Introduction

In the nineteenth century, critical comments against the eighteenth century Kantian turn to Anschauung in mathematical science emerged.Footnote 1 Immanuel Kant’s concept of intuition (Anschauung) guaranteed mathematics’ ontological relevance when referring to the world. This was particularly the case as the basis for the experimental science ex datis of Galileo Galilei and Isaac Newton. Therefore, Kant interwove mathematics with the spatiotemporal ability of men to recognize objects.Footnote 2 However, such a concept of mathematics, as the captious critics would say, was naively based on the visuality of Euclidean geometry and its constructive nature of creating mathematical objects with tools such as compasses or inspired by counting with fingers, as Gottlob Frege sneered against Kant.Footnote 3 Most prominently Carl Friedrich Gauss argued in a letter to Friedrich Wilhelm Bessel on January 27th 1829 that the loss of Anschauung in his non-Euclidean geometry would cause hue and cry of the “Boeoter,” the Kantians of his time.Footnote 4 Gauss, afraid of being bashed by his philosopher colleagues for rethinking the “fifth postulate” of Euclidean geometry (parallel postulate), withheld the publication of his new geometry on which he had been working since the 1790s.Footnote 5 His fear was that the reputation of rethinking the fifth postulate might fall into common rank with the work on the quadrature of the circle and the perpetual motion machine, respectively.

This episode illustrates how contested the Kantian turn and how fierce the dispute on Anschauung was during the nineteenth century. It also reveals the struggle between Platonists and early constructivists in the Kantian notion later leading to the Grundlagenkrise of mathematics in the early twentieth century. A struggle that was transferred into physics in a different manner by Werner Heisenberg and Erwin Schrödinger fighting over Anschauung in early quantum theory.Footnote 6 Finally, the dispute on Anschauung and its loss, respectively, caused mathematician and phenomenologist Edmund Husserl to state that European science was in crisis in a lecture in 1935 on the eve of the collapse of European civilization.Footnote 7

This spectrum of debates on Anschauung opens up the epistemological ground for asking about the power of mathematics to envision worlds beyond human Anschauung. A power that can lead to victory as well as crisis, but generally speaking to new physical sciences. However, based on the loss of Anschauung in geometry, models came into play in order to replace Anschauung (intuition) by Anschaulichkeit (palpable visuality).Footnote 8

Immanuel Kant’s Philosophy of Applied Mathematics

The concept of Anschauung in Kant’s Critique of Pure Reason (Kritik der reinen Vernunft) is neither related to visuality nor to images.Footnote 9 Anschauung is the foundational ability of human beings to have sensory impressions. Sensory impressions, and this is the Kantian turn in epistemology, are not received passively but result actively from the ability of Anschauung to process sensory data and thus enable human beings to recognize objects. However, and this is the decisive point, the procedure of processing sensory data is not arbitrary but ordered: either simultaneously (space) or successively (time).Footnote 10 This formal procedure of ordering is called ‘synthesis’ in Kant’s work.

Anschauung, however, is only able to process sensory data, not to interpret them conceptually. The latter is the task of the mind—a separate ability in Kant’s concept. It is this separation of Anschauung and mind that provides the basis for criticizing pure reason for Kant. Pure reason is the empty—empty of sensory data—ability of producing ideas, sometimes illusive, and abstract concepts in the mind, as Kant argued. Only together, mind and Anschauung produce our experience of the world. It is Anschauung that gives ideas and concepts content, although it is arrived at only quantitatively and gradually that there is something (sensory data). Anschauung, in other words, indicates the existence of things but does not denote their meaning.

Although Kant introduced these two separate abilities of Anschauung and mind, he interconnected both by the procedure of transcendental schematism.Footnote 11 Transcendental schematism realizes the operation of interconnecting Anschauung and mind in time by synthesis; thus, creating synthetic judgments a priori. The various modes of synthesis define the categories of the mind like quantity, quality, relation, and modality, ordering and integrating sensory impressions in terms of various modes of temporality.Footnote 12 Thus, the term ‘schema’ does not refer to a visual schema or image. It is not true, as Philip Kitcher later accused Kant, that Kant had mental images or “cartoons” in mind in order to apprehend synthetic judgments a priori.Footnote 13 But what Kant had in mind were rule-based procedures for ordering sensory data simultaneously (space) and successively (time) and subordinating them to the temporal modes of the category. It was Edmund Husserl who advanced Kant’s theory of temporal modes.Footnote 14

However, this ordering of sensory data was conceived homogenously by Kant.Footnote 15 Homogenous and uniform ordering of time and space is a characteristic of Newtonian mechanics, which keeps the latter in the realm of uniform motion and linearity as well as in the realm of Euclidean geometry and its regular forms: lines, circles, spheres, cubes, triangles, etc. In the Critique of Pure Reason, Kant never argued, in any way, that spatial data must be three-dimensional. Although neither favoring three-dimensionality manifested in the constructible objects of Euclidean geometry nor limiting space to three dimensions, Kant’s homogenous form of synthesis enabled him to succeed in providing a concise epistemological ground for the positivistic science of his time. In particular, he was able to provide a concise epistemological ground for Newtonian physics.Footnote 16

By rooting mathematics in his formal concept of Anschauung—mathematical judgments are synthetic judgments a priori for Kant—he was overcoming the dichotomy of analytic (logical) and synthetic judgments (empirical ones). Furthermore, this root in Anschauung ensured the ontological relevance of mathematics when referring to the world resulting from the indexical ability of synthetic judgments a priori. In retrospect, Kant’s Critique of Pure Reason can be called the first textbook for the philosophy of applied mathematics.

Loosing the root in Anschauung exiled mathematics into the realm of pure reason and of solely analytic judgments; it cut the ties to the world and encapsulated mathematics in a purely formal endeavor of dealing with symbols—only obeying the criteria of being not contradictory. However, self-consistency is the logical precondition for the existence of mathematical objects but does not imply the existence beyond mathematics. Without Anschauung mathematical objects were not linked to the world anymore. The emergence of pure mathematics during the nineteenth century was the result of the loss of the Kantian link (synthetic judgments a priori) between synthetic (empirical) and analytic (logical) judgments, or in other words: of the loss of Anschauung.

The Loss of Anschauung in the Nineteenth Century and the Declaration of Anschaulichkeit as a Model in Geometry

The loss of Anschaulichkeit in nineteenth century geometry is not necessarily linked to a loss of Anschauung in the formal sense presented by Kant. Anschaulichkeit and Anschauung are usually mixed up in the mathematical literature of the nineteenth century by wrongly assigning visuality to Kant’s concept of Anschauung—as it is the case for Anschaulichkeit.Footnote 17 However, the loss of Anschaulichkeit in nineteenth century geometry resulted from an increasing process of abstracting and systemizing geometry which led to the development of mathematical group theory. By questioning Euclid’s ‘fifth postulate’ (parallel postulate) new objects were introduced into geometry. Already in 1766, Johann Heinrich Lambert contested that the sum of the angles of a triangle has to be always 180° as in Euclidean geometry.Footnote 18 By introducing triangles with less than 180° (imaginary geometry) and more than 180° (spherical geometry) Lambert invented non-Euclidean geometry. Mathematicians like Adrien-Marie Legendre, Jean-Baptiste D’Alembert, Gauss and others followed. It was Gauss in 1829 who coined the term “Nicht-Euclidische Geometrie” in his letter to Bessel,Footnote 19 although Nikolai Lobatschewski und János Bolyai had already axiomatized these early versions of non-Euclidean (hyperbolic) geometry in the 1820s.Footnote 20

The loss of Anschaulichkeit in nineteenth century mathematics resulted from various developments; questioning Euclid’s parallel postulate was only one aspect.Footnote 21 Questioning three-dimensionality as well as geometrical properties of figures in the emerging projective geometry added an even more abstract view to geometry. Furthermore, replacing the synthetic-constructive method of geometry by the analytic-algebraic method—a paradigmatic change in mathematical media—opened up the door for describing unanschauliche geometries. ‘Unanschaulich’ thereby refers to geometries which are not constructible with compasses anymore.Footnote 22 It was René Descartes in 1637 who developed analytic geometry replacing geometric constructions by algebraic descriptions.Footnote 23 However, Descartes rejected terms like a4 and thus stuck to the translatability of Euclidean geometry into algebra and vice versa—a limitation which was rearticulated as the ‘dictum of geometric construction’ by Jean-Victor Poncelet in the beginning of the nineteenth century;Footnote 24 a dictum which dominates descriptive geometry for engineers and architects until today but was given up by projective geometers in the mid of the nineteenth century.

The development of projective geometry into abstract group theory has been retrospectively described by Felix Klein,Footnote 25 who successfully used group theory to reunite the scattered field of geometry.Footnote 26 It has been reconstructed by the historian Hans Wussing.Footnote 27 From a group theoretic perspective, geometric figures become manifolds and ‘constructing’ turns into operating algebraically with symbol transformations on these manifolds. In group theory, geometric research is focused on attributes which remain invariant under transformations of space (e.g. geometric transformations such as translation, mirroring, rotation). Groups were defined “by means of the laws of combination of its symbols […] in dealing purely with the theory of groups, no more concrete mode of representation should be used than is absolutely necessary.”Footnote 28 Euclidean geometry, in this respect, became one of many groups, called the ‘main group.’ The main group describes the geometric attributes which remain invariant: dimension, orthogonality, parallelism, and indices. However, for the most general group, the projective group, only indices remain invariant under transformation.

The price for reuniting the scattered field of geometry by introducing abstract group theory was not only the irreversible loss of Anschaulichkeit but also of Anschauung for geometry. The translatability of geometry into algebra and vice versa had to be given up. Thus, as Klein argued, what was left over was Anschaulichkeit as a “model” for non-Euclidean geometry; for instance, the model of a constructible, plane model of elliptic non-Euclidean geometry.Footnote 29 Therefore, Klein differentiated two forms of geometry: an abstract form of geometry (projective group) for which Anschaulichkeit demonstrates some aspects by using visual models for pedagogical reasonsFootnote 30; and “eigentliche” geometry as the mathematical science of space and thus spatial Anschauung.Footnote 31 The increasing amount of engineering students in the second half of the nineteenth century, as Anja Sattelmacher has outlined in her study on material models in mathematics, led to a collection of models for teaching engineers such as wire models of functions.Footnote 32 In his volume on Anschauliche Geometrie, David Hilbert provided several photographs of material models, for instance of a wire model of a hyperboloid and a wood model of ellipsoids.Footnote 33 However, as Hilbert outlined in his introduction, his Anschauliche Geometrie is more a popular science book than an academic, which would provide “Freude” (joy).Footnote 34 Luis Couturat in 1905 described this development in mathematics as a paradigmatic shift from Anschauung as the old medium of mathematical discovery to the new medium of logic and deduction in formal symbol systems.Footnote 35

Matrices as New Tools for Compensating the Loss of Anschauung in Physics

The new medium of mathematical discovery solely belonged to the realm of analytic judgments, dismissing Kant’s synthetic judgments a priori. However, dismissing Kant’s synthetic judgments a priori released mathematics, and in particular geometry, from its ties to reality.Footnote 36 Thus, for physics, the “Unreasonable Effectiveness of Mathematics in the Natural Sciences” became an unsolvable miracle.Footnote 37 As Eugene Wigner stated in his essay: “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”Footnote 38 However, it was not a miracle but the consequence of exiling mathematics into the realm of pure reason and, thus, of cutting its ties to the world.

What this cut would mean for physics can be learned from another fierce debate about Anschauung; this time for the emerging quantum theory in the 1920s between Werner Heisenberg and Erwin Schrödinger.Footnote 39 Following Albert Einstein’s dictum that every entity of a physical theory had to be observable, i.e. measurable,Footnote 40 Heisenberg pleaded for giving up the traditional idea of trajectories of electrons because such trajectories were not measurable. This un-observability resulted from the elimination of the relation between position and momentum (Heisenberg’s uncertainty principle). Furthermore, instead of a continuous movement from one position to the other electrons jump between energy plateaus (quantization). However, such an uncertain and discontinuous behavior could not be conceptualized by using the old vocabulary of physics: differential equations, because movement was no longer conceivable as a function of time like in traditional physics. Thus, in Heisenberg’s opinion, quantum physics had to develop a new tool in order to conceptualize this strange quantum behavior properly. His achievement was to describe the movement of electrons as the square schema of transitions resulting from the rule of multiplication of quantum mechanical quantities.Footnote 41

Heisenberg’s approach to conceptualize the strange quantum behavior and, thus, to compensate the loss of Anschauung consisted of introducing matrices as new tools for physics. However, it was not Heisenberg but his collaborators Max Born and Pascual Jordan who introduced matrices for conceptualizing the discontinuous behavior of electrons.Footnote 42 They identified Heisenberg’s rule of multiplication of quantum mechanical quantities as the well-known rule of multiplication of matrices shown along the main diagonal of the matrix (Fig. 1).

Fig. 1
figure 1

Graphics: © Gabriele Gramelsberger, all rights reserved

Square matrix.

Full size image

However, matrices have already been invented in the mid of the nineteenth century by Arthur Cayley. In his “A Memoir on the Theory of Matrices,” Cayley defined a matrix as a set of quantities arranged in the form of square arrays.Footnote 43 He realized “that the square arrays themselves actually had algebraic properties.”Footnote 44 In particular, square matrices combine properties of tables with diagrams. As tables they provide a definite order of quantities in lines and columns, as diagrams they have algebraic properties for calculation. Although “Hamilton must be regarded as the originator of the theory of matrices, as he was the first to show that the symbol of a linear transformation might be made the subject-matter of a calculus,”Footnote 45 it was Cayley who had developed the characteristic form of matrices and who had articulated the rules for using matrices as algebraic tools. Related to Euclidean space, square matrices describe the geometric transformations of the main group like translation, rotation, and mirroring.

Based on these tools, Heisenberg, Born, and Jordan were able to articulate the states of electrons organized in lines (p position) and columns (q momentum). Along the main diagonal one can follow the probability of phase transitions of the electrons. Based on this new matrix analysis, “the laws of the new mechanics have been completely articulated. Every other law of quantum mechanics must be derived from these basic laws.”Footnote 46 As outlined above, it was Heisenberg’s aim to introduce only such elements to his matrix analysis which were observable, i.e. measurable. p and q are conjugated operators which cannot be measured at the same time. However, comparing the measurements of both states is possible but only statistically as it is displayed in the diagonal of the matrix. It is this unique experimental circumstance, called the uncertainty principle, which led to the development of new tools as Dirac highlighted in accordance with Heisenberg:

Heisenberg puts forward a new theory, which suggests that it is not the equations of classical mechanics that are in any way at fault, but that the mathematical operations by which physical results are deduced from them require modification.Footnote 47

Early Twentieth Century Debate on Anschauung and Anschaulichkeit in Physics

Opposed to Heisenberg’s matrix mechanics, Schrödinger stated that it would be much more helpful to use the traditional idea of oscillation modes merging into each other and, thus, conceptualizing phase transitions instead of jumping electrons.Footnote 48 Schrödinger’s version of wave mechanics draws on the old vocabulary of physics: differential equations. His reason was the desire for Anschaulichkeit, as Henk W. de Regt has pointed out aptly:

The association of visualisability with understanding rather than with realism may be elucidated by considering the German word Anschaulichkeit, which is the term Schrödinger used in his writings. This word does not only mean ‘visualisability’ but also ‘intelligibility.’Footnote 49

However, the debate between Heisenberg and Schrödinger can be reconstructed as a mismatching debate on Anschauung (Heisenberg) and Anschaulichkeit (Schrödinger). While Heisenberg put the emphasis on the loss of spatiotemporal Anschauung in the Kantian notion, thus asking for new concepts of Anschaulichkeit, Schrödinger ignored the loss of Anschauung in favor of the traditional spatiotemporal concept of (Newtonian) Anschaulichkeit. Of course, Schrödinger was aware that his vote for a familiar style of Anschaulichkeit did not describe quantum reality but was a helpful ‘image’ or ‘model.’Footnote 50 Nevertheless, he rejected Heisenberg’s approach because of its ugly Unanschaulichkeit. Schrödinger introduced his 1935 essay on the current situation of quantum mechanicsFootnote 51 with a section on entitled “The physics of models.” In this section, he argued for the imagination of models which are more accurate than empirical knowledge can ever be. Mathematics and geometry are the languages for articulating such models. However, as Schrödinger pointed out, while geometry is static these models must include temporal behavior. Thus, if enough elements and relations between these elements are known, then these models can be used for predicting future states of temporal behavior. If these predictions can be validated empirically by observation, the underlying hypotheses were valid. If not, it leads to progress of knowledge by adapting the underlying hypotheses and models. “This is the way how slowly a better adaptation of the image, i.e. of our thoughts, to the facts can be achieved.”Footnote 52

This ‘model style of physics’ was new and was later called the ‘hypothetic-deductive research style’ by philosophers of science.Footnote 53 Different than the inductive-deductive research style of Newtonian physics, hypothetic-deductive thinking was firstly introduced by Einstein’s unique theory of relativity in 1905.Footnote 54 As the term ‘hypothetic-deductive’ reveals, scientific thinking starts with a hypothesis that has to be evaluated afterwards. In case of the predictions of Einstein’s general theory of relativity,

three ‘crucial tests’ are usually cited as experimental verifications […]: the red shift of spectral lines emitted by atoms in a region of strong gravitational potential, the deflection of light rays that pass close to the sun, and the precession of the perihelion of the orbit of the planet Mars.Footnote 55

From a philosophical point of view, the transformation from the inductive-deductive into the hypothetic-deductive research style is the pivotal point where theories have to turn into models and were explanations have to turn into predictions. From now on, a ‘good science’ is a model-based predictive one which derives computable models of its theories. In turn, a model is only scientifically valuable if it can compute some predictions which can be evaluated empirically afterwards. However, models usually predict spatiotemporal behavior based on partial differential equations. For instance, Einstein’s general theory of relativity has to be re-articulated as a mathematical model involving ten coupled partial differential equations in order to compute any predictions. However, such a complex model is neither exactly computable nor does it give exact predictions; it can only be numerically simulated, i.e. approximated.Footnote 56

Surreality of the New Physics

The loss of Anschauung also caused major disputes in philosophy at this time. The new physics was seen as a problem by Edmund Husserl but embraced by Gaston Bachelard. Husserl, in his lecture on “The Crisis of European Sciences and Transcendental Phenomenology,” argued that the loss of Anschauung also means the loss of the subject for science and this, in turn, is accompanied by losing reference to everyday lifeworld.Footnote 57 Science, and first of all physics, has turned into an abstract endeavor unrelated to humankind and its concerns. For Husserl, rationality has failed because the link to humans as rational beings has been given up. Science has turned into a remote endeavor ignoring most aspects of rationality such as values and aesthetics by highlighting only instrumental and logical rationality.

Bachelard, however, celebrated the new physics and exactly this highlighting of instrumental and logical rationality in his book Le nouvel esprit scientifique.Footnote 58 Einstein’s and Heisenberg’s new physics is turning the naïve imaginations of everyday experiences—an epistemological obstacle in Bachelard’s opinionFootnote 59—into mathematical activities constructing the world anew and enabling new (observational) experiences. Mathematics’ operational lucidity is replacing the Cartesian model of lucidity. For this paradigmatic shift in scientific thinking Bachelard gives the example of conceiving the scattering of light. In classical physics, scattering was conceived as deflection of particles by a mirror based on Newton’s laws concerning moving bodies and their interactions. In 1871, John Strutt (Lord Rayleigh) developed a first equation of scattering of lightFootnote 60 and this equation, as Bachelard stated, mirrored the naïve imagination of deflection geometrically-anschaulich. However, the new equation of Hendrik Kramers and Werner Heisenberg from 1924Footnote 61 overcame these naïve epistemological obstacles. For Bachelard, the equation of Kramers and Heisenberg marks a paradigmatic shift in scientific thinking, although Heisenberg, Born, and Jordan had developed this equation in 1925 just before the introduction of matrix mechanics.Footnote 62 Nevertheless, it involved the new quantum mechanical concept of interaction between matter and radiation. Although for Strutt the path of a beam of light is deflected differently depending on the wavelengths of the light, the deflection was conceived deterministically; while for the equation of Kramers and Heisenberg radiation was modified by the influence of matter. Thus, not the Anschauung of elastically vibrating electrons but the abstract idea of an inelastic scattering of light with certain frequencies, where the frequencies correspond to possible transitions within the atom, characterized the new approach. In order to express these possible transitions Kramers and Heisenberg used vector analysis as a mathematical notation system for expressing scattering as sums of all contributing absorption and emission amplitudes. For Bachelard, not Anschauung but the use of vector notation was guiding the new scientific thinking as a complex interplay of notational and numerical conventions.Footnote 63 This new scientific thinking provides a more general and, thus, realistic view of the world. Today, the modern equation of scattering of light, considering all possible transitions, is using operator analysis.Footnote 64 Jun John Sakurai’s transition operator T probabilistically describes all possible transitions between the initial and final states of a scattering atom. Reality, in Bachelard’s interpretation, turns into a special case of possibility.

This development from naïve, geometrically-anschaulich inspired imaginations to reality as a special case of possibility is owed to mathematics and its increasing complex notations: from vectors over matrices to tensors. Or, put differently: Reality becomes the model! Reality turns into the hypothetical-deductive program of realizing notational predictions empirically. Bachelard called this new scientific thinking ‘surrational,’ that means it goes beyond the rational view of classical determinism.Footnote 65 Just as the microscope has opened up access to new worlds, the new mathematical notations opened up new worlds because they allowed for new generalizations. Rationality tries to understand given reality, while surrationality tries to understand all possibilities of reality. Of course, the prerequisite of surrationality is the overcoming of Anschauung.

Conclusion

Celebrating new physics means celebrating matrices. Matrices, like vectors and tensors, are powerful epistemic tools for mathematics as well as other sciences.Footnote 66 They can be modified by basic algebraic operations like addition, scalar multiplications, and transposition. Matrices can be used to express vectors and tensors as well as to articulate and work with linear equations which are widely applied in physics and engineering, for instance to approximate the behavior of non-linear systems. Furthermore, matrices are used for carrying out linear transformations in group-theoretic geometry, for representing complex numbers by particular real 2 × 2 matrices, and for providing normal-form representations of games in game theory, for instance the payoff matrix of the prisoner’s dilemma.Footnote 67

Usually, mathematics, science, as well as philosophy do not pay much attention to their semiotic media for expressing thoughts like notational systems. But when these media become the basis for achieving new insights into new worlds—turning induction into hypothetical-deduction—then some thoughts should be invested in the impact of these semiotic media on science. For instance, Mark Steiner has asked in his study The Applicability of Mathematics: “How, then, did scientists arrive at the atomic and subatomic laws of nature?” His answer: “By mathematical analogy.”Footnote 68 If there is no Anschauung only mathematical analogy based on symbol manipulations remains as a tool for scientific imagination, research becoming an investigation of “representational systems [… rather] than nature.”Footnote 69 In Steiner’s reading, Schrödinger also turned away from Anschaulichkeit:

In sum, Schroedinger began with a sine wave of fixed frequency, based on an analogy to an optical wave, where the frequency is given by a fixed energy field. In writing down the ‘wave’ equation by taking derivatives, Schroedinger completely abstracted away from this intuition, ending with an equation having no direct physical meaning; one with superposed solutions; one with solutions having no ‘wavelike’ qualities at all.Footnote 70

When Heisenberg in 1932 postulated the existence of positrons and neutrons by analyzing the local SU(2) rotation symmetry with a \(2\times 2\) matrix,Footnote 71 he employed a mathematical analogy based on a fictitious isospin space. In this fictitious isospin space, the rotation of 180° obtains a neutron from a proton and a full 720° rotation returns a neutron or a proton to its initial isospin state. “It seems clear,” as Steiner stated, “that the mathematics is doing all the work in this analogy.”Footnote 72 Finally, Steiner concluded:

It is the formalism itself (and not what it means) that is the fundamental subject of physics today. […] discoveries made this way relied on symbolic manipulations that border on the magical. I say ‘magical’ because the object of study of physics became more and more the formalism of physics itself, as though the symbols were reality.Footnote 73

However, as long as the hypothetic-deductive research style requires experimental verification, ‘magic’ can turn into reality.Footnote 74 But if this is not possible as in current cosmology of multiverse theories and string theory, respectively, science can either criticize the empirical necessity as an “overzealous Popperism”Footnote 75—signing away its spirit on the miracle of a purely mathematical isomorphismFootnote 76—or, it can pose the question anew about the role of Anschauung for applying mathematics in science.