Philosophically Specified Types of Methods Important for Theoretical Natural Science^*

Jaroslav Kubrycht

Department of Physiology, Second Faculty of Medicine, Charles University, Prague, Czech Republic.

DOI: 10.4236/ojpp.2024.142029                

Abstract

In accordance with current philosophical opinions, four classical and one more recently proposed types of methods frequently used in theoretical natural science are specified here together with the corresponding sources of inspiration. More precisely, abstract models, thought experiments, mathematical hypotheses and metaphors are dealt with here as classical types of methods, whereas hybrids of mathematical hypotheses and thought experiments represent more recent methodic group. In addition, this paper describes the relationships of the introduced types of methods to the (i) three-floor hierarchy of scientific theories, (ii) examples of ancient or recent discoveries and (iii) recent usage of computers.

Keywords

Model (s), Hypothesis, Metaphor, Theoretical, Thought Experiment (s)

1. Introduction

Four types of frequently used theoretical methods (TTM) can be denoted as classical TTM, i.e. as many centuries used groups or families if not categories of methods. Three of TTM represent important synthetic forms of ideas, i.e. abstract models (AM), thought experiments (TE) and mathematical hypotheses (MH) (Černík et al., 1980; Ado et al., 1981) . In addition, intuitionists add to the centre of theoretical investigations metaphors (Mp), used elsewhere but exhibiting considerable importance in theoretical science (Stachová, 1993) . Hybrids of MH and TE (HMT) then represent more recent TTM (Míček, 1981) .

The aim of this paper consists in contribution to better and more frequent usage and creation of TTM. This could to be important, because progress usually occurs when genius attempts like many well-known TTM become the school routine. In addition, the knowledge about TTM can sometimes help us to clarify trends in theoretical and empirical science, our own considerations or important points of school lectures. I show here the corresponding definitions or specifying notes, various subset classifications, ways of origin, multiple important structural features (chapters 2-4), some sources of the corresponding inspiration (chapter 7), relationships to computers (chapters 5 and 8), hierarchy of theories (chapter 6) as well as TTM interrelationships (chapters 2-4 and 10). In addition to examples from history (chapter 8), I deal with here the result of the first historical reconstruction performed in case of TE (Černík, 1972; Míček, 1981; chapter 2) and proposed also by another scientist (Ernst, 2015) . Though the specific methodological natural-science-related approaches described here differ from related general informatics-based classification of theoretical science expressed in two papers (Nilsen, 2015; Bergeron et al., 2017) , the author of the first of these papers consider importance of epistemological view in his Conclusions, i.e. view specifically discussed here.

2. Thought Experiment as a Half-Brother of Laboratory Experiment and Sci-Fi

According to historical reconstruction (Černík, 1972; Míček, 1981) of thought experiment (TE) development, TE arose in Antique Greece from germinal (precursor) form denoted as “speculative fiction”. Somewhat dialectical, consensual and artistic approaches present in Fragments of Heraclitus (2009) and Tao-te-tiang of Laozi (1997) then possibly represented ideal forms near boundary line between “speculative fiction” and TE. Though the first TE arose in antique Greece (possibly Atomic theory of Leucippus and Democritus), their maturated forms appeared only during Modern Period (Míček, 1981) . More precisely, the two physicists and natural philosophers markedly contributed to development of TE. Galileo Galilei used his exactly formulated TE in 16th and 17th century, whereas H.C. Oersted was the first who tried to retrospectively define TE in the first quarter of 19th century (Witt-Hansen, 1976; Míček, 1981) . TE represented a considerably differentiated form of exact scenario-like speculation comprising reasonably acceptable versatile images concerning an investigated act, which specifically participates as an important dynamical component in many observed processes. However during later part of 19th century, usage of TE expanded to other areas with educated people occurring outside critical voices of philosophers and exact scientists. This even resulted in the reverted situation where almost any speculation was wrongly recognized as original scientific TE. In contrast, masterly formulated TE in natural science maintained their exact forms in the following history, e.g. those formulated by J.C. Maxwell in the third quarter of the nineteenth century or A. Einstein, E. Schrödinger and R.P. Feynman in the twentieth century (Brdička & Dvořák, 1977; Jost et al., 2009; Apertet et al., 2014) .

According to Míček (1981) , younger sci-fi differentiated from the same “speculative fiction” like TE only in Modern Period. This means that sci-fi differs from TE in (i) purpose, including among others frequent less exact (“liberated”) relationships to reality and (ii) its “second parent”, i.e. older fantasy. Consequently, inspiring artistic sci-fi represents ideal form distinct from TE. Nevertheless, sometimes but rarely, TE or its germinal forms can occur inside some sci-fi novels mostly written by authors having their own scientific experience (e.g. A.C. Clark) or deep knowledge about scientific or technical progress (e.g. J. Verne; in agreement with Wiltsche, 2019 ). Except for some illustrative TE or TE with freely eligible typical objects (see below), recent TE and sophisticated experimental designs (SED) terminally diverged. This followed from difficult, expensive, extensive and sometimes only partially or completely impossible (e.g. in evolutionary approaches and under inaccessible conditions) concretization of TE to current decisive networks of laboratory/empirical experiments and/or observations (Ado et al., 1981) . Nevertheless, certain parallelism between TE and SED still exist and often concerns new or unusual (frequently hypothetical), unexplored, surprising, possibly unacceptable, model, unclear, uncertain, suspiciously too sure or only partially verified limiting situations.

Abstract-object-related specification of TE deduced from the historical reconstruction of TE development (Míček, 1981) yields better defined TE or opportunely also HMT (see below) including consensual, i.e. typical or ideal, objects (cf. “abstract objects” in Table 1). In fact, TE-specific abstract objects mostly reflect some necessary conditions and simplifications of modeled reality. More archaic but still valid typical objects represent variously strictly selected examples of actual objects. The strict selection of the typical object then reflects low degree of freedom in the investigated process. For instance, Gallileo’s law of inertia was derived with the help of typical objects, when describing the motion of masterfully crafted ivory sphere on Venetian mirror (Míček, 1981) . The second type of abstract objects, i.e. ideal objects, arise in processes called as (i) idealization based on simplifying scientific conventions (e.g. the “point particle” defined in Newtonian mechanics in 17th century; Šolcová, 2017 ), (ii) statistic processing appearing in 19th century (e.g. kinetic theory of gases, Brdička & Dvořák, 1977 ), (iii) professional and systemic evaluation of sufficient volumes of knowledge (e.g. knowledge-based objects representing idiotypes, epitopes, mimotopes known in immunology and interactomics; Kubrycht, 1985; Kubrycht et al., 2012; Landmann et al., 2017 ) and (iv) rare cases of sophisticated symbolic abstraction or artificially modified objects (cf. section “abstract objects” in Table 1). The important example of TE with sophisticated symbolic ideal objects is Maxwell’s demon from 19th century (Brdička & Dvořák, 1977) . Hidden exactness of this TE consists in substitution of the demon by more realistic objects or in its consistent step-by-step rejection (e.g. investigation of Maxwell’s demon in the context of quantum thermodynamics of information; Cottet et al., 2017 ). The specification of TE including abstract objects does not mean that we have to start formation of TE only after object formation. The alternative ways of TE development can consist in skeptical object reformation after: (i) considerations near TE (see e.g. section “specificity” in Table 1) or (ii) processing of ideas or fundamental questions following from inspiration sources including those in chapter 7.

Table 1. Some connections important for thought experiments.

¹STES—schemes restricting TE subsets; TE—thought experiment(s); *—indicator of alternative notes concerning the same table section.

²AM—abstract model(s) (see chapter 3); FOM—formalized object models.

Independently of considerations following from the discussed historical reconstructions, graphic clarification (i.e. the corresponding drafts pictures, power point presentations etc) and understanding to TE diversity appear to be important for TE creation. An important type of diversified TE classification follows from different purposes of TE (cf. Table 1; Míček, 1981 ). TE can (i) lead to the formation of MH described below (e.g. derivation of Maxwell-Boltzmann distribution of molecular rates in gas; Brdička & Dvořák, 1977 ) or AM (in agreement with the immediately following point), (ii) enable to formulate non-trivial questions or considerations important for further organizational (Aguinis et al., 2023) , experimental or theoretical work (e.g. Maxwell’s demon; Brdička & Dvořák, 1977 ), (iii) reasonably warn in cases of contradictions in theory ( Míček, 1981 ; cf. also counter-thought-experiment in Yeates, 2004 ; differently from decisive critical empiric experiments initiating crisis of paradigm, these warning/critical TE are not solely destructive for paradigms), (iv) compose combinations or more complicated arrangements of differently intellectually-based processes including TTM (e.g. derivation of molecular orbitals for more complex molecules; Brdička & Dvořák, 1977 ), (v) forming set of alternative TE representing starting points for possible future research (Burge et al., 1979) , and (vi) simply illustrate considerably tortuous problems. Certain problems with authenticity can sometimes follow from the fact that the last purpose-related variants of TE (illustrative TE) cannot be accurately distinguished from the first one (generative TE, i.e. TE frequently generating MH). More detailed structural classification of TE was based on schemes restricting TE subsets (STES). This important concept appeared online only after millennium (Yeates, 2004) and was also recently presented in Wikipedia page concerning TE. Several opinions and glosses concerning STES are displayed in Table 1. Recent approximate classification correlates age of theory (only young and old theories have been yet distinguished) with relationship of TE-forming author to this theory (Aguinis et al., 2023) . This classification and accompanying logical scheme could help us with orientation when building at least some TE or the related AM.

3. Description of Additional Classical TTM

Differently from TE and fundamental MH (see chapter 6), abstract models (AM) constitute as much as credible but still simple descriptions of concrete processes to approximate their complexity and simultaneously to avoid frequent errors, respectively. In the cases of approximating/preliminary model descriptions we can distinguish four subtypes of AM. (i) Non-living physical 3D models are AM as possible plausibly imitating necessary space properties of modeled actual objects (e.g. 3D model of DNA described by Watson & Crick (1953) ). (ii) Gross models are based on very limited structural knowledge represented by schemes, formulas, symbolic chain changes or 2D projections. These AM currently enable us to get initial orienting information about the tortuous problem or as yet little known reality (e.g. Kubrycht & Novotná, 2014 ; in this case we met the problem of prevailing difference between autoepitopes containing mainly aliphatic amino acids and well predictable epitopes). (iii) Black box analysis comes from knowledge of behavior only (Sarbaz & Porakbari, 2016; Terayama et al., 2021) . (iv) Gray box analysis then uses weak but existing knowledge concerning both structure and behavior of modeled objects (Oussar & Dreyfus, 2001; Sarbaz & Porakbari, 2016) . Modern advanced mostly computer assisted complex AM descriptions frequently comprise systemic analysis involving rules from graph theory (Kolář, 2009) , object-oriented programming using abstract objects denoted as formalized object models (FOM) and sometimes also design patterns or universal modeling language (Gamma et al., 2003; Fowler, 2009; Bohmer, 2012) .

Mathematical hypotheses (MH) constitute TTM important for recent physics and some interdisciplinary areas of natural science. MH consistently reflect quantitative relationships observed in nature without sensory-based visualization using usually formulas or equations (Ado et al., 1981) . In contrast to mathematics dealing with consistent description of any postulated, estimated or idealized shapes and their frequently countable or logically derived relationships, MH describe real processes occurring under certain justified background conditions and use experimentally correlated constants (cf. Barrow, 1997 ). This requires not only sufficient quantitative and structural agreement of MH, but also certain knowledge about contemporary scientific progress in investigated areas. MH frequently arise via at least three following mechanisms. (i) MH can be generated based on relationships present in the corresponding TE, certain AM or Mp, valid schemes or several approximate empiric formulas (Míček, 1981; Brdička & Dvořák, 1977) . (ii) If new suitable calculus or transform is generated in mathematics, it is necessary to specify important relationships in which this novelty will be successfully used (e.g. usage of infinitesimal calculus or Fourier transform; see chapter 8; Štecha, 2003; Klíč et al., 2012; Šolcová, 2017 ). (iii) Two distinct MH can be sometimes unified to form a more general solution, e.g. when formulating Schrödinger equation (cf. chapter 9) or local biothermodynamic version of the first law of thermodynamics (Dvořák et al., 1982) .

Freely considered term scientific metaphor(s) (Mp) concerns not only symbolic objects, but also structurally or dynamically intended judgments following from analogy (Brdička & Dvořák, 1977; Stachová, 1993) . Such Mp can considerably contribute to the generation of some TE, because they (i) participate in the formation of typical objects and can influence the formation of ideal objects, (ii) sometimes help with building TE-related scene or specification of scenario (Míček, 1981) or (iii) enable us to select mathematical description based on structural or dynamic similarities (see chapter 9). In addition, Mp yield qualitative predictions, orienting concepts (cf. chapters 8) and sometimes enable usage of existing approaches for other purposes, i.e. metaphorical transference (e.g. lytic unit of NK cytotoxicity as generalized Michealis’ constant for enzyme kinetics; Pross & Maroun, 1984 ). It can be noted that poets as the most frequent producers of Mp create some of their poems in intervals of seconds or minutes, whereas writing up other poems may take even years (statement of the Czech poet M. Holub). This implies the question whether scientific Mp can be similarly to poetic ones formed by different forms of intellect.

4. Newly Proposed Hybrids of MH and TE

Hybrids of MH and TE (HMT) represent a more recent TTM proposed for theoretical natural science (Míček, 1981) . The first pattern of HMT appeared during solution of black-body radiation by Max Planck in 1900 (Kleppner & Jackiw, 2000) . This discovery started the era of quantum physics (Brdička & Dvořák, 1977) . In case of HMT, TE and MH compose combined or more complicated arrangement of procedures (cf. purposes of TE in chapter 2). In comparison with TE, we can observe wilder (sometimes almost surrealistic) scene composing HMT. This scene opportunely includes modified abstract objects of TE, local rules, transformations (e.g. transformations from Euclidean space to space reflecting string theory; Barrow, 1997 ), various mathematical projections (Lakatos, 1976) , stochastic schemes (Unčovský, 1980) , numbers and characters. Characters play an important role in many bioinformatics considerations (occurring sometimes on boundary lines of TE and HMT), because words above alphabet (e.g. motifs, consensi and synonymous functionally related segments, etc.) frequently form primary components of the corresponding objects (Chytil, 1984; Alberts et al., 2008; Kubrycht et al., 2013) . Such bioinformatics words represent mostly nucleotide and protein sequences, whereas the corresponding observed or potential changes can be denoted as formal grammar-like records and classified as existing or potential statistical events of various structural or functional importance, respectively (Rogozin & Kolchanov, 1992; Hatina & Sykes, 1999; Kubrycht et al., 2006, 2016; Duquette et al., 2007) . Diversification of scientific scenography in HMT appears to be interesting with respect to the possibility of reverse abstraction (i.e. structurally based depicting abstraction close to modern painting) important with respect to intended future applications of artificial intellect. This could make more clear formalistic or only executing schemes sometimes necessary for understanding to some theoretical papers or development of proposals.

5. Computer-Assisted TTM

Provided that substantial decision/evaluation is not performed out of selected set of programs, we can speak about computer-assisted AM. On the other hand, if outer substantial decision/evaluation is based on TE, MH or HMT, we can refer computer-assisted MH, TE, HMT, respectively (cf. e.g. Möller & Schenck, 2008 ). At least the two last preceding alternatives and some AM (mostly simulations) are sometimes alternatively denoted as computer experiments (CE). The question concerns existence of autonomous group of CE different from other TTM. Computer-assisted Mp appear to be interesting topic for recent and future research of artificial intellect.

6. Floors of Theoretical Science

Three stages related to grading of theoretical science are denoted as T0, T1 and T2. T0 constitutes more likely a group of candidates for theory forming thus frequently criticized basement of theoretical science. These candidates include gross estimations, immature hypotheses, guesses, presumptions etc. Nevertheless, some of these theoretical candidates can represent starting points of subsequent important scientific research. T1 comprises correct empiric hypotheses (frequently described in methodology of empiric science), AM, some MH and abstract objects used in theoretical science. In T1 stage, theoretical principles develop from empiric data and pieces of knowledge and analytical character of knowledge predominates (Černík et al., 1980) . New hypotheses and designs for experimental verification appear. AM become to be continuously perfected, better formalized and generalized when adding parameters and new conditions or refining formalized object models. Ideal objects mature, relatively accurate Mp and typical objects are proposed or reconsidered. Numbers of approximate empiric formulas or more versatile MH and concrete AM are increasing. TE are proposed, whereas some failed attempts to create an adequate TE are replaced by HMT to attain important solutions. Newly formulated laws then represent specific manifestation of T2 stage (only T2 in further text). Laws often expand to consistent theoretical systems via hypothetical-deductive way (for details see below) in which the synthetic character of knowledge prevails (Černík et al., 1980) . As follows from the Theory of Paradigm, laws are pragmatically substituted by their more precise and general followers during crisis of paradigm. This substitution requires formation of new T0- and/or T1-related representatives (Kuhn, 1962; Lakatos, 1970; Míček, 1981; Fajkus, 1997) .

In T2a level of T2, factual/empiric laws constitute the most frequent theories (Černík et al., 1980) . For instance, such laws include (i) geographical relationship indicating movement of continents (A. Wegener), (ii) qualitative important table-related arrangement of data (Mendeleev’s periodical system) or (iii) descriptions of typical morphology accompanied with structurally-functional descriptions (cell theory; M.J. Schleiden and T. Schwann). Factual/empiric laws can arise via different manners, i.e. based on observations, experimental experience or when using TE or Mp, various forms of abstraction, data representations, data processing including statistical, systemic or structural analysis. These laws can expand to systems of logical or fuzzy-logical rules associated with knowledge-based topologies, hierarchies, networks or lists of statistical linkages sometimes forming factual ontological data system (cf. Devkota et al., 2022 ).

Only theories in physics and some interdisciplinary branches combined with physics, mathematics or informatics can be classified by means of the level T2b (only T2b in further text; cf. Černík et al., 1980 ). For instance in biology, this concerns biomatematics, biophysics, bioinformatics and biothermodynamics. Idealized laws are most frequent entities in T2b. Idealized law begins its own existence at the time, when unique so-called fundamental MH or several such MH expressing this law is/are formulated. Fundamental MH usually represents substantial and widely usable MH frequently (cf. Ado et al., 1981 ). Differently from factual/empiric laws, expansion of idealized laws yields mathematically consistent systems (networks) of MH, i.e. formulas and equations. Generalized laws comprise mostly generalized idealized laws. Like current idealized laws, generalized idealized laws achieve T2b level and are mainly expressed using differential equations. The first generalized law, i.e. Euler-Lagrange equations, arose in 18-th century whereas most of such laws appeared in last hundred years (Lehner & Wendt, 2017) . Generalized laws concern also biological processes, e.g. law of generalized diffusion (Murase & Matsuo, 1991) and local biothermodynamic version of law of energy conservation (Dvořák et al., 1982) . The question is whether at least some advanced and verified descriptions comprising formal grammar-like developmental processes in special animal models draw near T2b (e.g. in the corresponding research of Caenorhabditis elegans; Larsson et al., 2011; Tarkhov et al., 2019; Ewe et al., 2022 ).

In summary, the development of theoretical research consists in formulation of still more versatile, flexible and better abstracted methods describing reality. Nevertheless, the usage of these methods for the description of complex events sometimes needs the return to graphic representations and schemes, more simple procedures and searches for new completing views. Accessible consistent mathematical description and formation of quantitative networks constitute undisputed advantage of theories.

7. Inspiration Sources for TTM creation

Certain objects in real world and some abstracted constructs can be considered as inspiringly interesting patterns (IIP) when forming abstract objects usable in certain AM, TE and HMT. This concerns mainly IIP reflecting principles important for progress in history of knowledge and science. The list of such IIP comprises quantum computer, representing recent paradigmatic object (Barrow, 1997) , water and related more general and “more chaotic” theoretical constructions of events occurring in superfluids (Kapitza & Lifshitz, 1969; Scott, 2022) , language as medium/means of speech and reflection of the world (including e.g. biological sciences), Turing machine, cellular automata (Ermentrout & Edelstein-Keshet, 1996) , translating, decoding or decrypting programs or hardware components, astronomical clocks, etc.

Dual chimeras separately approximate double-dealing behavior of observed objects (Table 2). More precise dual descriptions then usually comprise unifying attempts, i.e. (i) solutions of differential equations like Schrödinger equation (i.e. MH; Brdička & Dvořák, 1977 ), (ii) well interpretable TE, HMT, (iii) some co-routine-based computer simulations concerning delayed, diffusive or sometimes almost immediately (infinitesimally) reciprocally communicating or responding processes (i.e. AM or MH), (iv) verified formal grammar-like relationships (mostly AM or MH; cf. Chytil, 1984 ), (v) valid structurally-statistic linkages reassessing or specifying AM, MH or HMT (Janout, 1995; Lepš & Šmilauer, 2016) and (vi) considerations based on observed dialectical relationships, cyclic processes or more specific system of changes near life comprising structure associated with information, fluctuation linked to noise and function requiring energy (Prigogine, 1978) .

Due to an extensive usage of computers, the number of various graphical presentations increased including current graphs, 2D (frequently map-like pictures) or 3D (space or space-like) representations. These attempts evoke an image of futurological scene in which very intelligent computer offers to tested or only scared scientist its cleverest variants of representations: “Please, make your choice”.

Table 2. Examples of interesting dualities in natural science.

¹DC—dual chimeras separately approximate double-dealing behavior of observed objects; FC—formal description of any two components forming dual chimeras, when using lucid symbolic characters A and B. For abbreviations or terms E-loops, EL-coordination, M-cells, NO, PO see inner space of this table.

²For possible variants of duality descriptions see chapter 7.

³For details see chapter 9 and Table 3.

⁴This type of dualities requires more complicated descriptions which makes difficult to use mathematical hypotheses.

Though IT principles represent important inspiring source for natural science, the inspiring processes and structures in nature and in computers somewhat qualitatively and quantitatively differ. This means that mechanistic transfers of rules and algorithms do not always hold being sometimes even waiting for further progress in informatics. This is the reasons why some important philosophical questions concerning natural processes are still rather solved using biothermodynamics or reaction kinetics though using computers (for details see chapter 9).

Among the effective manners of less exact but still important inspiration, we can find (i) our own or acquired by reading (mainly modern fantasy and sci-fi constitute topical sources) imagination, (ii) certain principles, connections and events described by architects and mystery scientists, (iii) various stories, myths and fairytales, (iv) certain games (e.g. chess, Go, Sudoku) or (v) shapes observable in maps, music, ornaments, building constructions and products of plastic arts.

8. Concise History of TTM and Their Implementations

The simple MH and AM were possibly formulated in time of beginnings of astronomy and geometry in Sumer and Egypt (Asimov, 1994; Steele, 2019) . Lately, Greeks and their Byzantine descendants became serious and most frequent authors of TTM during Antique period and early Middle Ages, respectively. In addition to examples displayed in Table 3, we have to mention new rules, if not rediscoveries, in geometry (e.g. Thales of Miletus, Pythagoras and Euclid) and logic (mainly Aristotle) important for further development of TTM (Aristotle, 1961; Kessidy, 1976) . Subsequently, Persians and Arabians contributed with their new opinions to development of theoretical natural science (e.g. Al Kindi (Alkindus), Ibn Miskaway, Ibn Sina (Avicena), and Alhazen; for Ibn al-Nafis see Table 3; Ado et al., 1981; Hehmeyer & Khan, 2007 ). In the 13th and the 14th centuries, the reports about Chinese discoveries (e.g. Million of Marco Polo) and immigration of Byzantine scholars contributed to Italian Renaissance (Martin, 2023) . Multiple physical 3D models and AM-related drafts of Leonardo da Vinci (e.g. Richardson (2019) or Marusic & Broomhall, (2021) ), rediscovery of heliocentric system by Italian student and Polish scientist Nicolas Copernicus as well as some opinions of young Galileo Galilei constituted in fact initial manifestations of European rationalism in the end of the 15th and during the 16th centuries (Table 3). New Organon published by Francis Bacon in 1620 then became to be turning point in history of science. Hence this book specified inductive methods, requiring experimental verification of the proposed hypotheses concerning actual world (Ado, 1981). Further development of TTM was markedly influenced by Rene Descartes. His contribution among others included: (i) important and perhaps dialectical relationship between radical skepticism and constantly verified experience (“Experientia”), sometimes accompanied by reasonable doubt about radical skepticism (Major & Sobotka, 1977) and (ii) a new useful and lucid manner how to record explicitly defined mathematical functions (Šolcová, 2017) . Discovery of infinitesimal calculus performed independently by Isaac Newton and Gottfried W. Leibniz in the second half of the 17th century represented strategic point for further development of fundamental MH namely in area of physics (Šolcová, 2017) . This revolutionary calculus enabled for mulation laws of Newtonian mechanics (in the 17th century), Euler-Lagrange equations representing generalized law of mechanic (in the 18th century), Maxwell’s equations representing law describing electromagnetic field (in the 19th century) and later fundamental MH mentioned in Table 3 and chapter 9. In the 18th century, the multiple new formal mathematical models were derived and lately used in more concrete AM in several disciplines ( Šolcová, 2017 ; Table 3). During 19th century, TE were successfully used not only in physics but also in chemistry, whereas MH were formulated in biology (Table 3). The first HMT appeared only in the year 1900 (see Table 3 and chapter 4).

Table 3. Historical examples of the discoveries implemented using TTM.

¹Three main parts of this table represent three important eras, i.e. Antiquity, Middle Ages and Modern Period.

²Country—country of author’s origin; England/Denmark—Danish immunologist born in London.

³AM—abstract model(s); AM/MH—Veneziano’s model represented AM tending to become fundamental MH; AM/TE—conventional AM close to TE based on empirical knowledge; HMT—hybrids of MH and TE (cf. chapter 4); MH—mathematical hypothesis/hypotheses; MH/HMT—additional usage of TE and data re-evaluation, i.e. formulation of HMT, can be sometimes important in cases of molecular orbitals related to complex molecules Mp—metaphor(s); T2b—the highest level of theory (see chapter 6). For additional abbreviations see Table 1 or Introduction.

⁴ME—Maxwell’s equations; SE—Schrödinger equation.

In the first half of the 20th century, TTM appeared in novel quantitatively investigated braches of natural science such as biochemistry (namely enzyme kinetics), theory of relativity and quantum physics (see chapter 9 and Table 3). Certain philosophical aspects of theoretical natural science became to be analyzed by E. Husserl and N. Bohr (Ado et al., 1981) . E. Husserl dealt with post-epochal processing of arising hypothesis including formulation of MH or AM. N. Bohr formulated methodological philosophy concerning quantum physics. The first materialistic evaluation of MH as a type of scientific method was subsequently performed by physicist S. I. Vavilov (co-author of Vavilov-Cherenkov radiation honored with Nobel Prize; Ado et al., 1981; James et al., 2011 ). In fiftieth, the structure of DNA was successfully modeled using specific physical 3D model based on data obtained in X-ray diffraction, i.e. by means of AM. This discovery represented the law explaining molecular reproduction of genomes ( Watson & Crick, 1953; Alberts et al., 2008 ; Table 3). Later tracing of protein interactions of steroid hormones employed scientific Mp. In accordance with the results with individual steroid hormones, many analogous or even evolutionarily related transport and regulatory proteins were found, except for several imperfections (cf. Harper, 1977; Ganong, 2005; Wang et al., 2014 ). TE appeared more frequently in biology only in last decade of 20th century and later (e.g. Boregowda et al., 1997; Möller & Schenck, 2008; Falissard, 2011; Yamamoto et al., 2019; Krauzlis et al., 2023 ).

Early computer processing of natural scientific theoretical approaches occurred in fiftieth, sixtieth and seventieth of the 20th century. This processing comprised mainly current enumeration following form TE, MH and HMT systemically pre-processed to their computer-assisted forms (e.g. in cases of quantum mechanics, astronomy and enzyme kinetics; Brdička & Dvořák, 1977; Horák & Kotyk, 1977; Barrow, 1997 ). Lately in eightieth and ninetieth, further progress in programming, increasing rates of computer processing and large-volume memories including newly built knowledge-based databases (Table 3) enabled marked progress in physics and molecular biology. The progress in molecular biology concerned mainly comparison and classification of still increasing numbers of protein and nucleotide sequences (e.g. Altschul et al., 1997 ), building of sequence-based trees important for molecular evolution (Felsenstein, 1981; Tateno et al., 1982; Saitou & Nei, 1987; Sourdis & Nei, 1988; Philippe, 1993 ), predictions of secondary and 3D structures and even the possible interactions of the corresponding molecules (Godzik et al., 1993; Rodionov & Johnson, 1994; Dunbrack, 1999) . After millennium computer programs offer hybrid comparisons via crossing several independent methods (Kaur & Raghava, 2004; Standley et al., 2010) , advanced molecular dynamics enabling more precise quantitative models of reactions or interactions (Lakhani et al., 2017; Sanapalli et al., 2022) , similarly intended knowledge-based homologous 3D modeling (Evers et al., 2003; Clark & van Vlijme, 2008; Zhu et al., 2014; Arcon et al., 2021) and interactomic databases (Gemovic et al., 2019) . Recent pandemic of COVID19 led to molecular dynamic studies of interactions between proteins necessary for reproduction of SARS2 virus and their potential high affinity natural inhibitors of plant origin (mainly certain flavonoids were selected; Ali & Kunugi, 2021; Chapman & Andurkar, 2022; Kashyap et al., 2022; Rahman et al., 2022; Toigo et al., 2023 ). The studies contributed to further rationalization of traditional medicine and selected molecules interesting for possible future therapies of coronaviral and other viral diseases using or combining existing or newly prepared nutritional supplements if not sophisticated diets.

Though artificial intelligence (AI) formerly applied in weather prediction already in sixtieth, its boom in natural science came only after millennium (Šnorek, 2002; Guo et al., 2006; Wang et al., 2011) . In fact AI represent very simplified, but selectively and thus efficiently acting models of brain or certain abstract cognitive activities, i.e. AM. As well known special forms of AI like neural networks (Šnorek, 2002) or Gaussian fuzzy logics (see Wang’s theorem; Wang, 1992; Jura, 2003 ) sufficiently imitate almost any mathematical function defined on a compact set. It is a question, whether we can restrict some more communicable description of the functions generated by AI. This means the possibility of converting these functions into the corresponding schemes or scenes related to TE, HMT or consistent differential equations or formulas.

9. Examples of MH Forming Recent Theories Important for Natural Science

The analogy (i.e. Mp) between sound and light evoked the idea of the wave substance of light pronounced by C. Huygens (Table 3). This idea was then generalized to all electromagnetic waves by J.C. Maxwell ( Ado et al., 1981 ; Table 3). It is a historical question, whether these textbooks interpretations inspired analogous derivation of Schrödinger equation, representing fundamental MH in recent structural chemistry More precisely, this means the substitution of the wave length in acoustic (i.e. sound-related) equation by the wave-length-determining right part of De Broglie formula ( Brdička & Dvořák, 1977 ; see also Schrödinger in Table 3).

Turing (1952) was the first scientist, who tried to model non-mechanistic chaos using MH (cf. Maxwell J.C. in Table 3). His model concerned cell differentiation and comprised diffusion phenomena. In the end of the same decade, the first messages about deterministic (non-mechanistic) chaos appeared and were lately strongly expressed by the equation of Lorenz (1963) investigating weather (see Table 3). Deterministic chaos was continuously substituted as physical paradigm (see below) by its more precise followers after more than twenty years, i.e. by (i) quantum chaos (Steeb, 1985) and (ii) generalized diffusion specified for understanding certain events in embryogenesis and cell differentiation (Murase & Matsuo, 1991) similarly to diffusion-based Turing’s considerations. As in case of generalized diffusion, quantum chaotic events are accompanied by diffusion. We can speak about “quantum smoothened” description, which lead to the loss of singularities typical for deterministic chaos (Altland & Haake, 2012) . Consequently, a question arises how the new chaotic MH influence fashionable philosophical opinions reflecting or supported by deterministic chaos (see also Table 3).

String theory is a unique theory which perhaps unifies four types of known natural forces (i.e. electromagnetic, weak or strong nuclear attraction and gravity) to one concept of association and dissociation of different strings substituting point particles in Euclidian space (Long et al., 2003; Trevors, 2006) . String theory was formulated in sixtieth (Table 3). One time-related and ten space-related dimensions were proposed for our current space when assuming necessary external dimensions (Damour et al., 2002; Maartens, 2004) . In addition, twenty six space-related dimensions were designed for bosons due to additional considerations concerning string orientations (Nojiri, 1987; Clavelli & Jones, 1989; Becker, & Schwarz 2007; Park & Sugimoto, 2020) . In agreement with quantum and string theory, certain minimal possible lengths were restricted as Planck length (1.616 * 10⁻³⁵ m) and string lengths (in the range 10⁻³⁴ - 10⁻³³ m), respectively (Amelino-Camelia, 2001; El Naschie, 2004; Burgess & Quevedo, 2007) . These limits thus constitute interesting weight maximum for photons as separated energy transferring or transforming particles. This maximum corresponds to the spherical drops of water with the diameter 0.16 - 0.64 mm. Deciding majority of the other energy transforming and energetically unified units, i.e. cells, achieves lower sizes than these drops (Kubrycht & Sigler, unpublished data). The sizes of protozoans and different types of somatic eukaryotic cells forming multicellular organisms (TSEC) are even closer to the sizes of the model drops. The considerable part of the sizes related to vertebrate TSEC then falls into the interval limited by the values about ten times lower than corresponds to the drop-related interval. This raises the question whether the described size relationships are somehow significant.

Eight levels of biothermodynamic descriptions (expressed by means of MH), whose complexity increased with decreased physical volumes, were known even in the seventies (Dvořák et al., 1982) . Besides the most complicated thermodynamic description of the processes in the smallest volumes, Liouville equation allowed also the simulation of electron paramagnetic resonance (Balescu, 1975; Dvořák et al., 1982; Misra, 2007) . This equation was even generalized in this century (Tarasov, 2004; Keller et al., 2011) . Moreover, biothermodynamics and reaction kinetics participated in the solution of philosophically important questions, i.e. searches for (i) principles describing departures from thermodynamic equilibrium such as origin of life (Prigogine, 1978; Attard, 2006; Hordijk, 2017) , (ii) general thermodynamic criterion of evolution (Dvořák et al., 1982; Hochberg & Ribo, 2021) and (iii) rational general description of biological self-organization (Killingback & Doebeli, 1998; Karl, 2012; Busseniers et al., 2021) .

10. Conclusion

In spite of certain progress, some unsolved questions and problems appear to be important for further investigation of the commented theme. This concerns (i) better identification of TTM, i.e. clarification of detailed specifications or definitions, correct usage of the terms and knowledge about history and prehistory of TTM, (ii) learning or teaching abilities to form TTM representatives, (iii) reasonable and correct usage of different TTM in experimental science, (iv) importance of dualities for formation of new TTM representatives and (v) better understanding interrelationships between individual TTM (for approximate summarizing scheme see Figure 1).

Due to importance of the introduced five points, several the corresponding comments associated also with this paper have to be introduced. As regards the preceding point (i), unspecific usage of the term TE in nineteenth century was commented and some attempts to specify TE were described in chapter 2. Since historical reconstruction represents important procedure in the corresponding investigation, the deduced abstract consensual (typical and ideal) objects appear to be serious at least as entities defining standard subset of TE denoted as consensual abstract-object-containing TE (coTE). The classifications including coTE and non-coTE can be thus considered as independent dimension additional to the dimensions described in the corresponding section of Table 1. For possible competitive terms denominating some non-coTE as non-TE TE-like entities see section “Specificity” in Table 1. The formation of new TTM representatives or adequate usage of existing TTM (cf. the pre-selected points ii, iii and iv) depends on correct understanding to investigated process and formalization. This comprises not only knowledge about TTM and the corresponding topical systemic relationships, but also record or outline of possible critically re-evaluated relevant abstractions, and sometimes also similarities, patterns or simulations (cf. chapters 2, 3, 4 and 7). To develop, clarify or better present opinions in both experimental and theoretical science, illustrative TE, reverse abstraction and different data representations can be suitable in agreement with chapters 2, 4 or 7 (cf. the points ii and iii). Dualism in description of natural processes (cf. the point iv) represent important stimulus to look for unifying theoretical solutions enabling better understanding to observed events (cf. chapter 7 or Tables 2 and Tables 3). In my opinion, the proposed group of inter-growing dualities (see examples displayed in Table 2) could constitute important form of dualities occurring in biology. This however needs further discussions philosophical and biological aspects of this proposal. In accordance with Figure 1 corresponding to the pre-selected point v, certain TTM can sometimes cooperate or even step-by-step participate in grading searches for solutions.

Though considerable part of the recent theoretical papers comprise usage of computer assisted AM, certain TTM without or before computer applications or even without preformed systemic analysis still exist (chapters 2-4, 8 and 9). These papers include TE as well as some HMT, MH and approximate AM, which indicates continuing possibility of old conventional theoretical work. Recent trends of computer applications comprise predictions based on widely-usable quantum computers (Barrow, 1997) , AI (chapters 7 and 8) and perhaps also AI connected with knowledge-based systems. Consequently, it is a question, whether we wait to see novel computer programs markedly helping with formulation of TTM.

Figure 1. Simplified scheme dealing with the relationships between TTM described here. In addition to separate formation of individual TTM-related representatives, part of these entities participated in formation of the other TTM-related forms during history (cf. Table 3 and chapters 2,3,4,8 or 9). In this scheme, TE are reduced to the subset of TE containing consensual abstract objects (i.e. coTE) mentioned also in chapter 10. If distinct TE complementary to coTE (i.e. non-coTE) indeed exist, their better specification will enable their correct integration to the proposed scheme. Abbreviations: AM – abstract models; FOM – formalized object models (abstract objects used in object-oriented programming); HMT - hybrids of mathematical hypotheses and thought experiments (more frequently using modified ideal objects; cf. chapter 4); MH – mathematical hypotheses; Mp – metaphors; TE – thought experiments; TTM – types of theoretical methods described in this paper. Colors of lines: black – advisory or specifying relationships; blue - generative interrelationships between TTM subsets indicating via arrows which TTM subset can form the subsequent TTM subset (cf. chapters 2-5 and Table 3); orange – components of the pointed types of methods. Forms of lines: dashed – relationships of lower (but still significant) frequency than those represented by full lines; dotted – relationships in question; full – frequent relationships. Proofs: ochre – classical TTM, yellow – more recently proposed TTM.

Since the world of theoretical science is indeed heterogeneous and variously investigated, many theoretically important personalities were not mentioned here. Consequently, I would appreciate, if anybody will substantially complete or correct the above views and opinions.

Acknowledgements

The author thanks his family for a patience and ing Karel Sigler DrSc for the help with processing of English version of the paper.

NOTES

*Types of theoretical methods in natural science.

Conflicts of Interest

The author declares no conflict of interest regarding the publication of this paper.

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