An innovative approach to understanding mathematical proofs
A groundbreaking philosophical approach to understanding proving in mathematics is on the horizon. Sorin Bangu does research that can change how we teach mathematics, making it more engaging.
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“The idea is to encourage students to ask questions and express doubts, fostering a deeper understanding rather than just memorizing steps,” Sorin Bangu says. He is doing research that aims to articulate a new way of understanding what it means to provide evidence for a mathematical claim.
Challenging traditional views on mathematical proofs
“There are many ways to prove in mathematics, but the rigor of proofs is not sufficiently well-studied. Our study aims to address this by exploring what can be called 'reasonable' objections to mathematical proofs,” Bangu says.
His research aims to redefine how we understand evidence for mathematical claims, traditionally seen as proof. It addresses three main issues: the meaning of “rigorous proof”, the variety of proof methods, and the concept of "reasonable doubt" in relation to proofs.
The research tackles a range of issues, especially focusing on objections to proofs. It distinguishes between legitimate and illegitimate objections and emphasizes the importance of agreed-upon meanings of mathematical terms. This ensures that mathematical evidence is debated within a specific context (framework), rather than unconditionally, or universally.
“We argue that what ultimately makes a difference, and characterizes the legitimate objections, in other words reasonable doubts, is that the meanings of the mathematical terms involved in challenging the proof are agreed upon by both the proponents and the critics of the proof,” Bangu says.
Mathematical evidence and proof, in general, are typically considered to have a unified nature. However, this project explores the idea that this is not necessarily the case, Bangu says. According to him, there are several ways in which one can become convinced of a mathematical claim.
“Evidence can be more fragmented and varied than it is often assumed; even empirical considerations can be relevant”, Bangu says.
The Pythagorean theorem and the nature of mathematical proofs
Bangu explains that mathematics is often seen as timeless, meaning that once an argument is accepted as a proof, it remains a proof forever. This research challenges that idea. It also argues that providing mathematical evidence is a collective effort involving communities of mathematicians across different times and places. Therefore, whether an argument is considered a proof depends not on a single genius, but on the ongoing consensus of mathematical communities.
“My favourite example is the Pythagorean theorem. It originally states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. However, if we consider a right triangle on a sphere, the proof doesn't work. This raises the question: what does ‘triangle’ mean? is a spherical triangle really a triangle? If not, the objection is not legitimate”, Bangu explains.
One can find similar puzzles in advanced mathematics, but another simple example involves negative numbers, Bangu says.
“Up to high school level, we teach that negative numbers don't have square roots because squaring any number yields a positive result. The proof is correct, but only if we mean by numbers real numbers. If we expand this meaning to include complex numbers, the claim is false, and the proof doesn't work as it turns out that even negative numbers have square roots. This shows that the validity of a proof depends on the framework we use.”
New perspectives on teaching mathematics
Bangu notes that the impact of this research is meant to have long-term significance, as it is tackling fundamental, unsettled conceptual matters, which are somewhat remote from one's everyday concerns.
“This kind of research addresses profound issues not immediately visible in daily life and is something that will influence our deepest thinking structures over time.”
However, the research’s findings could significantly impact teaching by encouraging a more interactive and dialogical approach to learning mathematics, Bangu says. “We hope that our approach to proving could be the start of a new way of teaching maths, encouraging students to voice their confusions and discuss objections to proofs.”
Bangu says that this method promotes deeper understanding and confidence, recognizing that different students learn at different paces.
“It aims to show that mathematics can have a ‘human’ face, making it less intimidating and more approachable.”