Four fingers on a hand and four points on the compass: from the analytic aspect there are two 4s here, because it differentiates between fingers and compass, but from the quantitative aspect there is only one 4. It is not so much number that exists in the quantitative aspect, as 'numberbess', e.g. *4-ness*. Functioning in the quantitative (hand 'has' four fingers) aspect feels, not like the dynamic agency found in most aspects, but more like possessing an attribute or property, or 'being' of a certain amount; dynamism enters with the kinematic aspect.

The fundamental 'good' possibility that the quantitative aspect introduces is **reliable amount and order**. Quantitative amount is reliable because each amount, other than infinity, always and in all situations retains the same quantitative meaning, different from all others. Order is reliable; for example 4-ness is always after 3.9-ness and before 4.1-ness. This is so fundamental that we usually take it for granted, yet functioning in all other aspects relies on this.

- Discrete amount (Dooyeweerd's kernel)
- "One, several many; more and less. Introduces Reliable amount" (Basden's intuitive rendering)
- Quantity
- Countability
- Think of this as not so much 7 as 7-ness, not so much 283756y365 as 283756y365-ness, and you'll get close to grasping what the kernel is.
- 'Numberness' (rather than 'numbers')

rather than:

- Number expressed as digits, which is Lingual
- Continuous number, which Dooyeweerd sees as within the spatial aspect.
- It is not even number as such, so much as 'amount', of which an idea may be grasped intuitively; see below.
- The act of discretization, which Dooyeweerd sees as within the analytical aspect.

- Progression. Which gives infinite series.
- Irrational numbers are, Dooyeweerd would argue, not genuine numbers but rather an anticipation of the spatial aspect. Our tendency to assume they are numbers of equal status with rational numbers comes from some arithmeticians having merged the subject side (numbers) into the law-side (arithmetical laws).
- Maximization, minimization of numeric attributes.
- Numeric comparison
- Numeric operations, functions and algebra
- Discreteness of quantity differs from 'distinction' as found to be the kernel of the analytical aspect.

But unity-and-multiplicity seems different. I searched throughout Dooyeweerd's New Critique, all four volumes for "multiplicity" and discovered that it is often used in the context of analytical counting rather than of quantity as such.

The study: I divided the 144 occurrences of "multiplicity" in NC,I-IV into those that accompanied "unity" (69) from those that did not (75). In the former, "logical" was found 65 times. In the latter, "logical" was found only 20 times. In the occurrences without "unity", "multiplicity of" was mostly merely saying "many" of some kind of thing, e.g. of historical forms. Ergo, it seems that "unity-multiplicity" is about analytical, not pure quantitative.

- That quantity can be continuous; continuity is of the spatial aspect; see below.

- Infinity. Infinity is a number where "X-ness is never Y-ness" breaks down.
- Randomness of the occurrence of prime numbers. From the reliability one might expect laws that determine where each numberness occurs, for example evenness is every-other-integerness. But the distribution of prime numbers seems completely random, i.e. without law. No way has been found yet of reliably predicting the next prime number.

- Arithmetic
- Statistics
- Note that Mathematics, as it is practised as a whole, includes much more than quantity in mathematics, since it is an activity that involves e.g. analysis and distinction, symbolic communication, etc.

- That we can rely on a quantity always being that quantity until it is acted upon. 7-ness does not suddenly become 6-ness.

- National targets - as the U.K. government seems keen on. It was reported today that, because the government has given general practitioners a target of a waiting list of no greater than 48 hours, some surgeries are refusing to book appointments more than 48 hours in advance. We can see this as an elevation of the quantitative aspect over those of health (biotic) or care (ethical). (Of course, targets also involve achievement, which is of the formative aspect.)

"Number theory is the elaboration of the properties of all structures of the order type of the numbers. The number words to not have single referents." Paul Benacerraff in What Numbers Could Not Be. (Philosophical Review, 74, 1965:47-73)Benacerraf argues so elegantly--from a PCI viewpoint about the breakdown of linguistic and logical concepts posing as the numerical. The relationship between truth, identity and reference are in such tension in analytical texts. Just something I thought to share for whatever it is worth.

In the other direction, all laws have some dependency on those of this aspect. Indeed, that seems to be what we find.

- Size is spatial amount.
- Speed is perhaps kinematic amount.
- Music is what convinced the Pythagoreans that that the universe ran on numbers - some link with the aesthetic aspect? Probably an analogical rather than dependency link.
- The computer's ability to perform arithmetic by using binary logic might be seen as a retrocipation from the logical (analytical) aspect to the quantitative. But this needs checking out more precisely.
- Bergson saw duration as the unfolding of number in action.
- etc.

Analogies of other aspect in the quantitative might include:

- That each number (greater than 1) may be said to be simultaneous with all the factors that make it up (whether multiplicative or additive) might be an analogy of the spatial aspect. It is not simultaneity in the way we find in the spatial aspect, being non-necessary. Hence Goldbach's conjecture, that every even number is the sum of two primes.

First, we have simple discrete quantity, which counts things; this however is targeting rather than full anticipation.

Second, we have ratios of these ('rational numbers') from dividing groups of things into parts. So far, only in the quantitative aspect. And only positive (or non-negative) rationals.

Then we look at the spatial aspect, and look at a right angle triangle with sides of one unit, and find that the hypotenuse has a length whose amount is not a rational number, that is cannot be constructed as a ratio. So, by looking at the spatial aspect, we discover a new type of amount in the quantitative aspect, namely irrationals. Square roots are often of this kind.

Next, look at the kinematic aspect, and we encounter movement. Now, let us 'move' among the numbers themselves, and we find that as well as moving to larger numbers we can move to smaller ones, and move through zero to negative numbers. We also find a special type of number, the imaginary or complex number, which is the square root of a negative number. So, by moving up to the kinematic aspect, we discover yet another type of number.

And so on. So, different things in the quantitative aspect seem to anticipate different things in later aspects, and we can determine which aspect by asking "Which aspect makes this meaningful rather than merely an academic curiosity of mathematics?":

- Integers and rationals: stay in this aspect. As do the notions of 'more' and 'less'.
- Irrationals: the spatial aspect [NC, II:185]; Approximation [NC, II:185]
- Negative numbers (as movement), complex numbers [NC, II:170,172], and differentiation [NC, II:185]: the kinematic aspect.
- Fibonacci series anticipates biotic.
- Logarithms: the sensitive aspect (e.g. decibels for sound level, piano octaves, perception of brightness) (suggested by the late Arie Dirkzwaager)
- Set theory anticipates the analytical aspect in which we distinguish individuals; see below. Also enumerated numbers, used to identify elements in a list.
- (I believe, against Dooyeweerd, that) numerical order anticipates the formative aspect, because without the latter there is no reason to place them in order; see below.
- Double-entry book-keeping anticipates the economic aspect.

See this in fuller, tabular form to illustrate the notion of anticipation as a whole.

The root of the problems is in using numbers as legal limits, which is a type of reduction. Though it might not be as severe a type of reduction as others, it nevertheless leads to problems. What is happening is that the real differentiator of legal from illegal has been transducted into a numerical measure and limit. For instance, the real problem of driving under the influence of alcohol is one of irresponsible, selfish behaviour and also of dulled responses when in charge of a powerful, dangerous artifact. For instance (as some, including myself, would argue) the differentiator of sexual activity should be a serious, volitional act of commitment to the other person (called 'marriage'), rather than an arbitrary age limit.

The problem is that transducing something to number might be convenient, and might give the appearance of precision, but it fundamentally misses the real point and purpose. And when we start to **rely** on such a transduction then we have a **reduction**.

I have recently been discussing integer and 'real' (continuous) numbers with a mathematician. The discussion centred on two types of infinity. That related to real numbers is larger than the infinity related to integers! Due to the continuous nature of spatial numbers (continuous extension). So, he was saying, 'reals' are a fundamentally different kind of number, and require a different kind of mathematics. This recognition of a fundamental difference is indicative of crossing an aspectual boundary. So, since the main use of reals is to cope with spatial factors, indirectly if not directly, then it might seem that they are attached more to the spatial aspect.

Andrew Hartley sent me the following expansion on this theme:

"I went back to some of your Dooy web pages and felt I must sometime soon come to grips with Dooy's idea that the real numbers belong to the spatial mode and not the quantitative one. ... I have an idea that it is all related to the concept of "countability," e.g., as in the language of Nancy McGough who said 'In 1874 Georg Cantor discovered that there is more than one level of infinity. The lowest level is calledcountable infinityand higher levels are calleduncountable infinities. The natural numbers are an example of a countably infinite set and the real numbers are an example of an uncountably infinite set. In 1877 Cantor hypothesized that the number of real numbers is the next level of infinity above countable infinity. Since the real numbers are used to represent a linear continuum, this hypothesis is calledthe Continuum Hypothesisor CH.'"

It is interesting to find that Dooyeweerd, a lawyer, understood this deep mathematical idea, and saw the kernel of the spatial aspect as continuous extension rather than shape, position, distance, curvature, or whatever.

**AM**: The numerical dimension we term as related to amount with the kernel numeric value. The modality is visible in anything that can be expressed in numbers and that refers to amount and/or quantity, e.g. number of produced goods, number of rooms in a house (but not the size of the rooms which refers to spatial), number of walls, number of articles one buys in a shopping situation, amount of money to pay for the articles etc.

**AB**: Interestingly, I have recently been discussing integer and 'real' (continuous) numbers with a mathematician. I used to think that real numbers were part of the quantitative modality, but have now changed my mind. While integers and rationals are part of the quantitative modality, real numbers as part of the spatial. The discussion centred on two types of infinity. That related to real numbers is larger than the infinity realted to integers! Due to the continuous nature of spatial numbers (continuous extension). It is interesting to find that Dooy understood this deep mathematical idea.

**AM**: I have problems to understand this modality and have to tell myself that this is the definition I will adopt. I see numbers as symbolic representations (lingual) and therefore the numeric dimension becomes very confusing. I have not been able to differentiate between numbers as 'numeric value' and as symbols for language.

**AB**: Yes, you will find it confusing if you cannot grasp the concept of number without its conventional symbolic forms. To help me I take two steps.

- 1. Recognise I am dealing with 'amount' rather than the cardinality of a set (number of objects in a set).
- 2. Think of 'amount' as on the amount of sand I have in my hand. (Or sugar, anything with grains, if you prefer.)

So, what we have is the raw, intuitive appreciation of number-as- amount as opposed to the conceptualisation of number-as-distinct-symbol.

This is part of The Dooyeweerd Pages, which explain, explore and discuss Dooyeweerd's interesting philosophy. Questions or comments would be welcome.

Copyright (c) 2004 Andrew Basden. But you may use this material subject to conditions.

Number of visitors to these pages: . Written on the Amiga with Protext.

Created: by 1 December 1997. Last updated: 1 July 1998 added re. reduction to legal limits. 30 August 1998 rearranged and tidied. 19 April 1999 Added anticipating other aspects. 28 June 1999 added retrocipation from logical. 7 February 2001 new contact and copyright. 8 February 2001 added bits strengthening our understanding of the aspect, after reading NC II:90-94 and thereabouts. 14 February 2001 counter. 3 January 2003 added Andrew Hartley's bit about Cantor and countability. 29 April 2005 targets as harm; .nav. 16 May 2005 incorporated several anticipations, and rewrote some bits at start. 18 May 2005 bit more on logarithms. 25 May 2005 link to anticipation table. 24 August 2005 nav to aspects. 22 September 2010 Dooyeweerd's and Basden's kernel. 13 January 2011 absoluteness and reliability. 12 April 2012 number de Wet. 10 October 2013 Bergson. 21 September 2016 briefly. 2 January 2021 analogy of simultaneity. 19 March 2021 Unity and Multiplicity. 29 May 2021 more on that; deification of qtv.