# Inner voice when reading mathematics

I am writing some notes for my students, and am curious about how others read mathematics: do you have an inner voice when reading mathematics? What about when you come across a symbol which you don't know the name of? Do you make something up in your head, or just skip over the inner enunciation of that symbol and remember the picture?

• Are you asking this because you're wondering whether to provide a "sounds like" pronunciation tip for symbols they might not know? Commented Apr 27 at 16:10
• Yes, I did think of that. There's also the question of how important is it to use familiar symbols.
– Ben
Commented Apr 27 at 16:53
• I've always made up names for my inner voice; for example $\xi$ has always been 'squiggle'. Now that I am an ignorant overseas expat, by extension when I am looking for street signs written in Chinese characters, it's 'diode, stack of books, vicegrips...' I do it for humans too ('noisy one, smart one, quiet one, helpful one')
– uhoh
Commented Apr 28 at 1:16
• @uhoh: Nuh.. you should learn to pronounce it correctly.. ξ is "cksi" (yes it starts with a stop).. Commented Apr 28 at 8:25
• math.stackexchange.com/q/4581006/96384 seems to be almost a duplicate of this question. Commented Apr 28 at 15:27

My impression is that an inner voice reading symbol-by-symbol is fairly common in primary and secondary education and for many undergraduates. Primary school students certainly have a strong verbal association between expressions like $$2 + 5$$ or $$10 - 7$$ and their readings like "two plus five" or "ten minus seven". That trend seems to continue up through secondary education into undergrad; here in the US I know a number of students who can recite $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ in an almost sing-song "$$x$$ equals minus $$b$$ plus or minus the square root of $$b$$ squared minus four $$a$$ $$c$$, all over $$2$$ $$a$$", and even into calculus ideas like "plus $$C$$" have tight verbal bindings.

But, at least for me, sometime in my undergrad as a math major or perhaps early in graduate school, I switched to processing math expressions more visually, with my inner voice taking on a commentating role. For example, if you show me this:

$$0 \rightarrow A \xrightarrow{f} B \xrightarrow{g} C \rightarrow 0$$

then my inner voice is acting more like someone giving a conference talk, albeit in rather more detail: roughly, I hear "We have a short exact sequence with $$B$$ in the middle. $$A$$ is on the left, $$C$$ is on the right, and our homomorphisms are called $$f$$ and $$g$$." (Or, if the details aren't super important, the commentary might just be "It's an SES with the usual variable names.") The difference is especially notable when reading expressions in the midst of a lot of context, like when reading research publications. For example, I might look at a very long equation, but my inner voice would be making comments like "Ah, with that substitution, now the second and third cases look almost identical, except that down here the third term is $$c$$ with the long subscript, not $$1$$." (Real example; I just grabbed a paper to test this out.)

While I can't give you proper empirical data to back this up, I suspect most people with an inner voice who get into graduate-level mathematics undergo a similar change. At some point, there are just too many pauses as one digests or nonlinear jumps to other parts of the text or chunkings of subexpressions into single ideas for symbol-by-symbol reading to still make sense.

As for internally vocalizing unknown symbols, it's harder to speak from recent experience because running into something new happens less often now, but generally I'm focused on the meaning more than the shape, which, if I can identify it, can get me by without knowing the symbol. So instead of "squiggly", I'm more likely to recognize a particular squiggly and read it as something specific to the context, like "degree of branching" or "quasi-dual" or whatever. If, however, I can't recognize the meaning right away, then my inner voice will say things like "that weird symbol", which I assume is comparable to the descriptions others already posted about saying "squiggly".

For your purposes of writing notes, I would say that using familiar symbols and giving information about how to pronounce unusual symbols are both important at all levels, but the name of a symbol is more important at lower levels, whereas at higher levels your readers are probably more likely to be able to get by as long as the context gives them a way to verbalize each symbol's meaning, even if that verbalization isn't exactly the name of the squiggle.

• You very nicely expressed the thoughts I had on reading the question. I don't read mathematical symbols at all like I read words - in particular nothing is "pronounced", i.e. there are no sounds that correspond to the symbols. Instead, they produce images, sometimes still, sometimes moving. Commented Apr 28 at 3:30

I took at honors calculus course at the University of Michigan during my first semester of college. The textbook (by Joseph Kitchen) used the full Greek alphabet, or most of it. I hadn't encountered it before, and would say squiggle for many of the letters. I was not able to remember the pictures, not to keep straight which was which. It definitely interfered with my learning of the mathematics.

Because of that bad experience, I've always been careful to write out the names of new symbols and how to say them.

• Yes. If I can't pronounce a symbol, it impedes my reading. If I cannot easily write it, it impedes my note-taking. Worst-case: about 70 years ago, C. Chevalley gave a course on classfield theory in Japan (I have a printed copy), and used Japanese/Chinese characters to denote many objects... I couldn't cope. :) Likewise, in the earlier versions of Langlands' notes on Eisenstein series and automorphic forms, "fraktur" letters were drawn by hand, with a similar cryptic effect. :) Commented Apr 27 at 17:39
• a fellow squiggler! :-)
– uhoh
Commented Apr 28 at 1:20
• I hand out a sheet with the Greek alphabet, including the letter's names, in each math class I teach. Commented Apr 28 at 15:28

I remember when I first encountered "Euler" in textbooks. My inner voice pronounced it "uler"...rhymes with "ruler." The professor in my discrete math class kept talking about "Oiler" this and "Oiler" that. I realized that Oiler must be an important person. It took me about a week to realize this was the same person I had encountered in the textbook. When it dawned on me, I remember raising my hand to correct the professor's pronunciation. As soon as I did this, I realized what a fool I was! It's been 45 years, and I am still embarrassed.

• That's a nice anecdote but I don't think it answers the question. Commented Apr 28 at 19:04
• Sometimes when something is emotionally charged, it's hard to see past that. So maybe give him some slack? Commented Apr 29 at 10:10
• That's nothing to be ashamed of.  I had almost the same misunderstanding, except that I assumed it was pronounced ‘Yooler’ — after all, the name looks like it could be Greek, and there's ample precedent in names like ‘Euclid’! Commented Apr 29 at 15:43
• I had an electromagnetism lecture once which wasn't really improved by my lecturer having a ten minute long debate with one of the students in the front row about the nuances of the pronunciation of "beta". I remember how stupid it all was. They brought up ancient Greek (which no one speaks any more), modern Greek (which we weren't speaking), how it was taught at various posh English schools (which most of us never attended), including in 18th-century (none of us that old), how it was pronounced in the US (which we weren't in). Nothing to do with what we should have been learning.
– Dan
Commented Apr 30 at 20:11

This question receives a very detailed treatment by J. Hadamard in his book An Essay on the Psychology of Invention in the Mathematical Field (1945, available on archive.org; repr. ed. as The Mathematician's Mind: The Psychology of Invention in the Mathematical Field, 1996, if you prefer dead tree books), which I recommend extremely highly. For the book, Hadamard has interviewed, among his other great contemporaries, Lévi-Strauss, Pólya and Einstein.

In particular, ch. VI is dedicated to the subject of the mathematician's ‘inner voice’, or the cases of complete absence thereof (Einstein, Hadamard himeslf, …).

• Can you summarize the salient points related to the OP's question? Commented Apr 28 at 20:47
• @Kimball Abridge Hadamard's understanding of the thought process of Pólya and Einstein when they worked on maths??? No way, I'm no Poincaré, and not qualified indeed. As Hadamard wrote a book on the topic, not a short paper, I believe he had a reason. I'm sorry, but the abridgement that you're asking for is a work of art on its own, and may only be done by a mind on par with the Hadamard's, not me. It's short, some 70‒80 pages in the 1996 reprint on 12mo paper; he called it ‘an essay’. The archive.org 1945 copy has a fairly small page size. And I wouldn't mind at all if it were longer! Commented Apr 29 at 0:24
• @kkm-stillwaryofSEpromises I think the concern @‍Kimball is expressing is that StackExchange answers are supposed to be self-contained, so even a just run-down of the main points would be helpful. You could still include a disclaimer that the source text has a lot of detail you aren't able to capture here. Commented Apr 29 at 1:57
• @audiation I understand, but a summary of Hadamard's essay may only be my own interpretation. There isn't a word in it that is unnecessary—he's a mathematician above all. Except perhaps direct quotations from the respondents, but then I'm going to be summarising Einstein… I'm really not qualified. I may try, it will take a week of work¹, but I'm afraid the result won't be satisfactory without me even noticing. Do you think the answer is unsalvageable? / ¹v. the joke with the punchline: "I need about 2 weeks to prepare a 10-min talk. Oh, not at all, a 3-hr lecture I can begin right now". Commented Apr 29 at 4:47
• @audiation, concretely, I must have a clear understanding of the essay as a whole; e.g., v. the two discussions (by the same OP) about a 4-paragraph-long quote from it: math.stackexchange.com/q/4083340, hsm.stackexchange.com/q/13044. I think I've grossly underestimated the effort required calling for just one week of work. :) The comments and the A in HS&M refer to 3 books and 2 papers without summarising them. I'm beginning to doubt that the self-containment requirement is satisfiable in this case. Is the reference to Hadamard really inadmissible while anecdotes are? Commented Apr 29 at 5:59

Good question. I think when I see an unfamiliar symbol I mentally voice it like "glyph" or "thingy" or something. Remembering the symbol visually doesn't do any good. (I assume that deep-reading a natural alphabetic-type language is ultimately closer to the verbal part of the brain than the visual.)

My own anecdote is that my first college math course had an excellent instructor, always careful to define symbols, and I recall that one of the smaller questions on the final exam was something like "list 5 Greek letters with their names that we've used in this course".

As audiation said, "At some point, there are [just too] many [...] chunking of subexpressions into single ideas for symbol-by-symbol reading to still make sense.". This actually causes every professional mathematician to use pronouns like "this" and "that" while reading mathematics mentally, and the mental articulation of "this" or "that" actually comes with a mental link to the referent. $$\def\code#1{\overline{#1}}$$

For example, I am pretty sure that no logician reads every symbol of "Since $$¬R$$ is computable, its execution is captured by a $$4$$-parameter $$Δ_0$$-sentence $$ψ$$, meaning that for any $$Q,y,Q'$$ we have that $$¬R(Q,y) = Q'$$ iff $$∃t\ ( ψ(\code Q,\code y,\code{Q'},t) )$$ is true.". Instead, I think something like "Since negation R is computable, its execution is captured by a 4-parameter delta-0-sentence, meaning that for any Q y Q prime we have that negation of R Q y equals Q prime iffff there is some t such that that sentence is true when you plug in all their codes and t.", and here I am thinking of "Q,y,Q'" while saying "their".

In some cases, I might even talk to myself like "If this is true, then this this this, but then we have this, so contradiction.", where each "this" refers to a different thing, but merely saying the word "this" helps the head.

Also not yet mentioned, we may speed up or slow down to mentally tell ourselves of grouping. For example "exp(x)·(1−x) ∈ (1+x+O(x^2))·(1−x) ⊆ 1+O(x^2) as x → 0" turns into "exponential x times one-minus-x is.. one plus x plus O x squared.. times.. one-minus-x.. which is 1 plus O x squared.. as x goes to zero".

I think I don't have an "inner voice" that reads symbols aloud to me, and probably I don't have an inner voice for equations (as I don't have an inner voicer for maps or chemical formulas). I just recognize the symbols and understand the expressions that use them.

My problem is that sometimes I don't realize that I don't know the names of the symbols I use until I try to actually read them aloud, and that usually happens as a surprise when I'm in front of the blackboard, which is annoying at least. Therefore, I need to be careful when studying new material

For example, I can't see any problem with solving the equation $$월+1=2$$ despite not knowing that Hangul character name nor being able to draw it. Having to read the equation aloud would be a different problem.

• +1 "I think I don't have an "inner voice" that reads symbols" Of course the OP should understand that not everyone does it the same way ... and in particular not all of the OP's students do it the same way. Commented Apr 29 at 15:41
• +1 more. I never had an inner voice. I'm thinking maths in silent abstractions, sometimes putting them into spatial relations representing their specific, not necessarily inherently geometric relations. But what I'm putting in space, I don't really know. “These things”, I don't visualise them. A colleague of mine had remarked when we were discussing the Hadamard's essay that she mentally operates with symbols as if they were written on a piece of paper. I'm certainly wanting visual memory to do such a feat. :) Yes, people math differently indeed! Commented May 6 at 6:46
• You may be interested in anauralia. See The Silent Mind and Its Association With Aphantasia or anauralia.com/anauralia (U. Auckland). It's an emerging field of research, and it's not clear to me exactly what is settled and what is in dispute. -- My inner-voice experience is similar to yours. I frequently whisper to myself, when I want to voice thoughts. Commented May 13 at 15:46

One thing I like about some algebra or geometry books is they will have a list of the Greek symbols, uc and lc, along with names in the inner cover or front materials.

I normally sneer at the fetish for team training and interdisciplinaryness. But a little session on learning the Greek alphabet seems like it would be kinda fun and beneficial. Learn to write them, their names, correlation to Latin letters, and most common usage. Wish I had had it. And kinda gives the non math loving kids a tiny break from math itself.

I do recall that Navy Nuclear Power School actually semiofficialy used the terms squiggle and then...other squiggle for the Greek letters that look like that. They also called Y0 and J0 Bessel functions Yo, as in Yo, Adrian, and Jo as in the girl in Little Women.

To this day, I am repulsed by the use of Fraktur letters...especially when not needed.

Also I prefer vectors to have arrows or slanty hats...not bold or italics, which is a hassle to replicate in handwriting.

Let me just explain you why I think your question is important: once I have done an intelligence test during a job interview, and after quite some deciphering, it turned out that there were following values:

I succeeded the test, managed getting the job, and afterwards I had a talk about it with a colleague, who said "Well, once you understood that 's' stands for 'square', 'r' for 'rectangle', 'c' for 'circle' and 'e' for ellipse, the test was not that hard.". I was shocked because I had never even thought of that (for me, it were just meaningless letters).

So, indeed: people might give meanings to mathematical symbols and it might be interesting to consider this while teaching those symbols. Hereby another typical example:

$$\cup \text{ stands for "union" and indeed, it looks like a letter "U".}$$ $$\cap \text{ stands for "intersection" but it does not look like a letter "I" at all.}$$

• That's interesting. I similarly just see those 4 lines as an image, s/r/c/e being "variable names" of no significance. I suppose that if, on the other hand, in a plodding way, one were reading it to oneself - "square, rectangle, ..." - then it might jump out. Commented Apr 29 at 10:23
• I still remember from primary school being taught that the greater than/less than symbols were greedy crocodiles that "ate" the larger numbers :) The U for union was defn another one. Commented Apr 29 at 23:42
• I always saw the intersection symbol as "n"tersection (it looks a lot like the letter "n")
– D.R
Commented Apr 30 at 2:40

If you really want to help your students, then keep an eye out for those who struggle. It is easy to believe that everyone have the same capabilities as yourself, but that is not true.

There are people that don't have a "Mind's eye" (the capacity to visualize). It is called "Aphantasia". Visualize in this context will include imagining doing anything regarding your senses, including having internal speech.