Infinity – When too much is never enough, Dose 1 – Teaching Mathematics: What, When and Why

June 5, 2026

Infinity – When too much is never enough, Dose 1

Much of modern mathematics is the result of centuries of wrestling with infinity. Even in school mathematics, we find the concept pops up here and there. Each time it points to some interesting part of mathematics. This is a guided tour of some of these occurrences. As usual I focus the ideas through my personal experience.

It’s a large topic, so I will make several instalments. Each begins with school mathematics, then will point into some part of the mathematics that you meet at university. These latter parts are too complex to explain in detail. I hope the reader will be content with sniffs of some of the infinitely many sub-worlds of mathematics.

Uncategorized

Posted by:

tom

11 responses to “Infinity – When too much is never enough, Dose 1”

Tom’s First Dose of Infinity – BAD MATHEMATICS

June 6, 2026 at 8:46 am

[…] Peachey, who is apparently swanning it up in Thailand, has alerted me to his new post on his Teaching Mathematics blog, on infinity and Winnie the Pooh, and indefinable numbers. Please […]

LikeLike

Reply
ΣίΓΜα

June 6, 2026 at 9:31 am

I love that you made it to the “undefinables”, a much underappreciated set of numbers IMHO.

LikeLike

Reply
1. tom
  
  June 6, 2026 at 11:53 am
  
  Thanks Sigma. In all seriousness, I hesitated to release the post, as I was not sure whether the set of definables has measure zero. But then convinced myself that the same argument that shows that the rationals have that property, could be modified to work for definables.
  
  LikeLike
  
  Reply
Potii

June 6, 2026 at 3:42 pm

Awesome read. Thanks Tom.

LikeLike

Reply
1. tom
  
  June 6, 2026 at 6:13 pm
  
  Hi Potii. Perhaps I should make my posts more contraversial to get a discussion going …
  
  LikeLike
  
  Reply
  1. Potii
    
    June 24, 2026 at 10:06 am
    
    I suppose you did threaten the reader with death if we revealled these mysteries. Though, I assumed just like the Pythagorean story, the mean was about being careful with what you think you know and how you use it – you might just misused it because you actually don’t really know it.
    
    LikeLike
  2. tom
    
    June 25, 2026 at 2:20 pm
    
    That sounds about right, Potii. Then again, perhaps I related that tired story about the Pythagoreans just so I could make a feeble joke at the end.
    
    LikeLike
Jon

June 6, 2026 at 8:42 pm

Very nice indeed – good for extension kids (and others). The bit that I only recently learned was that set theory, the basis of the formalisation of maths as I understand, arose from Cantor’s infinity stuff. That filled in the gap as to why we studied sets and I think is useful for kids to realise – hope I’ve got this right!

LikeLike

Reply
1. tom
  
  June 7, 2026 at 12:13 am
  
  Hi Jon
  
  All you say is true. But as an obligate contrarian I am opposed to any set theory at school level, except maybe for those few able to enjoy abstraction. Running the engineering syllabuses at Swinburne in the 80s I saw no need for set theory. I recall teaching complex variable to 5th year engineering students, where I mentioned the word “domain” which actually means something different at that level. I saw the class scratching an ancient memory. “What was all that domain and range stuff?” Good question.
  
  LikeLike
  
  Reply
  1. Jon
    
    June 7, 2026 at 9:27 am
    
    Very interesting indeed- I hadn’t thought of that Tom. When was set theory introduced to high school maths?
    
    LikeLike
  2. tom
    
    June 7, 2026 at 11:49 am
    
    I’m guessing late sixties. It was 1966, I was a temporary teacher, filling in for an absentee at Swinburne, while waiting to hear if I had a position as a trainee actuary. For light reading on the train home I would pull a maths book down from the staff room shelves. On one occasion it was a tiny book published by the Open University. In a few pages I discovered that functions can only map to single values, that there was this obsession with defining things called domain and range, and just before alighting at the 4th stop, that with enough conditions, the inverse function existed.
    
    The next day I mentioned to a colleague, Henry Yeo, that I had been reading some “New Maths”. He related how the head of department had asked him to give an MAV course to teachers. This head wanted him to make the subject relevant to school children. He said he found that impossible and just taught the abstract beginnings, the stuff we see in schools now. I have spent the rest of my life looking for that relevant approach.
    
    BTW: As you probably know, in advanced function theory, we allow functions to be multivalued, just as I was taught in school. This is needed to access Riemann’s beautiful theory of analytic continuation. And in point set topology, without apology, the books let $f^{-1}$ to be multivalued, or to be an empty set. I can assure you that maths is easier in that framework.
    
    LikeLike