2 = ∞: The Incompleteness of the Standards for Mathematical Practice

SMP.1 encapsulates three distinct practices: problem solving, sense making, and perseverance. Accordingly, each is paired with its own complement.

First, SMP.1 frames the work of mathematics as addressing pre-formed and prescribed problems. But mathematicians also pose their own questions and make their own problems. Mathematicians are constantly asking, “What if?”  Cardano and Bombelli wondered if the square roots of negatives could be used in arithmetic just like integers and the foundation for complex numbers emerged. Ada Lovelace wondered what else an Analytical Engine might do and was the first to imagine possibilities for computers beyond the realm of mathematical calculation. Von Neumann and Morgenstern asked if mathematics could help us understand games, economics, and strategic thinking. Mathematicians are constantly dreaming, wondering, and posing new questions.

These questions are not pursued just as an exercise in endurance. Mathematics can be joyful and exhilarating. Francis Su asserts that the basic human desire for play is deeply connected to the practice of mathematics. While many mathematicians find the appeal of math to be in its thrill and beauty, SMP.1 highlights a sense of difficulty and hardship – a math that must be persevered. Developing persistence is good, but not necessarily for its own sake and certainly not as a gatekeeper. Finding joy can feed the desire to persist. Finding joy is also worthy on its own.

Joy in math is also not always found in solutions or answers. Math is teeming with confounding ideas whose power cannot be fully grasped without an appreciation their mystery. The Banach-Tarski Paradox shows how a single ball can be disassembled and then re-assembled into two balls that are identical copies of the first. Cantor’s revelation that there is more than one size of infinity baffled even his most notable contemporaries. Mathematics has more meaning when we are able to hold both how something can seem not to be and also how it is. Additionally, like Cantor’s detractors, math is not always understood the first time around, or the second, or third. Feeling a sense of mystery, confusion, and uncertainty is a normal and necessary part of mathematics.

Math is more than a cold and distant logic described with numbers. Mathematics can be done with and about very real and tangible objects. Mathematics can also involve working with space and shapes, from basic to advanced geometry to three-dimensional calculus. Shapes and spaces can be assembled, dissected, and transformed. The spatial is just as mathematical as the numerical.

Mathematicians also use their intuition and gut. William Byers writes that a mathematician can often feel or sense that “something is going on here” before they delve in the logic of it.¹ Mathematicians might feel that something does not sit right before fully being able to articulate a flaw in an argument. Or, they might have a gut instinct of a strategy to try. Mathematicians also decide which of the many unsolved problems and unasked questions to pursue and which direction to go with them. Perhaps informed by evidence and experience, their choice is nevertheless into some degree of the unknown. Byers also writes, “math research is characterized by an instinct for the right problems: those that are significant yet accessible [italics added].”¹

The history of mathematics is full of people who made great contributions but who also held wrong convictions. Pythagoras has the perhaps apocryphal legend of refusing to acknowledge the existence of irrational numbers and then having someone killed when they dared to argue otherwise.  Less apocryphally, Hilbert spent over a decade of work and argument on his program to create formalized and axiomatic mathematics. Gödel’s Incompleteness Theorem, however, would show that Hilbert’s vision was not possible.

And yet even though Hilbert’s arguments proved inviable, was he not still was practicing and contributing to mathematics? Mathematicians often pursue ideas that do not work out or are later found to be incorrect or incomplete. Moreover, even correct and viable ideas are often only last in a long line of failed attempts. Proposing incorrect, incomplete, and inviable arguments and ideas is an as common, if not more common, occurrence than posing correct, complete, and viable ones. And yet each incomplete, incorrect, inviable attempt still develops the field of mathematics.

Accepting their fallibility and limited perspective requires mathematicians to approach their work with humility. Appreciating the reasoning of others (originally coined by Gutierrez, 2017) not only expands our own thinking but also enlarges the possibilities for mathematics itself. Unfortunately, the history of math more easily illustrates this truth in counterexample. Many Europeans initially refused to accept the concept of zero already use in India and many other places (see here and here). As mentioned above, Cantor’s ideas about infinity where initially resisted by his peers. Math has the potential to be so much richer when we are truly open to understanding other ways of thinking, seeing, and doing. And, perhaps more importantly, our treatment of people and cultivation of our collective humanity is enriched when we seek to first understand and appreciate others.

SMP.4 frames mathematical modeling as a non-invasive act of an outside observer. Yet, mathematics and mathematical models have had real and sometimes deleterious effects on people.  Cathy O’Neil has documented how mathematical models have facilitated the mistreatment of workers, helped determine who gets policed, affected hiring practices, and contributed to the widening wealth gap.³

Mathematics should not be practiced under the assumption that the work is abstract and without tangible and human consequence. It is not just used to understand the world; it also affects the world. Ethical and moral mathematicians take responsibility and ownership for the impact their work and their field has on people. This includes taking action to correct and prevent harm and oppression. For example, a group of mathematicians recently called for their colleagues to stop work on predictive policing and other algorithms that have perpetuated racism and police brutality.

This practice cannot be fully realized unless the diversity of mathematicians and mathematical practices reflects the diversity of our world and community.  Thus, this practice also requires mathematicians to actively broaden both the mathematical community and the practice of math itself.

Mathematicians regularly develop tools to meet their needs. Today, the coordinate grid might be considered a useful tool in certain situations. But for Rene Descartes it was a new idea. And so he developed the now common tool for identifying location. The development of mathematical tools did not stop hundreds of years ago – it continues today. In the 1930s, Mary Clem invented the Zero Check method to detect errors. Even as recently as 2019, a new method was developed for solving a quadratic equation.

Mathematicians also use tools in ways that they were not necessarily intended. In doing so they often reveal useful and even unexpected connections. Trigonometric functions, Exponential functions, and complex numbers seem like distinct tools for understanding different phenomena. But they find a surprising connection in Euler’s Formula. The arithmetic of complex numbers was also found to model geometric rotations. Galois used groups to draw conclusions about the solutions polynomial equations. Groups then became a study unto themselves and now appear in multiple fields of mathematics. Integration as a tool for finding area and differentiation as a tool for finding velocity are bound together in the Fundamental Theorem of Calculus. Mathematicians practice creativity including finding new applications for and insightful connections between mathematical tools.  

It can often be useful to ignore or flaunt precision. Scientific notation of course rounds off the digits that have little meaning in the context of very large and very small numbers. In geometry, imperfect diagrams are drawn that do not attend to scale, congruence, or angle measure. Interpreting what is rigid and what has flexibility in these diagrams is a skill in and of itself. In calculus, functions are graphed with a few key points connected by a rough curve.

More generally, many problems are first tackled with slapdash sketch. And, of course, estimation often requires forgoing precision in favor of rough approximation. Abandoning precision can facilitate understanding, highlight important features, and simplify complex work.  

It could be argued that ‘attending to precision’ includes also knowing when to ignore precision. In a denotative sense this may be true, but the title and description of SMP.6 clearly intends to emphasize detail, accuracy, and specificity over the practices described above.

Mathematicians find new ideas by questioning or ignoring aspects of previous defined structure. Euclid found a structure for a geometry built on five postulates. Then mathematicians negated his 5^th postulate and opened the door to multiple non-Euclidian geometries. In a similar sense, the study of topology finds no difference between the shape of a coffee cup and a donut or the shape of an ellipse and a circle, thus abandoning many of the geometric and physical structures normally associated with those shapes.  Consider also that the study of statistics depends on the notion of randomness, a complete lack of structure. 

Of course, some might argue that randomness has structure. Thus, in the sense, the word structure is taken to encompass both itself and its opposite.  SMP.7, however, does not attempt to define structure in such a broad sense. SMP.7 defines the practice of mathematics as strictly about looking for order and organization. SMP.7 does not position mathematicians as having the freedom to wonder what happens when certain aspects of order and organization are ignored.  It does not describe structure as something made or defined, only found. 

Moreover, the existence of structure in one area inherently implies the lack of structure elsewhere.  How can structure be noticed or defined other than as a boundary, or distinction, between itself and the thing it is not? Thus, the ‘thing it is not’ must also exist and is an interesting and valid mathematical consideration.

Mathematics is not just about pattern and repetition. Mathematicians also notice what is unique. For instance, the defining features of a shape or function that make it distinct and different. Mathematicians do not just develop rules to demonstrate expectations – they also find exceptions. The discovery of single counterexample can disprove a generalized conjecture. The search for uniqueness also produces interesting results. There are only five Platonic solids. There are only 17 wallpaper groups.

Consider also that Fermat’s Last Theorem asserted that that integer solutions to aⁿ + bⁿ = cⁿ only exists when n=1 or 2. The generalization that no solutions exists for n>2 is inextricable from the special cases where solutions do exist.  

Fundamental Theorem of Arithmetic exemplifies this tension between the value of the unique and the expectation of pattern, repetition, and generalization. Every integer is either prime or can be represented by a unique product of primes. What is more essential to this insight: the regularity that this property holds from one integer to the next or the uniqueness of the set of prime factors?