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# Mathematical images of time

Introduction

The human sense of time is qualitative: we have the feeling that time ‘flows’ continuously. In contrast, real events such as birth, death, the daily rising and setting of the Sun, the annual cycle of seasons, the monthly phases of the Moon, and other memorable phenomena, all of these punctuate that continuity. Memory and the needs of society to record and predict such events undoubtedly led humans to develop a quantitative and discrete description of time using counting numbers (integers). We still count years, days, hours, and minutes in those terms.

Integers are of fundamental importance to mathematicians. They are regarded as the basis of mathematics by constructivists and intuitionists, those mathematicians who seek a ‘natural’ basis to mathematics. A fundamental constraint on all mathematicians regardless of their persuasion is that every aspect of mathematics is based on the taking of discrete, countable steps: no theorem is proved ‘continuously’ but step by step, line by line, even though mathematicians think they themselves are operating in continuous time. When Cantor developed his controversial theory of transfinite numbers, he was constrained to take countable numbers of steps, using discrete lists (ordered sets) of numbers in his ‘diagonal’ method; when Alan Turing [1912-54] began to theorize about and then build his ‘machines’, they were designed on discrete principles that eventually led to the modern computer [Hodges, 2014].12

Discreteness is also the basis of the physical world despite our subjective feeling to the contrary. Quantum phenomena underpin physical reality and are predicated on discreteness. When any experiment is analysed realistically, in terms of what observers actually do in the laboratory rather than what is imagined they do, information, the basis of everything to do with time, always comes in discrete amounts. It took humans a very long time to come to that perspective however: the atomic hypothesis was finally settled only just over a hundred years ago.

The language of integers penetrates all aspects of human existence, even to the extent of our mental processes: a thought is a discretization, an objectivization of

12 Before the Second World War started, Turing was seen making metal cogs in his Cambridge College rooms: cogs with teeth are the embodiment of discreteness [Hodges, 2014].

Images of Time. First Edition. George Jaroszkiewicz.

© George Jaroszkiewicz 2016. Published in 2016 by Oxford University Press.

unimaginably complex processes going on inside our heads. A decision is a definite, recognized end to a set of thoughts. We count events, thoughts, and decisions with integers: thoughts without conclusion lead to indecision.

We saw in Chapter 4 that there is evidence for Paleolithic calendars dating as far back as 30,000 years ago [Pasztor, 2011]. This proves that Stone Age humans were dealing with integers even if they did not appreciate that fact. As humans became more sophisticated, they began to fill in the mathematical gaps between the integers: halves, thirds, and so on, until finally they invented[1] the continuum of real numbers, denoted by R. Currently, the dominant mathematical model of time in science is based on R, which has all the properties required to model a continuum. We shall discuss continua in more detail further on in this chapter. Other mathematical representations of time have been invoked besides the reals: we shall mention some of them also.

• [1] Or discovered, according to the Platonists.

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