How to make empirical sense of Newton’s second law?

Tim Ludwig, Philosophy for Physics, 15 June 2025.
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Abstract

Newton’s second law is at the core of classical mechanics. And experienced physicists have no difficulty in applying it. Yet, its logical status remains somewhat unclear. Is it just a definition of force? An actual law of nature? Or a law governing our description of nature? Before we can answer such questions, we first need to know the empirical content of Newton’s second law. To that end, we discuss how to make empirical sense of it. Thereby, we develop a deeper understanding of the central concepts of force and mass and of Newtonian mechanics as a whole. This deeper understanding, ultimately, will help us to also improve our understanding of other physical theories, as we can compare and contrast them to classical mechanics.

Introduction

Classical mechanics is an essential theory of physics and, arguably, Newton’s second law is at its core. Despite its importance, however, the logical status of Newton’s second law remains somewhat unclear [1]. Is it just a definition of force, as Mach, Kirchhoff, and Boltzmann used to believe [2]? Is it an actual law of nature, whatever that might mean [3]? Or is it a law governing our description of nature by structuring the research program of classical mechanics [4, 5]? To give meaningful answers to those questions, we first need to understand whether Newton’s second law has any empirical content and, if it does, what its empirical content is. For that purpose, in this article, we will discuss how to make empirical sense of Newton’s second law.

The key difficulty in making empirical sense of Newton’s second law arises from its mathematical underdetermination; there is one equation (the second law) but two new empirically unknown quantities (force and mass). In general, we can make sense of such underdetermined laws by making use of the qualitative aspects of the unknown quantities [6]. Applied to Newton’s second law, we need to make use of the qualitative aspects of force and mass. Thereby, we not only develop a deeper understanding of Newton’s second law and the concepts of force and mass but also for how to make sense of physical laws in general.

In section 1, strongly inspired by references [7, 5], we show how the qualitative aspects of force and mass can be used to make empirical sense of Newton’s second law. To also provide an alternative, in section 2, we discuss a standard trick by which the underdetermination problem is circumvented rather than solved.

1. Newton’s second law

In the beginning of the Principia, Newton states eight definitions and three laws [8]. Arguably, the most important of those laws is his second law [9]:

“The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.”

Nowadays, it is usually presented in a mathematical form as
\[\mathbf F = m \mathbf a\ , \tag{1}\]
where the “impressed force” is represented by \(\mathbf F\) and the “alteration of motion” is represented by \(m \mathbf a\) with mass \(m\) and acceleration \(\mathbf a\).

While Newton considers acceleration to be well known from kinematics [10], he newly introduced “mass” and “impressed force” [11]. Here, “newly introduced” must be understood in relation to kinematics. The actual historical development is much more involved [12, 13]. From Newton’s first definition and the explanatory text following it, we learn that mass is a quantity associated to an object [11]; see also references [7, 14]. From Newton’s fourth definition and the explanatory text following it, we learn that an impressed force is external to an object but acting upon it [11]. That is, instead of being associated to an object, an impressed force is associated to an object’s environment or, more precisely, to how that environment affects the object’s motion [15]; see also references [7, 14]. Note that Newton does not give us a direct way to quantify mass or impressed force [16].

1.1. Newton’s second law is mathematically underdetermined

Let us, for a moment, take the perspective of someone who is familiar with kinematics but not yet with Newtonian mechanics. From that perspective, it is clear how to quantitatively determine acceleration \(\mathbf a\); but it seems that we have no way to quantitatively determine the force \(\mathbf F\) or the mass \(m\). In turn, looking back at equation \((1)\), we might start to feel a bit uneasy. We have one equation but two unknowns. So, from a mathematical perspective, Newton’s second law is simply underdetermined.

As an underdetermined equation does not have a unique solution, it seems that Newton’s second law cannot be used to make empirical predictions. If this was true, Newton’s second law—usually considered to be a law of nature at the core of classical mechanics—would be empirically empty. And we might have to doubt whether classical mechanics really was a scientific theory. Luckily, however, all this is only true from a mathematical perspective. And, as it turns out, a purely mathematical perspective is just too narrow to understand the empirical content of Newton’s second law.

1.2. Interplay between quantitative equality and qualitative difference

We can make empirical sense of Newton’s second law, when we take the qualitative aspects [17] of mass and impressed force into account. Note that, despite their quantitative equality in Newton’s second law, impressed force, \(\mathbf F\), and mass times acceleration, \(m \mathbf a\), are qualitatively different: the mass is associated to the object whose motion we describe; the impressed force is associated (in addition) to that object’s environment; and the object’s acceleration provides the empirically accessible basis to which the combination of mass and impressed force have to fit.

By using the interplay between quantitative equality expressed in equation \((1)\) and qualitative difference between force, mass, and acceleration, we can indirectly determine values for force and mass from the observed accelerations. Explicitly, we can make use of this interplay by bringing one object into different environments and different objects into one environment [7]. As the mass is associated to an object, it will remain the same, when the object is brought into a different environment. Similarly, at least in the simplest case [18], the impressed force is associated to an environment, such that it will remain the same, when a different object is brought into that environment. So, the number of independent parameters we need to consider corresponds to the number of objects, which are characterized by mass parameters, plus the number of environments, which are characterized by force parameters [19]. Now, the key point to realize is that the number of combinations of objects and environments grows faster than the sum of the number of objects and the number of environments. And because the acceleration for each combination is described by an equation of the form of Newton’s second law with the corresponding mass and force parameters, we can increase the number of equations beyond the number of independent parameters. Thus, by considering various combinations of objects and environments, instead of having to deal with a single underdetermined equation, we can generate an overdetermined set of equations and, in turn, make some empirical predictions.

While this approach might appear a bit abstract at first, it can be easily understood by considering an example.

1.3. Example: mechanical springs

To illustrate how we can make use of the interplay between quantitative equality and qualitative difference, we consider a simple example of mechanical springs roughly along the lines of reference [7]. Let us start by considering a bob attached to a mechanical spring in the earth’s gravitational field and investigate the bob’s oscillations around its equilibrium position; see figure 1 (a). For simplicity, we assume the motion to take place in only one dimension (up and down) and consider downward motion as positive, because it elongates the spring.

The bob’s equation of motion is found by combining Newton’s second law with the force of gravity and the force exerted by the spring. The forces of gravity and the spring can be combined into an effective restoring force, \(F_r =\, – k x\), where \(k\) is the spring constant and \(x\) is the deviation of the bob from its equilibrium position [20]. In turn, for the one-dimensional motion, we find the equation of motion
\[m \ddot x =\, – k x\ , \tag{2}\]
where we used that the acceleration is the second-order time derivative of the position, \(a = \ddot x\). The general solution is given by
\[x(t) = x_0 \cos \left( \omega_0 t \right) + \frac{v_0}{\omega_0} \sin \left( \omega_0 t \right)\ , \tag{3}\]
where we introduced the initial position \(x_0 = x(0)\), the initial velocity \(v_0=\dot x(0)\), and the oscillation frequency \(\omega_0 = \sqrt{k/m}\); see figure 1 (b).

While the (linear) form of the restoring force determines the form of the trajectory \(x(t)\), the problem of mathematical underdetermination is still present in the oscillation frequency. In an experiment, the oscillation frequency \(\omega_0\) can be determined by observing the bob’s position at different times; see figure 1 (b). However, our theoretical prediction for the oscillation frequency \(\omega_0 = \sqrt{k/m}\) involves two parameters: the bob’s mass \(m\) and the spring constant \(k\). And there is no way to uniquely determine two independent theoretical parameters from a single experimental parameter. So, it seems that, as a consequence of the mathematical underdetermination of Newton’s second law, we cannot find the bob’s mass \(m\) or the spring constant \(k\) from experimental measurements of the oscillation frequency. Even worse, it seems that we cannot make any real prediction for the oscillation frequency, because any experimental result could be ‘explained’ by an appropriate choice of the bob’s mass and the spring constant. Note, however, that we did not yet make much use of the qualitative aspects of force and mass.

To resolve the problem of mathematical underdetermination, we now use the interplay between quantitative equality and qualitative difference, as described in section 1.2. To that end, consider three mechanical springs to each of which a bob is attached. At first glance, that seems to only worsen the situation, as we now have six independent parameters on the theoretical side (three masses of the bobs and three spring constants) but only three independent experimental parameters (three oscillation frequencies). By switching the bobs around, however, we can generate up to nine different combinations of springs and bobs, each of which oscillates at its own frequency; see figure 2. With nine experimentally observable oscillation frequencies, three mass parameters, and three spring constants, we have converted a mathematically underdetermined situation into a mathematically overdetermined situation, which has some empirical content; namely, it predicts that the frequencies \(\omega_{ij}\) of the nine different combinations of bobs and springs can be accurately described by only six different parameters (the bobs’ masses \(m_1, m_2, m_3\) and the spring constants \(k_1, k_2, k_3\)) according to
\[\omega_{ij} = \sqrt{k_i/m_j}\ .\tag{4}\]
Note that the qualitative aspects of mass and force (spring constant)—that is, their association to the bobs and springs respectively—were crucial to reach this overdetermined situation. We made use of these qualitative aspects by implicitly assuming that: (i) the mass of a bob does not change when the bob is attached to another spring; (ii) the spring constant of a spring does not change when a different bob is attached to it.

2. A standard trick to circumvent the underdetermination problem

Having resolved the underdetermination problem of Newton’s second law in the previous section, we might wonder how it is typically addressed. Actually, physicists rarely acknowledge the problem and, consequently, provide no explicit solution. But physicists clearly know how to apply classical mechanics. So, there must be another way to deal with the underdetermination problem. And indeed, while not a proper solution, there is a standard trick by which we can circumvent the problem.

When Newton’s second law is supplemented by a specific force law (with pre-determined parameters), the combination of both laws is no longer underdetermined [21]. Then, declaring one specific type of force, usually gravity, to be a reference force, all other forces can be determined by comparison to that reference [22]. By this trick, one can get around having to answer the question of how to make empirical sense of Newton’s second law alone; it just never stands on its own. However, it is only a trick and not a real solution, as it creates a new problem: how can one find the force law for whichever type of force one chooses as reference? Or, more specifically, how did Newton find his law of gravity?

2.1. Newton’s law of gravity

Newton found his law of gravity, according to references [23, 24], roughly along the following lines. Starting kinematically, for an object’s uniform motion on a circle, the centripetal acceleration is directed towards the circle’s center with the magnitude of [25]
\[a = 4 \pi^2 \frac{r}{T^2}\ , \tag{5}\]
where \(r\) is the radius of the circle and the period \(T\) is the time it takes the object to travel around the circle once. From Kepler’s third law, Newton knew that for planets the period \(T\) is proportional to \(r^{3/2}\). So, combining Kepler’s third law with the result from kinematics, we find
\[a \propto \frac{1}{r^2} \tag{6}\]
for the centripetal acceleration of planets. As this acceleration is independent of a planet’s mass, the corresponding gravitational force must be proportional to the planet’s mass, such that it cancels out in the equation of motion. At the same time, however, for the gravitational force between two objects of mass \(m_1\) and \(m_2\), Newton’s third law requires equality in magnitude between (i) the force the object with mass \(m_1\) exerts onto the object with mass \(m_2\) and (ii) the force the object with mass \(m_2\) exerts onto the object with mass \(m_1\). Thus, the gravitational force between two objects must be proportional to the masses of both and, in turn, we arrive at Newton’s universal law of gravity
\[F_G = G\, \frac{m_1 m_2}{r^2}\ , \tag{7}\]
where \(G\) is some proportionality constant (the gravitational constant) and \(r\) is the distance between the two objects. For more details, see references [23, 24].

For our discussion, it is essential to note that the derivation of Newton’s gravitational force, equation \((7)\), relies on Newton’s second law of motion. So, we should not rely on Newton’s law of gravity to make empirical sense of his second law. Instead, we should first know how to make empirical sense of Newton’s second law, for example as in section 1, before we use it to derive Newton’s law of gravity from a kinematic description and Kepler’s laws. With Newton’s second law and law of gravity both well established, we can then indeed take the shortcut suggested by Sommerfeld [22] and use the force of gravity as a reference to which all other forces are compared. But, if we were to include Newton’s law of gravity into the core of classical mechanics, as Nagel indicates [1] and Lakatos suggests [4], we could not make adjustments to the law of gravity without changing classical mechanics as a whole; however, see reference [26]. We should note, in addition, that learning only the shortcut would come at some costs: we would not learn how to make empirical sense of mathematically underdetermined laws; and we would not develop an explicit qualitative understanding of mass and force, two key concepts of classical mechanics.

2.2. Methodological difference between the use of Hook’s law and Newton’s law of gravity

After the discussion of Newton’s law of gravity, we might wonder how this differs from the use of Hook’s law in the mechanical-spring example of section 1.3. There are two essential differences, which we will briefly discuss.

In the laboratory, we have our systems under control. For example, for the mechanical springs discussed in section 1.3, we can take one of the bobs from one spring and attach it to another spring. We could even go further and restrict the extension of a string or cut one of the bobs in half before reattaching it to a spring. Having our systems under good control is essential to make use of the interplay between qualitative difference and quantitative equality. In the solar system, however, we have no control over the objects (planets, moons, comets, …) or their environment (sun, planets, …). Because of this lack of control, we cannot make explicit use of the interplay between qualitative difference and quantitative equality. The key difference is that we can make experiments in the laboratory but we can only make observations in the solar system.

Furthermore, we used the linear restoring force, or Hook’s law, in the mechanical-spring example only for simplicity. In some situations, for example for large extensions of a spring, the form of the restoring force will be more complicated; and it might have more independent parameters to determine than just the spring constant. So, in general, we not only have to determine some unknown parameter but also the form of the force function that we use to describe the motion of objects in some type of environments. In strong contrast, Newton’s law of gravity is meant to be universal; that is, it should apply to all objects with the same gravitational constant \(G\) and with the same functional form, as described by equation \((7)\).

Conclusion

Reviewing the previous discussion, we should start by noting a few key points. Classical mechanics relies on kinematics as a pre-theory [27], which allows us to consider the position, velocity, and acceleration of moving objects as empirically accessible quantities. With force and mass, Newton introduces two new quantities that, in combination with Newton’s second law, are used to describe the motion of objects. While mass is a quantity associated to the object whose motion we describe, force is a quantity associated to that object’s environment or, more precisely, to the environment’s effects on that object’s motion. Note that the term “quantity” is a bit unfortunate, as it is the qualitative aspect of a quantity, which is essential to give mathematically formulated laws their empirical content; compare reference [6].

Regarding Newton’s second law, interpreting reference [6], we might say that its empirical content is as follows: two hidden quantities are sufficiently many to describe the motion of objects in relation to their environments. Those hidden quantities, however, are not arbitrary; instead, they are specified by their qualitative aspects. Only the qualitative aspects of force and mass allow us to turn Newton’s second law, usually expressed as the single equation \((1)\), into an overdetermined set of equations with empirical content; compare figure 2. This overdetermination is at the core of what makes classical mechanics ‘better’ than kinematics: we need less independent quantities or parameters to describe the motion of various objects in various environments; compare figure 2. Note, however, that the resolution of the mathematical underdetermination is not necessarily via other springs (as in section 1.3); it can also be via other physical systems, as long as we can generate more different empirical situations than we have independent unknown quantities.

Finally, let us address the opening questions about Newton’s second law: Is it a definition of force? A law of nature? Or a law governing our description of nature? Interestingly, we can answer yes and no to all three questions. Having some empirical content, Newton’s second law lacks the arbitrariness one usually expects from a definition; yet, it is used as definition in the sense that it gives a quantitative meaning to force. Since Newton’s second law on its own cannot fully describe the behavior of any individual physical system, it cannot really be a law of nature; yet, it can be viewed as a law of nature in the sense that its empirical content tells us something about the world. Having some empirical content and, therefore, also being falsifiable, Newton’s second law cannot be part of the unfalsifiable hard core of a research program, as suggested by Lakatos [4]. Nevertheless, it structures our description of nature by demanding us to find suitable force functions for the different types of environments; compare reference [5]. Being able to answer yes and no to all three questions indicates that, in philosophy of science, we have not yet found the right concepts to address the complexity involved in an empirical interpretation of Newton’s second law [28]. In conclusion, while the precise logical status of Newton’s second law remains somewhat unclear, we were at least able to demonstrate how to make empirical sense of it by using the qualitative aspects of force and mass.

Acknowledgements

I thank Mathias Gutmann very much for many fruitful discussions, for exceptionally useful literature recommendations, and for valuable feedback to the manuscript.

License and citation.
This work was published by Tim Ludwig on 15 June 2025 in the Philosophy for Physics blog, accessible under https://timludwig.de/blog-philosophy-for-physics/, and it is openly licensed via CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). To give appropriate credit, in academic contexts, please cite as follows: Tim Ludwig, “How to make empirical sense of Newton’s second law?” Philosophy for Physics (blog), June 15, 2025, https://timludwig.de/philosophy-for-physics-2025-june-15/.

Footnotes

[1] Ernest Nagel, The Structure of Science: Problems in the Logic of Scientific Explanation (Indianapolis, IN: Hackett Publishing Company), 185-202.

[2] Nagel, The Structure of Science, 187.

[3] Andreas Hüttemann, “Naturgesetze,” in Wissenschaftstheorie: Ein Studienbuch, 2nd ed., ed. Andreas Bartels and Manfred Stöckler (Paderborn, DE: mentis Verlag, 2009), 135-153.

[4] Imre Lakatos, The methodology of scientific research programmes: Philosophical Papers Volume 1, ed. John Worrall and Gregory Currie (1978; 7th repr., Cambridge, UK: Cambridge University Press, 1999), 47-52.

[5] Holm Tetens, Experimentelle Erfahrung: Eine wissenschaftstheoretische Studie über die Rolle des Experiments in der Begriffs- und Theoriebildung der Physik (Hamburg, DE: Felix Meiner Verlag, 1987), 63-70.

[6] Michael Heidelberger, “Über eine Methode der Bestimmung theoretischer Terme,” in Aspekte der physikalischen Begriffsbildung: Theoretische Begriffe und operationale Definitionen, ed. Wolfgang Balzer and Andreas Kamlah (Braunschweig, DE: Friedr. Vieweg & Sohn Verlagsgesellschaft, 1979), 37-48.

[7] Günther Ludwig, Einführung in die Grundlagen der Theoretischen Physik, vol. 1, Raum, Zeit, Mechanik, 2nd ed. (Braunschweig, DE: Friedr. Vieweg & Sohn Verlagsgesellschaft, 1978), 144-151.

[8] Isaac Newton, The Principia, trans. Andrew Motte (Amherst, NY: Prometheus Books, 1995), 9-19.

[9] Newton, The Principia, 19.

[10] Newton, The Principia, 13-18.

[11] Newton, The Principia, 9-10.

[12] Max Jammer, Concepts of Mass: in Classical and Modern Physics (1961; repr., Mineola, NY: Dover Publications, 1997).

[13] Max Jammer, Concepts of Force: A Study in the Foundations of Dynamics (1957; repr., Mineola, NY: Dover Publications, 1999).

[14] Tetens, Experimentelle Erfahrung, 65.

[15] The important difference between “an object’s environment” and “how that environment affects the object’s motion” is that, in the latter case, force may also depend on the object itself. While not relevant for the simple example of mechanical springs discussed in section 1.3, the gravitational force acting on an object also depends on that object via its mass; compare section 2.1. Similarly, to give another example, the Coulomb force acting on an object depends on that object’s charge.

[16] Strictly speaking, Newton does define mass as “quantity of matter” by the product of “density” and volume [11]; mathematically, \(m = \rho V\), where \(\rho\) and \(V\) respectively represent density and volume. However, while the volume is assumed to be known [10], the density \(\rho\) is not defined quantitatively; though, qualitatively, it becomes clear that density is associated to a material [11]. So, while not literally true, it is effectively true that mass \(m\) is quantitatively undefined.

[17] The “qualitative aspects” of the concepts mass and impressed force are what Heidelberger [6] would refer to as pre-understanding (“Vorverständnis”) of those concepts. However, I prefer qualitative aspects or qualitative content. The “pre” in pre-understanding (“Vor” in “Vorverständnis”) suggests that we can forget about it, when we have developed a full understanding; but the qualitative aspects remain relevant even then.

[18] Here, with “the simplest case” we mean that the force acting on an object does not depend on the object itself but only on its environment; compare [15].

[19] In many cases, the effect of environments on an object’s motion will be characterized by a full force function rather a simple parameter. Yet, the general argument remains the same, such that—to keep the discussion simple—we can disregard the difference between force function and force parameter. Note that the discussion of gravity in section 2.1 provides an example for how to determine a force function (rather than a mere parameter) from a kinematic description. However, note also that gravity has some specific methodological difficulties, as discussed in section 2.2.

[20] In one dimension, Newton’s second law becomes \(F= m a\). The (Galilean) gravitational force is given by \(F_g = m g\), where \(g\) is the gravitational acceleration constant. And the force exerted by the spring, for simplicity assumed to be described by Hook’s law, is given by \(F_s =\, – k (l – l_0)\), where \(k\) is the spring constant, \(l\) is the spring’s length, and \(l_0\) is the spring’s equilibrium length without load. Combining everything, we find the equation of motion \(m \ddot l = m g\, – k (l – l_0)\), where we used that the bob’s acceleration is given by the second-order time derivative of the spring’s length \(a=\ddot l\), because the bob is attached to one end of the spring. Note that the net force vanishes for \(l = l_e\), where \(l_e = l_0 + mg/k\) is the spring’s equilibrium length with load. For that length, the forces acting on the bob balance each other and define an equilibrium position for the bob. We can then simplify the equation of motion a bit by describing the bob’s motion in relation to its equilibrium position as \(x = l\, – l_e\) or, equivalently, \(l=l_e + x\) for which we find the equation of motion \(m\ddot x =\, – kx\), where the right hand side is the effective restoring force \(F_r =\, – kx\), as it brings the bob back to its equilibrium position at \(x=0\) or \(l = l_e\).

[21] Ludwig, Einführung in die Grundlagen der Theoretischen Physik, vol. 1, 150.

[22] Arnold Sommerfeld, Vorlesung über Theoretische Physik, vol. 1, Mechanik, 8th ed., ed. Erwin Fues (Thun, DE: Verlag Harri Deutsch, 1994), 5-6.

[23] Malcolm Longair, Theoretical Concepts in Physics: An Alternative View of Theoretical Reasoning in Physics, 3rd ed. (Cambridge, UK: Cambridge University Press, 2020), 48-71.

[24] K. Simonyi, Kulturgeschichte der Physik: Von den Anfängen bis heute, trans. Klara Christoph and Károly Simonyi, 3rd ed. (Frankfurt am Main, DE: Wissenschaftlicher Verlag Harri Deutsch, 2004), 252-266.

[25] With a uniform motion on a circle, we mean that the object moves on a circle with constant radius \(r\) and a constant angular frequency \(\omega_c\). To describe this motion kinematically, we use a planar coordinate system that coincides with the circle’s plane. The coordinate system is chosen such that (i) its origin coincides with the center of the circle and (ii) the object lies on the \(x\)-axis at \(t=0\). We can then describe the object’s position by the vector \(\mathbf x(t) = r\, (\cos \phi(t), \sin \phi(t))\), where the time-dependent angle is given by \(\phi(t)=\omega_c t\). From the position, by taking the derivative with respect to time, we can determine the velocity \(\mathbf v(t) = \mathbf{\dot x}(t) = r \omega_c (-\sin \phi(t), \cos \phi(t))\). In turn, by taking the derivative with respect to time again, we find the acceleration \(\mathbf a (t) = \mathbf{\dot v}(t) = \mathbf{\ddot x}(t) =\, – r \omega_c^2 (\cos \phi(t), \sin \phi(t))\). Note that the object’s acceleration is always directed towards the center of the circle. Its magnitude is given by \(a = |\mathbf{a}(t)| = r \omega_c^2\). By definition, the period \(T\) it the time in which the object travels around the circle exactly once, such that \(\phi(T) = 2 \pi\) or, equivalently, \(T = 2\pi/\omega_c\) and \(\omega_c = 2\pi/T\). In turn, we find \(a = 4 \pi^2 r/ T^2\), which is the same as equation \((5)\). For a more geometric derivation, see references [23, 24].

[26] Shahan Hacyan “What does it mean to modify or test Newton’s second law?” American Journal of Physics 77, no. 7 (July 2009): 607-609. https://doi.org/10.1119/1.3120121.

[27] Here, pre-theory (“Vortheorie”) is used in the sense of Ludwig, Einführung in die Grundlagen der Theoretischen Physik, vol. 1, 48-49 and 137-142. In that sense, ‘pre-theory’ is a relational term that is used to describe how one physical theory relates to another physical theory. If a physical theory \(T_2\) takes the results of another theory \(T_1\) for granted, then \(T_1\) is called a pre-theory to \(T_2\). In particular, kinematics is a pre-theory to classical mechanics: in classical mechanics, we neither question the validity of distance and duration measurements nor the validity of calculations used to determine velocities and accelerations from those measurements.

[28] The difficulty in developing appropriate philosophical concepts to discuss the empirical interpretation of Newton’s second law and the difficulty in clarifying the role of the physical concepts of force and mass in the empirical interpretation of Newton’s second law can both be better addressed and understood, when we use the distinction between concept (“Begriff”), object (“Gegenstand”), and thing (“Ding”) developed in reference [29]. This distinction, very roughly speaking, can be viewed as a clarification, extension, and generalization of the more widely used type-token distinction that can be found in many places in the literature (for example, see reference [30]). However, a discussion of these distinctions and their application in the context of Newton’s second law would go way beyond the current article and, therefore, is deferred to another time.

[29] Mathias Gutmann, Leben und Form: Zur technischen Form des Wissens vom Lebendigen (Wiesbaden, DE: Springer VS, 2017), pt. 1.

[30] Julian Baggini and Peter S. Fosl, The Philosopher’s Toolkit: A Compendium of Philosophical Concepts and Methods, 2nd ed. (Chichester, UK: John Wiley & Sons, 2010), 187-189.