Open research questions in History and Theory of Mathematics

What the literature leaves open

• The gap between Huygens' geometric approach and Abel's analytic approach. The need to understand the transformation of mathematical thinking from constructive geometry to abstract analysis.

  The tautochrone of Huygens and Abel: from constructive geometry to fractional calculus · 2026 · DOI

• The complexity of geometric behavior in algebraic curves of degrees 3-6. The need to bridge the gap between mathematical theory and engineering applications. The challenge of analyzing and applying these curves in various engineering domains.

  Higher Degree Algebraic Curves Degrees · 2026 · DOI

• The immaterial nature of mathematical objects makes it challenging to describe mathematical objectivity. The lack of a formal guarantee that a mathematical model can be applied to all empirical phenomena. The difficulty of finding mathematical structures in all empirical phenomena.

  On Certain Aspects of Mathematical Objectivity · 2026 · DOI

• The paper identifies the challenge of addressing gender disparities in STEM disciplines. It highlights the need to promote visibility and foster mentorship networks for women mathematicians.

  Empowering Equations: The Transformative Role of Indian Women Mathematicians in Higher Education · 2026 · DOI

• The paper identifies the gap in traditional proof frameworks as the lack of a geometric primitive mechanism that intrinsically connects prime number distribution and zero topological structure. The paper identifies the need for a rigorous proof of the Riemann Hypothesis based on classical mathematical axioms without extra unproven assumptions.

  A Rigorous Proof of the Riemann Hypothesis Based on the Prime Octahedral Spiral Topology and Π-Spinal Topological Field Theory · 2026 · DOI

• The origin of the Gibbons conjecture is less transparent than its frequent citation suggests. There is a lack of understanding of how mathematical terminology develops through transmission and collective usage.

  On the origin of the Gibbons conjecture · 2026 · DOI

• The parity obstacle. The need for a novel sieving method. The challenge of distinguishing between primes and semiprimes.

  How to Bypass the Parity Obstacle——A Perspective from the Tower Sieve Method · 2026 · DOI

• To coordinate overlapping output rises across scales. To prove the centered maximal-variation conjecture.

  Monthly Mathematics Conjectures: Open Problems, Evidence, and Complete Results · 2026 · DOI

• The endpoint-corrected centered maximal-variation conjecture is still open. The unresolved mathematical task is now sharply localized.

  Monthly Mathematics Conjectures: Open Problems, Evidence, and Complete Results · 2026 · DOI

• The Riemann Hypothesis remains unsolved. The paper identifies a gap in the traditional approach to the hypothesis.

  Tawhid of the Zeros: The Riemann Hypothesis (RH) as Qadar at One-Half. A Theological Formulation of the Riemann Hypothesis · 2026 · DOI

• The need for a deterministic engine approach to the Ramanujan Challenge. The lack of a unified method for research-level mathematical problems.

  A Deterministic Engine Approach to the Ramanujan Challenge v2 (With Visualiser) · 2026 · DOI

• Traditional methods of teaching the times tables can be boring and ineffective. There is a need for alternative methods that make learning more enjoyable and effective.

  Chasing Away the Times Tables Blues · 1983 · DOI

• The challenge of presenting complex mathematical ideas in a comprehensible manner. The need to balance rigor with accessibility in mathematical proofs.

  Structuring Mathematical Proofs · 1983 · DOI

• The traditional linear method of presenting mathematical proofs may not be suitable for communication. There is a need for an alternative method that can increase comprehensibility.

  Structuring Mathematical Proofs · 1983 · DOI

• The gap in understanding is how Cauchy was able to put the calculus on a rigorous basis, - The prior work of mathematicians such as Gauss and Bolzano is not sufficient to explain Cauchy's innovation

  Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus · 1983 · DOI

• The paper identifies a gap in the understanding of symmetric functions, highlighting the need for a more general principle. The authors note that existing approaches are limited and do not provide a comprehensive understanding of symmetric functions.

  Do Symmetric Problems Have Symmetric Solutions? · 1983 · DOI

• The paper identifies a gap in the understanding of bijection and its relation to infinite sets. The paper suggests that this gap is due to a lack of understanding of the concept of bijection and its implications.

  L'obstacle du dedoublement des objets mathematiques · 1983 · DOI

• The paper identifies the challenge of teaching geometry using traditional methods. The paper notes the challenge of introducing new geometrical textbooks and teaching methods. The paper highlights the challenge of balancing the need for a child-centered and age-graded approach with the need for a formal and systematic approach to teaching geometry.

  Geometry and the Universities: Euclid and his Modem Rivals 1860–1901 · 1975 · DOI

• The lack of singular value decomposition in Pearson's time. The problem of defining a scale to compute correlation coefficients.

  On the prehistory of Correspondence Analysis · 1983 · DOI

• The lack of singular value decomposition prevented Pearson from completing the discovery of correspondence analysis.

  On the prehistory of Correspondence Analysis · 1983 · DOI

• Further examination of the implications of the study's findings on various fields, such as physics and engineering. Investigation of the applications of the new perspective on positive and negative areas and volumes.

  The Time-Long Misconceptions in Classical Mathematics · 2026 · DOI

• The lack of strict logical or physical justification for many mathematical conventions. The need for a systematic analytical re-examination of mathematical conventions.

  The Time-Long Misconceptions in Classical Mathematics · 2026 · DOI

• Systemic barriers, including educational exclusion and gender bias, have limited women's participation and visibility in mathematical science. The 'leaky pipeline' phenomenon contributes to the underrepresentation of women in leadership positions. Stereotype threat and implicit bias can affect women's performance and self-confidence in mathematics.

  The Trans-formative Role of Women in Mathematical Science: Contributions, Structural Challenges, and Future Directions · 2026 · DOI

• The historical narrative of mathematics has often obscured women's contributions. Systemic barriers have limited women's participation and visibility in mathematical science. The challenges women face in mathematical science have not been fully recognized or addressed.

  The Trans-formative Role of Women in Mathematical Science: Contributions, Structural Challenges, and Future Directions · 2026 · DOI

• The lack of understanding of the conditions under which the repeating decimal length equals the denominator minus 1 - The lack of insight into the possible existence of fractions with infinite repeating decimals

  The Length of the Repeating Decimal · 2026 · DOI

Most-cited papers in History and Theory of Mathematics

• The number sense represents (rational) numbers · Behavioral and Brain Sciences · 2021 · 66 citations
• On understanding mathematical problem-posing processes · ZDM · 2023 · 50 citations
• Mathematics and crises · Educational Studies in Mathematics · 2021 · 34 citations
• Mathematical connections from a networking of theories between extended theory of mathematical connections and onto-semiotic approach · International Journal of Mathematical Education in Science and Technology · 2021 · 32 citations
• Groundwork for a Fallibilist Account of Mathematics · The Philosophical Quarterly · 2020 · 29 citations
• The Birth of Period Three · Mathematics Magazine · 1995 · 29 citations
• History of mathematics in mathematics education: Recent developments in the field · ZDM · 2022 · 27 citations
• Disentangling the Research Literature on <i>Number Sense</i> : Three Constructs, One Name · Review of Educational Research · 2020 · 25 citations
• Multiliteracies: A Short Update · The International Journal of Literacies · 2023 · 24 citations
• The introduction of real numbers in secondary education: an institutional analysis of textbooks · Research in Mathematics Education · 2013 · 23 citations

Most recent work

• The contribution of Czech mathematicians to the reform movement in mathematics education at the beginning of the twentieth century · British Journal for the History of Mathematics · 2026
• A Cultural History of Mathematics in the Modern Age · The Mathematical Intelligencer · 2026
• The Eulerian Numbers can D.I.E. · Mathematics Magazine · 2026
• Some Brief (and Not-so-Brief) Ideas for Bringing History into the Mathematics Classroom · MAA Convergence · 2026
• The Time-Long Misconceptions in Classical Mathematics · Indian Journal of Advanced Mathematics · 2026
• The Trans-formative Role of Women in Mathematical Science: Contributions, Structural Challenges, and Future Directions · Zenodo (CERN European Organization for Nuclear Research) · 2026
• The Length of the Repeating Decimal · Numerical Mathematics: Theory, Methods and Applications · 2026
• Did Euler found continuum mechanics on balance laws? “ <i>Argumentum ab auctoritate est infirmissimum</i> ” · Mathematics and Mechanics of Solids · 2026
• On biquadratic fields: when 5 squares are not enough · Periodica Mathematica Hungarica · 2026
• 45-F Global Secondary Mathematics Systematic Analysis Based on PFUS/PFUSR Unified Model · Zenodo (CERN European Organization for Nuclear Research) · 2026

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