100 Questions: A Mathematical Conventions Survey
"The leading cause of murder among mathematicians in the United States." - Sun Tzu

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Written on: March 11th, 2024
Last updated: March 13th, 2024

Contact: thomas.lam at cims.nyu.edu

Overview

Is $0 \in \mathbb{N}$? Are constant functions increasing? How do you pronounce LaTeX? What the **** is a tensor?

From basic arithmetic to advanced real analysis, we mathematicians can't seem to agree on anything. The Mathematical Conventions Survey (MCS) aimed to highlight this while simultaneously both entertaining and pissing everyone off at the same time.

I never expected this to get so popular, and with the MCS now having over 2500 responses at the time of writing, I do feel some obligation to make good use of the data. I hope you find great joy and/or seething rage in looking through the results. If nothing else, I believe that the survey's results can help expose math enthusiasts around the world to popular mathematican notations or conventions that may be unknown to them. For example, numerous respondents were surprised that $0$ could be natural, or that there was any debate whatsoever on whether $f(x) = 1/x$ is continuous.

If you have about 5 minutes of spare time and energy, I'd highly recommend checking out the questions before seeing the results!

Some Credits:

• My very cool puzzle associates, for suggesting around 10 of the questions.
• Me, for coming up with the other 90.
• Several respondents, for reminding me that vector bundles go like $E \to M$ instead of the other way.
• Numerous angry physicists, for convincing me to include $A^\dagger$ as an explicit option for the conjugate transpose of $A$.
• A good friend, for confirming the Japanese opinion on Q55.
• Many people, for sharing the survey around.
Lastly (but far from least), thank you to everyone that filled out the survey!

Methodology

This isn't really that important so you can just skip to results if you want.

Oh wow, you're still reading. I'm flattered.

Ok, here's some background. The survey opened up on February 3rd, 2024, 20:20:11 EST. The results on this page include all responses submitted up to March 5th, 2024, 8:33:24 EST, totalling exactly 2500 non-empty responses.

This project was supposed to be a joke, so I did not have high standards of scientific rigor in mind when conducting this survey. Nevertheless, there are a few semi-important kinks concerning the survey. First, there were actually two different copies of the survey sent out!

• The first survey ("Survey A") was initially sent out to my puzzle associates, various CMU Discord servers, an NYU math Discord server, and the CIMS graduate math mailing list.
• The second survey ("Survey B") is completely identical to Survey A, except the word "fuck" was censored to "f***". This was initially sent out to a few math competition Discord servers, the AoPS forums, r/math, and a Facebook group called ">implying we can discuss mathematics".
The surveys then spread around from their repsective origins. Cloning the survey was primarily motivated by wanting a better sense of what sorts of people were responding to the survey (with questionable effectiveness). It also helped me feel less iffy about Survey B reaching minors.

If you wrote down a university affiliation, or if I was able to confidently guess your university affiliation, then I marked you down as a university-affiliated respondent.

The above chart shows most of the universities. For the full list, please see the data sheet. Also included on the data sheet include a collection of some of the online communities and social media platforms that the survey reached.

A few more details and design decisions:

• The survey is completely anonymous, but there is an optional prompt at the end asking for the respondent's affiliation and/or where they found the survey.
• A few small changes were made to the survey at the onset. These included adding the $A^\dagger$ option, clarifying what the second option of Q28 should look like (since it doesn't look like centered dots on some screens), adding some more "Other" options, adding "...feel free to [complain] about insufficient answer choices" at the end, and adding contact info on the survey for clout in case anyone really wanted results but didn't know where they'd be posted.
• Exactly one extremely offensive response was deleted.
• There were numerous complains about not enough "Other" options. This design was intentional, to encourage decisiveness wherever possible. Conventions such as "sometimes I do $X$ but I would do $Y$ if we're in context $Z$" are perfectly valid! But it's hard to include into the data analysis. Encouraging complaints in the last prompt was included as a sort of compromise.
• As for the questions themselves: There was a wide range of complexity in the questions. All questions were optional. As some respondents correctly observed, I am an analyst. I tried my best to diversify topics, but I admit that the MCS would have benefited from some more variety, particularly combinatorics.
• Google Forms does not support LaTeX (as far as I'm aware), so I had to do my best to write the questions using unicode. An exception was made for the Fourier Transform question.
• In the forthcoming charts, an "Other" slice was only included if I felt that the write-ins were sufficiently significant in proportion or interest. I also do not claim that all the specific write-in tallies are exact (you'll see what I'm referring to), but they should be close enough.

Possible sources of error include:

• Trolling
• Finger slips
• Not reading the "All questions are optional" part of the instructions (Multiple people!)
• Exhaustion due to the length of the survey
• Choosing a favorite answer among what is listed instead of bothering to write in a custom answer
• Some universities or communities or countries being more represented than others

Without further ado, here are the results!

Results

"I found this survey left on my doorstep in the dead of night. The rain was pouring, the wind howling and the lightning threatening, and I had no choice but to bring the poor soul in. Damned be I for feeling such sympathy." - Respondent #2482

Q1: Where does $\mathbb{N}$ start?

A slight majority for 0!

There was a notable difference between the surveys: In Form A, there were 24% more 0'ers than 1'ers, whereas in Form B there was a tie (!), with no significant difference. If we restrict to university-affiliated respondents, there were 72% more 0'ers than 1'ers, compared to only 13% among the other respondents. It's an interesting discrepancy!

This is the first and last time I'll mention any discrepancies between groups since it's gonna take more effort than I signed up for to repeat this analysis for every question. Most of the work is done for you in the data sheet though, so do check that out if you want to analyze the data yourself!

Q2: $\pi$ or $\tau$?

An overwhelming majority for $\pi$.

If you're unfamiliar with the debate here, check out the original "$\pi$ is wrong!" article by Palais, the noted "Tau Manifesto" by Hartl, this counterargument to the Tau Manifesto (preserved by the Wayback Machine), and, if you're feeling adventerous, this sacriligous "compromise".

Q3: What is $0^0$?

Chaos! A great $\approx 50\%$ majority adopts $0^0 = 1$, with "indeterminate" and "undefined" each taking about a quarter of the pie.

Interesting responses include:

• "It depends on context." - A few dozen respondents
• "$0^{\text{the natural$0$}} = 1$, but $0^{\text{the real$0$}}$ is undefined." (x2)
• "$0^0$ is forbidden by God"

The argument for $0^0 = 1$ is typically that adopting this value makes a bunch of formulas much nicer, especially in discrete settings. For example, it allows us to write the Binomial Theorem without worrying over $0$ as a special case.

Those that say that $0^0$ is indeterminate are typically interpreting each "$0$" as a limiting value. That is to say, the information $\lim_{x \to x_0} f(x) = \lim_{x \to x_0} g(x) = 0$ is insufficient for determining $\lim_{x \to x_0} f(x)^{g(x)}$. To my knowledge, this is mostly discussed in Calculus courses.

As I understand it, the case for taking $0^0$ to be undefined is that there is no natural value for it in the realm of continuous mathematics. Indeed, I do not think there is a natural way to justify assigning a value to $0^0$ when constructing the real exponential. It could also be the case that $0^0$ is undefined because it corresponds to an indeterminate form.

I am not aware of any compelling argument for $0^0 = 0$.

Naturally, there is a Wikipedia on this issue: https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero (Check out the talk page for some "entertainment"...)

Q4: Are manifolds second-countable?

Note to the colorblind: The labels always correspond to the slices in clockwise order, with the first label corresponding to the slice starting at $90^\circ$ (i.e. the top) and proceeding clockwise.

The rationale for second-countability is largely to ensure that manifolds are paracompact, which allows for the usage of useful tools such as partitions of unity. Requiring second-countability excludes objects such as the long line from being considered manifolds. As Topospaces puts it, "the long line fails to satisfy only the second condition for a manifold; it is simply too long."

One respondent commented that in their definition, manifolds are paracompact but not necessarily second-countable. As far as I'm aware, this doesn't really break much.

Q5: Is the Euler totient function $\varphi$ or $\phi$?

"i'm having an identity crisis i think i've used both" - Respondent #1259

The curlier $\varphi$ wins over $\phi$ by a significant margin.

Q6: What is $\sqrt{-1}$?

Unfortunately this question was potentially a bit vague. Let me explain the options and their intended meanings:

• Option 1: $\sqrt{-1} = i$ by the definition of $i$. That is, the constant $i$ is defined to be $\sqrt{-1}$.
• Option 2: $\sqrt{-1} = i$, but this is merely an (indirect) consequence of how complex square root and $i$ are defined. $i$ is instead defined as, say, the point $(0,1) \in \mathbb{R}^2$ where $\mathbb{R}^2$ is endowed with a multiplication operator to form $\mathbb{C}$.
• Option 3: $\sqrt{-1}$ is not defined because you cannot take the square root of negative numbers, even in the complex plane.
The reason why this question could be vague is because "by definition" could be interepreted as "by definition of $\sqrt{\cdot}$" rather than "by definition of $i$". Fortunately, no matter the interpretation, the chart still shows the split between those that say $\sqrt{-1} = i$ and those that leave $\sqrt{-1}$ undefined.

Other responses include:

• $\sqrt{-1} = \pm i$ (x20)
• The value of $\sqrt{-1}$ depends on the branch cut (x4) (Though, I must object --- unless otherwise specified, $\sqrt{\cdot}$ should be interpreted as the principal square root, however you define that.)
• Judging from some responses, $\sqrt{-1}$ could also be interpreted as being a placeholder for a number that squares to $-1$, rather than a function called "square root" being applied to $-1$, making the question difficult to answer.

I offer my insincere apologies to the engineers that would have preferred $j$.

Q7: How do you denote the empty set?

"anyone who uses phi for the empty set should be shot on sight" - Respondent #459

A few dozen respondents were fine with either $\{\}$ or $\emptyset$. I am aware that $\emptyset$ and $\varnothing$ are distinct, and it's unfortunate that this slipped my mind. Unicode can't tell the difference between them, so let's consider them the same for this question.

Q8: $m+n$ or $n+m$?

After years of writing $n+m$, my reality was shattered when I realized that $n$ does not, in fact, come before $m$ in the alphabet.

Q9: $m/n$ or $n/m$?

The previous two charts differ slightly. Naturally this demands further analysis. Here's a chart of how people responded to both questions, among all those respondents that answered both questions.

$N = 2271$

 $m/n$ $n/m$ $m+n$ 64% (1456) 11% (247) $n+m$ 6% (141) 19% (427)

Q10: Is the empty graph connected?

This question can be the subject of very long arguments. At first glance, it may be obvious that the empty graph is connected because it is vacuously true that any pair of points in the empty graph can be connected by a path. But there are also rather compelling arguments for adopting the convention that the empty graph is not connected! Some of these include:

• A graph is connected iff it has exactly one connected component. The empty graph has zero connected components, therefore it is not connected.
• Rejecting the connectedness of the empty graph ensures that the decomposition of graphs into maximal connected components is unique.
• In view of the above points, the empty graph being not connected is the analogue of $1$ being not prime. So the empty graph is both not connected and not disconnected, just as $1$ is both not prime and not composite.
The empty graph is an example of an object which is too simple to be simple.

Q11: How do you denote the set of complex numbers?

A largely unsurprisingly result, and there were no significantly common write-ins.

Q12: "Let ($E$ or $V$?) be a (vector or linear?) space."

According to Wolfram MathWorld, the term "linear space" is most commonly seen in functional analysis. The preference of $E$ over $V$ for vector spaces is also prominent in this field, from what I can tell. (See e.g. Functional Analysis, Sobolev Spaces, and Partial Differential Equations by H. Brezis.)

Q13: Is $f(x) = 3$ increasing?

Proponents of "increasing" are using the definition that a function $f$ is increasing if $f(x) \leq f(y)$ whenever $x \lt y$. If instead $f(x) \lt f(y)$ for all such $x,y$, then $f$ is "strictly increasing".

Those who instead take the latter $f(x) \lt f(y)$ definition to mean "increasing" may characterize the former $f(x) \leq f(y)$ definition as "non-decreasing".

Q14: How do you denote the inner product of two vectors $v$ and $w$?

The bra-ket notation $\langle v | w\rangle$ was very prominent in the write-in responses, so it is included in the chart. Other responses include:

• $v^iw_i$
• $v \lor w$
• $v^Tw$
• $\operatorname{inner}(v,w)$
• $\langle\langle v,w\rangle\rangle$
• $vw$
• $g(v,w)$
• $(v|w)$

Q15: Fill in the blank: "$E = \{n \in \mathbb{Z}\ \_\_\_ \ \text{$n$is even}\}$"

"Honest to god it's either $:$ or $|$ depending on my mood I'm sorry" - Respondent #1868

I got a few comments preferring notation such as $E = \{n | n \in \mathbb{Z}, \text{$n$is even}\}$. Common write-ins include indecision between $|$ and $:$, and fierce hatred towards $|$.

As jarring as it may be, $\ni$ is apparently a real symbol used by real people.

Q16: What does $\subset$ mean?

This data sugests that we may want to avoid writing $\subset$ if the difference between "subset" and "strict subset" is important. To dodge the ambiguity of $\subset$, I believe it is common to write $\subsetneq$ or $\subsetneqq$ to denote a strict subset.

Q17: Is the Laplace equation $\Delta u = 0$ or $-\Delta u = 0$?

The rationale for $-\Delta u = 0$ is that $-\Delta$ is somehow a more "natural" operator than $\Delta$. In particular, integration by parts shows that $\int_\Omega F \cdot \nabla u\,dx = \int_\Omega (-\operatorname{div} F)u\,dx$ for $u,F$ compactly supported, so the adjoint of the gradient operator is $\nabla^* = -\operatorname{div}$, and now $\nabla^*\nabla = -\operatorname{div}\nabla = -\Delta$. Also, $u$ is subharmonic iff $\Delta u \geq 0$, which is easier to remember as $-\Delta u \leq 0$.

I received at least one comment that takes this philosophy to the extreme, defining $\Delta$ to be $-\operatorname{div}\nabla$. Several respondents would have liked to see $\nabla^2$ instead of $\Delta$.

Q18: What is the heat equation?

This was just a matter of how you denote time derivatives in PDEs. It would appear that $\partial_tu$ is the majority favorite, with the "dot" notation losing favor.

Q19: What is the usual variable for an arbitrary category?

"please never use \mathbb for categories" - Respondent #859

My category theory professor should steer clear of Respondent #859.

Q20: Do numbers exist?

"my views on if numbers exist can and will fill a thesis" - Respondent #1385

Mathematical philosophy is fun! But I am absolutely not an authority in the field, so I'll instead direct you to the Stanford Encyclopedia of Philosophy. See e.g. Platonism and Fictionalism.

Q21: How do you denote the divergence of a vector field?

Q22: How do you denote the curl of a vector field?

"$\operatorname{div}$ is a perfectly reasonable choice for many applications... but $\operatorname{curl}$ is never acceptable" - Respondent #868

It's time for another table! How many people chose each possible combination for denoting divergence and curl?

$N = 1815$

 $\operatorname{curl} F$ $\nabla \times F$ $\operatorname{div} F$ 35% (642) 3% (51) $\nabla \cdot F$ 4% (73) 58% (1049)

Q23: Which octant is the 5th octant?

$(+,+,+)$ was not an option since it's hard to imagine how it could be anything other than the 1st octant. Anyways, here are two alternative visualizations of the data, depending on whether you think up is $y$ or $z$.

Q24: What is $\mathbb{Z}_p$?

Many respondents were indecisive.

Several respondents stated that $\mathbb{Z}_p$ should be the "localization of the integers at $p$". I don't know what that means and I'm too afraid to ask.

Q25: Does "up" go along the $y$-axis or along the $z$-axis?

The axes should comply with the right-hand rule, but besides that, there seem to be a bunch of different ways that people like to orient the axes. Having "up" be the positive $y$-axis is quite common in 3D videogames, with the view that "the third axis ($z$) adds depth to the 'right/up' plane". Having "up" be the positive $z$-axis is from the view that "the third axis adds height to the flat horizontal/vertical plane".

Apparently, some people that fly airplanes take "up" to be the negative $z$-axis.

Q26: What is the existence and uniqueness theorem for ODEs?

Other responses include:
• "Cauchy"
• "Lindelöf"
• "Existence and uniqueness theorem" or "Theorem of ODEs" (x12)
• "Lindelöf-Picard"
• "Cauchy-Picard"
• "Cauchy-Lipschitz-Picard"

Q27: Do you accept the axiom of choice?

Several angry set theorists have remarked that I should have put down "dependent choice" instead of "countable choice". Whoops.

Q28: What is the set of the first $n$ positive integers?

This question was about where you like to align the dots. Should they be centered ($\cdots$) or lowered ($\ldots$)? The overwhelming consensus appears to be lowered.

A number of respondents complained that neither answer was correct because $0$ is positive. I forgot that people do that.

Q29: Is lowercase epsilon $\varepsilon$ or $\epsilon$?

Once again, the curlier option is superior!

Q30: What symbol do you use to signal that your proof is done, if any?

"You don't have to have anything at the end of a proof, some vertical space is enough. I know that this upsets a lot of people." - Respondent #822

Total chaos! The "Other" responses include:

• Nothing (x8)
• // (x8)
• ✔ (x5)
• *mic drop* (x3)
• # (x2)
• 🎉 (x2)
• Three dots in a triangle (x2)
• "whatever the tex environment does" (x2)
• wwwww ("Which Was What We Wanted")
• "Heckin gottem lmao"
• [jazz hands]
• [picture of a bat]
• "a triangle facing left, matching the triangle at the start of the proof"

"You guys actually prove stuff?" - Respondent #652

Q31: Are rings commutative?

The majority rules that rings do not necessarily satisfy $ab=ba$ for all elements $a$ and $b$...

Q32: Are rings unital?

...but are assumed to have an identity element.

Chart time!

$N = 1515$

 Rings are... unital not necessarily unital commutative 27% (406) 7% (101) not necessarily commutative 38% (580) 28% (428)

Q33: What is the (1D) Fourier transform?

I pulled these four options from the Mathematica documentation for the Fourier Transform (See "Details and Options", where the page explains FourierParameters). But there are still a bunch of different variants not listed.

Q34: What is the group of symmetries of a regular $n$-gon?

The rationale for $D_n$ is that $n$ is just the number of sides, whereas the rationale for $D_{2n}$ is that the number of elements in the group is $2n$.

Q35: If $T$ is a linear map on a vector space containing an element $x$, how do you denote the result of applying $T$ to $x$?

The possibly jarring $\langle T, x\rangle$ notation is seen in functional analysis. It's interesting to see the "rebellious" $Tx$ being so evenly matched with $T(x)$.

Q36: What is the mnemonic for the order of operations?

Some have no such mnemonic, since this is country-dependent. I believe PEMDAS is US-centric whereas BODMAS is used at least in the UK and India.

Apparently there are numerous other mnemonics. The write-ins include:

• BEMDAS (x5)
• BIDMAS (x3)
• GEMS (x3)
• PEJMDAS (x3)
• GEMDAS (x2)
• GEMA
• BOMA
• BIMA
• Hate and disdain for all order-of-operation acronyms

Q37: What is the mnemonic for the signs of trig functions?

This is again quite country-dependent, and in particular such trigonometric mnemonics are probably heavily US-centric.

The idea is that these letters, when placed on the four quadrants of the unit circle, outline which of the three trig functions $\sin(x),\cos(x),\tan(x)$ are positive on each quadrant. For instance, the second letter of ASTC is S, indicating that on the second quadrant $\pi/2 \lt x \lt \pi$, only $\sin(x)$ is positive. "ASTC" is memorized with funny phrases such as "All Students Take Calculus" or "All Students Try Cheating". Though, I sense that the latter may be in bad taste...

Q38: What is the second tetration of $x$?

"Up-arrow" notation wins!

The $n$th tetration of $x$ is essentially the value of the "power tower" formed by $n$ $x$'s. So the second tetration of $x$ is just $x^x$. Wikipedia lists some other notations for tetration.

Q39: What is the symbol for the Lebesgue measure?

$\mu$ seems to have the edge, but there is no clear consensus. Write-ins include:

• $|\cdot|$ (x4) (...I assume the differential is just $dx$ then?)
• $\operatorname{Leb}$ (x3)
• $\operatorname{vol}$

Q40: How is the Fibonacci sequence indexed?

A dozen or so respondents complained that the second option should have a zeroeth term, $F_0 = 0$. They have a good point, my bad.

A nice thing to note about the second option $F_1=1,F_2=1,F_3=2,\ldots$ is that it gives the aesthetic identity $\gcd(F_m,F_n) = F_{\gcd(m,n)}$. This property is also consistent with $F_0 = 0$.

Q41: What are the dimensions of the Jacobian matrix of a function $f:\mathbb{R}^n \to \mathbb{R}^m$?

I am not familiar for reasons to take the Jacobian to be $n \times m$.

Q42: How do you denote the Jacobian matrix of a function $f$?

Other responses include:

• $df$ (x3)
• $f'$ (x3)
• $J$ (x2)
• $\operatorname{Jac}(f)$ (x2)
• $J[f]$ (x2)
• $\frac{\partial f}{\partial(x_1,\ldots,x_n)}$
• $\frac{\partial f^i}{\partial x^j}$
• $\partial_if_j$
• $f_{i,j}$
• $\partial f$

Q43: What is the inverse function of $\sin(x)$?

The unpopular, unlabeled slices of the pie are $\operatorname{Arcsin}(x)$ (in green) and $\operatorname{Asin}(x)$ (in orange).

Q44: Let $a$ and $x$ be elements of a group. What is the conjugation of $x$ via $a$?

Q45: What is a tensor?

"I DON'T KNOW!!!!" - Respondent #12

"dude I've been asking people this for ages and I've never gotten a straight answer" - Respondent #813

"no one knows, and anyone who says they do is bullshitting" - Respondent #864

"no-one in the history of maths has been able to give a concise answer to this question and we should just accept that (it's an array of vectors and covectors which remains the same under a change of basis even if its components themselves transform. It behaves analogous to a generalisation of a matrix (where a (1,1) tensor is a matrix, but so also are some other forms of tensors, but not all). a tensor is a tensor if it has the vibes of a tensor and it is useful to consider it as such. any object can be written as a tensor, and all of mathematics is simply different elements within a tensor, including that tensor itself. a tensor is whenever you have ten of whatever a sor is.)" - Respondent #1319

This survey has confirmed what we all knew: nobody actually knows what a tensor is.

Other responses include:

• An element of a tensor product of modules (x7)
• A bifunctor with associativity and unitality up to isomorphism
• A bifunctor in a monoidal category
I don't even remotely understand what those last two responses mean, I just copied and pasted them.

Q46: What is the base of $\log$?

Taking the base to be $e$ becomes more prevelant upon pursuing a math degree. Taking the base to be $2$ is favored by some computer scientists.

Q47: Is it "global maximum" or "absolute maximum"?

Q48: Is it "local maximum" or "relative maximum"?

Time for another chart!

$N = 2117$

 local relative global 85% (1800) 2% (50) absolute 9% (186) 4% (81)

Q49: What is three times four?

There were several respondents that preferred $3(4)$ over $(3)(4)$. That probably would've been a good option.

Q50: What's the answer to the Sleeping Beauty problem?

For those unfamiliar, see Wikipedia. It's quite the conundrum! The survey confirms that the problem's answer is highly contested with pretty much no consensus.

A dozen or so respondents say that the problem is not well-defined.

The question was worded slightly differently in the survey ("In the Sleeping Beauty problem, what is the probability that the coin flipped heads?"), causing a few respondents to harp on the lack of conditioning on the event of being woken up. I think that's a bit pedantic --- the intent of the question should be obvious.

Q51: What is the set of all $x$ satisfying $1 \lt x \lt 2$?

For those that are mortified at "$]1,2[$", I believe this is a convention that is seen in France (and maybe other countries?).

Q52: For $x \in \mathbb{R}^n$, what is the notation for the norm of $x$?

While $\|\cdot\|$ is the standard notation for a norm, it is not uncommon to write the "shortcut" $|\cdot|$ when in Euclidean space.

Q53: What is the $L^p$ space of functions on the open interval $(0,1)$?

"its $L^p((0,1))$ but im not happy about it" - Respondent #983

This was a question about two different conventions:

1. Should the $p$ be a superscript or a subscript?
2. The $L^p$ space over an interval $I$ is denoted $L^p(I)$, so if $I = (0,1)$ then surely we should write $L^p(I) = L^p((0,1))$. Unfortunately this looks a bit strange. Should we keep the double parentheses or shorten it to $L^p(0,1)$?
The survey shows that the superscript $L^p$ dominates, but we can't quite decide between $L^p((0,1))$ and $L^p(0,1)$.

There were some write-ins, but they varied quite a bit.

Q54: What is the type of a connection?

I included this question for purely selfish reasons. I struggled for an eternity last semester trying to understand what a connection is since there was seemingly a ton of conflicting definitions. One reason for this is because there are like a billion different types of connections. At least as it pertains to smooth manifolds, a general sort of connection is this thing, which can be used on vector bundles other than the tangent bundle. Then there's the affine connection, where the $X$ and $Y$ in $\nabla_XY$ both need to be vector fields. Then there's the covariant derivative which I'm pretty sure is just the same bloody thing as an affine connection except in "local form". Speaking of which, you can define a connection in either a "global" or "local" sense (see here). AAAAAAAAAAAAAAAA.

Anyways, figuring this out was painful and I wanted to know if there was any consensus. It would appear that the "local form" for the affine connection, or perhaps the "covariant derivative", with type $\nabla:TM \times \Gamma(TM) \to TM$, is the clear winner. Yay!

Other resposnes include:

• $TM \to \operatorname{End}(E)$
• $\Gamma(TM) \to \Gamma(T^*M \otimes TM)$
• $\Gamma(E) \to \Gamma(T^*M \otimes E)$
• $\Omega^0(M;E) \to \Omega^1(M;E)$

Q55: Is it "Arzelà–Ascoli" or "Ascoli–Arzelà"?

My Italian advisor has always written "Ascoli–Arzelà" and I refuse to change.

Other languages that put Ascoli before Arzelà include Ukrainian, Japanese, and Russian. Interestingly, the French exclude Arzelà's name from the theorem, whereas Kazakhstan excludes Ascoli.

Q56: How do you write "$3$ is congruent to $5$ modulo 2"?

Among the write-ins, variants in where the "2" is placed include:

• $3 \equiv_2 5$ (x10)
• $3 \equiv 5\ [2]$ (x7)
• $3 \equiv 5\ (2)$ (x7)
• $3 \stackrel2\equiv 5$ (x3)
• $3 \equiv^2 5$
Variants in what is written in place of $\equiv$ include:
• $=$ (x7)
• $===$ (x2) (This might just be a javascript joke. Please contact me if you find anyone that actually writes this in math.)
• $\cong$ (x2)

Q57: What is the range of $f:X \to Y$?

Both definitions for the range are widely taught. Neither option seems necessarily superior: If the range is $f(X)$, then $Y$ can be called the codomain, and if the range is $Y$, then $f(X)$ can be called the image.

In the current century, defining the range to be $f(X)$ is said to be more popular, as the survey shows. It also makes the name of the Closed range theorem sensible. It's certainly not an overwhelming majority though, so it should be good practice to clarify the meaning of "range" if we ever choose to use the term.

Q58: What does $\log^2(x)$ mean?

Ambiguous superscripts bring endless joy.

You've made it more than halfway! Here's a joke: What do mathematicians burn in their fireplaces?

Q59: What is the quantity $(A,B;C,D)$ called?

This is a fundamental quantity in projective geometry. Essentially, when $A$, $B$, $C$, and $D$ are on the same line in 3D space, then the value of $\frac{CA}{CB} \div \frac{DA}{DB}$ stays the same no matter where you "view" the points, which lets you compute some lengths in photographs. I recall this technique being helpful in tracking down Osama bin Laden, but I'm unable to procure a source for this.

Q60: How many holes does a straw have?

This is a popular controversial question. At last, we have the authoritative mathematical consensus: A straw has one hole!

Q61: What is the golden ratio?

Namely, is it $\frac{1+\sqrt{5}}{2} \approx 1.618$ or $\frac{\sqrt{5}-1}{2} \approx 0.618$? The question may seem pointlessly obvious, but it was inspired by a news story about a decade ago which was caused by differences in the exact definition of the golden ratio. See here for a news article. Essentially, the fuss is that both $\frac{1+\sqrt{5}}{2}$ and $\frac{\sqrt{5}-1}{2}$ have the same idea, being reciprocals of each other. It's just a matter of whether you want the bigger number in the numerator or the denominator.

Well, we've now confirmed that $\varphi = \frac{1+\sqrt{5}}{2}$ is the overwhelming consensus.

Q62: What is the floor function?

My current probability professor writes $[x]$. It exists. Fear it.

I wonder what $[x]$ users write for the ceiling?

Q63: What is the set of limit points of $\{1\}$?

The definition of a limit point is not universally agreed upon (though I admit that I had to do some digging for that link). The less ambiguous terms in question are accumulation point (every neighborhood of $x$ has a point in the set different from $x$) and adherent point (every neighborhood of $x$ has a point in the set, and it could be $x$). So does "limit point" mean accumulation point or adherent point? If you thought "accumulation point", then you chose $\{\}$. If you thought "adherent point", then you chose $\{1\}$.

It appears that a decent majority chose $\{1\}$, allowing to take the point itself in the definition of a limit point, which disagrees with Wikipedia, Wolfram Mathworld, Proof Wiki, and many MSE posts. Fascinating.

Q64: "One-to-one" or "injective"?

Every time someone says "one-to-one", I cry a little.

Q65: How should three real numbers $a,b,c$ be given an order without loss of generality?

Essentially, suppose you have a question like "Prove that $\frac{a+b+c}{3} \geq \sqrt[3]{abc}$ for all $a,b,c \geq 0$", where the roles of each of the variables $a$, $b$, and $c$ are symmetric. Then you can assume some order on them. Is it aesthetically preferable to assume $a \leq b \leq c$ or $a \geq b \geq c$? A very strong majority has chosen the former.

Q66: How do you write the integral of $x+1$ over $(0,1)$?

A number of physicists wanted to have $dx$ before the integrand, as in $\int_0^1 dx\,(x+1)$, but for this question I only cared about whether or not you enclose the integrand in brackets when the integrand has multiple terms.

I believe the rationale behind brackets is that the integrand is being multiplied by the differential, so writing $\int_0^1 x+1\,dx$ wouldn't mean what you want it to mean because the order of operations dictates that the product $1\,dx$ happens before you add $x$. But it would appear that a majority either don't care or don't think that way.

Q67: For a matrix $A$ with complex entries, what is the notation for the conjugate transpose of $A$?

$A^\dagger$ was not originally among the default choices, which was quite clearly a major blunder on my part.

The $H$ in $A^H$ stands for Hermitian, which is a synonym for the conjugate transpose.

Q68: A smooth map $f:M \to N$ between manifolds induces a smooth map $df(x)$ between tangent spaces. What is $df$ called?

Q69: What symbol do you use to signal that your proof has arrived at a contradiction, if any?

Most do not have a symbol, with the lightning bold and the "clashing implication arrows" being popular options among those that use a symbol. Other responses include:

• $\rightarrow\!\leftarrow$ ("two arrows colliding") (x15)
• ("an x inside a horizontal segment") (x6)
• ☠ (x5)
• ⚔ (x3)
• # (x3)
• (x2)
It's fun to notice how some of these may have evolved from each other!

Q70: Is the entire function $f(z) = e^z$ conformal?

The clash here stems from the fact that there are two non-equivalent definitions of a conformal map. Most commonly (as suggested by the survey), a holomorphic $f:U \to \mathbb{C}$ is said to be conformal if its derivative $f'(z)$ never vanishes. In this case, $e^z$ is conformal. However, some say that $f$ is conformal if it is holomorphic and injective. This is a stronger property due to complex analysis shenanigans, and under this definition $e^z$ is not conformal because it is not injective.

Q71: What is the notation for the expected value of a random variable $X$?

It's interesting to see $E(X)$ and $E[X]$ both being more popular than $\mathbb{E}(X)$.

Other responses include:

• $\langle X \rangle$ (x9)
• $\mathbf{E}[X]$ or $\mathbf{E}(X)$ (x4)
• $\langle \langle X \rangle \rangle$
• $\mu(X)$
• $E\{X\}$

Q72: Is $f(x) = \frac1x$ continuous?

Many people are understandably flabbergasted that there is any debate here. $\frac1x$ obviously "jumps" at $0$. How could it be continuous?

The motto for the "$\frac1x$ is continuous" camp is "since $0$ isn't in the domain, it can't cause issues for continuity". Indeed, the standard definition for a function $f$ being continuous is that $f$ is continuous at every point in its domain, and this cannot be denied for $f(x) = 1/x$.

In certain parts of math, you might not consider $1/x$ to be continuous because $\{0\}$ is a null set, so $1/x$ is essentially a function $\mathbb{R} \to \mathbb{R}$ with no continuous representative.

Q73: For a vector field $F$, what is the name for the property that $\frac{\partial F_i}{\partial x_j} = \frac{\partial F_j}{\partial x_i}$ for all $i,j$?

The popularity of "irrotational" is not too far off from that of "curl-free"!

Q74: What is the notation for the cardinality for a set $S$?

Some respondents said they use different notations depending on context. The write-ins were otherwise not very noteworthy.

Q75: Are these questions about math or maths?

"How American are you?"

Q76: Let $E = \mathbb{Q} \cap (0,1)$, as a subset of the metric space $\mathbb{Q}$ endowed with the usual metric. Is $E$ precompact?

Intuitively, precompctness is the property of being well-approximated by finite sets. Unfortunately, "precompactness" refers to two different definitions.

Precompactness could mean relatively compact, meaning that the closure is compact. The closure of $E$ is $\mathbb{Q} \cap [0,1]$, which is not compact because given an enumeration $\{q_n\}_n$ of $\mathbb{Q} \cap [0,1]$ we may construct the open cover $\bigcup_{n=1}^\infty \left(q_n - \frac{1}{100^n}, q_n+\frac{1}{100^n}\right)$ which has no finite subcover. So $E$ is not precompact under this definition. It is important to note that the closure is not $[0,1]$ because the ambient metric space is $\mathbb{Q}$.

Precompactness could also mean totally bounded, meaning that for every $\varepsilon > 0$ there exist a finite number of balls of radius $\varepsilon$ that cover the set. It is not hard to show that $E$ is totally bounded, so $E$ is precompact under this definition.

The survey suggests that there is no clear consensus on what "precompact" means. Fortunately, the two definitions are equivalent in any complete metric space. So, for instance, Ascoli–Arzelà can be phrased in terms of precompactness without any ambiguity.

Q77: What character is used to denote the indicator function?

Standards may differ depending on exactly which field of math you're using indicator functions in. $\chi$ stands for "characteristic", and the other three notations have clear motivations. Other responses include:

• $\mathbf{1}$ (x6)
• $I$ (x6)
• The Iverson bracket $[\cdot]$ (x3)
• $\perp$ (x2)
Another interesting thought is when or if you tend to write something such as $1_A(x)$ for the function $x \mapsto \begin{cases}1, & x \in A \\ 0, & x \not\in A\end{cases}$ versus something like $1_P$ which is $1$ if $P$ is true, and otherwise $0$, so e.g. "$1_A(x) = 1_{x \in A}$".

Q78: What is the shorthand for a sequence $a_1,a_2,a_3,\dots$?

The write-ins were total chaos. Many respondents expressed a preference for multiple options, and if so then I assigned $0.5$ for every such choice, even if there were more than two. I combined different forms if they were essentially the same. With all that in mind, the top "Other" choices include:

• $a_n$ (x12.5)
• $(a_n)_{n \in \mathbb{N}}$ (x12)
• $(a_n)$ (x11.5)
• $\{a_n\}$ (x8)
• $\{a_n\}_{n=1}^\infty$ (x6)
• $a$ (x3)
• $a_\bullet$ (x3)
• $(a_n)_{n=1}^\infty$ (x2)
• $(a_i)_n$ (x2)
• $(a_n)_{n>0}$ (x2)
• $n \mapsto a_n$ (x2)
• $\langle a_n \rangle$ (x1.5)

Q79: What is the shorthand for the $n$-volume of the unit ball in $n$ dimensions?

Among the write-ins, a few suggested "$V_n$", but most said they had no notation for this.

Q80: When integrating, should the "d" be upright like $\mathrm{d}$ or slanted like $d$?

Someone should make a follow-up survey about how much space is included between the integrand and $dx$.

Q81: What is the set $[n]$?

A popular complaint was the lack of the option $\{0,1,\ldots,n\}$. The motivation for this comes from the study of simplices, perhaps in algebraic topology. In the simplex category, $[n]$ refers to the object corresponding to the totally-ordered set $\{0,1,\ldots,n\}$. See also simplicial set. At a basic level, I believe the idea is that you want to be able to label the vertices of an $n$-simplex with the set $[n]$. For example, a 3-simplex is a tetredron with 4 vertices, which we can label as $0,1,2,3$.

Q82: Are signed measures allowed to take values in $\{-\infty,+\infty\}$?

If we allow a signed measure $\mu$ to take values $\pm\infty$, then it can only take on one of them, otherwise there will be a set with measure $+\infty-\infty$, which is quite bad.

As per the survey, we apparently can't decide whether or not to let signed measures be infinite. As far as I can tell, the choice does not matter too much. The main results for signed measures, such as Hahn and Jordan decomposition, all hold for signed measures that can take values $\pm \infty$. So they also hold for real-valued signed measures, since this is a smaller class of measures.

Q83: Is Brownian motion continuous, or only continuous almost surely?

Brownian motion is basically that super duper zig-zaggy thing you see at the stock market. Essentially, if you were to choose a random such zigzaggy path, must you always get a continuous trajectory? Or will this trajectory only be continuous with probability 1? Survey says most people say the latter. As usual, this choice is quite inconsequential.

Q84: What is the notation for the complement of a set $A$?

According to Wikipedia, "$A^\complement$" is the standard notation for the complement of $A$.

Sorry but this is the one time I'm going to explicitly voice an opinion on this results page. The reason why $A^c$ is written in the above chart instead of $A^\complement$ is because I think $A^\complement$ is a horrid abominiation that needs to be nuked from orbit. $\complement$ is a visually appalling manifestation of the devil. It is absolutely mortifying to exist in the same plane of reality as the hideous "$\complement$". Just look at it. It has no idea what it wants to be. Its vertical elongation is absolutely jarring. It looks completely different from the letter styles of everything else in LaTeX. $c$, $C$, $\mathcal{C}$, $\mathbb{C}$, $\mathbf{C}$, $\mathsf{C}$, $\mathscr{C}$, and $\mathfrak{C}$ are all beautiful. What the hell is $\complement$??? I'll tell you what it is: It's an atrocity. It's a war crime. It is NOT welcome in the math typesetting family and needs to be exiled to the black hole from whence it came. I yearn for the day when $\complement$ will be abolished. With your help, we can make it happen.

Other responses include:

• $X \setminus A$ where $X$ is the universal set (x32)
• $A'$ (x5)
• $\sim A$ (x3)

There are more interesting notations listed here.

Q85: What is the notation for the differential when integrating over a 2-dimensional surface in Euclidean space?

A brief explanation of where the options come from: $dS$, $d\sigma$, $d\Sigma$ likely stem from "Surface", $dA$ surely stems from "Area", and $d\mathcal{H}^2$ indicates the 2-dimensional Hausdorff measure. It would appear that this is mainly a fight between $dS$ and $dA$.

Other responses include:

• "$dA$ if the surface is a plane, $dS$ otherwise" (x4)
• $d[\text{INSERT NAME OF SURFACE}]$ (x2)
• Nothing (x2)
• $ds$
• $d^2\sigma$

Q86: Are partitions of positive integers represented via Young tableaux or Ferrers diagrams?

A popular complaint is that it should be Young diagrams, not tableaux. That's my bad. Technically "Young tableaux" are what you get when you fill in the boxes of Young diagrams with numbers or other objects.

Q87: Is the unit outward normal $n$ or $\nu$?

$n$ wins over the Greek $\nu$. Other responses include:

• $\hat{n}$ (x8) (Though I suspect that some that would have preferred this just chose "$n$")
• $\vec{n}$ (x3) (Ditto)
• $\mathbf{n}$ (x2)
• $\mathfrak{n}$
• $\frac{\partial}{\partial\nu}$

Q88: What is the usual variable name for the Yoneda embedding?

None of the choices are considerably unpopular, but the Japanese hiragana よ is the clear winner! The undergrads reading should consider taking a course in category theory if only because it's the only time where you get a very good excuse to use a Japanese character for a variable name.

Two respondents wrote "$h$". That's interesting.

Q89: What is your go-to dimension for a Euclidean space?

"3" - Respondent #1487

I think one reason for $N$ is so that $n$ is freed up for indices. Four respondents preferred $d$.

Q90: Is it the Frobenius Coin Theorem or the Chicken McNugget Theorem?

I'm not aware of any one person responsible for popularizing the name "Chicken McNugget". It originated from a time when McDonalds sold their nuggets in boxes of either $9$ nuggets or $20$ nuggets. Using number theory, this means that the largest number of chicken nuggets that you cannot buy is $(9)(20) - 9 - 20 = 151$.

Q91: Is the convolution of $f$ and $g$ denoted by $f \ast g$ or $f \star g$?

"i didnt realize these were different until now" - Respondent #442

Apparently, as noted by one respondent, there exist people that write $f \times g$. If you don't believe me, look at page 228 of $C^*$ Algebras and their Automorphism Groups by G. K. Pedersen. Quoting Pedersen on that page, "We use $\times$ for convolution instead of the usual $\ast$ to avoid confusion with the adjoint operation." (Yes, I scrolled through a textbook in a subject I know nothing about for half an hour just for this bit. You're welcome.)

Q92: How do you pronounce LaTeX?

"typesetting software is usually not a great conversation topic. lets see the weather for today." - Respondent #946

I didn't write these in IPA because I can't read IPA (and I imagine a bunch of respondents wouldn't have bothered to figure it out on the spot either; I wanted to maximize accessibility). So I hope these approximations were clear enough in intent. It appears that the popular choices are "lay-tek", "lah-tek", and "lay-teks", in that order.

Several respondents wrote something about an "unvoiced velar fricative". Several mention a "German -ch". A few respondents pronounce the "TeX" part as "-tesh".

Q93: What is the symbol for the independence relation between random variables?

Most of the "Other" responses stated they did not have a symbol, and several were quite passionate that there should be no symbol.

Q94: What is the notation for the larger number among two real numbers $x$ and $y$?

A few people preferred to use $x \lor y$. $\land$ and $\lor$ for min and max might be more common in probability, since it makes the stopped process $X_{n \land \tau}$ (for $\tau$ a stopping time) neater to write.

Q95: What is the notation for the collection of morphisms between two objects $A$ and $B$?

About a dozen preferred $C(A,B)$ where $C$ is the name of the category in question.

Q96: If $A$ and $B$ are points in the Euclidean plane, what is the notation for the length of the segment connecting $A$ and $B$?

I specified the Euclidean plane because I intended for this to be an elementary geometry question, not a "normed space" question. So I did not consider write-ins such as $\|A-B\|$.

The write-ins bore interesting fruit nevertheless. A couple dozen respondents wrote $\overline{AB}$ and about a dozen wrote $|\overline{AB}|$. I believe $\overline{AB}$ may be quite vomit-inducing to many Americans, that define $\overline{AB}$ to be the segment itself rather than its length. See e.g. Math Open Reference.

Q97: Let $f(x,y)$ be a function in two variables. How do you denote the function that sends $y$ to $f(x,y)$ for a fixed value of $x$?

In my experience, I see $f(x,-)$ more often in algebraic-flavored fields, whereas I see $f(x,\cdot)$ more among the analysts. I don't know if this generalizes.

Other responses include:

• $f_x(y)$ (x7)
• $\lambda y.f\ x\ y$ or similar (x6)
• $f(x,\_)$ (x5)
• $y \mapsto f(x,y)$ (x5)
• $f(x,\ast)$ (x2)
• $f(x,.)$

• Q98: Is $f(x) = 3x+1$ linear?

In grade school, it is widely taught that "linear functions" are functions of the form $f(x) = mx+b$. So it is no surprise that the majority says $3x+1$ is linear.

What may surprise quite a lot of people is that a significant fraction of the voters say that $3x+1$ is not linear. Why? This is because in a wide array of contexts in higher math, a linear function or map refers to a function $f$ that satisfies the linearity condition $f(sx+ty) = sf(x)+tf(y)$ for all $x$ and $y$ and for all scalars $s$ and $t$. The first time this comes up is usually in linear algebra, where the (finite dimensional) linear maps are essentially what matrices are. In this view, the only linear functions $f:\mathbb{R} \to \mathbb{R}$ are those that take the form $f(x) = ax$ for some $a \in \mathbb{R}$. Functions such as $3x+1$ are then called affine instead.

Or maybe you knew all of that and answered "linear" anyway, which is totally reasonable. After all, the real villain here is the word "linear" itself, which really shouldn't be the sort of term with contradictory definitions... and yet here we are!

Q99: Does $\left\lfloor e^{e^{e^{79}}}\right\rfloor$ exist?

This may seem like a pointless question, with the obvious answer being "yes". You may be surprised to find that not everyone agrees!

The question was about ultrafinitism, which is a matter of mathematical philosophy. Numbers such as $3$ and $4$ exist because they can be represented in the real world --- say, by having 3 apples or 4 grains of sand. But is a number such as $\left\lfloor e^{e^{e^{79}}}\right\rfloor$ too big to exist?

It's difficult to quantify precisely what fraction of respondents were actually ultrafinitists due to sources of error. Nevertheless, the results and comments do seem to suggest that there exist a few ultrafinitists among the respondents. A few respondents commented that $\left\lfloor e^{e^{e^{79}}}\right\rfloor$ exists, but with some... caveats? *shrug*

(A reader pointed out an odd observation when Q20 and Q99 are compared: There are clearly people that chose both "numbers do not exist" and "$\left\lfloor e^{e^{e^{79}}}\right\rfloor$ exists". My guess is that this is because such people interpreted the words "exists" differently in each question (philosophically in the former, mathematically in the latter). You can find a breakdown table in the 91-100 page of the data sheet.)

Q100: What is the value of $6 \div 2(1+2)$?