Abstract
In this chapter, I investigate whether dimensions in physics are analogous to quality dimensions (in the sense of Gärdenfors P, Conceptual spaces: the geometry of thought, 2nd edn. MIT Press, Cambridge, MA, 2000, 2004) and whether phase spaces are to be considered as conceptual spaces (as proposed by Masterton G, Zenker F, Gärdenfors P, Eur J Philos Sci 7:127–150, 2017). To this end, I focus on the domain of force in classical physics and on the dimension of time from classical to relativistic physics. Meanwhile, I comment on the development of abstract spaces with non-spatial dimensions, such as conceptual spaces, which is itself part of a long history of conceptual development.
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Notes
- 1.
I use this example because I know it best (Douven et al. 2017), although I acknowledge that a drawback of relying on a single example is that it might be atypical in some ways.
- 2.
Some of the concepts are also central to other natural and applied sciences, such as chemistry and engineering. We will consider physics as a common ground for all examples in this chapter.
- 3.
As suggested by a referee, the concept ‘classical states of matter’ could be modelled by a CS as follows: a discrete Euclidean space with values 0 and 1 along dimensions solid, liquid, gas, and plasma. Equilibrium states and critical points would be indicated by vectors that are not aligned with a particular axis. It is possible to distinguish more states (such as amorphous or crystalline solid) and phases thereof (although that becomes a very long list if the material isn’t specified upfront), or to include modern states (such as Bose-Einstein condensate) and very high energy states (such as quark-gluon plasma), which leads to a higher dimensional CS.
- 4.
Hence, a possible disanalogy between phase diagrams and colour CS is that the same dimensions can be used to represent phases of different materials. This seems different from the way ‘the’ CS of colour is discussed in the literature, although we could in principle consider multiple colour CSs to distinguish, e.g., between different human individuals or groups of humans belonging to different language communities, etc.
- 5.
We can feel temperature and pressure, but our crude bodily perception of them does not factor into the organization of the phase diagram.
- 6.
Newton (1697, 1999: 416): “A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.”
- 7.
He did not need calculus, since his formulation of the second law only concerned impulsive forces, rather than continually acting ones (Newton 1697, 1999: 113).
- 8.
This is one of the concepts that are unlikely to be represented by a CS, because it belongs to the meta-theory of such spaces.
- 9.
See Bramwell (2017) for a brief overview of dimensions in physics.
- 10.
Although analytic geometry is used to represent lengths, its symbols represent lengths implicitly divided by some reference length; this yields dimensionless scalar values, such that inhomogeneous terms can be added or equated (Schulman 2010).
- 11.
See also Sterrett (forthcoming): “Similarity of systems is always relative to some kind of behaviour (e.g., kinetic, dynamic, electrical, thermal, magnetohydrodynamic).”
- 12.
Maxwell considered units of length, time, and mass as fundamental (Bramwell 2017).
- 13.
SI does not require fractional powers of basic units, but to represent unit systems that do (such as CGS), we need a richer mathematical structure.
- 14.
Applying Toa’s view to SI, DA is the representation theory of the structure group {(M,L,t,T,I,n,li) | M,L,t,T,I,n,li ∈ℝ+}; see also Table 8.1.
- 15.
Since we use the colour CS as our guiding example, it is interesting that observe that Sonin (1997: 15) excludes colour as a physical quantity because the addition operation is not uniquely defined.
- 16.
Besides physical operations that can be represented numerically, purely numerical operations can be defined, such that monomials of base quantities are defined.
- 17.
See also Sect. 8.2.3.1.
- 18.
Bridgman’s principle is the rationale behind the aforementioned requirement that physical equations and terms therein should be homogenous. Recall that Galileo and Newton expressed equalities (verbally) in ratio form, which yields dimensionless quantities. Physically relevant dimensionless quantities are invariant under unit changes; hence, the importance of such quantities (like Reynolds number in fluid mechanics) in DA.
- 19.
Euclidean geometry resulted from the observation of empirical regularities, which have been ubiquitous from the dawn of civilization that it is hard to recognize them as discoveries (Blåsjö 2016). It is conceivable that the physical world could have been such that the area of a rectangle does not (approximately) equal the product of its length and width, as, say, in a curved space where a non-Euclidean geometry applies.
- 20.
Tao (2012) has commented that, while it may be “geometrically natural to multiply two lengths to form an area” (but see footnote 19 for a comment on ‘geometrically natural’), “it is less clear how to apply it for more general [dimensions]”, such as the product appearing in the righthand-side of Einstein’s formula E = mc 2.
- 21.
In this sense, all our empirical knowledge is comparative only and coherentism rather than foundationalism is the epistemology that fits naturalism best.
- 22.
Sonin (1997: 32) points out that “only a few of the available universal laws are usually ‘used up’ to make base quantities into derived ones of the second kind.” “There are many laws left with universal dimensional physical constants that could in principle be set equal to unity: the gravitational constant G, Planck’s constant h, Boltzmann’s constant k B, the speed of light in vacuum c, etc.” Although this leads to so-called natural units (e.g., c = 1 as used in some branches of physics), the choice is far from unique.
- 23.
A revision of SI is due in 2018, such that the seven units will be defined in terms of seven constants that are defined rather than measured. Until the planned reform, the definition of the kilogram still depends on an artefact (the International Prototype of the Kilogram), as was previously the case for other units. Nothing in this exposition depends on the planned change. In this paragraph and in Table 8.1, the SI convention for the notation of dimensions is used (BIPM 2006, Sect. 8.2.3).
- 24.
Various suggestions of this kind can be found in the DA literature. I not aware a standard approach, but an early suggestion along these lines can be found in Moon and Spencer (1950): they considered dimensions as concepts and called their geometric approach ‘idon theory’ or ‘idonics’, but these names did not catch on. These authors have also developed a theory of ‘holors’ (mathematical quantities that are made up of one or more other independent quantities – a generalisation of tensors), which I want to flag here as potentially relevant for CSs.
- 25.
It should be noted that the number of baSIc dimensions (seven) is not a fundamental property of the physical world, but a matter of convention. It is possible to distinguish fewer basic dimensions; in fundamental physics, it is common to consider all quantities as scalar (dimensionless; see, e.g., Duff et al. 2002: part III).
- 26.
Since unit systems are closely related to metrology, in practice considerations of accuracy, reliability, and reproducibility come into play as well.
- 27.
This is reminiscent of Raubal’s (2004) formalization of psychological CSs as vector spaces but suggests opting for affine spaces instead. Affine spaces are generalizations of Euclidean spaces, which have no specific origin: they can be used to represent displacements (translations) or lengths, durations, etc. rather than absolute positions, points in time, etc.
- 28.
Moon and Spencer (1949) distinguished two basic length concepts (tangent and radial) to get rid of ambiguities (different concepts having the same dimensions if only one length dimension is considered).
- 29.
In the case of discrete dimensions, the corresponding phase space is called a ‘state space’. A two-dimensional phase space is called a ‘phase portrait’.
- 30.
If we read this assuming implicit SI units, there are tacit ratios involved to arrive at real values (cf. footnotes 10 and 18).
- 31.
See also Gärdenfors 2000: 215–219.
- 32.
The extant literature on CSs also considered velocity, acceleration, momentum, energy, and work as derived quantities, claiming that to define them, we need to combine magnitudes belonging to different domains. Again, DA adds a helpful distinction between merely defined quantities and derived quantities of the second kind.
- 33.
We should not expect to reveal time’s true nature by studying it as a physical dimension. As Sonin (1997:14) reminds us: “What time ‘is’ has no relevance in the present context, only the defining operations matter.”
- 34.
Huggett and Wüthrich (2013) provide an introduction suitable for philosophers of science.
- 35.
For example, we can present Euclidean geometry as a special case of curved geometry, with the curvature equal to zero. However, this is not a suitable reconstruction of the conceptual change involved in develo** the first non-Euclidean theory, which required recognizing spatial curvature as a valid concept in the first place. And once curvature has been accepted as a variable, it is equally hard to imagine reasoning without it or to appreciate how counterintuitive its introduction must have been starting out from a strictly Euclidean viewpoint.
- 36.
See de Courtenay (2015) for an account of how Fourier’s treatment of dimensional analysis and the standardisation of physical units have profoundly changed the meaning of mathematical equations in physics.
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Acknowledgments
I am grateful to the organizers and participants of the workshop “Conceptual Spaces at Work” (2016) for giving me the opportunity to present and discuss my early ideas on this topic. I also thank Joel Parthemore, Danny Vanpoucke, and an anonymous referee for their helpful comments on an earlier version of this text. Part of this project was supported by a grant from the FWO (Research Foundation – Flanders) through grant number G0B8616N.
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Wenmackers, S. (2019). Lost in Space and Time: A Quest for Conceptual Spaces in Physics. In: Kaipainen, M., Zenker, F., Hautamäki, A., Gärdenfors, P. (eds) Conceptual Spaces: Elaborations and Applications. Synthese Library, vol 405. Springer, Cham. https://doi.org/10.1007/978-3-030-12800-5_8
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