Temporal freedom in mathematical thought: A philosophical–empirical enquiry

• The paper outlines the difference between temporality as lived human experience and ‘clock-time’.
• The paper considers high-school students’ explorations of the symmetry group of the square.
• The paper argues that the notion of temporal freedom is key in our conceptualisation of learning mathematics.

This paper seeks to introduce and elaborate on the notion of ‘temporal freedom’ in the context of mathematical thinking.The intention is to merge a philosophical and empirical discussion on this topic, and findings will be presented from a qualitative study in which 29 high-school students were invited to explore the symmetry group of a square during clinical interviews. The data indicate how the interviewees – who were given the opportunity to freely explore an unseen mathematical system – repeatedly brought past and future together in a tangible present during their investigations. In doing so they were exceptionally successful in engaging with advanced abstract algebra, far beyond what might have been expected at the outset.By using empirical data to exemplify an otherwise highly philosophical discussion, I hope to draw explicit attention to the temporal aspect of the human mind in mathematical action which instructional methods frequently fail to recognise, value and utilise.