# Towards Good Practices in Self-supervised Representation Learning

Srikar Appalaraju, Yi Zhu, Yusheng Xie, István Fehérvári
Amazon Inc.
{srikara, yzaws, yushx, istvanfe} @ amazon.com

###### Abstract

Self-supervised representation learning has seen remarkable progress in the
last few years. More recently, contrastive instance learning has shown
impressive results compared to its supervised learning counterparts. However,
even with the ever increased interest in contrastive instance learning, it is
still largely unclear why these methods work so well. In this paper, we aim to
unravel some of the mysteries behind their success, which are the good
practices. Through an extensive empirical analysis, we hope to not only
provide insights but also lay out a set of best practices that led to the
success of recent work in self-supervised representation learning.

##  1 Introduction

Self-supervised representation learning (SSL) has been a hot area of research
in the last several years [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The allure of
SSL is the promise of annotation-free ground-truth to ultimately learn a
superior data representation (when compared to supervised learning). More
recently, a type of contrastive learning method based on instance
discrimination as pretext task [12, 13] has taken-off as it has been
consistently demonstrated to outperform its supervised counterparts on
downstream tasks like image classification and object detection [14, 15, 16,
17, 18, 19, 11].

Putting the progress in contrastive instance learning aside for a moment, many
recent observations seem to contradict what we have known from supervised
learning. For example, [20, 19] have shown that simply adding a nonlinear
projection head, i.e., one fully-connected (fc) layer and one activation
layer, can significantly improve the quality of learned representations.
Quantitatively, the nonlinear projection head can help to improve top-1
classification accuracy on ImageNet by over $10\%$ in [19] and $5.6\%$ in
[21]. However, adding such a shallow multilayer perceptron (MLP) head is often
not effective in supervised learning. Take another example, recent methods
[17, 19] adopt aggressive and strong data augmentation during contrastive pre-
training. Although data augmentation has been proven to be useful, overly
aggressive augmentations often lead to worse results in supervised or other
self-supervised learning methods. Then why contrastive instance learning does
not suffer from strong data augmentation? At this moment, there is no concrete
evidence to answer these questions.

After closely observing recent contrastive instance learning work, it becomes
apparent that what seem to be design choices are in-fact good practices which
are largely responsible to their disruptive success. Furthermore, some of
these good practices can effectively be transferred to other non-contrastive,
unsupervised learning methods [10]. Hence in this work, we focus on the
importance of these design choices using extensive experimental evidence. We
hope to provide insights to the self-supervised learning community, with the
potential impact and application even beyond it. Specifically, our
contributions include: 1) We empirically show why MLP head helps contrastive
instance learning and visualize it using a feature inversion approach. 2) We
present the semantic label shift problem caused by strong data augmentation in
supervised learning and study the difference between supervised and
contrastive learning. 3) We investigate on negative samples and find that good
practices can help to eliminate the need of using large number of them,
thereby could simplify the framework design.

##  2 Nonlinear Projection Head

Figure 1: A generic visual depiction of recent contrastive instance learning
methods.

Method | Top-1 Acc ($\%$)
---|---
MoCov2 | $64.4$
(a) no MLP head | $59.2$
(b) fixed MLP head | $62.9$
(c) deeper MLP head | $63.9$
(d) narrower MLP head | $62.2$
Figure 2: Investigation of nonlinear projection head in MoCov2 [21] on
ImageNet.

Figure 3: Visualization of feature inversion results by DIP [22]. Each example
contains: (a) original image, (b) reconstructed image using $h$ from a model
trained without MLP head, (c) using $h$ from a model trained with MLP head and
(d) using $z$ from a model trained with MLP head.

Before diving into the details, we first revisit recent contrastive instance
learning methods and present a generic visual depiction of its framework in
Figure 2. More formally, given a set of images $\mathcal{X}$, an image $x$ is
uniformly sampled from $\mathcal{X}$ and is augmented using various
augmentation techniques $t\in\mathcal{T}$ to generate the positive pairs
$x_{1}^{+}$ and $x_{2}^{+}$. $f_{\theta}$ and $f_{\phi}$ are encoders which
map the images to visual representations $h_{1}$ and $h_{2}$. These visual
representations are then projected via $g_{\theta}$ and $g_{\phi}$, which are
often MLP heads, to lower-dimensional features $z_{1}$ and $z_{2}$ for
similarity comparison. By optimizing InfoNCE loss [14], the model learns to
map similar instances closer and dissimilar instances farther apart in the
embedding space.

A main goal of unsupervised representation learning is to learn features that
are transferable to downstream tasks. Typically the outputs of the penultimate
layer $h$ are considered for transferring to other tasks. Recently, [20, 19,
21] have shown that simply adding a MLP head as shown in Figure 2 can
significantly improve the quality of learned feature representation. However,
adding such a shallow MLP head is often not so effective in supervised
learning regime. So the question arises, why adding a nonlinear projection
head is so important for contrastive instance learning?

In this paper we attempt to answer this question by designing experiments to
explore different aspects of the nonlinear projection head in MoCov2 during
unsupervised pre-training. First, as shown in Table 2 (a), removing the
projection head $g$ significantly degrades the classification accuracy
compared to baseline. Next, we initialize the nonlinear projection head $g$
with a uniform distribution and freeze its parameters during the unsupervised
pre-training. Interestingly, as we can see in Table 2 (b), we obtain better
representations compared to removing the projection head. This indicates that
the nonlinear projection head is useful beyond its learning capability offered
by the extra two layers. In fact, it is the nonlinear transformation itself
that somehow benefits the learning process even if the parameters of this
transformation is randomly initialized. To strengthen our observation, we
investigate two other model variations. We first deepen the nonlinear
projection head with more hidden layers, i.e., fc $\shortrightarrow$ ReLU
$\shortrightarrow$ fc $\shortrightarrow$ ReLU which in theory adds more
learning capacity. As shown in Table 2 (c), this seems not to bring extra
benefits. We then narrow the nonlinear projection head by reducing the
embedding dimensionality, e.g, $2048\shortrightarrow 128$ for a ResNet50. As
shown in Table 2 (d), such a drastic dimension reduction indeed lowers the
performance compared to baseline, but still outperforms setting (a) by a large
margin. With these insights we are more confident to conclude: it is the
transformation of the projection head, which separates the pooled
convolutional features from the final classification layer, that helps the
representation learning. But why is such separation beneficial?

We argue that the nonlinear projection head acts as a filter separating the
information-rich features useful for downstream tasks (i.e. color, rotation,
or shape of objects) from the more discriminative features that are more
useful for the contrastive loss. This conjecture was previously introduced in
[19] and verified by using features to predict transformations applied during
the pre-training. In this work, we provide further visual evidence to support
this hypothesis.

Inspired by deep image prior (DIP) [22], we perform feature inversion to
obtain natural pre-images. By looking at the natural pre-image, we can
diagnose which information is lost and which invariances are gained by the
network. Specifically, we invert the features before and after the nonlinear
projection head, $h$ and $z$ respectively. As we can see in Figure 3, using
features before MLP projection head gives the best reconstruction result. Even
though we use globally pooled features without spatial dimension, they are
able to generate decent image reconstructions, maintaining most color, shape,
location and orientation information. However, features learned without
projection or features after projection head only preserve the most
discriminative information to make classification. This observation supports
our claim that layers close to loss computation will lose information due to
invariances to data transformations induced by the contrastive loss.

|  |
---|---|---
(a) | (b) | (c)
Figure 4: $t$-SNE plots from 10 randomly chosen ImageNet classes. We use the
1000-d features from a ImageNet-trained ResNet50. The sub-figures use the same
images but under: (a) no; (b) weak; (c) strong augmentation.

##  3 Strong Data Augmentation

Data augmentation is an important regularization technique in training most
deep learning models, ranging from AlexNet [23] to cutout [24] and autoAugment
[25]. Empirical experience however shows that too strong augmentations (i.e.,
hard positives) are sometimes counterproductive in the supervised setting. In
this section, we explore why unsupervised contrastive instance learning can
benefit from hard positive samples.

Table 1: Investigation on different data augmentation settings’ effect on
MoCov2 [21] versus supervised training. The setting of color jittering
strength follows SimCLR [19].

Method | baseline | medium | extreme
---|---|---|---
| 20%/100% | 20%/50% | 2%/10%
MoCo | $64.4$ | $63.7$ | $47.1$
v2 |  | $-0.7$ | $-17.3$
Super- | $75.5$ | $74.8$ | $52.0$
vised |  | $-0.7$ | $-23.5$
(a) Rand cropping strength (min/max size)

Method | from weak to strong
---|---
| 1/8 | 1/4 | 1/2 | 1
MoCo | $63.1$ | $63.9$ | $64.2$ | $64.3$
v2 |  | $+0.8$ | $+1.1$ | $+1.2$
Super- | $75.8$ | $75.7$ | $75.6$ | $74.6$
vised |  | $-0.1$ | $-0.2$ | $-1.2$
(b) Color jittering strength

In contrastive learning without class labels, each image becomes its own class
and as a result, there are no clear semantic class boundaries like they exist
in supervised training. In order to illustrate how this hurts supervised
learning, we provide a t-SNE visualization [26] in Figure 4 to show how the
class boundaries in ImageNet break down when strong augmentations are applied.
We term this phenomenon as semantic label shift problem. However, during
instance discrimination, strong augmentation turn samples into hard positive
samples that are recently found to be quite helpful in discriminating instance
from instance [27].

Quantitatively, from Table 1(a), we can see that as the cropping augmentation
becomes more “extreme”, MoCov2 performance suffers less than its supervised
counterparts. The reason why performance of MoCov2 also drops might be MoCov2
is learning occlusion invariant features, a view that is corroborated by [28].
Such occlusion invariant learning process is hindered by aggressive cropping
augmentation. In Table 1(b), we show that stronger color jittering benefits
MoCov2 but hurts supervised learning which is consistent with the findings
from SimCLR [19].

##  4 Negative Samples

Computing contrastive loss requires sampling negative pairs to avoid learning
collapsed representations. Recently, [13, 17, 19] have empirically shown that
using a large number of negative samples is beneficial to learn good features
in contrastive instance learning. However, one needs to design sophisticated
mechanisms to store the negative examples and ways to update them. So we ask,
is it possible to use less negative examples during contrastive loss
computation without performance degradation?

Figure 5: Comparison of different algorithms with varying number of negative
samples. $K$ denotes the number of negative samples. We show that MoCov2
performs the same ranging from $K=512$ to $K=65536$.

Figure 6: Illustration of how we perform dynamic hard-negative sampling on
MoCov2 during training time for each mini-batch. We sort the entire dynamic
queue based on similarity with incoming positive samples. We use positive
skew-normal distribution to select more of hard and semi-hard samples and less
of “easy” positive samples.

Quantity of Negative Samples: In order to answer the question, we first run
experiments on recent contrastive instance learning approaches using different
number of negative examples. We choose InstDisc [13], MoCo [17], SimCLR [19]
and MoCov2 [21] as illustrating methods. Although they use negative examples
in different manners, e.g., memory bank in InstDisc, queue in MoCo and MoCov2,
and large mini-batch in SimCLR, they all use large number of negative
examples. We adopt their official released code and follow the same hyper-
parameter setting.

We find, that the performance of InstDisc and MoCo drops when number of
negatives decreases unlike for SimCLR and MoCov2 where the impact is less,
consistent with [13, 17, 19]. In fact, MoCov2 performs consistently with
respect to the number of negatives and its accuracy on ImageNet stays the same
ranging from $K=512$ to $K=65536$, where $K$ denotes the number of negative
samples. Hence, the quantity of negative samples during contrastive loss
computation has little impact on final linear probe performance. We also find
that AP on PASCAL VOC object detection task does not regress due to lower
negatives, indicating the quality of the learned features remain the same no
matter how many negatives were used ($512$ or $65536$) during contrastive
instance pretraining.

Quality of Negative Samples: What if we alter the quality of negative samples
by performing dynamic hard-negative mining in the MoCov2 queue? Specifically,
for a positive batch of images $B$ and an existing dynamic queue $Q$, we sort
$Q$ for each batch based on cosine-similarity between the query-branch feature
$q$ and the latent features in $Q$. We mine hard and semi-hard negative
exemplars from dynamic $Q$ using a skew-norm distribution. Note, we over-
sample in order to keep the overall queue size same as MoCov2. The intuition
for this experiment can be seen in Figure 6. Contrasting among these hard-
negative samples intuitively should improve the learned representation.
However, as shown in Figure 5 (blue line), in spite of over-sampling hard-
negative samples, performance of MoCov2 does not change for various sizes of
$K$.

To summarize, in spite of quantity and quality of negative samples in queue
$Q$, MoCov2 performance appears to be stable. We hypothese that the good
practices introduced in MoCov2 (MLP head, GaussianBlur augmentations, cosine
learning rate scheduling) are responsible for this effect. Thus, we conduct
ablation studies to find out the which good practice(s) helps in eliminating
the need for large number of negatives.

Table 2: Investigation of good practices’ impact on the number of negative
examples (K) being used in computing contrastive loss (wo: without, w: with).

Method | K=512 | K=65536
---|---|---
MoCov2 | $67.3$ | $67.5$
(a) wo MLP projection head | $61.7$ | $63.6$
(b) Data Aug: wo GaussianBlur | $65.5$ | $66.4$
(c) wo cosine learning schedule | $67.2$ | $67.3$
(d) w smaller mom. (0.5) in mom. encoder | $59.8$ | $64.5$

Good Practices: From Table 2, we can see that the MLP head and momentum
encoder have the biggest impact (row a and d). Without these two techniques,
the performance quickly drops as the number of negative samples decreases.
This also explains why SimCLR is robust to the number of negatives with
$K=4096$ and $K=65536$, however performance quickly degrades with $K=256$ due
to the lack of such mechanisms. We hope our empirical evidence can provide
insights and better understanding of recent progress, as well as advance
future development of self-supervised representation learning.

## References

  * [1] Gidaris, S., P. Singh, N. Komodakis.  Unsupervised representation learning by predicting image rotations.  In _International Conference on Learning Representations_. 2018.
  * [2] Noroozi, M., H. Pirsiavash, P. Favaro.  Representation learning by learning to count.  In _Proceedings of the IEEE International Conference on Computer Vision_ , pages 5898–5906. 2017.
  * [3] Doersch, C., A. Gupta, A. A. Efros.  Unsupervised visual representation learning by context prediction.  In _Proceedings of the IEEE international conference on computer vision_ , pages 1422–1430. 2015.
  * [4] Pathak, D., P. Krahenbuhl, J. Donahue, et al.  Context encoders: Feature learning by inpainting.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 2536–2544. 2016.
  * [5] Noroozi, M., P. Favaro.  Unsupervised learning of visual representations by solving jigsaw puzzles.  In _European Conference on Computer Vision_ , pages 69–84. Springer, 2016.
  * [6] Zhang, R., P. Isola, A. A. Efros.  Colorful image colorization.  In _European conference on computer vision_ , pages 649–666. Springer, 2016.
  * [7] Caron, M., P. Bojanowski, A. Joulin, et al.  Deep clustering for unsupervised learning of visual features.  In _Proceedings of the European Conference on Computer Vision (ECCV)_ , pages 132–149. 2018.
  * [8] Donahue, J., P. Krähenbühl, T. Darrell.  Adversarial feature learning.  _arXiv preprint arXiv:1605.09782_ , 2016.
  * [9] Agrawal, P., J. Carreira, J. Malik.  Learning to see by moving.  In _Proceedings of the IEEE international conference on computer vision_ , pages 37–45. 2015.
  * [10] Grill, J.-B., F. Strub, F. Altché, et al.  Bootstrap Your Own Latent: A New Approach to Self-Supervised Learning.  _arXiv preprint arXiv:2006.07733_ , 2020.
  * [11] Caron, M., I. Misra, J. Mairal, et al.  Unsupervised Learning of Visual Features by Contrasting Cluster Assignments.  _arXiv preprint arXiv:2006.09882_ , 2020.
  * [12] Dosovitskiy, A., J. T. Springenberg, M. Riedmiller, et al.  Discriminative Unsupervised Feature Learning with Convolutional Neural Networks.  In _NeurIPS_. 2014.
  * [13] Wu, Z., Y. Xiong, S. Yu, et al.  Unsupervised Feature Learning via Non-Parametric Instance-level Discrimination.  In _CVPR_. 2018.
  * [14] Oord, A. v. d., Y. Li, O. Vinyals.  Representation learning with contrastive predictive coding.  _arXiv preprint arXiv:1807.03748_ , 2018.
  * [15] Tian, Y., D. Krishnan, P. Isola.  Contrastive multiview coding.  _arXiv preprint arXiv:1906.05849_ , 2019.
  * [16] Misra, I., L. v. d. Maaten.  Self-supervised learning of pretext-invariant representations.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pages 6707–6717. 2020.
  * [17] He, K., H. Fan, Y. Wu, et al.  Momentum Contrast for Unsupervised Visual Representation Learning.  In _CVPR_. 2020.
  * [18] Khosla, P., P. Teterwak, C. Wang, et al.  Supervised Contrastive Learning.  _arXiv preprint arXiv:2004.11362_ , 2020.
  * [19] Chen, T., S. Kornblith, M. Norouzi, et al.  A Simple Framework for Contrastive Learning of Visual Representations.  In _ICML_. 2020.
  * [20] Bachman, P., R. D. Hjelm, W. Buchwalter.  Learning Representations by Maximizing Mutual Information Across Views.  In _NeurIPS_. 2019.
  * [21] Chen, X., H. Fan, R. Girshick, et al.  Improved Baselines with Momentum Contrastive Learning.  _arXiv preprint arXiv:2003.04297_ , 2020.
  * [22] Ulyanov, D., A. Vedaldi, V. Lempitsky.  Deep Image Prior.  In _CVPR_. 2018.
  * [23] Krizhevsky, A., I. Sutskever, G. E. Hinton.  ImageNet Classification with Deep Convolutional Neural Networks.  In _NeurIPS_. 2012.
  * [24] DeVries, T., G. W. Taylor.  Improved regularization of convolutional neural networks with cutout.  _arXiv preprint arXiv:1708.04552_ , 2017.
  * [25] Cubuk, E. D., B. Zoph, D. Mané, et al.  Autoaugment: Learning augmentation strategies from data.  In _2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , pages 113–123. 2019.
  * [26] van der Maaten, L., G. Hinton.  Visualizing Data using t-SNE.  _IMLR_ , 2008.
  * [27] Khosla, P., P. Teterwak, C. Wang, et al.  Supervised contrastive learning, 2020.
  * [28] Purushwalkam, S., A. Gupta.  Demystifying Contrastive Self-Supervised Learning: Invariances, Augmentations and Dataset Biases?  _arXiv preprint arXiv:2007.13916_ , 2020.


# On definitions of “mathematician”

Ron Buckmire  Carrie Diaz Eaton  Joseph E. Hibdon, Jr  Katherine M. Kinnaird
Drew Lewis  Jessica Libertini  Omayra Ortega  Rachel Roca  Andrés R. Vindas-
Meléndez

###### Abstract

The definition of who is or what makes a “mathematician” is an important and
urgent issue to be addressed in the mathematics community. Too often, a
narrower definition of who is considered a mathematician (and what is
considered mathematics) is used to exclude people from the discipline—both
explicitly and implicitly. However, using a narrow definition of a
mathematician allows us to examine and challenge systemic barriers that exist
in certain spaces of the community. This paper explores and illuminates
tensions between narrow and broad definitions and how they can be used to
promote both inclusion and exclusion simultaneously. In this article, we
present a framework of definitions based on identity, function, and
qualification and exploring several different meanings of mathematician. By
interrogating various definitions, we highlight their risks and opportunities,
with an emphasis on implications for broadening and/or narrowing participation
of underrepresented groups.

As we reflect on our mathematical communities’ journeys towards social
justice, we have been thinking about how to use mathematics and data to keep
these communities accountable to and focused on this goal; specifically, we
are interested in data about who is included, who is excluded, who is
overrepresented, and who is underrepresented. As we began shaping possible
research projects, we entered into conversations about data availability and
data collection, and a natural question emerged: We began to ask, “How do we
define the mathematical community?”

Thinking about definitions is a common first act of mathematical inquiry.
Thinking about how we are defining “mathematical community” led us to ask for
a specific definition for “mathematician.” The co-authors reflected on how
designating who is and who is not a mathematician is itself a complex social
justice issue. It is difficult to consider the question of defining a
mathematician without also considering the question of what is mathematics. Of
course, one can reasonably define mathematics in numerous ways.

For the purposes of this paper, we will use “mathematics” as an umbrella term
for the mathematical sciences, broadly construed. Additionally, while we, the
authors, believe the “mathematics community” is the set of all people who
identify as mathematicians, throughout the paper, we will often use the term
“mathematics community” or “mathematicians” to mean the set of mathematicians
that results from explicitly employing a particular definition. We understand
that a “set of mathematicians” does not imply “community,” but use this word
to intentionally humanize our language. This approach allows us to explore the
impact of different definitions of mathematician on the make-up of the
community that results when a given definition is used to form the boundary
between the in-group and out-group.

This paper seeks to explicitly uncover and unpack assumptions and ideologies
associated with the term “mathematician.” This paper will do three things: 1)
provide a framework based explicitly on three different ways to define the
term “mathematician” (identity, function and qualification) informed by the
co-authors’ lived experiences that complements existing, broader mathematical
identity frameworks found in the literature; 2) analyze and discuss the
definitions, assumptions and implications of the term “mathematician”; and 3)
act as an entry point for readers of this paper to engage in self-reflection
about their identities as mathematicians and the importance and ramifications
of different definitions of the term. The goal is to provide a framework for
others who work with data on the “mathematics” community to clarify their
definitions and the reasoning for their definition choice for the specific
issue they are addressing.

We note that we write this paper from the perspective of people grounded in
various mathematical backgrounds and with different lived experiences, rather
than from a grounding in social science research around identity. In Section
2.1, we provide more detailed positionality statements.

The rest of the paper is organized as follows. We conclude this section with a
brief discussion of the relevant literature related to mathematics identity.
In Section 1 we provide our definitional framework for “mathematician” based
on three aspects of the term: qualification, identity, and function, along
with broad and narrow examples of each definition-type. This is followed by
our analysis of the implications of our definitional framework in Section 2
that begins with our positionality statements and includes reflections on them
and features a visualization of how the co-authors reflect our definitional
framework given in Table 1. This section also includes our implementation of a
polarity analysis that discusses the risks and opportunities associated with
the broad and narrow versions of each of the definition-types in our
“mathematician” definitional framework along with a visualization of the
analysis in Figure 1. The paper ends with two short sections with conclusions
in Section 3 and discussion of future work in section4.

### Review of Mathematics Identity Scholarship

The question of defining the term “mathematician” is intimately related to the
idea of mathematics identity. Mathematics identity is well-studied in the
mathematics education literature, particularly in K-12 education and with
respect to the formation of mathematical identity, while the definition of
“mathematician” is often treated as an axiom or universally accepted given.
Our paper builds on the mathematics identity scholarship, analyzing these
identity constructs to define “mathematician” with a focus on practicing
mathematicians.

In an extensive literature review of the topic, Darragh [7] broadly
categorizes definitions of mathematics identity into five groups:
participative, narrative, discursive, psychoanalytic, and performative. A
participative definition is one that defines identity through participation
and engagement with a social group. A narrative approach defines identity
through the stories we tell about mathematics. Sfard and Prusak [19] go as far
as to say that identities are stories. A discursive perspective of identity
views it through the language people use when they interact. This can be
viewed either as the language people use when they do mathematics, or to refer
to broader societal narratives about mathematics. A psychoanalytic definition
describes identity in terms of interactions between conscious and unconscious
processes. A performative conceptualization holds that identity is constituted
through one’s actions, repeated over time.

Black et al. [4] use three categories of definitions. The first two are
discursive and psychoanalytic, as above, while they term their third category
as sociocultural, which is a broadening of the participative definition above.

Gee [9] outlines four overlapping ways to view identity. A nature identity
captures an intrinsic state of being. An institutional identity is where
identity granted by an institution. Discourse identities are, how one
discourses and engages with others, and how others discourse about one.
Affinity identities are centered around communities of shared identity to
which one chooses to belong, similar to Darragh’s participative definition.

Darragh, following Gutiérrez [10] and Gee [9], further highlights a dichotomy
between mathematics identity as something intrinsic that an individual has
acquired, and mathematics identity as a set of actions, i.e., things that an
individual does.

Gutiérrez [10] highlights how (mathematics) identity and power are
intertwined. Being granted access to the community of “mathematicians,”
however defined, grants one access to these power structures. We now turn our
attention to definitions of “mathematician,” building upon the above
literature on mathematics identity and power. We then analyze the
opportunities and risks of various definition choices will integrate these
perspectives to focus both on on implications for identity formation,
maintenance, and power.

##  1 Definitions

In this section, we present our definitional framework for the term
“mathematician.” This framework is based on three kinds of definitions, each
of which can be tuned across a spectrum. These definitions with their
accompanying ranges provide opportunities for a wide range of insights in
exploring one’s identity from multiple perspectives. Building from these
definitions, we produce another set of definitions by constructing unions and
intersections of the first three types of definitions.

From our lived experiences and discussions, we recognized the need to
explicitly design a framework that more specifically serves to define the
“mathematical community” and a “mathematician.” This framework
below—highlighting function, identity, and qualification—is not meant to
contradict previous work done on identity, but to complement and focus it on
the unique structures and politics surrounding mathematics. Our framework,
like others, explicitly recognizes complexities in the social nature of
identity, while addressing specific issues the mathematics community faces.

We further present a variety of definition types for who is and what makes a
mathematician. Of course, this list is not exhaustive. For each type—function,
identity, and qualification—we offer three example definitions, including
narrow and broad example definitions for mathematicians as well as a “hybrid”
example that is somewhere between the broad and narrow extremes. Recall that
we defined “mathematics” in the context of this paper as the mathematical
sciences, but we note that the way mathematics is defined could also broaden
or narrow the pool of those that satisfy the definitions given below. We
emphasize that these are examples of function-based, qualifications-based, and
identity-based definitions that we, as the authors, have developed. We
encourage others to explore other definitions that also align with the given
framework.

###  1.1 Function-based Definitions

A function-based definition for a mathematician is a definition that hinges on
what they do (or how they “function”) on a daily basis. Function-based
examples may include a collection of specific actions or a more general list
of habits or holistic activities.

Below we provide both broad and narrow examples of function-based definitions
followed by a hybrid example definition, which we emphasize are our examples
of function-based definitions that we, as authors, have developed.

######  _Function-based Definition_ 1 (Broad).

A mathematician is a person who uses mathematical concepts, tools or
techniques to study and solve problems.

Function-based Definition 1 may include all humans. If counting is considered
a mathematical technique, then anyone who counts for a problem-solving purpose
can then be defined as a mathematician. This definition is often invoked in
books such as Keith Devlin’s The Math Instinct [8] in order to make the case
that “we are all mathematicians.” This use is related to a broader effort to
boost confidence in more people to empower them to be successful in their
mathematical work and to combat the prevalent and pernicious idea that “Some
people are not ’math’ people.”

######  _Function-based Definition_ 2 (Narrow).

A mathematician is a person who proves theorems using proof techniques.

Function-based Definition 2 is often implied in the construction of
mathematics majors. It manifests in, and is reinforced by, the requirement of
a proofing course as the foundational “methods” courses in the curricular
design of the undergraduate mathematics major at some institutions.

######  _Function-based Definition_ 3 (Hybrid).

A mathematician is a person who, as part of their daily work, employs
mathematical techniques and tools to solve mathematical and other problems.

There are many function-based example definitions that fall between the narrow
and broad example definitions. However, Function-based Definition 3 might be
used to identify jobs held by such mathematicians, in order to advise students
about career options beyond academia.

###  1.2 Qualifications-Based Definitions

In a qualification-based definition, a mathematician is someone who has
cleared a normative bar established to certify knowledge or accomplishment.
This bar or “qualification” is something that can be moved depending on what
someone is trying to measure. The construction of this definition implies the
existence of an external body that certifies that the qualification has
occurred.

######  _Qualification-Based Definition_ 1 (Broad).

A mathematician is someone who has completed a course in mathematics at any
level.

In Qualification-Based Definition 1 the external body that has certified the
qualification (the “completion”) would be the teacher of the course. We could
be more specific in this definition, by choosing an example like “being able
to count to ten” or “can correctly multiply two-digit integers” or “has taken
a course in a specific mathematics topic like trigonometry.”

######  _Qualification-Based Definition_ 2 (Narrow).

A mathematician is someone who has a Ph.D. in mathematics and currently holds
a research position in mathematics.

Qualification-based Definition 2 is often operationalized in order to make
hiring decisions consistent with terminal degree expectations for post-
secondary accreditation review.

######  _Qualification-Based Definition_ 3 (Hybrid).

A mathematician is someone who holds a Bachelor’s degree or higher in
mathematics or mathematics coursework, where the mathematics coursework allows
for one to teach secondary education mathematics.

The teacher-credentialing process in mathematics uses a qualifications-based
definition which equates a set of post-secondary coursework with sufficient
pedagogical content knowledge for teaching. Guidelines of these criteria are
given by the National Council of Teachers of Mathematics [1]. However the
Qualification-Hybrid definition is a moving definition based on some arbitrary
criteria. Many of these qualifications have been set initially, but not
updated enough to stay in line with current societal needs.

###  1.3 Identity-based Definitions

Unlike the qualifications-based and the function-based definitions which lie
on a narrow-broad spectrum, we can consider identity-based definitions along
an internal-external spectrum. An internal identity-based definition centers
the individual perspective, while the external identity-based definition
relies on the collective perception of others on the individual’s
mathematician status.

These identity-based example definitions highlight the overlap between Gee’s
affinity-identity and discursive-identity [9].

######  _Identity-Based Definition_ 1 (Internal).

A mathematician is anyone who says they are a mathematician.

Bottom line, if you decide you are a mathematician, by Identity-Based
Definition 1, you are. If you decide you are not, then you are not. This
definition invokes the idea of self-determination of identity, which can be a
particularly powerful activity to those who experience oppression because of
their identities [16].

######  _Identity-Based Definition_ 2 (External).

A mathematician is anyone whom others say is a mathematician.

Identity-Based Definition 2 requires others to invoke their identity as a
mathematician, and then requires enough consensus so that the title is
formally or informally bestowed. Formal versions of this type of identity
validation could include earning an award given only to mathematicians.

######  _Identity-Based Definition_ 3 (Hybrid).

A mathematician is anyone who someone else says is a mathematician.

Like Identity-Based Definition 2, Identity-Based Definition 3 is still
external, but only requires one person’s criteria for mathematician is met.
Therefore, it might be more likely to be satisfied than the ones above. For
example, if someone introduces you as a mathematician, then you will be given
the identity of mathematician. In this case, this identity might be ephemeral,
lasting a mere conversation.

###  1.4 Generating Further Definitions

In the above sections, we have presented three different axes under which to
establish who is (and who is not) a mathematician. Much like the generating
set of a group that can build other elements, each of the definitions we have
given above could lead to further definitions about whether someone is a
mathematician or not. Most individuals use a combination of these definitions
to self-define or define others as “mathematicians.” In this subsection, we
will now consider the previous three definitions as a generating set, then
look to see if some union, intersection, or other combination is satisfied.
Because this new definition refers to sets generated from the other
mathematician definitions, we refer to this new generated definitions as
_compuesto_ definitions.

######  _Compuesto Definition_ 1 (Union/broad).

A mathematician is someone who satisfies any definition of a mathematician.
Effectively, this definition is the union of all definitions.

######  _Compuesto Definition_ 2 (Intersection/narrow).

A mathematician is someone who satisfies every definition of a mathematician.
Effectively, this definition is the intersection of all definitions.

It is entirely possible that this narrow definition would result in the empty
set. We investigate this possibility by examining the identities of the co-
authors through their positionality statements in Section 2.1, with the
results summarized in Table 1.

##  2 Analysis

In this section, we analyze the example definitions provided above to explore
the similarities and to tease apart the differences between both the types of
example definition (function-based, qualification-based, and identity-based)
as well as the dichotomous variation available for each one (i.e., either
narrow/broad or internal/external). Our analysis takes two forms. Our analysis
moves from the micro to the macroscale, first by applying these definitions to
individual co-authors of this paper and then moving to the larger discipline-
specific implications of these example definitions. This section is organized
as follows. First, in Section 2.1, we offer positionality statements for each
of this paper’s co-authors and then situate their relationships to each
example definition type (and axis). Then, in Section 2.2, we use a polarity
mapping analysis to explore implications of using the example definitions from
Section 1 when defining a “mathematician”. Specifically, we used a
participatory process to capture the authors’ perspectives on the
opportunities and risks associated with the application of extremal versions
of each definition type. Finally in Section 2.3, we bridge from this
broadening analysis of the example definitions (Section 1) towards future
conversations and implications on the whole field of mathematics.

###  2.1 Positionality Statement

Here, we offer a series of personal examples of self-definition via individual
positionality statements to help readers contextualize our work. We then use
the definitions from Section 1 to analyze these positionality statements and
provide a table that summarizes this analysis.

#### Ron Buckmire

I am a cisgender, gay man of Afro-Caribbean descent who has been a member of
the mathematics faculty of a small liberal arts college in Los Angeles,
California since 1994. I earned all three of my degrees in mathematics at
Rensselaer Polytechnic Institute, an Engineering-oriented institution. Perhaps
influenced by where I was educated, when I do mathematics, I often think about
it in the context of solving a specific problem, and almost always use
computation to assist me in the solution process. I believe that “mathematics
is for solving problems” and that “mathematics is a human endeavor.”
Mathematics is done, taught, learned, discovered and created by people, and
that the identities and prior experiences of people who do math, are
important.

#### Carrie Diaz Eaton

I am a mathematician by degree. Each higher degree I hold is in mathematics,
but all my degrees were very interdisciplinary with biology. I am not a
mathematician by job title, having had appointments in biology and
computational studies departments after graduation. I used to joke that I was
the biologist in the room of mathematicians and the mathematician in the room
of biologists, so my disciplinary identity feels fluid and relative at times,
much like my queer and multi-ethnic identities, respectively. My primary
research activity is in STEM education. My methods are statistical,
computational, and qualitative. I do not partake in formal proofs in any of my
scholarship, yet all of my work requires sound logic. I am deeply involved in
my mathematics community, but also involved in the biology education community
as well. As a Latina, I resonate with the idea of Nepantla [11] and see
boundary spanning as my superpower. If I relied on all definitions of a
mathematician or any discipline to be satisfied, then I would have no
disciplinary identity. Instead I choose to embrace all my boundary-spanning
disciplinary identities as a strength.

#### Joseph E. Hibdon, Jr.

I am a cisgender Native American man, who is the descendant of the Luiseño
Band of Mission Indians. My Ph.D. is in Applied Mathematics. On a daily basis
I teach and do research in applied mathematics at a public, four-year,
university that is a Hispanic Serving Institution. Much of my research is in
the realm of mathematical education and I do not do proofs; beyond the courses
I am teaching. In general, I identify myself to others as a practicing
mathematician who works to make mathematics available to all.

#### Katherine M. Kinnaird

On a daily basis, I think about computational tasks. This can include: scoping
a project to leverage existing (or soon-to-be-collected) data, drafting an
algorithm, creating a method for collecting data, coding an algorithm,
analyzing data, or writing up results. My bachelors, masters, and Ph.D. are in
mathematics. In my Ph.D., I did prove a lemma and a few corollaries, but I
have yet to publish a proof in a peer-reviewed journal. While others might
call me a mathematician due to my qualifications, I would no longer call
myself a mathematician. I identify as a computational scientist, by which I
mean a person who exists in the intersection of data science and computer
science.

#### Drew Lewis

I am a cishet white male mathematician. My early career work was squarely in
the realm of “pure” mathematics (algebraic geometry), but I am now probably
better described as a mathematician who does education research. This latter
work has shown me how who is perceived as a “mathematician” has real
implications, as often the views of mathematicians (including me) are given
disproportionate weight in math education spaces.

#### Jessica Libertini

My day-to-day activities include overseeing interdisciplinary research efforts
that use creativity science, design thinking, and operational planning to
address global challenges. While these activities rarely are “math” in the
strictest sense, I often draw on mathematical, modeling, computational, and
logic frameworks to advance progress. I do not write proofs and have limited
experience with proofs, as I am trained in mechanical engineering (B.S. and
M.S.), applied mathematics (Sc.M. and Ph.D.), and international relations
(MAS). I do not consider myself a mathematician; rather, I identify as an
undisciplinarian who challenges disciplinary boundaries.

#### Omayra Ortega

My day-to-day activities as an applied mathematician include teaching
mathematics and statistics courses, conducting research with my collaborators
at all levels (undergraduates, grad students, post-docs, faculty, and other
professionals). I rarely write proofs unless it is in the classroom setting.
My research focuses on solving real world problems and relies heavily on data
analysis and model fitting. I have a Ph.D. in Applied Mathematics &
Computational Sciences and degrees in music, “pure” math, biostatistics and
public health. Broadly, I identify as an Afro-Latinx non-binary women of
Panamanian descent with strong New York roots.

#### Rachel Roca

On a day-to-day basis, outside of my course work, I spend much of my time
reading and discussing articles (mostly from applied math and STEM education),
working with code, and TAing a computational methods course. My work, thus
far, has been focused primarily on course work, some of which are proof-based,
while my current research is more computational based. I graduated with a
bachelors in mathematics from a small liberal arts school in 2021 while also
minoring in computer science and Spanish. Currently, I am a second year Ph.D.
student in computational mathematics, science, and engineering at Michigan
State University. My research interests span topological data analysis,
computing education, and math for social justice. In a broader sense, I
identify as a queer white woman mathematician (in training). Within my
department, I am a mathematician, but do not feel like one among those in a
traditional math department. Due to being a newcomer to the world of academia,
coupled with my interdisciplinary work, I am still in the process of exploring
and developing my own identity.

#### Andrés R. Vindas-Meléndez

I am a queer, chronically-ill, Costa Rican-American mathematician raised in
South East Los Ángeles, California. I am a first-generation college graduate
having, earned my bachelor’s degree in mathematics and minored in Philosophy
and Chicanx/Latinx Studies. I also earned a master’s and Ph.D. in mathematics.
While pursuing a Ph.D, I earned a graduate certificate in Latin American,
Caribbean, & Latinx Studies. All of my schooling has been through public
institutions. My research is primarily in combinatorics and I have been
expanding my research interests to include applications of data science to
combinatorics and mathematics for social justice. In particular, for my
combinatorial work I prove results on lattice-point enumeration and
triangulations of lattice and rational polytopes. In mathematics for social
justice, I am interested in data analysis, interdisciplinary study (e.g.,
social science, history, economics), and development of mathematical
techniques to investigate racial/social issues.

I believe mathematics is as diverse and dynamic as those who use it. My
mathematics is influenced by my experiences and I bring to it my own
perceptions. I aim to build meaningful and empowering experiences with
mathematics, while also challenging others to think about the power structures
that are present in and outside mathematical spaces.

####  2.1.1 Reflections on the Co-Authors’ Positionality Statements

We examined our positionality statements through the lenses of each of the
definitions outlined in Section 1. When we analyzed the positionality
statements given above, written by each of this article’s authors, and
collapsed the findings into a table, we found some interesting patterns. We
would expect this group of co-authors to have many similarities, since we
began this work as research fellows at the Institute for Computational and
Experimental Research in Mathematics (ICERM), a prestigious mathematics
institute, in Summer 2022. In fact, all of the co-authors were classified as a
“mathematician” under the following seven example definitions: Functional-
Broad, Functional-Hybrid, Qualifications-Broad, Qualifications-Hybrid,
Identity-External, Identity-Hybrid, and _Compuesto_ -Broad; therefore those
columns are not included in Table 1. In these seven common definitions there
are both kernels of what it generally means to be a mathematician and
qualities unique to individuals who have been entrenched in, and validated by
this field for many years. We include Table 1 to highlight the diversity
within our group while, by omission, also highlighting the commonalities.

Author |  Functional-Narrow |  Qualifications-Narrow |  Identity-Internal |  _Compuesto_ -Narrow
---|---|---|---|---
RB |  |  X |  X |
CDE |  |  X |  X |
JEHJ |  |  X |  X |
KMK |  |  X |  |
DL |  X |  X |  X |  X
JL |  |  |  |
OO |  |  X |  X |
RR |  X |  |  X |
ARVM |  X |  X |  X |  X
Table 1: This table highlights the differences among the authors alignment
with each Example Definition.

Many of us identified at one time as applied mathematicians, who are often
excluded by narrow function-based definitions due to a lack of proof writing,
and we explicitly mentioned this in our positionality statements. While,
intellectually, we know that one narrow functional definition does not need to
define us, as a group, many of us leaned into other definitions of
mathematician. Perhaps we were just speaking to the myriad of definitions
described, but for some of us, there is trauma around whether external
entities would identify us as mathematicians, particularly when we hold other
marginalized social identities in mathematics.

When narrow functional definitions were not satisfied, many suggested using
broader functional definitions (e.g., RB and DL), either explicitly or
implicitly through describing how they use mathematics (e.g., JEH, CDE, and
OO). Most co-authors also leveraged other definitions of mathematician such as
qualification-based and identity-based (see Table 1). RB, JH, and AVRM
describe beliefs and future hopes that mathematics should be accessible for
all or should used in service of humanity. Redefining boundaries through
futurism is a revered tradition among scholars who study marginalized
identities and power (i.e.[2, 15]).

Those of us with pure mathematics training addressed power differences based
on identity in other ways by thinking about power between mathematics and
mathematics education or in the field of mathematics more generally. For
example, CDE used to say “I am a mathematician in a room of biologists and a
biologist in a room of mathematicians.” This is an example of identity as
relative to the context, if alternatives are available.

This same quote also invokes another observation on identity as expressed in
the positionality statements – that they are fluid over time. Early career co-
authors readily acknowledged that their explorations of their identities as
mathematicians are ongoing. KMK, a mid-career co-author, expressed leaving a
mathematician identity for that of a computational scientist, reflecting an
evolution in research area over her career.

Finally, we see some co-authors willingly accept multiple simultaneous
disciplinary identities. This could be expressed as a subdisciplinary
identity, such as ARVM describes, or could be from different disciplines, such
as CDE describes. Alternatively, JL feels limited and constrained by
disciplinary labels and therefore rejects any disciplinary identity in favor
of “undisciplinarian.”

Another challenge to the analysis of the positionality statements is the
difference between the definition of a mathematician that an individual may
hold personally, versus the definitions that an individual may feel beholden
to from outside. Many acknowledged that their positionality statement listed
the qualifications that they believed most in the mathematical sciences would
accept as the correct qualifications or qualities for someone to be considered
a mathematician. This internal versus external duality adds a level of
complexity to the analysis of an individual’s positionality statement.

###  2.2 Implications of Definitions: A Polarity Analysis

During the authors’ time at ICERM in Summer 2022, we used a polarity framework
to help us explore the tension between inclusive (broad or internal) and
exclusive (narrow or external) definitions of the term “mathematician.” This
framework was introduced by Barry Johnson in the 1990s and has been widely
adopted in the business leadership and design thinking communities [13]. A
polarity analysis is a participatory methodology that allows a group to
explore the tensions that exist when there is value and wisdom in two sides of
seemingly polar opposite perspectives or approaches. In this case, we used the
polarity analysis to understand how those of us working to advance social
justice in mathematics might find value in all of the definitions, depending
on our purpose.

The author team’s approach was very similar to the Polarity Mapping exercise
outlined by University Innovation Fellows
(https://universityinnovation.org/wiki/Resource:Polarity_Mapping). First we
defined each pole; in this case the two poles are the broad definition(s) of
mathematician on one end and the narrow definition(s) of mathematician on the
other. (Note that in the case of identity-based definitions, we classify
internal as broad, as the autonomy is given to the individual, and external as
narrow, as the power to bestow the title relies on others who may apply a
definition that would exclude a self-identifying mathematician.)

Then we spent time individually ideating on the opportunities of focusing on
each pole as well as the risks of over-focusing on a pole at the expense of
energizing the other pole. Next we discussed the results of the individual
ideation in small teams, and ultimately, we shared these with a larger group
that included our co-author team as well as other participants in the ICERM
program. Each idea was captured on a sticky note (see figure 2 in the
appendix), and this allowed the team both to appreciate areas of high synergy
(where multiple team members had identified a common theme) as well as to
highlight those creative ideas at the margins that may have only been thought
of by one individual. We then discussed what had been revealed in this initial
analysis as an entire group.

Lastly, the team also explored action steps that could be used to energize the
positives of one pole, as well as early warning signs that a pole could be
over-emphasized at the expense of the other pole. This portion of our polarity
analysis was not as well-suited to our efforts, as we are not beholden to
adopting a single definition; in fact, we argue for intentionality in
selecting, stating, and justifying a definition unique to the purpose at hand.
Therefore, the action steps and early warning analysis has been excluded from
this paper.

Based on our polarity exercise as well as our work in exploring different
types of definitions, we specify potential motivations and the implications of
using narrow or broad definitions through the lens of opportunities and risks
below. We acknowledge that some team members entered the exercise with
skepticism, voicing a strong preference for more inclusive definitions.
However, the analysis revealed opportunities and risks for each type of
definition. We also remark that due to the participatory nature of this
qualitative research approach, the results reflect the perceptions, beliefs,
and values of the participants. Since this work was done in the context of a
research program on social justice and mathematical sciences, the co-authors’
brought a social justice lens to the polarity mapping exercise and its
analysis.

####  2.2.1 Opportunities of Narrow Definitions

A narrow definition may initially seem limiting, but there are purposes and
settings that justify using a more narrow definition to support social justice
efforts. Below, we explore each narrow definition and its utility towards
social justice.

The narrow Function-based Definition 2 (“A mathematician is a person who
proves theorems using proof techniques”) allows for there to be a well-defined
cutoff for membership in the class of mathematicians, so that measurements,
data collection, and statistics are conducted in a relatively narrow context.
Such studies could leverage the narrowness of the definition to highlight
gender, racial, and other representational disparities, study the impacts of
interventions designed to address the disparities, and compare the disparities
in the mathematical community with those in other narrowly defined professions
such as medical doctor or trial lawyer.

The narrow qualifications-based Definition 2 (“A mathematician is someone who
has a Ph.D. in mathematics and currently holds a research position in
mathematics.”) also has the benefit of having a well-defined boundary that
could be used for social science research about the field of mathematics. For
example, one could use this definition to conduct surveys or interviews with
individuals that have a Ph.D. in mathematics and also do active research in a
defined mathematical field. Such a study could help reveal insights about how
welcoming (or unwelcoming) the narrowly-defined community is to persons from
underrepresented communities who meet the membership criteria; a similar study
could explore the differences in experiences between Ph.D. research
mathematicians who are male and white with those who are not.

Similarly, from a justice-oriented perspective, using narrow function-based
and qualification-based definitions can allow us to more easily identify who
is being excluded from the mathematics community. Using a qualification-based
definition of earning a Ph.D. in mathematics, it is widely documented that
woman and people of color are underrepresented in this version of “the
mathematics community” as compared to white men [5]. As the mathematics
community is heterogeneous with many disciplinary areas contained within it,
we might also consider what kinds of individuals are most likely to be
excluded from different subareas of mathematics when using a narrow function-
based definition of “mathematician” (like “a mathematician is someone who
publishes papers with proofs of theorems”).

For the identity-based definitions, one could consider the external definition
to be the equivalent analogue of the narrow modality for the function-based
and qualifications-based definitions. Recall, the external identity-based
definition, 2, is “A mathematician is anyone whom others say is a
mathematician.” We found it difficult to develop a social justice benefit to
having groups excluded in a potentially arbitrary way from the mathematical
community. However, we noted that there was at least the potential for
transparency with an opportunity to gain insights into the perspectives of
those who draw a tighter circle around who they consider to be a
mathematician. For example, in a paper, the definition used by the authors may
be clearly stated. Alternatively, even in casual conversations, insights and
contexts can be gleaned to reveal the dynamics of social perception about who
counts as a mathematician by different individuals and groups. As we
participate in formal and informal conversations about who is a mathematician,
those working to promote social justice within the field of mathematics can
monitor the social dynamics in various ecosystems throughout the mathematical
community. This has the potential to illuminate which barriers are eroding and
which are not, allowing more focused efforts in distributing the power of
mathematics.

This subsection has demonstrated that there can be social justice benefits to
adopting a narrow definition of the term “mathematician” under certain
conditions, despite the exclusive tone that is inherent in a narrow
definition. Specifically, narrow function-based and qualification-based
definitions offer clarity for researchers seeking to shine a light on
inequities. With the potential for systemic problems to be identified through
the use narrow definitions, they may lead to the creation of a more equitable,
just mathematics community.

####  2.2.2 Risks of Narrow Definitions

As shown above, there are some benefits to the adoption of a narrow definition
of the term “mathematician,” but there are also risks. Below we explore social
justice risks of employing the extremal (narrow and external) function-based,
qualification-based, and identity-based definitions of a mathematician.

The narrow Function-based Definition 2 (“A mathematician is a person who
proves theorems using proof techniques.”) does not allow for an inclusive
interpretation of the mathematics community beyond some subjective definition
of what a mathematician “should’ be able to do (i.e., prove theorems). Those
that may not have had the privilege of having a proof-based course or are just
getting started in the exploration of mathematics would not be considered part
of the mathematics community under Function-based Definition 2. It is likely
that the people who are excluded from the community through the application of
Function-based Definition 2 are more likely to be people of color, women, and
people who have less access to proof-based courses and may have attended
lower-resourced institutions. Thus, the narrow function-based definition would
generally tend to reduce the diversity (and maintain the overrepresentation of
maleness and whiteness) in a community defined by this definition. This
particular effect is a substantial negative outcome of the use of a narrow
function-based definition.

Another risk of the use of the narrow Function-based Definition 2 is that this
definition fails to include the students and graduates that increasingly
populate data science and other programs that are very applied, but may not
have a proofs requirement. By exclusion of these individuals, we also lose the
opportunity to count (as mathematics) the work that they do, which has the
potential to contribute more tangibly to the social fabric, as data scientists
are explicitly trained to use data to understand the world around us.

Examining the narrow Qualification-based Definition 2 (“A mathematician is
someone who has a Ph.D.in mathematics and is currently doing research in
mathematics”) also exposes several risks to the advancement of social justice
in mathematics because it imposes barriers on who is part of the in-group and
leaves many in the out-group. If, for example, we employ an alternative narrow
qualification-based definition, “A mathematician is someone who has earned a
Ph.D. in Mathematics at an Ivy League institution,” then those who meet the
criteria may feel a strong community bond, which can exacerbate the sense of
exclusion for those who do not belong. As another example, research in faculty
hiring in mathematics (and other disciplines) demonstrates a cycle of elitism
and exclusion through the application of a narrow qualification-based
definition, such as “A mathematician (worthy of being hired at a prestigious
university) is one who currently holds a mathematical research position at
a(nother) prestigious university.” The result is that high prestige
universities exchange faculty with one another or export faculty to less
prestigious schools but rarely hire faculty from the less elite universities
[20]. Furthermore, this definition may dissuade future students from selecting
a mathematics major, as this definition of mathematician sharply limits what
careers they could pursue, despite the data that shows mathematics majors
being successful in a broad range of career choices beyond those meeting the
narrow qualification-based definition 2.

In analyzing the external Identity-based Definition 2 (“A mathematician is
anyone whom others say is a mathematician”), we reveal risks similar to those
of the 2. These definitions require an external review, and unlike the
function-based definition, this review of who qualifies to be a mathematician
can lead to subjective barriers of belonging in the mathematical community,
which can breed elitist and exclusive mentalities that marginalize those who
do not meet the criteria. This means that these types of narrow/external
definitions, if employed without care, could perpetuate the marginalization of
those who do not satisfy the external criteria of a selected definition.

The negative implications of a narrow definition are not merely static, as
they provide a feedback loop that can exacerbate exclusivity. For example,
leveraging the theory of possible selves, potential future members of the
community may self-select out if they do not see people with identities
similar to their own in the in-group [18].

####  2.2.3 Opportunities of Broad Definitions

Here, we will discuss the social justice opportunities that emerged through
our polarity analysis as we examined function-based, qualification-based, and
identity-based broad definitions definitions of “who is a mathematician.”

Recall our broad Function-based Definition 1, “A mathematician is a person who
uses mathematical concepts, tools or techniques to study and solve problems.”
Such a definition can foster a mathematical community that is welcoming by
including and engaging those who are otherwise excluded from the conversation
of “who is a mathematician.” When we apply a broad function-based definition,
we see that the mathematical community that results now includes those who are
not necessarily writing proofs but are using mathematical tools in contextual
ways – as data scientists, statisticians, and quantitative social scientists.
The broader the circle, the more people are included, which provides an
opportunity to put the power of mathematics (and the power of the title
“mathematician”) into the hands of people with a collectively broader
perspective than the one dominated by a smaller (and largely white and male)
community. This approach also allows for broader recruitment and engagement
and could support a migration from an environment where certain folks are
relegated to disciplinary silos to a more collaborative interdisciplinary
environment.

By exploring Broad Qualification-based Definition 1 (“A mathematician is
someone who has completed a course in mathematics at any level”), we see that
this definition is highly inclusive. This level inclusion can be used to
empower people of all ages, but it has the potential to be particularly useful
in planting seeds of possibilities into the minds of young children who are
just starting to consider their future careers. Using a broad qualification-
based definition offers the opportunity to rebrand mathematics as something
that everyone can do, invest in a growth mindset, and challenge the societal
norm that “math is hard.” By investing in the mindsets and perceptions of
young children and others who have not previously considered themselves to be
mathematicians and inviting them into the circle of mathematicians, there is
the opportunity to begin dismantling some of the barriers that exclude people
from seeing their future self as a mathematician - because they have been told
they already are one. Such an approach has the potential to challenge the
hegemony of white males in mathematics community that can be created through
the application of a broad qualification-based definition.

When considering a “broad” identity-based definition, we selected the Internal
Identity-based Definition 1, “A mathematician is anyone who says they are a
mathematician.” This definition inherently empowers individuals to opt-into
the community of mathematicians, and this has the potential to increase the
diversity of that community. Recall that one of the risks of narrow
definitions in general was that of driving away potential community members
who do not see themselves represented in the group. Conversely, a broad
definition casts a wider net and therefore paints a picture of “who is a
mathematician” that can be more diverse. And that increased diversity has the
potential to be more inviting to more members who may have otherwise opted out
of engaging with the community when it was defined narrowly. The people who
would be most affected by such a change include persons of color, women, and
people from other underrepresented identities. Note that this concept is the
contrapositive to the point on possible selves and the non-static impacts of
an exclusive or narrow definition given in the previous section 2.2.2. So, in
this case broadening the definition of mathematician makes it more likely that
the community will continue to grow in diversity.

####  2.2.4 Risks of Broad Definitions

There are also downsides to the application of the broad version of the
definitions of “who is a mathematician.” In this section, we explore the
downsides to broad function-based, qualification-based, and identity-based
definitions of “who is a mathematician.”

As previously noted, the Broad Function-based Definition 1 (“A mathematician
is a person who uses mathematical concepts, tools or techniques to study and
solve problems”) is very inclusive by design. However, it could be so
inclusive as to include nearly everyone to the point of dilution of the value
of the term at all. For example, is a farmer who is estimating their yield a
mathematician? Using a broader definition of mathematician may dilute the
power of those who enjoy this access under the current definition; this
potential loss of power may explain why certain groups may cling to narrower
definitions of the term and oppose broader ones. On the other hand, using a
broader definition could a rhetorical choice to intentionally dismantle this
power.

For this reason, it may be appropriate to discuss the Hybrid Function-based
Definition 3, “A mathematician is a person who, as part of their daily work,
employs mathematical techniques and tools to solve mathematical and other
problems.” However, even this definition could be so inclusive that it may
result in people who do very different kinds of mathematics (i.e., who use
very theoretical and very applied techniques) being lumped together under the
same umbrella. When people who view themselves as being very different are
given the same label face practical challenges when their internal sense of
who they are is incongruent with the external impression of how they are
viewed by others, such as more limited access to resources ([12]) and more
time taken to cultivate external identity ([14]). The negative consequences of
an identity crisis fueled by use of a broad function-based definition could
include: lack of confidence to do mathematics, reduced identification with the
mathematics community and increased fragmentation into disciplinary silos.

The broad Qualification-based Definition 1 (“A mathematician is someone who
has completed a course in mathematics at any level”) also has risks. If the
definition of mathematician is broad to the point of near universality through
the application of the Internal Identity-based Definition 1 (“A mathematician
is anyone who says they are a mathematician”), then the barriers that have
excluded people, particularly those who are not white and male, are
artificially erased. The barriers to entry and the pressures to leave the
mathematics community may still be very real and marginalizing to some, so the
adoption of such a broad definition risks erasing the perception of those
barriers and pressures, reducing access to resources to dismantle those
barriers and pressures, and minimizing the acknowledgment of harm to those
affected by those barriers and pressures. In other words, while such a broad
definition might work in a utopia, there is risk of exacerbating harm if
applied in our current world.

The Internal (Broad) Identity-based Definition 1 allows anyone to self-
identify as a mathematician. While this definition supports individual agency,
it also risks the dilution of a sense of community in mathematics as it is
difficult to assess the commonalities of the resulting body of people. This
also could impact the resources available to the community, as there is a risk
of less homogeneity in the definition of who a mathematician is, what a
mathematician does, and therefore, what a mathematician needs to be successful
in their pursuits. What is more, this kind of dilution is reminiscent of the
risks that Gutiérrez[10] discusses in terms of dilution of power.

####  2.2.5 Overview of Polarity Analysis

The polarity analysis revealed wisdom and opportunities, but also risks as we
explored the extrema of the continuum of definitions along the narrow-broad
(external-internal) continuum. The overall findings are collectively captured
in Figure 1. As we look across these findings, we note some key themes.

Figure 1: A summary of some of the opportunities and risks associated with
broad, narrow, internal, and external dimensions of the definitions proposed.

In the pursuit of social justice, narrow definitions provide those working
towards social justice in the field an opportunity to perform research that
illuminates systemic barriers and pressures that disproportionately exclude
certain demographics from inclusion under those narrow definitions. However,
they also carry the risk of implying that “real mathematicians are …”, causing
new harm and exacerbating existing harm. Therefore, narrow definitions should
be used with caution and with clear explanation about why they are being
employed.

Conversely, broad definitions, by their very nature, are more inclusive, which
invites more diversity into the community and provides a positive feedback
loop welcoming in future members who see themselves represented. However,
these definitions could lead to a dilution of identity cohesion in the
mathematical community (loss of an understanding of what/who is a
mathematician), reduce available resources, and risk fragmenting the community
into smaller, more homogeneous groups. Perhaps even more harmfully, if used
without care, broad definitions could imply that the systemic barriers and
pressures that have historically been used as gatekeepers against certain
groups (e.g., women and persons of color) have simply disappeared because,
well, now anyone can be a mathematician. This could actually make things
worse, as it could minimize the lived experiences of those encountering
marginalization. Therefore, broad definitions should also be used with caution
and with clear acknowledgement that some may continue to face barriers.

We also see that there is a benefit to using hybrid definitions. For example,
when we explored the functional definition, the narrow definition carried the
risk of excluding those working in mathematics beyond the world of proofs, but
the broad definition was so wide-ranging it risked defining someone who uses
mathematics to calculate the tip on a restaurant check as a mathematician. If
there is a need for a cutoff for the definition of “a mathematician,” but one
needs to, simultaneously, cast a large net, then the Hybrid Function-based
Example Definition 3 (“A mathematician is a person who as part of their daily
work, employs mathematical techniques and tools to solve mathematical and
other problems.”) is one appropriate definition that could be employed. Many
people getting degrees in mathematics and related fields will not go on to
become professors of mathematics and will not spend a career writing proofs,
but they will be using mathematical skills in their careers. Having students
realize that, even if they pursue careers outside of academia, they will still
be part of the mathematical community has significant benefits in that it
enlists many more people to the profession and thus solidifies the future of
the mathematical community. At the same time, drawing a boundary has the
potential to create a sense of community for this larger group of those who
meet this hybrid function-based definition.

Lastly, we note that the application of any definition of a mathematician is
shaped by the motivation behind its selection. For example, in this article we
are engaging in an open conversation about defining the term “mathematician”
as a means to encourage others to join the mathematical community. But people
with other motives could use these definitions to accomplish different ends.
We therefore advise those involved in this discourse to be mindful of the
implications of their own definition choices and to use a critical lens when
examining the definitions of others.

###  2.3 From Definitions to Future Impacts

Above we have given examples of how various definitions may be used both by
individuals and applied to larger groups of people. As with all mathematical
definitions, these definitions reflect choices made by people, and we have
explored some of the implications of those choices. Importantly, a choice of
definition depends heavily on context and the motivations of the person/people
choosing that definition. Thus, when presented with a definition, it is
important to interrogate: who is making the definition? In what ways does this
choice work towards or against justice? What groups benefit from the choice
being made, and who is harmed?

In addition to asking readers to apply a critical lens to how mathematician is
defined, by themselves and by others, the authors encourage readers to engage
in self-reflection as well as discussions within their own communities - with
students, colleagues, or at mathematics conferences. Perhaps set aside time
for yourself (and others) to explore the following series of questions:

  * •

What is my positionality statement? Do I consider myself a mathematician, and
how is the answer to this question important to my own identity?

  * •

Using a function-based definition, do I meet the narrow definition? If not,
how might I develop a hybrid definition that captures what I do? And under
that new definition, who is included, and who is excluded?

  * •

Using a qualification-based definition, do I meet the narrow definition? If
not, how might I develop a hybrid definition that captures what I do? And
under that new definition, who is included, and who is excluded?

  * •

Using an identity-based definition, who would define me as a mathematician, in
what settings or situations, and why?

  * •

How do I define mathematician? How does my institution, employer, and/or
social group define mathematician? Who is included and who is excluded with
each of these definitions, and what does that mean both for the excluded
individuals and for the resulting community?

  * •

As we see a surge in programs in data science and other similar fields, how
might intentional decisions about including or excluding these groups in the
definition of mathematician shape the mathematical community?

##  3 Conclusion: Identity as Power

Mathematics and the epistemology of mathematics is socio-cultural by nature,
rather than absolutist, argues Burton [6]. As we prepared to move into
research projects in which we would collect and analyze data on
mathematicians, we were forced to try to make assumptions about who is a
mathematician, in order to provide guidance in determining whose data, which
data and what data was worth collecting for our various social justice and
data science projects. Our search for the right definition of a mathematician
eventually came to an unsatisfying, but possibly predictable answer: it
depends. Your choice of definition for mathematician will depend on why you
are collecting the data and which questions you are answering. It will depend
on what data is available to you for collection purposes. It will depend on
whether your purpose is to use the definition to exclude individuals from
opportunities, to welcome people into a more inclusive spaces for all
mathematical identities, or to illuminate the barriers that have make it
difficult for some individuals and groups to become and persist as
mathematicians.

In determining who counts as a mathematician, Burton highlights the power of
Eurocentric male dominance in the mathematics community, as well as the
exploitation and erasure of those who do not share those identities. Those who
are “mathematicians” wield the power to define and identify “important”
mathematical areas, where value is accorded to some results rather than others
and decisions are taken on what should or should not be published in a society
determined by ”power relationships” [6]. In the context of our own research
group, in order to create the field of data science for social justice within
mathematics, we had to assert ourselves as mathematicians (and a computational
scientist and an undisciplinarian) to define the field. Therefore, the
rationale for determining who is a mathematician moved beyond one for data
research purposes and became directly connected to defining the worth of our
research program.

All of these decisions and mathematical contributions are products of people,
who may include or exclude others from the mathematics community. Possible
exclusion from the traditionally elite math community, what Bartholomew et
al.[3] call the “maths club,” threaten identity formation. This “maths club,”
utilizes fear and narrow definitions of mathematicians to keep power to some
and disempower others. Even advanced undergraduate mathematics students
express uncertainty about their identity in being a mathematician [3]. In
addition, research has repeatedly demonstrated that co-construction of social
identity and mathematics identities are tightly related. For example, a case
study of a student’s experiences navigating and overcoming racialized
treatment that aimed to exclude her from the mathematics community by Oppland-
Cordell [17], highlights how racial, gender, and class identities affect
mathematics participation and learning. Therefore, we want to bring attention
to how crafting mathematical environments that embrace and showcase the
strengths of multiple social identities is just as important as creating
environments that nurture a variety of mathematical identities.

##  4 Future Work

In the preceding, we have explored various definitions of “mathematician” and
analyzed the implications of these kinds of definitions. While our definitions
emerged from the perspectives of a relatively diverse group of mathematicians,
we are only a small subset of the mathematical community. As such, we
anticipate this work as a first step for future work exploring how the broader
mathematics community interprets the term. Some questions we anticipate
answering are:  1) To what extent does the mathematics community have a shared
definition of “mathematician”?  2) Do members of the mathematics community
switch among multiple definitions to suit particular contexts, or persist with
a rigid idea of who is a mathematician?  As discussed above, different answers
to these two questions have different consequences for the profession and the
people who are or would be part of it.

What is more, we hope this work will inspire the mathematical community to
think both locally and globally about the definitions of “mathematician” we
assume and use, and the consequences these definitions can have in the broader
community as well as in our classrooms, majors, and institutions. With this in
mind, we invite readers join us in examining identity in mathematics either
individually (in one’s sphere of influence) or collectively (perhaps in
collaboration with the authors of this paper). Mathematicians, as those in
other disciplines, often accept the terms of their profession without question
or thought. We hope this paper has inspired you to question these assumptions.
If we are more thoughtful about how we define ourselves, our surroundings, and
the community that we work in, we can influence the social contract of this
community. Put another way, this is a call to action to create the mathematics
community that you wish to see. If you are interested in joining our working
group, please reach out to the authors on this paper.

## Authorship Contributions

All the listed co-authors contributed equally to all portions of the paper.

## Acknowledgements

This work was supported in part by the National Science Foundation under Grant
No. DMS-1929284 while the authors were in residence at the Institute for
Computational and Experimental Research in Mathematics in Providence, RI,
during the Data Science and Social Justice: Networks, Policy, and Education
program. ARVM is partially supported by the National Science Foundation under
Award DMS-2102921. RB acknowledges the support of the Occidental College
Office of the Dean of the College for funding his sabbatical in the 2022-2023
academic year. KMK is the Clare Boothe Luce Assistant Professor of Computer
Science and Statistical & Data Sciences at Smith College, this work has also
been partially supported by Henry Luce Foundation’s Clare Boothe Luce Program.

## References

  * [1] National Council of Teachers of Mathematics, December 2022.  https://www.nctm.org/ncate/.
  * [2] Gloria Anzaldúa et al.  Borderlands/la frontera.  na, 1987.
  * [3] Hannah Bartholomew, Lisa Darragh, Fiona Ell, and Jeanette Saunders.  ‘I’m a natural and I do it for love!’: Exploring students’ accounts of studying mathematics.  International Journal of Mathematical Education in Science and Technology, 42(7):915–924, 2011.
  * [4] Laura Black, Heather Mendick, and Yvette Solomon.  Mathematical relationships in education: Identities and participation.  Routledge, 2011.
  * [5] Ron Buckmire.  Who does the math?: On the diversity and demographics of the mathematics community in the united states.  Improving Applied Mathematics Education, pages 1–12, 2021.
  * [6] Leone Burton.  Moving towards a feminist epistemology of mathematics.  Educational Studies in Mathematics, 28:275–291, 1995.
  * [7] Lisa Darragh.  Identity research in mathematics education.  Educational Studies in Mathematics, 93(1):19–33, 2016.
  * [8] Keith Devlin.  The math instinct: Why you’re a mathematical genius (along with lobsters, birds, cats, and dogs).  Basic Books, 2009.
  * [9] James Paul Gee.  Identity as an analytic lens for research in education.  Review of Research in Education, 2000.
  * [10] Rochelle Gutiérrez.  The sociopolitical turn in mathematics education.  Journal for research in mathematics education, 44(1):37–68, 2013\.
  * [11] Rochelle Gutiérrez.  Living mathematx: Towards a vision for the future.  North American chapter of the international group for the psychology of mathematics education, 2017.
  * [12] Shang Jing.  External identities of directors, board functions and firm performance, 2009.  Master’s Thesis, National University of Singapore.
  * [13] Barry Johnson.  Polarity management: Identifying and managing unsolvable problems.  Resource Development, 1992.
  * [14] Alice Lam.  Hybrids, identity and knowledge boundaries: Creative artists between academic and practitioner communities.  Human Relations, 73(6):837–863, 2020.
  * [15] Diana Maldonado.  Futuristic traditions: Rethinking “hybrid” identities on the southern side of the us-mexican border.  TEMPORALITIES AND TRADITIONS, pages 67–103, 2014.
  * [16] Terrell R. Morton and Eileen C. Parsons.  #blackgirlmagic: The identity conceptualization of black women in undergraduate STEM education.  Science Education, 102(6):1363–1393, 2018.
  * [17] Sarah Oppland-Cordell.  Mathematics and racial identity co-construction in multiple sociopolitical contexts: A case study of a Latina undergraduate student from an urban community.  Journal of Urban Learning, Teaching, and Research, 9:89–99, 2013\.
  * [18] Daphna Oyserman, Deborah Bybee, and Kathy Terry.  Possible selves and academic outcomes: How and when possible selves impel action.  Journal of Personality and Social Psychology, 91:188–204, 2006\.
  * [19] Anna Sfard and Anna Prusak.  Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity.  Educational researcher, 34(4):14–22, 2005.
  * [20] Hunter K. Wapman, Sam Zhang, Aaron Clauset, and Larremore Daniel B.  Quantifying hierarchy and dynamics in us faculty hiring and retention.  Nature, 610:120–127, 2022.

## Appendix

Figure 2: Documentation of the polarity analysis conducted by the authors.