The testimony of numbers is a dubious witness. We trust numbers at our peril. All numbers outside of the formal simplifications of mathematics...being the supposed consequence of a temporal counting process...are subject to error and the bias of the inclusion criteria. To begin with the numerical result of a simple count takes time and there may be no assurance whatsoever that the entities that were counted at the start of the process will still exist at the end. Try counting bubbles. Also, if a measurement is being made and dimension units being counted, the instrument being used for measuring may be entirely uncalibrated and error prone. Measuring the pressure inside a bubble with a sphygmomanometer won't be too convincing. And as well, inclusion criteria are rarely free of bias. A census of people eligible to vote will include individuals or not according to the rules of the census-takers. The number of faeries identified on an extensive research program will depend upon who thinks what qualifies as a faerie.

Number depends upon the capacity to become aware of ontological entities. To exist as a number is to be the concept of the cumulative total at the completion of a cumulative counting of the elements of a set of entities. Number is a concept resulting from a counting process. The concept of a specific number is that concept which develops as a consequence of attention being drawn to the possibility of grouping certain ontological entities. If the entities are counted as they are classified and a termination condition is realized, then the final accumulation is the number of entities. When beancounters count beans, they arrive at the concept of nine after deciding to group three different varieties of beans under one category of phaseolus.

Numbers do not exist in some absolute space-time continuum. They are not immutable entities. A number is not an absolute reality existing independently of thought but rather a concept which develops due to the consequences of accumulating entities. If during some time interval the mind is not contemplating that number then that number does not exist for that mind. Once it is accepted that the counting process provides the means whereby any number may be realized then it is no longer necessary to ascribe continual existence to all numbers. It is not necessary to ascribe a disembodied existence to red in order to preserve the sensory capacity to see red beans. We see red whenever we perceive red objects. In the same way, we can be aware of the number nine whenever we apply the counting process to nine entities.

A numeral is a distinguishing symbolic sign for a cumulative counting total. A numeral is an entity and is thus countable itself. For the large part it is not numbers that are manipulated but numerals. Unless some sophisticated mental technique has been developed to visualize large accumulations, numbers represented by numerals like 157,345 are unlikely to be contemplated as a number by any conscious mind and as a consequence will not exist. However, the symbolic numeral for the above number now exists and this can be contemplated and manipulated by skilled arithmeticians. It is only when an attempt is made to grasp the existential significance of the numeral, that the intellectual problem of actually conceptualizing the number will be encountered. One suspects that in fact few humans could conceptualize a number like 157,345 unless assisted by the physical presence of 7868 other humans lined up with no more than 15 missing fingers or toes.