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I am reading this article by Donald E. Knuth and get stuck by this sentence:

Our mathematical language continues to improve, just as “the d-ism of Leibniz overtook the dotage of Newton” in past centuries.

I know I can get some hints from the reference 4 but I don't think I can fathom the history clearly. Could anyone please help explain what that means in simple English? Thanks in advance.

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  • $\begingroup$ It is interesting that European authors still use the dot to indicate time derivatives. $\endgroup$
    – ACR
    Commented May 29, 2019 at 19:39
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    $\begingroup$ @M.Farooq So do American authors and Wikipedia, primes are also routinely used. Knuth had a wishful thinking moment, I am afraid. $\endgroup$
    – Conifold
    Commented May 30, 2019 at 3:46

3 Answers 3

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It is a play of words by Charles Babbage. Deism was a religious belief or rather a movement promoting the idea that God exists but it does not interfere with whatever happens in this world. This old philosophy according to the Wikipedia "...asserts God's existence as the cause of all things, and admits its perfection (and usually the existence of natural law and Providence) but rejects divine revelation." Now, Babbage writes d-ism which sounds the same, but now the reference is Leibniz who was one of the founders of calculus (there was a major controversy whether Newton invented calculus or Leibniz). We still use his d's in differential calculus. Newton on the other hand was using dots in his calculations for symbolic purposes. Dotage also means someone who is becoming senile (see comments for other meanings). Basically, the writer is making fun of Newton, who is an Englishman himself, and asking his peers to adopt the symbolism of Leibniz.

P.S. In some European textbooks, one might see dots over a variable to show a differential operator usually with respect to time. No "d" is used then.

Google Books on The Methodist Review

Google Books, The Methodist Review

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    $\begingroup$ " Dotage also means someone who has lost his mind." - That's not quite correct. en.oxforddictionaries.com/definition/dotage defines it as "the period of life in which a person is old and weak". In other words, Newton was born 170 years before 1812, but according to the Analytical Society members, English mathematics had not progressed during that time. $\endgroup$
    – alephzero
    Commented May 30, 2019 at 10:45
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    $\begingroup$ The primary meaning of dotage listed in the unabridged version of Oxford is " The state of one who dotes or has the intellect impaired, now esp. through old age; feebleness or incapacity of mind or understanding; infatuation, folly; second childhood; senility. " The second meaning is "The action or habit of doting upon any one; foolish affection; excessive love or fondness." All these seem to fit in the discussion above. $\endgroup$
    – ACR
    Commented May 30, 2019 at 10:53
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    $\begingroup$ Becoming feeble-minded due to age is the primary definition, yes, but “feeble-minded” and “lost their mind” are two very different states of mental degeneration, at least in connotation. The former implies someone is slow, often forgetting things and having a difficult time grasping concepts in front of them and adapting. The latter implies ranting and raving, quite possibly hallucinations or delusions, and so on. $\endgroup$
    – KRyan
    Commented May 30, 2019 at 13:49
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    $\begingroup$ It should be Leibniz and Newton, you have the first consistently wrong and the second at least once. $\endgroup$
    – Lutz Lehmann
    Commented May 30, 2019 at 18:42
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The first answer is excellent but just for context on the actual math:

Newton notation for derivative of f(x): $ \dot f(x) $

Leibniz notation for derivative of f(x): $ \frac {df}{dx} $

Newton's notation is fine for very basic single variable derivation and generally the derivative for the described cases but the Leibniz notation is general purpose and demonstrates the process of finding complex derivatives/partial derivatives very clearly. So the newton notation was abandoned (in most cases other than a single variable) in favor of the more consistent and concise notation. Thus our programmer friend Knuth insinuates that we should judge something's value with logical evaluation and not emotional attachment (England's mathematicians went on to use Newton's notation for nationalistic reasons since Newton was an English lad but Leibniz was German/Prussian/Subject of the holy roman empire /Not English so their calculus development stagnated compared to the rest of Europe according to an old math professor of mine, don't know the validity of this statement though).

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    $\begingroup$ More concise? You can't be more concise than a single dot. $\endgroup$
    – Peter Taylor
    Commented May 30, 2019 at 7:15
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    $\begingroup$ It's not quite accurate to ascribe the notations you wrote to Newton and Leibniz, since neither used the "parenthesis x" part. That came later with Bernoulli and Euler. Moreover, I suspect that Newton would have written $\dot{y}/\dot{x}$, for what Leibniz would have written $dy/dx$. $\endgroup$
    – Michael Bächtold
    Commented May 30, 2019 at 7:15
  • $\begingroup$ Speaking of TeX, your example nicely indicates why $\dot x$ and $\dot y$ are good candidates for dotting, but $\dot f$ isn't, namely, that the dot on the latter, while physically centred, appears to be floating off somewhat to the left. I thought \Dot f might fix this, but MathJax appears not to recognise it. $\endgroup$
    – LSpice
    Commented May 30, 2019 at 14:13
  • $\begingroup$ @PeterTaylor you need a single dot only in the case of a single input variable function, which is hardly the most common case in much of applied science, and I would assume science too. A great example is the total derivative. The Leibniz notation carries over perfectly for this case, while for the Newton notation you would have to do something like $\{\dot{f}(x,y)\}(x)$ instead of simply $df(x,y) / dx$ $\endgroup$
    – ShinyDemon
    Commented Feb 1, 2023 at 9:11
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I think the two other answers here when combined offer a complete explanation of Knuth's word game.

He calls Leibniz notation "d-ism" to combine the English word "deism" with Leibniz's use of the prefix $d$ in $dx$ to suggest an infinitesimal change in $x$. So Leibniz would write $dx/dt$ for the rate of change of $x$ with respect to time $t$. Believing in infinitesimals is akin to believing in a deity - but writing with infinitesimals is good notation because it is extraordinarily productive mathematically - it helps you understand what is really happening and suggests new theorems. Then if you wish you can provide formal proofs that replace the theological infinitesimals with limits. (It is also possible to make the reasoning with infinitesimals rigorous.)

That notation prevailed over Newton's use of $\dot{x}$ for the rate of change of $x$ with respect to time (in part) because it was so suggestive. The pun for Newton is the connection between the dot over the $x$ and "dotage", suggesting senility or old age. Newton's notation (mostly) faded away.

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