The Topos

2,100 words

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Editor’s Notes: 

  1. Thank you to Boian Etropolski, former intern at the Qualia Research Institute, for telling me about the topos.
  2. I am not a mathematician, so please take everything I say below with a very heavy grain of salt. 

In Our Mathematical Universe, physicist Max Tegmark argues for a four-level multiverse [1]:

Level I: “bubbles” or Hubble volumes of space created by cosmic inflation, in which, according to Tegmark, “a tiny subatomic speck of space is blown to dimensions much greater than the entire currently observable region” in “a fraction of a second.” While inflation ended in our neighborhood of the Level I multiverse over 13 billion years ago, it is occurring in other regions in the infinitely large expanse of cosmic space, resulting in the perpetual birth of new universes. 

Level II: other “bubbles” that we in principle cannot travel to, since the space between our Level I multiverse and its neighbors is expanding faster than the speed of light. 

Level III: branches of the multiverse that correspond to different measurements of quantum phenomena. Quantum mechanics describes reality in terms of a wavefunction, which resides in a potentially infinite-dimensional vector space known as a Hilbert space. The (probabilistic) state of a quantum system can be characterized by superpositions of wavefunctions, implying that a single particle can be in multiple places or have multiple velocities at the same time. Whereas the Copenhagen interpretation of quantum mechanics claimed that the act of measurement collapses the wavefunction, the Many-Worlds Interpretation suggests that no such collapse ever occurs. Rather, all possible outcomes are realized in parallel universes.

Level IV: the realm of mathematical structures, such as the empty set, the dodecahedron, the group of symmetries of an equilateral triangle, the Klein bottle, bipartite graphs, and so on.

Level IV is a direct affirmation of mathematical Platonism. This is the view that mathematical objects are not mere constructs of the human mind, but are rather objectively real entities that belong to an abstract realm that is fundamentally separate from this universe and exists outside of space and time. (In other words, mathematical Platonists believe that mathematics is discovered, not invented.)

I briefly argued against mathematical Platonism in my blogpost “Naturalizing Mathematics”. But in this essay, I would like to suppose the opposite and entertain the idea that mathematical Platonism is, in fact, true. If so, then what is the nature of the Level IV multiverse? What does a multiverse of mathematical structures “look like”? 

Tegmark has given serious thought to these questions, of course, but he has surprisingly little to say about the underlying structure of the Level IV multiverse. While he notes that there are “many interesting relations” between the various mathematical structures “at the meta-level,” such as the fact that one structure can be composed of another, he concedes that the structures “aren’t connected in any physically meaningful sense.” His only systematic characterization of the Level IV multiverse is a list of computer programs that encode all the mathematical relations within each structure. Each program is a bit string, or a finite arrangement of zeros and ones, so each mathematical structure is uniquely identified “by the number whose bit string is the shortest computer program whose functions define all the relations of the structure,” according to Tegmark. But this computational description of the Level IV multiverse is merely a method of labeling each mathematical structure; it tells us nothing about how these structures are related to one another in the multiverse, aside from indicating the relative length of the programs that are required to encode each of the structures. (It is akin to assigning a number to each belief in the political spectrum. While some numbers are closer to each other than others, this method does not actually explain the relationships between the various political beliefs.) 

Tegmark, then, seems to suggest that there is no single system that relates all the mathematical structures of the Level IV multiverse. At best, an individual structure may bear some relation to a select subset of the others. In other words, the Level IV multiverse is merely a collection of structures; it lacks a broader meta-structure that would impose a set of relationships between each of the structures. By virtue of this fact, Level IV stands in contrast to Level I, which is characterized by the meta-structure of spacetime. Everything in Level I is related to every other thing by a particular distance in space and time. No two phenomena lack a spatial or temporal relation to one another. To be more precise, the meta-structure of Level I is supplied, among other things, by a so-called metric, which captures a generalized notion of distance between any two points in spacetime. [3] Indeed, according to Wikipedia, the spacetime metric formulated by Einstein’s theory of general relativity “captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past.” Level III is also a metric space, as the distance between quantum wavefunctions in the Hilbert space is measured through a computation known as the “inner product.” [2] But Tegmark makes no indication that Level IV is a metric space, as it has no meta-structure. In fact, Tegmark states directly that metric spaces are just one type of mathematical structure within the broader Level IV multiverse. 

Is it a problem for Tegmark’s ontology that the Level IV multiverse does not define a metric? Tegmark would say no; Level IV does exist outside of space, according to mathematical Platonism, whereas any metric presupposes some notion of space. The absence of any formalized notion of meta-structure, however, does undermine Tegmark’s ontology (a metric is only one example of structure). This is true for two reasons: 1) Tegmark admits some degree of meta-structure in Level IV, even though there is no global structure that relates all of the mathematical objects to one another. For instance, as stated above, one structure can contain another. Since Level IV is purely mathematical in nature, then any meta-structure, even if it is only local, must also be formalized as a mathematical object. In other words, meta-relations like containment of one structure by another must themselves correspond to mathematical objects. But Tegmark does not clarify the mathematical nature of these meta-relations. 2) The notion of independently existing mathematical structures is undercut by the Buddhist metaphysics of co-dependent origination, which I have alluded to in past essays. Co-dependent origination claims that nothing has an independent essence; everything affects something else, and is affected by other things. In the case of our mathematical multiverse, co-dependent origination implies that no single mathematical structure can be the fundamental ground for all others, thereby existing independently; even seemingly rudimentary structures are dependent on some other phenomena. For instance, suppose I claimed that all of mathematics can be based in set theory, as many 20th-century mathematicians believed before they were disproven, and that the most fundamental structure is therefore the empty set, i.e. the set that contains no elements. (A set is defined as a container of elements.) Every other mathematical object can be built on the foundation of the empty set; everything else is an empty set that is populated with elements, or a union or intersection of such sets, and so on. But co-dependent origination would suggest that the notion of the empty set as fundamental is dependent on another (mathematical) phenomenon. Namely, we humans subscribe to this notion because of our natural predisposition as intelligent creatures to divide the world into interior and exterior phenomena. We are programmed to split the universe into “self” and “environment,” into what lies within us and what lies beyond us; after all, if we confused our own limbs for the limbs of a predator, we would not survive. Set theory is a direct product of this ingrained tendency; the set serves as a container of elements, which implicitly constructs a boundary between objects that are inside the set and those that are outside. (See my other post, “Category Theory and the Participatory Ontology”, for further explanation.)

This brief digression into cognitive science has highlighted the idea that there is no objectively fundamental mathematical object; what we view as fundamental is dependent on our own cognitive structures, which are themselves mathematical. Any true ontology would be incomplete if it were not to recognize this deep interrelationship between mathematical structures. Thus, the Level IV multiverse cannot be one of independently existing structures, dwelling separately in the Platonic ether; there must be a larger meta-structure that relates them to one another, and furthermore one that acknowledges the participatory role that intelligent beings have in determining the math of the universe.

In abstract math, the most generalized notion of structure is not a metric but rather a topological space. As Wikipedia says, a topological space “is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence.” (Note that the formulation of a topological space is rooted entirely in set theory, but as I will explain later, an ontology grounded in topological spaces does not presuppose that set theory is fundamental. See more later.) Continuity, connectedness, and convergence are all very basic forms of structure. Continuity refers to the idea that there is a continuously nested series of neighborhoods surrounding a point in a space; connectedness to the property that there are no divisions between points; and convergence to the fact that sequences of points can tend towards a limit. 

One other structural phenomenon that is relevant to our discussion of topological spaces is the “sheaf.” The sheaf essentially equips a space with a well-defined notion of what is local and what is global. Essentially, as one Redditor explains, the sheaf translates between local and global data such that global data determines its local counterpart, but not the other way around. More accurately, given global data, the interpretation of multiple local fragments of data should always yield the same result as the interpretation of a single, but larger, collection of local fragments. However, the interpretation of global data depends on the local data that is given; there can only be a unified global interpretation of fragments of local data if the overlaps between those segments all share the same interpretation.

With sheaves and topological spaces, one can construct a mathematical structure known as a topos. The significance of the structural properties of the topos may not seem apparent from the discussion above, but it turns out, for reasons beyond me, that they actually have tremendous mathematical power. Indeed, the topos could serve as the meta-structure of the Level IV multiverse because it defines a very generalized space in which each mathematical structure inhabits a different region. As mathematical physicist John Baez elaborates,

“The word “topos” means “place” in Greek. In algebraic geometry we are often interested not just in whether or not something is true, but in where it is true. For example, given two functions on a space, where are they equal? [The] topos … is roughly a category that serves as a place in which one can do mathematics. Ultimately, this led to a concept of truth that has a very general notion of “space” built into it!”

For instance, suppose you are a mathematician who wishes to study a particular set. “Then,” as Baez writes, “you want to work in the topos Set, where the objects are sets and the morphisms are functions.” Or “suppose you know the symmetry group of the universe, G. And suppose you only want to work with sets on which this symmetry group acts, and functions which are compatible with this group action. Then you want to work in the topos G-Set.”

In other words, the mathematician chooses the region of the topos that is most well-suited as a workspace for the problems that she is seeking to solve. There is no objectively fundamental mathematical structure, and hence no independently existing structure either, since a structure becomes fundamental only once it begins to serve as the foundation of a mathematician’s research: as the basic tool that is needed to carry out a proof, for instance. Thus, metaphysically, the topos relates mathematical structures to the mathematician, and mathematically, it creates a map that places individual structures in relation to each other, through the meta-structural devices of a topological space and a sheaf.


Endnotes

[1] Thank you to Mike Johnson, CEO of the Qualia Research Institute, for recommending that I read Our Mathematical Universe

[2] The inner product is a generalization of the dot product, which may be more familiar to readers.

[3] In Principia Qualia, Mike Johnson speculates that the state space of consciousness, i.e. the map of all experiential phenomena, is a metric space.

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