Infinity and Impossibility
Posted: May 17, 2024 Filed under: Philosophy | Tags: epistemology, mathematics, TROM Leave a commentMathematicians define a singularity as a point where a given mathematical object becomes ‘pathological’ rather than ‘well-behaved’ and cannot be assigned any definite value. Spencer-Brown’s imaginary Boolean values (neither 1 nor 0) and Stephens’ IP (impossibility point or insanity point) are both examples of singularities.
So is the elusive concept of infinity, which has had philosophers and mathematicians chasing their tails for two millennia or longer.
The reciprocal function f(x) = 1/x has a singularity at x = 0 because division by zero implies that no numerical value of the function can be defined. At best, we can say that as the divisor approaches 0 the quotient approaches infinity. Infinity is not a vast number, nor it is a small number; it’s not a number at all. The largest number possible falls short of + infinity, and the smallest number possible is still greater than – infinity.
Infinity can be called the point where the factor we are considering ceases to be relevant to the result. For example, the size of a cake becomes irrelevant to the number of slices that can be cut from it if the size of each slice is reduced to zero (Mary Boole,1891). In Stephens’ terminology, the postulate no longer constrains the result.
Having previously posed the question of the relation between (1) imaginary Booleans, (2) IPs and (3) Hofstadter’s strange loops, I would like to suggest a simple answer: The first two are equivalent to infinity, and the third is a cycle that contains one or more of these singularities.
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Infinite series have the surprising property that a subset can be put in direct correspondence with the whole. For example the set of all real, positive integers can be paired off with its subset of real, positive, even integers. So one way to study infinity is through the unlimited recursion that happens when a function f is re-entered into itself. In the calculus of indications this is done simply by enclosing f in a nested series of spaces (Spencer-Brown,1969), which are indicated here by square brackets since html does not provide a symbol for Spencer-Brown’s mark.
[ [ [ [ [ [ [ f ] ] ] ] ] ] ] …
The part of the expression enclosed by the brackets at every even-numbered interval of levels such as two is identical to the whole expression, which may be regarded as re-entering its own space at any of these even depths.
If the function re-enters at an even depth, the equation f = [ [ f ] ], where f is the whole expression, can be satisfied either by f = [ ] or f = , the marked and unmarked states which can be translated by 1 and 0 respectively in Boolean algebra.
George Boole (1854) had called 0/0 the indefinite class symbol which “admits of the numerical values 0 and 1 indifferently”, while 1/0 has no real numerical value but is infinite – just like the cake cut into zero-sized slices, and consistent with the concept of infinitesimal in the differential calculus. In a logical equation used to define a class, any subclass with the ambiguous coefficient 0/0 might be wholly or partly present or absent, but the coefficient 1/0 explicitly excludes a subclass.
In Spencer-Brown’s algebra, 1/0 is the imaginary Boolean produced by re-entering the function into itself at an odd-numbered interval such as one. The equation f = [ f ] is satisfied neither by f = [ ] nor by f = , so its solution can only be an imaginary value which might be represented by an oscillation such as “if 1 then 0, if 0 then 1”.
But f = [ f ] is the same as “f is not-f ”, which we recognise as the postulate x = (1 – x) of the insanity loop defined by Stephens. It puts the function f into an IP, making an imaginary Boolean and consequently a strange loop in a tangled hierarchy. Consequently, the insanity loop is an instance of a strange loop while the sanity loop is not.
Boole also noted that as x approaches infinity, it departs from the fundamental law of his algebra, x (1 – x) = 0. That law is part of the sanity loop and defines the universe where we live, the universe that we call real. We can now recognise that infinity is an IP, an impossibility point. In an essay written in 1856 but not published until 1997, Boole explicitly linked 1/0 with impossibility when he listed a tetralemma of four terms: 1, 0, 0/0 (indefinite) and 1/0 (impossible).
Stephens formalised the theory that all activities can be understood as games – conflicts between opposing postulates that are worked out in time and space – by analysing games in a framework of Boolean algebra. Just as Boole intended his algebra to codify the laws of thought and Spencer-Brown was a student of both mathematics and psychology, Stephens turned to this basic branch of pure maths to fulfill his original interest in human psychology. It led him to the concept of an IP as the end point of a game, where one of the postulates is reduced to zero, leaving the other as the ended game’s whole universe of discourse.
Colloquially, infinity is “where things happen that don’t”, such as parallel lines intersecting. It’s a nowhere, off the edge of the game board, across the boundary, outside the universe. And the same can be said of an IP, which is beyond the distinction between what is and what isn’t.
Infinity and IP fit within the definition of a static as something without mass, motion, wavelength or location in space and time. An object at rest in an inertial frame of reference is not a true static as it has a location that may be moving in relation to other parts of the universe. This definition might have been foreshadowed by Porphyry when he began his Sententiae by distinguishing physical objects, which necessarily have separate locations, from anything lacking a physical form and therefore omnipresent without discontinuity. If something has no location in space or time it cannot be moving; it cannot have any velocity, which is change of location in space over time.
Mass is one of the things that a static lacks. It may be objected that Stephens (1994) has explained the sensation of mass as a consequence of contradictory postulates at an IP. But the IP itself does not have the mass, as what we perceive as a virtual mass comes from those opposing postulates. Try to move an object that someone has made big and heavy so that it will stay in one place: it’s your postulate pushing against theirs.
‘Static’ is a broader category than IP, imaginary Boolean and even infinity as discussed here. The privative definition given above would also include all abstractions and anything else that is not tied down to a location by physical form.
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References
Boole, M.E. (1891) Philosophy and Fun of Algebra. (Daniel: London).
Boole, G. (1854) An Investigation of the Laws of Thought. (Walton & Maberley: London).
Boole, G. (1997) On the Foundations of the Mathematical Theory of Logic. In: Grattan-Guinness, I. & Bornet, G. (eds) George Boole, Selected Manuscripts on Logic and its Philosophy. (Springer: Basel).
Spencer-Brown, G. (1969) The Laws of Form. (Allen & Unwin: London).
Stephens, D.H. (1994) Sensation. Tapes of 27 July 1994.
More loops – but not necessarily strange
Posted: March 6, 2024 Filed under: Philosophy | Tags: epistemology, mathematics, TROM Leave a commentA postulate, what it allows within a given universe, and what it forbids there, are three logically equivalent statements. Knowing one of them, it is possible to deduce the other two as if they were connected in a loop. For example, the material implication ‘if avocado then green’ would allow green avocados, green things that are not avocados, or things that are neither avocados nor green; but it excludes avocados that are not green.
Stephens (1994a) defined such a loop. It is like the ‘trinity’ mentioned by Spencer-Brown (1972), where the inside (what is included), the outside (what is excluded), and the boundary between them (the postulate) are all one form. He compared a sanity loop with an insanity loop.
Sanity loop
What Stephens called the sanity loop is the default condition in the real universe we inhabit, a universe defined by the law that a thing cannot both exist and not exist simultaneously. In logic, ‘simultaneously’ does not imply time, but means ‘in the same universe of discourse’. Logical propositions may refer to time as subject matter, but are themselves outside of time, and each instant may be considered as a complete and self-consistent universe.
This is not a strange loop because, being logically equivalent, the three do not represent different levels of a hierarchy.
X = X a thing is itself 1
X + (1 – X) = 1 it either exists or it doesn’t 2
X (1 – X) = 0 it cannot both exist and not exist simultaneously 3
These three equations can be proved equivalent by using Boole’s algebra:
X = X 1
X – X = X – X subtract X from each side
X – X = 0 simplify RHS
X – X + 1 = 1 add 1 to each side
X + (1 – X) = 1 reorder 2
X2 + X (1 – X) = X multiply by X
X + X (1 – X) = X X2 = X
X (1 – X) = 0 subtract X from each side 3
X – X2 = 0 multiply out LHS
X – X = 0 X2 = X
X = X add X to each side 1
The same proof can be written in a reverse sequence, just like going around a loop in a reverse direction:
X = X 1
X – X = X – X subtract X from each side
X – X = 0 simplify RHS
X – X2 = 0 X = X2
X (1 – X) = 0 factorise LHS 3
X + X (1 – X) = X add X to each side
X2 + X (1 – X) = X X = X2
X + (1 – X) = 1 divide by X 2
X – X + 1 = 1 reorder
X – X = 0 subtract 1 from each side
X = X add X to each side 1
In the calculus of indications (Spencer-Brown, 1969), equations 1 and 2 can both be written as:
[X] X = []
and equation 3 as:
[[X][[X]]] = [[X] X] =
Since HTML cannot display Spencer-Brown’s mark as nested single brackets, I have used the convention of replacing it with pairs of square brackets.
Insanity loop
On the other hand, the insanity loop (Stephens, 1994b) is the condition at an IP (impossibility point), which by definition is not part of the real universe.
The three equations making up the insanity loop are the negations of equations 1 to 3:
X = (1 – X) a thing is not itself 4
X + (1 – X) = 0 it can neither exist nor not exist 5
X (1 – X) = 1 it both exists and does not exist simultaneously 6
From 5 we see that 1 = 0 in this imaginary universe of discourse which contains the single class X , 1 – X. If drawn as a Venn diagram it would be one shaded space without any division. Everything is nothing; and nothing is everything.
Substituting 4 into 4,
X = (1 – X)
(1 – X) = (1 – (1 – X)) simplifies to (1 – X) = X
(1 – (1 – X)) = (1 – (1 – (1 – X))) simplifies to X = (1 – X)
(1 – (1 – (1 – X))) = (1 – (1 – (1 – (1 – X)))) simplifies to (1 – X) = X
(1 – (1 – (1 – (1 – X)))) = (1 – (1 – (1 – (1 – (1 – X))))) simplifies to X = (1 – X)
This is a re-entering expression that oscillates between the two equivalents however far it is extended.
Substituting 4 into 5,
X + (1 – X) = 0
(1 – X) + (1 – (1 – X)) = 0
(1 – (1 – X)) + (1 – (1 – (1 – X))) = 0
(1 – (1 – (1 – X))) + (1 – (1 – (1 – (1 – X)))) = 0
This is a re-entering expression that always simplifies to 0 (since 1 = 0) however far it is extended.
Substituting 4 into 6,
X (1 – X) = 1
(1 – X) (1 – (1 – X)) = 1
(1 – (1 – X)) (1 – (1 – (1 – X))) = 1
(1 – (1 – (1 – X))) (1 – (1 – (1 – (1 – X)))) = 1
This is a re-entering expression that always simplifies to 1 however far it is extended.
If the same substitutions are done with equations 1 to 3 the results are trivial since we are just substituting X with X. The re-entry of equation 4 at an odd depth introduces an imaginary Boolean value into the insanity loop. And because the insanity loop is based on an imaginary Boolean, it contradicts the rules of Boole’s algebra which then cannot be used to prove the equivalence of its component equations.
However, the RHS of equations 4 to 6 are the negation of the RHS of equations 1 to 3 respectively: 1 – X, 1 – 1 and 1 – 0. And the LHS of equations 4 to 6 are identical to the LHS of equations 1 to 3. Since 1 to 3 have been proved equivalent, 4 to 6 must necessarily be equivalent also.
X and (1 – X) are coextensive (equation 4), therefore their conjunction X (1 – X) must equal 1 (equation 5). Equation 6 is the dual of equation 5, just as 2 and 3 are duals.
The relationship between the two loops can be demonstrated by writing them in the calculus of indications. We see that this simply means placing an additional mark over each expression in the sanity loop to get the insanity loop. Equations 4 and 5 both become:
[[X] X] =
and equation 6 (sharing the form of equation 3) is:
[[[X][[ X]]]] = [X][[ X]] = []
As with the NAND, this is another instance of a high-level abstraction having the same mathematical form as a subjective human experience. The concept of impossibility matches the phenomenon of insanity because an insane individual believes the impossible statement that X = (1 – X).
Impossibility and insanity seem to have a mathematical form that is as rigorous as the form of possibility and sanity. Such a form cannot be understood by considering it only as a negative, a departure from reality. It is also a return to a condition more ‘primitive’ or fundamental than the default state of this universe represented by the sanity loop. This may be the real reason insanity is so unconfrontable to those following the universe’s postulates. It’s a look behind the scenes, a pointer to how we create this universe; a condition where one instant is not separated from another and nonexistence equals existence as envisioned by Nāgārjuna and the other Mahāyana masters.
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References
Boole, G. (1854) An Investigation of the Laws of Thought. (Walton & Maberley: London).
Spencer-Brown, G. (1969) The Laws of Form. (Allen & Unwin: London).
Spencer-Brown, G. writing as James Keys (1972) Only Two Can Play This Game. (Julian Press: New York).
Stephens, D.H. (1992) The Unstacking Procedure. Tape of 3 November 1992.
Stephens, D.H. (1994a) The Loop. Tape of 6 April 1994
Stephens, D.H. (1994b) Insanity. Tape of 30 June 1994.
Strange loops and alternative denial in logic
Posted: February 25, 2024 Filed under: Philosophy | Tags: mathematics, Spencer-Brown, TROM Leave a commentDouglas Hofstadter coined the term ‘strange loop’ for a feedback cycle consisting of a series of stages that shift from one level of abstraction to another in an apparent hierarchy, yet paradoxically return to their starting point. He included many intriguing examples from the drawings of Escher and the music of Bach in his first book on this subject; but perhaps the most blatant example of a strange loop is a self-contradicting statement such as “This sentence is false.”
We experience a strange loop subjectively as a mental knot analogous to a Möbius strip. More formally, it’s a mathematical function that re-enters its own space, in the sense of Spencer-Brown (1969) who showed that some functions re-entering at odd (rather than even) depths give rise to imaginary Boolean values that are not stable as either 1 or 0. These imaginary Boolean values are closely related to the self-contradictory logical classes that Stephens (1994) called impossibility points: classes violating the basic rule of logic that a thing must either be or not-be.
An implication or single bonding such as A ⇒ B is not a loop. A biconditional or double bonding of A ⇔ B is a loop but not necessarily strange; it becomes strange if A and B are at different levels in a conceptual hierarchy. A series of implications that return to the first term such as A ⇒ B, B ⇒ C and C ⇒ A could also be a strange loop.
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There’s another form of strange loop that can arise when A and B are mutually exclusive and therefore their conjunction A,B is a null class. In informal English, mutual exclusivity means that you can choose either of two alternatives on its own, or you can have neither, but you can’t have them both together.
Such a loop is formed using the logical relation known as alternative denial or NAND, which can be written as ¬ (A ∧ B), or simply as A↑B using the Sheffer stroke. This is the negation of conjunction, the logical relation that only allows A and B together but not the absence of one or both. If A and B are in alternative denial, their conjunction A,B is necessarily a null class, but since not-A and not-B are not mutually exclusive their conjunction is not a null class. In fact, ¬ (A ∧ B) says explicitly that all possible classes except A,B are allowed.
Notice that B resembles the negative of A, written as not-A, but they are not exactly the same thing. A and not-A are a true dichotomy; but A and B are not. Nor is A co-extensive with not-B, or B coextensive with not-A, because they are not exact synonyms. Yet they are not unrelated: they are mutually referential, and so can form a self-referential loop.
A language is any system used to represent thoughts, and a word is a particle of meaning in such a system. Verbs expressing postulates that something will be done or happen may be called the root words of those postulates. And at this juncture we’ve stepped from pure abstract logic into the field of human will, human experience and human values. When A and B are mutually exclusive verbs, their mismatch may be experienced as a problem, a knot that needs to be untangled.
To take a common example, A might be ‘Accept’, and B might be ‘Reject’. Rejection is one way of not accepting, but not the whole of it, and vice versa. There is always a middle road, however narrow, between accepting a thing and rejecting it. It’s not a true dichotomy because there’s a third alternative to A or B, the middle way represented by not-A, not-B.
Some other such pairs are Display and Hide, or Love and Hate. It’s very easy to fall into the misconception that display and hide are a dichotomy, so that if we do not display something we are hiding it, and if we do not hide it we must be displaying it. That is the two-valued logic of philosophers like Ayn Rand. But are you displaying or hiding the Eiffel Tower at this moment? Probably neither, unless you’re a Parisian tour guide or the owner of a very big tarpaulin.
And check out almost any politician’s rhetoric for examples of spurious dichotomies like these. “Anyone who is against capitalism must be a communist.” It’s almost as if they were trying to gaslight us, perish the thought.
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The four postulates A, B, not-A and not-B can be arranged in a 2 by 2 matrix:
1. A 3. B
2. Not-A 4. Not-B
paralleling the basic game matrix described by Stephens (1979):
1. Be known 3. Know
2. Be not-known 4. Not-Know
but with an important difference since A and B are such that B implies not-A and A implies not-B. In formal logic this condition is written A↑B : B ⇒ not-A, A ⇒ not-B
The root words in the basic game matrix are Be Known and Know. When they are replaced by other root words such as Accept and Reject, a junior goal is formed within the basic goal. Any junior goal can be reduced to a special case of the basic goal of Know/BeKnown, since everything we try to do in life is a means of either knowing or making something known. Seeing, owning, learning are ways of knowing. Creating an effect in the world is a way of being known. In this context, the words ‘postulate’ and ‘goal’ are equivalent.
Stephens demonstrated the value of the basic game matrix in making decisions in life, and as a therapy. This property of the basic game is a consequence of Know and Be Known not being mutually exclusive. They are complementary conditions that satisfy each other, represented in English by the active and passive forms of the same verb. Neither of the classes Be Known, Know or Be not-known, Not-know is a null class.
But in the NAND matrix, B is a subset of not-A and A is a subset of not-B. Therefore B becomes a negative of A and not-B becomes a double negative, effectively a positive but on a lower level to the starting point of A. This matrix functions as a descending spiral if the four postulates are followed in sequence.
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I first encountered NAND matrices 35 years ago when studying the theory of implant GPMs. Now, that term needs some explanation. Implants were believed to be mental control operations that had been done to us in long-vanished civilisations, and consisted of postulates enforced by painful energy. They would constrict our ability to think just as malware reduces the efficiency of a computer. GPM stands for Goals Problem Mass, the apparent mass generated by the problem of two opposing goals rather like a standing wave between two forces meeting head-on. GPM implants are particularly severe because they consist of balanced pairs of conflicting postulates, logical knots such as the NAND.
According to the chronology of Hubbard (1963) the earliest implant GPMs were the ones now called the Invisible Picture implants. These have the logical form described above, the form of A↑B : B ⇒ not-A, A ⇒ not-B, being a sequence of contradictory items in NAND relationships.
The basic game matrix in TROM Level Five undercuts any implanted GPM because it is closer to truth (Stephens, 1992). We now know that each item in these implants is reducible to one of the four legs of the basic game matrix. Must Be Known (MBK) is an postulate to put something there to be known and Must Not Be Known (MNBK) is the opposing postulate – to not do this. So a sequence can run MBK, MNBK, MNBK, MBK over and over, pairs of true dichotomies that stand in alternative denial to each other. Or it may change to MK (Must Know), MNK (Must Not Know), MNK, MK. One word in each item, the root word, varies through the sequence.
Here’s a made-up example (I’m not plagiarising from Hubbard’s published materials):
Grow MBK
Don’t grow MNBK
Shrink MNBK
Don’t shrink MBK
Build MBK
Don’t build MNBK
Demolish MNBK
Don’t demolish MBK
Find MK
Don’t find MNK
Lose MNK
Don’t lose MK
Note that in each pair of verbs in a NAND relationship, the negative of one is similar (but not identical) to the positive of the other. Thus ‘don’t find’ and ‘lose’ both fit within the broader negative concept of ‘must not know’, while ‘find’ and ‘don’t lose’ are within the positive ‘must know.’ The structure has a regular + + – – + + – – pattern down the page.
An impossibility point (Stephens, 1994) would occur between each pair of opposing items, meaning either MK/MNK or MBK/MNBK, generating the perceived mass that holds the charge in the GPM. This generated mass would tend to persist because – unlike the basic game matrix – a GPM lacks complementary pairs of items that would enable the release of charge.
This may all sound like science fiction, but the structure of implant GPMs makes them resemble artefacts composed by someone who knew about the basic game matrix and was deliberately perverting it. Whether they are really artefacts, or just consequences of the basic game, or something else, I can attest that they have real power that feels like mass or charge in the mind when first contacted. They have been found on many cases around the world; they’re not something dreamed up one sunny afternoon at Saint Hill.
Going even further into speculative fiction, suppose the structure of the Invisible Picture implant was copied by later implanters who devised new GPMs as variations on that original sequence. Hubbard also discovered many other implant GPMs that are structured with different forms of contradictory postulates instead of the NAND. It’s as if there had been several lines of development from several original implant authors.
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The previous section is rather a digression. My main point is that Spencer-Brown’s imaginary Boolean values are closely related to Stephens’ impossibility points within the less well-defined popular concept of strange loops.
Stephens and Spencer-Brown were pioneers of that basic area of knowledge where mathematics and psychology converge. Perhaps someone with more mathematical insight than this blogger will be able to clarify the formal relationship between strange loops, impossibility points and imaginary Booleans.
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References
Hofstadter, D. (2007) I Am a Strange Loop. (Basic Books: New York).
Hubbard, L.R. (1963) Routine 3N line plots. HCO Bulletin of 14 July 1963, Saint Hill.
Spencer-Brown, G. (1969) The Laws of Form. (Allen & Unwin: London).
Stephens, D.H. (1979) The Resolution of Mind. Public domain text.
Stephens, D.H. (1992) The Unstacking Procedure. Tape of 3 November 1992.
Stephens, D.H. (1994) Insanity. Tape of 30 June 1994.
Genetic diversity of introduced weeds
Posted: February 9, 2024 Filed under: Botany | Tags: genetics, taxonomy Leave a commentThese notes are over two decades old, from a time when I was working on the declaration of weeds under legislation, but they may still be worth a glance. There has been a tension in biosecurity policies between the demand for minimal restraints on trade and a precautionary principle that warns against allowing the entry of any new material of species that have already demonstrated weedy behaviour. The free trade lobby argues that if a weed is already so abundant in Australia that eradication or containment programs have ceased, then restrictions on its entry to the continent (particularly as a contaminant in goods) can no longer be justified. Their position is based on a naïve, pre-Darwinian concept of species as homogeneous entities.
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Introduced populations of a plant may be either more or less uniform than those in its native range. This can be attributed partly to high or low phenotypic plasticity of the species and the broad or narrow range of habitats in which it is naturalised. Species which can produce a wide range of phenotypes, in adaptation to the range of local environments, are pre-adapted to become weeds. Naturalised Oxalis pes-caprae is quite uniform in genotype because it has depended on asexual reproduction for its spread in Australia, but it adapts widely in size according to microhabitat and seasonal conditions.
But genetic variation may also increase or decrease with naturalisation, depending on the biology and breeding system of the plant and the type of selection to which it is exposed.
We might expect naturalised plants to show less genetic variation than the same species in its native range, because the whole naturalised population has descended from the few individuals that were its founders. There has been a “bottleneck” in genetic diversity at the time when the species was transported to its new habitat.
This appears to be demonstrated by Reseda lutea in South Australia. The disjunct distribution of the first records around the coast suggest that there were several successive introductions from overseas. In its native range in Europe and the Mediterranean two subspecies and five varieties are recognised, distinguished by such things as the presence or absence of an aril on the seed, seed size, number and arrangement of ovules, leaf shape and division, and the dissection of the petals. However, collections of this species from throughout its South Australian range show no significant variation in these or other morphological characters (e.g. flower colour, vestiture) which vary in European populations. It is a very uniform species here, suggesting that, even if it was introduced more than once, all the founders of the population came from a similar source. Reseda lutea was introduced in ships’ ballast around the turn of the century, and our material is a good match for collections from close to the shoreline in Britain and western Europe.
The bottleneck may be beneficial to the introduced species in the longer term, helping it become better adapted after an initial lag phase. Due to the phenomenon of dominance complementation, a period of inbreeding could purge deleterious recessive alleles from a population since these will have an increased chance of appearing in homozygotes than can be eliminated by selection (Barrett & Charlesworth, 1991).
On the other hand, the genetic variance of a plant may increase suddenly as it spreads in a new habitat during the logistic growth phase that follows the lag. Fisher (1930) predicted that mutant alleles, even if they did not increase fitness, were more likely to become established in an expanding population than in a stable one. Populations of an r-strategist weed invading a new habitat are an extreme case, expanding even more rapidly than the populations considered by Fisher. The pressure of natural selection against mutant alleles is proportionately lower in that situation and genetic variation might be expected to rise. And such was the case with many of our common annual weeds of broadacre agriculture, which dispersed as quickly as farming land was opened up in the early 19th century and built up large populations in a new vacant niche.
Naturalised populations of salvation Jane (Echium plantagineum) appear to have a high level of genetic variability (Brown & Burdon, 1983), and at all 16 isozyme loci they examined were as diverse as the European source populations (Burdon & Brown, 1986). This may be due to the amount of variation that was present in the original parent plants which may have come from more than one region, or to mutations that had occurred in the 140-odd generations since the species was introduced.
Even when the population has reached a stable size, if the plant has occupied a range of habitats natural selection can be expected to preserve that portion of the new genetic variability which increases fitness in these habitats; this has also been found in salvation Jane which shows clinal variation across south-eastern Australia (Wood & Degabriele, 1985) in morphology as shown by measurements of leaves and flower parts and also in its physiology – in its response to different temperature and moisture regimes.
One reason for the difference in behaviour of Echium and Reseda may be that the former is an annual but Reseda is a perennial and therefore has passed through fewer generations since arriving in Australia with proportionately less opportunity for the establishment of new alleles.
The difference in breeding systems of the two species would tend to reinforce this effect. Echium is an outbreeder but Reseda is an inbreeder whose flowers self-fertilise if not pollinated by insects, therefore there has been less opportunity for recombination in Reseda. Additionally, Echium exists in populations many orders of magnitude larger than those of Reseda, especially if we count the clonal patches of the latter as unit individuals. Again, Fisher predicted that greater levels of genetic variance are produced and maintained in abundant species than in rare ones.
Another naturalised perennial, Myrsiphyllum asparagoides, also shows much less variation in Australia than in its native southern Africa (Cooke & Robertson, 1990). The stapeliad Orbea variegata is extremely variable in its floral and vegetative morphology across its wide native range (White & Sloane, 1937) but the material naturalised in South Australia is quite uniform in morphology, implying that it was introduced from only a small part of its range, most likely close to Cape Town.
On the other hand, the annual Rumex hypogaeus also introduced from South Africa shows little variation in isozymes across its introduced range in Australia. This may simply reflect the similarly low level of variation in these enzymes its native range (Panetta, 1990).
Weeds and other invasives are species released from fitness constraints and fortuitously adapted to their new home. This adaptation may come about only after hybridisation and recombination between successive introductions from different source populations (Ellstrand & Schierenbeck, 2000). Consequently, it is important that the introduction of additional genotypes of the weed species that are already established here be prevented. Any new addition of germplasm may turn out to be one that can significantly improve the adaptation of the species to local habitats, and so increases its invasiveness and/or impact as a weed.
The same general principles could be applied to other pest organisms of biosecurity concern.
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References
Barrett, S.C.H. & Charlesworth, D. (1991) Effects of a change in the level of inbreeding on genetic load. Nature 352: 522-524.
Brown, A.H.D. & Burdon, J.J. (1983) Multilocus diversity in an outbreeding weed, Echium plantagineum L. Aust. J. Biol. Sci. 36: 503-510.
Burdon, J.J. & Brown, A.H.D. (1986) Population genetics of Echium plantagineum L. – target weed for biological control. Aust. J. Biol. Sci. 39: 369-378.
Cooke, D.A. & Robertson, M. (1990) Bridal creeper, Myrsiphyllum asparagoides, in South Australia. Proc. 9th Aust. Weeds Conf. Adelaide 113-115.
Ellstrand, N.C. & Schierenbeck, K.A. (2000) Hybridization as a stimulus for the evolution of invasiveness in plants? Proc. Nat. Acad. Sci. 13: 7043-7050.
Fisher, R.A. (1930) The Genetical Theory of Natural Selection. (Oxford University Press).
Panetta, D. (1990) Isozyme variation in Australian and South African populations of Emex australis Steinh. Aust. J. Bot. 38: 161-167.
White, A. & Sloane, B.L. (1937). The Stapelieae. edn 2 (Abbey San Encino: Pasadena).
Wood, H. & Degabriele, R. (1985) Genetic variation and phenotypic plasticity in populations of Paterson’s curse (Echium plantagineum L.) in south-eastern Australia. Aust. J. Bot. 33: 677-686.
Inflorescence cormlets in Watsonia meriana
Posted: February 6, 2024 Filed under: Botany | Tags: Iridaceae, Watsonia Leave a commentAll corms and cormlets produced by a Watsonia are axillary. Sympodial branching is a diagnostic character of the genus, with the main corm being exhausted by producing an annual flowering stem and one or several new lateral underground corms each subtended by a basal leaf. Additional smaller corms, variously called cormlets, cormils or bulbils are typically produced in the axils of lower cauline leaves in many species including W. aletroides (Burm.f.)Ker Gawl., W. humilis Miller and W. meriana (L.)Miller.
The Watsonia inflorescence is a spike with each solitary, sessile flower subtended by an outer bract and a less robust inner bract. The weedy variety W. meriana var. bulbillifera (J.Mathews & L.Bolus)D.A.Cooke is distinguished by having the lower flowers in its inflorescence replaced by clusters of cormlets. This variety is usually a sterile triploid depending on above-ground cormlets for dispersal, and is known as bulbil watsonia.
However, some genotypes of diploid W. meriana var. meriana can produce small solitary cormlets or cormlet clusters in the inflorescence, at least in cultivation. The extent of cormlet development is variable from one year to another; they are sometimes absent and are never as large or numerous as in var. bulbillifera.
Accession 183. At the right of the picture is a typical solitary cormlet 3mm wide produced in the axil of the reduced leaf subtending an inflorescence branch – but on the outside of that branch. This has been observed on all branches in six out of eight years. To the left is a cluster of six cormlets replacing the lowermost flower on the main axis; this has only been observed once.
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Accession 187. A cluster of three cormlets replacing the lowermost flower of the inflorescence. This has only been observed once in eight years.
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I used to wonder if the cormlet clusters of var. bulbillifera might be homologous to the twelve organs (six tepals, three stamens, three carpels) in a flower. This hypothesis was attractive because, although the number of cormlets in each closely packed cluster is variable, twelve is a typical number. The observations reported here make the hypothesis much less likely. There is a gradient transition from stem cormlets through inflorescence cormlets to clusters. The position of the cormlets in accession 183, axillary to the phyllome subtending an inflorescence branch but abaxial to that branch as if an extra axillary growing point, is rather surprising.
The consistent development of inflorescence cormlet clusters is still the morphological character separating var. bulbillifera from var. meriana. However, the difference should be understood as one of degree rather than a clear dichotomy.
Accession 183 was a gift from Mr Graeme Dallimore who collected it on Mornington Peninsula, Victoria. Accession 187 is said to be Cronin material discarded from Melbourne Botanic Garden in 1995. Both readily produce seed, implying that they are diploids.
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References
Conran, J.G., Wilson, P.A. & Houben, A. (2004). Pollination and ploidy changes in South Australian populations of bulbil watsonia, Watsonia meriana (L.)Miller var. bulbillifera (J.Mathews & L.Bolus)D.A.Cooke (Iridaceae). Herbertia 57: 57-70.
Cooke, D.A. (1998) Bulbil watsonia is a variety of Watsonia meriana (L.)Miller (Iridaceae). J. Adelaide Bot. Gard. 18: 5-7.
Goldblatt, P. & Manning, J.C. (2020) Iridaceae of southern Africa. Strelitzia 42. (South African National Biodiversity Institute: Pretoria).
The I Ching reveals more symmetries
Posted: January 31, 2024 Filed under: Philosophy | Tags: mathematics, Spencer-Brown Leave a commentThe classic Chinese text Zhouyi, known in the west as the Book of Changes or I Ching, has existed for three millennia. It is organized around 64 six-line hexagrams, each constructed from pairs of the eight three-line trigrams. Around this core, layers of verbal content accreted over the centuries: first proverb-like divinatory messages, then philosophical commentary.
Apart from the traditional ordering of the 64 hexagrams attributed to King Wen, another sequence may be produced by reading each hexagram from the top down as a six-place binary number, reading unbroken lines as 1 and broken lines as 0. The 64 hexagrams are found to code uniquely for the numbers 0 to 63. So kūn is the first hexagram corresponding to the number 0; bō is the second, corresponding to 1; bǐ, 2 through to qián, 63. This is not a new idea; the understanding of the hexagrams as binary numbers goes back to Shao Yong during the Song dynasty period, and the King Wen sequence was not universally used in early Chinese manuscripts of the Zhouyi (Redmond, 2017).
Since the work of Spencer-Brown (1969) we can now reduce each hexagram to one of two states called marked or unmarked by considering them as arrangements in his calculus of indications. An unbroken or ‘positive’ line is read as the mark and a broken or ‘negative’ line is read as absence of a mark. The whole hexagram is simplified by the rule that when one mark is placed inside another they cancel out to the unmarked state. Consequently, the 32 hexagrams with an even number of unbroken lines take the unmarked state; the others with an odd number are marked. However, we would obtain the same result if we chose to identify unbroken lines as marked and broken as unmarked because a hexagram with an odd number of broken lines must also have an odd number of unbroken lines.
Taking the hexagrams in the binary sequence outlined above and counting unmarked as 0 and marked as 1, we get the following string.
0110100110010110100101100110100110010110011010010110100110010110
This string is a palindrome consisting of the two shorter palindromic units, 0110 and 1001, each occurring eight times. It can be seen that these two units are complementary. Further, if the first half of the string is written alongside the second half, these are also seen to be complementary, i.e.
01101001100101101001011001101001
10010110011010010110100110010110
Looking more closely at how the two shorter four-digit units are arranged, replace 0110 with ‘A’ and 1001 with ‘B’. This makes it easier to recognise another palindrome ABBABAABBAABABBA, which consists of two complementary palindromic units ABBA and BAAB, each occurring twice. Naturally, the second half is the complement of the first.
Incidentally, when the eight trigrams are treated in the same way by arranging them according to the three-place binary numbers they represent and simplifying each to marked or unmarked, they yield a string containing one instance of each of the two complementary palindromic units, i.e. 01101001, but not an overall palindrome.
Of course, this structure was not arbitrarily designed by the old sages. Not does it necessarily have any connection to the verbal content of the Zhouyi or its original use as a divination tool. Rather it is an inherent property of the binary numbers the hexagrams represent, and is evidence in support of Shao Yong’s binary sequence. It is also another example of the power of Spencer-Brown’s calculus of indications to penetrate beyond words and numbers to the simpler deep structure of things.
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References
Redmond, G. (2017) The I Ching (Book of Changes). (Bloomsbury: London).
Spencer-Brown, G. (1969) The Laws of Form. (Allen & Unwin: London).
Herbertia lahue in Australia
Posted: December 19, 2023 Filed under: Botany | Tags: Iridaceae Leave a commentNow that Deble (2021) has clarified the nomenclature of this group of related species, it’s clear that the Herbertia commonly grown in Australian gardens is H. lahue (Molina)Goldblatt, which is remote from H. pulchella Sweet.
Its closest relatives include H. amoena Grisebach and H. caerulea (Herbert)Herbert. These three are sympatric in South America, but H. lahue is the commonest in central Chile, and so may be better adapted to grow autumn to winter in a Mediterranean climate where it continues flowering through spring until dryness forces it to go dormant for the summer. The epithet lahue comes from a Chilean common name.
In my experience of growing H. lahue in Adelaide, it is self-compatible or possibly apomictic, with every flower producing a full capsule of seed after opening for one day. Seed germinates readily after sowing in autumn and plants may flower in their second year. Although perennial, the bulbs do not multiply rapidly and may die after a few years. Like many other r-strategist irids, it depends on seed to maintain a population.
The true Herbertia lahue is recognised by the combination of characters:
– a single flower per spathe (instead of two)
– outer tepals 10–18 mm wide
– anthers straight, 5–7 mm long
– apices of the 3 style branches only shortly bifid and straight (not recurved)
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Reference
Deble, L.P. (2021) Herbertia lahue (Iridaceae) and its allies. Balduinia 17: 2-13.
An Iris resembling Iris aphylla
Posted: November 27, 2023 Filed under: Botany | Tags: Iridaceae Leave a commentMany rhizomatous bearded irises are loosely called “Iris germanica” in Australia, and I have to admit I was guilty of this in the Flora of Australia (1984) and the Flora of South Australia (1986). Even the complex garden hybrids known as Tall Bearded Irises are sometimes lumped under this binomial, although the real Iris germanica L. is highly unlikely to have contributed to any of them, being difficult to use in a breeding program.
Among them are several distinctive cultivated clones; you may find them in your garden, or your neighbours’ gardens, or growing feral on waste ground from plantings several (human) generations ago. It seems approriate to call them clones as they represent a small number of quite uniform phenotypes with no intermediate forms, and rarely if ever produce seed. One of these irises that is frequent around Adelaide resembles I. aphylla L. at first glance, but can only be included in the broadest concept of this species.
It’s clear that the epithet “aphylla” used by Linnaeus in Sp. Pl. 38 (1753) did not mean that the plant is leafless at flowering time, although that may be the case in the colder parts of its historical range, which extends from the Balkans north to Germany and east to Russia. He actually wrote “scapo nudo longitudine foliorum” – with a naked scape the same length as the leaves – implying that he saw leaves and flowers together, as shown in the habit photo at Plants of the World Online. By “scapo nudo” he surely meant that the scape did not bear any leaves apart from the bracts.
Rhizomatous perennial herb to 70 cm high. Leaves in a basal fan of 5-12, ensiform, glaucous, to 37 mm wide, 50 cm long, apices acute and straight or gently incurved, present throughout the year. Scape elliptic in cross-section, to 50 cm tall, naked apart from a bract subtending each of the 0-2 short branches. Bracts slightly inflated, cymbiform, herbaceous but densely purple-streaked. Flowers solitary or paired within the bracts. Perianth tube to 3 cm long. Perianth lobes obovate, to 6 cm long, indigo to dark violet, the outer series (falls) patterned with a dark network on paler ground in the lower part, with a beard of white hairs, those near the haft yellow-tipped. Anthers 15 mm long, white, on 15 mm dark blue filaments. Style branches indigo, shading to almost white at base. Ovary fusiform, 6-angled, to 15 mm long. Fruit and seeds not seen. Flowers in August in Adelaide, long before I. germanica and the Tall Bearded Irises.
Thus it differs from I. aphylla sens. strict. in the leaves, which are seasonal, fewer, broader, and strongly falcate in that species (Dabrowska et al., 2019).
There are several irises in southern Europe close to I. aphylla and included in it by Service (1997) that have also been given names at species level. They may be of hybrid origin (Colosante & Mathew, 2008) but none of them matches the one described above.
Many of the supposed species in Iris Section Iris could be ancient garden hybrids that have been given binomials as if they were actual species that consist of wild populations. It’s a similar story in many horticultural genera such as Citrus, where our familiar oranges and lemons are ancient hybrids bred from less familiar Asian species. Long before the well-documented hybridisation of irises over the past two centuries (Darlington, 1973) there may have been earlier cycles of hybridization, selection, and the survival of some forms as feral populations that botanists in recent times have interpreted as species. My father used to say that history is a lot longer than most people think, and this is surely true of the history of plant domestication.
I. aphylla sens. strict. is a tetraploid with 4n = 48 chromosomes, while I. albicans Lange and I. germanica are reduced tetraploids with 44. All three behave as obligate outbreeders in cultivation; seed can only be produced by hand pollination with compatible pollen. When documented hybridisation began in the 19th century the first Tall Bearded Iris cultivars were diploids produced from I. variegata L. and I. pallida Lam. with 2n = 24, and the contemporary tetraploids and hexaploids are derived from them.
For comparison, here is a plant matching Iris germanica L. in morphology and pigmentation – although I haven’t seen its chromosomes.
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References
Colasante, M. & Mathew, B. (2008). Species of natural hybrid origin and misinformation in the Irises: A reappraisal of the presence of I. aphylla L. in Italy. Plant Biosystems 142: 172-178.
Dabrowska, A., Smigala, M. Denisow, B. & Winiarczyk, K.(2019) Biology of flowering and insect visitors of Iris aphylla L. (Iridaceae). Turk. J. Bot. 43: 798-808.
Darlington, C.D. (1973) Chromosome Botany and the Origins of Cultivated Plants. 3rd edn (Allen & Unwin: London).
Service, N. (1997) Section Iris. In The Species Group of the British Iris Society A Guide to Species Irises. 17-56. (Cambridge University Press: Cambridge).
Watsonia fulgens
Posted: May 1, 2023 Filed under: Botany | Tags: Iridaceae, Watsonia Leave a commentWatsonia fulgens (Andrews)Pers. based on Antholyza fulgens Andrews was regarded as a nomen confusum by Goldblatt (1989) because the type illustration could not be matched to any wild population. Andrews’ description of this plant whch had been introduced to England in 1792 was little more than a diagnosis differentiating it from Antholyza ringens (= Babiana ringens): it had much longer glabrous leaves that remained green until new growth appeared, and bright scarlet, curved trumpet shaped flowers with large spreading lobes.
Ker Gawler (1802) treated it as a distinctive variety of Watsonia iridifolia (Jacq.)Ker Gawl., which is another name of uncertain application. An illustration by Planchon (1856) under W. iridifolia var. fulgens matches a clone that is still widely grown in Melbourne although apparently not commercially available. Planchon noted that it flowered in autumn with a scape to 1-2 metres long, far exceeding the leaves, simple or sometimes branched in vigorous specimens. Plants of this name were being sold in England by 1820 (Loddiges, 1820). In New South Wales, Macarthur (1843) had a plant he called Watsonia iridiflora fulgens and presented material to the Sydney Botanic Gardens in 1831.
The following description is based on accession 180 in my collection:
Evergreen, proliferating, to 150 cm tall. Basal leaves about 4, to 60 cm long, 35 mm wide, bright green with faint glaucous striations and thin green margins. Stem leaves 2, bract-like, slightly inflated. Flowers 24-28 (to 4 open at once) on a brown axis plus 0-2 short branches. Bract acute, to 19 mm long, exceeding the internode, brown-herbaceous. Bracteole subequal, obtuse or notched at apex. Perianth intense orange-red, with a paler star inside throat, alternating red and pale stripes inside tube. Tube to 49 mm long; basal part to 23 mm long; distal part cylindric, curved, to 26 mm long, 8 mm wide at mouth. Ridges absent. Lobes semi-flared with flat margins; outer acute, oblanceolate, to 27 mm long, 11 mm wide; inner elliptic, obtuse, to 28 mm long, 14 mm wide. Stamens closely arcuate with style, anthers 11 mm long, purple with purple pollen. Style branches far exceeding anthers, red with paler stigmas. Capsule cylindric, truncate, to 25 mm long, brown. Seeds with two short wings, 8-10 mm long, dark brown.
Unlike Watsonia tabularis and W. fourcadei, this plant is undamaged by full summer sun in Adelaide as long as it gets enough water. New shoots appear in January while the previous year’s leaves are still green. Flowering is irregular any time from April to September.
There is a superficial resemblance to photos of wild W. zeyheri in colouring: orange-red flowers on a dark axis. But accession 180 is clearly separated from this species by its size, truncate capsules, autumn-spring flowering season, non-thickened leaf margins and the rather characteristic pale star marking in the flowers.
One possible origin could be a garden selection from random hybrids between W. tabularis and W. zeyheri or W. angusta, with strong, hardy growth in cultivation due to F1 vigour. An irregular flowering season is common in Watsonia hybrids between parents with differing phenologies. It also resembles my hybrids of typical W. tabularis pollinated by W. fourcadei in such features as size, flower colour and capsule shape. The four species mentioned in this paragraph are closely related and were treated as the Subsection Angustae in Goldblatt’s revision.
Below is Planchon’s illustration. The prominent leaf venation may be the artist’s interpretation of the striated glaucous bloom emphasising the longitudinal veins.
And the type illustration from Andrews. Assuming it is the same plant as Planchon illustrated, this is less informative. Perhaps it was grown in shaded or otherwise unfavourable conditions, as he described it as only 3 feet tall.
The plant known as Watsonia fulgens has been a “thing” for over 200 years. If it does not match any wild population, perhaps it should be treated as a cultivar. Unfortunately the name has been loosely applied in horticultural literature, for example to W. angusta by Campbell (1986). Watsonia fulgens sensu Montague (1930) was probably a hybrid cultivar; it was described as having pale-rose flowers appearing early in spring. It was distributed by Law Somner (1933) and may have been identical to the Watsonia fulgens described as a deep pink in Brunning’s 1905 and 1918 catalogues.
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References
Andrews, H.C. (1801) Botanist’s Repository 3: t.192.
Brunning, F.H. (1905) Manual of Seeds, Bulbs, Horticultural Sundries. (F.H. Brunning Pty Ltd: Melbourne).
Brunning, F.H. (1918) Winter Flowers, Bulbs, Spring Flowering Sweet Peas. (F.H. Brunning Pty Ltd: Melbourne).
Campbell, E. (1989) Watsonia. In Walters et al. (eds) The European Garden Flora 1: 385-386. (Cambridge University Press: Cambridge).
Goldblatt, P. (1989) The genus Watsonia. (National Botanic Gardens: Kirstenbosch).
Ker Gawler, J.B. (1802) The Botanical Magazine 17: t.600.
Law Somner Pty Ltd (1933) Law Somner Catalogue 1933-34. (Law Somner Pty Ltd: Melbourne).
Loddiges, C.L., Loddiges, G. & Loddiges W. (1820) Catalogue of Plants which are sold by Conrad Loddiges and Sons, nurserymen, at Hackney, near London. (Loddiges: London).
Macarthur, W. (1843) Catalogue of Plants Cultivated at Camden.
Montague, P. (1930) The new watsonias should be freely grown. The Australian Garden Lover 6: 33.
Persoon, C.H. (1805) Synopsis Plantarum 1: 42.
Planchon, J.E. (1856) Flore des Serres et des Jardins de l’Europe. 11: 1.
‘The Compulsion to Move’ by Dennis Stephens
Posted: April 29, 2023 Filed under: Philosophy | Tags: TROM Leave a commentThis is a new transcription of a talk by Dennis Stephens as discussed in a previous post.
You can download the 108Kb pdf file from this link.
Stephens chose the chess term Zugzwang for a situation where a player is obliged to take one of two (or more) actions knowing that either will result in a loss. There are many colloquial English terms for this situation – in a cleft stick, between a rock and hard place, damned if you do and damned if you don’t, caught between the devil and the deep blue sea – but none of them explain how the dilemma arises.
Zugzwang happens because the game on which the person is focused exists within a broader encompassing game. For example, a businessman is playing the game of making profits, but he is within the larger game of the society where he operates and the laws of that society.
Perhaps the most uncomfortable thing about being in a Zugzwang situation is that the ball is in your court. It’s your move. You’re free to choose although every option is a losing one. The player still has their self-determinism and can be held responsible for whichever losing choice they make.
Their only solution is to change their own postulates, their own aims, so that the outcome is no longer considered to be a loss.
The original audio can be found online at Tromology and TROM World.
ādarśajñāna 