Watsonia schlechteri L.Bolus grows in the montane veld of the winter-rainfall parts of the Cape, South Africa. Plants grown from seed imported via Silverhill Seeds first flowered this year and are a good match for the lectotype of W. schlechteri.
The flowers are orange-vermillion with perianth lobes to 23 mm long. The buds end in a slightly downcurved point. There are no staminodal ridges in the perianth tube, a character that distinguishes it from the closely related W. pillansii. However, the fresh leaves of these specimens lack the strongly thickened margins and midvein that are used as another distinguishing character; these are only evident in dried material.
W. schlechteri is one of the smaller watsonias, usually much less than 1 metre tall with leaves about 40 cm long. It flowers in late December to January, resuming growth from offsets soon after flowering while the previous season’s shoot may still be green. Thus it has some leaves all year, or non-flowering plants may be briefly leafless before the new growth starts in late summer. Goldblatt notes that flowering in the wild is conditional on the plants not being shaded out by surrounding vegetation.
Like other watsonias native to high altitudes, it is at risk of damage in the agonisingly hot, dry summers we get here at sea level in Adelaide. The problem is to give the plants sufficient light without excess heat, a tall order on days of 42°C with northerly winds.
Goldblatt, P. (1989) The genus Watsonia. 148 pp. (National Botanic Gardens: Kirstenbosch).
A dichotomy is the division of a whole into two parts that are mutually exclusive and jointly exhaustive.
For example, the unity of 1 may be divided into two classes of entities, X and 1 – X, defined by an elective function x. George Boole showed that expanding the function f(x) gives the two constituents x and 1 – x only.
The two classes are mutually exclusive; this means that X (1 – X) = 0, or nothing can simultaneously be both X and not-X. They are also jointly exhaustive, X + (1 – X) = 1, as everything must be either X or not-X.
If the universe of discourse is limited to “birds” for the sake of a simple example, then x might be the definition of the group of bird species known as penguins. Then the class X contains all penguins, and the class 1 – X contains all the other birds that are not penguins. Note that one side of a true dichotomy always has a privative definition, it is defined only by what it is not.
A language such as English has many spurious dichotomies, pairs of words that are “antonyms” in the broad sense as offered by a thesaurus but do not satisfy the two equations above.
To define a dichotomy or decide whether it is a true or spurious one, it is necessary to first define the universe of discourse. For practical purposes, the universe of discourse is the real physical universe or a subset of it.
The two ends of a gradient such as pure black to pure white are not a true dichotomy in the real world even though black and white are mutually exclusive. But they are not jointly exhaustive because most of that gradient is neither black nor white, but grey. Not to mention all the other colours. Of course, in some hypothetical universe that consisted only of pure white and pure black things, black/white would be a true dichotomy.
Opposed pairs of concepts such as ‘freedom’ and ‘slavery’ are not true dichotomies either, except in a hypothetical universe where they are the only two things that exist so that x = (1 – w) and w = (1 – x). In the real world, their mathematical relationship can be expressed by relational statements of the form x ⇒ (1 – w); a slave is not free, but it does not necessarily follow that everyone who is not a slave is free.
On the other hand, direct opposites such as ‘known’ and ‘not known’ or ‘penguin’ and ‘not penguin’ are true dichotomies, as they are mutually exclusive and jointly exhaustive.
If antonyms are pairs of words with exactly opposite meanings, then the precise antonym of any word is formed by simply adding the prefix ”not”. This may seem trivial, but any attempt to invent more sophisticated antonyms leads to imprecision and confusion.
For example, the antonym of ‘accept’ is ‘not accept’. A thesaurus may suggest ‘reject’ as an antonym, but that is not an exact antonym, just an approximation. The pair accept/reject are more like the gradient of black to white because they are separated by a grey zone of neither accepting nor rejecting. This is because the concept of rejection is within not-acceptance, it is a narrower concept because there are many ways of not accepting something without actually rejecting it. Again it is a mathematical relation of the form x ⇒ (1 – w).
Sets of entities, of any kind, can be linked in logic by bonding postulates of the form A ⇒ B (meaning that A is a subset of B, implies B, is within B). The same statement can be written in reverse as B ⇐ A (meaning that B is a superset of A, is implied by A, includes A). A is called the antecedent, and B is the consequent. This does not imply either a causative or a temporal sequence between antecedent and consequent, but simply a logical relationship.
In each case, the set of entities classed as A is completely included within the set of B. A is never found without B although B may occur without A. This situation is described in Boolean algebra as a (1 – b) = 0, or a = ab
Uppercase letters here refer to sets of actual entities, whereas the postulates (in other words, the elective functions or decisions) that define those sets are indicated by the corresponding lowercase letters, following the usage of Boole (1847).
Any relationship that exists between two entities or two postulates can be exposited as a nested hierarchy of A ⇒ B relations. It’s hardly an exaggeration to call this relation the basis of all logical thought.
If A ⇒ B, a pair of conditions holds:
B is necessary for A: A needs B in order to exist, although B can exist without A. eg, water is necessary for plant growth.
A is sufficient for B; the presence of A guarantees B, although B might also exist under alternative conditions not involving A. eg, seeing plants growing is sufficient evidence to assume the presence of water.
Stephens (1994) pointed out that the necessity of B for A and sufficiency of A for B together form a tautology that arises from the way we have circumscribed A and B such that A ⇒ B. For example, if we agree that all dogs are mammals, or dog ⇒ mammal, then being a mammal is one of the necessary qualifications for being a dog, but being a dog is by itself sufficient to qualify an animal as a mammal. This type of tautology is ubiquitous in the systematic classification and naming of plants and animals. Thus species A may be assigned to genus B as one of its members so that A ⇒ B, and that genus is in turn assigned to a family. Thus the classification system of the plant kingdom is a nested hierarchy of A ⇒ B relations with A sufficient for B, and B necessary for A, at each level.
By the same logical process, a taxonomist may assign species M to another species, N, as a synonym if he considers them too similar to merit separate names. A synonymy is an example of what Boole (1854) called an abstract proposition as it is a proposition about species concepts, which are in turn propositions about actual, tangible specimens. Every scientific name of a species refers ultimately to one specimen, known as the type specimen. It will be seen from the paragraphs above that if name ‘M’ is a taxonomic synonym of ‘N’ they cannot be at precisely the same level in the hierarchy: M must be within N as a name applying to a subset of the whole set of individual organisms comprising species N. Therefore two names cannot both be taxonomic synonyms of each other.
The same issue arises with synonyms in ordinary language. There is always an asymmetry in rank, a difference in level between one word and another that is considered to be its synonym. The meaning of the latter must always be a subset within the former. A thesaurus might glibly suggest ‘vehicle’ as a substitute word for ‘car’. But ‘vehicle’ is a more inclusive concept than ‘car’: all cars are vehicles but not all vehicles are cars. Therefore cars are a subset of all vehicles, and the word ‘car’ is within ‘vehicle’ as a synonym.
However, the codes of biological nomenclature were drafted without reference to Boolean algebra. They can add a little confusion since the principle of priority mandates that the earliest-published name be used for the merged species, although this may not be the name associated with the most inclusive set. This arbitrary rule may give the paradoxical impression that a larger M can reside within a smaller N. For instance, many garden plants from China such as the Banksian rose (Rosa banksiae) and the weeping willow (Salix babylonica) were given their botanical names based on the selected horticultural forms first introduced into Europe, but those names must now apply to all wild populations of these species as well.
All the examples above are single bondings where A ⇒ B but not B ⇒ A. This can be expressed in Boolean algebra as a (1 – b) = 0 and b (1 – a) ≠ 0.
However, if A is both necessary and sufficient for B, then B exists if, and only if, A exists. This is the state of equivalence A ⇔ B, meaning that A and B are co-extensive, and either can be called the antecedent or the consequent. This is quite distinct from the taxonomic tautology mentioned above (where the antecedent is necessary for the consequent to be true, and the consequent is sufficient to prove the truth of the antecedent). An example of equivalence would be the relation between the concepts “the 4th of July” and “USA’s Independence Day”; then a statement that “July 4 is Independence Day in the USA” is quite true but adds no new information. If it is agreed that A and B refer to exactly the same things, they may be called nomenclatural synonyms rather than taxonomic synonyms as they differ only in name, not in the sets of entities to which they refer.
On the other hand, a mutual bonding of two non-equivalent entities – that is, A ⇒ B and B ⇒ A where A ≠ B – represents a logical contradiction. They cannot each be contained wholly inside the other if they are different in any way. This is what Stephens (1994) called a double bonding, and may be expressed in Boolean algebra as a (1 – b) = 0 and b (1 – a) = 0.
Therefore every double bonding contains a fallacy. Either one of the bonding postulates is untrue, or they do not both belong to the same logical type in the sense that Whitehead & Russell (1910) used this term, or the same level in the sense of Polanyi (1968).
Boole, G. (1847) The Mathematical Analysis of Logic. (Macmillan: Cambridge).
Boole, G. (1854) An Investigation of the Laws of Thought, on which are Founded the Mathematical Theories of Logic and Probabilities. (Macmillan: London).
Polanyi, M. (1968) Life’s irreducible structure. Science 160: 1308-1312.
Stephens, D.H. (1994) Relationships – Bonding. audio recording of 21 February 1994.
Whitehead, A.N. & Russell, B. (1910) Principia Mathematica. Vol.1 (Cambridge University Press: Cambridge) .
The surest way to mess up someone’s mind is to force them to do something that they already want to do. Or to forcibly prevent them doing something that they don’t want to do. Either way, they have to make an intolerable choice between submitting to force, or acting against their own will.
This is the essence of insanity, this is a double bind. As L. Ron Hubbard said in one of his better moments, fundamental aberration is the enforcement of basic truth.
But the converse is also true. If your mind is doing things outside your control, for example bringing up unwanted images or emotions, the solution is to do deliberately and consciously whatever the mind has been doing automatically. This will bring that automaticity under control so it can be used, or not, as appropriate. If someone is haunted by an unpleasant memory, they need to recall that memory by their own free will until its apparent power over them vanishes.
The rediscovery of Mendel’s principles of heredity at the beginning of the 20th century inspired a surge of ornamental plant breeding by researchers, commercial nurserymen, and perhaps most importantly by individual gardeners.
John Cronin, Director of the Royal Botanic Gardens in Melbourne from 1909 to 1923, had a personal hobby of experimenting with the improvement of garden flowers. He aimed to demonstrate the application of Mendel’s laws to flower breeding, and encourage gardeners to make their own hybrids. He worked with Dahlia and other genera, but particularly the winter-growing South African watsonias, which he recognised as “everyone’s flower” – easy to grow, attractive, a natural for southern Australian gardens.
In a previous publication I lamented that the exact pedigrees of his Watsonia cultivars were lost with the destruction of his papers after his death in 1923. But now the National Library of Australia has come to the rescue with their wonderful resource of newspaper files at Trove. Cronin was a tireless populariser and communicator, speaking at the evening meetings of horticultural societies around the suburbs of Melbourne and giving interviews to journalists.
In the spring of 1904, while employed by William Guilfoyle at the Botanic Gardens, he crossed a pink Watsonia borbonica with W. borbonica ‘Arderne’s White’. This cross may be represented by the following formula (but note that the order is arbitrary, it is not known which was the pollen parent and which the ovule parent in any of the crosses discussed here):
borbonica × Arderne’s White
He noted that pink flowers were dominant over white in the F1 generation, as has been confirmed by other researchers. In spring 1907 he selected one F1 plant with tall stature, dense branching and large flowers. He crossed this with a purple Watsonia meriana and the widely grown red Watsonia aletroides, and also backcrossed it to W. borbonica ‘Arderne’s White’ to create three lines for further breeding:
1. meriana × (borbonica × Arderne’s White)
2. aletroides × (borbonica × Arderne’s White)
3. Arderne’s White × (borbonica × Arderne’s White)
Cronin’s appointment as Principal of Burnley Horticultural College in 1908 seems to have interrupted this work, and in the following year he succeeded Guilfoyle as Director of the Botanic Gardens. By 1913 he had time to resume his watsonia experiments, and on 20 March sowed seeds from his three 1907 crosses at the Botanic Gardens nursery. Six years is not an inordinately long time to store Watsonia seeds, but there would be some loss in viability which may have unintentionally favoured some genotypes over others. Cronin’s management of the plants was another possible source of selection pressure to produce watsonias adapted to Melbourne gardens: he left the corms in the ground over summer, and gave the plants no fertiliser or watering even though 1913-14 was a drought period.
This generation produced their first flowers in October 1914; Cronin stated that these resembled the 1907 selection in size and colour, and were inbred that year. I interpret this to mean that he produced an F2 generation in each of the three lines by cross-pollinating siblings, since selfing would have produced little or no seed due to incompatibility. Thus,
1. (meriana × (borbonica × Arderne’s White)) × (meriana × (borbonica × Arderne’s White))
2. (aletroides × (borbonica × Arderne’s White)) × (aletroides × (borbonica × Arderne’s White))
3. (Arderne’s White × (borbonica × Arderne’s White)) × (Arderne’s White × (borbonica × Arderne’s White))
Large numbers of these seedlings were raised in the main nursery of the Botanic Gardens. By October 1916 Cronin saw the first flowers of the inbreds, which had a wider range of colours than their parents. Some whites showed up, as would be expected from recombination, including some with flowers of improved size and form compared to the original ‘Arderne’s White’. The watsonias commercially released in the 1920s as the Commonwealth hybrids or “Watsonia Cronini” were selections from this generation.
Line 1 would have produced the many Cronin cultivars with a mixture of characters from W. meriana and W. borbonica. These often have subtle tertiary flower colours due to genes from both species influencing anthocyanin pigment production. Floral bracts are typically well-developed and obtuse, compared to the shorter acute bracts of W. borbonica. Examples include ‘Lilac Towers’, which is the most widely grown Watsonia in Australia today and may be the same as Cronin’s ‘Sydney’, and the one illustrated below which may be his ‘Maitland’.
Line 2 would have yielded flowers with long tubes and small lobes like Watsonia aletroides. The one illustrated here was discussed in a previous post.
It’s significant that Cronin did not use a long breeding program: the cultivars released were no more than three generations away from the original genotypes that had been imported from Africa in the 19th century. As he was working with a perennial that is normally propagated vegetatively, he could stop at the F2 with its fixed heterozygosity. I have bred watsonias four generations on from these and other old cultivars, and can attest that hybrid breakdown soon appears. Some of the resulting plants had interesting extremes of flower shape or colour, many were dwarf or weak in growth, but few were gardenable.
In the spring of 1917 Cronin presented this data to the horticultural correspondent of The Leader, and was lecturing on flower hybridisation to amateur horticultural societies with his new watsonias as exhibits. The following year he gave an interview to The Argus, repeating that his new watsonias were produced by first crossing and then inbreeding on Mendelian lines.
Anon. (1917) Melbourne Botanic Gardens – New colors in flowers – The laws of Mendel. The Leader (Melbourne), Saturday 10 November 1917 pp.13-14.
Anon. (1917) Horticultural society. The Advertiser (Footscray), Saturday 15 December 1917 p.3.
Anon. (1918) Botanic Gardens Experiments. The Argus (Melbourne), no.22,553. Monday 11 November 1918 p. 6.
Cooke, D.A. (1998) Descriptions of three cultivars in Watsonia (Iridaceae) J.Adelaide Bot. Gard. 18: 95-100.
Pescott, E.E. (1926) Bulb Growing in Australia. (Whitcombe & Tombs: Melbourne).
Take a look at whatever you call your “self”. Is it an entity with hard, permanent boundaries?
In my own experience, awareness extends across conceptual spaces that could be called fields, divided by boundaries that might be called discontinuities in awareness.
Michael Polanyi described a particular type of boundary in a series of publications in the 1960s. He introduced the theory of tacit knowledge, where information tacitly known at one level of reality is the basis of explicit understanding at a higher level. For example, whenever we read a text we are tacitly perceiving all the letters but normally notice only of the words or sentences that they spell. Many such levels may exist in a hierarchy, such as letters forming words according to rules of spelling, that form sentences according to the rules of grammar, that in turn carry meanings according to semantic rules.
Each level is a field containing a consistent set of concepts that is incomplete in that it allows its boundary to be ruled by the next higher level. The lower or proximal field contains things known tacitly but the distal field consists of things that are known explicitly, or are still unknown. The proximal field is experienced as self, the distal field as not-self or in other words the external world. For the purpose of this discussion I’ll call these Polanyi boundaries.
The old truism that anything has both an inside and an outside aspect is rediscovered from time to time. For example, the botanist Agnes Arber wrote that “The fact that each organism is both a unity intrinsic to itself, and also an integral part of the nexus which is the Whole, informs it with a basic duality.”
The subjective experience of being a self and separate from an external world – that is, the rest of the universe – was analysed by Gerbode in terms of the theory of tacit knowing. We tacitly know such things as the movement of our voluntary muscles, ideas with which we have identified, skills that have been learned and experiences internalised. All these things are within the aggregate that we think of as self. The other things that we perceive are considered to be separate from the self and therefore parts of an external world.
Another type of boundary that exists between opposing postulates in the mind was described by Stephens as occurring where postulate pairs such as “must know” and “must not be known” meet head-on like opposite flows forming a ridge, a mass that we experience as sensation. Such ridges might be called Stephens boundaries. Moreover, since one self-consciousness cannot simultaneously hold contradictory postulates, the boundary may effectively divide the mind into two fields that function as if they were independent entities.
Please note that I’m using the term postulate here to mean a causative thought, following the usage of Stephens and Gerbode, and before them of Hubbard. This isn’t quite the usual meaning of the term in English. Unfortunately, English doesn’t have any word that captures this concept exactly, and the Buddha’s Pali term saṅkhāra would be more precise. In Buddhist philosophy, saṅkhāra does not depend on self-consciousness but is actually a precondition for that consciousness.
At first sight, a Stephens boundary appears to separate a pair of entities that are both on the same level. The pair of postulates that define their boundary are not immediately recognisable as a rule imposed from a higher level that defines the boundary of the lower one.
But a Stephens boundary can also be seen as an instance of a Polanyi boundary. Both types of boundary represent an inconsistency that marks the limit of an internally consistent field. In fact, the contradictions between postulates are the source of the incompleteness or inconsistency that marks the boundary.
A pair of exactly opposed postulates forms a unity, just like the two ends of the same stick. More importantly, any Stephens boundary actually has higher and lower sides like a Polanyi boundary. The stick has a proximal and a distal end relative to the observer’s viewpoint.
The proximal field is experientially a self, which is normally a lower level field than the corresponding not-self. Self (the field of what we tacitly know) is a small portion of the whole universe (the field of what we explicitly know + what we tacitly know + everything that exists beyond our knowledge). In our everyday experience, the universe of discourse is whatever we perceive as the whole world. Any thing that we can readily view, including ourselves, is much smaller than the universe. Brotherhood with the universe can be a heady feeling when meditating under the summer stars, but taking that feeling too literally is the road to megalomania.
Any thing that we call our tacitly known “self” is an instance of what Stephens called a junior universe – an object that is selected as one side of a dichotomy, leaving the rest of the universe on the other side. Compulsive game playing compartmentalises a person into progressively smaller junior universes by successive dichotomies.
Could it be that a subjective sense of self arises from opposed postulates? If one being cannot hold both postulates simultaneously, there would be a division into self and not-self. The field of not-self can then be subdivided into various objects and even other living beings known as “them” or “you”.
Conversely, resolving the postulate opposition would resolve the perceived boundary of a self. An experimental test of this hypothesis would be to erase some contradictory postulates from one’s mind and observe what happens to the sense of self. Does it expand?
Arber, A. (1954) The Mind and the Eye: A study of the biologist’s standpoint. (Cambridge University Press).
Gerbode, F.A. (2013) Beyond Psychology: an Introduction to Metapsychology. 4th edn (Applied Metapsychology International Press: Ann Arbor).
Polanyi, M. (1968) Life’s irreducible structure. Science 160: 1308-1312.
Stephens, D.H. (1979) The Resolution of Mind: A Games Manual. (privately published: Sydney).
Further to the previous post, I’d been thinking about why the basic pair of complementary postulates in the mind is Know and Be Known, instead of anything else. Is this an arbitrary? Could it be something different in another universe?
But, of course!
Know and Be Known are the two essential properties of Life, the one thing that ultimately exists. They add up to BEING.
” … yet always a twoness in that many. And that twoness so near unite to oneness as sense to spirit, yet so as not to confound to unity the very heart and being of God, who is Two in One and One in Two.” – E.R. Eddison, The Mezentian Gate.
The duality of knowing and being known can explain the duality of self and not-self. Why am I sure that I exist? Because I can know (sense, see, feel, understand) things. Why am I sure that those things exist objectively outside me? Because they can be known (sensed, seen, felt, understood).
These are also are the two sides of the communication cycle: that is, receipt point and source point.
In Eddison’s novel The Mezentian Gate, Life has created a universe of experience by dividing itself into Love and Beauty, the knower and the known. Similarly, all our experience in this universe that we inhabit depends on a division into self and not-self. That division opens the door to the possibility of games, aberration and all the states of woe. But if the two were collapsed into one without division there would be no consciousness.