Does Mathematics Progress?
Joe MalkevitchYork College (CUNY)
Introduction
With the beginning of a New Year—2026 in one system of counting, though not all societies use the same calendar—many people take the opportunity to examine how their lives changed or progressed during the course of the prior year. Some relate this to the observation that an unexamined life is not worth living. Not only can one examine one's own life for how it progresses with time, one can also examine the different areas of knowledge that humans have named and have studied both from the perspective of history, and how this knowledge can assist us in leading our lives. Here I take a small look at how mathematics, viewed as a body of knowledge has progressed, changed, or evolved.
Progress viewed historically
There are currently about 8.3 billion humans inhabiting the earth, a dramatically different number from what was the case only 100 years ago, and in the more distant past, 2026 years ago. Think for our purposes of a mathematics aficionado who we'll nickname Tangent Sphere Projection (TSP for short). But wait, isn't TSP short for the Traveling Salesperson Problem—that is, finding the shortest-measure (in time, distance or cost) route that starts and ends in the same location and visits a collection of sites once and only once? Sorry, Tangent!
Tangent and I often go off on tangential thinking, say thinking about where we fit into this bigger picture of 8.3 billion people. A person might start with thinking about where they fit into the world in geographical terms. TSP might think about their current location—starting with small geographical terms rather than large ones. They live in an apartment, in a building with 12 floors, and 6 apartments per floor. The apartment building is in a neighborhood where there are few other apartment buildings, there being mostly one family houses in this neighborhood. But if the person lives in New York City, that large city is made of 5 parts called boroughs, so perhaps they live in the neighborhood called Bayside in the borough of Queens. But why stop at the city one lives in? New York City lies within the state of New York, which at the current time is one of 50 states making up the United States.
But let us not stop with a country, but situate the USA in North America, which lies in the Western Hemisphere on Earth, which is one of 9 planets (old timers like me still think of Pluto as a planet even if some scholars have now downgraded it from that status), in the solar system of a small star, which often is referred to as the Sun, but which is a star in the Milky Way galaxy. The closest star to earth after the Sun is Proxima Centauri. So TSP lives in various domains, some small and some gigantic.
The purpose of this perhaps tangential (to the issue of Mathematical Progress) discussion is to try to hint that when thinking about the issue of mathematical progress, one has to set the question in time and context. As the Feature Column tries to suggest, mathematics is a complex and multifaceted subject.
What do I reference with this term multifaceted? I am alluding to the idea that polyhedra have different surface regions known as facets, and that polyhedra with more facets are more complex than the simplest polyhedron. This simplest polyhedron is the tetrahedron, which has only 4 facets. In the case of the regular tetrahedron all four of the faces are congruent equilateral triangles.
Five polyhedra, known as Platonic Solids, made up of congruent regular polygons, shown as possible dice for use in a game. The tetrahedron shown on the left consists of 4 equilateral triangles. Image courtesy of Wikipedia.
Different human societies use different languages to communicate among themselves - there are approximately 7100 different languages being used in the world today. All societies have at different times in history found it convenient to count, but the systems used to represent numbers has changed. Thus, for the Romans, 2026 was not written that way but as MMXXVI. Roman numerals are not a place notation system of the kind that is now universally used in carrying out mathematics. In a place notation system there are a fixed number of symbols, often called digits; by way of example, say 4 digits (say *, #, & and $) where the symbolic expression #**& represents a different number from &***# because the symbols read from left to right vary. The number of digits used in the system is referred to as the "base" of the number system. The number represented by #**& is # multiplied by the base times itself 3 times added to * multiplied by the base times itself 2 times added to * multiplied by the base times itself once added to &. Today we use the bases 10 and 2 almost exclusively; for example, 1345 in base ten stands for the number in words one thousand three hundred forty five, while the same number in binary would be written as 10101000001. At the hardware level, typically computers use a binary system of representation. For large numbers, the binary system uses more symbols than decimal but the rules for adding and multiplying the symbols is simpler than it is for decimal. A multiplication table in binary is a 2x2 table (matrix or array) while in decimal the multiplication table is a 10x10 table.
What bases were used by different cultures over the centuries? The Mayan culture used base 20 and in Babylon base 60 was used. The base 10 numeral system we use today was invented in India and further refined by Islamic mathematicians. These choices of base are probably related to the facts that humans have 10 fingers, humans have 20 fingers and toes, and that 60 has many divisors: -1, 2, 3,4, 5, 6, 20, 12, 15, 30, and 60. In the development of counting systems a milestone was to have a special symbol for no objects—zero. In some early positional number systems where a symbol had not been developed for zero, a gap or a space was used to indicate that the symbol zero was being implied for that that position.
Progress in mathematics education
Throughout the world it is now standard for children to be required to go to school to learn information that will make it possible for them to lead fulfilling lives and pay back to society the benefits that accrue to them due to society's investment in educating them at public expense. In the United States, learning about mathematics and learning about numbers, shapes and patterns starts with kindergarten (and sometimes preschool) and runs through high school. At times particular groups of children have been classified as unable to benefit by learning mathematics and have been put in tracks that make careers that require a rich collection of mathematical skills difficult to pursue. On the other hand, the curriculum has often been structured so that the mathematics taught speeds the path for those students who want to pursue careers needing mathematical skills. This approach structures mathematics curriculum for all students so that they can study calculus in grade 12.
Though the number of students who study calculus in high school has grown tremendously since opportunities to take this college level course in high school and get college credit for it have broadened, the number of students who major in mathematics in college has not grown by equivalent amounts. Furthermore, the emphasis on learning calculus before college has muddied the waters about what mathematically based skills are needed for disciplines other than mathematics itself, where learning calculus is one of the major threads for courses required of mathematics majors in college. Thus, the subject that has come to be known as discrete mathematics does not get as much attention in K-12 education as skills related to mastering calculus, which involves ideas in continuous mathematics. Yet discrete mathematics has made possible many technologies Americans quickly came to love and endorse using!
Mathematical progress
Did mathematics progress on planet Earth from 2025 to 2026? Those steeped in the issues of mathematics and its history would immediately wonder what I might mean by "mathematical progress," and, once the term is defined, how I would measure it—assuming that what I defined had aspects that allow one to carry out measurements! Here I hope, by using informal language, intuitive ideas, and the occasional tangent (a tangent to a Euclidean circle is a line that has one point in common with the circle), to try to convey a sense of the properties I associate with mathematical progress.
Today, new mathematics is produced by a large range of people. Some of these are people who majored in mathematics in college and then pursued advanced degrees in mathematics. Loosely speaking, to get a doctoral degree in mathematics, one is expected to produce some "significant" new mathematics, but many doctoral degrees consist of synthesizing old knowledge in a novel way or organizing prior mathematical work. Scholars whose advanced degrees are in subjects such as physics, chemistry, biology, economics also produce new mathematics as a consequence of their study of issues they investigate in their own subjects. New mathematics is also produced by people with varied educational attainments who work on problems that come up in the jobs that they have and cause them to study issues as varied as fluid flows, how to distribute funds fairly that were made available to people who suffered a natural emergency such as a hurricane, or how to design a new linkage to help deal with an industrial production problem.
New mathematics varies in its importance, which again could be measured in many different ways, as well as to what extent it raises ideas that had little presence in earlier work. There is also a certain amount of new mathematics that arises from the work of amateurs and students. For example, for many years Martin Gardner (1914-2010), who had little formal education in mathematics, wrote columns for Scientific American where he raised questions and puzzles for the readers of the magazine to ponder. Over time, these columns augmented with Gardner's comments appeared in books addressed to the general public, though many people in the mathematics community also read the columns and bought his books. Some of these questions were posed by professional mathematicians (e.g. John Conway (1937-2020) and Ronald Graham (1935-2020)) but sometimes came from not mathematically trained readers of the column.
The amateur mathematician Marjorie Rice (1923-2017) discovered four classes of pentagons which would tile the plane with no holes or overlaps. Her work, together with other work on pentagonal tiling resulted in there being a complete enumeration of the types of pentagons which would tile the plane, where in some of the tiling the pentagons met only along edges, but in some of the tilings the pentagons don't fully match up along their edges. Given the rules involved there turn out to be 15 types of pentagonal tiling; one of the types has only one pentagon in that category but for many of the types of pentagons there are infinitely many different non-congruent pentagons in each type.
It is often observed that every time some mathematical question is answered it generates new mathematical issues to think about. In this spirit, it seems to me that new insights in the idea of tiling the plane with convex (no dents or holes) pentagons might result by looking at the different partition types of pentagons that can arise. Thus, a {5} ; {5} pentagon is one where there are five edges all of the same length and 5 internal angles of equal size (measure) while a {4,1} ; {5} pentagon would have 4 edges of one length and one edge of another length and 5 angles of the same measure. There are pentagons of the first type but such a pentagon cannot tile the plane while there are no pentagons of the second type. One could make a table of all the possible pairs of side length and angle measure patterns, rows for side length partitions and columns for angle size. Since there are 7 partitions of 5 (for example 3,2; 2,2,1, and 1,1,1,1,1) the table described would have 49 entries and it might be of interest to study if the table has any interesting patterns. For the equivalent question about quadrilaterals it turns out that if the quadrilateral corresponding to the entry in the $i$th row and $j$th column exists, then the entry in the $j$th row and and $i$th column also exists ($i$ not equal to $j$).
The question of which convex pentagons tile the plane is of relatively recent origin. Different mathematicians differ about which previously suggested lines of mathematical investigation are the most important, and for what reasons, but here are two problems that have attracted much attention.
a. Goldbach's Conjecture, named for Christian Goldbach (1690-1764)
Can every even number at least 4 be written as the sum of two prime (2, 3, 5, 7, .....) integers?
While known to be true for many millions of cases, no proof is known that it is always true.
b. P = NP?
Intuitively, this question asks whether two collections of mathematical problems consist of exactly the same questions. One collection consists of problems that can be solved using an algorithm in an amount of time that is given by a polynomial function of the size of the input of the problem; the other collection consists of problems where a proposed answer to the problem can be checked in polynomial time.
For example, the Traveling Salesman Problem mentioned above is a very important practical problem and all known algorithms to solve it optimally do not run in polynomial time. On the other hand, if one offers up a solution of the TSP as being optimal, one can check if that is the case in polynomial time.
Many would judge the P=NP problem to be much more important than the Goldbach Conjecture, especially if the solution is positive. Finding a way to solve the TSP in polynomial time would make it possible to solve problems we can only find approximately optimal more efficiently. However, it is conceivable that if Goldbach's Conjecture could be settled in some way, this would lead to progress on the P=NP problem.
As an example of recent mathematical progress, Ryan Williams, currently at MIT, and others who helped him have made a significant discovery related to issues of the complexity of computation. When designing an algorithm to solve a problem one typically needs two kinds of resources - memory and time. Memory is used to store calculations which are made on the way to solving a problem and time is the "total" time for finishing the "job." Williams provided new insight into the tradeoff between using space and time in algorithms to solve certain problems.
Repositories of progress
In this digital age where information comes to our attention in many ways (podcasts, TV, radio, books, etc.) it is easy to forget that in the Egypt of 6000 years ago many of the technologies just listed did not exist. The Egyptians had a written language, hieroglyphics and later other systems. The famous Rosetta stone helped scholars understand ancient scripts because the same information on that physical stone was written down in several different scripts. We take for granted cell and other kinds of phones and computer screens and paper. However, in ancient Egypt if one one wanted to write something down it was either chiseled into stone or written out on papyri, which was a form of paper that was made from reeds. Because the Babylonians used clay tablets to write their cuneiform script much more of what constituted mathematics in Babylon compared to Egypt is available today. Despite the difficulty of understanding what was being recorded on clay tablets, it is even harder to infer what mathematics was being used from engineered objects (roads, houses, pyramids, etc.) that have survived the thousands of years since they were made. It is my understanding that plans for certain military hardware of less than 100 years of age are not available because machines to read the software on which these plans exist are no longer available—or perhaps I am repeating an urban legend—an untrue story that keeps getting repeated even though untrue. Scholars have turned to journals to record their mathematical accomplishments and there are now journals related to dozens of subpieces of the broad area of mathematics known as geometry (e.g. Discrete Geometry, Computational Geometry, Algebraic Geometry, Differential Geometry).
In terms of volume of newly published scholarship more and more mathematics and computer science is being published. Some of this work while new may not contribute much of "importance." However, sometimes putting together lots of small results and obtaining more examples of different kinds in some areas of mathematics provides a platform on which to make a much bigger leap of progress possible.
A major development whose significance is being hotly debated is the development of artificial intelligence tools (AI) to assist humans in doing and progressing in mathematics and computer science. While for a long time now humans have used computers and computation in clever ways to make mathematical progress in the form of new theorems and conjectures for results that appear to be true, AI offers what appears to be the possibility of of giant leaps forward. Could a human request an AI system to prove the Goldbach Conjecture and could the AI spit out a genuine proof? Could an AI system report that, "I have found a wonderful new theorem about prime numbers? Here is the new result not currently mentioned in the mathematics literature and here is its proof."
Already AI is assisting humans in mathematical progress. The downside is that some humans might use AI to do mathematics and pretend it is their own work when it is not. Some researchers are trying to require that work generated with AI assistance be watermarked with a tag that shows that AI was used in generating the work.
Promoting Mathematical Progress
In the America of 2026, our society should and can do more to educate its citizens in the myriad benefits that putting the mathematical knowledge attained by people from around the globe can make for human societies. We should and can do more to promote mathematical progress, for doing so will lead to better lives for all humankind.
References
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Dedication
Dedicated to the memory of my friend and mentor Claudi Alsina (1952-2025) distinguished geometer, mathematics educator and author of splendid mathematics books.
