Kant's Pre-Critical Thought Experiments and the Vis Viva Controversy

Fuentes Gonzalez Sergio Alberto ORCID: 0000-0003-4200-2605 Postgraduate student; Graduate School of Philosophy, History and Social Sciences; Immanuel Kant Baltic Federal University (BFU) Researcher; Kantian Academy; I. Kant Baltic Federal University. 236041, Russia, Kaliningrad region, Kaliningrad, Alexander Nevsky St., 14 Gfuentes1@kantiana.ru

The study of the early or precritical period of Immanuel Kant's work may be motivated by historical and scientific interest. This period opens with the essay "Thoughts on the true assessment of living Forces" (hereinafter: "Living Forces") ^[1]1, where Kant analyzes the postulates and proofs put forward by both followers of Descartes and supporters of Leibniz (as well as Newton), seeking to identify the basis of their concepts of force. In addition, in his essay "Universal Natural History and the Theory of Heaven" ^[2], Kant formulates and develops the so-called nebulous hypothesis about the formation of the Universe from an initial nebulous state in accordance with the principles of Newtonian physics. These early works are of particular interest to researchers dealing with the history of the development of physics in the 17th and 18th centuries and cosmological theories – and these are just two possible directions among many. Another important motive for turning to Kant's early work is the desire to trace the origins of philosophical problems that led him to mature critical philosophy, primarily the "Critique of Pure Reason." Kant scholars ^[8-10] agree that such writings as the essay on "living forces" had a significant impact on Kant's later philosophy, especially at the level of methodology. Thus, Dlugach ^[8] points out some difficulties that were unresolved during the subcritical period, which eventually contributed to Kant's transition to the critical phase. Among them is the question of the difference between the method of metaphysics and the method of other sciences. This problem has occupied Kant since the 1760s, although its roots can be found already in the essay on "living forces" ^{[8, p. 39]}.

Descartes considered mechanics as kinematics, where force was reduced to the amount of motion. Leibniz, on the contrary, interpreted force dynamically as an inherent quality of matter, while its quantitative expression receded into the background. Newton, unlike Leibniz, believed that the concept of force does not require any metaphysical justification: the attention of mechanics, in his opinion, should be focused on the effects of force, not on its causes. It is in this respect that Kant departs from Newton's position. Dlugach illustrates this position by noting that "the force turns out to be a kind of "black box" hiding what is happening inside" ^{[8, p. 40]}. In addition, she emphasizes that the limitations of Newtonian mechanics were added to this problem in the phenomenological description: "... the phenomenological description of Newtonian mechanics is not enough to understand the conclusions arising from the principle of inertia, but this is revealed only when the conclusions are brought to such a logical limit, when it becomes clear that they are paradoxical and they should be abandoned ..." ^{[8, p. 43]}. These paradoxes include, for example, the idea of multiple spaces, which contradicts the idea of absolute space in Newtonian physics. It was the presence of such paradoxes that led Kant to question the adequacy of the methodology underlying his own metaphysical research.

Against the background of the limited explanatory possibilities of the Cartesian and Newtonian concepts of force, Kant turns to a more fundamental question — about the nature of space itself. In the essay "On the first basis of the difference of sides in space" (1768) ^[3], Kant formulates a thought experiment that became a turning point in his rethinking of the nature of spatial relations and played a significant role in the formation of his new concept of space as a form of sensuality. "If we imagine that the first thing that was created [in the world] was a human hand, then it was necessary to have either a right or a left hand" (II: 382-383)2 ^{[3, p. 378]} — with this reasoning, Kant tried to substantiate the inconsistency of Leibniz's relational concept of space, according to which space there is only an external relationship between the parts of matter. The key question posed in the imaginary scenario is whether such a hand will be right or left. One hand, Kant explains, is an incongruous likeness of the other, i.e., "A body that is completely equal and similar to the other in everything, although it cannot be enclosed within the same boundaries with it" (II: 382-383) ^{[3, p. 376]}. Since there are no other objects in the imaginary world under consideration, it is impossible to determine whether the hand is pointing to the right or to the left. In other words, "... there is no difference in the relation of the parts of this hand to each other" (II: 382-383) ^{[1, p. 378]}, regardless of its orientation. It follows from this that such a hand could correspond equally to both the right and left wrists of the body — which is a contradiction and, therefore, impossible.

V. V. Vasiliev emphasizes the importance of this argument for the development of the Kantian concept of space.: "... the impossibility of reducing relations in space to the laws of reason meant for Kant the inevitability of abandoning the old understanding of space" ^{[11, p. 8]}. Instead, Kant argues, there is an internal difference, an essential quality (i.e., directionality) that gives space its own reality.3 In the end, "Kant decided to generally accept the theory of absolute space ..., but to consider it not as an independent reality, but as a subjective form of sensuality" [11 p. 9]. Vasiliev also points out that Kant's reflections on incongruent similarities influenced not only his theory of space, but also his understanding of the soul: having understood space as a form of sensuality, Kant comes to the idea that the soul cannot have an extended location and is present in space only virtually ^{[12, p. 239]}.

Kant's scientific and philosophical works abound with such experiments. Moreover, the term "thought experiment" (dat. "Tankeexperiment"; German. "Gedankenexperiment") was introduced in 1811 ^{[13, p. 296]} by the Danish scientist and philosopher Hans Christian Oersted (1777-1851) to describe certain types of arguments revealed by Kant in the Metaphysical Principles of Natural Science (1786) ^[6]. The fact that Kant never used this term and did not distinguish between "real" and "thought" experiments does not mean that his philosophy lacks the elements necessary for at least a preliminary outline of the concept of a thought experiment. In the 21st century, Kant's works have become increasingly relevant in philosophical and historical discussions on this issue ^[14-20]. These works present Kant as an experimenter and/or philosopher of thought experiments. Of course, such statements depend on the specific tasks of historical and rational reconstruction of the many experiments proposed by Kant at various stages of his philosophical development — in such diverse fields of knowledge as natural science, metaphysics, ethics and political philosophy. The present work contributes to this field by focusing primarily on Kant's philosophy of science and epistemology. The main task is to reconstruct some of his arguments as thought experiments with their subsequent careful analysis within the framework of Kantian epistemology. The first section examines the strategies undertaken by critical literature to reconstruct Kant's approach to thought experiments, as well as offers methodological observations on the importance of his subcritical work for the development of the relevant approach. Section 2 is devoted to Kant's assessment of the vis viva dispute in Thoughts on the True Assessment of Living Forces; in particular, it asks the question of the methodological role of ME in Kant's intermediate solution to the contradiction between the quantitative Cartesian concept of force and its Leibnizian qualitative counterpart. The last section analyzes four hitherto ignored thought-experimental arguments contained in "Living Forces" in order to deepen the understanding of the role that subcritical thought experiments played in Kant's philosophical development.

1. Methodological aspects of the Kantian approach to thought experiments

The general strategy adopted in modern literature aimed at developing Kant's philosophy of thought experiments is to show how Kant's epistemology answers such questions as, for example, the so-called informativeness problem: "Thought experiments give us new information. But where does such information come from?" ^{[21, p. 89]}. This question became one of the central ones in the discussions between Norton and Brown, which, following Mouet et al., can be characterized as the tension between the "platonic rationalism" of James R. Brown and the "classical empiricism" of John Norton ^{[22, p. 69]}4. The emergence of Kantian approaches as an attempt to synthesize rationalistic and empirical ideas seems quite natural in this context. More precisely, we are talking about neo-Kantian interpretations, which, in particular, offer alternative readings of the concept of a priori. Marco Buzzoni's transcendental-operational approach, for example, adopted a functional, non-hypostatic reading of Kant's a priori ^{[16, p. 97]}. I. Fehige ^[17], meanwhile, moves further away from Kant and chooses a relativized a priori in the spirit of Michael Friedman ^[24]. As for the epistemological status of knowledge obtained through thought experiments, V. P. Filatov ^[25] also rejects the possibility of a material a priori in the Kantian sense. Analyzing Galileo's famous thought experiment, he emphasizes that such experiments alone are not capable of providing strictly a priori synthetic knowledge about nature. Instead, their main role is to clarify and refine the conceptual tools and schemes used in scientific cognition, as well as to verify the stability of our ideas when conditions and prerequisites change mentally ^{[25, p. 25]}.

Now, among the attempts at a rational reconstruction of Kant's philosophy of thought experiments, Virvidakis' work ^[19] is perhaps the only one that most closely follows both the spirit and the letter of Kant's writings. Among the examples considered are Kant's objections to the principle of the identity of indistinguishable Leibniz in the Appendix to the "Analysis of Principles", as well as criticism directed against the hyperbolic doubts that Descartes defended with the help of thought experiments, such as the so-called dream argument and the malevolent genius hypothesis. An assessment of these and other cases leads Virvidakis to conclude that Kant's subcritical assessments of thought experiments proposed by his contemporaries demonstrate a recurring "... concern about a proper account of the possibility of experience." ^{[19, p. 137]} Perhaps his reading goes too far in stating that "Kant was concerned about the idea of a thought experiment in philosophy" ^{[19, p. 127]}, since it presupposes a broader concept of "experiment" that is absent in Kant's philosophy of science. Nevertheless, this reading allows us to draw attention to Kant's objections to philosophical statements that his subcritical opponents defended by referring to imaginary situations. The limitations imposed by Kant on the use of thought experiments by his subcritical interlocutors represent an early formulation of a critical position characteristic of his subsequent transcendental enterprise.

This article follows the approach of Virvidakis ^[19], except that it focuses on Kant's precritical works on natural science. The role of thought experiments at this stage remains largely outside the field of view of critical literature; meanwhile, a number of interesting examples can be found already in Kant's "Living Forces". Perhaps the reason for this neglect is that Kant's intervention in the vis viva controversy has historically often been viewed as unsuccessful and methodologically flawed. However, the purpose of this article is not to review the meaning of Kant's participation in the discussion, but only to identify some aspects of this controversy that are necessary for contextualizing the analyzed thought experiments.

2. The role of thought experiments in Kant's proposed solution to the dispute about living forces

The "vis viva dispute" is an anachronistic name for the confrontation between the proponents of the Cartesian concept of force as momentum (the product of the velocity of a body and its mass [mv]) and the Leibnizian concept of force as kinetic energy (the product of the mass of an object and the square of its velocity [mv 2]). This designation is anachronistic, because, firstly, before the concept of "mass" (Latin "massa") was introduced and consolidated in the works of Newton, the term "volume" (Latin "moles") was used— the amount of solid matter pressed down by a certain ethereal substance, to which it was supposed to be proportional (that is, in the ratio of weight and gravity). Secondly, instead of symbols such as "m" or "v", geometric constructions were used in the 17th and 18th centuries to express mechanical concepts ^{[26, p. 31]}.

For Descartes, force was an untrue entity; after the inert mechanical interaction of bodies, the only possible definition of force, vis mortua, would be the amount of body movement expressed by the formula mv. Forces, according to Descartes, cannot have a "life of their own", since the only dynamic cause is God. And since God is unchangeable, the initial amount of movement invested in the world must be preserved when transferred from one body to another. These ontological obligations, combined with the concept of force as a quantity of motion, led Descartes' formula to serious inconsistencies when compared with empirically supported laws, especially Galilean mechanics. Leibniz, for his part, argued in a number of works that “the most famous Cartesian proposal” leads to the absurd conclusion that the effect of an acting force can be more powerful than its cause.5 For Leibnizians, despite the fact that force is expressed quantitatively, it is primarily a quality of matter. Therefore, if something must remain constant in the mechanical interaction of bodies, it is the amount of force.

As Schoenfeld explains, the debate about vis viva was largely the result of terminological confusion between two different approaches to the concept of force, which, in turn, relied on different ontological grounds.:

"Cartesian force can be reduced to a number, and what appears as dynamis is actually a kinematic quantity of motion. In the Cartesian painting, movement was more real than force.

In Leibniz's picture, movement and force change their ontological places. In Leibniz's philosophy, force becomes a genuine essence, and movement becomes only a relation between phenomena" ^{[10, p. 24]}.

In other words, the Cartesian quantitative approach asserts that force is reducible to a number, while Leibniz's qualitative approach is a philosophical attempt to clarify the nature of force as an ontological quality, the mathematization of which is only one of the possible ways of expression. Accordingly, Kant's interim solution in the vis viva dispute was to try to demonstrate that both sides are right to a certain extent and, in fact, complement each other. Eric Watkins explains this by emphasizing that "...Kant uses the distinction between natural reality and mathematical form to reconcile Leibniz's "natural" dynamics and Descartes' "mathematical" kinematics" ^{[7, p. 688]}. The structure of the Living Forces vividly illustrates this position: in the first two chapters, Kant apologetically expounds, respectively, the natural scientific aspect of the Leibnizian approach and the mathematical aspect of the Cartesian tradition, while in the third chapter he attempts to conceptually reconcile them.

It is worth dwelling on Kant's attempt to find a compromise between Descartes' mathematical kinematics and Leibniz's natural dynamics. How is it possible to evaluate quantitative demonstrations and evidence based on qualitative arguments so that both approaches are commensurate and complementary? As will be shown, it is enough to look at some of the main arguments of the debate to notice that they are thought experiments. By analogy with the third paralogism, where Kant builds a mental counterexample against Descartes' hyperbolic doubt, in "Living Forces" he, figuratively speaking, "knocks out a wedge with a wedge", seeking to refute both Leibnizians and Cartesians, using modified versions of their own thought experiments.Some of these will be discussed in the next section: the elastic collision of two bodies tending eternally and omnidirectionally to move; the thought experiment of Jacob Hermann (1678-1733) with an infinite spring on falling bodies; Kant's counter thought experiment with five springs; as well as Kant's evaluation of the thought experiment of James Jurin (1684-1750) with a ball on a raft, developed against Descartes' concept of dead power.

Before proceeding to consider our examples, it is necessary to make two observations: first, although Kant evaluates the real experiments 6 (RE) that influenced the debate about vis viva, most of his arguments are based not on observations, but on hypothetical reasoning and counterfactual assumptions. Secondly, it should be emphasized that Cartesian mathematical kinematics, both in Descartes and Kant, is developed in the context of the vis viva dispute not through symbolic "notation", that is, syntactic operations with formulas, but through geometric constructions. A number of questions arise: if we are not talking about real experiments, then in what sense are Kantian proofs experimental? What gives us the right to call them thought experiments rather than, say, geometric proofs? As will be shown in the examples that will be analyzed later, Kant himself thought and called his proofs "experiments" and constructs them accordingly. In addition, his reasoning demonstrates the key features of a thought experiment: conducting logical operations based on empirical background knowledge, the non-necessity of empirical implementation, and the assumption of counterfactual and idealized conditions.

3. Kant's Proofs in the Dispute about Living Forces as Thought Experiments

For further analysis, let's turn to the thought experiments presented in Living Forces. In this work, Kant begins by asserting one of the postulates of Leibnizian dynamics, namely that "... the body has an inherent essential force that belongs to it even before extension" (I:17) ^{[1, p. 22]}. He then shows that force, as a quality inherent in every substance, manifests itself externally in motion, thereby being expressed quantitatively, although there are other forms of its manifestation (§3). At this stage, the apology of the Leibnizian approach requires an explanation of such manifestations as, for example, the representations that matter evokes in the soul (§6), or the three-dimensionality of space (§10). Kant, however, adopts a dual strategy: on the one hand, he seeks to show the explanatory potential of the Leibnizian concept vis viva, on the other — to reveal the limitations of the Cartesian approach, resorting to thought experiments. In the latter case, Kant refutes the concept of Georg Erhard Hamberger (1697-1755), who defines force as an inner, eternal, omnidirectional striving (Bestrebung) for action (vis insita) (§12): "Modern philosophy has certain ideas about the essential forces of bodies, but these ideas are unacceptable. One [Hamberger] calls this force the eternal striving for movement" (I:25) ^{[1, p. 29]}. To better understand how, according to Hamberger, vis insita should have worked, one can imagine a body — say, a sphere — such that each point on its surface is constantly subjected to the same pressure, that is, it tends to move evenly in all directions. In this case, the rest will be the result of the equilibrium of pressure exerted on all pairs of opposite points (monads) on the surface of the sphere. Since the urge to act can be transferred to other spheres, a blow at one point can lead to a decrease in pressure at the opposite point, which in turn causes movement in the direction of a larger trend. Kant illustrates this point by referring to how motion is transmitted from one bowl to another in a pair of classical scales — with two suspended bowls evenly spaced on either side of the central axis.

Against this conceptual background, Kant's destructive thought experiment unfolds (§13). Let's assume that we accept Humberger's thesis: power is an inner, eternal, omnidirectional urge to act. According to Kant, this idea "satisfies the imagination in the case when a moving body continues to move simultaneously with the body it sets in motion" (I:26) ^{[1, p. 29]}. This idea corresponds to intuitive expectations: the force acts continuously and generates a smooth, continuous movement. The next step is to construct a counterexample that does not satisfy the imagination. Kant suggests considering a case in which a body, having transferred motion to another, stops completely. Based on the experience of the interaction of the bowls in a pair of classical scales, Hamberger's hypothesis leads to the conclusion — contrary to experience — that the second body should also stop moving. If force is equal to action, and action is equal to movement, then as soon as the first body stops acting, the second body must also stop. This conclusion clearly contradicts both empirical observations and the principle of conservation of motion. Therefore, Kant argues, Hamberger's concept of vis insita not only does not explain the phenomenon of motion transfer, but also makes it impossible — even at the level of imagination — to imagine a case of elastic collision of two bodies of equal mass, in which one is in motion and the other initially at rest.

Kant's second objection is aimed at showing with the help of a thought experiment: Humberger's theory allows the construction of such collision cases in which a moving body continues to move simultaneously with a body set in motion, but in such a way that it "does not satisfy the imagination" (I:27; §14) ^{[1, pp. 29-30]}. According to the diagram given by Kant himself (Fig. 1), imagine two bodies A and B of the same mass, each of which has a tendency to move (F: vis insita), such that F A = 3F B, or, equivalently, F B = 1/3 F A. After a collision, A loses 1/3 of its velocity, and B gains 1/3. Thus, A is moving at a speed of 2 units, and B is moving at a speed of 1 unit.

Figure 1 (I: 28) ^{[1, p. 30]}

According to Hamberger's concept, "the striving of the body's power in all directions is inseparable from the very substance of the body" (I:27) ^{[1, p. 29]}. This means that body B is moving because there is a pressure on it at one point that exceeds the "pressure" at the opposite point. Thus, the bodies turn out to be "connected" to each other, like scales: if A is "moving forward", then B is moving; if A stops, then B. stops. This, however, implies a metaphysical absurdity: body A would have to penetrate body B, so that A would reach point C, while B would remain at point D.

The thought experiment (§14) is followed by a double classification of force (§§15-18): vis mortua (dead force) is passive pressure, as in the case when a body is pulled; vis viva (living force) is a force in which movement is maintained by itself, as in the case of a projectile in flight. The first chapter of Living Forces concludes with Kant's recognition of the limitations of purely metaphysical reasoning and the need to turn to mathematics to resolve the dispute (§19): "I cannot expect to achieve anything decisive and indisputable in reasoning that is only metaphysical, and for this reason I move on to the next chapter, which, perhaps, thanks to the application of mathematics, will have more grounds to claim to be convincing" (I: 30-31)] [1 pp. 32-33].

In the first two paragraphs of the second chapter (paragraphs 21-22), Kant formulates his methodological position: the desire to find a middle position between the "strong and reasonable" arguments of both Cartesians and Leibnizians. This is followed by a broader critique of Leibniz's views. In general terms, Kant argues that vis viva — the product of the amount of matter per the square of velocity — is applicable to cases of free, self-sustaining motion, but unsuitable for describing dead pressure, let alone a complete transition from vis mortua to vis viva. As a result of this conceptual flaw (§28), Kant concludes that "mathematics can never serve as a proof in favor of living forces" ^{[1, p. 40]}. Paragraph 31 begins with a more detailed analysis of the specific arguments of the Leibnizian side. This section also sets the thematic direction for the rest of the chapter. At first, the criticism focuses on the example of falling bodies, where the key question is whether the effect produced by the weight of a body in free fall is commensurate with time or with the distance traveled.

In the Leibniz camp, Jacob Hermann tried to prove that forces are determined not by space, but by time. Kant describes the proof, which is a thought experiment, as follows (Fig. 2): "Let the infinite spring AB represent the weight following the body as it falls from A to B" (I: 44) ^{[13, p. 43]}. An infinite spring is introduced to represent the pressure at each point of the trajectory traversed by the body. Thus, each rectangle represents the pressure exerted at a specific point, so that the force at each point is defined as the sum of all the "accumulated" pressures; for example, the force at point D corresponds to rectangle ACFB.

Figure 2 (I: 44) ^{[1, p. 43]}

Consequently, according to Kant, Hermann comes to the conclusion that loads act on bodies in free fall depending on distance, not time. Further, Kant's strategy (§32) (I:45) ^{[1, p. 44]} is to change the variables "time" and "height" in an experimental situation similar to Hermann's experiment, so that the force can be described not in terms of distance and weight of springs, but as a function of time, during which they remain compressed or stretched. Kant's mental counterexperiment (Fig. 4)8 includes five identical springs A, B, C, D, E and the body M. The experimental scheme aims to demonstrate that "the body of M...It exerts the same pressure to keep one spring A in a compressed state that would be required to compress five such springs sequentially over the same period of time" (I: 45) ^{[1, p. 44]}.

Figure 3 (I: 44) ^{[1, p.43]}

The reasoning develops as follows: let the time t during which the spring A is in a compressed state be divided into five equal intervals, so that t1 = t2 = t3 = t4 = t5 = t/5. Now suppose that the body M exerts pressure on spring A at t1, but is then replaced by spring B at t2. Since all five springs are identical, that is, they have the same tension, it makes no difference whether spring A or B continues to exert pressure in t ₂, and the same applies to B and C in t 3, C and D in t 4, and so on by analogy. To draw a correct analogy with Hermann's experiment with an infinite spring, Kant emphasizes that the result persists "even if the number of springs is increased to infinity" (I:44) ^{[1, p. 44]}. In conclusion, Kant argues that the pressure exerted on spring A in the closed state during time t is equivalent to that required to successively compress five springs over the same period of time. Thus, the total effect of the body M should be estimated not by the space traveled, but by the duration of the effect of gravity on the springs.

One could argue that the high degree of abstraction in representing the relationships between variables makes the proof not so much experimental as mathematically kinematic. In this context, it is useful to compare the above cases with the famous thought experiment by which Galileo refuted the Aristotelian concept of free fall. 9 Just as Galileo's bodies are assumed to fall in a vacuum, Kant's springs are idealized.: they are exactly the same, devoid of friction and mass. However, with all the idealization, springs remain not just lines or abstract directions in space; in other words, there really are objects in nature that behave like springs, and it is this empirical acquaintance with them that forms the background experience that makes Kant's proof understandable or incomprehensible, convincing or controversial. At least four conditions are fulfilled that allow Kant's proof to be qualified as a thought experiment: i) the presence of empirical operational knowledge about the behavior of springs; ii) the non-necessity of actually conducting an experiment to convince the experiment; iii) the use of the method of variations; iv) the assumption of counterfactual and idealized experimental conditions, for example, an infinite number of exactly identical springs. Moreover, and this is no less important, Kant explicitly calls his proof "Versuch" and declares that it (§32) is the first of a series of experiments presented in the second chapter of The Living Forces: "This, therefore, is the first [Versuch] experiment, which, as I believe, confirms what I said above, namely: Descartes' opinion surpasses the law of Herr von Leibniz in mathematical proofs" (I:45) ^{[1, p. 44]}.

The essay on living forces is full of experiments of this kind. A vivid example of a thought experiment with springs is presented in section 36, where Kant describes "pressures caused by gravity and transmitted to the body... an infinite number of metal springs..." (I: 47-48) ^{[1, pp. 46-47]}. It is noteworthy that he indicates the material of the springs — metal — although this, in fact, does not matter for the output. Another interesting case is contained in section 110: a raft is imagined on which there is a ball set in motion by a spring. This scenario was proposed by James Jurin in order to show that doubling the velocity of a projectile entails a doubling of force (mv2 according to Descartes), but not a fourfold increase (mv2 according to Leibniz).:

Figure 4 (I: 123) ^{[13, p. 108]}

"He suggested a movable raft, for example, a barge AB, which moves with one unit of speed in the direction of BC and simultaneously carries the ball E with the same movement. Due to the movement of the raft, the ball, respectively, has one unit of speed and one unit of force. Next, he suggested the presence of a spring R on this raft, which is triggered by the stop D and gives the imaginary ball E another unit of speed, which means another unit of force. As a result, ball E has two units of velocity and, therefore, two units of force. Thus, a doubling of speed entails only a doubling of force, and not a quadrupling of it, as the Leibnizians mistakenly believed" (I:123) ^{[1, p. 108]}.

Although the author of this thought experiment was not Kant, it is worth noting that he highly appreciated Jurin's work, noting it as "in a simple way... with crystal clear clarity" (I: 122) ^{[1, p. 107]}. Consideration and critical evaluation of examples like Jurin's experiment can be considered as an indirect way to formulate Kant's understanding of thought experiments. Already at the beginning of The Living Forces, Kant prefers the Cartesian position for its appeal "only to simple cases in which the definition of truth and error is easy and obvious," whereas the Leibniz camp, on the contrary, "makes its proofs as confusing and unclear as possible" (I:15) ^{[1, p. 20]}. What is this clarity and distinctness when it comes to evaluating thought experiments? In the case of Hermann's infinite spring and Kant's counterexample with five springs, it would be possible, hypothetically, to replace the springs with uniformly distributed molecules or ethereal matter between the points of the body's trajectory in free fall. However, the "world" in which the experiment takes place must be comparable to ours in its essential characteristics; otherwise it loses its explanatory power and becomes uninformative.

Conclusion

The functional a priori interpretation proposed by Marco Buzzoni suggests a connection between real experiments and thought experiments, similar to the one that Kant establishes between contemplation and concepts.: To paraphrase Kant: (empirical) ME [mental experiments] without RE [real experiments] are empty; RE without ME are blind... Their connection should be considered as a connection between elements that, although not reducible to each other, nevertheless cannot be separated without losing any definite meaning" ^{[16, p. 100]}. As Buzzoni emphasizes, from a neo-Kantian point of view, "... the specific content of any empirical ME should, in principle, ultimately be reduced to sensations (or, more precisely, to empirical and operational interventions in reality, that is, to experiment)" ^{[16, p. 100]}. Kant's use of imagination in his proofs often suggests hypothetical situations, including, for example, a marginal change in magnitudes or an infinite number of ideal springs. Without this empirical-operational connection with the world, the experimenter runs the risk of mistaking the empirical use of reason for the transcendental. Virvidakis ^{[19, p. 23]} reminds us that Kant raises a similar objection to Leibniz, pointing out the inapplicability of the principle of the identity of the indistinguishable outside of sensory intuition. As shown, the flexibility of mental experimentation allows Kant to evaluate both the natural dynamics of Leibniz and the mathematical kinematics of Descartes from a single methodological position. Ultimately, his accusations of methodological ambiguity against Cartesians conceal a broader requirement for thought experiments: they must be both logically consistent and operationally representable.

Comments on:

1. Quotations and references to Kant's texts are given in the Russian edition in six volumes ^[2-6], with the exception of "Thoughts on the True Assessment of Living Forces", which is quoted in the Cambridge edition ^[1]. Accordingly, all excerpts from this essay have been translated by the author from English. The reason for this choice is that the Russian edition in six volumes lacks most of the discussed sections of the essay on living forces (from Chapter 1, §13 to Chapter 2, §87). On the contrary, the Natural Science volume ^[7] in the Cambridge edition contains a complete translation of the work, provided with numerous factual and linguistic notes necessary for an accurate understanding of the arguments presented by Kant.

2. The additions in parentheses indicate the volume and page of the Academic Edition of Kant's Works (Gesammelte Schriften, Kant, 1900–). For example, the designation "(IX:18)" means: Logik Jäsche, Academic Edition, volume IX, page 18.

3. It should be emphasized that this thought experiment fits into the logic of reasoning that would eventually lead Kant to a transcendental concept of space.: "absolute space is not an object of external perception, but is one of the basic concepts that alone make all such objects possible" (II:383) ^{[3, p. 378]}. The existence of incongruent similarities, an example of which is the right-left hand paradox, was mentioned by Kant several years later in his Dissertation (1770) in order to support the claim that space should be known through contemplation (II: 403; §15) ^{[4, pp. 403-404]}. Later, in Prolegomena to Any Future Metaphysics (1783), Kant uses the same paradox to show that space (hence time) is a form of external intuition (IV:285-286; §13) ^{[5, pp.101-102]}.

4. An overview of the debate between Norton and Brown can be found in Hitchcock ^{[23, cc. 21-66]} (chapters 1 and 2).

5. A concise summary of Leibniz's demonstration can be found in Schönfeld (2000) [10. p. 26]; see also Leibniz ^{[27, p. 391]}.

6. Indeed, there is an experiment with a shot, which, according to Kant, he conducted himself, and the description of which is used to illustrate that "a freely and uniformly moving body does not yet have maximum force at the beginning of its movement, but rather, this force increases when the body is in motion for some time. movement" (I: 152; §130) ^{[1, p. 132]}. As Watkins notes: "With the exception of a few suggestions about experiments ..., the test with a shot is quite possibly the only experiment that Kant ever conducted" ^{[7, p. 688]}; see also ^{[28, p. 76]}.

7. Brown proposes a classification of thought experiments, distinguishing between constructive ones aimed at substantiating or formulating theoretical ideas, and destructive ones, the purpose of which is to undermine theory by revealing contradictions or leading to absurd consequences ^{[29, cc. 32-35]}

8. Drawings serve as markers of thought experiments, which is especially important when reading Kant's scientific texts, given that in none of his writings does he refer to his arguments with the term Gedankenexperimente; even his commentators often refer to them with other words such as "case" or "example." Obviously, the most common markers of thought experiments are verbs expressing modality (to think, to believe, to assume, to allow, to imagine, to imagine, to think), as well as the corresponding modal adverbs (perhaps, presumably, conceivably) and modal adjectives (possible, imaginable, conceivable). For an insightful analysis of the interaction of images and thought experiments in physics, see Meynell ^[30].

9. Aristotle argued that heavy bodies fall faster than light ones, so if an object with a certain weight covers a distance in a certain time, then another object with twice as much weight will cover twice as much distance in the same time. Galileo contrasted the principle of Aristotle with the following reasoning: suppose that Aristotle's law is true and that, in addition, two objects A and B fall in a vacuum, and B is heavier than A. Let's also assume that these two bodies are connected to each other. It follows, firstly, that system A + B will fall faster than body B alone (A + B> B). However, secondly, since A is lighter than B, it should slow down B, acting as a "parachute", which means that the A + B system will fall slower than B (A + B < B). Thus, a contradiction arises: simultaneously [A + B > B] and [A + B < B].