# The things in proofs are weird: a thought on student difficulties

By Ben Blum-Smith, Contributing Editor

“The difficulty… is to manage to think in a completely astonished and disconcerted way about things you thought you had always understood.” ― Pierre Bourdieu, Language and Symbolic Power, p. 207

Proof is the central epistemological method of pure mathematics, and the practice most unique to it among the disciplines. Reading and writing proofs are essential skills (the essential skills?) for many working mathematicians.

That said, students learning these skills, especially for the first time, find them extremely hard.[1]

Why? What’s in the way? And what are the processes by which students effectively gain these skills?

These questions have been discussed extensively by researchers and teachers alike,[2] and they have personally fascinated me for most of my twenty years in mathematics education.

In this blog post I’d like to examine one little corner of this jigsaw puzzle.

#### Imported vs. enculturated

To frame the inquiry, I posit that there are imported and enculturated capacities involved in reading and writing proofs. Teachers face corresponding challenges when teaching students about proof.

Capacities that are imported into the domain of proof-writing are those that students can access independently of whether they have any mathematics training in school or contact with the mathematical community, let alone specific attention to proof.[3] Capacities that are enculturated are those that students do not typically develop without some encounter with the mathematics community, whether through reading, schooling, math circles, or otherwise. Examples of imported capacities are the student’s capacity to reason, and fluency in the language of instruction. Enculturated capacities include, for example, knowledge of specific patterns of reasoning common to mathematics writing but rare outside it, such as the elegant complex of ideas behind the phrase, “without loss of generality, we can suppose….”

For imported capacities involved in proof, the teaching challenge is to create conditions that cause students to actually access those capacities while reading and writing proofs.

For enculturated capacities, the prima facie teaching challenge is to inculcate them, i.e., to cause the capacities to be developed in the first place. But there is also a prior, less obvious challenge: we have to know they’re there. Since many instructors are already very well-enculturated, our culture is not always fully visible to us. If we can’t see what we’re doing, it’s harder to show students how to do it. (This challenge has the same character as that mentioned by Pierre Bourdieu in the epigraph, although he was writing about sociology.)

When my personal obsession with student difficulty with proofs first developed, I focused on imported capacities. I had many experiences in which students whom I knew to be capable of very cogent reasoning produced illogical work on proof assignments. It seemed to me that the instructional context had somehow severed the connection between the students’ reasoning capacities and what I was asking them to do. I became very curious about why this was happening, i.e., what types of instructional design choices led to this severing, and even more curious about what types of choices could reverse it.

My main conclusion, based primarily on experience in my own and others’ classrooms, and substantially catalyzed by reading Paul Lockhart’s celebrated Lament and Patricio Herbst’s thought-provoking article on the contradictory demands of proof teaching, was this: It benefits students, when first learning proof, to have some legitimate uncertainty and suspense regarding what to believe, and to keep the processes of reading and writing proofs as closely tied as possible to the process of deciding what to believe.[4]

I stand by this conclusion, and more broadly, by the view that the core of teaching proof is about empowering students to harness their imported capacities (in the above sense) to the task, rather than learning something wholly new. That said, in the last few years I’ve become equally fascinated by the challenges of enculturation that are part of teaching proof reading and writing. If I’m honest, my zealotry regarding the importance of imported capacities blinded me to importance of the enculturated ones.

What I want to do in the remainder of this blog post is to propose that a particular feature of proof writing is an enculturated capacity. It’s a feature I didn’t even fully notice until fairly recently, because it’s such a second-nature part of mathematical communication. I offer this proposal in the spirit of the quote by sociologist/anthropologist Pierre Bourdieu in the epigraph: to think in a completely astonished and disconcerted way about something we thought we already understood. Naming it as enculturated has the ultimate goal of supporting an inquiry into how students can be explicitly taught how to do it, though this goes beyond my present scope.

#### The things in proofs are weird

I recently encountered an article by Kristen Lew and Juan Pablo Mejía-Ramos, in which they compared undergraduate students’ and mathematicians’ judgements regarding unconventional language used by students in written proofs.[5] One of their findings was that, in their words, “… students did not fully understand the nuances involved in how mathematicians introduce objects in proofs.” (2019, p. 121)

The hypothesis I would like to advance in this post is offered as an explanation for this finding, as well as for a host of student difficulties I’ve witnessed over the years:

The way we conceptualize the objects in proofs is an enculturated capacity.

These objects are weird. In particular, the sense in which they exist, what they are, is weird. They have a different ontology than other kinds of objects, even the objects in other kinds of mathematical work. An aspect of learning how to read and write proofs is getting accustomed to working with objects possessing this alternative ontology.[6] If this is true, then it makes sense that undergraduates don’t quite have their heads wrapped around the way that mathematicians summon these things into being.

The place where this is easiest to see is in proofs by contradiction. When you read a proof by contradiction, you are spending time with objects that you expect will eventually be revealed never to have existed, and you expect this revelation to furthermore tell you that it was impossible that they had ever existed. That’s bizarro science fiction on its face.

But it’s also true, more subtly perhaps, of objects appearing in pretty much any other type of proof. To illustrate: suppose a proof begins,

Let $\Lambda$ be a lattice in the real vector space $\mathbb{R}^n$, and let $v\in \Lambda$ be a nonzero vector of minimal (Euclidean) length in $\Lambda$

Question. What kind of a thing is $v$?

[The camera pans back to reveal this question has been asked by a short babyfaced man wearing a baseball cap, by the name of Lou Costello. His interlocutor is a taller, debonair fellow with a blazer and pocket square, answering to Bud Abbott.]

Abbott: It’s a vector in $\mathbb{R}^n$.
Costello: Which vector?
Abbott: Well, it’s not any particular vector. It depends on $\Lambda$.
Costello: You just said it was a particular vector and now it’s not a particular vector?
Abbott: No, well, yes, it’s some vector, so in that sense it’s a particular vector, but I can’t tell you which one, so in that sense it’s no particular vector.
Costello: You can’t tell me which one?
Abbott: No.
Costello: Why not?
Abbott: Because it depends on $\Lambda$. It’s one of the vectors that’s minimal in length among nonzero vectors in $\Lambda$.
Costello: Which vector?
Abbott: No particular vector.
Costello: But is it some vector?
Abbott: Naturally!

Costello: You said it depends on $\Lambda$. What’s $\Lambda$?
Abbott: A lattice in $\mathbb{R}^n$.
Costello: Which lattice?
Abbott: Any lattice.
Costello: Why won’t you say which lattice?
Abbott: Because I’m trying to prove something about all lattices.
Costello: You mean to say $\Lambda$ is every lattice???
Abbott: No, it’s just one lattice.
Costello: Which one?!

For any readers unfamiliar with the allusion here, it is to “Who’s on First?”, legendary comedy duo Abbott & Costello’s signature routine.[7] What’s relevant to the present discussion is that the skit is based on Costello asking Abbott a sequence of questions about a situation to which he is an outsider and Abbott is an insider. Costello becomes increasingly frustrated by Abbott’s answers, which make perfect sense from inside the situation, but seem singularly unhelpful from the outside. Abbott for his part maintains patience but is so internal to his situation—enculturated, as it were—that he doesn’t address, or even seem to perceive, the ways he could be misunderstood by an outsider.[8]

My goal with this little literary exercise has been to dramatize the strangeness of the “arbitrary, but fixed” nature of the objects in proofs. Most things we name, outside of proof-writing, don’t have this character. Either they’re singular or plural; one or many; specific or general; not both. Every so often, we speak of a singular that represents a collective (“the average household”, “a typical spring day”), or that is constituted from a collective (“the nation”), but these are still ultimately singular. They are not under the same burden as mathematical proof objects, to be able to stand in for any member of a class. Proof objects aren’t representative members of classes but universal members. This makes them fundamentally unspecified, even while we imagine and write about them as concrete things.

There’s an additional strangeness: proof objects, and the classes of which they are the universal members, are themselves often constituted in relation to other proof objects. We get chains, often very long, where each link adds a new layer of remove from true specificity, but we still treat each link in the chain, including the final one, as something concrete. I was trying to hint at this by posing the question “what is it?” about $v$, rather than $\Lambda$, in the example above. As consternated as Costello is by $\Lambda$, $v$ is doubtless more confounding.

I think there are at least two distinct aspects of this that students new to proof do not usually do on their own without some kind of enculturation process. In the first place, the initial move of dealing with everything in a class of objects simultaneously by postulating a “single universal representative” of that class just isn’t automatic. This is a tool mathematical culture has developed. Students need to be trained, or to otherwise catch on, that a good approach to proving a statement of the form “For all real numbers…” might begin, “Let $x$ be a real number.”[9]

But secondly, when we work with these objects, their “arbitrary, but fixed” character forces us to hold them in a different way, mentally, than we hold the objects of our daily lives, or even the mathematical objects of concrete calculations. When you read, “Let $f$ be a smooth function $\mathbb{R}\rightarrow\mathbb{R}$,” what do you imagine? A graph? Some symbols? How does your mental apparatus store and track the critical piece of information that $f$ can be any smooth function on $\mathbb{R}$? Reflecting on my own process, I think what I do in this case is to imagine a vague visual image of a smooth graph, but it is “decorated”—in a semantic, not a visual, way—by information about which features are constitutive and which could easily have been different. The local maxima and minima I happen to be imagining are stored as unimportant features while the smoothness is essential. Likewise, when I wrote, “Let $\Lambda$ be a lattice in the real vector space $\mathbb{R}^n$,” what did you imagine? Was there a visual? If so, what did you see? I imagined a triclinic lattice in 3-space. But again, it was somehow semantically “decorated” by information about which features were constitutive vs. contingent. That I was in 3 dimensions was contingent, but the periodicity of the pattern of points I imagined was constitutive. I’m positing that students new to proof do not usually already know how to mentally “decorate” objects in this way.[10]

#### Examples

Here are some specific instances of student struggle that seem to me to be illuminated by the ideas above.

• In the paper of Lew and Mejía-Ramos mentioned above, eight mathematicians and fifteen undergraduates (all having taken at least one proof-oriented mathematics course) were asked to assess student-produced proofs for unconventional linguistic usages. The sample proofs were taken from student work on exams in an introduction to proof class. One of these sample proofs began, “Let $\forall n\in \mathbb{Z}$.” Seven of the eight mathematicians identified the “Let $\forall$…” as unconventional without prompting, and the eighth did as well when asked about it. Of the fifteen undergraduate students, on the other hand, only four identified this sentence as unconventional without prompting, while even after being asked directly about it, six of the students maintained that it was not unconventional. I would like to understand better what these six students thought that the sentence “Let $\forall n\in \mathbb{Z}$” meant.
• Previously on this blog, I described the struggle of a student to wrap her head around the idea, in the context of $\varepsilon$$\delta$ proofs, that you are supposed to write about $\varepsilon$ as though it’s a particular number, when she knew full well that she was trying to prove something for all $\varepsilon>0$ at once.
• A year and a half ago, I was working with students in a game theory course. They were developing a proof that a Nash equilibrium in a two-player zero-sum game involves maximin moves for both players. It was agreed that the proof would begin by postulating a Nash equilibrium in which some player, say $P$, was playing a move $A$ that was not a maximin move. By the definition of a maximin move, this implies that $P$ has some other move $B$ such that the minimum possible payout for $P$ if she plays move $B$ is higher than the minimum possible payout if she plays $A$. The students recognized the need to work with this “other move” $B$ but had trouble carrying this out. In particular, it was hard for them to keep track of its constitutive attribute, i.e., that its minimum possible payout for $P$ is higher than $A$‘s. They were as drawn to chains of reasoning that circled back to this property of $B$ as a conclusion, as they were to chains of reasoning that proceeded forward from it.
• In the same setting as the previous example, there was a student who, in order to get her mind around what was going on, very sensibly constructed some simple two-player games to look at. I don’t remember the examples, but I remember this: I kept expecting that when she looked at the fully specified games, “what $B$ was” would click for her, but it didn’t. Instead, I found myself struggling to be articulate in calling her attention to $B$, precisely because its constitutive attribute was now only one of the many things going on in front of us; nothing was “singling it out.” I found myself working to draw her attention away from the details of the examples she’d just constructed in order to focus on the constitutive attribute of $B$. My reflection on this student’s experience was what first pointed me toward the ideas in this blog post: I mean really, what is $B$, anyway, that recedes from view exactly when the situation it’s part of becomes visible in detail?
• This semester I taught a course on symmetry for non-math majors. It involved some elementary group theory. An important exercise was to prove that in a group, $xz=yz$ implies $x=y$. One student produced an argument that was essentially completely general, but carried out the logic in a specific group, with a specific choice of $z$, and presented it as an example. Here is a direct quote, edited lightly for grammar and typesetting. “For example [take] $x\cdot R90=y\cdot R90$; if we will operate on both sides the inverse of $R90$ we will get $(x\cdot R90)R270=(y\cdot R90)R270$. As we have proven that $(AB)C$ always $= A(BC)$, we can change the structure of the equation to $x(R90\cdot R270)=y(R90\cdot R270)$, [which] shows that x has to be equal to y.” The symbols $R90$ and $R270$ refer to one-quarter and three-quarters rotations in the dihedral group $D_4$. From my point of view as instructor, the student could have transformed this from an illustrative example to an actual proof just by replacing $R90$ and $R270$ with $z$ and $z^{-1}$, respectively, throughout. What was the obstruction to the student doing this?

#### Conclusion

My claim is that the mathematician’s skill of mentally capturing classes of things by positing “arbitary, but fixed” universal members of those classes, and then proceeding to work with these universal members as though they are actual objects that exist, is an enculturated capacity.[11] I think it’s a little bit invisible to us—at least, it was so to me, for a long time. My purpose in advancing this claim is that making the skill visible invites an inquiry into how we can explicitly lead students to acquire it. I hope those of you who have given attention to how to train students in this particular aspect of proof (reading and) writing will offer some thoughts in the comments!

#### Acknowledgement

I would like to thank Mark Saul and especially Yvonne Lai for extremely helpful editorial feedback.

#### Notes and references

[1] I trust that any reader of this blog who has ever taught a course, at any level, that serves as its students’ introduction to proof, has some sense of what I am referring to. Additionally, the research literature is dizzyingly vast and there is no hope to do it any justice in this blog post, let alone this footnote. But here are some places for an interested reader to start: S. Senk, How well do students write geometry proofs?, The Mathematics Teacher Vol. 78, No. 6 (1985), pp. 448–456 (link); R. C. Moore, Making the transition to formal proof, Educational Studies in Mathematics, Vol. 27 (1994), pp. 249–266 (link); W. G. Martin & G. Harel, Proof frames of preservice elementary teachers, JRME Vol. 20, No. 1 (1989), pp. 41–51 (link); K. Weber, Student difficulty in constructing proofs: the need for strategic knowledge, Educational Studies in Mathematics, Vol. 48 (2001), pp. 101–119 (link); and K. Weber, Students’ difficulties with proof, MAA Research Sampler #8 (link).

[2] Again, I cannot hope even to graze the surface of this conversation in a footnote. The previous note gives the reader some places to start on the scholarly conversation. A less formal conversation takes place across blogs and twitter. Here is a recent relevant blog post by a teacher, and here are some recent relevant threads on Twitter.

[3] This and the following sentence should be treated as definitions. I am indulging the mathematician’s prerogative to define terms and expect that the audience will interpret them according to those definitions throughout the work. In particular, while I hope I’ve chosen terms whose connotations align with the definitions given, I’m relying on the reader to go with the definitions rather than the connotations in case they diverge. I invite commentary on these word choices.

[4] This is an argument I have made at length in the past on my personal teaching blog (see here, here, here, here, here), and occasionally in a very long comment on someone else’s blog (here). Related arguments are developed in G. Harel, Three principles of learning and teaching mathematics, in J.-L. Dorier (ed.), On the teaching of linear algebra, Dordrecth: Kluwer Academic Publishers, 2000, pp. 177–189 (link; see in particular the “principle of necessity”); and in D. L. Ball and H. Bass, Making believe: The collective construction of public mathematical knowledge in the elementary classroom, in D. Phillips (ed.), Yearbook of the National Society for the Study of Education, Constructivism in Education, Chicago: Univ. of Chicago Press, 2000, pp. 193–224.

[5] K. Lew & J. P. Mejía-Ramos, Linguistic conventions of mathematical proof writing at the undergraduate level: mathematicians’ and students’ perspectives, JRME Vol. 50, No. 2 (2019), pp. 121–155 (link).

[6] Disclaimer: although I am using the word “ontology” here, I am not trying to do metaphysics. The motivation for this line of inquiry is entirely pedagogical: what are the processes involved in students gaining proof skills?

[7] Here’s a video—it’s a classic.

[8] One of the keys to the humor is that the audience is able to see the big picture all at once: the understandable frustration of Costello, the uninitated one, apparently unable to get a straight answer; the endearing patience of Abbott, the insider, trying so valiantly and steadfastly to make himself understood; and, the key idea that Costello is missing and that Abbott can’t seem to see that Costello is missing. I’m hoping to channel that sense of stereovision into the present context, to encourage us to see the objects in a proof simultaneously with insider and outsider eyes.

[9] Annie Selden and John Selden write about the behavioral knowledge involved in proof-writing, and use this move as an illustrative example. A. Selden and J. Selden, Teaching proving by coordinating aspects of proofs with students’ abilities, in Teaching and Learning Proof Across the Grades: A K-16 Perspective, New York: Routledge, 2009, p. 343.

[10] The ideas in this paragraph are related to Efraim Fischbein’s notion of “figural concepts”—see E. Fischbein, The theory of figural concepts, Educational Studies in Mathematics Vol. 24 (1993), pp. 139–162 (link). Fischbein argues that the mental entities studied in geometry “possess simultaneously conceptual and figural characters” (1993, p. 139). Fischbein’s work in turn draws on J. R. Anderson, Arguments concerning representations for mental imagery, Psychological Review, Vol. 85 No. 4 (1978), pp. 249–277 (link), which, in a broader (not specifically mathematical) context, discusses “propositional” vs. “pictorial” qualities of mental images. The resonance with the dichotomy I’ve flagged as “semantic” vs. “visual” is clear. I’m suggesting that the particular interplay between these poles that is involved in conceptualizing proof objects is a mental dance that is new to students who are new to proof. (Actually, it is not entirely clear to me that the dichotomy I want to highlight is “semantic” vs. “visual” as much as “general” vs. “specific”; perhaps it’s just that visuals tend to be specific. However, time does not permit to develop this inquiry further here.)

[11] Because this circle of skills involve taking something strange and abstract and turning it into something the mind can deal with as a concrete and specific object, they strike me as related to some notions well-studied in the education research literature: Anna Sfard’s reification and Ed Dubinsky’s APOS theory—both ways of describing the interplay between process and object in mathematics learning—and the more general concept of compression (see, e.g., D. Tall, How Humans Learn to Think Mathematically, New York: Cambridge Univ. Press, 2013, chapter 3).

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### 8 Responses to The things in proofs are weird: a thought on student difficulties

1. Sylvia Wenmackers says:

Dear Ben,
This is very interesting! It reminds me of some work in philosophy of mathematics on “arbitrary objects”. Kit Fine worked on this in the 1980s and Leon Horsten published a book about it in 2019, called “The Metaphysics and Mathematics of Arbitrary Objects”. For a very brief intro to Fine’s work, see Horsten’s summary with some criticism (2019, p. 131): https://books.google.be/books?id=hmSbDwAAQBAJ&pg=PA131.
Best wishes, Sylvia

• Ben Blum-Smith says:

Thank you Sylvia! I’m delighted, if unsurprised, to learn of philosophical work on these types of objects. I’m excited for the opportunity to have consideration of these teaching questions be informed a metaphysics lens in addition to a cognitive psychology lens as in the references discussed in note [10].

2. Paul Pearson says:

Hi Ben,

I love the Abbott and Costello routine!

Our language of formal proof uses loaded phrases (WLOG, let, etc.) that students initially don’t know how to understand because these phrases don’t explain *why* particular choices are being made or what some of the alternative choices are, which means they’re not grounded in something that makes sense to the student.

Students need to unpack the meaning of phases such as “Let v in Lambda…” and explain their context and purpose to themselves and the reader. For example, if a student wrote, “There are potentially many nonzero vectors in the lattice of minimal length. We know such a vector exists because the lattice has at least one nonzero vector. Further, there could be more than one vector in the lattice of minimal length because such vectors of minimal length could all lie on the same n-sphere. For the sake of having a general argument that does not rely on any particular properties of one such minimal vector, but only relies upon the fact that the vector is of minimal length among all the vectors in the lattice, we will choose one minimal length vector v in the lattice Lambda and only use that it is minimal moving forward.” Then, it would be clear that the student fully understood the context and purpose (the what and the why) of the situation. The fact that we write “Let v in Lambda…” and expect our students to know what that means and to be able to unpack it for themselves is kind of absurd. We should be developing fluency with formal mathematics skills in our students. We should have students explain things in depth and in detail in ways that make sense to themselves and others, rather than having them produce concise math without meaning and purpose.

Should professors and students be this verbose all of the time? Probably not. But, we should at least initially (and perhaps periodically) require very thorough explanations. An over reliance on symbols (upside down A, backwards E, etc.) and abbreviations (WLOG) can be an indicator that someone is faking understanding, which is why students learning to write proofs should be encouraged to explain things rather than rely on compact notation that doesn’t carry a lot of meaning for them yet.

Thank you,
Paul

• Ben Blum-Smith says:

Thanks for this Paul! Your unpacking of “Let v in Lambda…” is a beautiful illustration of the large amount of implicit cultural knowledge we have condensed into the linguistic conventions used in proof writing.

I’d love to hear a little more about concretely what it looks like in your (or others’ – chime in!) teaching, as far as the pedagogical process to support students in writing down all of that beautiful exposition.

3. Japheth Wood says:

Thanks for educating me about “imported vs. enculturated” – I hadn’t quite thought of it that way, but found it useful. The naming of some of the specific weird objects that we handle in proof goes to the heart of the matter in addressing students’ difficulties in learning proof, as well as professors’ difficulties in teaching it.
I already commented on your epsilon-delta post that I’m a fan of Professor Susanna Epp’s writing on teaching and learning proof. She has a nice article titled “The Language of Quantification in Mathematics Instruction” on her webpage that offers similar ideas.

• Ben Blum-Smith says:

Thanks for pointing out this reference! For interested readers, here’s the url for convenience:

https://condor.depaul.edu/~sepp/NCTM99Yrbk.pdf

Looks like it has a nice discussion of differences between the linguistic conventions of mathematical writing and everyday language.

4. Michael Bächtold says:

This “encultured” mathematical reasoning isn’t fundamentally different from reasoning in everyday life. First of all, saying “let v be an arbitrary but fixed vector…” is redundant, in the sense that nothing would change in the remainder of the proof, if we simply said “let v a vector…”. Second, we encounter such hypothetical objects in everyday reasoning. For instance you might say to a child, “suppose I throw a stone into a window…”. The child might then ask “which stone” and “which window”, but I think that rarely happens, since we learn hypothetical reasoning with “unspecified” objects early on. The main difference between mathematical and everyday reasoning seems to be the habit of naming/labeling these objects with made up names (like single letters). I guess that has to do with the fact that we commonly deal with several objects of the same type in mathematical arguments (like numbers x,y,z) and naming them makes further reasoning easier. In everyday language we might use other lables like “suppose I throw two stones into a window, the first and the second…” and then use the lables “first” and “second” in further arguments.

• Ben Blum-Smith says:

Thanks for this comment; it advances my thinking. The example of “suppose I throw a stone into a window…” is illuminating.

You’ll be unsurprised that I disagree (rather strongly) that this “isn’t fundamentally different” from proof objects; I think the difference is fundamental enough to write a blog post about. But perhaps this type of hypothetical language is the (what I would call) imported capacity from which the enculturated capacity of thinking in proof objects is built! I look forward to thinking more about this.

Of course adding the words “arbitrary, but fixed” before an object in a proof changes nothing mathematically. Indeed, I don’t think I have ever seen this phrase in a proof. The phrase “arbitrary, but fixed” is used in the blog post (and in the article by Selden & Selden mentioned in note [9], and undoubtedly many other places) not as something you’d actually write in a proof but as a descriptor of the (ontological? psychological?) character of proof objects. Elsewhere, Susanna Epp uses the descriptor “generic particular” for the same phenomenon (I thank Japheth Wood for directing me to Dr. Epp’s writings and this phrase in particular).

The question is whether or not this “arbitrary but fixed”/”generic particular” character of proof objects is distinct from the character of the mental objects of every day life. The argument of the blog post is that it is very distinct, with the implication that it’s useful for educators to focus our attention on how students come to master the technique of working mentally with objects possessing this character.

You disagree that this involves learning anything new; if you’re right, then there’s nothing to see here. Your example does not convince me of this (more below). But the continuity you see, between proof objects and the objects of everyday speculative hypotheticals, strikes me as useful to the project I’m proposing, in at least two ways: (1) this continuity directs us toward a *place to look* for an understanding of the path students take toward competent handling of proof objects; and (2) it also gives a clue for how one might build on students’ (what I would call) imported capacities in taking them along this path.

Now for the substance of my disagreement. I think there is a great psychological distance between “suppose I throw a stone into a window” and “let v be a vector of minimal length in a lattice…,” and it goes far beyond the habit of naming. To me the key point is this: the stone and the window are not under a burden to be universal within the classes of stones and windows. Say the grownup says, “Don’t throw a stone into a window because it’ll break,” and the child replies, “What if it’s is made of bulletproof glass?” or better yet, “What if it’s the window of a giant’s house and it’s the size of Pluto?” In all likelihood, all involved will understand that the child is being cheeky. In the rare case they are being earnest, the grownup will at least understand they are having a different conversation than the intended one. In contrast, when the analogous thing happens in a proof context, then the audience is making a good point! And perhaps the hypotheses of the theorem, or the proof strategy, need to be adjusted in response.

This is happening because the “stone” and the “window” of the everyday hypothetical are not precisely defined categories. It’s okay to have the context cue us about what kind of stone, what kind of window. We don’t have to handle our conversation in a way that encompasses everything that could rightly be called a stone and everything that could rightly be called a window. Context plays an important role in mathematical communication too; still, we make a great effort to minimize that role, with our lovingly crafted definitions. Because of this, mental images of proof objects have a much brighter line delineating their constitutive attributes from their contingent ones. When I imagine a window, I wouldn’t even know where to begin in sorting out the part of my mental image that “makes it a window” from everything else in my mind’s eye. (Well, it’s a hole in a wall of a house. Well, but so is a door. Ok, well, its purpose is to look out of. Ok well but is it still a window if it’s too high for my head to reach? In any case, why a house? What about a train window? Ok fine, it’s a hole in the wall of a dwelling or vehicle that’s designed to let light in. Well, but is it still a window if it’s on a subway train that never gets ambient light? Etc.) When I imagine a vector of minimal length in a lattice, on the other hand, it’s very easy for me to identify what makes it a vector of minimal length. It’s the bright line separating the constitutive from the contingent that makes the proof object different.