Geometry with Proofs, Quantum Supremacy, and the American Survival

An open letter to the US Secretary of Education Linda E. McMahon

Preface: The text below is a version of an open letter sent on April 8, 2021, by Maxim Olchanyi, Alioscia Hamma, and 18 other academics who wished to remain anonymous to then-Science Advisor to the President, Eric Lander. Since then, little has changed, except for the recently published National Report Card by the National Assessment of Educational Progress. It reveals that our 12th-grade math skills are the lowest since 2005, when the current assessment scheme was introduced. In absolute terms, only 22% of 12th graders are proficient in math. This decline is a direct result of K-12 mathematics curricula being stripped of what mathematics is: proofs. This must change, radically and urgently. Three years of Euclidean geometry with proofs (grades 6 through 8) should be the absolute minimum.

Maxim Olchanyi (UMass Boston), Alioscia Hamma (UMass Boston and University of Naples Federico II), and Nurit Haspel (UMass Boston)

Euclid's Elements, Book I, proposition 43

Dear Madam Secretary,

The national security and economic well-being of our country are at risk. We feel it is our duty to bring the danger to your attention as swift, sustained, and focused actions are needed to remedy the situation.

As you are aware, the race for quantum supremacy is a high-stakes international competition for global influence and world standing. It is widely perceived as a race to acquire paradigm-shifting technology, but ultimately, it is a race to cultivate the deeply educated minds and the highly refined expertise needed for a national quantum enterprise.

As such, it is a race we are losing. What is worse, as with the tortoise and the hare, we do not seem to realize we are losing.

Individuals that possess the rigorous mathematical foundation needed to fathom quantum technology are a dwindling few in American society. Mathematical deficiencies among youth are well documented and, in our experience, far more profound than is appreciated. They begin at early ages, continue through high school and into college. First-year college students fail mathematical gateway courses at alarming rates and math remediation programs, once a rarity, are now ubiquitous, systemic elements of American colleges and universities. What cannot be solved by remediation is customarily addressed through the active and tacit educational conspiracy of our age—grade inflation.

Mathematical rigor—the degree to which mathematical argument is sound—is essential to quantum physics and other scientific disciplines. Learning quantum science without rigor is akin to boating without a compass: one can make one’s way down a river, but one cannot navigate an ocean. Without intensive experience with logical systems, subsequent mathematics and science subjects become but a collection of facts, not a mental workplace where a person can expand, generalize, and improve.

It is the acquisition of expert performance—another type of rigor—that creates coherence among facts and synergy among concepts. At least a decade of focused and dedicated practice is required to refine one’s proficiency to a level where one can begin to perceive the expert’s world [Ericsson and Charness, Expert performance: Its structure and acquisition (1994)]. It is not enough to be ‘exposed’ to mathematical logic, one must assimilate it over time, adopt it, value it, and employ it as one’s thinking philosophy.

This mathematical way of thinking is also at the foundation of independent thought in free societies. Understanding a mathematical proof is a way of acquiring knowledge that does not rely on anyone’s authority, on anyone’s testimony, on anyone else’s faith. It provides the human mind with the individual autonomy and freedom necessary to pursue creative endeavors. This is a freedom that is personal; that is cultivated internally within individuals and which no one can take away.

For twenty-three centuries, the Euclidean axiom-theorem-proof system, preserved for the modern world by the Arab mathematicians, was the foundation of that thinking philosophy [see e.g., Mathematics and Logic: From Euclid to Modern Geometry, online catalog, Hillsdale College (2025)]. It served as the mind’s gymnasium for logical reasoning and abstract thinking. Isaac Newton’s Principia is a triumph of the methods and spirit of Euclid’s Elements. Abraham Lincoln, as a young lawyer, “…read Euclid by the light of a candle after others had dropped off to sleep” [Carl Sandburg, Abraham Lincoln: The Prairie Years (1926)] and structured his political arguments as Euclidean propositions. Thomas Jefferson constructed the American Declaration of Independence using Euclidean form, garnering faith in his argument by conveying the logical inevitability of his conclusions. [Judith V. Grabiner, The Centrality of Mathematics in the History of Western Thought (1988)]

This foundation of all foundations was de-emphasized or disappeared altogether from our schools some fifty years ago. Ironically, it retains its sanctified place in other countries – countries that rank well above the US in mathematical proficiency rankings.

We seek to arrest the relentless diminution of expertise in mathematical abstract reasoning and the ongoing dismantling of the educational foundations of such expertise. Many believe that we are on the right track; that a first step approach—encouraging all to embrace science and technology—will, by itself, insure our future. This is a fallacy. The foundational pathway has been swept away; and what appears to be opportunity for all is actually a no man’s land of narrowed options and impeded growth.

We advocate for the restoration of rigorous mathematical logic and proof to our American educational system and to our American society. Restoration means moving beyond initial exposures or brief enrichments. It means that American students pursue logical reasoning and abstract thinking to high levels of refinement and expertise. It means that American students view logical reasoning and abstract thought as a standard by which truth is verified and reality agreed upon. And, for the sake of our country, it means that Americans who can fathom advanced ideas and technology are not a dwindling few but a thriving populace.

We suggest the following broad steps:

• Inculcate Euclidian Proof Solving Skills Early and Comprehensively: Students should begin Euclidian Proof at the earliest, developmentally appropriate age and become facile axiom-theorem-proof solvers while still of a young age.

• Develop Expertise in the Mathematical Rigor of Reasoned Proof: From the age of introduction forward, promote an uninterrupted, coherent, and increasingly sophisticated skill development in logical reasoning and abstract thinking.

• Provide Entry Points for All: Create multiple entry points to this logic and abstract thought system for students of different ages and circumstances. No one should be shut out for a late or unconventional start, and everyone should have the benefit of rational thought for life-long learning. This part will require substantial government funding.

Though not experts in politics or governance, we respectfully submit that we are experts in fields that yield technologies and scientific breakthroughs of national importance. We are keenly aware of the mathematical framework that underpins our disciplines and the years of intense, focused study necessary to achieve expertise. We know with certainty that there are no shortcuts, no timesavers, no substitutes. In particular, the absence of mathematical rigor and proofs not only deprives young Americans of fundamental tools for the understanding of the world but also conceals the structural socio-economic inequalities that are responsible for different performances in different sectors of society. As such, it perpetuates and magnifies these inequalities, while being wrapped in the mantle of, ironically, inclusion.

We also feel fervently that mathematics belongs to everyone and to no one. Generations of Americans are being deprived of a thought system that underlays scientific achievement and that guided our founders’ passions for the concepts of justice and freedom.

Please take steps now to ensure a sustainable, bright future for America.

Sincerely,

Maxim Olchanyi, Alioscia Hamma, and Nurit Haspel