Don't Make Me Think!
The brain is an organ for learning. If you do not feed it new ideas, it will atrophy.
If you never use your muscles, they atrophy, too. Some people do not like to go to the gym or play sports, and that is fine, as long as they at least walk regularly to stay healthy. Unfortunately, many people do not learn much after they leave school or even university. It is as if they scream: “Please, don’t make me think!”
With infinite feeds, it is tempting to indulge in content that does not make you think. Doomscrolling is the path of least resistance. The average person spends three-quarters of an hour each day at work and more than two hours overall on any given day doomscrolling. But it offers no benefits: it wastes your time and causes anxiety, depression, sleep problems, and headaches.
What if you read a book or article with fresh ideas instead? Both fiction and non-fiction are fine, as I shall show.
New words
When you read prose, you are not merely turning pages, but encountering new language and ideas. Even for adults with large vocabularies, any decent text contains unfamiliar words that represent novel concepts. Linguists estimate that the 8,000–9,000 most common word families cover 98% of general texts. A word family includes inflections and derived words. For instance, under the dictionary entry to play you will find play, plays, played, and playing, yet the larger word family also includes player and (un)playable.
For academic texts (e.g. journal articles and textbooks) that figure needs to be bumped to 10,000 word families or more. Note that television and film only require half of that. So, if you know 9,000 word families, you’ll understand nearly all tokens or words in almost any article and book, but 2% still slip through.
If you do not care for mathematics, please skip the next section at your leisure.
A mathematical detour
Word frequencies follow power law distributions: the word frequency is inversely proportional to rank of a word. This is known as Zipf’s law. Phrased differently, the probability of the word at rank \(r\) is \(\mathbb{P}(r) \propto r^{-s}\) with \(s\) specific to each language. To properly normalize that, we need to sum over all ranks up to \(K\), which is done with generalized harmonic numbers:
\[H_{n,m}=\sum_{k=1}^{n}{\frac{1}{k^{m}}}.\]So, let’s say we wanted to know the coverage of \(K\) word families in a language with \(N\) word families in total, we could simply compute \(H_{K,s}/H_{N,s}\) for any given \(s\).
We can use HarmonicNumber[n,m] in Mathematica (or Wolfram Alpha) or harmonic(n,m) in Maple.
In Python, we can employ the fact that \(H_{n,m} = \zeta(m,1)-\zeta(m,n+1)\), the Hurwitz zeta function, though the standard implementations in scipy.special.zeta or mpmath.zeta often lead to numerical problems.
In reality, English has a double power law with a much steeper drop-off after the 10,000 most common words. The head slope is more or less unity. Note that these estimates are based on Google Ngram data, which contains tokens, not lemmas (i.e. dictionary entries) or word families.
This means that in reality, we cannot rely on a single exponent for the probability:
\[\mathbb{P}(r) = \begin{cases} a r^{-1}, &1\leq r \leq \rho,\\ b r^{-s}, &\rho < r \leq N. \end{cases}\]Here, \(\rho\) is where the drop-off becomes steeper (\(s>1\)), and \(a,b\in\mathbb{R}\) are normalization constants.
Obviously, we demand \(a\rho^{-1}=b\rho^{-s}\) for continuity, from which we find that \(b=a\rho^{s-1}\). The total probability must be unity, so \(a\sum_{r=1}^{\rho}{r^{-1}} + a\rho^{s-1}\sum_{r=\rho+1}^{N}{r^{-s}}=1\). We can factor out \(a\) and rewrite that with the generalized harmonic numbers as:
\[a \left[ H_{\rho,1} + \rho^{s-1}\left( H_{N,s}-H_{\rho,s} \right)\right] = 1,\]from which we can read off the normalization constants \(a\) and \(b\). The coverage \(C(K)\) for \(K\leq \rho\) is straightforward: \(C(K)=aH_{K,1}\). To obtain the coverage \(C(K)\) with \(K>\rho\), we must calculate \(a H_{\rho,1} + b\left(H_{K,s}-H_{\rho,s}\right)\). The first term is merely the complete head and the second one the partial tail up to \(K\).
How can we estimate \(s\)? With a fixed \(\rho\), we can calibrate the model to fit through \((K, C(K))=(9000,0.98)\). The key unknown is \(N\), the number of word families.
How many word families are there? The Oxford English Dictionary (OED) offers 301,100 canonical forms. The OED itself mentions half a million definitions, though these include compound words, multi-word phrases, archaic spellings, obsolete words and phrases, dialectal variants, and many (place) names. From these 300k headwords, we can estimate there to be 80,000–200,000 word families. That is because an adult native speaker of English knows about 42,000 lemmas from 11,100 word families. Figures for the family-to-lemma ratio go from 1.5 to 3.8, depending on whether or not to include less common derivations. So, 9,000 words families amount to 13,000–34,000 lemmas, which is a wide range indeed, though often the lower end is understood for the 98% coverage mentioned earlier. For instance, in the earlier example of play, we can also include airplay, cosplay, horseplay, overplay, playback, and so on. Let’s settle for 140,000 word families, the mean of the range.
Let’s summarize what we have in terms of word families rather than words:
- \(N=140,000\) word families in the OED.
- \(\rho=4,000\) word families, which is based on the middle range of family-to-lemma ratio and the typical inflection point around 10,000 words.
- \(K=9,000\) word families that lead to \(C(K)=0.98\) coverage.
With a little bit of tinkering we can estimate \(s\approx 2.51\).
The effect of reading one hour a day
Now we can calculate the coverage and number of novel word families in a text of 10,000 words, which is roughly an hour of reading:
| Vocabulary (\(K\)) | Coverage \(C(K)\) | New words families |
|---|---|---|
| 9,000 | 0.980 (98.0%) | 201 |
| 15,000 | 0.991 (99.1%) | 91 |
| 20,000 | 0.994 (99.4%) | 58 |
| 30,000 | 0.997 (99.7%) | 30 |
| 50,000 | 0.999 (99.9%) | 12 |
Even people with a massive vocabulary of 50,000 word families (or 190,000 lemmas) encounter up to a dozen new word families in an hour of reading prose. Phrased differently, if you remember one word per day, then after only one year of reading one hour per day instead of doomscrolling, such a person’s vocabulary is 1% richer. For a person with a vocabulary of only 9,000 who only retains a single new word family per day, that’s a 4% boost to their vocabulary. A few new words a day keeps the brain rot away.
The key insight is that getting smarter by just a fraction each day, let’s say 0.01%, compounds. After one year, you are \(1.0001^{365}\approx 1.04\) or 4% smarter. That may not sound like much, but after five years that’s a whole 20%, and after a decade that is 44%. A little bit of dedicated reading can make a huge difference, especially if you mix up genres where you are more likely to encounter new words and concepts.
A word-a-day calendar won’t work, though. For a new word to stick around in your mind, you need to see it 10–20 times in context.
Sadly, more than half of all adults have not picked up a book in the last year; one in ten has not even read a book in the last decade. The generation that has grown up addicted to social media on smartphones is now teaching their children to never pick up a book because it does not provide instant dopamine hits.
Why many do not read one hour a day
Humans default to mental short cuts, which turn them into cognitive misers who prefer the instant gratification of social media through our bias for negativity, which doomscrolling taps into. There are also metabolic reasons why people abhor thinking: it costs the brain a bit more to read or think, but not as much as you might expect: about 5%. The basic brain activity saps most of the energy. In fact, evolution has made it preferable to avoid spending extra energy on the brain, because at only one-fiftieth of the body’s mass, it consumes 20% of the energy. For children that figure is almost half, which shows why children and adolescents are easy targets for mindless social media usage: their brains try to conserve as much energy as possible, as they already have a much higher energy expenditure than adults.
Why care?
Evolution’s stingy voice inside our head whispers continuously: “Don’t make me think!” But you can occasionally pause scrolling and engage your mind, whether by reading, questioning, or wrestling with a tough problem. It does not cost much more energy or effort, it’s just evolution’s voice from thousands of years ago, so feel free to silence it once in a while.
But why should you really care? First, there is Alzheimer’s. Keeping your brain stimulated enough to stave off that awful disease is worth it. It is heartbreaking to see loved ones’ minds deteriorate: people have pages of their past torn from their minds until they have no more memories or recollections of their loved ones left. Families suffer immensely in such situations.
Second, while you might love to see a six-pack on your abdomen and a bulge on your upper arms, your abs and biceps are very unlikely to change the world, but your mind might make a difference. As an overly complex 20W breathing apparatus, it is doubtful your brain can have any positive impact on the world. So do not waste the power between your ears.
Once in a while, you just might stumble upon a decent idea. After all, thinking does not hurt, except when you discover you might have been wrong, and that pain is far less in the long run than the stagnation born of ignorance. So, put your phone down right now and read a few pages. Your brain thanks you.