What does our experience have to do with mathematics?
This is something of a follow-up to my last post. I checked out a series of links related to Max Tegmark in the last few days, having heard about the release of his first book Our Mathematical Universe. But I was also motivated by having observed that the latest conference organized by the Foundational Questions Institute (for which Tegmark is one of the directors) included prominent neuroscientists, Christof Koch and Giulio Tononi. This is not the first time a FQXi conference has included neuroscientists among their list of speakers. There are a series of threads that one can follow through Tegmark’s and Tononi’s work, but I would like to make a particular observation. Tegmark’s thesis in Our Mathematical Universe, and Tononi’s strategy in his 2008 paper on ‘Consciousness as Integrated Information,’ each rely on the significance of pure ‘relations,’ in how we analyze our experience as well as in how our experience is produced.
Tegmark has been arguing that the universe itself is a mathematical object or structure. His book is a full treatment of this idea. One of the keys to his defense of this idea is the claim that, as theories in physics have developed, their content has become more and more purely relational. In a 2007 paper that preceded the recent book, Tegmark explains that all of the physical theories that have been produced thus far have two components: mathematical equations and what he calls “baggage,” or the words that we give to the relations when we describe them.
However, could it ever be possible to give a description of the external reality involving no baggage? If so, our description of entities in the external reality and relations between them would have to be completely abstract, forcing any words or other symbols used to denote them to be mere labels with no preconceived meanings whatsoever.
A mathematical structure is precisely this: abstract entities with relations between them.
He then says later:
In other words, our successful theories are not mathematics approximating physics, but mathematics approximating mathematics.
With a more recent paper (Jan. 2014), Tegmark takes on the nature of consciousness. In Consciousness as a State of Matter, he brings principles of physics into a discussion of consciousness. He proposes the possibility that consciousness can be understood as a state of matter, like the states of matter we call a liquid, a solid and a gas and then begins an analysis of the properties that such a state of matter would have. When enormous numbers of particles are brought together, he explains, new and interesting emergent phenomena begin to happen. And while there are a large number of kinds of gasses, there is an independent substrate that they all share. These kinds of ideas can be brought to an analysis of the states of matter that define consciousness as well. One of the properties of memory, for example, is that it has many long-lived stable states. It also has dynamic properties. So the question becomes, can one take the ideas in neuroscience and use them to say something interesting about the physical world? Why do we perceive ourselves, for example, as living in a 3-dimensional space with a hierarchy of objects? How do we get there from the fundamental properties of matter described by modern physics?
Using some of the mathematics that describes physical systems, Tegmark tries to find the way that our experience would emerge from (he actually says “pop out”) of the mathematics. He calls it the ‘physics from scratch problem.’ Tegmark’s paper means to extend Tononi’s work on consciousness to more general physical systems by using information theory and Tononi’s idea of integrated information. He is convinced that the problems of neuroscience and the problems of physics are very strongly linked.
Can a deeper understanding of consciousness breathe new life into the century-old quest to understand the emergence of a classical world from quantum mechanics, and can it even help explain how two Hermitean matrices H and ρ lead to the subjective emergence of time? The quests to better understand the internal reality our mind and the external reality of our universe will hopefully assist one another.
Tononi’s paper finds experience to be the mathematical shape given to integrated information. Information is defined as the reduction of uncertainty. And it is the discrimination among alternatives that generates information. Tononi proposes a way to characterize experience with a geometry that describes informational relationships. The integration of information produces a ‘shape’ in what he calls qualia space, and a particular shape is a particular experience. The ‘space’ is defined using a set of axes each labeled with probabilities related to the states of a system in the brain (like visual systems) and the interactions among elements in the system. When a large number of elements and connections are at play, the dimension of the quailia space far exceeds three. For example, four elements with nine connections among them is a simple system, but it produces a 16-dimensional space. About these shapes Tononi writes that they are
often morphing smoothly into another shape as new informational relationships are specified through its mechanisms entering new states. Of course, we cannot dream of visualizing such shapes as qualia diagrams (we have a hard time with shapes generated by three elements). And yet, from a different perspective, we see and hear such shapes all the time, from the inside, as it were, since such shapes are actually the stuff our dreams are made of— indeed the stuff all experience is made of.
And then there’s this bit of poetry in the paper:
If one accepts these premises, a useful way of thinking about consciousness as a fundamental property is as follows. We are by now used to considering the universe as a vast empty space that contains enormous conglomerations of mass, charge, and energy—giant bright entities (where brightness reflects energy or mass) from planets to stars to galaxies. In this view (that is, in terms of mass, charge, or energy), each of us constitutes an extremely small, dim portion of what exists—indeed, hardly more than a speck of dust.
However, if consciousness (i.e., integrated information) exists as a fundamental property, an equally valid view of the universe is this: a vast empty space that contains mostly nothing, and occasionally just specks of integrated information —mere dust, indeed—even there where the mass-charge–energy perspective reveals huge conglomerates. On the other hand, one small corner of the known universe contains a remarkable concentration of extremely bright entities (where brightness reflects high levels of integrated information), orders of magnitude brighter than anything around them. Each bright “star” is the main complex of an individual human being (and most likely, of individual animals). I argue that such a view is at least as valid as that of a universe dominated by mass, charge, and energy.
In a talk given by Tegmark for the “Philosophy of Cosmology” project, he makes the claim that perhaps physical existence and mathematical existence are the same. The view of mathematics proposed by Tegmark and supported by Tononi seem to reverse the embodiment ideas first presented by George Lakoff and Raphael Nunez in their book, Where Mathematics Comes From. The idea analysis in the Lakoff/Nunez book rests on the claim that mathematical concepts develop, through effective metaphors, from fairly simple, fundamental, physical experience. In Tegmark’s world, at least, the mathematics comes first.