Spaghetti Universe Hypothesis (2017)

Walter Tay Ann Lee

Associate Professor Kuldip Singh

UNL2210 – Mathematics and Reality

Due 20th November 2017


In this paper, I will show how ontic structural realism (OSR) aims to solve the issues that plague traditional scientific realism. I will then look at Max Tegmark’s (2008) radical take on OSR, his Mathematical Universe Hypothesis (MUH), and attempt to draw an analogy between his argument for MUH and the “no-miracles” argument typically made by scientific realists. Finally, I will show how we can refute his “no-miracles” argument, and lead to our Spaghetti Universe Hypothesis.



Scientific realism is the view that scientific theories correctly describe the nature of our mind-independent world. This is arguably the most commonsensical view of science and is usually supported by the “no-miracles” argument. The idea behind this argument is that, the best explanation for why our scientific theories are so successful at describing and predicting phenomena, is that they correctly describe the nature of the world, or at least, are very close to doing so. It would be a miracle for our theories to be able to describe and predict phenomena, without what the theory says about the nature of the world to be true. And we shouldn’t accept miracles, especially if there is a non-miraculous alternative (Putnam, 1978).

However, scientific realism has since been discarded by many philosophers of science because of problems that it is unable to resolve or explain. In the following sections, I will show one of the biggest issues that plagues scientific realism, as well as how ontic structural realism (OSR) seeks to resolve it.



Firstly, what is a scientific realist being realist about? He is being a realist about the entities in nature that are picked out by the language of our scientific theories. That is, the realist believes that whatever the theory says about those entities is true. By true, we mean that what the theory says about the entities is exactly, or approximately, what is happening in the external world.

Scientific theories typically make claims about both observable and unobservable entities. Now, it is easy (and even, intuitive) to believe in the existence of observable entities. Observable entities exist, and they affect us in such a way that we are confident, by virtue of their affecting us (us perceiving them), that they exist (Chakravartty, 1998). However, is there a good reason for us to believe in the unobservables picked out by our scientific theories?

Note that the antirealist is not necessarily saying that unobservable entities do not exist. Merely, he is posing the question, do our scientific theories really give us epistemic access to unobservable entities (Ladyman, 1998)? Let us consider a few examples. The luminiferous Ether which used to be the propagation medium of electromagnetic waves is now believed to be non-existent. What used to be gravitational forces acting on two bodies over a distance is now, instead, better described by the curvature of space-time. Even more remarkable is that our idea of an electron has changed drastically over the years from being interpreted as a particle, a wave, and even both (as it exhibits both wave and particle properties). Is it meaningful to ascribe a reality to these unobservable entities if we were wrong about their existence in the past, and may very well be wrong now as well? And if our meaning of an unobservable entity (like the electron) changes, to what extent can we be said to be discussing the same entity (Chakravartty, 1998)?



Ontic structural realism (OSR), as presented by Ladyman (1998), seeks to resolve this issue with scientific realism. Ladyman’s (1998) argument begins with a tribute to Worrall (1989), who originally introduced structural realism.

A good summarizing statement of Worrall’s (1989) view is from his paper, “What we can know for sure on the basis of observation, at most, are only facts about the motions of macroscopic bodies, the tracks that appear in cloud chambers in certain circumstance and so on”. Worrall is a skeptic about the nature of the unobservable entities picked out by scientific theories, and is a realist about the structure of scientific theories. He argues, what retains across theory change is not the unobservables, but the structure. This allows him to seemingly solve the problem of unobservables changing, or ceasing to exist, as we make progress in science. That is, we can believe in the existence of the structure that remains, and it does not matter what we say about the unobservables because we are, fundamentally, interested in the structure.

However, this solution brings in a different problem. This ‘version’ of structural realism seems to presuppose a distinction between the form and content of a theory (Psillos, 1995). Are we able to tell the difference as to whether our theories are talking about unobservable entities or the structure of our world? For example, what gives us an idea of the electron is the description of its behaviour in terms of its laws, interactions, and so on in our theories. However, can we tell the difference between what the theory is saying about the properties of the electron, and what the theory says about the structure of the world in which these properties interact with? Is this any different from being a scientific realist? Scientific realists, too, advocate a structural understanding of the unobservable entities. This is what it means for the language of our scientific theories to “pick out” objects in nature. Thus, there seems to be no meaningful difference between structural realism and scientific realism.

Now, one can argue that there is a retention of structure across theory change. And it is this unchanging structure that we should be a realist about, not the ever-changing unobservables. However, it gains no advantage over scientific realism because it tells us nothing about how we should distinguish between what we should believe, and what we shouldn’t believe about our current scientific theories. It is easy to look at older theories and explain which parts are the structures that are retained, and which are thrown out in the form of unobservable entities, however, this distinction is dubious when it comes to our current theories. How do we know which parts about our current theories will be retained, and which parts will be discarded in future? As Papineau (1996) puts it, “restriction of belief to structural claims is in fact no restriction at all”. Similarly, the scientific realist can easily posit that the some properties about an unobservable are retained across theory change, and it is these properties that the realist is being realist about (Chakravartty, 1998). But this doesn’t solve the epistemological problem with current theories at all!

Another problem is the issue of metaphysical underdetermination. That is, even without considering theory change, we do not seem able to agree on the metaphysical implications of a theory. For example, there are multiple formulations of Newtonian mechanics (action at a distance, variational, field-theoretic, and curved spacetime formulations) that describe different realities (Roger Jones, 1991). Another example is French and Redhead’s (1988) paper, which concludes that quantum physics does not determine whether quantum objects are individuals or non-individuals. This is an alarming issue. Despite similarities in the structure of the theory, the reality and existences that we believe to exist are different depending on how we formulate those structures.

To resolve these problems, Ladyman (1998) advocates for a complete shift on how we understand existence. He argues that, we should not take objects to be the fundamental building blocks of reality, but should “shift to a different ontological basis altogether, one for which questions of individuality simply do not arise”. That is, he is arguing that we should take the ontology of quantum physics (and even fundamental physics in general) to be one of only relations.

What sort of relations? Weyl (1931) and Ladyman’s (1998) approach is a group-theoretic definition of the symmetry relations of objects. We can have various representations which may be transformed or translated into one another, but we still have an invariant state under such transformations which represents the objective state of affairs (Ladyman, 1998). The way to look at what we believe to exist, and what we do not believe to exist, is to look at the invariance with respect to the transformations relevant to it. Thus, on this view, unobservable entities (such as elementary particles) are just sets of quantities that are invariant under their symmetry groups (Ladyman, 1998). This seems to resolves the problem of metaphysical underdetermination.

Notice how this also serves to solve our earlier problem of theory change. With Ladyman’s (1998) ontic structural realism, we do not have to posit a difference between what we believe to be the unobservables and what we believe to be the structures of our world that are laid out by a scientific theory. What we believe, instead, is that the fundamental building blocks of reality is not unobservables, but structures. What structures? We should choose to believe in the existence of what remains invariant under a set of transformations, and be agnostic about everything else.

Now, I would like to reiterate how radical this view is. Ladyman (1998) is redefining what it means for something to exist, and not merely saying that what exists is just relations (with our previous notion of existence). With Ladyman’s (1998) view, what we typically understand as an object that ‘exists’ is an illusion, we have to understand existence in structural terms because of the problems that can’t be explained with scientific realism or Worrall’s structural realism. And thus, it seems that the only objects that exists are just structures.



    With an understanding of Ontic Structural Realism (OSR), we are now better equipped to talk about Max Tegmark’s (2008) radical take on OSR, his Mathematical Universe Hypothesis (MUH).

The idea behind MUH is the same as OSR, the only difference being that Tegmark explicit embraces a Pythagorean form of OSR, whereas Ladyman (1998) is not so committed to a mathematical universe. To elaborate, Tegmark similarly believes that the world is just made up of structures, because objects are structures, which are then related to each other by more structures, and turtles all the way down. However, he takes it a step further, the world is not only described in structural terms mathematically, but it is itself a mathematical structure.

His main argument for MUH starts with a description of scientific theories as having two components: mathematical equations and “baggage”. Tegmark (2008) describes “baggage” as words that explain how our theories are connected to what we humans observe and intuitively understand. He further argues, if a future physics textbook contains the Theory of Everything (TOE), which would be able to describe and predict all phenomena in the universe, it would be of mathematical form. Since the external world is isomorphic to the mathematical structure described by the TOE, he argues, there is no meaningful sense in which they are not one and the same. From the definition of a mathematical structure, it follows that if there is an isomorphism between a mathematical structure and another structure, then they are one and the same (Tegmark, 2008).



Tegmark (2008) makes other assumptions and arguments for his Mathematical Universe Hypothesis (MUH), however, I shall choose to focus on the parts of his arguments as laid out above and attempt to draw an analogy to the “no-miracles” argument put forth by Putnam (1978).

Recall that scientific realists typically argue that, antirealism makes the success of science a miracle. The antirealist has no explanation as to why ‘electron calculi’, ‘space-time calculi’ and ‘DNA calculi’ are able to correctly predict observable phenomena if, in reality, there are no electrons, no curved space-time, and no DNA molecules. If those objects don’t exist at all, then it is a miracle that our theories are successful at predicting phenomena (Putnam, 1978).

Now, let us take a look at Tegmark’s (2008) “isomorphism” argument for MUH. He argues that since the external world is isomorphic to the mathematical structure described by the TOE, there is no meaningful sense in which they are not one and the same. You could say that, it would be a miracle for our scientific theories to be isomorphic to a mathematical structure, and yet not be a mathematical structure. How would one explain that?

I argue that because Tegmark’s argument is so similar to the no-miracles argument, we are able to use a similar argument typically made against the no-miracles argument: The Argument from Underdetermination. It goes as follows:

Two isomorphic groups can differ from each other; they just do not differ as groups, there is no difference if we take only the group operations into consideration. Symmetry, isomorphism, relevant sameness are all context-dependent notions (van Fraassen, 2006). Now, it is also possible to imagine an infinite number of structures that are isomorphic to the mathematical structure of the universe, but is itself not a mathematical structure. What good epistemic warrant, then, do we have to believe in the existence of a purely mathematical structure over a physical one?

What sort of structures can exist other than mathematical ones? One radical alternative to MUH is that the physical structure of the universe is a structure made of spaghetti that is isomorphic to the mathematical structure described by the possible Theory of Everything assumed by Tegmark (2008). What would this mean? Here, I would like to introduce the two different ways of viewing the external physical reality proposed by Tegmark (2008) himself: the outside view or bird perspective of a mathematician studying the mathematical structure of the spaghetti and the inside view or frog perspective of an observer living in it. In the four-dimensional spacetime of the bird perspective, these particle trajectories resemble a tangle of spaghetti. If the frog sees a particle moving with constant velocity, the bird sees a straight strand of uncooked spaghetti. If the frog sees a pair of orbiting particles, the bird sees two spaghetti strands intertwined like a double helix. To the frog, the world is described by Newton’s laws of motion and gravitation. To the bird, it is described by the geometry of the pasta, obeying the mathematical relations corresponding to minimizing the Newtonian action. What about the observer? The frog itself must be merely a thick bundle of pasta, whose highly complex intertwining corresponds to a cluster of particles that store and process information (Tegmark, 2008).



What does this mean for us? Do we really live in a world made of spaghetti? I doubt so. Merely, I wish to show that when construed in such a way, Tegmark’s arguments for MUH can be attacked using the argument for underdetermination, or even other arguments that typically counter the no-miracles argument.


[2500 words]


Works Cited

Aguirre, Anthony, Brendan Foster, and Zeeya Merali. “Trick or Truth?.” (2016).

Chakravartty, Anjan. “Semirealism.” Studies in History and Philosophy of Science Part A 29.3 (1998): 391-408.

Ghins, Michel. “Putnam’s no-miracle argument: A critique.” Recent themes in the philosophy of science. Springer Netherlands, 2002. 121-137.

Halvorson, Hans. “What scientific theories could not be.” Philosophy of Science 79.2 (2012): 183-206.

Ladyman, J. “What is Structural Realism?.” Studies In History and Philosophy of Science Part A 29.3 (1998): 409-424.

Putnam, Hilary. “What is mathematical truth?.” Historia Mathematica 2.4 (1975): 529-533.

Tegmark, Max. “The mathematical universe.” Foundations of physics 38.2 (2008): 101-150.

Van Fraassen, Bas C. “Structure: Its shadow and substance.” The British Journal for the Philosophy of Science 57.2 (2006): 275-307.

Worrall, John. “Structural realism: The best of both worlds?.” dialectica 43.1‐2 (1989): 99-124.


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