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.
rather than:
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.
sqrt(2)
) actually exist, or can they only be approximated? (Note: I use the word "number" for them since that is customary, but strictly I should use "quantity", to rid us of any connotation that "number" has of conceptuality; we mean quantity without conceptualization of it.)
It does of course depend on what we mean by "exist" (see page on Existence). Let us take the following view. We must begin with a view of existence that allows numbers to exist as numbers. Does it require that humans have seen or thought about it (as in the tree falling unheard)? No; that makes existence totally dependent on human psychical functioning, and would rule out the numbers that humans have never thought about. Does it require physicality? No; that would immediately rule out even asking whether numbers exist. Existence requires meaningfulness in reference to at least one modality of meaning (aspect) - in our case, the quantitative aspect.
By reference to the quantitative aspect, the most basics are amount and the functionings (or relationships) of more and less (and equal). So quantities like 5, 6, 7, 9779273 exist by virtue of the quantitative aspect, and we can apply more and less to them meaningfully (without resorting to analogy). So, such integers exist - and express countability. History shows gradual widening of what quantities can exist beyond countability - to fractions between 6 and 7 (such as those we denote by decimal points like 6.5, 6.25), zero, negative numbers. All these - integers, fractions, zero and negatives - can be discovered by the application of the laws of arithmetic, which include addition and subtraction and things like multiplication and division that can be derived from them, and then things like primeness, which can be derived from these.
But irrational numbers can never be discovered by these means. They can be approximated more and more closely, but must remain little more than a fascinating curiosity of imagination or speculation. For example, for the square root of 2, we can say it is more than 1 but less than 2, then we can say it is more than 1.4 but less than 1.5, then more than 1.41 but less than 1.42, and so on. But we never find the precise square root of two however many decimal places we add. So does it exist?
We can take two answers to this question:
Which we choose is never given by the rationality of the quantitative aspect, but by pistic functioning of belief and commitment. Having acknowledged this, I tend to choose the Yes, because it is the option based solely on quantitative meaningfulness or functioning. The No option imports the idea of discovery, which is (at least) analytical functioning. The fact that we cannot find something, though it could mean it does not exist, it does not necessarily do so.
If we bring in meaningfulness from the spatial aspect, then we discover what the meaningfulness of the square root of two can be. Without importing spatial meaningfulness. (We can find similar importation of meaningfulness from later aspects in other aspects, for example when trying to differentiate justice with and without mercy.)
That is different from impossible numbers, which are those that break the laws of the quantitative aspect, such as a-quantity-that-is-both-less-and-more-than-6. sqrt(2)
does not break any laws of the quantitative aspect.
But imaginary numbers like the square root of minus one (sqrt(-1)
), might break these laws, hence might be impossible and a mere fiction arising from our analytical functioning with curiosity. There is a law of the quantitative aspect that since if we multiply any quantity by itself (square it) we always get a positive quantity, so sqrt(-1)
breaks this law and therefore is impossible. However, it may be that this law is not be as fundamental as we suppose. It is not as fundamental as the laws of addition and multiplication are, but arises from the law of multiplication together with the lack of discovery of any quantity whose square is negative. Where it might be impossible might depend on whether it is meaningful to say sqrt(-1)
is more or less than numbers we know exist. It is meaningless to compare it with 0.9, 1.0, 1.1. But it operates in a second dimension of quantity-order. I know imaginary numbers are useful in conceptual and theoretical thinking if we presuppose the quantitative meaningfulness sqrt(-1)
, denoted as i
(mathematicians) or j
(engineers), and use it as a multiplier of any 'proper' number, and treat this as obeying quantitative laws, such that for instance 7i
is greater than 6i
. But notice the word "dimension" - revealing that we are hereby importing meaningfulness from the spatial aspect. The debate does not get resolved here, so I come to no conclusion about whether imaginary numbers exist or not, except that I know how to treat them as though they do.
On this basis, I choose to believe that irrational numbers exist as genuine quantities that can never be discovered by processes purely quantitative, but that imaginary numbers might or might not exist. For transcendental numbers like pi
and e
I leave the reader to work it out.
"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.
Much more to be added here.
In the other direction, all laws have some dependency on those of this aspect. Indeed, that seems to be what we find.
Analogies of other aspect in the quantitative might include:
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?":
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 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.
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.
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. 31 Jan 2025 canon, bgc. 23 July 2025 irrational numbers; disclaimer.