In this lecture Professor Hart showed some mathematical illusions: “proofs” that 1 = 0, that fractions don’t exist, and more. There are curious and important implications behind what’s going on.
These “proofs” reveal some very common logical slips that can go unnoticed when we are trying to prove more plausible statements. And the stakes are high. Professor Hart showed that once you have “proved” one false claim, you can prove absolutely any statement at all.
https://www.gresham.ac.uk/sites/default/files/transcript/2024-05-14-1300_Hart-T.pdf
Further Reading
If you want to explore geometric constructions such as testing the isosceles triangle argument, try Geogebra https://www.geogebra.org/
Unfortunately, in terms of spurious proofs, there isn’t a great deal of further reading to recommend. Professor Hart has been collecting them for many years. All the proofs in the lecture, and many others, seem to be part of the mathematical folklore and impossible to attribute to any individual. Professor Hart has certainly never seen any of them associated to any one person. The first such proof she ever saw was when she was 14 years old and attending a mathematics masterclass – it was the 1-1+1-1⋯ “proof” with which this talk began, and it was shown to the students by a mathematician called Heather Cordell. The proof that all triangles are isosceles was told to Professor Hart by her mother, who was a maths teacher. Since then, she has been shown some by mathematician friends, has found more online, and encountered others in the one book she is aware of having been published in the UK on this topic, which she picked up second-hand some years ago: Fallacies in Mathematics, by E. A. Maxwell, Cambridge University Press (1959).
© Professor Sarah Hart 2024
Professor Sarah Hart
Professor of Geometry (2020 – )
Sarah Hart is the first woman Professor of Geometry at Gresham College, and was appointed in 2020. She is Professor Emerita of Mathematics at Birkbeck, University of London.
She studied at Oxford and Manchester, gaining her PhD in 2000. Postdoctoral research and teaching followed, including a prestigious Engineering and Physical Sciences Research Council Fellowship, before she was appointed to a lectureship at Birkbeck in 2004. She became Professor of Mathematics there in 2013, and served in various management roles including as Head of Mathematics and Statistics, Assistant Dean, and Programme Director for the MSc Mathematics.
Her academic publications have been mainly in the area of pure mathematics known as group theory, which has many applications both inside and outside of mathematics, for example in coding theory and cryptography. She is actively involved in the British Society for the History of Mathematics, and has served a three-year term as President of the Society from 2021-2023.
Professor Hart is passionate about communicating mathematics and is a sought-after public speaker. She is particularly interested in the links between mathematics, culture and creativity: many of her public lectures and talks in schools relate to these topics. Her book Once Upon a Prime: the Wondrous Connections between Mathematics and Literature, was published in 2023, and has been positively reviewed in the press, including The Sunday Times, The Observer, The Economist, The New York Times, The Washington Post and the Wall Street Journal.
Read more about Sarah Hart in Qantat Magazine
helen the hare 