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The intention of this question is to find practical examples of improved mathematical notation that enabled actual progress in someone's research work.

I am aware that there is a related post Suggestions for good notation. The difference is that I would be interested especially in the practical impact of the improved notation, i.e. examples that have actually created a better understanding of a given topic, or have advanced actual research work on a given topic, or communication about results.

I would be interested in three aspects in particular

(1) Clarity and Insights: Improved and simplified notation that made structures and properties more clearly visible, and enabled insights for the researcher.

(2) Efficiency and Focus: Notation that created efficiencies (e.g., using less space and needed less time, dropped unnecessary or redundant details).

(3) Communication and Exposition: Improved notation that supported communicating and sharing new definitions and results. And notation that evolved and improved in the process of communication. Would you have any practical examples of this evolving process, including dead-ends and breakthroughs?

Edit: Have received great examples in the answers that illustrate what I am interested in. Very grateful for that!

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    $\begingroup$ I don't completely understand how different this is from the previous question you linked. That already has plenty of answers. $\endgroup$ May 21, 2020 at 13:58
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    $\begingroup$ @DonuArapura It seems like this one is asking for specific personal examples rather than general suggestions. $\endgroup$
    – JoshuaZ
    May 21, 2020 at 14:37
  • $\begingroup$ @DonuArapura thanks a lot for you comment. Indeed, as JoshuaZ says, my key interest is actually in examples that had an actual and practical impact. Maybe let me rephrase my question a bit to make that clearer $\endgroup$
    – Claus
    May 21, 2020 at 18:14
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    $\begingroup$ @ClausDollinger, I edited the title to remove “your”, since the question in the body is in not specific about whose research is impacted, and since you say that the Feynman diagrams are a great example, even though that answer says nothing about that responder’s research. $\endgroup$
    – Matt F.
    Jun 13, 2020 at 12:04
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    $\begingroup$ I remember reading somewhere that J W Gibbs was quite obsessed with notation. Does anybody know if this is indeed right and to what extent? $\endgroup$
    – lcv
    Jun 15, 2020 at 5:40

12 Answers 12

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There is a notation that had an immediate and profound impact on research in algebraic topology, later algebraic geometry, and was eventually adopted by all areas of mathematics: the introduction of arrows to denote mappings. Compare $f \colon X \to Y$ with $f(X) \subset Y$, which is what was used previously. It meets all three criteria mentioned by the OP and is recognized by every mathematician.

Just as importantly, the use of arrows led to commutative diagrams, without which many parts of modern mathematics are now inconceivable. I mentioned this before in an answer here.

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  • $\begingroup$ that’s a great example, thanks a lot. And thanks for the link, I was surprised to see how relatively “young” this notation is $\endgroup$
    – Claus
    Jun 15, 2020 at 3:48
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This might be too old to qualify but I have always felt that the decimal notation is a wonderful thing!

From a modern perspective, when it is taught to everyone at a very young age, it might be hard to appreciate how interesting and useful it is. The fact that it is at all possible relies on the following simple but non trivial theorems: $$\sum_{k=0}^n(a-1)a^k = a^{n+1}-1$$ and for $|x_k| \leq a-1$ $$\sum_{k\leq n}x_k a^k = 0 \implies x_k =0.$$

The notation is, in fact, mathematically optimal for representing numbers in terms of information theory!

I think it is a hallmark of the greatest notations to encode non trivial theorems so that they seem like trivialities. This leads to great cognitive savings in my experience.

It is very clear that it led to vast improvements, not only in mathematical research but human society as a whole.

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    $\begingroup$ really great example. I never thought about it! Thanks a lot for Bringing it up! $\endgroup$
    – Claus
    Jun 15, 2020 at 3:37
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May I offer the four-vector notation as an example from physics? Quoting Feynman:

The notation for four-vectors is different than it is for three- vectors. [...] We write $p_\mu$ for the four-vector, and $\mu$ stands for the four possible directions $t$, $x$, $y$, or $z$. We could, of course, use any notation we want; do not laugh at notations; invent them, they are powerful. In fact, mathematics is, to a large extent, invention of better notations. The whole idea of a four- vector, in fact, is an improvement in notation so that the transformations can be remembered easily.

The Feynman Lecture on Physics, Volume 1, Chapter 17.

A trigonometric notation that Feynman invented in his youth did not catch on. And then of course Feynman diagrams are perhaps the most celebrated example of an impactful notation in physics.

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    $\begingroup$ thanks a lot for providing a great example and the link. The quote from Feynman is spot on and resonates a lot with me $\endgroup$
    – Claus
    May 21, 2020 at 18:16
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Richard Stanley’s symbol for number of ways to make choices with replacement. Looks like a binomial coefficient but with double parentheses. (More in my blog.)

It's a calculation that comes up frequently -- I became more aware of just how frequently it comes up when I started giving it it's own symbol -- and it helps to give it its own notation, even though it reduces to a simple expression in terms of binomial coefficients.

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  • $\begingroup$ Great example. I like this a lot! Thank you $\endgroup$
    – Claus
    May 23, 2020 at 20:58
  • $\begingroup$ I might write $\displaystyle \left(\!\!\binom n k \!\! \right) $ rather than $ \displaystyle \left( \binom n k \right). \qquad $ $\endgroup$ Dec 15, 2021 at 17:53
  • $\begingroup$ I might write $n^k$ rather than $( \binom n k )$.... this answer is suggestive, but it hasn't convinced me yet that this is a useful change of notation... $\endgroup$
    – Tim Campion
    Dec 15, 2021 at 23:49
  • $\begingroup$ @TimCampion I’d say your notation is much worse in that case, given that $n^k$ counts something very different…! $\endgroup$ Dec 16, 2021 at 15:12
  • $\begingroup$ (To be more specific since it took me a minute: $n^k$ counts the number of ways of making an ordered choice of $k$ items from a set of $n$ (with replacement); the Stanley symbol counts the number of ways of making an unordered choice of $k$ items with replacement; e.g., $[1, 2, 1, 4]$ is to be treated as identical to $[1, 1, 4, 2]$.) $\endgroup$ Dec 16, 2021 at 16:06
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Might Hindu–Arabic numerals belong in this list?

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What might be called the Gelfand Philosophy notation has become popular in the field of 'Woronowicz' quantum groups in the past decade.

The idea starts with the Gelfand theorem that a commutative $\mathrm{C}^*$-algebra $A$ is isometrically isomorphic to $C_0(X)$, for $X$ a particular topological space, certainly compact and Hausdorff if $A$ is unital, in which case $A\cong C(X)$. Restricting now to the unital case, this Gelfand Philosophy says that a noncommutative $\mathrm{C}^*$-algebra $A$ should be thought of as the algebra of continuous functions on a compact quantum space, $\mathbb{X}$, and so we write $A=:C(\mathbb{X})$. Of course $\mathbb{X}$ is not a set, let alone a topological space but a so-called virtual object. A more radical (if only stylistically) approach is not to use blackboard bold to signify that $\mathbb{X}$ is a virtual object but just to use $X$.

For examples of where this goes I want to talk about compact quantum groups. Compact quantum groups are spoken about through what are called algebras of functions on the compact quantum group. For example, a compact quantum group $G$ might be spoken about via it algebra of continuous functions $C(G)$, a (Woronowicz) $\mathrm{C}^*$-algebra. These algebra of functions have Haar states $h$ that are precisely integration against the Haar measure whenever $C(G)$ is commutative/$G$ is classical. Playing a little fast and loose with issues of null sets, in the classical case we can define $$L^2(G)=\left\{f\in C(G)\mid \int_G |f(t)|^2\,d\mu(t)<\infty\right\},$$ and via $|f|^2:=f^*f$ for $f\in C(G)$ non-commutative, we can also define $L^2(G)$ spaces for compact quantum groups $G$: $$L^2(G)=\left\{f\in C(G)\mid h(|f|^2)<\infty\right\}.$$ This kind of thing can go in all kinds of directions, the basic principle is if you have a notation for something in commutative algebras of functions/classical groups that makes sense for noncommutative algebra of functions/quantum groups, use that same notation for quantum groups.

This also allows you to, in a strictly nonsensical but useful way, to talk about the quantum group as if it really exists. For example for finite groups at least, with full algebra of functions $F(G)$, there is a bijective correspondence with representation $G\rightarrow L(V)$ and corepresentations $V\rightarrow V\otimes F(G)$. Through this lens one can talk about a representation of a quantum group or in a similar way the action of a quantum group.

To actually answer the question asked I will quote from a recent preprint:

When, for example with the representation theory of compact quantum groups, the noncommutative theory generalises so nicely from the commutative theory, it can be useful to refer to a virtual object as if it exists: this approach helps point towards appropriate noncommutative definitions, and sometimes even towards results, such as the Peter-Weyl Theorem, that are true in this larger class of objects. Even when commutative results do not generalise to this larger class, the Gelfand Philosophy gives a pleasing notation, helping readers from the commutative world understand better what is going on in the noncommutative world.

Some examples of this from my own work on finite quantum groups, say given by a $\mathrm{C}^*$-algebra $A$ include:

  • referring to $A$ as the algebra of functions on the finite quantum group $G$, and denote it by $F(G)$
  • referring to the unit $1_A$ in the algebra of functions as $\mathbf{1}_G$
  • referring to the set of states $\mathcal{S}(A)$ as $M_p(G)$, the set of probability measures on the group
  • I have started using $f\in F(G)$ for a general "function" rather than the usual $a\in F(G)$ or before $a\in A$
  • I have used the notation $2^G$ for the set of projections in $F(G)$

Some of these are pushing the envelope a little on this notation... we will see what the Reviewers say!

I note in this preprint that:

This philosophical approach ramped up in the 2000s, and into the 2010s, and up to 2020.

HOWEVER, when I got my hand on the 1967 paper of Kac and Paljutkin (highly recommended if you can get a copy), the famous eight-dimensional algebra of functions on a finite quantum group, the smallest of which is neither commutative nor cocommutative, the authors refer to it by $\mathfrak{G}_0$ --- not the algebra but the virtual object!

I assume similar notations are at play in other fields.

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    $\begingroup$ great example, thanks a lot. Find it intriguing that this notation then allows you to talk about action of a quantum group! $\endgroup$
    – Claus
    Jun 13, 2020 at 9:52
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"Why didn't Romans invent algebra?"

"Because for them X always evaluated to 10."

IMHO the biggest impact on development of Mathematics was the Viete's invention of symbolic algebra, the idea that you can represent arbitrary numbers with symbols such as Latin letters. Before Viete solutions to algebraic equations were described mostly with specific examples rather than general formulae.

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  • $\begingroup$ Interesting... I thought I'd heard the invention of "variables" attributed to Aristotle... but I wouldn't be surprised if it really took this long for the notion to be applied to mathematics! $\endgroup$
    – Tim Campion
    Dec 15, 2021 at 23:57
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An important historical example is the introduction of our typical $a \equiv b$ (mod $m$) notation. Thinking about remainders when people divided by numbers was early on done in a haphazard fashion. For example, one had this sort of thinking in Gersonides's proof that 8 and 9 are the largest power of 2 next to a power of 3. And similarly, Euler's theorem that an odd perfect number $n$ must be of the form $n= q^e m^2$ where $q$ is a prime and $q \equiv e \equiv 1$ (mod 4), and $(q,m)=1$. But it wasn't until Gauss introduced the modern systematic notation that it became really helpful in all sorts of number theory. While much of the Disquisitiones Arithmeticae, is dedicated to proving facts we now consider basic about modular arithmetic, having good notation for it was itself a major step forward.

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I like to use figures to represent quantities. For example, in this recent preprint, my coauthor and I use simple diagrams to represent certain weighted sums (polynomials). Writing out the sums explicitly would be extremely cumbersome to parse, and any sane reader would just convert it back to a generic figure anyway, in order to understand the sum.

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  • $\begingroup$ @PerAlexaderson thank you for a great example. It is really good $\endgroup$
    – Claus
    May 23, 2020 at 21:02
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The notation that I found extremely useful to support my intuition is from Grothendieck $\begin{array} \\{\cal F} \\ \hspace{0.05in}\mid \\X \end{array}$ to denote a sheaf $\cal F$ over a topological space $X$.

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The ZX-calculus introduced by Bob Coecke and Ross Duncan in 2008 is a graphical language for reasoning about quantum processes using string diagrams. Quantum teleportation has an exceptionally elegant formulation in the ZX-calculus (see for instance Section 5.4 in ZX-calculus for the working computer scientist by John van de Wetering). In particular, the Bell state $\frac{1}{\sqrt{2}}(\left|00\right> + \left|11\right>)$ is represented in the ZX-calculus simply as a cup $\cup$. For me at least, this shows the ZX-calculus meets the three requirements in the question.

According to (2) on page 3 of Kindergarden quantum mechanics graduates ... or how I learned to stop gluing LEGO together and love the ZX-calculus by Bob Coecke, Dominic Horsman, Aleks Kissinger, Quanlong Wang,

'For certain quantum circuit optimisation problems, ZX-based methods now outperform the state of the art'

This is justified by a reference to Fast and effective techniques for $t$-count reduction via spider nest identities by N. de Beaudrap, X. Bian, and Q. Wang. Therefore the efficiency of the ZX-calculus as a graphical language can translate directly into efficiency (as measured by gate count) of quantum circuits.

The ZX-calculus is closely related to string diagrams for braided monoidal categories and to Penrose's string notation for tensor operations. For instance, the dual of the Bell state $\cap$ is, up to some bother with conjugation, tensor contraction. Either of these notations could merit a separate answer. For an nice example see the exercise on page 5 of Coecke's Quantum pictorialism, in which the output of a chain of four projectors on a tensor product of three Hilbert spaces is given by simple string rules.

For a textbook account see Picturing quantum processes by Bob Coecke and Aleks Kissinger. Also the website zxcalculus.com has many papers.

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I'm a bit surprised that no one mentioned the Einstein summation convention

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