{"id":2617,"date":"2026-02-01T00:01:03","date_gmt":"2026-02-01T05:01:03","guid":{"rendered":"https:\/\/mathvoices.ams.org\/featurecolumn\/?p=2617"},"modified":"2026-02-01T16:02:30","modified_gmt":"2026-02-01T21:02:30","slug":"does-mathematics-progress","status":"publish","type":"post","link":"https:\/\/mathvoices.ams.org\/featurecolumn\/2026\/02\/01\/does-mathematics-progress\/","title":{"rendered":"Does Mathematics Progress?"},"content":{"rendered":"

Every time some
\nmathematical question is answered, it generates new
\nmathematical issues to think about….<\/em><\/span><\/p>\n

Does Mathematics Progress?<\/h1>\n

Joe Malkevitch
\nYork College (CUNY)<\/b><\/p>\n

Introduction<\/h2>\n<\/p>\n

With the beginning of
\na New Year—2026 in one system of counting, though not all societies use
\nthe same calendar—many people take the opportunity to examine how their lives changed or progressed during
\nthe course of the prior year. Some relate this to the observation that
\nan unexamined life is not worth living. Not only can one examine one’s
\nown life for how it progresses with time, one can also examine the
\ndifferent areas of knowledge that humans have named and have studied
\nboth from the perspective of history, and how this knowledge can assist
\nus in leading our lives.
\nHere I take a small look at how mathematics, viewed as a body of
\nknowledge has progressed, changed, or evolved.<\/p>\n

Progress viewed
\nhistorically<\/h2>\n<\/p>\n

There are currently about 8.3 billion humans
\ninhabiting the earth, a dramatically different number from what was the
\ncase only 100 years ago, and in the more distant past, 2026 years ago.
\nThink for our purposes of a mathematics aficionado who we’ll nickname Tangent Sphere
\nProjection (TSP for short). But wait, isn’t TSP short for the Traveling
\nSalesperson Problem—that is, finding the shortest-measure (in time, distance or
\ncost) route that starts and ends in the same location and visits a
\ncollection of sites once and only once? Sorry, Tangent!<\/p>\n

Tangent and I
\noften go off on tangential thinking, say thinking about where we fit into this bigger picture of 8.3 billion people. A person
\nmight start with thinking about where they fit into the world in
\ngeographical terms. TSP might think about their current location—starting with small geographical terms rather than large ones. They live
\nin an apartment, in a building with 12 floors, and 6 apartments per
\nfloor. The apartment building is in a neighborhood where there are few
\nother apartment buildings, there being mostly one family houses in this
\nneighborhood. But if the person lives in New York City, that large city
\nis made of 5 parts called boroughs, so perhaps they live in the
\nneighborhood called Bayside in the borough of Queens. But why stop at
\nthe city one lives in? New York City lies within the state of New York,
\nwhich at the current time is one of 50 states making up the United
\nStates.<\/p>\n

But let us not stop
\nwith a country, but situate the USA in North America, which lies in the
\nWestern Hemisphere on Earth, which is one of 9 planets (old timers like
\nme still think of Pluto as a planet even if some scholars have now
\ndowngraded it from that status), in the solar system of a small star,
\nwhich often is referred to as the Sun, but which is a star in the Milky
\nWay galaxy. The closest star to earth after the Sun is Proxima Centauri.
\nSo TSP lives in various domains, some small and some gigantic.<\/p>\n

The purpose of this perhaps tangential (to the issue of
\nMathematical Progress) discussion is to try to hint that when thinking
\nabout the issue of mathematical progress, one has to set the question in
\ntime and context. As the Feature Column tries to suggest, mathematics is
\na complex and multifaceted subject.<\/p>\n

What do I reference with this term
\nmultifaceted<\/em>? I am alluding to the idea that polyhedra have different surface regions known as
\nfacets, and that polyhedra with more facets are more complex than the
\nsimplest 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
\ncongruent equilateral triangles.<\/p>\n


\n\"Dice
\nFive polyhedra, known as Platonic Solids, made up of congruent regular polygons, shown as possible dice for use
\nin a game. The tetrahedron shown on the left consists of 4 equilateral
\ntriangles. Image courtesy of Wikipedia.<\/DIV><\/p>\n

Different human
\nsocieties use different languages to communicate among themselves –
\nthere are approximately 7100 different languages being used in the world
\ntoday. All societies have at different times in history found it
\nconvenient to count, but the systems used to represent numbers has
\nchanged. Thus, for the Romans, 2026 was not written that way but as
\nMMXXVI. Roman numerals are not a place notation system of the kind that
\nis now universally used in carrying out mathematics. In a place notation system
\nthere are a fixed number of symbols, often called digits; by way of
\nexample, say 4 digits (say *, #, & and $) where the symbolic
\nexpression #**& represents a different number from &***# because
\nthe symbols read from left to right vary. The
\nnumber of digits used in the system is referred to as the
\n"base" of the number system. The number represented by #**& is # multiplied by the base times itself 3 times added to *
\nmultiplied 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
\nalmost exclusively; for example, 1345 in base ten stands for the number in words one thousand three hundred forty
\nfive, while the same number in binary would be written as 10101000001.
\nAt the hardware level, typically computers use a binary system of
\nrepresentation. For large numbers, the binary system uses more symbols
\nthan decimal but the rules for adding and multiplying the symbols is
\nsimpler than it is for decimal. A multiplication table in binary is a
\n2×2 table (matrix or array) while in decimal the multiplication table is
\na 10×10 table. <\/p>\n

What bases were used by different cultures over
\nthe centuries? The Mayan culture used base 20 and in Babylon base 60
\nwas 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
\nrelated to the facts that humans have 10 fingers, humans have 20 fingers and toes, and
\nthat 60 has many divisors: -1, 2, 3,4, 5, 6, 20, 12, 15, 30, and 60. In
\nthe development of counting systems a milestone was to have a special
\nsymbol for no objects—zero. In some early positional number systems
\nwhere a symbol had not been developed for zero, a gap or a space was
\nused to indicate that the symbol zero was being implied for that that
\nposition.<\/p>\n

Progress in mathematics
\neducation<\/h2>\n<\/p>\n

Throughout the world it is now standard for
\nchildren to be required to go to school to learn information that will
\nmake it possible for them to lead fulfilling lives and pay back to
\nsociety the benefits that accrue to them due to society’s investment in
\neducating them at public expense. In the United States, learning about
\nmathematics and learning about numbers, shapes and patterns starts with
\nkindergarten (and sometimes preschool) and runs through high school. At
\ntimes particular groups of children have been classified as unable to
\nbenefit by learning mathematics and have been put in tracks that make
\ncareers that require a rich collection of mathematical skills
\ndifficult to pursue. On the other hand, the curriculum
\nhas often been structured so that the mathematics taught speeds the path
\nfor those students who want to pursue careers needing mathematical
\nskills. This approach structures mathematics curriculum for all students
\nso that they can study calculus in grade 12.\n<\/p>\n

Though the number of
\nstudents who study calculus in high school has grown tremendously since
\nopportunities to take this college level course in high
\nschool and get college credit for it have broadened, the number of
\nstudents who major in mathematics in college has not grown by equivalent
\namounts. Furthermore, the emphasis on learning calculus before college
\nhas muddied the waters about what mathematically based skills are needed
\nfor disciplines other than mathematics itself, where learning calculus
\nis one of the major threads for courses required of mathematics majors
\nin college. Thus, the subject that has come to be known as discrete mathematics does
\nnot get as much attention in K-12 education as skills related to mastering
\ncalculus, which involves ideas in continuous mathematics. Yet discrete mathematics has made possible many
\ntechnologies Americans quickly came to love and endorse using! <\/p>\n

Mathematical progress<\/h2>\n<\/p>\n

Did mathematics progress on planet
\nEarth from 2025 to 2026? Those steeped in the issues of mathematics
\nand its history would immediately wonder what I might mean by
\n"mathematical progress," and, once the term is defined, how I
\nwould measure it—assuming that what
\nI defined had aspects that allow one to
\ncarry out measurements! Here I hope, by using informal language,
\nintuitive ideas, and the occasional tangent (a tangent to a Euclidean circle
\nis 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.\n<\/p>\n

Today, new
\nmathematics is produced by a large range of people. Some of these are
\npeople who majored in mathematics in college and then pursued advanced degrees in mathematics. Loosely
\nspeaking, to get a doctoral degree in mathematics, one is expected to
\nproduce some "significant" new mathematics, but many doctoral
\ndegrees consist of synthesizing old knowledge in a novel way or
\norganizing prior mathematical work. Scholars whose advanced degrees
\nare in subjects such as physics, chemistry, biology, economics also
\nproduce new mathematics as a consequence of their study of issues they
\ninvestigate in their own subjects. New mathematics is also produced by
\npeople with varied educational attainments who work on problems
\nthat come up in the jobs that they have and cause them to study issues
\nas varied as fluid flows, how to distribute funds fairly that were made
\navailable to people who suffered a natural emergency such as a
\nhurricane, or how to design a new linkage to help deal with an industrial
\nproduction problem.<\/p>\n

New mathematics varies in its importance, which again could be measured in many different ways, as well as to what extent
\nit raises ideas that had little presence in earlier work. There is also
\na certain amount of new mathematics that arises from the work of
\namateurs and students. For example, for many years Martin Gardner (1914-2010), who
\nhad little formal education in mathematics, wrote columns for Scientific
\nAmerican where he raised questions and puzzles for the readers of the
\nmagazine to ponder. Over time, these columns augmented with Gardner’s
\ncomments appeared in books addressed to the general public, though many
\npeople in the mathematics community also read the columns and bought his
\nbooks. Some of these questions were posed by professional mathematicians
\n(e.g. John Conway (1937-2020) and Ronald Graham (1935-2020)) but
\nsometimes came from not mathematically trained readers of the column.<\/p>\n

The amateur mathematician Marjorie Rice (1923-2017)
\ndiscovered four classes of pentagons which would tile the plane with no
\nholes or overlaps. Her work, together with other work on pentagonal
\ntiling resulted in there being a complete enumeration of the
\ntypes of pentagons which would tile the plane, where in some
\nof the tiling the pentagons met only along edges, but in some of the
\ntilings the pentagons don’t fully match up along their edges. Given the
\nrules involved there turn out to be 15 types of pentagonal tiling; one
\nof the types has only one pentagon in that category but for many of the
\ntypes of pentagons there are infinitely many different non-congruent
\npentagons in each type.<\/p>\n

\"One
\nPosted by Tomruen on Wikipedia<\/a>, CC BY-SA 4.0<\/a><\/div>\n

It is often observed that every time some
\nmathematical question is answered it generates new
\nmathematical issues to think about. In this spirit, it seems to me that
\nnew insights in the idea of tiling the plane with convex (no dents or
\nholes) pentagons might result by looking at the different partition
\ntypes of pentagons that can arise. Thus, a {5} ; {5} pentagon is one
\nwhere there are five edges all of the same length and 5 internal angles
\nof equal size (measure) while a {4,1} ; {5} pentagon would have 4 edges
\nof one length and one edge of another length and 5 angles of the same
\nmeasure. There are pentagons of the first type but such a pentagon
\ncannot tile the plane while there are no pentagons of the second type.
\nOne could make a table of all the possible pairs of side length and
\nangle measure patterns, rows for side length partitions and columns for
\nangle size. Since there are 7 partitions of 5 (for example 3,2; 2,2,1,
\nand 1,1,1,1,1) the table described would have 49 entries and it might be
\nof interest to study if the table has any interesting patterns. For the
\nequivalent question about quadrilaterals it turns out that if the
\nquadrilateral corresponding to the entry in the $i$th row and $j$th column
\nexists, then the entry in the $j$th row and and $i$th column also exists ($i$
\nnot equal to $j$).<\/p>\n

The question of which convex pentagons tile
\nthe plane is of relatively recent origin. Different mathematicians differ about which
\npreviously suggested lines of mathematical investigation are the most
\nimportant, and for what reasons, but here are two problems that have
\nattracted much attention.<\/p>\n

a. Goldbach’s Conjecture, named for
\nChristian Goldbach (1690-1764) <\/p>\n

Can every even number at least
\n4 be written as the sum of two prime (2, 3, 5, 7, …..) integers? <\/p>\n

While known to be true for many millions of cases, no proof is known
\nthat it is always true.<\/p>\n

b. P = NP? <\/p>\n

Intuitively, this
\nquestion asks whether two collections of mathematical problems consist
\nof exactly the same questions. One collection consists of problems that can be solved
\nusing 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
\npolynomial time.<\/p>\n

For example, the Traveling Salesman Problem mentioned above is a very
\nimportant practical problem and all known algorithms to solve it
\noptimally do not run in polynomial time. On the other hand, if one
\noffers up a solution of the TSP as being optimal, one can check if that
\nis the case in polynomial time. <\/p>\n

Many would judge the P=NP problem to be much more important than the Goldbach Conjecture, especially if the solution is positive. Finding a
\nway to solve the TSP in polynomial time would make it possible to solve
\nproblems we can only find approximately optimal more efficiently.
\nHowever, it is conceivable that if Goldbach’s Conjecture could be
\nsettled in some way, this would lead to progress on the P=NP problem.<\/p>\n

\nAs an example of recent mathematical progress, Ryan Williams, currently
\nat MIT, and others who helped him have made a significant discovery<\/a>
\nrelated to issues of the complexity of computation. When designing an
\nalgorithm to solve a problem one typically needs two kinds of resources
\n– memory and time. Memory is used to store calculations which are made
\non the way to solving a problem and time is the "total" time
\nfor finishing the "job." Williams provided new insight into
\nthe tradeoff between using space and time in algorithms to solve certain
\nproblems.<\/p>\n

Repositories of progress<\/h2>\n<\/p>\n

In this
\ndigital age where information comes to our attention in many ways
\n(podcasts, TV, radio, books, etc.) it is easy to forget that in the
\nEgypt of 6000 years ago many of the technologies just listed did not
\nexist. The Egyptians had a written language, hieroglyphics and later
\nother systems. The famous Rosetta stone helped scholars understand
\nancient scripts because the same information on that physical stone was
\nwritten down in several different scripts. We take for granted cell
\nand other kinds of phones and computer screens and paper. However, in
\nancient Egypt if one one wanted to write something down it was either
\nchiseled into stone or written out on papyri, which was a form of
\npaper that was made from reeds. Because the Babylonians used
\nclay tablets to write their cuneiform script much more of what
\nconstituted mathematics in Babylon compared to Egypt is available today.
\nDespite the difficulty of understanding what was being recorded on clay
\ntablets, it is even harder to infer what mathematics was being used from
\nengineered objects (roads, houses, pyramids, etc.) that have
\nsurvived the thousands of years since they were made. It is my
\nunderstanding that plans for certain military hardware of less than 100
\nyears of age are not available because machines to read the software on
\nwhich these plans exist are no longer available—or perhaps I am
\nrepeating an urban legend—an untrue story that keeps
\ngetting repeated even though untrue. Scholars have turned to journals to
\nrecord their mathematical accomplishments and there are now journals
\nrelated to dozens of subpieces of the broad area of mathematics
\nknown as geometry (e.g. Discrete Geometry, Computational Geometry,
\nAlgebraic Geometry, Differential Geometry). <\/p>\n

In terms of volume
\nof newly published scholarship more and more mathematics and computer
\nscience is being published. Some of this work while new may not
\ncontribute much of "importance." However, sometimes putting
\ntogether lots of small results and obtaining more examples of different
\nkinds in some areas of mathematics provides a platform on which to make
\na much bigger leap of progress possible.<\/p>\n

A major development
\nwhose significance is being hotly debated is the development of
\nartificial intelligence tools (AI) to assist humans in doing and
\nprogressing in mathematics and computer science. While for a long time
\nnow humans have used computers and computation in clever ways to make
\nmathematical progress in the form of new theorems and conjectures for
\nresults that appear to be true, AI offers what appears to be the
\npossibility of of giant leaps forward. Could a human request an AI
\nsystem to prove the Goldbach Conjecture and could the AI
\nspit out a genuine proof? Could an AI system report that, "I have
\nfound a wonderful new theorem about prime numbers? Here is the new
\nresult not currently mentioned in the mathematics literature and here is
\nits proof."<\/p>\n

Already AI is assisting humans in
\nmathematical progress. The downside is that some humans might use AI to
\ndo mathematics and pretend it is their own work when it is not. Some
\nresearchers are trying to require that work generated with AI assistance
\nbe watermarked with a tag that shows that AI was used in
\ngenerating the work.<\/p>\n

Promoting Mathematical Progress<\/h2>\n<\/p>\n

In the America of 2026, our society should and can do
\nmore to educate its citizens in the myriad benefits that putting the
\nmathematical knowledge attained by people from around the globe
\ncan make for human societies. We should and can do more to promote
\nmathematical progress, for doing so will lead to better lives for all
\nhumankind.<\/p>\n

References<\/p>\n<\/h2>\n

Alonso, O. (2009).
\nMaking sense of definitions in geometry:
Metric-combinatorial
\napproaches to classifying triangles and
quadrilaterals. Unpublished
\ndoctoral dissertation, Teachers
College, Columbia University of New
\nYork City, NY. <\/p>\n

Alonso, O. and J. Malkevitch, (2013)
\nClassifying Triangles and
Quadrilaterals. NCTM Mathematics Teacher
\nJournal, (March,
2013 issue), 06 (7), 541-548 3 <\/p>\n

Beineke,
\nL. and B. Toft, R. Wilson, Milestones in Graph Theory A century of
\nProgress, American Mathematical Society, 2025.\n<\/p>\n

Gowers,
\nG.T. Mathematics A Very Short History, Oxford U. Press, 2002.<\/p>\n

\nGrünbaum, B., (1995). The angle-side reciprocity of quadrangles.
\n
Geombinatorics, 4, 115-118. <\/p>\n

B. Grünbaum, Side-angle
\nreciprocity – a survey. Geombinatorics, 2 (2011) 55-62.<\/p>\n

\nRao, Michael., Exhaustive search of convex pentagons which tile the
\nplane, arXiv preprint arXiv:1708.00274 (2017).<\/p>\n

Dedication<\/h2>\n<\/p>\n

Dedicated to the memory of my friend and mentor Claudi Alsina
\n(1952-2025) distinguished geometer, mathematics educator and author of
\nsplendid mathematics books.<\/p>\n

\"Photo<\/p>\n

Image by Victoria Alsina, Wikipedia. CC BY-SA 4.0<\/a><\/DIV> <\/p>\n","protected":false},"excerpt":{"rendered":"

Every time some mathematical question is answered, it generates new mathematical issues to think about…. Does Mathematics Progress? Joe Malkevitch York 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 examineRead More →<\/a><\/span><\/p>\n","protected":false},"author":2,"featured_media":2037,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"advanced_seo_description":"","jetpack_seo_html_title":"","jetpack_seo_noindex":false,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,12,15,17],"tags":[217,216,72],"class_list":["entry","author-uwhitcher","post-2617","post","type-post","status-publish","format-standard","has-post-thumbnail","category-213","category-discrete-math-and-combinatorics","category-history-of-mathematics","category-joseph-malkevitch","tag-artificial-intelligence","tag-pnp","tag-tilings"],"jetpack_featured_media_url":"https:\/\/i0.wp.com\/mathvoices.ams.org\/featurecolumn\/wp-content\/uploads\/sites\/2\/2024\/07\/cropped-FC1380x500x2.png?fit=1380%2C288&ssl=1","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/posts\/2617","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/comments?post=2617"}],"version-history":[{"count":5,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/posts\/2617\/revisions"}],"predecessor-version":[{"id":2627,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/posts\/2617\/revisions\/2627"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/media\/2037"}],"wp:attachment":[{"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/media?parent=2617"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/categories?post=2617"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/tags?post=2617"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}