I do not know my answer relevan to your question or not. Before a lion to become a lion, because it can not get a victim to be ate, it may make a rigorous imaginative thought :” If I have sharp fang and claw, I can get victim easily.” Or, before a horse to become a horse, because it must run from a lion, may be it make a rigorous imaginative thought: “If I have strong hoof, I can run more quickly.” So, I think rigorous imaginative thought is one that determine the course of evolution.

Daniele:

For Constructional and Functional Morphology studies, Mathematics is important but not indispensable. Power of innovative Imagination is a powerful tool to make some fundamental discoveries and therefore, should be viewed with positive optimism.

Best

Syed

I believe the answer to your question is yes! There are proponents of Goethean science working today who proceed as you describe:  Craig Holdrege, Martin Lockley, Mark Riegner and Wolfgang Schad.  Their results are qualitative and holistic. Here are some references:

Holdrege, C. (2003). The flexible giant: Seeing the elephant whole. Nature Institute.

Holdrege, C. (2005). The giraffe's long neck: From evolutionary fable to whole organism. Nature Institute.

Holdrege, C. (2013). Thinking like a plant: a living science for life. Lindisfarne.

Lockley, M. G. (2000). The eternal trail: a tracker looks at evolution. Basic Books.

Lockley, M. G. (2006). The morphodynamics of dinosaurs, other archosaurs, and their trackways: holistic insights into relationships between feet, limbs, and the whole body. SPECIAL PUBLICATION-SEPM, 88, 27.

Riegner, M. F. (2008). Parallel evolution of plumage pattern and coloration in birds: implications for defining avian morphospace. The Condor, 110(4), 599-614.

Riegner, M. F. (2013). Ancestor of the new archetypal biology: Goethe’s dynamic typology as a model for contemporary evolutionary developmental biology. Studies in History and Philosophy of Science Part C: Studies in History and Philosophy of Biological and Biomedical Sciences, 44(4), 735-744.

Schad, W. (1977). Man and mammals: toward a biology of form (p. 306). Garden City, NY: Waldorf Press.

Schad, W. (1993). Heterochronical patterns of evolution in the transitional stages of vertebrate classes. Acta biotheoretica, 41(4), 383-389.

Mathematics is of course very important in solution of some biological problems, less so in many others, and not at all in still others. On the other hand, it may be decidedly harmful if - what is increasingly the case nowadays... - treated as a panacea or formal obligation (slogans like "a study not applying mathematics is not scientific"), provoking to "switch off" our brain, substitute its function (thinking)  by a computer, and leading to the kind of "recipe for modern science" "proposed" in one of my papers [2010. Is paraphyly indication of poor taxonomy? – open letter to Drs. Carvalho and Ebach.  Munis Ent. Zool. 5, Suppl.: 825-829]: "Non-luddite recipe for a modern efficient taxonomy [compiled from commentaries of funding agencies, editors, peer-reviewers &c.]: Collect one-one specimen of two-three (out of 100 known) species, send to a laboratory to obtain partial sequence of this or that gene, download from GenBank more or less equivalent sequences of yet three allegedly (even if uncontrollably) well determined species, put them into computer; after performing the alignment with PerhapsFit2009 choose the Abrakadabra version 3.ac, command preprapro*xy to obtain 123456 equally parsimonious trees: apply Tratratretre 5.2a for a full consensus tree, then Hurumburum-test with Maybegood correction for Whynotthis model selected by Iknowbetter(spec) will tell you that the Makebelieve’s sigma-tt is 0.73 what corresponds to 132% support for the clade Q; print it out with Hokeypokey ZW, describe the cladogram with words and call the result “classification” (don’t bother with information content or predictive power: the assumedly apomorphous curvature of seta on last antennal joint is “predictable” from the cladogram, all the other characteristics of involved taxa being by definition not interesting). Don’t think (thinking is subjective), don’t take external evidence into consideration (this would be preconception), don’t try to reconstruct ancestors (ancestors do not exist, only descendants do!), this will release you from any temptation to ponder whether these (non-reconstructed because non-existent) ancestors could have ever been viable or where they might have occurred (modern scientist does not dabble at subjective ad hoc conclusions), don’t present any own interpretation (speculative scenarios are unscientific), don’t question results disagreeing with commonsense (commonsense? – oh, how primitive...), proudly proclaim that you have just falsified the old-fashioned view (resulting from two centuries of useless archaic activity of “morphological” taxonomists) that tiger is a member of Felidae (Mammalia), as your rigorous analysis has shown that it cladistically belongs to flatworms where – together with green hydra and swallowtail – it should be grouped into a new phylotaxon Paranormalomorpha. Now submit the paper to SkyreachingImpactFactor journal from the newest Philadelphia-list, and wait for the well deserved Nobel Prize!". Caricature? – of course! But already long ago LEVINS & LEWONTIN (teste WILLIAMS 1988: 417) stated that “… most science in the western world is already merely a caricature of what science should be, …, and that in the non-western world is simply a caricature of a caricature”; now the difference between West and East largely disappeared: everywhere dominates a caricatural version of the caricature of a caricature...

To conclude: use mathematics where and as far as it is really needed, but only there!

I am very hapyy for your very interesting and precious answers. I follow both Goethe and Steiner since 80'years. I personally know the researchers cited by Stephen William Wood. In particular I find very important the work by Jochen Bockemühl on the plant Lepidium sativum. I think that the mesocosmic sphere represents the level where thought and perception can meet to bring to light the form considered as a symbol within consciousness. In the sense used by Goethe, the archetypal phenomena are those events that immediately reveal the ideal content of nature to us. They can be found directly in nature, as the magnet, or they must be brought to reveal themselves through the scientific experiment, whose object is to arrange them, as the signs of an ancient language, in runes apt to express their symbolic content directly. As for the runes, also for the forms of nature the means by which they are produced do not matter so much; their arrangement rather is relevant, the semantic relationships that constitutes them. This is why E. Cassirer  pointed out that, for Goethe, morphogenesis is not historical, but ideal and symbolical. Following this view, the term morphodynamics is better suited to express the transformations in nature that are brought to light with this method. In organic nature the point is in fact not only to set up genealogical trees, but to describe the ideal dynamics of the form, acknowledging its laws, which do not result from the analysis of the constituents involved in the physical genesis. In the organic sphere, Goethe meant the form as "Bildung" (and not as "Gestalt"), that is a dynamic totality that "wavers in incessant movement" (6) and reveals itself in the metamorphosis. From this point of view, evolution and metamorphosis are both real events. However, the first represents a plurality of physical processes, while the second consists of a totality of ideal relations. Therefore, typological study does not exclude phylogenetic analysis, but it deals with morphodynamic and not with morphogenetic relations. The task of typological study is to make evident the synthetic connections that relate the type with the elements of its organic manifestation according to a temporal scheme which turns out to be, as Cassirer has noted concerning Goethe (5), more similar to the temporal schemes of legends than to that of history. This kind of temporal scheme, in fact, can connect the type, as a purely ideal thing, with the level of phenomena which otherwise would be fragmented in a chaotic way. The task of morphological thinking is, therefore, that of thinking this temporal mode and of reproducing its metamorphosis. Consequently, inthe world of living things, comparison turns out to be the method that can distinguish the morphodynamics that are characteristic of an organism. These dynamics, which can be perceived only by thought, can be considered, however, more real than the single stages of the perceptible manifestation itself. Goethe and Schiller defined the method of this ideal science as rational empiricism. Differently from Newton and Bacon's empirical induction, rational empiricism does not pretend to generate theories and models, it deals instead with ideas and phenomena. Morphology in the sense of rational empiricism is a way that must develop using also some technology we can today find in the science of normal paradigm.

Dear Colleagues,

I'm working on the taxonomy and systematics of recent taxa and I came across the problem that more and more researcher finding the morphological examination ineligible on its own by taxonomic description and revisionary works.

My recent experiences and preliminary findings stress that "mathematics", particularly geometric morphometrics can provide a good background in solving taxonomic questions (see the attached conference paper). E.g. if there's a structure which is

can be good subject of such a study which is based on statistical methods.

The problem is that some of the researchers treat these result "omnipotent" on their own. You can observe the same phenomenon by molecular taxonomic methodology where results based on DNA-barcoding often published without "classical" knowledge on the studied taxa.

The results of these methods should be interpreted together. The possibility to involve further methods (e.g. GIS, DNA-barcoding) in the work is given as well. If the circumstances are inadequate for the use of some of them then those methods can be "left out". The emphasis is on the approach. We should observe a problem from all of the possible perspectives in order to get the possibly most precise and timeless results. To decide the spectrum of applicable methods the problems should be walked around carefully and none of the techniques should be treated "almighty".

Summarized, nowadays taxonomy and systematics requires an integrated approach which uses classical methods and the applicable novelties in the solution of problems and questions.

Best regards,

Peter

Conference Paper A morphometric approach to delimit groups in Heterogastridae...

I totally agree with Peter with regard to an integrated approach.  D'Arcy Thompson's on growth and form is an example of that. He did study methods used by Albrecht Dürer.

Thinking of the aristotelian principle that philosophy originates from θαυμάζειν (thaumazein) a Greek word that means simultaneously astonishment and dismay (thauma), I am every time disappointed faced with the lack of pathos of the majority of scientists and philosophers who have become bureaucrats and officials. Morphology, in the sense of Goethe, is a way to rediscover the face of nature and its "excess of splendour". In addition to a science, morphology is a therapy of the soul... 

Daniele:

With the advent of magnifying devices and Computer the classical concept of Morphology contained in works, notably by D'Arcy Thompson, Leonardo Da Vinci, Keppler and several artists including Albrecht Duerer and formally proposed as "Morphology" by J.W.Goethe at the end of Eighteenth Century acquired new dimension. Simulation studies by Computer and application of Cartesian coordinates for example, for biological entities provided a major breakthrough in understanding Morphotype variations with strong element of predictability. Individual cognitive abilities coupled with rigorous brainstorming could yield results of fundamental importance with simple tools of mathematics.

Best

Syed

Yes Syed I agree witk you, computer science can be very useful and rigorous. But you cannot feel the scent of a rose... You can study anatomy using computer science but you cannot feel by this way what did feel Andreas Vesalius and you cannot design the pathos of an human body. Morphology in the sense of Goethe and Saint Hilaire is art and science at the same time. I know that for many it is more "cool" obtain strictly practical results and produce technology. For Goethe and Saint Hilaire science was episteme and not techne. You should not confuse the limb with the prosthesis...

Mathematics, computer programs, &c., while certainly very useful in many situations, have - like any other "rigorous" methodology - two immanent Achilles' heels. First is what computer specialists often formulate as "garbage in, garbage out": to start a procedure we must put some data in, and if these data are wrong, wrongly interpreted or wrongly selected, then the results would have usually rather little in common with the truth; selection, formulation and interpretation of the data to be put in must be done based on the observable "external" (ecological, geographical, or any, but in taxonomy most frequently morphological) evidence, so our morphological knowledge and intuition is in most cases primarily responsible for the "input" not being "garbage". Second is the fact that complicated (and sometimes even relatively simple) mathematical or computerized procedures are almost invariably (at least for the "average": not specially trained in mathematics and/or computer programming) biologist something like a "black box": we put something (garbage or not) in and receive something out, but typically are not able to check (and understand!) step-by-step what happens in between, so we cannot easily (if at all) discover potential "bugs" (e.g. analogical to those on which the popular mathematical or logical "tricks" - "proofs" that 3 is less than 2, that swift-footed Achilles would never overcome a turtle, &c. - are based, or resulting from the application of the procedure in a situation where the preconditions of its reliable functioning are not met); such pitfalls are not always easy to see, and formalized procedures are typically not "self-verifying": they will not by themselves warn us if the result is wrong, here again the help of observable facts are needed. So, even if we recourse (as we must frequently do) to mathematics or computer-program, the external, directly observable (typically morphological) evidence remains the "alpha and omega" of our study: this assures that input is not "garbage", and this enables us to check whether the output makes biological (ecological, geographical, and especially morphological) sense. Only in this way we can avoid acceptation of such "rigorously derived" results as those obtained by Giribet & Ribera [2000. A review of arthropod phylogeny: new data based on ribosomal DNA sequences and direct character optimization. Cladistics 16: 204-23] where any of the 7 versions of molecular-phylogenetic reconstruction consistently placed e.g. Daphnia among Myriapods or Onychophora, or in BUHAY’s (2009. “COI-like” sequences are becoming problematic in molecular systematic and DNA barcoding studies. J. Crust. Biol. 29, 1: 96-110) somewhat surrealistic but highly symptomatic experiment “I used a subset of” a dataset from a published phylogenetic analysis “and added my favourite recipe for pumpkin pie (imagine it is a numt sequence or junk DNA) to the nexus file ... executed the file and”... “demonstrated” (with 100% bootstrap support!) that her pumpkin pie belongs to the genus Orconectes (Crustacea: Decapoda: Cambaridae) and is the sister species of O. burri! [that, of course, neither Buhay nor Giribet & Ribera have accepted the result of "rigorously performed analyses", was just because the morphological control ["The polyphyletic status of hexapods is certainly unacceptable from a morphological point of view, although it has been repeatedly obtained in molecular analyses of ribosomal sequence data ... [Giribet & Ribera: 222]"]. Not only taxonomy, but science in general, is - as it was for Goethe and Saint-Hilaire - "art and science at the same time", and it must remain so if we do not wish to reduce it to a kind of formal game in which (to use the famous mathematician, Kornél Lánczos words) "interesting is not what the world is, but what it should be" - I am biologist, not mathematician, for me interesting is what the world is, not what it (according to somebody or some rigorous formula) "should" be!

Roman your answer is very interesting and meaningful and I totally agree with you.

I really would like to know what does Maria Bettencourt Pires think of this problem, because I hold her in great esteem for her works as anatomist.

By these last years I'm trying to compare the morphology of the objects of nature and that one of the language of mathematics, in search of a symbolic language in the sense of the Mathesis Universalis of Leibniz and of Pavel Florenskij's Symbolarium. For Florenskij the symbol was a transparent intermediary between the antinomies that dominate the reality of being as well as the central platform for his theory of cognition. In the objects of mathematics and geometry, we find “ideal” forms, whereas in nature we find only “individuals”. We can cross maths and individuals as in Newton's method, otherwise we can observe the entities  of mathematics as ideal forms to which individuals tend asymptotically. The objects of nature that are closest to the ideal forms represent symbols in the sense of Goethe where infinity is revealed in the finished. They are archetypal forms. In this way the mathematics is reflected in the morphology without crossing it.

Daniele,

''In the objects of mathematics and geometry, we find “ideal” forms, whereas in nature we find only “individuals”.''

Mathematics is a language and word/symbol in any language are ideals which are usefull for refering individual which never can be perfectly reduce to these ideals.  Take the word ''chair'', this is a sort of ideal that does not exist in itself, ''something you can use to sit'' but which we can recognize instance of.  We can recognize instances of the number ''2'' although nobody has seen a ''2'' in itself.  That symbol ''2'' is only a sign pointing to that concept and that concept is this capacity to recognize an instance of it.  Object of mathematics and words in languages are means of communication between humans.  We have to say a lot in small sentences such ''There is two apples on the table''.  It takes three seconds to describe a reality that has meaning for English speakers that know to count.  Not a lot can be explicitly said in 3 seconds but at the same time a lot is said with only very few words.  So these words are very powerfull methods to describe the worlds, not in all its infinite complexity but in a very condensed finite simplicity that is very usefull for the type of doing being like us do.

Dear Louis I agree with you, but I would like to go a little bit more deeply. For that I have attached, to answer, a short article about the evolution of Mathematical thinking. Is a not-finished work but it is important for me to explain better my previous considerations about the researh of a mathesis universalis in the sense of Leibniz.

Hi Daniele,

I quickly browse the document. I agree with what I read but I am only seeing the trees and not the forest.  Where are you trying to go?

i am trying to go to the forest

I will answer more clearly later

It is very difficult to explaine in few words what I mean an what I am looking for. Therefore I will try to be synthetic also entrusting myself to your capacity for insight. ". According to Salthe, for the viewpoint of formal analysis the present mathematical formalism (algebra, equations etc.) would not allow to express with enough autonomy and precision the macrodetermination of the higher levels (ibid.). The world of mechanics and in general the inorganic sphere of nature is formalized with more success than the organic sphere. In the world of plants, where metamorphosis is prevailing we must have recourse for example to the theory of categories with its collection of objects and concepts as morphisms, groups, transformations and so on. Until today the vegetable world is formalized with the same mathematical instruments that apply to the physics. In the theory of categories, for example, we are out from contamination caused by Newton in its equations in which empirical elements, then inductive, appear,. As stated by Hegel in the lectures on the history of philosophy, while Kepler "treated the sensitive things as concepts", Newton "treated the concepts as sensitive things". In the dissertatio de orbitis planetarum (Part 4)Hegel criticizes the union of physics with mathematics made by Newton and warns against confusing  purely mathematical relations with physical reports.  It is different to deduce "reports" between phenomena, from "deducing" phenomena, i.e. to give an empirical  meaning to  mathematical Entities.

 What I am looking for? I try to develop what Raimondo Lullo and Leibniz, and more recently Folerenskij were looking for, i. e. a mathesis universalis. I (and collaborators of our Goethean group in Italy) look for highlight "pure" mathemamatical factors or mathematical archetypes that exist on the ideal plane in which the various levels of nature, plants, animals, human beings tend asymptotically, trying to develop a relationship between pure logico-mathematical sphere and morphology without contaminating the one with the other.