#ghhardy — Public Fediverse posts

“Beauty is the first test: there is no permanent place in this world for ugly mathematics”*…

Is mathematical beauty real? Or is it just a subjective, human ‘wow’ that is becoming redundant in an AI age? Rita Ahmadi explores…

It is a hot July day in London and I take the bus to Bloomsbury. I often come here for the British Library, the British Museum or the London Review Bookshop. More than a location, Bloomsbury feels like stepping into a work of art – maybe one of Virginia Woolf’s stories, or Duncan Grant’s paintings.

This time, I am here for mathematics: the Hardy Lecture at the London Mathematical Society (LMS), named after G H Hardy, a professor of mathematics at the University of Cambridge, a member of the Bloomsbury Group, and a president of the LMS. You may know him from the film The Man Who Knew Infinity (2015), in which he’s played by Jeremy Irons.

The 2025 lecture is by Emily Riehl of Johns Hopkins University in Baltimore, who is talking about a complex mathematical ‘language’ known as infinity category theory: could we teach it to computers so that they could understand it? If successful, computer programs could verify proofs and construct complex structures in this area.

A few seats to my left, I recognise Kevin Buzzard, wearing the multi-coloured, patterned trousers he’s known for among mathematicians. Based at Imperial College London, Buzzard is working on a computer proof assistant called Lean. His interest is personal: after long disputes with a colleague over a flawed proof, he lost trust, as he often puts it, in ‘human mathematicians’. His mission now is to convince all mathematicians to write their proofs in Lean. In the Q&A after one of his talks, he said of the debate between truth and beauty in mathematics: ‘I reject beauty, I want rigour’ – though his vibrant sense of fashion suggests otherwise.

Interest in an AI-driven approach to mathematics has been exponential, and many mathematicians have left traditional academic research to explore its potential. Recently, one group of distinguished mathematicians designed 10 active, research-level questions for AI to tackle. At the time of writing, various AI companies and researchers had claimed to find solutions, which were under evaluation by the community.

Sitting in the room in Bloomsbury, I stared at the Hardy plaque and wondered: would Hardy find proofs generated by AI beautiful? I wasn’t sure. He believed there should be a strong aesthetic judgment in mathematics, drawing parallels with poetry, and argued that beauty is the first test of good mathematics. He went as far as to say that there is no permanent place in the world for ugly mathematics.

If asked, many mathematicians today still talk about the aesthetic appeal of one approach over another.

Yet we live in a different century to Hardy and his Bloomsbury peers, with different technologies and techniques, so perhaps we need a clearer definition of what mathematical beauty actually is. Over the history of mathematics, we can find examples where both rigour and the pursuit of beauty have shaped mathematics itself. So, if we’re completely replacing this with a computer-assisted quest for truth and rigour, we ought to know what we’d be abandoning, if anything. Is mathematical beauty like the beauty in literature and art – or is it something else?…

[Ahmadi explores the idea of “beauty,” generally and in mathematics; traces the rise of AI as a tool, and concludes…]

… my own definition of beauty in mathematics would be as follows:

“Asimplemathematical structure that surprises even the most experienced mathematicians and transfers a sense of vitality.”

But is an AI-assisted proof simple or surprising? How do we define vitality in a machine? On these questions, the jury is out. Myself, I am torn. Maybe models just need more training to match our creativity. But I also wonder whether our limbic system is required. Can we write proofs without emotional kicks? I am also unsure if perfectly efficient brains can come up with novel revolutionary ideas.

Ultimately, this debate is about more than aesthetics; it is closely tied to the development of AI-assisted mathematics. If AI models can produce novel mathematical structures, how should we direct them? Is it a search for beautiful or truthful structures? A question that possibly guides the years to come.

Some mathematicians say they prefer the ‘truth’ and only the ‘truth’. However, my recent discussions with mathematicians showed me that most immediately recognise, enjoy, and even wholeheartedly smile at a beautiful piece of maths. In fact, they spend their whole lives in search of one…

Fascinating: “The eye of the mathematician,” from @ritaahmadi.bsky.social in @aeon.co.

* G. H. Hardy

###

As we embrace elegance, we might send garcefully-calculated birthday greetings to Eduard Heine; he was born on this date in 1821. A mathematician, he is best remembered for his introduction of the concept of uniform continuity, for the Mehler–Heine formula, and for the Heine–Cantor theorem… all of them, quite beautiful.

source

“Juggling is sometimes called the art of controlling patterns, controlling patterns in time and space”*…

A skill for our times…

The Library of Juggling is an attempt to list all of the popular (and perhaps not so popular) juggling tricks in one organized place. Despite the growing popularity of juggling, few websites are dedicated to collecting and archiving the various patterns that are being performed. Most jugglers are familiar with iconic tricks such as the Cascade and Shower, but what about Romeo’s Revenge or the 531 Mills Mess? The goal of this website is to guarantee that the tricks currently circulating around the internet and at juggling conventions are found, animated, and catalogued for the world to see. It is a daunting task, but for the sake of jugglers everywhere it must be done.

For every trick found in the Library, there will be an animated representation of the pattern created via JugglingLab, in addition to general information about the trick (siteswap, difficulty level, prerequisite tricks, etc.). If I am able to run the pattern, then I will provide a text-based tutorial for the trick with the help of animations. I will also include links to other tutorials for the trick that can be found online, ranging from YouTube videos to private sites like this one. If I am unable to provide my own tutorial, there will still be a short description of the trick in addition to outside tutorials and demonstrations…

… if you have come to the Library looking to find out how to start juggling, than it would be best to begin with the Three Ball Cascade pattern. If you are a juggler who is already familiar with the basics, then the various tricks included in the Library can be accessed via the navigation tree on the left, or you can click here to view all of the tricks by difficulty…

Enjoy “The Library of Juggling.”

And see also: “The Museum of Juggling History,” the resources at the International Jugglers’ Association, and “The world cannot be governed without juggling.”

* mathematician (and juggler) Ronald Graham

###

As we toss ’em up, we might send carefully-calculated birthday greetings to G. H. Hardy; he was born on this date in 1877. A mathematician who made fundamental contributions to number theory and mathematical analysis, Hardy juggled other interests as well– for example his  Hardy–Weinberg principle (“allele and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences”) is now a basic principle of population genetics.

In Hardy’s own estimation, his greatest contribution was something else altogether: from 1917, Hardy was the mentor of the Indian mathematician Srinivasa Ramanujan, a relationship that has become celebrated.  Hardy almost immediately recognised Ramanujan’s extraordinary (albeit untutored brilliance), and the two became close collaborators. When asked by a young Paul Erdős what his greatest contribution to mathematics was, Hardy unhesitatingly replied that it was the discovery of Ramanujan, remarking that on a scale of mathematical ability, his own ability would be 25, Littlewood would be 30, Hilbert would be 80, and Ramanujan would be 100.

source

I just checked how many commits I had made to a certain software project (to be posted publicly soon):

$ git log --oneline | wc -l
1729

Mathematicians will know immediately what I thought about. :-)

‘I had ridden in taxi-cab no. 1729, and remarked that the number (7⋅13⋅19) seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways.”’
— G.H. Hardy, in conversation with Ramanujan, as related in his obituary of the latter [https://archive.org/details/pli.kerala.rare.28155/page/n34/mode/1up]

#GHHardy #Ramanujan #TaxicabNumber

Big update to the annotated edition of G.H. Hardy's ‘A Mathematician's Apology’: the PDF is now *tagged* (plus various minor improvements).

(A #TaggedPDF contains extra semantic information to assist screen-reading software etc.)

The new version is available (#OpenAccess as always) at https://archive.org/details/hardy_annotated

The LaTeX tagging system is still under development [https://latex3.github.io/tagging-project/], but I have been making progress in adapting my LaTeX styles to be tagging-compatible.

For the annotated ‘Apology’, I also had to re-implement a subset of the functionality of the "manyfoot" package (which is currently not tagging-compatible), because there are two different kinds of footnotes in the annotated ‘Apology’. (Hardy's original footnotes and the annotations.)

Although the PDF passes VeraPDF validation, there may of course be mistakes in the tagging. Feedback, especially from users of screen readers, would be much appreciated, especially because I now hope to add tags to ‘Form & Number: A History of Mathematical Beauty’ [https://archive.org/details/cain_formandnumber_ebook_large].

#GHHardy #HistPhil #Mathematics #MathematicalBeauty #TeXLaTeX #Accessibility

Big update to the annotated edition of G.H. Hardy's ‘A Mathematician's Apology’: the PDF is now *tagged* (plus various minor improvements).

(A #TaggedPDF contains extra semantic information to assist screen-reading software etc.)

The new version is available (#OpenAccess as always) at https://archive.org/details/hardy_annotated

The LaTeX tagging system is still under development [https://latex3.github.io/tagging-project/], but I have been making progress in adapting my LaTeX styles to be tagging-compatible.

For the annotated ‘Apology’, I also had to re-implement a subset of the functionality of the "manyfoot" package (which is currently not tagging-compatible), because there are two different kinds of footnotes in the annotated ‘Apology’. (Hardy's original footnotes and the annotations.)

Although the PDF passes VeraPDF validation, there may of course be mistakes in the tagging. Feedback, especially from users of screen readers, would be much appreciated, especially because I now hope to add tags to ‘Form & Number: A History of Mathematical Beauty’ [https://archive.org/details/cain_formandnumber_ebook_large].

#GHHardy #HistPhil #Mathematics #MathematicalBeauty #TeXLaTeX #Accessibility

Big update to the annotated edition of G.H. Hardy's ‘A Mathematician's Apology’: the PDF is now *tagged* (plus various minor improvements).

(A #TaggedPDF contains extra semantic information to assist screen-reading software etc.)

The new version is available (#OpenAccess as always) at https://archive.org/details/hardy_annotated

The LaTeX tagging system is still under development [https://latex3.github.io/tagging-project/], but I have been making progress in adapting my LaTeX styles to be tagging-compatible.

For the annotated ‘Apology’, I also had to re-implement a subset of the functionality of the "manyfoot" package (which is currently not tagging-compatible), because there are two different kinds of footnotes in the annotated ‘Apology’. (Hardy's original footnotes and the annotations.)

Although the PDF passes VeraPDF validation, there may of course be mistakes in the tagging. Feedback, especially from users of screen readers, would be much appreciated, especially because I now hope to add tags to ‘Form & Number: A History of Mathematical Beauty’ [https://archive.org/details/cain_formandnumber_ebook_large].

#GHHardy #HistPhil #Mathematics #MathematicalBeauty #TeXLaTeX #Accessibility

Big update to the annotated edition of G.H. Hardy's ‘A Mathematician's Apology’: the PDF is now *tagged* (plus various minor improvements).

(A #TaggedPDF contains extra semantic information to assist screen-reading software etc.)

The new version is available (#OpenAccess as always) at https://archive.org/details/hardy_annotated

The LaTeX tagging system is still under development [https://latex3.github.io/tagging-project/], but I have been making progress in adapting my LaTeX styles to be tagging-compatible.

For the annotated ‘Apology’, I also had to re-implement a subset of the functionality of the "manyfoot" package (which is currently not tagging-compatible), because there are two different kinds of footnotes in the annotated ‘Apology’. (Hardy's original footnotes and the annotations.)

Although the PDF passes VeraPDF validation, there may of course be mistakes in the tagging. Feedback, especially from users of screen readers, would be much appreciated, especially because I now hope to add tags to ‘Form & Number: A History of Mathematical Beauty’ [https://archive.org/details/cain_formandnumber_ebook_large].

#GHHardy #HistPhil #Mathematics #MathematicalBeauty #TeXLaTeX #Accessibility

Big update to the annotated edition of G.H. Hardy's ‘A Mathematician's Apology’: the PDF is now *tagged* (plus various minor improvements).

(A #TaggedPDF contains extra semantic information to assist screen-reading software etc.)

The new version is available (#OpenAccess as always) at https://archive.org/details/hardy_annotated

The LaTeX tagging system is still under development [https://latex3.github.io/tagging-project/], but I have been making progress in adapting my LaTeX styles to be tagging-compatible.

For the annotated ‘Apology’, I also had to re-implement a subset of the functionality of the "manyfoot" package (which is currently not tagging-compatible), because there are two different kinds of footnotes in the annotated ‘Apology’. (Hardy's original footnotes and the annotations.)

Although the PDF passes VeraPDF validation, there may of course be mistakes in the tagging. Feedback, especially from users of screen readers, would be much appreciated, especially because I now hope to add tags to ‘Form & Number: A History of Mathematical Beauty’ [https://archive.org/details/cain_formandnumber_ebook_large].

#GHHardy #HistPhil #Mathematics #MathematicalBeauty #TeXLaTeX #Accessibility

‘Bertrand Russell & Trinity: A college controversy of the last war’ is a pamphlet written by G.H. Hardy in 1942, giving an account of the dismissal of Russell from his Trinity College lectureship in 1916 following his criminal conviction for anti-war political activity. (Hardy was at the time a fellow of Trinity and opposed Russell's ouster.)

I have uploaded a re-typeset version of Hardy's pamphlet to the Internet Archive [https://archive.org/details/hardy_russellandtrinity].

Creating this version was an exercise for me to learn how to (start to) adapt my LaTeX styles to the new interfaces created by the ongoing LaTeX tagging project [https://latex3.github.io/tagging-project/]. (A #TaggedPDF contains supplementary information to assist screen reading software, data extraction, etc.)

Also, Hardy's work is in the public domain; there seemed to be no downloadable ebook version of this pamphlet; and I thought that it would be a useful companion to the ‘Annotated Mathematician's Apology’ [https://archive.org/details/hardy_annotated].

I have made available the #LuaLaTeX source code on #Codeberg, in case anyone is interested in how the style was created: https://codeberg.org/ajcain/hardy_russellandtrinity

#GHHardy #BertrandRussell #WorldWarI #HistPhil #TeXLaTeX #Accessibility

‘Bertrand Russell & Trinity: A college controversy of the last war’ is a pamphlet written by G.H. Hardy in 1942, giving an account of the dismissal of Russell from his Trinity College lectureship in 1916 following his criminal conviction for anti-war political activity. (Hardy was at the time a fellow of Trinity and opposed Russell's ouster.)

I have uploaded a re-typeset version of Hardy's pamphlet to the Internet Archive [https://archive.org/details/hardy_russellandtrinity].

Creating this version was an exercise for me to learn how to (start to) adapt my LaTeX styles to the new interfaces created by the ongoing LaTeX tagging project [https://latex3.github.io/tagging-project/]. (A #TaggedPDF contains supplementary information to assist screen reading software, data extraction, etc.)

Also, Hardy's work is in the public domain; there seemed to be no downloadable ebook version of this pamphlet; and I thought that it would be a useful companion to the ‘Annotated Mathematician's Apology’ [https://archive.org/details/hardy_annotated].

I have made available the #LuaLaTeX source code on #Codeberg, in case anyone is interested in how the style was created: https://codeberg.org/ajcain/hardy_russellandtrinity

#GHHardy #BertrandRussell #WorldWarI #HistPhil #TeXLaTeX #Accessibility

‘Bertrand Russell & Trinity: A college controversy of the last war’ is a pamphlet written by G.H. Hardy in 1942, giving an account of the dismissal of Russell from his Trinity College lectureship in 1916 following his criminal conviction for anti-war political activity. (Hardy was at the time a fellow of Trinity and opposed Russell's ouster.)

I have uploaded a re-typeset version of Hardy's pamphlet to the Internet Archive [https://archive.org/details/hardy_russellandtrinity].

Creating this version was an exercise for me to learn how to (start to) adapt my LaTeX styles to the new interfaces created by the ongoing LaTeX tagging project [https://latex3.github.io/tagging-project/]. (A #TaggedPDF contains supplementary information to assist screen reading software, data extraction, etc.)

Also, Hardy's work is in the public domain; there seemed to be no downloadable ebook version of this pamphlet; and I thought that it would be a useful companion to the ‘Annotated Mathematician's Apology’ [https://archive.org/details/hardy_annotated].

I have made available the #LuaLaTeX source code on #Codeberg, in case anyone is interested in how the style was created: https://codeberg.org/ajcain/hardy_russellandtrinity

#GHHardy #BertrandRussell #WorldWarI #HistPhil #TeXLaTeX #Accessibility

‘Bertrand Russell & Trinity: A college controversy of the last war’ is a pamphlet written by G.H. Hardy in 1942, giving an account of the dismissal of Russell from his Trinity College lectureship in 1916 following his criminal conviction for anti-war political activity. (Hardy was at the time a fellow of Trinity and opposed Russell's ouster.)

I have uploaded a re-typeset version of Hardy's pamphlet to the Internet Archive [https://archive.org/details/hardy_russellandtrinity].

Creating this version was an exercise for me to learn how to (start to) adapt my LaTeX styles to the new interfaces created by the ongoing LaTeX tagging project [https://latex3.github.io/tagging-project/]. (A #TaggedPDF contains supplementary information to assist screen reading software, data extraction, etc.)

Also, Hardy's work is in the public domain; there seemed to be no downloadable ebook version of this pamphlet; and I thought that it would be a useful companion to the ‘Annotated Mathematician's Apology’ [https://archive.org/details/hardy_annotated].

I have made available the #LuaLaTeX source code on #Codeberg, in case anyone is interested in how the style was created: https://codeberg.org/ajcain/hardy_russellandtrinity

#GHHardy #BertrandRussell #WorldWarI #HistPhil #TeXLaTeX #Accessibility

‘Bertrand Russell & Trinity: A college controversy of the last war’ is a pamphlet written by G.H. Hardy in 1942, giving an account of the dismissal of Russell from his Trinity College lectureship in 1916 following his criminal conviction for anti-war political activity. (Hardy was at the time a fellow of Trinity and opposed Russell's ouster.)

I have uploaded a re-typeset version of Hardy's pamphlet to the Internet Archive [https://archive.org/details/hardy_russellandtrinity].

Creating this version was an exercise for me to learn how to (start to) adapt my LaTeX styles to the new interfaces created by the ongoing LaTeX tagging project [https://latex3.github.io/tagging-project/]. (A #TaggedPDF contains supplementary information to assist screen reading software, data extraction, etc.)

Also, Hardy's work is in the public domain; there seemed to be no downloadable ebook version of this pamphlet; and I thought that it would be a useful companion to the ‘Annotated Mathematician's Apology’ [https://archive.org/details/hardy_annotated].

I have made available the #LuaLaTeX source code on #Codeberg, in case anyone is interested in how the style was created: https://codeberg.org/ajcain/hardy_russellandtrinity

#GHHardy #BertrandRussell #WorldWarI #HistPhil #TeXLaTeX #Accessibility

Based on a book of the same name by Robert Kanigel, #TheManWhoKnewInfinity is the story of the mathematician Srinivasa Ramanujan (played by Dev Patel) from Tamil Nadu in India.
The film presents #mathematics as art and as a creative process of discovery. To Ramanujan, it reveals “thoughts of God”.

PS: 1729 is the smallest number expressible as a sum of two cubes in two different ways, the two ways being 13 + 123 and 93 + 103 💓

#Ramanujan #movies #India #love #GHHardy

https://youtu.be/Jji1zSwxstU?si=2Mqy705KY5FnyT5U