Lenneberg (1967)： The pages：243 ~ 245
(3) Transformations of Form and Function
Closely related to Huxley’s method of studying allometric growth is D’Arcy Thompson’s (1917) famous method of transformations in which he compares related forms such as shown in Fig. 6.6 by the superimposition of Cartesian coordinates. A rectangular system is drawn over a two-dimensional representation of one form so that the distortions of the coordinates may be studied that result from drawing lines through the homologous points on the second form. This method is purely descriptive and difficult to quantify. But it illustrates the topological relationships between certain forms and how certain differences in structure may be accounted for by a single principle, usually changes in growth gradients during ontogeny. In cases where specific dimensions can be compared allometrically, we would find different values for the parameters a and b (compare
However, there mat be intracellular genetic alterations such that ontogenetic histories are altered resulting in two different mature forms. The situation is diagrammed in Fig. 6.7. There are two molecular structures, Σ1 and Σ2, that are at the basis of two developmental histories H1 and H2. Σ1 and Σ2 are related to one another by the specification of a molecular transformation called Tm. The developmental histories H1 and H2, result in mature structures S1 and S2. In the case described by D’Arcy Thompson, an apparent transformation relation, Tα, persists that is characterized through the distorted coordinate systems. Notice, however, that the biological connection rests entirely in the molecular and “invisible” transformation Tm and that the apparent transformation Tα is more or less incidental—certainly not essential—for it is obvious that some or even most molecular transformations will alter the developmental histories in such a way that the corresponding two mature structure either lose their isomorphism (as in the isolated case of two-headed monsters or other deformities) or remain the same to the eyes of the unaided observer as in the case of certain inherited diseases such as hemophilia. Thus, D’Arcy Thompson transformations are probably special cases of a much more universal phenomenon.
FIG. 6.6 Morphological relations between selected species shown here as geometric transformations. (a) Argyropelecus olfersi and Sternoptyx diaphana; middle: Scarus sp. and Pomacanthus; (b) Diodon and Orthagoriscus. (From Thompson, 1917.)
A discussion of these transformations has some unsuspected relevance to the biological study of language, particularly the comparison of human language with animal forms of communication. We have said before that what is true of ontogeny and transformations of molecular structures is also relevant to the biological foundations of behavior because of the dependence of the latter upon the former. Thus, the emergence of a species-specific form of behavior has, essentially, a molecular transformational history. Just as in the case of mature structure, mature forms of behavior are the result of species-specific developmental histories H1 and H2, and the biological connections between any two forms of behavior must be sought for on the level of the molecular transformations Tm. What we have said of the apparent transformation Tα, holds a fortiori for the comparison of behavior.
Correspondences on this surface level will be special cases: in many more instances, all isomorphism will be lost to our eyes. We would expect mature behavior forms (that is, the homologues to S1 and S2) to vary with much greater freedom and into many more directions than gross structure, because the selection biases upon skeletal form are likely to be much more restraining than on behavioral modality, and it is also possible that epigenetic canalization (Waddington, 1956 and 1957) allows of fewer directional alternatives in the case of structural alterations than behavioral ones. Although these are speculations, it is a fact that there is greater variety in behavior among animals than in their types of Bauplan or structural pattern. In the light of this, the present thesis on the biological origins of language becomes very clear.
FIG. 6.7. Species are related to each other by transformations in molecular structure of genic material. These transformations affect the developmental histories of the animals in the course of which the original relationship may become obscured: the resulting mature structure may or may not bear resemblance to one another.
We assume that our potential for language has a biological history that is written in terms and on the level of molecular transformations Tm; but this belief commits us in no way to expect the occurrence of apparent transformations Tα. If human language be S2, we cannot even be sure, in fact, what may be token of S1. Similarly, if a superficial resemblance is pointed out to us between language and some behavioral aspect of another species, we cannot be certain how close or distant the underlying relationship Tm actually is, or for that matter, if there is any such relationship whatever. Because modifications of behavior may be freer and go into many more directions than modifications of structures, molecular transformations Tm may leave in many fewer cases apparent transformations Tα than is the case for skeletal structure and thus there is the danger of being misled by similarities that are in fact not objective but that are entirely due to anthropomorphic interpretation of animal’s activities. (Examples of this are not restricted to animal “language” but may be found in statements about animal “play”, or animal “families”, or animal “pleasures.”)
The transformational picture leads us to expect that molecular alterations indirectly caused changes in the temporal and spatial dimensions of the species’ developmental history and that the resulting alterations in structure and function brought with them prolonged and changed periods during which one function could be influenced by others, thus creating critical periods of special sensitivities and opening up new potentials and capacities. This is just the framework within which we would like to see our thoughts move; it is too vague to be a theory. Let us look at it as the direction for possible explanations that are yet to come.
Ⅲ. EVIDENCE FOR INHERITANCE OF LANGUAGE POTENTIAL
The inheritance of behavioral traits in man can never be definitively demonstrated because of our inability to do breeding experiments. Also, absolute control of the environment is difficult to achieve. If we are staunch believers in the sole determination of behavior by the social environment was held constant. It is always possible to argue that there might have been subtle differences in human relations so that even two individuals who are raised in the same home might have experienced different treatment, invisible to the observer, and that all differences in behavior might be due to these variations. Similar but converse arguments are also possible in the case of identical behavior in apparently different environments.
D'Arcy Wentworth Thompson：Sir D'Arcy Wentworth Thompson (May 2, 1860–June 21, 1948) was a biologist, mathematician, classics scholar and the author of the 1917 book, On Growth and Form, an influential work of striking originality. Nobel laureate Peter Medawar called On Growth and Form "the finest work of literature in all the annals of science that have been recorded in the English tongue". Born in Edinburgh, Scotland, Thompson was an early mathematical biologist, and a contemporary of Francis Galton and Ronald Fisher. He died in
D'Arcy Thompson's method of transformations Differences in body forms among organisms may be more simply explained by pattern transformations than by rearrangement of physical component parts. The form of the puffer fish (Diodon) can evolve into that of an ocean sunfish (Mola) by a transformation of the rectangular coordinate system in (A) (red dots) into a curvilinear system in (B) that "stretches" the posterior portion of the fish. [modified from Futuyma 1997, after Thompson 1917]
D'Arcy Thompson：In this 1917 classic, D'Arcy Thompson provides a mathematical analysis of biological processes, especially growth and form. D'Arcy believes that natural selection has a limited function in evolution: it removes the "unfit", but it does not account for significant progress. D'Arcy believes that new structures arise because of mathematical and physical properties of living matter, just like the shape of nonliving matter. Form is a mathematical problem, and growth is a physical problem. The form of an object is the resultant of forces. By simply observing the object, we can deduce the forces that have acted or are acting on it. This is easily proved of a gas or a liquid, whose shape is due to the forces that "contain it", but it is also true of solid objects like rocks and car bodies, whose shapes are due to forces that were applied to them. D'Arcy believes that living organisms also owe their form to a combination of internal forces of molecular cohesion, electrical or chemical interaction with adjacent matter, and global forces like gravity. The formative power of natural forces expresses itself in different ways depending on the "scale" of the organism. Mammals live in a world that is dominated by gravity. Bacteria live in a world where gravity is hardly visible but chemical and electrical properties are significant. D'Arcy investigates what physical forces would be responsible for the surface-tension that holds together and shapes the membrane of a cell, and then analogously for cell aggregates, i.e. tissues, and then skeletons. While his formulas have not stood up to experimental data, the underlying principle is still powerful: D'Arcy believes that genetic information alone does not fully specify form. Form is due to the action of the environment (natural forces) and to mathematical laws. D'Arcy was fascinated by the geometric shapes of shells and sponges and believed that their geometry could not be explained on the basis of genetics but would be explained in terms of mathematical relationships. (http://www.scaruffi.com/mind/thompson.html)
Cartesian Coordinates： Cartesian coordinates are rectilinear two-dimensional or three-dimensional coordinates (and therefore a special case of curvilinear coordinates) which are also called rectangular coordinates. The three axes of three-dimensional Cartesian coordinates, conventionally denoted the x-, y-, and z-axes (a notation due to Descartes) are chosen to be linear and mutually perpendicular. In three dimensions, the coordinates , , and may lie anywhere in the interval . In René Descartes' original treatise (1637), which introduced the use of coordinates for describing plane curves, the axes were omitted, and only positive values of the - and the -coordinates were considered, since they were defined as distances between points. For an ellipse this meant that, instead of the full picture which we would plot nowadays (left figure), Descartes drew only the upper half (right figure). The inversion of three-dimensional Cartesian coordinates is called 6-sphere coordinates. The scale factors of Cartesian coordinates are all unity, . The line element is given by and the volume element by The gradient has a particularly simple form, as does the Laplacian The vector Laplacian is = =The divergence is and the curl is = = The gradient of the divergence is = = (http://mathworld.wolfram.com/CartesianCoordinates.html)
細胞與組織結構方面的進展 有如D'Arcy Thompson1所觀察到的，細胞理論起步早而進步緩慢。其萌芽始於培根、Hooke、Grew、Malpighi（素描一已提過），以及墊基於萊布尼茲單子論（Monadology3）的一些思想家。細胞在植物（包括動物）生命上佔有很基本的任務。
D'Arcy Thompson對Huxley的allometric學習方法有密切的相關，這種方法完全描寫但難確定數量，在D’Arcy Thompson轉變的重要討論裡，Woodger指出這裡的現象必須按照遺傳學和胚胎學來理解，因為沒有一個成熟的形式能改變一個過程成為其他的。不過，可能有細胞內遺傳學的改變，因此個體發生的歷史被改變成兩個不同的成熟形式。
D'Arcy Wentworth Thompson從數學和物理學層面分析生命的進程，認為物種的演化可能是整個個體的主要改變，而不是各部位小改變的累積。將數學方法引入生物的形態問題，用微分方程構建理論模型或者用複雜的統計做邏輯實證論述。“關於大小”：其中心論點是面積／體積比隨生物體變大而下降。因此大動物和小動物生存在不同的作用力的領域。用兩種方式闡明復雜的形態事實上遵循了普遍的原理。 （1）即使沒有受到物理力的直接塑形，部分或者整體仍採用了理想幾何學的最優形態來解決形態學的問題。 （2）即使因為遺傳所賜，生物體必須接受複雜的原始型，但最起碼它們向相關形態的轉化仍然表現為整個系統的簡單物理變形—變換坐標理論。
Reference / tool：
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