The Quantitative Aspect
- Discrete amount (Dooyeweerd's kernel)
- "One, several many; more and less. Introduces Reliable amount" (Basden's intuitive rendering)
- 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')
- 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.
Note that probably this aspect covers integral and rational (ratio) numbers but not reals (continuous).
- 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.
Absoluteness has to do with reliability. We can rely on that which is absolute. One 'good' that this aspect introduces into created reality is reliability. For example, sevenness is always sevenness, never sixness, and this can be relied upon in all places and throughout all time. The sum of sevenness and sixness will always be thirteenness, never fifteenness. So the quantitative aspect seems absolutely reliable. For example, all physical functioning relies on this, whether quantum or macroscopic. But there seem to be at least two types of non-absoluteness [someone more knowledgeable than I needs to check them].
- 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.
- 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.
Pieter de Wet emailed (12 April 2012) the following:
- 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.
None - this is the earliest aspect, so its laws do not depend on those of any other aspect. However, as with all reality, they depend on the Living God, who upholds and sustains all creation.
In the other direction, all laws have some dependency on those of this aspect. Indeed, that seems to be what we find.
Sunflower seeds are arranged as a Fibonacci series. So are many other biological things. This is evidence for Dooyeweerd's concept that e.g. biotic laws depend on mathematical as an earlier aspect. We can see at least one reason for this (explained to me by an eminent mathematician whose name I cannot remember): it provides the best packing arrangement, with least waste of space. (Note the hint of a link with economic
We practise many analogies to the quantitative modality. Whenever we use a metric we are transducing to number, and thus making use of the in-built potential for analogy between each aspect and the quantitative. Here are some examples.
(Thanks to Arie Dirkzwager for making this clear.)
- 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.
First, we have simple discrete quantity, which counts things. 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.
Current finance and commerce and accountancy seems often to be reduced to this aspect. While keeping within a budget is truly of the economic whose kernel is frugality, the emphasis in business today is more in a simple numerical maximization e.g. of profits or efficiency and a minimization of costs. No idea of a limit here; it purely numeric.
In the same way, in many fields there is a desire or tendency to
reduce all things to numerical measures or metrics. This, it seems, is the only way we think we can make decisions. But such reduction (teleological) is harmful, as many now realise.
Many laws stipulate a numerical limit to separate legal from illegal activities. A common one is an age limit. These limits cause no end of trouble. For instance, just recently in the U.K. Parliament has debated whether the age at which consenting adults can engage privately in homosexual activity should be reduced from 18 to 16, to bring it in line with the heterosexual limit, on the grounds that it is unfair to differentiate. For instance, often we hear it argued that persons aged 16 years 1 day are allowed to do things that others only a few days younger cannot; the limit of 16 seems arbitrary. Especially when the chronologically younger person is actually more mature in various ways. There are many similar issues, such as limit of alcohol in blood.
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.
Dooyeweerd clearly states that the quantitative aspect has as its kernel discrete quantity (see his lengthy arguments in NC II:79-95ff). He places continuity within the spatial aspect, whose kernel is 'continuous extension', with a strong retrocipation to this aspect. For example, he argues that irrational numbers are not in fact true numbers (but rather functions). I found difficulty in accepting this, having been brought up to see number as essentially continuous. But I have now changed my mind.
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
e.g., as in the language of
Nancy McGough who said
Georg Cantor discovered that there is more than one level of infinity. The lowest level is called countable infinity and higher levels are called uncountable 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 called the Continuum Hypothesis or 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.
Why Numberic Order may be Formative
The usual reasons given for numeric order being an original quantitative thing are that if we have three amounts, A, B, C, such that A < B and B < C, then it is obvious that we should place the second after the first to get the order A, B, C. But I argue that, though in the quantitative aspect we have amounts and amount-comparisons, placing the second after the first to get the order A, B, C presupposes a reason why we should place the second after the first. So we deliberately employ formative power to choose to do so: formative aspect. This may be made clearer if we replace the second comparison by 'C > B' (which has the same meaning in the quantitative aspect).
Why Sets are Analytical
In the quantitative aspect we have discrete quantity (amount) which implies that we have counted things (e.g. people in a queue, grains of sugar in a pile, stars in a galaxy, air molecules in a room), but this aspect does not imply that we distinguish the items that are counted, one grain of sugar from another. To make such a distinction, in which the counted things become meaningful individuals to us implies making distinction, which is the kernel of the analytical aspect. In set theory, we make such distinctions. Therefore, in basing mathematics wholly on set theory, Russell and Whitehead were reducing the quantitative aspect tot he analytical.
(A discussion I had with AM over email. Retained exact wording
and spellings, but added html codes, refces, etc.)
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.
This helps in two ways.
First, when I hold sand in my hand I can immediately conceive of 'amount' without specific number. But, then when I ponder it I realise that it does also have a specific number that is at the basis of my intuitive conception. Namely, the number of grains of sand. It is just that thinking in this way moves me away from the confusion wrought by our tendency to separate numbers from one another, which is what symbolic representation of them as digits does. (NB. 'separation' is 'distinction', namely analytic modality .}
- 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.