Litrefs Articles: science

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Friday, 24 April 2015

Schrodinger's cat in literature

Schrödinger wrote: A cat is penned up in a steel chamber, along with the following device ...: in a Geiger counter, there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer that shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.

I suspect that allusions in literature to Schrödinger's cat aren't always understood correctly. Why is that situation any different to these?

A soccer referee tosses a coin before a match, covers it, asks a captain to call heads or tails. Until the coin's revealed, nobody knows what it is.
A prisoner's locked in a dungeon and the key thrown away. Sooner or later they'll die, but we won't know when.

At the sub-atomic level, strange things happen. We can predict probabilities of where a particle is likely to be with great accuracy, but we don't know where it actually is until we look. So far, this sounds like the coin situation - 50% chance of heads, 50% chance of tails. The difference is that whereas in the coin situation the coin's state is fixed before it's revealed, in the subatomic case the possible states simultaneously exist until the moment of observation, at which point the particle suddenly decides where it will be.

This may sound odd, but we're prepared to accept it for sub-atomic particles because experiments support it. Time for some terminology

"superposition" - the simultaneous multiple states
"interference" - the effect of the states on each other (i.e. proof that the states exist). "an individual particle, such as a photon (light particles) can cross its own trajectory and interfere with the direction of its path" (from http://whatis.techtarget.com/definition/quantum-interference)
"collapsing the probability function" - decision time
"The Copenhagen interpretation" - the idea that a system stops being a superposition of states and becomes a specific state when an observation takes place.

Suppose that instead of one particle there are several - a human-sized bundle of them in a room. Suppose that this bundle has the same indeterminacy as a single particle. We might be able to calculate probabilities of the person's position (50% likely to be in bed, say, and 50% looking out of the window). To us outside the room the person IS 50% in bed AND 50% at the window, but now that we have a sentient, self-conscious being instead of a sub-atomic particle we have difficulty putting ourselves in the situation of the person, who must surely be in only one place. However, being in one place would break one of the most successful theories ever.

Schrödinger's paradox has 2 extra features added to the scenario that I've sketched -

The stakes are raised. The indeterminacy concerns not position but life
Rather than saying that the cat behaves like a particle, its life depends directly on the behaviour of a particle. A machine translates probability from one scale to another.

Life and self-awareness are emergent qualities - ones that are hard to deduce, calculate or describe using the underlying concepts. The characters in these thought experiments may be self-consciousness, but what does that mean? At what precise moment is the cat dead? Maybe it could be revived?

The worldview of the people outside the box is hard to sustain (how can the cat in the box be both alive and dead?), but trying to imagine the worldview of the cat (or better still, a person) inside the box is harder still. One theoretical way out is multiverses - the idea that the universe splits into 2 with one copy having a live being and the other copy a dead one. But when does it split? Whenever an observation is made?

Translate the scenario from involving particles to involving thoughts, and the paradoxes disappear. Suppose someone's asked "Will you marry me?" They might have anticipated the question, knowing how they'd reply. That corresponds to the coin tossing situation. Alternatively, they may be uncertain, hoping they won't be asked. But the questioner insists on an immediate answer (they've just been offered a job abroad, maybe). Somehow, from the mess of conflicting thoughts, a single answer must be given. That corresponds more to the cat situation - the question forces a collapse of the probability function.

The idea's been used by several SF writers and some less illustrious poets, often for comic effect. The idea of large-scale superposition is I think the key feature of the concept. Examples used by writers may include

Replacing Death Row by cells like the cat's
Having more than one entity in the cell - perhaps all existing self-conscious entities except one - God.

Frayn's "Copenhagen" is considered a worthy artistic treatment of such issues.

The title of Heidi Williamson's poem "Schrodinger's pregnancy test" promises more superposition than the poem explicitly delivers. It begins with "For her, theoretical physics/ is a bird soaring next to a plane" ending with "All it takes/ is a broad mauve line in a window// for it to land, bang/ on her heart". Interestingly, a search online for "Schrodinger's pregnancy test" comes up with entries like that of naming the time between where the author describes what it's like to await the test results - "What makes this experience unique, compared to situations of waiting in unknown like waiting for results of a job application, is that the box of indeterminancy is located within your own body. That box that contains one truth and the experience of living with two, that is placed in the intimate space of your self.".

Saturday, 15 January 2011

Poetry about Science in the UK

Our world has been transformed by technology so one would expect cars, TVs and test-tube babies to appear in poetry. And as the percentage of scientists in the population grows, the lives and preoccupations of their profession feature more often in poems.

But the world we can't see directly has changed too. Relativity and (even more so) Quantum Physics have shown us that we live in a world beyond what common sense can cope with. And Mathematicians, dealing with topics like (and here I quote at random from the first maths journal at hand) 'Examples of tunnel number one knots which have the property 1+1=3' live in a stranger world still.

One might have thought that adventurous poets would have rushed into these newly opened territories, but it seems to me that poets over the last few centuries have withdrawn from trying to tackle the big questions about the Nature of the Universe. They tend not to deal with the moral complications that new technology engenders, and it's rare to see a Blakean anti-materialist piece attacking scientists.

In England the poetry of Prynne, Dorothy Lehane, etc sometimes includes a liberal sprinkling of science vocabulary. The risk with using science terms is that the poem is going to come over differently depending on how much science the reader knows. This risk applies to many types of allusions of course, but in the science case the reading communities are easier to define, and the material may more easily become out-dated.

Using science words is easy. Less frequently, poets deal with science (and maths) concepts. We are used to philosophical or religious poetry, poetry that presents an argument. Quantum theory and Relativity are common themes for those wanting to express scientific ideas poetically. More mainstream than Prynne and Lehane are Heidi Williamson and Lavinia Greenlaw who use science or (more often) scientists as subject matter. I think poets' attempts to write science poems fall into these main categories

Awe and Wonder - like children given a microscope for Christmas. Science is the new Exotic to some (mysterious trinkets from another land), to others it's the new Theology - deep truths masked by code. With its cornucopia of new ("X-ray") and re-used ("charm") vocabulary it's tempting to raid its word kitty. If the result sounds clever, the reader might think the poet's clever too. Few readers are going to be in a position to challenge from a scientific position, and in any case, what would it prove? A poem's not a thesis. However, I suspect that if poets appropriated the vocabulary of Art in similarly cavalier fashion, they wouldn't get away with it.
Effects on relationships - e-mail and the mobile phone
Science Fiction Poetry - there's a book of StarTrek poetry
Biographical pieces about famous scientists

These are topics that can easily be tackled by people who know little about science. And yet, more poetry is written by scientists nowadays than in any previous era. So why don't we have more poetry from the frontiers of science? On the international front, the internationally acclaimed Czech poet Miroslav Holub was also a serious immunologist. It is perhaps significant that he had doubts about twinning poetry and science. In "The Dimension of the Present Moment" (Faber and Faber, 1990) he wrote

"At first glimpse one might suspect that literature would be closer to the sciences than other art forms, because sciences also use words and depend on syntax for expressing their findings and formulating ideas. ... [but] There is no common language and there is no common network of relations and references. Actually, modern painting has in some ways come closer to the new scientific notions and paradigms, precisely because a painter's vocabulary, colours, shapes and dimensions are not congruent to the scientist's vocabulary." (p.130)
"In the use of words, poetry is the reverse of the sciences. Sciences bar all secondary factors associated with writing or speaking; ... poetry tries for as many possibilities as it can." (p.132)

In New Scientist (24 July 1999) Graham Farmelo (Science Museum, London) wrote "Be sceptical of any science-art initiative and you are liable to find yourself marked down as a narrow-minded reactionary. If a new work of art is based on a theme related to science, most critics will give it an easy ride... It seems that this flavour of political correctness encourages intellectual laziness, allowing shallow and sentimental nonsense about the relationship to pass for serious thought". So perhaps we should be wary of some recent initiatives.

In the UK ex physics/maths professors appear in small magazines, and Mario Petrucci, who's active in many areas of UK poetry, has a physics Ph.D. London's Science Museum sometimes has a poet-in-residence. The best known holder of that post is Lavinia Greenlaw. In 2000, she was awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts. She said she'd use the £67,000 to "undertake formal study" in science, and journey "to places with extreme perspectives - precarious and changing landscapes, or those which experience the natural phenomena of eclipses and equinoxes". For television, she has written a sequence of poems about the meaning of numbers for an Equinox documentary. Her WWW site is http://www.laviniagreenlaw.co.uk. Also worth a look is LUPAS (Liverpool University Centre for Poetry and Science).

Heidi Williamson was poet-in-residence for the London Science Museum’s Dana Centre in 2008 and 2009. On the back of her book it says that her "fascination with science leads her to explore less usual territories for poetry, including mathematics, chemistry, and computer programming, as well as space travel, electricity, and evolution". I think that more poets who write about science go into it nowadays with their eyes more pragmatically open than that -

"The main difficulty with 'Night Photograph' has been the “poetry about science” tag. I grew up in a family of scientists and have long been fascinated by time and space, so this is a natural source of metaphor for me. I only became conscious of how much science there is in the book when it was pointed out. Since then, I have resisted science like hell. It is mostly too seductive, incomprehensible and exciting to be anything other than borrowed" - Lavinia Greenlaw, interview in Thumbscrew (1997)
"I married an astronomer! ... I think initially I was trying to write metaphors for the science, based on human experience but that wasn’t working out so well, the science was present, but the poetry seemed dry. I didn’t think I was achieving anything more than representing the original idea, theory, or astrophotography I was looking at, or the paper I was researching. So I tried to do more than represent the original by using the science as a launch pad but moving away from it, by keeping a dialogue with human concerns at the same time" - Dorothy Lehane, interview in Annex (2013)

Andrew Duncan reviewing Lehane's "Ephemeris" in Litter magazine writes that "Lehane’s project has to do with combining poetry and science. The two are intertwined in a very specific way here: objective knowledge separates projections of feelings and wishes from the information provided by the eyes, but here the idea is to interfuse them. Her poems are intensely personal and highly coded: everything profound loves a mask".

He goes on to suggest that "The poetry-science project is likely to draw a great deal of attention in the next twenty years or so. It is quite hard to define what the purpose is; I think the core is the sense of opportunity, that there is a wilderness here, and that if you buy creative people time they will wander around that wilderness and bring back things never before seen. Part of the impetus is the wish of museum staff to make their holdings presented anew in visible or audible form."

If you want to read science poems, 2 UK anthologies of note are

"Poems of Science", eds J. Heath-Stubbs and P. Salman, Penguin, 1984.
"A quark for Mister Mark", eds Maurice Roirdan, Jon Turney: Faber, 2000.

Thursday, 13 January 2011

Science and the Arts

"The purpose of Art is to impart the sensation of things as they are perceived, and not as they are known" - Shklovsky [30]
"Art works because perception goes beyond the evidence and it ignores much counter-evidence" - Richard Gregory [13]
"Art evokes while science explains" - quoted by Richard Gregory [13]
"Science is a system of statements based on direct experience and controlled by experimental verification." - Carnap [5, p.42]
"Science, far from destroying the beauty and romance of the world as seen by artists, musicians and writers, enhances it by revealing the underlying reasons and purposes" - McConnell [21, p.2]
"Science is for those who learn; poetry for those who know" [23]

Newton received a mixed reception from poets. As the scientific revolution that he spurred took hold, the Romantics protested against its mechanistic abstractions. The protest was, in hindsight, valid though it wasn't until the 20th century that mainstream scientists realised the truth of the accusations [9, p.108]. The protest was also ill-informed - the later Tennyson was much better read in the sciences and subsequent generations of poets have always included a few professional scientists. Apart from the debate sparked off by C.P. Snow's "Two Cultures" [31] and sporadic fascination with polymaths like da Vinci, 20th century pure science and art continue in splendid isolation, though "the rise of science has had something to do with displacing [poetry] as a publicly important vehicle for those truths that people accept as being centrally important." [4]. The challenge of Positivism caused the Arts to retreat from making great claims about `truth'. Poetry, for example, was cut down to size by I.A. Richards and isolated/protected from other disciplines by its claim to be a unique mode of discourse. Meanwhile literary theory (as opposed to literary criticism) prides itself on being ever more scientific. Technology, with its invitation to control and change, has entered into all aspects of our everyday lives, making artists more aware of contemporary science than scientists are of modern art. This new openness from artists presents an opportunity to build greater understanding between the `cultures'. Connections and analogies have been advanced. Rather than give a general review of the (rather disappointing and often acrimonious) literature I shall pick out a few of the false resemblances and sweeping generalisations that recur, steering clear of sociological and historical discussion in an attempt to find some common ground.

Science is Materialist

Classical science was materialist - the real world was undeniably out there. Relativity was still a classical theory, though more phenomenologically based. With Quantum Mechanics the observer entered into the equations, exciting artists. Notions that the world was "out there" waiting to be discovered were questioned. Many quantum findings still feel counter-intuitive and their interpretation controversial; it's the area where science and common sense part company for good. Maths might be thought to have a simpler foundation, but in fact it encompasses a range of ontological outlooks similar to those in other science disciplines.

"The platonist metaphor assimilates mathematical enquiry to the investigations of the astronomer: mathematical structures, like galaxies, exist, independently of us, in a realm of reality which we do not inhabit but which those of us who have the skill are capable of observing and reporting on. The constructivist metaphor assimilates mathematical activity to that of the artificer fashioning objects in accordance with the creative power of the imagination.", Dummett, [9, p.225]
"In so far as the statements of geometry speak about reality, they are not certain, and in so far as they are certain, they do not speak about reality", Einstein [11, p.3]

Overall, the metaphysical basis of the sciences is closing in on that of the arts - science becoming more constructivist, art less platonic - confirmed early this century by a shared trend towards phenomenology led by Mach in science, and Pound in poetry. It may well be that, as Jon Corelis in private correspondence has suggested "science has made it impossible for poets to value their perceptions as having an intrinsic metaphysical validity - a validity which is proven by the desire, reverence, or exaltation they instill in us. Instead, poets like scientists must now use their perceptions as sense-images (analogous to observations) to which they must attempt to give value by poetic imagination (analogous to scientific theory.)"

Science is Reductionist and Closed

Though science doesn't hesitate to break the whole into parts, art does it too. Even in meditation, people sometimes concentrate on parts of the body first. And many 20th century art developments arose from separating form from content and recombining.

Of the sciences, maths is considered the one that's most simplified, derived from axioms, but Gödel showed that there are true statements which can't be proved. This finding has affected the grand designs of mathematicians. No longer do they look for a list of axioms and rules that can be used to generate all the theorems. It's made no difference to their other work because there's no way of knowing beforehand whether a proof that's being sought exists or not. Certainly no other fields of science have been affected.

Science seeks generalisations but ones which don't compromise. It seeks all-encompassing rules rather than rules which, though covering more cases, leave behind a slew of exceptions. "Reality is complex whereas truth is simple" says John Light [20]. But isn't truth an accurate statement of reality and thus at least as complex as its subject matter? The generalisations of science are simpler only because they're formulae, not lists of particulars.

Science is based on maths, and maths developed on Aristotlean logic where statements are either true or false. This excluded middle needn't permeate up through to science, and besides, new multivalue logics have been developed. Nevertheless it's true that science dislikes ambiguity, (which Empson considered a defining characteristic of poetry). At least when science is reductionist it usually states its assumptions.

The methods of science are popularly thought to be appropriate for only certain kinds of problems - truths of the world, not of people. Though "the part played by new observation and experiment in the process of discovery in science is usually over-estimated" [26, p.28] and most of the interesting results come from "reconsideration of known phenomena in a new context" [26, p.28], science depends on repeatable observation and prediction. In the main, science recognises these limits, perhaps anticipating that its time will come when new instruments and disciplines appear that can probe the previously hidden. For instance in the 70's interest grew in the study of chaos and complex systems, bridging, if only slightly, the gap between the domains. The use of neural nets in computing may signal progress towards a more human, pattern-matching type of computer.

Only in Science can works be wrong or outdated.

There's a feeling (expressed by Paul Mills [24], for example) that science has truth values that are denied to works of art, and that science theories have a limited lifespan. However, truth, even in science, is culture and context dependent, controlled by the reigning (Kuhn) paradigms. Both art and science are susceptible to the Zeitgeist (Rom Harré, [27]). Trends like subjectivity, symmetry, interaction and atomicity can come into favour (Philip Gell, [27], Halliwell, [15]), affecting notions of truth and beauty across the board.

Even if a science theory has been shown to present a fundamentally wrong model of reality, it can still be useful. Einstein's view of the universe superceded Newton's, but Newton's laws got us to the moon and back. Einstein's gravitation theory can't cope with quantum effects, but theoreticians still depend on it. Ultimate truth is not the only factor determining a theory's lifetime. The theoretical physicist Dirac said that when he had to choose between beauty and truth, he always chose beauty [8], expecting later experiments to prove him right.

Even in maths there is less certainty than is generally thought. In axiomatic systems (Euclidian Geometry, for example) there are sometimes disputes about which statements to use as axioms. In Number Theory the Axiom of Choice is contentious (thought by some to be not self-evident) but without it many other results couldn't be proved. And perhaps worse still, we know there'll always be true statements which can't be proved.

In science (and especially the arts) not all the past is discarded. "Both tend to stability by precedents from the past" [13]. An expression like e is an allusion to earlier work, just as much a shorthand as Eliot's burnished throne is.

Art is more natural than science

The methods of science have been compared to those used by children during normal development. Babies learn of the world using observation, experimentation and deduction. Mature science is forging more links with other fields of human involvement; witness Paul Davies' attempts to connect science and religion [7].

Perhaps it's time to look in more depth at those who've excelled in both fields to see if they get similar satisfactions out of their dual endevears. William Empson and Valery were once maths students. Holub, Primo Levi, Goethe, da Vinci, Danny Abse and William Carlos Williams all pursued dual careers. It seems to me that the incidence of scientific and artistic talent in the one person is no more than one would expect were the talents independent. There seems little cross-influence except that poet-scientists use their science experiences as subject matter and some of them (Edward Lowbury, for instance) attribute their dislike of obscurity in art to their scientific upbringing.

Art's a richer language than science

Maths and music make claims to be universal languages of sorts. Science's base metaphors are increasingly mathematical - building conceptual models from billiard balls is a thing of the past. Some theorists (for example Wimsatt) consider metaphor central to poetry. Colin Turbayne [33] thinks that that science is metaphor-laden too, the metaphors dead. Waismann [quoted in 17] argues that scientific concepts are only closed in specific contexts and that they are not different in kind to the metaphors of poetry. Indeed, his "open texture" concept was developed initially to deal with the language of science.

Natural language has room for various language games. Maths hasn't - games aren't maths any more. Maths is one of the games that can be played in language - and you've got to have a net. Its range of expression is limited and consequently maths requires a longer apprenticeship before creative work can be done. Children can write poems but even undergraduate maths students can't express themselves.

Appreciation of the arts also requires a long apprenticeship. The difference between the cultures is that people looking at a Constable can say "that's nice" because it's expected of them or because they like the countryside. Science rarely lets people off so lightly - scientists have to learn how appreciate as well as express, they have to learn the language. Poetasters can fake it.

I suspect that people with an impoverished appreciation of the arts (most of us) can only extract from art `truths' that they already know from lived experience. Playing Mozart to bushmen or reading Ashbery to almost anyone won't impress the audience. Art isn't clearly more natural or even richer than science.

Art and science have much in common

Prof Robert May [22] thinks that "the essential aim of science is to understand how the world works. This is also true of the arts ... the technical trappings of science obscure its underlying kinship with the arts". This is too low a common denominator to be of much interest even if it were true. I think the differences lie deeper than that, and that the aims of art are less clear. The arts also have their fair share of technical obscurity. Like I. A. Richards I'm unimpressed by such attempts to show that "the functions of science and poetry are identical" [28, p.62]. However I think practitioners share some heuristics: concepts may be held together in the mind to see if anything develops (for artists the concepts may differ widely); concepts grow by association (artists have more scope, fewer boundaries and can mix layers); experience builds templates so that future similar situations can be more easily dealt with by pattern-matching.

Intuition and imagination are highly valued in the sciences, though one can plod along quite merrily without them (but then, one can be a very technically accomplished and 'successful' concert pianist, I'm told, yet have little feel for music).

I.A. Richards thought that "the imaginative life is its own justification" [28, p.66]. More recently Holden in [19] says "Poetry is like Pure Maths, an end in itself" but the Arts can contribute to the Zeitgeist, which in turn affects scientists. On the whole it appears that science has been more useful to the arts than vice versa - it has provided materials (oil paints), artforms (cinema) and subject matter (science fiction). It has changed the world that some think is art's duty to describe.

Attempts have been made to find analogies between arts and sciences. Buchanan thought that "The symbolic elements of poetry are words, and the corresponding elements of mathematics are ratios" [2, p.18] and goes on to say that "The mathematician sees and deals with relations, the poet sees and deals with qualities. Functions and adjectives respectively are the symbols through which they see and with which they operate" [2, p.135]. This promising start isn't built on. Elsewhere, the analogies are neither surprising nor interesting:

Methodology - A case has been made (in [21]) that the processes of art and science correspond. I think the likeness rather tenuous. They split the process into 3 contentious stages

observation - true, this is done in both fields. Sometimes in the arts the thoroughness might be thought to match that in the sciences (Monet's multiple views of Rouen Cathedral, maybe, or Picasso and Braque's cubist experiments or a writer's research for a novel).
generalisation - this is seldom done in the arts. Perhaps an artist might extract the general features of sadness from the hundreds of faces seen, but such a result isn't used like generalisations in science. The aim of artists more to explore differences than find commonality.
testing - This stretches the metaphor too far. Comparing how scientific results are judged with the way that works of art are received by the public and critics begs too many questions. Apart from anything else, creators in the scientific world can propose ways that their work might be invalidated. In the arts there's nothing comparable.

The Scientific Method is only science's sense organs, extending mankind's natural ones. It's not the be-all and end-all of science.

Quantum Theory - In Quantum Theory, probabilities can be calculated but only when an observation is made can any certainty be established. Observation is said to "collapse the probability function." This has been used for an analogy to the way that a text is interpreted (dis-ambiguated) by the act of reading [24]. But texts can be re-read!

Relativity - Connections are made between Einstein's Special Relativity and analytic cubism. Awareness of the equal importance of world viewpoints, the impossibility of absolute motion and time perhaps permeated via the Zeitgeist to artists; the link came from no deep mutual understanding.

Gödel - Gödel's findings (see above) have helped soften artists' views on science and has removed an aim of classical science. They have only made maths more obviously like the other sciences. The gap between science and the arts hasn't thereby been reduced.

Geometry - Mondrian is heavily geometric and minimalist. This doesn't make him more appealing to mathematicians. Equally, the 4-colour problem in maths isn't appealing to artists.

The best scientists and artists may have more in common than Science and Art do. "Although scientists were found to share many similarities with artists (openness, flexibility, cognitive complexity, self-confidence, dominance and introversion) they differed in their view of power: scientists were found to be more accepting of authority whereas artists were more rebelious" (The Times Higher, 22/3/12, p.44)

References

Bush's and Nicolson's books and the introduction to "Poems of Science" give useful historical accounts. None of the other references are highly recommended. Green's book is interesting in that he's an arts person who took Snow to heart. Polanyi's book is worth a read too.

1. "Science and Human Values", J. Bronowski, London, 1961.
2. "Poetry and Mathematics", Scott Buchanan, Univ Press of Virginia, 1929.
3. "Science and English Poetry", D. Bush, OUP, 1950.
4. "Poetry in a Discouraging Time", Christopher Clausen, Georgia Review, Winter 1981.
5. "The Unity of Science", R. Carnap, London, 1934.
6. "Nature's Imagination: The Frontiers of Scientific Vision", John Cornwell, OUP, 1995.
7. "The Mind of God", Paul Davies, Simon & Schuster, 1992.
8. Paul Dirac in "Scientific American", CCVIII, May 1963.
9. "Truth and Other Enigmas", M. Dummett.
10. "The Concept of creativity in science and art" edited by Denis Dutton and Michael Krausz, Kluwer Boston, 1981.
11. "Geometry and Experience", Einstein.
12. "Art in the science dominated world: science, logic and art" by E.L. Feinberg; translated from the Russian by J.A. Cooper, New York: Gordon and Breach Science Publishers, 1987.
13. "Royal Society Conference on The Visual Culture of Art and Science", R Gregory, July 1995.
14. "Science and the shabby Curate of Poetry", M. Green, Longmans, 1964.
15. "Archadia, Anarchy and Archetypes", Jonathan Halliwell, New Scientist, 12th Aug 1995.
16. "Poems of Science", ed J. Heath-Stubbs and P. Salman, Penguin, 1984.
17. "Metaphor's Way of Knowing", P.L. Hagen, Peter Lang, 1995.
18. "Gödel, Escher, Bach: An eternal golden braid", D. R. Hofstadter, Penguin, 1980.
19. "Poetry and Mathematics", in "Style and Authenticity in Postmodern Poetry", Jonathan Holden, Univ of Missouri Press, 1986.
20. "Science, Truth and Art", John Light, Poetry Nottingham, V49, No 2, 1995.
21. "Art, science and human progress: the Richard Bradford Trust, lectures" edited by R.B. McConnell, London: John Murray, 1983.
22. Prof Robert May (Chief Scientific Advisor to the UK Government), "Daily Telegraph", Sept 6th, 1995, p.14.
23. "Meditations of a Parish Priest" Pt i, N. 71.
24. "The Quantum Uncertainty of the Narrator", Ian Mills, in "Poetry Review" V85.1, Spring 1995.
25. "Newton demands the Muse", M.H. Nicolson, Princeton Univ Press, 1946.
26. "Science, Faith and Society", Polanyi, Univ of Chicago Press, 1964.
27. "Common denominators in art and science", edited by Martin Pollock with the assistance of Keith Brown, Aubrey Manning and Barrie Wilson, Aberdeen: Aberdeen University Press, 1983.
28. "Poetries and Sciences", I.A. Richards, Routledge & Kegan Paul, 1970.
29. "Science and Truth", I.A. Richards, 1929.
30. "Art as Technique", Shklovsky, 1917.
31. "The Two Cultures: and a second look", C. P. Snow, Cambridge, 1964.
32. "Newton's Sleep", Raymond Tallis, P.N. Review, V17, Nos 3-6, 1991.
33. "The Myth of Metaphor", C. Turbayne, Univ of South Carolina Press, 1970.

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