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The limits of the mathematization of the living and the idea of formal morphology of the living world following Husserlian phenomenology

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Abstract

Through a presentation and a commentary of Husserl's little-known analyses of mathematization in the life sciences and on morphology, this article proposes three goals. First, it aims at establishing the real meaning and results of the critical analyses of the mathematization in natural sciences and of exactness put forth as a standard of scientific knowledge that we read in the Krisis. As a result, it will appear that these analyses belong to the perspective of a project of a formal morphology, understood as an extension of mathesis. It is then to explain why this project only makes sense in the larger framework of the description of the “correlational a priori,” i.e., the theory of constituting subjectivity, experiencing these morphologies, and engaging, theoretically, by induction, in the typification and categorial elaboration of possible explanatory models. After presenting the contours of this project and its achievements, we will conclude with some conjectural proposals concerning the profile of plausible mathematical structures likely to satisfy the minimal algebraic formal conditions for a model of stability and plasticity of the living and allowing to understand and express the dynamic stratification of morphological levels and the various forms of morphogenesis.

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Notes

  1. Letter from Husserl to Weyl, 10 April 1918, in Husserl/Weyl (1984: 229).

  2. cf. Lobo (2009: 205) sq. Lobo (2017c). Lobo (2008)

  3. Husserl (1974). Husserl (1969).

  4. Husserl (1966). On space, see: Lobo (2019c). About time, see Lobo (2021).

  5. Thom (1977: 42).

  6. Husserl (2001: 124–125).

  7. Husserl (2001: 95).

  8. Husserl (2001: 120).

  9. Husserl (2001: 203).

  10. Husserl (2001: 202). Husserl (1950b: 364) sq. Longo and Islami (2017: 186).

  11. Husserl (2001, 221–222).

  12. Husserl (2001: 124).

  13. Husserl (1975). Husserl (1996). For an overview, see Lobo (2019a, 2019b).

  14. Hausdorff (2006).

  15. Meinong (1915; 86, 313, 337, 398).

  16. von Kries (1886).

  17. Husserl (2001, 198).

  18. Husserl (1975: 236–237) See Lobo (2017a, 2017b, 2019a, 2019b, 2019c).

  19. Husserl (2001: 206).

  20. Husserl (1950b: 387–391). A first commentary on Husserl’s interpretation of Quantum Mechanics was given in New York, at the New School for Social Research, during a Husserl Circle, in 2009.

  21. It is known that Oskar Becker’s, Habilitationschrift (Becker 1923) was a deepening of the phenomenological underpinnings on Weyl’s presentation of non-Euclidean geometries and relativity. They had a correspondence well known among scholars (for ex. Tonietti) Geiger (1921) has developed a clear philosophical interpretation of relativity theory, understood as a theory of invariants.

  22. Weyl (1954: 134). Weyl (1949: 116–117), Weyl (2017, 205–206). See comments by Barlibar and Lobo in their introduction to Weyl (2017: 53–62). And Born (1923).

  23. Châtelet (2010), that I have commented in Lobo (2017b), and in Lobo (2019c).

  24. There are « nonempirical laws (Wesensgesetze), which express the way in which data and strata of consciousness are founded upon each other, but do not claim to involve statements of fact; this line of pursuit culminated in HusserI’s phenomenology, in which the a priori is much richer than in the Kantian system.» Weyl (1949: 134, 2017: 224).

  25. Husserl (1973a: 315, 319–321).

  26. “es ist das Urkoordinatensystem, durch das alle Koordinatensysteme Sinn erhalten “. “Jedes Ich findet sich als Mittelpunkt, sozusagen als Nullpunkt des Koordinatensystems vor, von dem aus es alle Dinge der Welt, die schon erkannten oder nicht erkannten, betrachtet und ordnet und erkennt. Jedes fasst aber diesen Mittelpunkt als etwas Relatives, es ändert z.B. leiblich seinen Ort im Raum, und während es immerfort „hier “ sagt, weiß es, dass das „Hier “ ein jeweilig örtlich anderes ist. Jedes unterscheidet den objektiven Raum als System der objektiven Raumstellen (Orte) von dem Raumphänomen als der Art, wie der Raum mit „hier und dort “, mit „vorn und hinten “, „rechts und links “ erscheint. Und ebenso in Ansehung der Zeit” (Husserl 1973b: 116–117).

  27. Cartan (1923: 326, 384). See my comment, in Lobo (2019c).

  28. Bachelard (2016: 122) passim.

  29. Among other places, in his “popular” writings, see Einstein, (1993: 198–199, 207, 215 passim, 225–226, 232–233).

  30. Husserl (1950b: 53), which alludes clearly to quantum mechanics, a point which is further developed in the appendices and complementary texts.

  31. See also complementary text: “Sience of reality and idealization”, Husserl, Krisis, (1950b: 290–291). See égalso lappendices II andIV, rHusserl, 1950b: 357–364 & 387–391).

  32. A few marks, in the French Epistemological tradition, are: Paulette Février, Déterminisme et indéterminisme, Paris, PUF, 1955. The special issue edited by Paulette Février, Mesure et théorie physique, de la Revue Philosophique de la France et de l'Étranger, d’Avril-juin, 1984, T. 174, No. 2. La querelle du déterminisme, Le débat, Paris, Gallimard, 1990. Physique et réalité, Dir. Bernard D’espagnat, Pergame, éd. Frontières, 1997.

  33. Husserl, (1950b: 357–364 & 387–391).

  34. For a definition of complexes in Husserl's sense, see below.

  35. Husserl (1950b: 337–388).

  36. About “the ‘pointless’ version of probability” and its affinity with “pointless topology” of Hausdorff, see Rota (1998: 60), and Rota (2001). It means equivalently the promotion of the algebraic presentation against the “unnatural” presentation in terms of “sample space.” My comments in Lobo (2018: 160 sq.).

  37. „ringförmige und sonstwie figurierte Verkettung oder allseitige kausale Verkettung” (Husserl 2012: 279).

  38. Weyl comments on the encroachment of the measurement operation performed by the subject and the "objective" phenomenon in quantum mechanics as follows: "This throws new light on the subject-object relation; they are more closely united than classical physics thought. It was said in §20 that the quantitative results derived from the observation of reactions of one body to other bodies were attributed to the body itself as inherent characters, whether these reactions actually take place or not. We now see that this 'Euler principle' has serious limitations." See on this subject WEYL, 2017, 263.

  39. Husserl (1950b: 389).

  40. Bachelard (1969: 153). See also Bachelard (2016)

  41. According to the famous formula of HUSSERL in Cartesian Meditations, (trans. PFEIFFER and LEVINAS, Vrin (1980, original edition, 1929, p. 39), but beyond the triviality and the equivocality of this formula, Husserl followed up on an aspect to which one was less inclined to pay attention: namely that the "something" is a "something". 39), but beyond the triviality and the equivocality of this formula, Husserl went on with an aspect to which one was less inclined to pay attention: namely that the "something" was always a synthetic or "syntactic" (noematic) unity of a multiplicity of noetic modalities, of which the doxic and positional modalizations were not the least (op. cit. p. 34 et 35).

  42. Husserl (1950b: 50–51).

  43. Husserl (1950b: § 9h, 49).

  44. Husserl, Krisis, Appendice II, au § 9a (1950b: 395).

  45. Bachelard (1969: 73).

  46. Bachelard (1969: 70). Here should be considered the complex and exciting history of the "Gauss-Laplace law" or normal law of errors.

  47. Bachelard (1969: 77).

  48. About this modal composition of the eidos and this modal Platonism, see Lobo (2011: 161–186).

  49. We refer here to the admirable critical commentary of Maurice HALBWACHS, La théorie de l'homme moyen, essai sur Quetélet et la statistique morale (ed. 1913). One may refer in particular to his conclusion and even more particularly to page 159 and passim.

  50. « D'ailleurs si nous ne connaissons pas a priori la loi de décroissance de la probabilité des observations aberrantes, si, en l'absence du centre absolu de référence nous ne pouvons pas donner a priori un coefficient de probabilité aux différents résultats d'une même mesure, nous savons tout au moins que les erreurs en deçà et les erreurs au-delà ont la même probabilité, vulgairement parlant; par conséquent le type expérimental que nous devons retenir doit être dégagé par élimination symétrique, et par cela même, dans son discontinu, la démarche proposée par M. Le Chatelier est légitimée.» Bachelard (1969: 120–121) (Emphasis mine).

  51. Bachelard warns against confusion. “In reality, we have passed from fact to type. The dangers would be great to reason about the first as about the second. We will see this in inductive reasoning, which appears pure as long as it links types or experimental symbols, but which, in its real application, when we consider the floating of facts around types, can lead to uncertainties that are difficult to appreciate.” Bachelard, 1969: 6).

  52. Husserl (1926), Ms. AIII12 153 <30a>.

  53. „Das ‚ ‘grün und einge­beult seiend ‘, das in der Urimpression auftritt, und der ganze Aspekt des Dings von der betreffenden Seite setzt die voran­ gegangene Erscheinungsreihe, die retentional noch bewußte, dem Sinn nach in einstimmigem Zuge fort, solange wir eben in der einen Schichte verbleiben “ Husserl (1966: 30).

  54. See Husserl's discussion of choice in his 1911 Lessons on Ethics, Husserl (1988, 2009).

  55. « Es wird uns hier wie überall sichtlich und immer besser noch sichtlich werden, daß sozusagen das Schicksal des Bewußtseins, all das, was es an Wendungen und Wandlungen erfahrt, in ihm selbst nach der Wandlung als seine ‘Geschichte’ niederschlagen bleibt “

  56. Husserl (1926), Ms. AIII12 <32a>. Husserl (2012: 282).

  57. Husserl (1926), Ms. AIII12 155, < 31a > 

  58. Husserl (2012: 284).

  59. See the recent developments of these ideas in Bailly and Longo (2008), Sarti et al. (2019), Sarti and Longo (2020); this idea inspired in the second by Deleuze finds thus a rigorous philosophical formulation in Husserl.

  60. Husserl (2001: 219–220).

  61. Husserl (1926), Ms. AIII12, 159.

  62. “The objective world has a dual universal legal structure. (1) A physical structure which grounds every other, […] formed of identical elements, of substrates with identical properties, retaining its identity as an absolute rigorous identity in all becoming, which is a causal becoming, which is entirely dependent on exact circumstances, so that there is no isolated becoming, but only a becoming in the dependence of other becoming, becoming in this thing in the dependence of other things; all this dependence being however governed by exact mathematical laws. (2) A universal morphological structure.” Husserl (1926), Ms. A III 12, 173.

  63. In particular, §§. 73–74 (Husserl 1950a).

  64. This is a group of unpublished manuscripts, but collated by Dirk Fonfara for publication, and partially included in the volume, Zur Lehre vom Wesen und zur Methode der eidetischen Variation, Texte aus dem Nachlass (1891–1935) (Husserl 2012).

  65. Husserl (1926), Ms. AIII12, 158.

  66. Husserl (1926), Ms. AIII 12, 161.

  67. Kant (1985).

  68. Husserl (2001: 219).

  69. Husserl (1926), Ms. AIII12, 189–190 < 68b > .

  70. Husserl (1926), Ms. AIII12, 190–191 < 69a-69b > .

  71. Husserl (1926), Ms. AIII12, 160–162.

  72. Husserl (1926), Ms. AIII12, 161.

  73. Which could correspond to the so-called biological pathways and a Boolean logical modeling in Watterson et al. (2008).

  74. Husserl (2012: 278–286).

  75. Husserl (1974: 424–425).

  76. Husserl (2012: 265).

  77. Husserl (2012: 264–286).

  78. Husserl (1926), Ms. AIII12, 191 < 70a > .

  79. The link between probability, collectives (or “aggregates”) and individuation is also underlined in penetrating reflections on combinatorics (Weyl 1949, Appendix B, 237 sq.; 2017, 337, note to the French edition).

  80. Husserl (2001: 55).

  81. Husserl (2001: 57).

  82. Husserl (2001: 58).

  83. See Islami, Longo (2017: 185).

  84. Points of which we have multiple “geometric” expressions: “point to infinity” of projective geometry, or “imaginary point,” at the origin of the shift from real affine analysis to complex analysis, “accumulation points” of topology, and of course the set of “irregular points” (“points of conflict” or “points of bifurcation”) of catastrophe theory. (See e.g., Thom 1983, 73 sq.).

  85. Which are phenomena, appearances, minus the presumptive sense that they are intersubjectively accessible. They correspond to appearances abstractly reduced to the “sphere of ownness” (the solipsistic sphere of appearances). This holds also for spatial and temporal determinations.

  86. Husserl (1926), Ms. AIII12, 161. (Je souligne).

  87. On René Thom's use of this Husserlian concept, see “Le problème des ontologies régionales en science,” Apologie du logos, pp. 458–467. Or again the “intelligible regions,” Thom (1988: 30).

  88. Husserl (1926), Ms. AIII12, 162.

  89. The explanation of a hierarchy of organizational levels requires a morphological approach of the Catastrophe theory type. (Thom 1983: 83).

  90. We use the term “semantics” in reference to formal theories: model theories (Tarski), but also the theory of possible worlds (with the modal logics developed in the wake of Kripke), which deal with the formal conditions of an interpretation of propositions: i.e., their truth and random modalities (necessary, possible, contingent truth, etc.).

  91. See Weyl (1949: 37, 74, 131, 220). This background can be perfectly orderly or, in the last instance, informal and chaotic.

  92. Husserl (1974: 296–297).

  93. Cf. Le Monde de Descartes Chap. VII, Des lois de la Nature de ce nouveau monde (From the laws of Nature of this new world): “je me sers de ce mot [nature] pour signifier la Matière même en tant que je la considère avec toutes les qualités que je lui ai attribuées comprises toutes ensemble, et sous cette condition que Dieu continue de la conserver en la même façon qu’il l’a créée.” Donc en tant que systèmes de règles suivant lesquelles se font les changements de la matière, i.e., “les lois de la Nature.” (Descartes 1996: 36–37).: “I use this word [nature] to mean Matter itself in so far as I consider it with all the qualities that I have attributed to it, understood together, and under the condition that God continues to conserve it in the same way he created it. Thus, as systems of rules according to which changes in matter are made, i.e., ‘the laws of Nature’. The expression goes back to Plato's Timaeus and his inquiry « péri phuseôs tou pantou», « péri phuseôs tou kosmou».

  94. Kant, Kritik der reinen Vernunft, [B 165], et Premiers principes métaphysiques de la science de la nature, trad. fr. F. de Gandt, Œuvres philosophiques, Tome II, Gallimard (1985: p. 362 passim).

  95. Husserl (2012: 286). (Emphasis mine.).

  96. Das Chaos in kosmischer Auslese (1898), p. 126. op. cit. Gesammelte Werke, Band V. p. 208, published in Hausdorff (2004).

  97. Hausdorff, Gesammelte Werke, Band V, p. 561.See Lobo (2020: 81–90).

  98. See Lobo (2000).

  99. This is mainly Kripke's model theory, one of which will find a criticism in Jean-Yves Girard, TRhe Blind, I, Foreword, xv: “The most trivial blind spot is this bad modal logic which is justified by bad Kripke semantics and vice versa”; and pp. 98–100. Girard (2009) and Girard (2011).

  100. Especially in the Krisis, and before that in the contemporary lessons of the manuscripts we are studying here, in particular, Natur und Geist, Chap. VI, §§ 23–24, Husserl (2001: 140–156).

  101. Husserl (2012: 287). The theory of induction must be understood in the larger frame of formal logic, which considers the “life” of modalizations, which to a certain extend is the core of life itself. See Husserl (2001: 134–140) and the whole Chapter VI, 140–156, results that have been obtained in the investigations on Passive Synthesis, Husserl (1966).

  102. Husserl (2012: 288). “But one must not neglect the fact that the imposition of a form, for example of Euclidean spatiality and temporality, leaves open an infinite number of possible natures of possible worlds and that an entirely free variation, which one can obviously accomplish here, and consequently a free variation of possible forms of natural legalities and, through them [i.e., of natural legalities, is possible. through them [i.e., by imaginary laws of nature] of possible forms of concrete realities that can be constructed in a purely mathematical way, constitutes a global system of possibilities, which are only composable under the idea of possible worlds, insofar as we thus traverse an infinity of different possible, but incompatible, worlds, as soon as we demand that one of these possible worlds be constructed and remain maintained in its fixed identity.”

  103. “For lack of a mathematics of concrete realities, we described our surrounding world of realities that appear there, by means of sensitive concepts and that, practising physics, we applied to it, so to speak, only surface physics and exact chemistry, that we should even do the same in chemistry, it is a necessity” (Husserl 2012: 290.).

  104. Husserl (2012: 291).

  105. Husserl (1926), Ms. AIII12, 12b.

  106. Husserl (1926), Ms. AIII12, 13 b.

  107. About the transfer of this relation of “Fundierung” into the realm of formal logic and its fecundity for a non-reductionist approach to multi-level phenomena, see among many instances in Rota’s work, Rota (1992: 171). Rota (1989, 1997).

  108. Noble (2006: 8; 70), Noble (2016: 61–62; 80–83).

  109. cf. Longo (2006: 123).

  110. “Consciousness in general, of whatever type and form it may be, is traversed by a radical division: first of all, any consciousness in which the pure self does not immediately live as the ‘accomplishing one’, and which therefore does not immediately have the form of the cogito, has in essence… the possibility of the modification that leads to this form. Now there are two fundamental possibilities in the mode of accomplishment of consciousness within the mode of the cogito; or, in other words: to each cogito belongs an exactly corresponding counterpart, so that its noema has in the parallel cogito its exactly corresponding counter-noema.” (Husserl 1950a, 277).

  111. The alleged Husserlian play on the word Wahrnehmung—wahr nehmen – nehmem-als-wahr-seiende (perception – really seizing — grasping-as-true-being) expresses in an exact and condensed manner the discovery made at the time of the Logical Investigations: the acts of simple representations always already have a doxical dimension (and, contrary to Brentanian theory, this dimension does not intervene afterward, when judgments are based on the representations). Perceptions are therefore immediately “judging” (“propositional”), although prelogical. It is easy to understand why, when we come to the sphere of expression the logical sphere in the proper sense of the term, we have the theory of nominalization, which is a theory of doxical nomination.

  112. On the distinction between doxic and non-doxic thetics, see. Lobo (2010: 137–154).

  113. Meinen, Meinung and consequently Intention in the phenomenological sense does not mean primordially aiming at (abzielen), directed toward (Gerichtet auf) or even striving for (Streben), but believing (Glauben), holding-for-true (possible, doubtful, plausible, etc.) Husserl (1966: 83–92).

  114. See. Beilage VII (zu § 20): < Glaube und Intention > . (Husserl 1966: 364–365).

  115. Husserl (1966: 225).

  116. Lobo (2005: 48–49).

  117. Emphasis mine. The analysis continues: “This means, for example, about the joy [given] by the existence of a fact, that not only, phenomenologically speaking, the quality of joy cannot present itself without the matter of joy being doxically qualified in a certain way, but that the presumption [Vermeinen] of the rejoicing being [Erfreulichkeit], as a ‘presumption’, presupposes a certain conviction of being, and this is a particular relation of qualities. The holding-for-rejoicing is based on the belief in being and includes it in itself, in a certain way. And where this ‘settling’ (which is a kind of modification) takes place, is the law according to which the whole act is, so to speak, wrong if the foundation is wrong, the rejoicing-oneself is wrong if the underlying conviction is wrong—not wrong in every respect, but precisely in view of this presupposition.», Husserl (1988: 127, 2009: 214).

  118. Husserl (1988: 118, 2009: 203–204).

  119. Concerning the lack of knowledge among psychologists and logicians of the “modalities of being” and the correlative “modalities of beliefs,” see also Husserl (1966: 225).

  120. Ideen I, § 109. Husserl (1950a).

  121. Husserl (1966: 365). “All our analyses were moving in the sphere of positionality. In the pure life of imagination, there is no belief, but only quasi-belief, imagination (Einbildung) of belief, just as there is no will, no evaluation, but only something like < an > imagining oneself in something like that.”.

  122. Paraboles et catastrophes, p. 85.

  123. Husserl (1966: 403).

  124. Lobo (2000). Cf. especially Chap. VII et VIII.

  125. Husserl (1966: 404).

  126. See the sample in Lobo (2019a, b, c, d: 533 sq.).

  127. Berthoz (2019: 15–18), Berthoz (2020).

  128. Boi (2008, 2011). About the three fundamental epigenetic mechanism which are “DNA methylation, histone modification», and non-coding RNA (ncRNA)-associated gene silencing.” (Berger et al. 2009; Nora et al. 2020).

  129. Longo et al. (1999).

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Lobo, C. The limits of the mathematization of the living and the idea of formal morphology of the living world following Husserlian phenomenology. Theory Biosci. 141, 175–202 (2022). https://doi.org/10.1007/s12064-021-00348-4

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