What is Mathematics?
Jenny Quinn
The title question has been on my mind for quite some time now. A thoughtful response could go a long way to communicate the beauty, power, and purpose of our discipline to anyone at any level. A careless response might increase misconceptions or anxiety related to mathematics. We owe it to our profession to craft a unified response that promotes a positive culture for mathematics in our society and then communicate it widely.
For me, two recent events heightened the need for action:
The first occurred at a press conference on April 18, 2022, when Florida Governor Ron DeSantis made the following statement:
Math is about getting the right answers, and we want kids to learn to think so they get the right answer.
My reaction was visceral and immediate. “This is wrong. The emphasis needs to be on understanding not the answer.” Most problems don’t have one right answer; but decisions need to be made to select solutions based on sound logic and what is good enough.
The second event occurred when I met a group of scientists who openly wondered what happened at math conferences. It was jarring to realize that these learned people, each having earned a PhD in their own field through a substantial piece of original research, didn’t realize mathematical scientists engaged in the same process of knowledge generation and communication. It was as if mathematics stopped with the last content they learned in school.
This reinforced for me that ideas about mathematics are informed by educational experiences and few people see beyond the computational aspects taught in school. In their book What is Mathematics? originally published in 1941, Richard Courant and Herbert Robbins wrote:
Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. […] For scholars and layman alike, it is not philosophy but active experience in mathematics itself that alone can answer the question: What is mathematics?
A beautiful sentiment that aligns with good teaching practices, but as a description of mathematics for the public, it leaves a little to be desired.
I don’t pretend to have the answer but I’m willing to start the conversation. So, I asked MAA members what they thought “mathematics was about” and got over 60 responses—some public on MAA Connect, others private and often anonymous through an online form. As you might expect, the responses were thoughtful and clever. Surprisingly, there was little agreement except that none talked about finding the right answer.
To give you a sense of the variety of replies, I tagged each response with key words and computed the frequency of each tag. Tags that occurred at least three times appear in Table 1. Most of the tags are self-explanatory but a few are broader than you might guess at first glance:
QUANTITY: Also includes numbers, number systems, numerical forms, and measurement.
DECISION: Includes problem solving, addressing challenges, decision making, or finding answers.
HUMAN: Includes references to humanity, human relationships, and life.
UNDERSTANDING: Includes all aspects of critical thinking—conceptualizing, applying, analyzing, synthesizing, or evaluating ideas.
Table 1: Key words used to code responses and their frequency of occurrence.
Thanks to Mark Huibregtse for starting the online conversation with the succinct response "Mathematics is about insight, not numbers." This got tagged with the key word UNDERSTANDING. Jeneva Clark’s submission of “Mathematics is about numbers, shapes, and making decisions” was tagged with QUANTITY, SHAPE, and DECISION. The anonymous response “Mathematics is about critical thinking and problem solving” was tagged with UNDERSTANDING and DECISION.
PATTERN was the most common tag, occurring in 28% of the responses. Charles Zahn first heard “Mathematics is the study of pattern" from Albert Tucker in 1955. Judith Grabiner credited G.H. Hardy in A Mathematician's Apology (1940) where he wrote “a mathematician, like a painter or poet, is a maker of patterns.” I became familiar with it through Keith Devlin’s book Mathematics: The Science of Patterns (1997). David Smith affirmed his appreciation for this short description of our subject with the caveat that it requires understanding the word ‘science.’
Others expanded on the PATTERN theme, adding more nuance and detail:
Mathematics is about patterns and relations. — Nuh (Noah) Aydin
Mathematics is about finding, describing, studying, confirming, and/or using patterns.
— Jackie Dewar
Mathematics is about discovering patterns and using logical and sequential reasoning to find solutions to problems in a variety of fields. — Peter Arvanites
Mathematics is about creating explanations for observed patterns (or dreamed up patterns) using clearly defined principles and accepted logical reasoning. — High School Mathematics Teacher for 50 years
Mathematics is about finding order, patterns, and meaning through generalized ideas and abstraction. — Anonymous
Mathematics is about making sense—in the truest form—of quantity, form, structure, and pattern, so as to make living in this world a richer and more meaningful experience for humans. — Danny Otero
Some focused more on LOGIC, DECISION, STRUCTURE, or UNDERSTANDING:
Mathematics is about synthesizing disparate, simplistic ideas to create something that is both logical and beautiful. — Anonymous
Mathematics is about problem-solving, and developing methods (often logical or computational) to apply towards solving challenging issues. — Cooper Jacob
Mathematics is about making sense of our world. — Kelli Wasserman
Mathematics is the celebration of implication.— Doug McKenna
Mathematics is about consequences. — Ryan Dunning
Mathematics is about discovery, whether that be discovery of the truths of the universe, or the discovery about oneself and being able to do hard things. — Anonymous Math Student
I agree with each of the previous statements but I might argue that there is nothing inherently mathematical in the words. Many sciences and social sciences could use the same language to describe themselves.
On the other hand, the next two contributions made me smile because they can only be applied to mathematics as they are circularly defined by the needs of mathematicians:
Mathematics is about solving problems to make life better for the mathematician (and all they might care about). — Allen G. Harbaugh
Mathematics is about what the community of mathematicians find intriguing. — John McCleary
Mark Sand gave a big-picture view by expressing a fundamental question for each science, math included:
Dave Renfro offered the friendly alternative of “What is provable?”
It was interesting to see that BEAUTY occurred with the same frequency as UNDERSTANDING and more often than SHAPE or TRUTH.
Mathematics is about the underlying structure of beauty in the universe. — Winston Ou
Mathematics is a subject of great intrinsic power and beauty, and is one of the crowning intellectual achievements of humankind. — Steven H. Weintraub
I particularly liked the metaphor sent in by Timothy Prescott, “Mathematics is art in the medium of ideas,” because of its beauty and recognition that mathematics is more than science. He further explains “Minimal proofs are those that use only the necessary ideas, and nothing more. Surprising proofs are those that incorporate ideas in unexpected ways. Definitions encapsulate ideas for their repeated use. Theorems tell us how one collection of ideas also implies another collection of ideas.“
Based on your input, any description of mathematics should purposefully incorporate ideas of PATTERN, LOGIC, QUANTITY, DECISION, STRUCTURE, HUMAN, UNDERSTANDING, and BEAUTY. Here is my attempt to synthesize the discussion in one sentence:
Mathematics is about understanding the world through studies of quantity, structure, pattern, and change to create logical solutions that make life more meaningful and more beautiful.
I hope it sparks continued conversation and improvement—both on MAA Connect and through the Google form (https://forms.gle/mogMJCckRsnT8VhQ6). Can we come to a consensus on “What is mathematics?” It would be invaluable in promoting mathematics to the public and a necessary step towards achieving MAA’s vision of a society that values the beauty and power of mathematics and fully realizes its potential to promote human flourishing.
Thank you to everyone who participated in the discussion so far. I appreciate your time, creativity, and support. Here’s hoping more voices join the continuing conversation.
Jenny Quinn is a Professor of Mathematics at the University of Washington Tacoma and MAA President for 2021-2022. She wants mathematics to be appreciated by and welcoming to all.
This is an adaptation of her President's Message in the Aug/Sep issue of FOCUS that can be found here.