“… for truly we live in shapes; and the clocks, with tiny steps, walk alongside our real days…” Rainer Maria Rilke, Sonnets to Orpheus, Part One, No. xii.

These living shapes are the ideas at the heart of mathematics. This is Living Mathematics, which originated in the school of Pythagoras and his wife, Theano, in Greece in the 5 th century BC.

The word, ‘mathematics’ (‘ta mathemata’ in Greek), means ‘those things which have been learned’. This was holistic learning: how to live mathematically, not just think mathematically. It accorded with the Hermetic doctrine, ’As above, so below; as without, so within’. The inherent complementarity of external and internal learning ensured a deep sense of responsibility for the use of knowledge of the external world. This was lost in the centuries that followed.

Mathematics became increasingly abstract. Powerful mathematical ideas came into being. In the absence of personal development and responsibility, these ideas were used to control nature (and people) with increasingly disastrous, unforeseen effects.

The task of Living Mathematics now is to revive those aspects that have been lost, incorporating the advances in technical mathematics that have taken place in the two and a half millennia since Pythagoras’ time. These can be summarised as:

The qualitative, complementing the quantitative;
The spiritual and personal developmental, complementing the material and technical;
The acoustic, musical, complementing the visual.

In this way, mathematics can return to its holistic, Pythagorean nature and help us, human becomings, to heal our relationships with ourselves and this exquisite world which we are blessed to inhabit.

“The real magic of discovery lies not in seeking new landscapes but in having new eyes”  Marcel Proust


You can read more here –

Human Mathematics – New Beginnings, 11pp, of which 2 are images and 2 are Notes.

God be in my Heart and in my Thinking, 20pp.

Who Carved Up the Integers? They Never Died, 89pp,

or the later, revised, arXiv version of this – arXiv:1705.02386  [pdf]  math.HO

Historical Changes in the Concepts of Number, Mathematics and Number Theory

Author: Nicola Graves-Gregory

Abstract: This essay traces the history of three interconnected strands. Firstly, changes in the concept of number, secondly, the study of the qualities of number, which evolved into number theory, and thirdly, the nature of mathematics itself, from early Greek mathematics to the 20th century. These were embedded in philosophical shifts, from the classical Greek ontologies through increasing pragmatism to formalism and logical positivism. Given that Goedel demonstrated the limitations of the latter as a foundation for mathematics, this essay explores phenomenology and Lakatosian ideas, which together offer a more sound epistemological and ontological basis for mathematics and a methodology for mathematical development.The question also then arises of the possible resurrection of earlier, neglected mathematical projects, including widening the domain of number theory to include integer qualities revealed in the growth of mathematics in general, which has predominantly been the growth of quantitative mathematics or logistike, the complement of arithmetike in classical Greece.