Monday Morning Math: ChatGPT “research”

Hello everyone!  Apologies for skipping a week – Naz’s Spring Break was a week ago, and I’m still catching up after accompanying a group of students to Hungary. Fortunately, the topic for this week’s Monday Morning Math fell into my lap (well, inbox – thanks Mark!) Last Monday (March 20), the BBC posted posted “The numbers that are too big to imagine”  about infinity.  Here’s a quote from the original article.

Some infinities, [Cantor] showed, are bigger than others.

How so? One of the simplest ways to understand why is to imagine the set of all the even numbers. This would be infinite, right? But it must be smaller than the set of all whole numbers, because it does not contain the odd numbers. Cantor proved that when you compare such sets, they contain numbers that do not match up, therefore there must be multiple sizes of infinity.

If you’ve taken an introduction to Proof class, you might remember the idea of sizes of infinity, but you might also remember that the whole numbers and even numbers are actually the same size of infinity.  The Cantor proof referenced in the paragraph above doesn’t exist.

The reason for the error?  Maybe Richard Fisher, the author, just made a mistake.  Or maybe the mistake was deeper than that.  At the end of the article was the quote:

The author used ChatGPT to research trusted sources and calculate parts of this story.

The error was found quickly, and corrected within a day.  The article now reads:

Some infinities, [Cantor] showed, are bigger than others.

How so? To understand why, consider the numbers as ‘sets’. If you were to compare all natural numbers (1, 2, 3, 4, and so on) in one set, and all the even numbers in another set, then every natural number could in principle be paired with a corresponding even number. This pairing suggests the two sets – both infinite – are the same size. They are ‘countably infinite’.

However, Cantor showed that you can’t do the same with the natural numbers and the ‘real’ numbers – the continuum of numbers with decimal places between 1, 2, 3, 4 (0.123, 0.1234, 0.12345 and so on.)

If you attempted to pair up the numbers within each set, you could always find a real number that does not match up with a natural number. Real numbers are uncountably infinite. Therefore, there must be multiple sizes of infinity.

Yay!  A nice explanation of the different sizes of infinity.  The explanation about ChatGPT also states, “For the sake of clarity, BBC Future does not use generative AI as a primary source or to replace the journalism needed for our articles.“

Indeed – as useful as Artificial Intelligence can be, it doesn’t replace the need for understanding and evaluating what it generates.

Monday Morning Math: 𝝅

Tomorrow, March 14, is 3/14 [in month/year form] and so we celebrate Pi Day!  And since I’m teaching History of Math for the first time in rather a lot of years, I’m thinking that the perfect topic is the history of the symbol.

But first, what is 𝝅?  The idea is that no matter how big or small a circle, the circumference is always a little more than 3 times as large as the diameter:  that ratio is about 3.14 in decimal terms.  Because it’s a ratio, the first symbols were also written as ratios:  William Oughtred called it 𝝅/δ in 1647, and while he didn’t explain what either of the terms meant, since 𝝅 is the Greek p it likely stood for periphery (according to my source, though I think of perimeter myself when I see it); likewise, δ is the Greek d and likely stood for diameter.  Several other mathematicians adopted this notation.

The first person to use a single symbol to represent this ratio was Johann Christoph Sturm, who in 1689 referred to it with the letter e.  Wait, what?  (Double check.)  Well that is something I didn’t know before.  Cool!  But using e didn’t catch on, and less than twenty years later, in 1706, William Jones used the symbol 𝝅 for this same ratio. No explanation as to why, and also no consistency – he used the same symbol to mean other things earlier in the same book.  This use of 𝝅 also didn’t catch on: other mathematicians continued to use other symbols for the ratio of a circumference to the diameter, and 𝝅 itself continued to be used for different mathematical numbers.  But eventually, over the 1700s, its use caught on and so we have the well known symbol today.

Source:  A History of Mathematical Notations by Florian Cajori

Sofia Kovalevskya

Good morning!  In honor of Women’s History Month we are featuring Sonia Kovalevskya, the first woman to receive a (modern) doctorate in mathematics.

Sofia Vasilyevna Korvin-Krukovskaya was born on January 3, 1850 in Moscow, Russia, although her birthday is sometimes listed as January 15 (the equivalent date in the Gregorian Calendar, which Russia adopted in 1918).  Her name, too, is written many ways: Sophie, Sofia, Sofya, Sonia.  She grew up literally surrounded by mathematics: the walls of her room were covered in her dad’s Calculus notes from when he was a student. She wrote later:

The meaning of these concepts I naturally could not yet grasp, but they acted on my imagination, instilling in me a reverence for mathematics as an exalted and mysterious science which opens up to its initiates a new world of wonders, inaccessible to ordinary mortals.

Her parents were well off and she and her siblings had private tutors, but Sofia liked math so much that she ignored her other subjects and her dad put stop to the math studying.  Or tried to, at least – she studied on the sly after her parents had gone to bed.

Sofie wasn’t able to go to university in Russia, what with being female and all, so she married Vladimir Kowalevski (aka Kovalevskij or Kovalevsky) and they moved first to Austria and then to Germany where, because of her continuing femaleness, she still couldn’t take classes. She was, however, able to take private lessons from the mathematician Karl Weierstrass. She wrote several papers and with Weierstrass’s support [and influence at the University of Göttingen] was granted a well-earned doctorate. 

Sophie had a daughter Sofia (who was called Fufa), moved multiple times, and after the death of her husband, became a professor at the University of Stockholm in Sweden. She wrote mathematical papers, non-mathematical works, and was recognized for her contributions even in her lifetime.  Short as it was – she was only 41 years old when she passed away from pneumonia, a complication of the flu.  Many schools have hosted Sonia Kovalevsky Days in her honor, bringing her love of mathematics to new generations.

Sources:

• The Potential to Inspire by Laura P. Schaposnick (written in verse in both English and Spanish)
• Wikipedia
• MacTutor, which itself referenced her autobiography: A Russian childhood: Sofya Kovalevskaya
• SK Days at the Association for Women in Mathematics