1. Introduction
Academics are often praised for displaying intellectual generosity. For example, sleep researcher Allan Rechtschaffen was described as “an affable, intellectually generous man” (Klinkenborg Reference Klinkenborg1997) and neuroscientist Leslie Ungerleider was praised for being “a world-class listener, intellectually generous, and unfailingly supportive” (Buffalo et al. Reference Buffalo, Courtney, De Weerd, Doyon, Jiang, Karni, Kastner, Pessoa and Rossi2021: 242). Mathematician Jean Bourgain was described as “kind, optimistic and intellectually generous” (Klarreich Reference Klarreich2019: 183), and historian of design Victor Margolin was characterized as “a man of immense intellectual generosity” (Brown et al. Reference Brown, Buchanan, DiSalvo, Doordan, Lee and Mazé2020: 5). These examples show that intellectual generosity is valued as a character trait and suggest it may be an intellectual virtue.
As virtue epistemologists have begun investigating individual virtues,Footnote 1 one might think they would be interested in intellectual generosity. Alas, it has unfortunately been relatively unexplored. To my knowledge, there are only two recent pieces of work that cover intellectual generosity in detail: Roberts and Wood’s chapter on this character trait in their 2007 book Intellectual Virtues: An Essay in Regulative Epistemology (Roberts and Wood Reference Roberts and Wood2007) and Morris’ 2021 article “Intellectual generosity and the reward structure of mathematics” (Morris Reference Morris2021b).Footnote 2
I can think of two reasons that may cause intellectual generosity to appear as less philosophically important than other virtues. First, it might be thought that sharing intellectual goods, an activity that is at the heart of intellectual generosity, is easy to do and straightforward to understand. After all, researchers share knowledge and ideas all the time, such as when they publish a paper or answer a student’s question. It may thus seem that intellectual generosity is relatively uninteresting and that there is little to be gained from subjecting it to philosophical analysis.
The second reason for a lack of interest in intellectual generosity may be because it seems to be less obviously impactful compared to other virtues. After all, we can intuitively grasp the impact of intellectual courage or humility and easily see how such virtues directly help researchers in their pursuit of truth and knowledge. For example, intellectually courageous people are not easily swayed from their research by risks to their personal safety or professional reputation. Moreover, intellectually humble people are willing and able to recognize that they could be mistaken in their beliefs and consider opposing viewpoints, helping them make progress in their research. In comparison, the consequences of intellectual generosity may seem less obvious and significant, perhaps because it is an “other-regarding” intellectual virtue (Kawall Reference Kawall2002). This, in turn, makes intellectual generosity appear as a less attractive target for philosophical investigation.
However, in what follows, I will challenge both these reasons for the lack of interest in intellectual generosity. More precisely, I will show that sharing intellectual goods can often be challenging and more complicated than it may first appear. I will also demonstrate that intellectual generosity can have significant positive consequences both for research and for communities of researchers.
I begin in section 2 by examining the work on intellectual generosity by Roberts and Wood and Morris. In section 3, I suggest that analyzing examples in which intellectual generosity is directed towards large groups, such as a research community, can highlight the complexity of this virtue as well as its significance. In sections 4 and 5, I present two new case studies of mathematicians who are intellectually generous (in Roberts and Wood’s sense) and who directed their generosity towards large groups. In section 6, I use these case studies to address the questions of the significance and complexity of intellectual generosity. Additionally, I present a tool from the field of design and human-computer interaction that could help agents cultivate this virtue. I then argue in section 7 that, due to the reward structure of mathematics, many mathematicians exhibiting intellectual generosity also exhibit intellectual courage. Finally, in section 8, I briefly discuss avenues for future work.
This paper is thus both a deepening and an extension of the work of Roberts and Wood and Morris. The conclusions that it arrives at complement the accounts of the previous authors. However, it also offers genuinely new insights about intellectual generosity that are absent from their accounts, such as suggesting a tool to help cultivate intellectual generosity and pointing to a connection between intellectual generosity and intellectual courage within the context of mathematics.
2. Intellectual generosity in the philosophical literature
Intellectual generosity is an intellectual virtue of the responsibilist tradition,Footnote 3 meaning, in rough terms, that it is an acquired character trait that promotes intellectual flourishing.
2.1. Roberts and Wood’s account
Roberts and Wood developed a detailed account of intellectual generosity in a chapter of their 2007 book Intellectual Virtues: An Essay in Regulative Epistemology. Their characterization makes use of the distinction between intrinsic (internal) and extrinsic (external) intellectual goods found in MacIntyre’s (Reference MacIntyre1981) work. Intrinsic intellectual goods are internal to an intellectual practice and include goods such as understanding and knowledge. Extrinsic intellectual goods are external to the practice and include things such as prizes, credit, and fame. Intrinsic intellectual goods are the kind of goods that must be aimed at to achieve excellence in a practice. Extrinsic intellectual goods, by contrast, are not. For example, to achieve excellence in philosophy, you must aim to understand arguments, but you do not need to aim to be rich and famous.
Roberts and Wood’s characterization of intellectual generosity is then as follows:
this virtue is a glad willingness to give intellectual goods, both intrinsic and extrinsic, to others, and … this willingness is based on a dominance of two kinds of concerns: an interest in the intrinsic intellectual goods of knowledge, information, confirmation (or disconfirmation) of hypotheses, understanding, and other such goods; and an interest in the intellectual well-being of other people. In particular, these two kinds of concerns dominate over the concern to have, for oneself, such extrinsic intellectual goods as position, honors, and wealth (Roberts and Wood Reference Roberts and Wood2007: 304).
Roberts and Wood illustrate their account of intellectual generosity with a variety of case studies, including that of Barbara McClintock, a cytogeneticist at Cornell who studied the role of chromosomes in inheritance. She developed an experiment to test whether genes are located on chromosomes, but instead of carrying out this important experiment herself, she shared it with Harriet Creighton, a graduate student. In doing so, she was not only sharing knowledge, understanding, and the opportunity to make a new scientific discovery with Creighton but also the credit and recognition that would come from the experiment. Indeed, the resulting paper “A correlation of cytological and genetical crossing-over in Zea mays” (Creighton and McClintock Reference Creighton and McClintock1931) won both women international recognition.
In sharing her important research project with Creighton, McClintock was emulating the policy, due to Cornell geneticist Rollins Emerson, of sharing important research problems with students (Roberts and Wood Reference Roberts and Wood2007: 299). Roberts and Wood note that this policy reflects a genuine concern for the intellectual well-being of others (Roberts and Wood Reference Roberts and Wood2007: 299).
Further, McClintock was much more concerned with intrinsic over extrinsic intellectual goods (Roberts and Wood Reference Roberts and Wood2007: 299–301). For example, Roberts and Wood point out that when describing her research “she spoke of the ‘real affection’ one gets for the ‘pieces that go together’” (Roberts and Wood Reference Roberts and Wood2007: 300). And when she won a Nobel prize, they report she viewed it as a mixed blessing as it served as a distraction from her work (Roberts and Wood Reference Roberts and Wood2007: 301).
Roberts and Wood further use this case study to draw attention to the benefits of intellectual generosity. They suggest that McClintock’s love of intrinsic intellectual goods allowed her to persevere where others would have failed, thus enabling her to make more discoveries and uncover new intrinsic goods (Roberts and Wood Reference Roberts and Wood2007: 301–302). Additionally, they point to how sharing projects can increase the speed at which new discoveries are made, as well as allowing researchers to pursue multiple projects at once, again increasing the yield of new intrinsic intellectual goods (Roberts and Wood Reference Roberts and Wood2007: 302). Finally, they note that sharing projects with graduate students brings science to a new generation and ensures they will go on to make new discoveries themselves (Roberts and Wood, Reference Roberts and Wood2007: 302).
2.2. Morris’ work on intellectual generosity
Morris (Reference Morris2021b) uses the mathematician William Thurston as a case study of intellectual generosity. Her discussion of intellectual generosity focuses on one episode of his life: his work on the geometrization conjecture. This is a conjecture Thurston made about mathematical objects called 3-manifolds and proved for the special case of Haken manifolds.
Thurston’s research on the geometrization conjecture was not widely understood at first, so he spent considerable time and effort explaining the concepts fundamental to his work. For example, he wrote up a set of widely circulated lecture notes aimed at graduate students and professional mathematicians to make his way of thinking more accessible. He was also deliberately slow to publish all the theorems he knew how to prove, because he wanted to give other mathematicians a chance to prove them and receive professional recognition, which he termed “theorem-credits,” for doing so.
Morris uses Thurston’s recollections in Thurston (Reference Thurston1994) as well as email interviews with his colleagues to argue that he exemplifies intellectual generosity in Roberts and Wood’s sense (Morris Reference Morris2021b, sec. 3.3). First, by explaining the important concepts and publishing his results slowly so that other mathematicians had the chance to publish theirs, Thurston was sharing both intrinsic and extrinsic intellectual goods with others. Second, Morris argues that his sharing of these intellectual goods was grounded both in a concern with the intellectual well-being of others as well as in a concern with intrinsic intellectual goods.
Morris further argues that generosity, such as Thurston’s, can overcome problems caused by the reward structure of mathematics. For example, the reward structure disincentivizes mathematicians from undertaking expository work, i.e., work that aims to make mathematical ideas accessible to a wider number of mathematicians (Morris Reference Morris2021b, sec. 5.2). This can hinder mathematical discoveries, which often come about by making connections between different subfields. However, intellectually generous mathematicians, concerned with intrinsic intellectual goods and the intellectual well-being of their colleagues, will write expository work that can facilitate collaborations between researchers in different fields and so promote new mathematical breakthroughs (Morris Reference Morris2021b: 363).
3. Analysis of existing work
There is an interesting difference between the case studies discussed by Roberts, Wood, and Morris: the number of recipients of intellectual generosity. In the McClintock case, the recipient of her generosity was a graduate student: Creighton. But in the Thurston case study, his generosity was directed towards the entire research community of mathematicians in the field of geometric topology.
It seems plausible to suggest that acting in an intellectually generous way toward a large and diverse group of people, such as a research community, is more difficult than acting in an intellectually generous way to a small number of people known to the giver. After all, an intellectually generous agent plausibly needs to know the intellectual wants and needs of their intended recipients, as well as knowing how to share intellectual goods with them effectively. Plausibly, it will be easier for them to meet these conditions when they are directing their generosity at a small number of specific recipients because they can, for example, communicate and work with them directly.
In the case where the intended recipients form a large, diverse group, like a research community, it will be harder to identify the wants and needs of the group, because different members of the group will have different backgrounds and levels of expertise, which will need to be weighed against each other.Footnote 4 Similarly, sharing intellectual goods with a large group will also be more challenging for an intellectually generous agent, because it is not feasible for them to communicate directly with or work together with all the members of the group.
However, the potential benefits of intellectually generous acts directed toward a large group appear to be more significant than those directed toward a small group. After all, in a large group, more people receive intellectual goods and can use them to improve their own intellectual situation and make new discoveries.
This suggests that studying intellectual generosity directed toward large groups can help us better understand both its significance and its complexity. I present two such case studies in sections 4 and 5 below. The case studies are from mathematics, but no mathematical knowledge is required to understand them. The case studies are more detailed than you would usually find in a philosophy paper and include email interviews with the mathematicians involved. I have presented the case studies with this level of detail to ensure they are realistic and allow for accurate insights into the complexity and significance of intellectual generosity.
In analyzing the case studies presented in sections 4 and 5, I consider the following questions, which will be discussed in detail in section 6:
Q1. In what ways can intellectually generous agents share intellectual goods with a large group?
The Thurston case study shows that publishing lecture notes aimed at graduate students and professional mathematicians, as well as being slow in publishing proofs of novel theorems, were effective ways of sharing intellectual goods with the research community of geometric topologists. But what other ways are there to share intellectual goods with large groups? And what challenges are associated with them?
Q2. What impact do intellectually generous actions aimed at large groups have in general?
Morris’ work showed that Thurston’s intellectual generosity had a large impact on the research community of geometric topologists. However, Thurston’s status as an incredibly talented and idiosyncratic mathematician may have contributed to the significant impact of his generosity. So, what is the impact of intellectual generosity when directed towards large groups in less extreme cases?
Q3. How can agents better cultivate intellectual generosity, especially directed towards large groups?
While Morris highlights Thurston as an exemplar of intellectual generosity, she does not directly discuss how we can better cultivate this virtue. So, what can we do if we want to become (more) intellectually generous? Are there any tools we can use that might help us direct our generosity towards large groups?
I’ll end this section with a brief preview of the case studies. As I’ll argue, the mathematicians in the examples are all intellectually generous in Roberts and Wood’s sense. In both cases, they share intellectual goods with a large group by writing, and one group invents a new form of writing specifically for this purpose. Both new case studies are less extreme than the Thurston example, yet the generosity of the mathematicians involved still went on to have a significant impact by stimulating more discoveries and improving the climate of research communities. Further, both cases highlight the importance of paying close attention to the intended recipients of intellectual generosity, which, as I discuss in section 6.3, suggests that a tool from the fields of design and human-computer interaction may be useful to agents wishing to cultivate this virtue. Finally, I argue in section 7 that the case studies reveal that the reward structure of mathematics means that many mathematicians who exhibit intellectual generosity also exhibit intellectual courage.
4. Case study: David Cox, John Little, and Donal O’Shea
4.1. Introduction
David Cox, John Little, and Donal O’Shea are mathematicians who work in the field of algebraic geometry. Together, they wrote a textbook called Ideals, Varieties and Algorithms to make a method of solving systems of equations more accessible to undergraduate students. The textbook was successful and popular beyond mathematics, with scientists using it to learn how to solve systems of equations as well. In this section, I argue that Cox, Little, and O’Shea exemplify intellectual generosity in the sense of Roberts and Wood. I then briefly discuss what this case study tells us about the questions from section 3, although I defer a full discussion until section 6.
4.2. The writing of Ideals, Varieties and Algorithms
As Cox, Little, and O’Shea explained in “The Story of Ideals, Varieties and Algorithms” (Cox et al. Reference Cox, Little and O’Shea2016), one of the goals they had in writing the textbook was to make Gröbner bases, a mathematical tool for solving systems of polynomial equations, accessible to undergraduates. They achieved this by writing the book in such a way that it did not assume knowledge of abstract algebra, which previous textbooks, aimed primarily at graduate students, all required.
But reducing the prerequisites was not the only way Cox, Little, and O’Shea made their textbook more accessible. In an email interview, Little explained that the trio “took special pains to include detail, and to make things clear, accessible and interesting” (Little Reference Little2022). Additionally, the trio included diagrams and pictures “to give vivid illustrations of the theory” (Cox et al. Reference Cox, Little and O’Shea2016: 627) to help make the subject matter more accessible.
Cox, Little, and O’ Shea’s efforts were more successful than they could have imagined. Ideals, Varieties and Algorithms was awarded the Steele Prize for Mathematical Exposition in 2016, in part for its “clarity of exposition” but also because of the “impact it has had on mathematics” (American Mathematical Society 2016: 418). Furthermore, the features that made the textbook accessible to undergraduates also made it accessible to scientists as well.
In an email interview, Little explained: “This was probably the biggest surprise for us – that our book would be used by lots of coding theorists, cryptographers, electrical engineers, mechanical engineers, physicists, and others who wanted to solve systems of polynomial equations coming from questions in their own areas and turned to our book for understanding of general methods and examples that would help them” (Little Reference Little2022).
When asked what helped the book have such an impact, Little responded, “From the comments I have heard over the years, I think most readers feel like the book is addressed to them. It’s not just an elegant presentation of a completed mathematical theory with no motivation for why or how people found these things out” (Little Reference Little2022). He added, “I think it also helped that all of us (David, Don, and I) really love this material and we got (and still get) excited to explain it when we get the chance” (Little Reference Little2022).
4.3. Intellectual generosity of Cox, Little, and O’Shea
From the above, we see that Cox, Little, and O’Shea were gladly willing to share intrinsic intellectual goods by making the theory of Gröbner bases more widely accessible. They took the time to write a textbook that was not only clear and understandable to those with a limited mathematical background but was also presented in an interesting and engaging way.
The textbook also shared intellectual goods in a more indirect manner by giving students more opportunities to engage in mathematical research. In an email interview, Cox explained: “We wanted to contribute to the idea that math majors can do more than just do homework and take exams – there are also course-based projects, senior theses, and REU [Research Experience for Undergraduates] programs, all of which could use IVA [Ideals, Varieties and Algorithms]” (Cox Reference Cox2022). This was something that Little also emphasized: “we were all naturally interested in finding ways to provide research experiences for undergraduates because we were teaching at liberal arts colleges with good mathematics programs” (Little Reference Little2022).
Providing more opportunities for research not only shares intrinsic intellectual goods in the form of the new mathematical knowledge the students uncover, but also provides them with extrinsic intellectual goods. More precisely, students who complete a research project, such as a REU program, often have the opportunity to write up their results and submit their paper for publication, for example, in an undergraduate mathematics journal. This allows them to earn professional recognition for their work in the form of, to borrow Thurston’s term, “theorem credits.”
For Cox, Little, and O’Shea to be intellectually generous in Roberts and Wood’s sense, however, their glad willingness to share intellectual goods must be grounded in a concern with the intellectual well-being of others, or with intrinsic intellectual goods, as opposed to their own extrinsic intellectual goods. And this is indeed the case. As we have seen above, Cox, Little, and O’Shea wanted to expand the mathematics available to undergraduates and give them more opportunities to participate in mathematical research. This all suggests that they were concerned about the intellectual well-being of their students.
Moreover, while Cox, Little, and O’Shea did receive extrinsic intellectual goods for their textbook, in the form of the Steele Prize for Mathematical Exposition and royalties, these rewards were not what motivated them to write the book. First, the authors had no expectation that their book, written for undergraduates, would win the Steele Prize. In fact, they describe how they were “stunned” to learn it had been awarded to them since, as far as they knew, “no book written explicitly for undergraduates had ever won the Steele Prize” (Cox et al. Reference Cox, Little and O’Shea2016: 626). And as to royalties, Little noted: “We took a cut in royalties from Springer so that the book would be more accessible (less costly)” (Little Reference Little2022).
Cox, Little, and O’Shea were thus gladly willing to share both intrinsic and extrinsic intellectual goods due to a concern for the intellectual well-being of mathematics students. They therefore exhibited intellectual generosity. Moreover, we see that they directed their generosity towards the large group of mathematics undergraduate students when they wrote their textbook. However, their generosity impacted an even larger group, including scientists and engineers, who also benefited from their work.
4.4. Summary
While Morris’s Thurston case study showed that writing lecture notes aimed at graduate students is one way to share intrinsic intellectual goods with a large group, the case study of Cox, Little, and O’Shea shows that writing an undergraduate textbook can also be effective. Moreover, by aiming their textbook at undergraduate students, the trio made the theory of Gröbner bases accessible to a much larger audience than they initially envisioned, including scientists working in fields such as engineering and physics. This meant Cox, Little, and O’Shea’s generosity had a wide impact on multiple fields, not just mathematics, as scientists used the theory of Gröbner bases to solve their own problems.
Finally, the case study shows how it is important for givers to pay close attention to the recipients of their intellectual generosity. For example, Cox, Little, and O’Shea knew how to make the theory of Gröbner bases more accessible to undergraduates (reduce prerequisites), how to best communicate the ideas to their audience (include diagrams and pictures), and to be mindful of their audiences’ limited financial resources (which is why they took a cut in royalties). Paying attention to their audience in this way helped Cox, Little, and O’Shea successfully exemplify intellectual generosity.Footnote 5 I come back to this in section 6, where I discuss a tool that can help potential givers pay attention to their audience.
5. Case study: the user’s guide project
5.1. Introduction
In 2014, mathematician Luke Wolcott started the User’s Guide Project (Larson et al. Reference Larson, Mazur, White and Yarnall2020). His initiative involved the creation of a new type of mathematical writing, called user’s guides, which participants wrote to make their own mathematical research more understandable. In this section, I argue that many of the participants in this project were intellectually generous, with the recipients of their generosity being graduate students in algebraic topology. I then briefly discuss what this case study tells us about the questions from section 3, although I defer a full discussion until section 6.
5.2. User’s guide
To understand the User’s Guide Project and its participants, we first need some background on user’s guides themselves. Essentially, a user’s guide is a less formal companion to a mathematical research paper, written by the author of the research paper. All guides are downloadable for free at the User’s Guide Project website: https://mathusersguides.com.
The purpose of a user’s guide is to make the corresponding research paper more understandable and accessible, as well as to document the research process the paper resulted from. In an email interview, Wolcott, the founder of the project, described the target audience of the guides as “that frustrated grad student struggling to make sense of a paper” (Wolcott Reference Wolcott2022).
Each user’s guide is structured in the same way, having four different sections:
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Section 1: Key insights and organizing principles. This provides the background for the research paper, giving an answer to the question “what is [its] essence […]?” (Enchiridion: Mathematics User’s Guides n.d).
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Section 2: Conceptual metaphors and mental imagery. This helps the reader answer the question “how should we think about” the research and often includes diagrams and pictures (Enchiridion: Mathematics User’s Guides n.d).
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Section 3: Story of development. Here, the authors provide an account of their research process, detailing, e.g., how they arrived at their results and any setbacks they experienced.
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Section 4: Colloquial summary. In this final section, the authors provide a summary of their research, aimed at non-mathematicians.
The project was designed so that the writing of the user’s guides was collaborative and structured. Participants in each volume of the project would spend a year working on their guides, writing one of their sections every 3 months. Each participant in that volume would then read and provide feedback on each section and, once the user’s guides were completed, the participants would peer review them in their entirety (Larson et al. Reference Larson, Mazur, White and Yarnall2020: 416–417).
In a reflective piece on the project published in the Journal of Humanistic Mathematics (Larson et al. Reference Larson, Mazur, White and Yarnall2020), four participants emphasized the benefits it had for the mathematical community. Specifically, they identified three problems within algebraic topology that it can help to overcome:
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1. The field relies on very technical tools that can be difficult for young researchers or other mathematicians to understand (Larson et al. Reference Larson, Mazur, White and Yarnall2020: 418).
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2. Folklore, i.e., “knowledge that is passed along through conversation that is never written down carefully,” is rife in algebraic topology (Larson et al. Reference Larson, Mazur, White and Yarnall2020: 418). This makes math less inclusive, as it can be difficult for early-career researchers, mathematicians from underrepresented groups, and those without significant travel budgets to be in a position where they can comfortably ask the “big shots” of the field what they need to know.
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3. The field has a culture of “hiding weakness in order to preserve reputation,” which can make early-career researchers feel like they are not cut out for mathematics when they struggle with their research (Larson et al. Reference Larson, Mazur, White and Yarnall2020: 418).
The information the user’s guides contain helps address each of these problems. By providing the reader with the key ideas of the corresponding research paper, as well as the right way to think about the concepts involved, the user’s guides make algebraic topology research more accessible, thereby overcoming the first problem.
The user’s guides help tackle the second problem by serving as an “open-access repository for mathematical folklore” that can make “algebraic topology and mathematics generally more inclusive” (Larson et al. Reference Larson, Mazur, White and Yarnall2020: 420). As an example, one of the key ideas in Don Larson’s research paper “came from an informal note listed in an “Other Documents” section of a leading researcher’s personal webpage. This idea is now explicitly stated as a Key Idea in the corresponding user’s guide” (Larson et al. Reference Larson, Mazur, White and Yarnall2020: 413). So, instead of being hidden in a relatively obscure location, the idea and its importance to Larson’s research have been made clear in the corresponding user’s guide.
Additionally, the User’s Guide Project can reduce epistemic injustice in mathematics. As Rittberg, Tanswell, and Van Bendegem have argued, certain kinds of mathematical folklore “promote secret knowledge and secret knowledge is a source of epistemic injustice in mathematics” (Rittberg et al. Reference Rittberg, Tanswell and Van Bendegem2020: 3890). They add that publishing these results “would help to dry out this source” (Rittberg et al. Reference Rittberg, Tanswell and Van Bendegem2020: 3890), which is exactly what the User’s Guide Project allows mathematicians to do.
Finally, the user’s guides often describe the authors’ setbacks, as well as how they eventually broke through them. This can help address the third problem by showing early-career researchers that everyone struggles with their research sometimes.
5.3. Intellectual generosity of the user’s guide project participants
Many participants in the User’s Guide Project appear to exhibit intellectual generosity. First, they were gladly willing to make intrinsic intellectual goods, in this case, their own original research and some of the tools and concepts important to its development, accessible to graduate students by voluntarily signing up for the project and writing and peer reviewing the guides. Further, by providing graduate students with intrinsic intellectual goods, the participants were also helping them to learn enough about algebraic topology to conduct their own research and receive professional credit for doing so. For example, the intrinsic goods the participants shared may have helped graduate students successfully complete their dissertations or publish research papers, thus earning “theorem credits.”
Second, the participants’ willingness to share intellectual goods was not grounded in a concern with extrinsic intellectual goods. As the user’s guides were expository, i.e., did not contain any new research, many of the participants’ institutions did not value their work on the project for tenure and promotion purposes. Six out of eleven participants who were asked said they would not receive credit towards tenure and promotion for their user guides, four were unsure, and only one indicated they would receive credit (Larson et al. Reference Larson, Mazur, White and Yarnall2020: 424). So, by participating, mathematicians had to give up time to work on other projects that would have earned them extrinsic intellectual goods.
Further, as Larson et al note in their reflective piece:
Most authors chose to participate in the User’s Guide Project because (1) they agreed with the mission of making algebraic topology more accessible and of humanizing the mathematical research process, and/or (2) they hoped to improve both their expository writing skills and their understanding of their research (Larson et al. Reference Larson, Mazur, White and Yarnall2020: 422).
Given the problems within algebraic topology that the User’s Guide Project addressed, those participants who found themselves agreeing with the project’s mission were presumably motivated to make their work more accessible out of a concern for the intellectual well-being of their fellow mathematicians, particularly less experienced researchers. In fact, Wolcott noted that “that frustrated grad student struggling to make sense of a paper […] was a really vivid image we all shared and we all came together around” (Wolcott Reference Wolcott2022).
Many participants in the User’s Guide Project, therefore, exhibit intellectual generosity. They were gladly willing to share intellectual goods with graduate students and did so not out of a concern for their own extrinsic intellectual goods, but out of a concern for the intellectual well-being of the students. Moreover, the participants directed their intellectual generosity towards a large group: that of graduate students in the field of algebraic topology.
5.4. Summary
We’ve already seen that existing forms of writing, such as lecture notes and textbooks, can be an effective way to share intellectual goods with large groups. The User’s Guide Project, however, took a different approach and invented a new form of writing called a user’s guide. This new format was specifically designed to make the corresponding research paper easier to understand and thus share intellectual goods with a wider audience.
The User’s Guide Project had a positive effect on the climate of algebraic topology by making technical tools more understandable, publishing folklore (and thus reducing epistemic injustice), and being honest about how difficult research can be. Wolcott remarked in an email interview that while he did not have any “concrete” examples of its impact, he did recall “contacts from strangers who found the user’s guides and read them and found them useful” (Wolcott Reference Wolcott2022).
The User’s Guide Project also highlights how important it is for givers to pay close attention to the recipients of their intellectual generosity. For example, participants in the project kept putting themselves in the shoes of their intended audience, frustrated graduate students, by asking questions such as “what do you wish you’d known at the start?” and “what tools do you wish you’d had?” (Wolcott Reference Wolcott2022). This allowed them to better explain the technical tools used in the field and communicate important intuitions about the objects of study, for example. In turn, this helped ensure that the intrinsic intellectual goods contained in the original research papers were made more accessible. I will discuss in section 6 how this observation about paying attention to the intended audience can be turned into practical advice for cultivating intellectual generosity.
6. What the case studies teach us
Having examined two detailed case studies in sections 4 and 5, I now offer tentative answers to the questions raised in section 3.
6.1. In what ways can intellectually generous agents share intellectual goods, especially with a large group?
The case studies discussed in sections 4 and 5 both focus on situations where intellectual goods are shared with a large group by simplifying high-level material and making it more understandable. Cox, Little, and O’Shea wrote an undergraduate textbook, while participants in the User’s Guide Project produced a new form of writing, user’s guides, that were specially designed to make high-level research more understandable.
While these case studies focused on making high-level research more understandable, there are other ways to share intellectual goods with a large group. For example, translating research into a different language or publishing it in an open-access journal can reduce barriers to the intrinsic intellectual goods it contains, thus opening it up to a larger audience. Often, a combination of approaches will have the biggest impact. For example, the user’s guides made current research more understandable and were published open access on the User’s Guide website to ensure they could be freely downloaded by anyone.
It is also worth noting that, in both case studies, the sharing of intellectual goods was done with written communication. Written communication is especially well-suited to sharing with a large audience because once a piece of writing has been completed, it can easily be shared with anyone. It will also persist, allowing new readers to find and engage with it years after it was created.
Oral communication, in contrast, does not have the same features as written communication and is not usually an effective way to share intellectual goods with a large group. In fact, as we saw in the User’s Guide Project, communicating information only orally can potentially lead to epistemic injustice. However, oral communication that has been recorded in a video can also be a productive way of sharing with a large group because it has similar characteristics to written communication. More precisely, video communication can be easily shared online, making it accessible to anyone, and if the recording is uploaded to a reputable video-sharing platform, it will persist and can be found by new viewers years after it was created. Thus, an intellectually generous mathematics professor wanting to make material in her field more understandable could make videos explaining concepts or conveying intuitions and share them on social media or her personal webpage.
Finally, the case studies highlight that sharing intellectual goods is not a passive process. It is an active one that often requires the giver to transform intellectual goods into a form that the intended recipients can receive and engage with. This means that the giver should not only have a deep knowledge of the intrinsic intellectual goods they want to share, but also insight into their audience, and knowledge of the best ways to communicate with them.
6.2. What impact do intellectually generous actions have in general?
The case studies examined in sections 4 and 5 show that intellectually generous actions aimed at large groups can have a big impact in two main ways. The first way is by generating new intrinsic intellectual goods, as shown by the case study of Cox, Little, and O’Shea. Recall that physicists, engineers, and cryptographers used Cox, Little, and O’Shea’s textbook to solve systems of polynomial equations that arose from questions in their own field, thus generating new intrinsic intellectual goods. This also suggests that acts of intellectual generosity directed at large groups have the potential to foster cross-fertilization between fields, which can lead to important breakthroughs (see e.g., Morris Reference Morris2021a).
The second way intellectually generous actions aimed at large groups can be impactful is by making the research community more intellectually edifying. This is exactly what the User’s Guide Project did for algebraic topology. By taking steps to make research in the field easier to understand, publishing folklore, and being transparent about difficulties in the research process, it made the field more approachable and inviting to newcomers and reduced the potential for epistemic injustice.
Making a research community more intellectually edifying can also indirectly promote the development of new intrinsic intellectual goods. After all, a welcoming and inclusive field will make it more likely to attract and retain practitioners who can uncover new intrinsic intellectual goods and help the field grow.
6.3. How can agents better cultivate intellectual generosity, especially directed towards large groups?
Recall that Cox, Little and O’Shea and the participants in the User’s Guide Project paid attention to the goals and difficulties of their audiences and used this knowledge to guide how they wrote their textbook and user’s guides.Footnote 6 This process was made explicit by Wolcott when he remarked in an email interview that he and other members of the project “were often talking about, “what do you wish you’d known at the start?” or “what tools do you wish you’d had?”.” He continued, “The target audience for the guides became that frustrated grad student struggling to make sense of a paper… and this was a really vivid image we all shared and we all came together around.”
Participants in the User’s Guide Project had essentially formed a proto-persona of their target audience. Personas originate from the field of design and human-computer interaction (see e.g., Cooper Reference Cooper1999) and are fictional representations of users who interact with a product, though they are developed from data obtained by real research, such as observational studies or interviews with users. Personas often include descriptions of the user’s goals, difficulties, and preferences, and are used to guide decision-making throughout the design process.
The proto-persona that Wolcott and others in the User’s Guide Project had formed, that of the frustrated graduate student, was not derived from observational studies or interviews with current graduate students. But it was derived from the experiences the participants had had when they themselves were graduate students. And it was something that they kept referring to during the writing process.
As many members of the User’s Guide Project exemplified intellectual generosity, personas may thus be a useful tool for agents who want to cultivate this virtue. I am not claiming that intellectually generous agents all use personas or must use personas, however. I am simply claiming that, if an agent wants to become intellectually generous by sharing intellectual goods with a large group, using personas might both help them share more effectively and develop intellectually generous motivations. Below, I will provide some examples of personas and discuss how they can help a mathematician better share intellectual goods with large groups. I will then draw on research from psychology to argue that personas also help agents to cultivate the motivational component of intellectual generosity.
Suppose Stacy wants to make algebraic topology more accessible by writing exposition aimed at a large audience. She first talks to graduate students, early-career professionals, and researchers in related fields such as algebraic geometry and number theory, to form personas for these different groups. For each persona, the name chosen is fictional, but the goals, pain points, and preferred content categories reflect data obtained from real conversations she had with multiple people in each category:
Name: Terry
Age: 20–25
Occupation: Graduate student
Goals: To finish dissertation in algebraic topology
Pain points: Overwhelmed by technical tools and concepts used in papers, existing textbooks are either too introductory or too advanced
Preferred content: An intermediate textbook, a free online workshop covering central tools for grad students in algebraic topology, blog posts summarizing recent research papers
Name: Abigail
Age: 25–30
Occupation: Postdoc
Goals: To keep up to date with research in algebraic topology and publish original research
Pain points: Keeping up with volume of very technical papers
Preferred content: Blog posts summarizing recent papers, textbook covering modern topics in algebraic topology
Name: Mark
Age: 35–45
Occupation: Mathematics professor (specializing in number theory)
Goals: To learn more about algebraic topology and how it could be useful in his work in number theory
Pain points: Feels intimidated by technical papers in algebraic topology, has difficulty identifying parts of algebraic topology that will be most relevant to his own work
Preferred content: A textbook focused on interdisciplinary applications of algebraic topology, a reading list for number theorists who want to learn algebraic topology
These personas can then help Stacy decide how to share intellectual goods in a way that most helps her intended recipients. For example, after creating the personas, she might realize that writing an open-access intermediate textbook with a significant chapter devoted to interdisciplinary applications could serve the needs of both Terry and Mark. Stacy may also realize that writing blog posts summarizing her recent research in an accessible way would be beneficial to both Terry and Abigail.
The personas Stacy created also help make the data she collected from her conversations more memorable and more likely to “stick” in her mind (Dykes Reference Dykes2025). Additionally, they help her avoid the trap of assuming the reader’s needs are “elastic” and will stretch to accommodate what she wants to write (Cooper Reference Cooper1999).Footnote 7 For example, if Stacy is writing an intermediate textbook but has recently found out about a new, advanced, but exciting technique, she may be tempted to include it. And she may try justifying it to herself by saying “the reader” would find it exciting, too. But the needs of “the reader” of an intermediate textbook have been stretched by including this advanced piece of research. Personas, however, make it more difficult to fall into this trap. For example, Stacy can ask herself, “Would this advanced technique help Terry or Mark?” and see that the answer is probably “no.”
Personas, therefore, provide an explicit, concrete way for a giver to pay attention to the goals, obstacles, and preferences of their intended recipients. This helps ensure they share intellectual goods, the core activity of intellectual generosity, in an effective way.
In addition to helping givers share intellectual goods effectively, personas can help agents cultivate intellectually generous motivations by deepening their empathy. For example, knowing more about the pain points and goals of potential recipients can help givers better understand and relate to their recipients’ emotions and can also help them feel concern for the recipients. In fact, personas were found to help graduate students in professional degree programs develop empathy towards the people they would be serving (van Rooij Reference Van Rooij2012) and promote empathy in instructional designers developing educational materials for a high school equivalency exam (Baaki and Maddrell Reference Baaki and Maddrell2020).
Moreover, psychological research suggests that empathy promotes altruistic motivation (Batson Reference Batson, Mikulincer and Shaver2010) and can lead to increased charitable donations (Smith et al. Reference Smith, Norman and Decety2020). Importantly, this suggests that by deepening empathy, personas can cause a giver to become motivated by a concern for the well-being of others, as well as making them more willing to share. And a concern for the intellectual well-being of others and a glad willingness to share are important motivational components of Roberts and Wood’s characterizations of intellectual generosity. This means that personas not only help a giver to share in a more effective way, but they also help them develop the motivational component associated with Roberts and Wood’s characterization of intellectual generosity.
6.4. Summary
Answering the three questions from section 3 helps us to see that intellectual generosity is both a more complex and more impactful virtue than it may first appear. It is more complex because sharing intellectual goods with others, especially with a large audience, can be deceptively challenging. The intellectually generous agent does not just passively pass on intrinsic intellectual goods to others. Instead, they must be willing to mold the intrinsic intellectual goods into a form that the intended recipients can readily receive and engage with. This requires them to have a deep understanding of the intrinsic goods themselves, as well as their recipients, and how to best communicate with them.
Intellectual generosity is also more impactful than it might seem because, when directed at large groups, many people can potentially benefit from the intellectual goods that are shared. This can directly lead to the generation of new intrinsic intellectual goods, as well as the development of a welcoming and inclusive atmosphere in the research community, which itself can further promote the development of new intrinsic intellectual goods over long periods of time.
Finally, recognizing the importance of the recipients of intellectual generosity suggests a way to cultivate this virtue using personas. Personas can help those who wish to develop intellectual generosity share intellectual goods in an effective way, as well as help them deepen their empathy. And increased empathy can, in turn, increase both concern for the well-being of others and the willingness to share intellectual goods, both of which are core features of Roberts and Wood’s characterization of intellectual generosity.
7. Intellectual generosity and intellectual courage in mathematics
In this section, I argue that there is a connection between intellectual generosity in Roberts and Wood’s sense and intellectual courage in the context of mathematics.Footnote 8 This connection is caused by significant risks some groups of intellectually generous agents face due to the reward structure of mathematics. I also address the effect that reducing this risk has on intellectual generosity as well as on its connection with intellectual courage.
According to Jason Baehr, “Intellectual courage is a disposition to persist in or with a state or course of action aimed at an epistemically good end despite the fact that doing so involves an apparent threat to one’s own well-being” (Baehr Reference Baehr2011a: 177). Mathematicians who exhibit intellectual generosity by writing expository work, such as those who participated in the User’s Guide Project, are clearly undertaking work that is directed “at an epistemically good end” and often face a threat to their own well-being for doing so. For example, many participants in the User’s Guide Project did not receive credit for their guides for the purposes of tenure and promotion, leaving them in a risky position. Don Larson, a participant in the project, summed up the precariousness of the situation as follows: “The act of making my prior original research more widely accessible did not itself constitute original research. Being on the tenure track as I was at the time (and still am), I couldn’t help but second-guess myself in that way. Was I investing my time prudently? In hindsight, I know the answer is an unequivocal “yes.” I suppose I’ll find out whether it was right for my career when I go up for tenure in a few months!” (Larson Reference Larson2022).
Moreover, this experience is not unique to participants in the User’s Guide Project. As Morris (Reference Morris2021a,b) has pointed out, the reward structure of mathematics primarily recognizes original research, not expository work. This means that early-career mathematicians who exhibit intellectual generosity by writing exposition face a genuine threat to their well-being: failing to achieve tenure and, potentially, losing their membership in the academic community. This significant threat to their well-being means that early-career mathematicians who write exposition are exemplifying both intellectual generosity and intellectual courage.
This threat to early-career researchers caused by the reward structure also raises an interesting question: Would reducing the risk associated with writing exposition, such as by recognizing such work for tenure and promotion purposes, increase the number of mathematicians exhibiting intellectual generosity?Footnote 9
First, recognizing exposition for tenure and promotion purposes would likely increase the number of mathematicians undertaking it. That does not mean, however, that these additional mathematicians would all exhibit intellectual generosity. Recall that intellectual generosity involves motivation, or, more particularly, a concern with intrinsic intellectual goods and/or the intellectual well-being of others over and above extrinsic goods one obtains for oneself. So, if more mathematicians undertake expository work but do so primarily because they desire the extrinsic goods it now provides, the number of mathematicians exhibiting intellectual generosity will not have increased.
However, some early-career mathematicians may want to engage in exposition because they have a genuine concern with intrinsic intellectual goods and the intellectual well-being of others. Nonetheless, they may not have written exposition previously because doing so would have put their tenure case at risk. After all, time spent writing exposition is time not spent working on research papers, which early-career researchers need to produce to gain tenure. And if an early-career researcher does not achieve tenure, they may be forced out of the field entirely and be unable to practice mathematics.
If exposition counts for tenure, however, then early-career mathematicians who genuinely want to write exposition will likely now feel safe to do so. In such a situation, these mathematicians will now no longer have to weigh doing work they are authentically interested in and motivated to pursue against the cost of risking their tenure case. And when they write exposition, they will be doing so out of a concern with intrinsic intellectual goods and the intellectual well-being of others, rather than the extrinsic reward the paper offers them. So, it seems reasonable to suggest that such early-career mathematicians exhibit intellectual generosity in Roberts and Wood’s sense.
It is true that, on Roberts and Wood’s account, an intellectually generous agent’s concerns for intrinsic intellectual goods and the intellectual well-being of others dominate the concern they have for their own extrinsic intellectual goods. And tenure is an extrinsic intellectual good. This may lead to the objection that the early-career mathematicians above do not exhibit true intellectual generosity on Roberts and Wood’s account, because their past inaction shows that their concern for intrinsic intellectual goods and the intellectual well-being of others does not fully dominate their concern with extrinsic goods like tenure. However, such an interpretation essentially forces intellectually generous agents to become victims of their own generosity. Such generosity is excessive and therefore arguably not virtuous. Further, in this situation, a concern with intrinsic intellectual goods and the intellectual well-being of others also requires a concern with staying within the mathematical community. That’s because an intellectually generous agent will be unlikely to contribute to the discovery of further intellectual goods or help improve the intellectual well-being of other mathematicians if they themselves are forced to leave the mathematical community. So we should resist interpretations of Roberts and Wood’s account that require mathematicians to risk their participation in the mathematical community to be intellectually generous.
While rewarding expository work in mathematics would thus very likely lead to more mathematicians undertaking it, only some of those mathematicians could be said to exhibit intellectual generosity in Roberts and Wood’s sense. Those writing exposition primarily for the rewards will fail to exemplify the virtue. But those with the appropriate motivations who previously did not feel safe writing exposition due to the risk of not getting tenure could reasonably be said to exhibit intellectual generosity.
Finally, note that rewarding expository work would dissolve the connection between intellectual generosity and intellectual courage in the context of mathematics. That’s because the risk to tenure, and hence the threat to early-career researchers’ well-being, disappears when expository work is recognized for tenure purposes. In such a situation, then, mathematicians writing exposition can still exhibit intellectual generosity, but would not be exhibiting intellectual courage.
8. Conclusion
This paper has aimed to deepen the work on intellectual generosity by Roberts and Wood and Morris by highlighting the complexity of sharing intellectual goods with others and documenting the wide-ranging impact this virtue can have.
The complexity of sharing intellectual goods, the core activity of intellectual generosity, comes from the fact that it is an active and creative process. Recall that the case studies have shown that a giver sharing intrinsic goods must often transform them so that the intended recipients can properly receive and engage with them. An intellectually generous agent should therefore ideally have deep knowledge of the intrinsic intellectual goods they wish to share, as well as insight into their intended recipients, and how best to communicate with them.
The case studies also revealed that intellectual generosity can have a large and lasting impact. Intellectual generosity can directly lead to the generation of new intrinsic intellectual goods as the recipients engage with and transform goods that were shared with them. But it can also shape the climate of a research community, creating a friendlier and welcoming environment. This atmosphere can help attract and retain practitioners in the community, allowing the field to flourish. This, in turn, ensures that new intrinsic intellectual goods will be generated over long periods of time.
This paper has also gone beyond the accounts of Roberts, Wood, and Morris in two ways. First, by suggesting that personas can be used as a practical tool to help agents develop the virtue of intellectual generosity. Second, by highlighting that the reward structure of mathematics means that many mathematicians who exhibit intellectual generosity also exhibit intellectual courage.
Intellectual generosity thus deserves further philosophical investigation. One such avenue for future exploration is the relationship between intellectual generosity and other intellectual virtues. For example, we saw that early-career mathematicians who exhibit intellectual generosity often also exhibit intellectual courage. Further, research in psychology has found that humility is a predictor of generosity (see e.g., Exline and Hill Reference Exline and Hill2012). This suggests that further investigation into the relationship between intellectual generosity and other virtues, such as intellectual humility or perseverance, would be fruitful.
A second avenue for exploration lies in the potential limitations and drawbacks of intellectual generosity. Although the case studies discussed here and in the existing literature have focused on the benefits of this virtue, we may wonder whether there are cases in which intellectual generosity has negative consequences, either for the giver or the recipients.Footnote 10 And, if there are such cases, we should consider what can be done to prevent or mitigate those harmful effects.
I hope that this paper stimulates investigation into these issues and intellectual generosity more generally.
Acknowledgements
I am very grateful to the editors and reviewers for their support and helpful feedback on previous versions of this paper. I am also deeply grateful to John Little, David Cox, Luke Wolcott, and Don Larson for agreeing to be interviewed for this work.
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