bounds on quantities like entanglement⁶.

We see, then, that a facility with practical calculation is not just a help for learning old concepts, but rather it has actually contributed much to the development of mathematics and physics as a discipline. Calculation helps understanding concepts and pushing the boundaries of existing methods of computation may generate new concepts or illuminate old ones in surprising ways.

Induced incompetence

Clearly, then, it is right to teach students computation by hand. It is not an obsolete skill. Teaching the logic behind computer algorithms would also be helpful. Göbner bases are a fruitful example of what may happen when a competent mathematician applies themselves to an algorithmic development, and so are theorem provers⁷.

What happens when one relies too much on tools and not enough on their own work? Incompetence. I am an incompetent horse rider and an even worse farmer because I have never practiced these skills.

I think that in the case of math and physics education – where understanding is to be prioritized over concrete results – there is a general decline in the ability of students as a direct result of the excessive availability of tools. Looking back at Finnish matriculation exams from 100 years ago, the questions are no easier than today, yet the students then had no possibility to use calculators of any kind. The scores have not increased, on the contrary, they keep decreasing. How is it that students with considerably better tools keep getting worse?⁸

Tools allow incompetent students to stay in college like a nice suit and a winning smile keep a conman in business. The ability to get answers from Google and ChatGPT does not mean you’ve learned anything of value.

Now that we have thoroughly discussed just one aspect of just one field, let’s try to extend this idea a little bit. The loss of some areas of mathematical competence in parts of the population is not, after all, that dangerous.

What other potentially significant skills are we losing as a consequence of technology? Some may find value in primitive methods in themselves – there are movements of anarcho-primitivists, for example. I will limit myself to those skills which I think almost everyone would value instead of talking about the intrinsic value of primitive techniques.

Other examples

Reading

There seems to be some evidence that heavy technology use impairs reading skill in kids ⁹. There’s a "decline by 9" in reading habits¹⁰

Not all of this is, of course, technology. The more readily available entertainment you have the less you’re going to reach for a book, though; at least this has been my experience. It is much harder even for me to focus on reading books, even though theoretically I am capable of self control¹¹.

Focus

Smartphones seem to reduce the ability to focus¹²¹³, though more thorough research would be welcome.

Regardless of studies, at least I personally experience distraction both by smartphone and funny Youtube videos when working on a computer. When a pleasurable distraction is readily available, it’s easy to take up on it. The ability to intentionally focus is, I should think, an intrinsic good; the availability of entertainment technology is eroding it.

Physical fitness

Modern life in a wealthy society is sedentary, which means that a chunk of the population doesn’t get enough physical exercise.

Manual labour used to be more common than it is today; in countries where it is still the norm, people are much more likely to meet physical exercise recommendations¹⁴. The problem is becoming global; more than 80% of adolescents do not meet WHO’s recommendation for physical activity¹⁵. A horrifying percentage of adults in the U.S. are obese¹⁶.

Physical fitness is linked to both physical and mental health. The ease of modern life – precipitated by technology – has made us sicker even as medicine has increased our lifespan. While it may be true that many aspects of physical labor are only valued for the ends they produce, the overall effect is to destroy something intrinsically valuable.

What then?

In general, it is bad when three conditions are satisfied:

1. Technology replaces a skill.

2. That skill was useful either directly (e.g. the math examples) or indirectly (physical fitness from manual labor).

3. The benefits of the skill aren’t obtained from the technological replacement or anywhere else.

Of course, condition (3) should not be taken to mean the benefits couldn’t be obtained anywhere else, it’s just that a good chunk of the population will not. Most of us could exercise more, we just choose not to.

In my view, the simplest way to reduce the harmful effects of technology is simply to moderate its use. Children can’t do this for themselves, so an adult has to. As for adults, I have to admit, to my shame, that I don’t have enough self control around computers; I work by pen and paper as much as possible, even writing drafts by longhand simply to avoid even the slightest temptation of technology. I find myself even wanting to look at the weather report when working. The weather report! Why? I don’t work outside! Whatever it says will virtually never change what I’m going to do, and yet since it’s so conveniently available, I take a quick glance, after which I check the news, and at that point we are not far away from a Youtube tutorial on how to roll a burrito.

Technology is a paradoxical thing: it makes the services we use (food, medicine, etc) better, but personal technology seems to make things in many respects worse. For now, my recommendation is to avoid technological incompetence simply by not using technology - I recommend using as little of it as possible, especially when trying to learn something like physics or mathematics.

1. Mathematica is often used by physicists for rote tasks such as computing complicated integrals. ↩

2. In the general sense of ordered calculation, not just computer algorithms. ↩

3. F. Klein, "Elementary Mathematics from an Advanced Standpoint", 1932 ↩

4. Al-Khwarizmi, c. 825; the title is translated from the original Arabic, and this version likely did not contain exactly the same content as Al-Khwarizmi’s original work. ↩

5. D.S. Jones, "The theory of generalised functions", 1982 ↩

6. F. Verstraete, "Criticality, the Area Law, and the Computational Power of Projected Entangled Pair States", Phys. Rev. Lett 96, 2006 ↩

7. See the "Lean" theorem prover by Microsoft, and the fun "Natural numbers game". ↩

8. There is an obvious selection bias here, since a hundred years ago few people were even allowed to get a high school degree. Yet you would expect that such a dramatic improvement in tools would yield competence at the high school level. ↩

9. https://www.edweek.org/teaching-learning/screen-time-up-as-reading-scores-drop-is-there-a-link/2019/11 ↩

10. https://www.theatlantic.com/books/archive/2023/03/children-reading-books-english-middle-grade/673457/ ↩

11. Debatable, but at least my brain should be as functional as it’s going to get at this age. ↩

12. Skowronek et al. "The mere presence of a smartphone reduces basal attentional performance", Scientific reports 13, 2023 ↩

13. "Smartphones and attention, curse or blessing? - A review on the effects of smartphone usage on attention, inhibition, and working memory", Comput. Hum. Behav. 1, 1–8 (2020) ↩

14. R. Guthold et al., "Worldwide trends in insufficient physical activity from 2001 to 2016: a pooled analysis of 358 population-based surveys with 1.9 million participants" The Lancet, 2018 ↩

15. World Health Organization, "Global status report on physical activity", 2022 ↩

16. https://www.niddk.nih.gov/health-information/health-statistics/overweight-obesity ↩