In the example at hand, this interpretation is an intension, because our comparison with another possible world in which there are imaginary charges does not need to consider contextual details, but only idealised details about the charges. Physical vs. But despite the above remark, that visualisation is logically independent of the question of physical reality: a second distinction needs to be made. Namely, whether a space is interpreted as physical in the actual world, or in some possible world or as merely mathematical , is an important question, which we will address.

This is because it is very hard and it would be contrived to formulate a theory that does not make any use at all of the mathematical concept of space, i. For example, the numerical parameters of physical theories all take values in fields, which are examples of spaces. So, in what follows, our concept of space will be physical not merely mathematical, and when we talk about visualisation we will mean visualisation through a space with physical properties such as particles, fields or forces defined on it.

And we submit that visualisation through physical spaces whether in our world, or in another possible world contributes to the intelligibility of the theory more than visualisation via merely mathematical spaces, because in the former case physicists can use physical arguments and intuitions, in addition to mathematical ones. Agreed, the distinction between merely mathematical and physical spaces can sometimes be blurry, if the mathematical spaces come with sufficient structure as, say, normed vector spaces, or even inner product spaces.

But this need not always be so: for example, the geometric interpretation of phase space in classical mechanics, as a symplectic manifold, is not physical: for the phase space is itself not deemed to be physically real, under the standard interpretation of classical mechanics. And yet it provides a visualisation which helps physicists to apply concepts such as the existence of forms on this space, the independence of the choice of coordinates, etc.

But in this paper we will not be concerned with such cases of mathematical but not physical spaces, since what is at stake in theories of quantum gravity is of course spacetime itself, and how theories without a physical spacetime can be interpreted. Summarising this discussion, a - b : the topic of visualisation in scientific understanding should in general also consider spaces which are mathematical and not physical. But for our specific question, of how theories without spacetime can be interpreted, only the question of whether there is a physical space is relevant second fork: b —but such a space need not be physically real first fork: a.

While we will not address the question of emergence here, it should be clear that we are considering cases in which physicists and philosophers alike do use the term see De Haro a , b , Crowther , Rickles Let us here explain the idea in simple terms. The idea is that T b is the theory of the strong interactions Yang-Mills theory and T t is a theory whose objects are strings.

The approximative theory is T t , i. The idea that strings held quarks together seemed to provide a reasonable explanation for this.

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As we remarked in the preamble to this section, the fact that these strings cannot be literally interpreted as holding quarks together is besides the point. For we are not concerned here with the literal physical truth of the constructions cf. Again, the interpretation is an intension: it abstracts away from the technically involved!

That is, T t is best interpreted as a string theory. This remarkable result was obtained through relatively simple physical methods: first, a slightly different notation for the Feyman diagrams, which use double lines, rather than wiggly lines, for the gluon propagators; quarks are denoted by single lines. Here, V is the number of vertices, E the number of edges, and F the number of faces, on any given triangulation of the diagram. The string interpretation of T t thus developed is a combination of Approximation which provides the initial interpretation in terms of Feynman diagrams, which are then reinterpreted as thickened diagrams , and the Similar -ity of the diagrams and expressions obtained in the approximative theory T t with string theories.

In terms of this new interpretation, it was easier to characterise the theory obtained in the limit: as a theory in which only planar diagrams those that can be drawn on the plane contribute. Namely, the original Feynman diagram expansion of the amplitudes of the theory can now be better and more significantly reorganised as a topological genus expansion, akin to the ones found in string theory:.

The interpretation of n is as the number of gluon loops. Notice that, for each power of N in Eq. These sorts of considerations have recently led more generally to relations between gauge theories and theories of gravity. Notice the close connexion between Approximation and visualisation in this example. So, this example instantiates the discussion in the preamble of this section, about starting with a space of one kind, in T b the one-dimensional worldline of a particle , and ending up with a space of a different kind, in T t the two-dimensional worldsheet of a string.

That this interpretation leads to the intelligibility of certain aspects of gauge theory can be seen from the further history of the subject. Furthermore, the exact amplitude can be completely reorganised as a topological expansion for stringlike surfaces. Thus the string interpretation gives insight into otherwise intractable calculations.

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The theory starts with an initial spacetime picture, and develops another spacetime picture in the approximation. For an introduction to random matrix models aimed at philosophers, and a discussion of emergence, see De Haro b. The physical idea is now as follows. Because of symmetry, the physical quantities do not depend on all N 2 components of the matrix, but only on its N eigenvalues.

And so, the dynamics of this model is the dynamics of the eigenvalues, subject to a potential. In principle, these eigenvalues can take any complex values, and so they correspond to N points on the complex plane. In other words: one started with a theory in which one had only matrices and no spacetime. It is this geometry which is extremely useful for the intelligibility the dynamics of the random matrix model.

And as before, all physical quantities have a topological expansion like Eq. The Approximation allows visualisation, because it fabricates a two-dimensional Euclidean space there is no time in which the theory is easy to visualise, and therefore easier to interpret. The Riemann surface turns out to be best interpreted as embedded in a 6-dimensional Calabi-Yau manifold, on which a closed topological string theory is defined.

This is because the original random matrix model described an open topological string theory. The cycles of the Riemann surfaces on the closed string side then correspond to projections of the 3-cycles of the Calabi-Yau, and the quantities in Eq. The basic quantity of the model is the partition function, as follows:. The measure is also U N -invariant. Thus one divides by the infinite volume of U N.

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This contribution can be added in the exponential, using the logarithm. The above integrals can be evaluated using the saddle-point approximation. The terms in the exponential potential plus the logarithm of the measure term are extremised, as follows:. This is a set of N equations for the eigenvalues, which are the independent variables of the theory. This equation can be interpreted as the classical equation of motion of the random matrix model.

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The term on the right-hand side, coming from the measure factor, induces a Coulomb-type repulsion between the eigenvalues. The eigenvalues then fill domains of the real axis; these domains are continuous, but may have several disconnected components. In the continuum limit, the discrete set of N equations, Eq.

However, it is more useful to first multiply Eq. The result of the manipulations described in the previous paragraph, starting from Eq. Notice that both x and y are complex. The above equation defines a hyperelliptic curve in the x , y -plane, because the right-hand side is a polynomial of degree 2 n with distinct roots. Clearly, the geometric interpretation gives a reoganisation of the amplitudes, like before, which allows a different interpretation of the random matrix model in terms of surfaces.

The interpretative lesson is as follows: Approximation , along with Similar and Internal , functions here as a tool like in the previous subsection, because it allows the development of a visual picture—a geometric interpretation in terms of a Riemann surface—which gives insight into otherwise intractable calculations.

## The Feynman Lectures on Physics Vol. II Ch. Curved Space

It thereby allows one to perform those calculations. This effective spacetime then allows physicists to visually interpret the theory. In this section, we move on to cases in which visualisation and effective spacetimes are not needed, and not used as interpretative tools—even if spaces might be present somewhere else in the theoretical structure. Therefore, in this section we focus on examples that do not contain any of the spaces of the disjunction a , and contain only the mathematical spaces of the disjunction b.

But we first make two warnings about merely mathematical spaces, in relation to the tools Similar - Internal which we will illustrate in the next three subsections.

The first warning is about the tools to be used in theories without spacetime; the second is about visualisation of merely mathematical spaces:—. Similar and Internal can be had in theories without spacetime, despite the presence of a spacetime somewhere in the theory. Ultimately, every realistic theory of quantum gravity attempts to recover the pseudo-Riemannian spacetime of general relativity, typically in an Approximation or an iApproximation : and so, every theory without spacetime, attempting to describe the world, must at one point or another develop a physical space.

But remember what our no-spacetime claim refers to: not to the presence , but to the use of spacetime as a tool for interpretation. More specifically, the claim is that these theories do not use Approximation when they develop their approximations; and that these theories do illustrate the use of Similar and Internal without reference to spacetime or visualisation. They also sometimes illustrate iApproximation , with its inverse relation between bottom theory and spacetime visualisation.

Merely mathematical spaces can be free from spacetime visualisation. We should say a bit more about our distinctions a - b recalled above, between physical and merely mathematical spaces, even if we do not attempt at a systematic treatment of this question here nor will we need it: for a discussion of a concrete case, cf. De Haro b : Section 4. And of course, even in the case of a set consisting of three elements, one can use visualisation, viz.

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But it would be foolish to mistake such a visualisation for a counterexample to the claim that we are dealing with a theory without spacetime. For, first: such a visualisation is not needed.