Are you suggesting coconuts migrate? or Where in the world did mathematics come from? (Part 2)

Introduction

In the previous article, we looked at the following periods in the development of mathematics.

• The Babylonians (ca. 1900 to 1600 BCE)
• The Egyptians (ca. 3000 BCE to 332 BCE)
• The Greeks (ca. 6 BCE to 4 CE)
• Middle Ages, the Islamic Golden Age (ca. 8 CE to 13 CE)

We observed how the following interconnected categories enhance our understanding of the history of mathematics and their influence on mathematical advancement.

1. Location: Where was mathematics advanced (e.g., Greece)?
2. Place: What were the characteristics of the location that fostered the development of mathematics (e.g., at the crossroads of major trade routes, facilitating the exchange of goods, ideas, and cultures)?
3. Interaction: How did the place engender the development of mathematics to fit the needs of the government, people, or both (e.g., a desire to preserve and expand upon earlier knowledge)?
4. Movement: What caused the decline and movement of mathematics from one location to another (e.g., as Islamic society became more conservative, the focus shifted toward religious studies at the expense of secular sciences, including mathematics)?
5. Region: How and why were places similar that allowed mathematics to flourish (e.g., trade, patronage, practical application, philosophical thought, and religion)?

This article will delve into the more recent periods to see how we arrived at mathematics that captivates us today.

• Renaissance (14 CE to 17 CE)
• 18th Century
• 19th Century
• 20th Century
• 21st Century

Let us continue our journey through each of the above categories as we explore how they influenced the development of mathematics.

Renaissance (14 CE to 17 CE)

The Renaissance was a fervent period of European cultural, artistic, political, and economic “rebirth” following the Middle Ages.

“The Renaissance was a cultural movement that profoundly affected European intellectual life in the early modern period. Beginning in Italy and spreading to the rest of Europe by the 16th century, its influence was felt in art, architecture, philosophy, literature, music, science, technology, politics, religion, and other aspects of intellectual inquiry. Renaissance scholars employed the humanist method in study, and searched for realism and human emotion in art.

The rediscovery of ancient texts and the invention of the printing press in about 1440 democratized learning and allowed a faster propagation of more widely distributed ideas. In the first period of the Italian Renaissance, humanists favored the study of humanities over natural philosophy or applied mathematics, and their reverence for classical sources further enshrined the Aristotelian and Ptolemaic views of the universe.” [1]

Mathematics

The European Renaissance saw a resurgence of interest in classical learning, including mathematics, and witnessed significant advancements in various mathematical disciplines that laid the groundwork for modern mathematics. Classical learning included a renewed interest in the works of ancient Greek and Roman mathematicians, such as Euclid and Archimedes. Scholars sought to translate and study these texts, integrating their ideas into contemporary thought. For instance, the introduction of Hindu-Arabic numerals and the spread of algebra transformed mathematical practices. Figures like Fibonacci popularized these concepts, making calculations more efficient.

The Age of Exploration demanded improved navigation techniques, which relied heavily on mathematical principles such as geometry and trigonometry. To meet the demands of information exchange and mercantile trade – patrons, including monarchs and wealthy merchants – funded mathematicians and scientists, providing resources for research and study.

Furthermore, the advent of the printing press allowed the spread of knowledge and the publication and wide distribution of mathematical texts. This increased accessibility encouraged more people to study mathematics and helped standardize mathematical practices.

Key Facts

Significant aspects of Renaissance mathematics include:

1. Revival of Ancient Mathematics: The Renaissance scholars rediscovered and studied ancient mathematical works, particularly those of Greek mathematicians like Euclid, Archimedes, and Ptolemy. The translation and dissemination of these texts reignited interest in classical mathematical principles and methods, inspiring new avenues of mathematical inquiry.
2. Development of Algebra and Symbolism: Renaissance mathematicians contributed significantly to algebra, introducing symbolic notation and algebraic methods that revolutionized mathematical thinking. Scholars like François Viète and René Descartes played pivotal roles in developing algebraic symbolism and formulating algebraic equations.
3. Advances in Geometry: Renaissance mathematicians expanded upon the geometric foundations laid by ancient mathematicians, incorporating new geometric concepts and techniques into mathematical practice. The works of mathematicians such as Johannes Kepler and Pierre de Fermat advanced the study of geometry and its applications in fields like astronomy and physics.
4. Mathematical Innovation and Experimentation: The Renaissance era witnessed a spirit of mathematical innovation and experimentation, with scholars exploring new mathematical ideas and problem-solving approaches. The development of calculus by mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz exemplified the period’s creative and groundbreaking mathematical work.
5. Influence on Scientific Revolution: The mathematical achievements of the Renaissance profoundly impacted the 16th and 17th centuries Scientific Revolution. Advances in algebra, geometry, and calculus provided the mathematical tools necessary for formulating new scientific theories and advancing empirical inquiry.

Renaissance mathematics marks a period of intellectual revival, creative breakthroughs, and exchanging ideas across different disciplines. This era laid the groundwork for modern mathematics and science. The mathematical innovations from the Renaissance continue to influence our understanding of mathematics and its applications in various fields today.

Transition

The decline of mathematics during the Renaissance from the 14th to the 17th century can be attributed to several interrelated factors. Institutions emphasized humanism, prioritizing literature, philosophy, and the arts over the sciences. School subjects included trivium (grammar, rhetoric, logic) and quadrivium (arithmetic, geometry, music, astronomy), but these liberal arts subjects did not advance mathematical inquiry. The influence of the Church and its doctrines sometimes stifled scientific inquiry, including mathematics, as specific ideas were deemed heretical or contrary to religious beliefs. Wars and political upheaval in Europe during this period diverted resources and attention away from scholarly pursuits, including mathematics.

Despite these factors, it’s important to note that the decline was not absolute. The late Renaissance eventually saw a resurgence of interest in mathematics, paving the way for significant developments in the 17th century with figures like Galileo, Descartes, and Newton, who revitalized mathematical thought and integrated it into the scientific revolution.

“Some of the greatest thinkers, authors, statesmen, scientists, and artists in human history thrived during this era, while global exploration opened up new lands and cultures to European commerce. The Renaissance bridges the gap between the Middle Ages and modern-day civilization.” [2]

18th Century

During the 18th century, elements of Enlightenment thinking culminated in the Atlantic Revolutions. Revolutions began to challenge the legitimacy of monarchical and aristocratic power structures. The Industrial Revolution began during the mid-century, leading to radical changes in human society and the environment. The European colonization of the Americas and other parts of the world intensified and associated mass migrations of people grew as part of the Age of Sail.

In Europe, philosophers ushered in the Age of Enlightenment. This period coincided with the French Revolution of 1789 and was later compromised by the excesses of the Reign of Terror. At first, many monarchies of Europe embraced Enlightenment ideals, but in the wake of the French Revolution, they feared loss of power. [18]

Mathematics

In the 18th century, France emerged as Europe’s leading center of mathematical progress and innovation. French mathematicians made significant contributions to various branches of mathematics, including calculus, number theory, geometry, and mathematical physics.

Calculus became essential in various fields, such as physics and engineering, which resulted in a surge of interest and further advancements. This progress led to the introduction of standardized symbols and notations, making complex mathematical concepts more accessible and easier to communicate. The adoption of standardized notation contributed to the rise of mathematical literature, which made knowledge more available, encouraged self-study and academic inquiry, and exchanged ideas and techniques.

Key Facts

Noteworthy elements of 18th century mathematics include:

1. Calculus: Mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz developed calculus, providing powerful tools for analyzing change and motion. French mathematicians such as Jean le Rond d’Alembert, Joseph-Louis Lagrange, and Étienne Bézout made noteworthy contributions to the development and application of calculus. Their work in differential equations, series expansions, and mathematical analysis helped establish France as a hub for calculus research and education.
2. Number Theory: Prominent French mathematicians like Adrien-Marie Legendre and Joseph-Louis Lagrange were vital in advancing number theory during the 18th century. Their investigations into prime numbers, Diophantine equations, and the theory of quadratic forms contributed to the growth of number theory as a distinct mathematical discipline.
3. Geometry: French mathematicians made significant geometric strides during the 18th century, with figures like Alexis Clairaut and Jean le Rond d’Alembert contributing to the development of analytic and differential geometry. Their work laid the groundwork for modern geometric theories and applications.
4. Probability Theory: Mathematicians like Pierre-Simon Laplace and Jakob Bernoulli made significant contributions to probability theory, establishing it as a formal discipline.
5. Mathematical Physics: France was at the forefront of mathematical physics during the 18th century, with scholars like Pierre-Simon Laplace and Joseph Fourier applying mathematical methods to solve problems in mechanics, astronomy, and heat transfer. Their contributions to mathematical physics helped bridge the gap between mathematics and the physical sciences.
6. Academies and Institutions: France’s prestigious academic institutions, such as the Académie des Sciences and the Collège de France, provided a fertile intellectual environment for mathematical research and collaboration. Mathematicians across Europe were drawn to France to study, teach, and exchange ideas, fostering a vibrant mathematical community.

Transition

While the 18th century saw significant developments, such as in calculus and probability, various factors contributed to a decline in some regions of mathematical exploration. While calculus flourished, much of the focus was on refining existing theories rather than exploring new areas, leading to a plateau in some mathematical developments. The increasing complexity of mathematics meant fewer individuals could master multiple areas, causing some fields to stagnate while others advanced. Ongoing conflicts, such as the French Revolution and the Seven Years’ War, disrupted academic institutions and research funding, hindering progress in various fields, including mathematics. Despite mathematics’ stagnation in the late 18th century, the subsequent 19th century witnessed a resurgence in mathematical innovation, further laying the groundwork for modern mathematics.

19th Century

The 19th century was characterized by vast social upheaval. Slavery was abolished in much of Europe and the Americas. Though the First Industrial Revolution began in the late 18th century, it expanded beyond its British homeland for the first time during the 19th century to continental Europe, North America, and Japan. The 19th century was an era of rapidly accelerating scientific discovery and invention, with significant developments in mathematics, physics, chemistry, biology, electricity, and metallurgy that laid the groundwork for the technological advances of the 20th century. Additionally, the 19th century saw the birth of science as a profession; the term scientist was coined in 1833 by William Whewell, which soon replaced the older term natural philosopher. [5]

Mathematics

“Throughout the 19th century, the Paris Academy of Science was no longer the mathematical powerhouse it had once been. It was overshadowed by Göttingen and the giants of German mathematics, Karl Friedrich Gauss and Bernhard Riemann.” [3] Germany was now the prominent hub of mathematical advancement and innovation. German mathematicians made significant contributions to a wide range of mathematical fields, shaping the landscape of mathematics during this period and influencing the development of modern mathematical theories and concepts.

The Second Industrial Revolution created a demand for mathematical tools in engineering, physics, and technology. This demand drove the further development and application of mathematical theories, and mathematics became more closely intertwined with other scientific fields, leading to innovative applications.

Key Facts

Some prevalent aspects of mathematics during the 19th century include:

1. Algebraic Geometry: The 19th century saw significant developments in algebraic geometry, a branch of mathematics that studies geometric objects defined by polynomial equations. Mathematicians like Évariste Galois, Bernhard Riemann, and Arthur Cayley made pioneering contributions to this field, exploring the connections between algebra and geometry.
2. Group Theory: The study of groups, a fundamental concept in abstract algebra, flourished in the 19th century thanks to the work of mathematicians such as Évariste Galois, Augustin-Louis Cauchy, and Camille Jordan. Group theory provided a powerful framework for understanding mathematical structure symmetry, patterns, and transformations.
3. Non-Euclidean Geometry: The 19th century witnessed the development of non-Euclidean geometries, challenging Euclidean geometry established by Euclid. Mathematicians like Nikolai Lobachevsky, János Bolyai, and Bernhard Riemann introduced new geometries that diverged from Euclid’s postulates, leading to revolutionary insights into the nature of space.
4. Analysis and Calculus: The 19th century saw further advancements in mathematical analysis and calculus, with mathematicians like Karl Weierstrass, Bernard Bolzano, and Augustin-Louis Cauchy refining the foundations of these fields. Rigorous definitions of limits, continuity, and derivatives laid the groundwork for modern analysis.
5. Number Theory: The study of number theory continued to thrive in the 19th century, with mathematicians like Carl Friedrich Gauss, Peter Gustav Lejeune Dirichlet, and Ernst Eduard Kummer making significant contributions. Topics such as prime numbers, congruences, and Diophantine equations were explored in depth, leading to new insights and theorems.

Transition

While the 19th century was primarily a period of growth for mathematics, certain factors contributed to challenges and a relative decline in specific areas or aspects of mathematical development. As the Industrial Revolution progressed, there was a growing emphasis on practical applications of mathematics in engineering and technology, sometimes overshadowing pure mathematical research.

As mathematics became more specialized, some mathematicians focused intensely on narrow fields, which could limit broader interdisciplinary collaboration and innovation. Additionally, these new areas and subfields sometimes lead to fragmentation, making it difficult for mathematicians to stay informed about developments outside their specialty. This led to educational systems in many regions struggling to keep pace with the rapid growth of mathematical theories, leading to gaps in teaching and understanding.

Overall, the 19th century was a period of profound mathematical discovery and innovation, with mathematicians exploring new frontiers, developing novel theories, and providing innovations that paved the way for the mathematical advancements of the 20th century.

20th Century

Technological advancements during World War I changed how war was fought, as new inventions such as tanks, chemical weapons, and aircraft modified tactics and strategy. These innovations and the quest for peace led to Germany’s defeat in World War I and II and the loss of German scientists and mathematicians to the United States.

Due to continuing industrialization and expanding trade, many significant changes of the century were, directly or indirectly, economic and technological. Inventions such as supertankers, airliners, motorways, radio communication and broadcasting, television, digital computers, air conditioning, nuclear power, frozen food, microcomputers, mobile telephones, the Internet, and the World Wide Web affected people’s quality of life across the developed world. Scientific research, engineering professionalization, and technological development—much of it motivated by the Cold War arms race—drove changes in everyday life. [6]

Mathematics

“World War One killed off most of the remaining [French] mathematical talent, and at the beginning of the 20th century, the next generation of French mathematicians found themselves without a father figure to inspire and guide them. Until one such mathematician, André Weil, wrote to a member of the Academy of Science to introduce a new colleague.” [3] That new colleague was Nicholas Bourbaki, not a person but a pseudonym chosen for a secretive math group founded by nine great mathematicians in France in the mid-1930s. Henri Cartan, André Weil, Szolem Mandelbrojt, Claude Chevalley, and Jean Dieudonne were some active founders who collectively wrote many academic papers under the Bourbaki pseudonym to represent the essence of a “contemporary mathematician.”

However, French mathematics did not regain its prominence in the world. During the 20th century, the United States emerged as a leading center for mathematical advancement and innovation. American mathematicians made significant contributions across a wide range of mathematical fields, shaping the landscape of mathematics and influencing the development of modern mathematical theories and applications.

Overall, the United States’ prominence in mathematics during the 20th century can be attributed to the groundbreaking work of its mathematicians, the support of academic and research institutions, and the country’s commitment to mathematical research and education. The contributions of American mathematicians during this period had a lasting impact on the development of mathematics and helped shape the modern mathematical landscape.

Key Facts

Key aspects of the United States’ mathematical advancement in the 20th century include:

1. Applied Mathematics: The United States was pivotal in developing applied mathematics during the 20th century. Mathematicians in the U.S. made significant contributions to areas such as mathematical physics, operations research, cryptography, and computational mathematics, applying mathematical principles to solve real-world problems and advancing technology.
2. Mathematical Physics: American mathematicians and physicists collaborated closely on the development of mathematical physics during the 20th century. Figures like Richard Feynman, John von Neumann, and John Archibald Wheeler made groundbreaking contributions to quantum mechanics, relativity theory, and other areas of theoretical physics, utilizing advanced mathematical tools to describe the natural world.
3. Computer Science and Algorithms: The United States was at the forefront of computer science and algorithm development in the 20th century. Mathematicians such as Alan Turing, John McCarthy, and Donald Knuth played key roles in shaping the foundations of computer science, artificial intelligence, and algorithmic complexity theory, revolutionizing how we process and analyze information.
4. Mathematical Education and Research Institutions: The U.S. boasted renowned research institutions and universities that fostered a vibrant mathematical community during the 20th century. Institutions like the Institute for Advanced Study, Princeton University, and the Massachusetts Institute of Technology (MIT) attracted top mathematicians worldwide, facilitating collaboration and innovation in mathematical research.
5. Pure Mathematics: American mathematicians contributed significantly to pure mathematics during the 20th century, exploring areas such as number theory, algebraic geometry, topology, and mathematical logic. Figures like Emmy Noether, John Tate, and Paul Cohen advanced our understanding of abstract mathematical concepts and laid the groundwork for modern mathematical theories. Andrew Wiles is best known for proving Fermat’s Last Theorem, a famous problem in number theory that remained unsolved for over 350 years. His solution to Fermat’s Last Theorem solved a historic problem and profoundly impacted mathematics, fostering new research directions and inspiring future generations.

Transition

The transition from the 20th to the 21st century marked a shift towards more computational, interdisciplinary, and collaborative approaches in mathematics, along with the rise of new theories and applications.

• Computational Mathematics
  • Computational methods became essential in research, leading to new branches like computational algebra and numerical analysis.
  • The rise of algorithms influenced various fields, including optimization and data analysis.
• Interdisciplinary Approaches: Mathematics increasingly intersected with fields like biology, finance, and social sciences, leading to new applications and methodologies (e.g., mathematical biology, financial mathematics).
• Mathematical Proofs and Theorems
  • Tools for formal verification of proofs became more prevalent, enhancing the rigor in mathematical research.
  • Large-scale collaborations, such as the proof of the Four Color Theorem, showcased the power of collective effort and computer assistance in mathematics.
• Development of New Theories
  • Homotopy Type Theory emerged as a new area of mathematics, bridging the gap between homotopy theory and type theory, leading to advancements in both logic and topology.
  • Category Theory continued to gain prominence as a unifying framework for various mathematical structures.
• Data Science and Statistics
  • Big Data analysis was developed to address the explosion of data sets that are too large or complex for traditional data-processing software to handle. Handling these data sets has led to significant advancements in statistics and probability theory, with new methods for data analysis and interpretation.
  • The rise of Machine Learning applications necessitated new mathematical models and statistical methods.
• Global Access and Collaboration: The internet facilitated global collaboration among mathematicians, leading to a more interconnected mathematical community.

21st Century

The 21st century saw the rise of a global economy. Third World consumerism marked the beginning of the century, along with increased private enterprise and deepening concern over terrorism after the September 11 attacks in 2001.

From the beginning of recorded history to 2003, the entire written works of humanity, in all known languages, are estimated to amount to five exabytes of data. Since 2003, with the beginning of social media and “user-generated content,” the same amount of data is created every two days. With the AI boom of the 2020s gaining international prominence, as of 2024, mass-produced AI-generated content comprised over half of the Internet.

Telecommunications in the early 21st century are much more advanced and universal than in the late 20th century. Only a few percent of the world’s population were Internet users and cellular phone owners in the late 1990s, while as of 2023, 67% of the world’s population is online, and 78% of all people aged 10 and above own a mobile phone. In the 2010s, artificial intelligence, mainly in the form of deep learning and machine learning, became more prevalent, and in the early 2020s, the AI boom began with the rise of generative AI. [11]

Mathematics

The 21st century in mathematics is marked by a significant focus on solving the Millennium Prize Problems, a collection of seven highly complex mathematical challenges, including the Poincaré conjecture, which was famously proven by Grigori Perelman in 2003; alongside this, advancements in areas like prime number theory (Yitang Zhang‘s breakthrough on prime gaps) and sphere packing problems (Maryna Viazovska‘s solution in dimension 8) highlight the ongoing vitality of mathematical research, with a strong emphasis on interdisciplinary applications and computational power to tackle complex problems across various fields. [7, 8, 9]

The achievements of many 21st mathematicians like Terence Tao, Peter Scholze, Maxim Kontsevich and Alain Connes are not as conspicuous as mathematicians of the past for two main reasons.

1. In the past two centuries, mathematical discoveries have grown at an exponential rate.
2. Much mathematical research today is conducted through collaboration, so credit is shared among several people.

The achievements of the most capable mathematicians have equaled those of the past. The proof of Fermat’s Theorem by Sir Andrew Wiles, and the proof of the Poincaré Conjecture by Grigori Perelman were achievements of a magnitude comparable to those of the past. [12]

What Did We Learn

‘Newton himself famously said, “If I have seen further it is by standing on the shoulders of giants.” The giants were the mathematicians and teachers that came before him and passed on their work to subsequent generations. This is always the case with any mathematical development. Progress in the subject is a long, winding walk toward truth and we should be mindful to take seriously the contributions of those who took steps along the way, not just those who took the final ones.’ [10]

As we have seen through a brief history of mathematics, the winding walk toward truth was aptly told by examining how location, place, interaction, movement, and region affected mathematical development. Note Kristin Blain’s words as she explains the history of mathematics.

“As we traverse this mathematical odyssey, it becomes evident that the history of mathematics is not just a sequence of isolated events but a continuum of ideas, discoveries, and innovations. From ancient times to the present day, mathematicians have shaped the way we perceive the world, unlocking the secrets of the universe one equation at a time.” [7]

What does the future hold for mathematics in the 21st century and beyond? We must watch as history unfolds each year and witness the hidden mysteries of mathematics be revealed.

 

References

[1] “Renaissance.” 2024. Wikipedia. Wikimedia Foundation. November 7. https://en.wikipedia.org/wiki/Renaissance.

[2] “Renaissance Period: Timeline, Art & Facts.” 2024. history.com. A&E Television Networks. Accessed September 2. https://www.history.com/topics/renaissance/renaissance.

[3] “A Brief History of Mathematics, Nicolas Bourbaki.” 2019. BBC Radio 4 Extra. BBC. August 24. https://www.bbc.co.uk/programmes/b00stcgv.

[4] “18th Century.” 2024. Wikipedia. Wikimedia Foundation. December 17. https://en.wikipedia.org/wiki/18th_century.

[5] “19th Century.” 2024. Wikipedia. Wikimedia Foundation. December 29. https://en.wikipedia.org/wiki/19th_century.

[6] “20th Century.” 2024. Wikipedia. Wikimedia Foundation. December 17. https://en.wikipedia.org/wiki/20th_century.

[7] Blain, Kristin. 2024. “Timeline of Mathematics: From Ancient Calculations to Modern Marvels – Smartick’s Data Visualizations.” Smartick’s Data Visualizations – Unleashing Data Insights for Smarter Learning. February 8. https://www.smartick.com/data/timeline-of-mathematics-from-ancient-calculations-to-modern-marvels/.

[8] ‘John Baez: “Mathematics in the 21st century”.’ 2025. YouTube. YouTube. Accessed January 1. https://www.youtube.com/watch?v=lUqqQXFTHUY.

[9] Moskowitz, Clara. 2024. “The 7 Coolest Mathematical Discoveries of 2024.” Scientific American. Scientific American. December 12. https://www.scientificamerican.com/article/the-7-coolest-mathematical-discoveries-of-2024/.

[10] Kitagawa, Kate, and Timothy Revell. The Secret Lives of Numbers: A Hidden History of Math’s Unsung Trailblazers. New York, NY: William Morrow, 2023.

[11] “21st Century.” 2025. Wikipedia. Wikimedia Foundation. https://en.wikipedia.org/wiki/21st_century.

[12] Kelly, Brendan. “Why are current mathematicians like Terence Tao, Peter Scholze, Makcim Konsevich and Alain Connes so weak and their achievements so weak compared to twentieth century mathematicians like Jean-Pierre Serre, Alexander Grothendieck and John Forbes Nash?” 2025. Quora. Accessed February 9. https://qr.ae/pYWBNQ.

---

The featured image on this page is from the online article The Mystery of Coconut Migration.