Errata: The Reflection of Life
1 Mapping Origins
p.15 l.15 (1) [subscript of the second product] (1,2,…,n) → {1,2,…,n}
p.25 l.24 single valued → single-valued
2 From Points to Sets
p.35 l.8 Pf → PF
8 Imminence of Life
p.141 l.8 Probabilitiés → Probabilités
This is an unpublished short note that I wrote in 2004: Multi-time [pdf]
Abstract
In the paper “Dissipation, Lorentz metric, and information: A phenomenological calculus of bilinear forms” [A. H. Louie & I. W. Richardson (1986) Mathematical Modelling 7, 227−240 sciencedirect.com/science/article/pii/0270025586900497], we described how our four- dimensional spacetime arose from a special bilinear form on our “description space”. For n=2, the splitting of the 4=22 dimensions into 3+1 of space and time is a natural consequence of the theory. For a general n≥2, the splitting of the n2 dimensions is ½n(n+1) for space and ½n(n−1) for time. In this paper, I revisit the theory that leads to universes with space of more than three dimensions and time of more than one dimensions.
Since 2004, I. W. Richardson and I have published two more papers in the phenomenological calculus sequence:
Louie, A. H. and Richardson, I. W. (2006)
A phenomenological calculus for anisotropic systems
Axiomathes 16(1−2), 215−243
springerlink.com/content/x2p11tg1984wq858
Richardson, I. W. and Louie, A. H. (2007)
A phenomenological calculus of Wiener description space
Chemistry and Biodiversity 4(10), 2315−2331
onlinelibrary.wiley.com/doi/10.1002/cbdv.200790189/abstract
We (in the Rashevsky-Rosen school of relational biology) have sometimes been asked by experimenters why we do not propose explicit experiments for them to perform, and subject our approaches to verification at their hands. We do not do so because it is precisely physicochemical particulars that are abstracted away in the process of generating relational models. There is no kind of one-to-one relationship between relational, functional organizations and the structures that realize them. A functional organization cuts across physical structures, and a physical structure is simultaneously involved in a variety of functional activities. So an (M,R)-system is not realized by identifying its components and maps in a ‘concrete’ biological example. To tackle the biological realization problem of (M,R)-systems, one ought not to be seeking physicochemical implementations of what the relations are, but, rather, one ought to be seeking interpretations of what the relations do.
In any case, the basic questions of biology, in our view, are not empirical questions at all, but, rather, conceptual ones. ‘Conceptual’ and ‘qualitative’ are not dirty words, but rather indicators of genericity.
As always, when one attempts to do theory, one is confronted with the trivia: Is it testable, and if so, how? People have been brought up with the prejudice that a scientific theory that is not testable is worthless. It is often considered part of the theorist’s job to make theory verifiable, in effect to construct some kind of experimental protocol for the sole purpose of falsification. As long as ‘experimental test’ exclusively takes on the conventional sense that prescribes ‘to verify some kind of specific physicochemical operation on individual systems’, there is in principle no way that the relational descriptions could in fact be ‘tested’.
A relational description of an organism is as valid, as realistic, a description as any conventional physicochemical one. But it is a description pertaining to a class of physically diverse (though functionally equivalent) systems. A well-constructed model creates a reality of its own: there is no model-independent test of reality. There are more kinds of experiments than physicochemical ones. Conceptual experiments are common in psychology and sociology, for example. Biology has a lot to learn from social sciences.
Relational biology is ‘decoding from formal system to realization’. Experimenters need something to verify, couched in terms of some specific observation, or physicochemical experiment, that they can perform. They need the ‘encoding’; that is to say, they need precisely what is not in our approach.
An act of observation is a quintessential act of abstraction. The observation of a single quality of a natural system is indeed the greatest kind of abstraction that can be made of that system, that of analytic modeling by a single real number. The development of theoretical science is thus a synthesis: an attempt to combine observations in such a way that our view of systems becomes less abstract than it could be if we were restricted to observation alone. There is no antagonism between ‘theory’ and ‘experiment’. That unfortunate mirage is an artifact of the antagonism between ‘theorists’ and ‘experimenters’.
Errata: More Than Life Itself
I fantasized the number of errors in More Than Life Itself to be zero, but hoped that it would be small, and that they would be slight and trivially fixable. My hopes were, fortunately, actualized. Tim Gwinn had found many of the errors (for which I am grateful), and he first posted his list on his website panmere.com. I also thank the readers who have been sending me corrections.
1 Praeludium
p.28 1.9 and write → and writes
2 Principium
p.49 l.-8 a1,…,an → a1,…,an+1
p.49 l.-7 xn Ran y → xn Ran+1 y
p.56 2.30 Let f and g be observable of X. → Let f and g be observables of X.
p.59 2.36 is and ordered pair → is an ordered pair
3 Continuatio
p.63 3.5 if it satisfy → if it satisfies
p.72 l.-8 Hesse diagrams → Hasse diagrams
p.73 3.27 Equivalently, lattice → Equivalently, a lattice
p.73 3.28 Equivalently, lattice → Equivalently, a lattice
4 The Modelling Relation
p.97 l.-8 The encoding arrow α → The encoding arrow ε
5 Causation
p.107 l.12 restrictive too encompass → restrictive to encompass
pp.122–123 The second half of Section 5.16 has been rewritten.
(An overly stringent application of “a mapping uniquely determines its domain and co-domain” led to the disallowance of two modes of connection of two mappings. These restrictions are my oversight, and turn out to be unnecessary. I have since lifted them in a 2010 paper “Relational biology of symbiosis”, Axiomathes 20(4), 495-509.
springerlink.com/content/h481521x7264626l.
p.124 l.-2 because diagram (35) my → because diagram (35) may
p.130 l.14 If in they heart → If in thy heart
6 Topology
pp.140–141 Section 6.10 has been rewritten.
This is to allow the acceptance of the disallowed connections (see note on pp.122–123 above).
p.146 The xi in (30) is larger than the other symbols. Reduced.
p.141 6.11 and (b) G has an Eulerian circuit → and (b) G has an Eulerian path
p.147 ll.6–7 Changed tab position so that ‘compositions’ fits in line 6.
p.156 The beginning of Section 6.24 has been rewritten
to mention pseudographs.
p.157 ll.8–9 it would force (65) → it would force (66)
p.157 l.10 but (65) contradicts → but (66) contradicts
p.157 l.-4 requires the some, → requires that some,
p.158 l.2 (61) → (67)
p.158 l.5 .. → .
pp.159–160 Discussion following Theorem 6.28 has been rewritten,
and a new Section 6.29 has been added.
Solid-headed arrow self-loops are discussed in detail.
7 The Category of Formal Systems
p.182 l.7 does not → do not
8 Simple Systems
p.213 l.4 a corresponding subsequence of program lengths
→ a corresponding subsequence with program lengths
11 Living Systems
p.259 Bertalanffy quote Hierarchical order is found → [Hierarchical order] is found
The actual line is “A similar hierarchy is found…”, where the similarity refers to the subject of the previous sentence: “One attractive scheme of hierarchical order (there are others) is that of Boulding (Table 1.2).”
p.260 11.2 “Nothing comes from nothing.” → “Ex nihilo nihil fit.”
p.281 11.16 Traversability as a Relation Diagram
→ Traversability as a Relational Diagram
13 Ontogenic
p.313 l.-4 my → may
p.314 l.-5 deploring → deploying
Appendix Category Theory
p.344 l.-1 picture (1) → picture (2)
p.345 l.1 A-morphism commutativity → A-morphism associativity
p.355 l.2 an B-object → a B-object
Bibliography
p.376 l.12 [2000] ‘Autobiographical…’ → [2006] ‘Autobiographical…’
Index
p.381 col.2 ll.-6 332 → 332–333
p.385 col.1 ll.2–3 213–216–219 → 213–216, 219
Journal of Cosmology, vol. 8, June, 2010
Abstract (by the editors)
On May 20, 2010, famed geneticist Craig Venter and colleagues published a landmark study in the emerging field of ‘synthetic biology’, the creation of an artificial bacterium genome (copied from DNA sequences of Mycoplasma mycoides) which was transferred into a closely related microbe which began to successfully reproduce, making over a billion copies of itself.
Venter’s achievement has drawn mostly enthusiastic praise, with some likening it to the ‘splitting the atom’ and deserving of the Nobel Prize. Yet others’ warn of a ‘Frankenstein monster’ and ‘genetic pollution’; fearing that artificial genes and artificial life may take over the world, and end life as we know it.
Scientists and bioethicists from around the world have been asked to comment and to explain. What is the real significance of this achievement, and is there any reason to feel fear?
A. H. Louie, Ph.D.,
Mathematical Biologist,
86 Dagmar Avenue, Ottawa, ON K1L 5T4, Canada
The achievement of The J. Craig Venter Institute, “Creation of a Bacterial Cell…” (Gibson et al, 2010), is an exceptional feat in biotechnology. And for that, the team must be congratulated. They did not, however, create a bacterial cell, and they did not produce synthetic life. Their work is one step closer to the telos of an artificial lifeform, perhaps, but the final goal remains out of reach.
They have clearly transformed one cell into another. But there is nothing new about the modification of existing organisms; people have been doing that for millennia, at least since the dawn of agriculture. In the 21st century, we simply have more sophisticated tools. The Venter achievement differs in degree, but not in kind. Instead of the now-commonplace partial modification of the genome by multiple insertions, substitutions, or deletions, they have synthesized the entire genome — but still just one component of the whole cell.
What it ultimately comes down to, as it does in most contention, is definitions. In Gibson et al (2010), they admit as much:
We refer to such a cell controlled by a genome assembled from chemically synthesized pieces of DNA as a “synthetic cell”, even though the cytoplasm of the recipient cell is not synthetic.
If it were truly and completely obviously a “synthetic cell”, no further justification of the selfevident usage would have been necessary. (But of course, they may simply define “synthetic cell” to be whatever they say it is, in which case its semantics become tautological.) Their ‘explanation’ that after “>30 divisions or >109 fold dilution”, “progeny will not contain any protein molecules that were present in the original recipient cell” is a very poor attempt in clutching at straws. Similarly desperate and silly is the next sentence “The properties of the cells controlled by the assembled genome are expected to be the same as if [italics mine] the whole cell had been produced synthetically (the DNA software builds its own hardware).”
Craig Venter reportedly said in an interview after the publication of Gibson et al (2010): “This is the first synthetic cell that’s been made, and we call it synthetic because the cell is totally derived from a synthetic chromosome, made with four bottles of chemicals on a chemical synthesizer, starting with information in a computer.” Note the unbridgeable gap between “synthetic cell” and “synthetic chromosome”. I think Craig Venter was closer to the mark in 2007, when (in a quote attributed to him) he likened the process to “changing a Macintosh computer into a PC by inserting a new piece of software”. In 2007 it was prophesied that artificial life would appear within months. Now, three years later (to continue the imperfect machine metaphor), the Venter group may have replaced the operating system, but they have not built a whole new computer from scratch — nor will they be able to in any foreseeable future.
Craig Venter has synthetic genome; George Church at Harvard has synthetic ribosome. Are we once again proverbially “within months” of a truly artificial lifeform? For fundamental logical reasons, this kind of ‘synthetic biology’ — a mechanistic, algorithmic, and by-parts fabrication of life — will not work. Biochemistry has progressed so far and so fast in the past century that people find it hard to imagine that the process cannot continue ad infinitum. The main problem is that the reductionist biology-is-chemistry approach has been so successful in solving biological puzzles, that although everyone can recognize that a living system is not just a machine, there is a great reluctance to admit that the two are different in kind and not just in degree.
Relational Biology
Biology is a subject concerned with organization of relations. A living system is a material system, so its study shares the material cause with physics and chemistry. But physicochemical theories are only surrogates of biological theories, because the manners in which the shared matter is organized are fundamentally different. Hence the behaviours of the realizations of these mechanistic surrogates are different from those of organisms. This in-kind difference is the impermeable dichotomy between predicativity and impredicativity.
The study of biology from the standpoint of this ‘organization of relations’ is a subject called relational biology. It was founded by Nicolas Rashevsky in the 1950s, thence continued and flourished under Robert Rosen (For a comprehensive exposition on relational biology, see More Than Life Itself (Louie 2009)).
The principles of relational biology may be considered the operational inverse of reductionistic ideas. The essence of reductionism in biology is to keep the matter of which an organism is made, and throw away the organization, with the belief that, since physicochemical structure implies function, the organization can be effectively reconstituted from the analytic material parts. Relational biology, on the other hand, keeps the organization and throws away the matter; function dictates structure, whence material aspects are entailed. Stated otherwise, an organism is a material system that realizes a certain kind of relational pattern, whatever the particular material basis of that realization may be.
The relational pattern that makes a natural system alive turns out to be the impredicativity that is ‘closure to efficient causation’. There is an alternative to physicochemical and algorithmic means in the quest for the fabrication of life.
The important and consequential Venter achievement is an impressive one in technology, but no synthetic life, alas, has been made. An achievement is diminished if it is accompanied by overreaching claims of success, when such hyperbolic ‘accomplishment’ is illusory, and not entailed from what has actually been done.
References
Gibson, D. G. et al (2010). Creation of a Bacterial Cell Controlled by a Chemically Synthesized Genome. Science [DOI: 10.1126/science.1190719] (Science Express Research Article, published 20 May 2010).
Louie, A. H. (2009). More Than Life Itself: A Synthetic Continuation in Relational Biology. ontos verlag, Frankfurt.
The correct term is ‘system biology’. Note the singular form system: not “systems biology”. This last usage is a solecism that became accepted when it had been repeated often enough, a very example of ‘accumulated wrongs become right’.
System biology may be defined as the study of life using the tools of system theory (not ‘systems theory’). Ludwig von Bertalanffy’s 1968 masterwork is called General System Theory. (In some of his later writings, the term “systems theory” did occasionally appear. I have in my collection some copies of his original typescripts, in which he had written “system theory”, but in the published versions they mysteriously mutated to “systems theory” — evidence of the handiwork of an over-zealous copy editor, perhaps…)
Consider the everyday terms ‘vegetable soup’, ‘ten-foot pole’, ‘train station’; NOT ‘vegetables soup’, ‘ten-feet pole’, ‘trains station’. In mathematics, one says ‘set theory’, ‘group theory’, ‘number theory’, ‘category theory’, ‘system theory’, etc; NOT ‘sets theory’, ‘groups theory’, ‘numbers theory’, ‘categories theory’, ‘systems theory’, …. Likewise, in biology, one uses ‘cell biology’, ‘population biology’, ‘system biology’; NOT ‘cells biology’, ‘populations biology’, ‘systems biology’. One may also note ‘computer science’, ‘plant science’, ‘system science’; NOT ‘computers science’, ‘plants science’, ‘systems science’.
Of course one studies more than one object in each subject! Indeed, one would say in the possessive ‘theory of sets’, ‘biology of populations’, …; one says ‘theory of systems’ and ‘biology of systems’ for that matter. But the point is that when the noun of a mathematical object (or indeed any noun) is used as adjective, one does not use the plural form.
Grammatically, a noun-used-as-adjective in a compound is called a noun adjunct (also attributive noun or noun premodifier). So in ‘system biology’, ‘biology’ is the noun and ‘system’ (singular!) is the noun adjunct. The rule is that in a noun adjunct, the singular form is used.
This is a brief introduction to the Rashevsky-Rosen school of relational biology.
Relational biology is the study of biology from the standpoint of ‘organization of relations’. It was founded by Nicolas Rashevsky in the 1950s, thence continued and flourished under his student Robert Rosen. And I was Rosen’s student.
Nicolas Rashevsky
(1899–1972)
Robert Rosen
(1934–1998)
Causality in the modern sense, the principle that every effect has a cause, is a reflection of the belief that successions of events in the world are governed by definite relations. Natural Law posits the existence of these entailment relations and that this causal order can be imaged by implicative order.
A modelling relation is a commutative functorial encoding and decoding between two systems. Between a natural system (an object partitioned from the physical universe) and a formal system (an object in the universe of mathematics) , the situation may be represented in the following canonical diagram:
The encoding ε maps the natural system N and its causal entailment c therein to the formal system F and its internal inferential entailment i ; i.e.,
ε : N → F and ε : c → i .
The decoding δ does the reverse. The entailments satisfy the commutativity condition
c = ε ► i ► δ .
Stated graphically, this equality says that, in the diagram above, tracing through arrow c is the same as tracing through the three arrows ε , i , and δ in succession. Thence related, F is a model of N, and N is a realization of F. In terms of the modelling relation, Natural Law is a statement on the existence of causal entailment c and the encodings ε : N → F and ε : c → i .
A formal system may simply be considered as a set with additional mathematical structures. So the mathematical statement ε : N → F , i.e., the posited existence for every natural system N a model formal system F , may be stated as the axiom
Everything is a set.
A mapping is an inference that assigns to each element of one set a unique element of another set. In elementary mathematics, when the two sets involved are sets of numbers, the inference process is often called a function. So ‘mapping’ may be considered a generalization of the term, when the sets are not necessarily of numbers. (The use of ‘mapping’ here avoids semantic equivocation and leaves ‘function’ to its biological meaning.)
Causal entailment in a natural system is a network of interacting processes. The mathematical statement ε : c → i , i.e., the functorial correspondence between causality c in the natural domain and inference i in the formal domain, may thus be stated as an epistemological principle, the axiom
Every process is a mapping.
Together, the two axioms are the mathematical formulation of Natural Law. These self-evident truths serve to explain “the unreasonable effectiveness of mathematics in the natural sciences”.
A living system is a material system, so its study shares the material cause with physics and chemistry. Reductionists claim this, therefore, makes biology reducible to ‘physics’. Physics, in its original meaning of the Greek word φύσις, is simply (the study of) nature. So in this sense it is tautological that everything is reducible to physics. But the hardcore reductionists, unfortunately, take the term ‘physics’ to pretentiously mean ‘(the toolbox of) contemporary physics’.
Contemporary physics that is the physics of mechanisms reduces biology to an exercise in molecular dynamics. This reductionistic exercise, for example practised in biochemistry and molecular biology, is useful and has enjoyed popular success and increased our understanding life by parts. But it has become evident that there are incomparably more aspects of natural systems that the physics of mechanisms is not equipped to explain. The overreaching reductionistic claim of genericity is thus a misrepresentation and renders it into a falsehood.
Biology is a subject concerned with organization of relations. Physicochemical theories are only surrogates of biological theories, because the manners in which the shared matter is organized are fundamentally different. Hence the behaviours of the realizations of these mechanistic surrogates are different from those of living systems. This in-kind difference is the impermeable dichotomy between predicativity and impredicativity.
The essence of reductionism in biology is to keep the matter of which an organism is made, and throw away the organization, with the belief that, since physicochemical structure implies function, the organization can be effectively reconstituted from the analytic material parts.
Relational biology, on the other hand, keeps the organization and throws away the matter; function dictates structure, whence material aspects are entailed.
In terms of the modelling relation, reductionistic biology is physicochemical process seeking models, while relational biology is organization seeking realizations. Stated otherwise, reductionistic biology begins with the material system and relational biology begins with the mathematics. Thus the principles of relational biology may be considered the operational inverse of (and complementary to) reductionistic ideas.
Any question becomes unanswerable if one does not permit oneself a large enough universe to deal with the question. The failure of presumptuous reductionism is that of the inability of a small surrogate universe to exhaust the real one. Equivocations create artefacts. The limits of mechanistic dogma are very examples of the restrictiveness of self-imposed methodologies that fabricate non-existent artificial ‘limitations’ on science and knowledge. The limitations are due to the nongenericity of the methods and their associated bounded microcosms. One learns something new and fundamental about the universe when it refuses to be exhausted by a posited method.
The above is a necessarily terse introduction to relational biology. The enthusiasts may want to explore the subject further. A good place to start is my 2009 book More Than Life Itself: A Synthetic Continuation in Relational Biology.
Louie Totem
Cyclic Entailment
Name Chop
[41] Renner, Ansel, Louie, A. H., and Giampietro, Mario (2021)
Cyborgization of Modern Social-Economic Systems
Dan Braha et al. (eds.), Unifying Themes in Complex Systems X. Springer, Cham, Switzerland.
link.springer.com/chapter/10.1007/978-3-030-67318-5_9
[40] Louie, A. H. (2020)
Relational Biology and Church's Thesis
Biosystems 197
www.sciencedirect.com/science/article/abs/pii/S0303264720300782
[39] Louie, A. H. (2019)
A Relational Theory of the Visible
Axiomathes, 32(5) (2022), 793–816
link.springer.com/article/10.1007/s10516-019-09416-3
[38] Louie, A. H. (2017) [monograph]
Intangible Life: Functorial Connections in Relational Biology
xxiii + 264 pp. (Anticipation Science, Vol. 2) Springer, New York. ISBN 978-3-319-65408-9
link.springer.com/book/10.1007/978-3-319-65409-6
[37] Louie, A. H. (2017)
Mathematical Foundations of Anticipatory Systems
Roberto Poli (ed.), Handbook of Anticipation. Springer, New York.
link.springer.com/referenceworkentry/10.1007/978-3-319-31737-3_21-1
[Louie MathFoundAnticipSys 2017 pdf]
[36] Louie, A. H. (2017)
Relational Biology
Roberto Poli (ed.), Handbook of Anticipation. Springer, New York.
link.springer.com/referenceworkentry/10.1007/978-3-319-31737-3_17-1
[Louie Relational Biology 2017 pdf]
[35] Louie, A. H. and Poli, R. (2017)
Complex Systems
Roberto Poli (ed.), Handbook of Anticipation. Springer, New York.
link.springer.com/referenceworkentry/10.1007/978-3-319-31737-3_3-1
[Louie Poli Complex Systems 2017 pdf]
[34] Albertazzi, L. and Louie, A. H. (2016)
A Mathematical Science of Qualities: A Sequel
Biological Theory, 11(4), 192–206
link.springer.com/article/10.1007/s13752-016-0248-0
[Louie Qualities 2016 pdf]
[33] Louie, A. H. (2015)
The Imminence Mapping Anticipates
Mihai Nadin (ed.), Anticipation Across Disciplines.
(Cognitive Systems Monographs, Vol. 29). Springer, New York. ISBN 978-3-319-22598-2, pp. 163-185.
link.springer.com/chapter/10.1007/978-3-319-22599-9_11
[Louie Imm Map Anticip 2015 pdf]
[32] Louie, A. H. (2015)
A metabolism–repair theory of by-products and side-effects
International Journal of General Systems, 44(1), 26-54
tandfonline.com/doi/full/10.1080/03081079.2014.937433
[31] Louie, A. H. (2013) [monograph]
The Reflection of Life: Functional Entailment and Imminence in Relational Biology
xxxii + 243 pp. (IFSR International Series on Systems Science and Engineering,
Vol. 29) Springer, New York. ISBN 978-1-4614-6927-8
link.springer.com/book/10.1007/978-1-4614-6928-5/page/1
[30] Louie, A. H. (2013)
Explications of functional entailment in relational pathophysiology
Axiomathes 23(1), 81-107
link.springer.com/article/10.1007/s10516-012-9189-9
[Louie Pathophysiology 2012 pdf]
[29] Louie, A. H. (2012)
Anticipation in (M,R)-systems
International Journal of General Systems, 41(1), 5-22
tandfonline.com/doi/abs/10.1080/03081079.2011.622088
[Louie AMR 2012 pdf]
[28] Louie, A. H. (2011)
Essays on More Than Life Itself
Axiomathes 21(3), 473-489
springerlink.com/content/71200q3v2867642h
[Louie Essays on ML 2011 pdf]
[27] Louie, A. H. and Poli, Roberto (2011)
The spread of hierarchical cycles
International Journal of General Systems, 40(3), 237-261
tandfonline.com/doi/abs/10.1080/03081079.2010.550579
[LouieP Hierarchical Cycles 2011 pdf]
[26] Louie, A. H. (2010)
Artificial claims about synthetic life: the view from relational biology
Journal of Cosmology 8, June 2010
journalofcosmology.com/ArtificialLife100.html#19
[Post html]
[Louie Venter 2010 pdf]
[25] Louie, A. H. (2010)
Relational biology of symbiosis
Axiomathes 20(4), 495-509
springerlink.com/content/h481521x7264626l
[Louie Symbiosis 2010 pdf]
[24] Louie, A. H. (2010)
Robert Rosen's anticipatory systems
Foresight 12(3), 18-29
emeraldinsight.com/journals.htm?issn=1463-6689&volume=12&issue=3&articleid=1864160
[Louie Rosen's AS 2010 pdf]
[23] Louie, A. H. (2009) [monograph]
More Than Life Itself: A Synthetic Continuation in Relational Biology
xxiv + 388 pp. (Categories Series, Vol. 1) ontos verlag, Frankfurt.
ISBN 978-3-86838-44-6
degruyter.com/document/doi/10.1515/9783110321944
[22] Louie, A. H. (2008)
Functional entailment and immanent causation in relational biology
Axiomathes 18(3), 289-302
springerlink.com/content/k7gh4648h6w5m636
[Louie Entailment 2008 pdf]
[21] Louie, A. H. and Kercel, Stephen W. (2007)
Topology and life redux: Robert Rosen's relational diagrams of living systems
Axiomathes 17(2), 109-136
springerlink.com/content/x17xx1734w82674w
[LouieK Topology 2007 pdf]
[20] Richardson, I. W. and Louie, A. H. (2007)
A phenomenological calculus of Wiener description space
Chemistry and Biodiversity 4(10), 2315-2331
onlinelibrary.wiley.com/doi/10.1002/cbdv.200790189/abstract
[RLouie Wiener 2007 pdf]
[19] Louie, A. H. (2007)
A Rosen etymology
Chemistry and Biodiversity 4(10), 2296-2314
onlinelibrary.wiley.com/doi/10.1002/cbdv.200790188/abstract
[Louie Etymology 2007 pdf]
[18] Louie, A. H. (2007)
A living system must have noncomputable models
Artificial Life 13, 293-297
mitpressjournals.org/doi/abs/10.1162/artl.2007.13.3.293?journalCode=artl
[Louie ALife_Noncom 2007 pdf]
[17] Louie, A. H. and Richardson, I. W. (2006)
A phenomenological calculus for anisotropic systems
Axiomathes 16(1-2), 215-243
springerlink.com/content/x2p11tg1984wq858
[LouieR Anisotropy 2006 pdf]
[16] Louie, A. H. (2006)
(M,R)-systems and their realizations
Axiomathes 16(1-2), 35-64
springerlink.com/content/e1503x8876611743
[Louie MR 2006 pdf]
[15] Louie, A. H. (2005)
Any material realization of the (M,R)-systems must have noncomputable models
Journal of Integrative Neuroscience 4, 423-436
worldscinet.com/jin/04/0404/S0219635205000926.html
[Louie JIN_Noncomp 2005 pdf]
[14] Richardson, I. W. and Louie, A. H. (1992)
Membranes and meters
Journal of Theoretical Biology 154, 9-26
sciencedirect.com/science/article/pii/S0022519305801845
[RLouie Meters 1992 pdf]
[13] Richardson, I. W. and Louie, A. H. (1988)
The metric structure of irreversible thermodynamics
Journal of Theoretical Biology 132, 125-126
sciencedirect.com/science/article/pii/S0022519388801978
[LouieR Metric 1988 pdf]
[12] Louie, A. H. and Richardson, I. W. (1986)
Dissipation, Lorentz metric, and information:
a phenomenological calculus of bilinear forms
Mathematical Modelling 7, 227-240
sciencedirect.com/science/article/pii/0270025586900497
[LouieR Bilinear 1986 pdf]
[11] Richardson, I. W. and Louie, A. H. (1986)
Irreversible thermodynamics, quantum mechanics, and intrinsic time scales
Mathematical Modelling 7, 211-226
sciencedirect.com/science/article/pii/0270025586900485
[RLouie Irreversible 1986 pdf]
[10] Louie, A. H. (1985) [tierce in the "Red House Book"]
Categorical System Theory
Theoretical Biology and Complexity:
Three Essays on the Natural Philosophy of Complex Systems
(Robert Rosen, ed.), Academic Press, Orlando FL, pp.69-163 (out of print)
amazon.com/Theoretical-Biology-Complexity-Natural-Philosophy/dp/0125972806
[Louie CST 1985 pdf]
[9] Louie, A. H. and Somorjai, R. L. (1984)
Stieltjes integration and differential geometry:
a model for enzyme recognition, discrimination, and catalysis
Bulletin of Mathematical Biology 46, 745-764
springerlink.com/content/20p754jk71815331
[LouieS Stieltjes 1984 pdf]
[8] Louie, A. H. (1983)
Categorical system theory and the phenomenological calculus
Bulletin of Mathematical Biology 45, 1029-1045
springerlink.com/content/m301h563w12v2wv7
[Louie CSTPC 1983 pdf]
[7] Louie, A. H. (1983)
Categorical system theory
Bulletin of Mathematical Biology 45, 1047-1072
springerlink.com/content/635736782532364m
[Louie CST 1983 pdf]
[6] Richardson, I. W. and Louie, A. H. (1983)
Projections as representations of phenomena
Journal of Theoretical Biology 102, 199-223
sciencedirect.com/science/article/pii/0022519383903600
[RLouie Projections 1983 pdf]
[5] Louie, A. H. and Somorjai, R. L. (1983)
Differential geometry of proteins: helical approximations
Journal of Molecular Biology 168, 143-162
sciencedirect.com/science/article/pii/S0022283683803271
[LouieS DGP Helical 1983 pdf]
[4] Louie, A. H. and Richardson, I. W. (1983)
Duality and invariance in the representation of phenomena
Mathematical Modelling 4, 555-565
sciencedirect.com/science/article/pii/0270025583900155
[LouieR Duality 1983 pdf]
[3] Louie, A. H. and Somorjai, R. L. (1982)
Differential geometry of proteins:
a structural and dynamical representation of patterns
Journal of Theoretical Biology 98, 189-209
sciencedirect.com/science/article/pii/0022519382902582
[LouieS DGP Patterns 1982 pdf]
[2] Louie, A. H., Richardson, I. W., and Swaminathan, S. (1982)
A phenomenological calculus for recognition processes
Journal of Theoretical Biology 94, 77-93
sciencedirect.com/science/article/pii/0022519382903307
[LouieRS Recognition 1982 pdf]
[1] Richardson, I. W., Louie, A. H., and Swaminathan, S. (1982)
A phenomenological calculus for complex systems
Journal of Theoretical Biology 94, 61-76
sciencedirect.com/science/article/pii/0022519382903290
[RLouieS Complex 1982 pdf]
A. H. Louie 