Unique Presentation Identifier:
54
Program Type
Honors
Faculty Advisor
Jeanine Myers
Document Type
Presentation
Location
Face-to-face
Start Date
9-4-2026 10:20 AM
End Date
9-4-2026 10:50 AM
Abstract
In the late nineteenth and early twentieth centuries, mathematics faced a foundational crisis: Greog Cantor’s set theory led to an interesting self-referencing paradox in math and questions about the logical consistency of mathematics. From this crisis, two opposing viewpoints emerged: the Formalists and the Intuitionists. The Formalists praised Cantor’s work as a way to place math on a secure logical foundation, ensuring the discipline’s purity, however the Intuitionists despised Cantor’s work and heralded Cantor as a charlatan and corrupter of the youth. The leader of the Formalists, David Hilbert, proposed a formal system of rigorous proofs to build a complete and consistent system of mathematics. Perhaps one of the most well known outcomes from Hilbert’s proposal was Russell and Whitehead’s Principia Mathematica where mathematics was rigorously defined from the ground up, famously taking hundreds of pages to justify the simple equation 1 + 1 = 2. Though analysis had existed prior to the 1800s, the crisis ultimately culminated in formalizing the use of axioms and the proof-based structure modern mathematics utilizes.
This project traces the Formalist reconstruction of the mathematical system, beginning with the definition of the number and Peano’s Axioms. From these axioms, we develop addition, multiplication, exponentiation, and the rational numbers, illustrating how increasingly sophisticated systems emerge from simple logical principles. Beyond this, developing further advanced systems is a matter of extending the logic. By reconstructing the system of mathematics from a few axioms, this presentation illustrates the beauty of logical construction and the level of analysis underlying the modern mathematical landscape.
Recommended Citation
Skaggs, Joy, "From Counting to Advanced Math: How Five Simple Axioms Shape the Mathematical Landscape" (2026). ATU Scholars Symposium. 24.
https://orc.library.atu.edu/atu_rs/2026/2026/24
Included in
From Counting to Advanced Math: How Five Simple Axioms Shape the Mathematical Landscape
Face-to-face
In the late nineteenth and early twentieth centuries, mathematics faced a foundational crisis: Greog Cantor’s set theory led to an interesting self-referencing paradox in math and questions about the logical consistency of mathematics. From this crisis, two opposing viewpoints emerged: the Formalists and the Intuitionists. The Formalists praised Cantor’s work as a way to place math on a secure logical foundation, ensuring the discipline’s purity, however the Intuitionists despised Cantor’s work and heralded Cantor as a charlatan and corrupter of the youth. The leader of the Formalists, David Hilbert, proposed a formal system of rigorous proofs to build a complete and consistent system of mathematics. Perhaps one of the most well known outcomes from Hilbert’s proposal was Russell and Whitehead’s Principia Mathematica where mathematics was rigorously defined from the ground up, famously taking hundreds of pages to justify the simple equation 1 + 1 = 2. Though analysis had existed prior to the 1800s, the crisis ultimately culminated in formalizing the use of axioms and the proof-based structure modern mathematics utilizes.
This project traces the Formalist reconstruction of the mathematical system, beginning with the definition of the number and Peano’s Axioms. From these axioms, we develop addition, multiplication, exponentiation, and the rational numbers, illustrating how increasingly sophisticated systems emerge from simple logical principles. Beyond this, developing further advanced systems is a matter of extending the logic. By reconstructing the system of mathematics from a few axioms, this presentation illustrates the beauty of logical construction and the level of analysis underlying the modern mathematical landscape.