Ultrafinitism

Karin Koller

“Infinity may or may not exist,
God may or may not exist, 
but there is no need for either
in mathematics” *

I do not claim to understand 
the half of it, 
or the quarter, 
or the eighth, 
or, indeed, the Nth value
taken to infinity 
or possibly beyond,  
to be returned
proven, done, dusted,
pinned down
to our earthly ground. 

But what I do claim
is respect for the 
minds of rebels
who compute, dispute
refute the use of 
endless tiny steps
which blur together –
the adding up of infinite
numbers of ‘almost nothings’
to the point where problems
appear to disappear. 

*quote by mathematician Doron Zeilberger at the Ultrafinitists Conference, Columbia University, New York, April 2025 as reported by physicist Abigail James, 3 December 2025 Mathematicians Discover a Strange New Infinity 

The Science

There are three types of infinity: the physical (e.g. is the universe finite or infinite?), the metaphysical (e.g. does God exist?) and the mathematical (symbolised by ∞). In mathematics the concept of infinity has been extremely useful, and includes the discovery of different hierarchies of mathematical infinities (e.g. countable and uncountable). For thousands of years it has been used almost like a normal number. However, in the 1960s, a breakaway group of rebels (the Ultrafinitists) claimed the use of infinity was undermining the foundations of mathematics, and infinity should be done away with altogether. They urged that mathematics should be brought back down to earth. My poem is an attempt to raise curiosity about this intriguing turn of events. For a brief overview of ultrafinitism I can recommend Sanjay Banu’s article Banishing Infinity.

The Poet

Karin Koller is a UK-based retired scientist with a long-standing interest in the use of poetry to communicate about science. Her PhD was in biophysics. She had an inspiring maths teacher at A-level, who explained how equations in calculus could be taken to infinity to prove their universal truth. She currently runs an open group for poetry enthusiasts in Oxfordshire.

Next poem: Uncountable by Robert Emmet