This past term I’ve been teaching a capstone class in which students are supposed to write a longer paper on some topic that means a lot to them. It’s meant to be a culminating event for their undergraduate work in philosophy. The class is always a fun exchange of ideas in which I can just participate rather than lead. It’s unfortunate that the COVID-19 virus came along – for many more serious reasons, of course, but also because it meant our seminar meetings were cut short, and we didn’t get to continue having the fun we were having.

I decided this term to write my own “undergraduate thesis on a topic that means a lot to me”, and came up with the following treatise on the nature of reality. I may as well post it here!

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Structure, energy, reality

(from the ICERM website at Brown University)

In this brief essay I will advocate the view that reality is a collection of possible mathematical structures infused with energy. There are many important questions I will not answer, such as what determines the range of mathematical possibility, what energy is, how a possible structure comes to be infused with energy, or whether there are any mathematical structures not infused with energy, in some universe or other. These are vital questions, but I do not know the answers to them. Still, one has to start somewhere. To provide a clear account of my view, I will divide this essay into three sections: (1) math as form; (2) energy as matter; and (3) the differences levels of interpretation make.

1. Math as form
Aristotle was right to think of substances as form united with matter, and right again to think of form as the more important of the two. When we seek to explain natural things or events, we always must turn to the form or structure of the things or events. When some atoms combine to form a molecule, what really matters are the structures of those atoms and their valences; the brute matter composing the atoms does not enter into the explanation, except as that matter is represented through structural and electrical properties. In this sense, materialism, understood literally as the view that everything is composed of matter, is false. If it were true, one would never be able to explain anything. There must be form as well as matter, and in any explanation, form matters more.

It might seem like form is not as real as matter, because form is usually not available to our senses except by being present in matter. We never see sphericality, but we see billiard balls and planets, and we might think of sphericality as a property that depends for its existence on some material substrate that has greater ontological weight. But this is an illusion that comes from the ways we are taught to talk about our sensory experience. When we become more serious about reality, we learn to talk about objects in the world independently of how they appear to us, and we try to talk about the world as it is in itself. As we do so, we begin to speak exclusively of the formal properties of objects: their structures, how they move, what other powers or properties they have, the range of ways they can affect other objects, and so on. The language of science is a language ranging over nature’s formal properties, and learning to speak it means leaving behind the ordinary supposition that material substrates are more important than the forms or structures they have.

Galileo rightly claimed that the book of nature is written in the language of mathematics. The central task of science is to try to get at the real properties things have, as opposed to the properties they may seem to us to have. So we learn that smells, colors, sounds, and tastes are merely subjective, and they are the effects of causes very dissimilar to them, such as shapes and motions. These shapes and motions can then be understood in austere, formulaic ways, to be sure we are divorcing our claims about them from any subjective biases. And thus, as we attempt to describe the world in itself apart from human sensory biases, the more we incline toward mathematical descriptions. We can call this claim the “nature is math” claim: the more that we try to isolate the real properties of nature, the more we are drawn to mathematical descriptions of those properties, because when a structure or property is described in close-to-pure mathematical form, then we can be relatively sure that we are leaving behind the biases and presuppositions suggested by our own sensory experiences, and getting at the features that are really in the world. Max Tegmark, in his book Our Mathematical Universe (2014), calls this “reducing the baggage allowance”, and connects it to the idea that our account of reality should not betray that fact that it is coming out of the mouths of human beings : “If we assume that reality exists independently of humans, then for a description to be complete, it must also be well defined according to nonhuman entities – aliens or supercomputers, say – that lack any understanding of human concepts” (255). The book of nature should be available to all audiences.

Tegmark follows Galileo and argues for “The Mathematical Universe Hypothesis (MUH)”, which states: “Our external physical reality is a mathematical structure” (Tegmark, p. 254). I prefer to remain agnostic about whether there is an entity to be called “the universe”, and about whether, if there is, it should have a single mathematical structure. It is possible that pluralism is true, and our reality consists of some number of mathematical structures which do not form some enduring, complete whole. But I agree with Tegmark and Galileo in the more general claim that our reality is fundamentally best described in mathematical terms, and that the real things in our world are mathematical structures that have been enlivened or infused with energy (more about that in the next section).

I should say a little more about mathematical structures. The most obvious thing about these structures is that they are mathematical, of course. But what does this mean? It would be very nice if we could articulate what it is to be mathematical in some short list of definitions and axioms, as if mathematics were a game defined by a finite rule set. But, for better and worse, the universe of mathematics is far richer than any finite rule set can determine, as Gödel proved in 1930. The question, “What makes mathematics true?” has been debated ever since. The view I incline to is called “structural realism” or more specifically “ante rem structural realism”. According to this view, all of the truths of mathematics, including the borders of what is and is not mathematically intelligible, arise from truths about mathematical structures. The “ante rem” part (which means “before the thing”) underscores the claim that these mathematical structures are more basic than any actual objects we may encounter. To dramatize the view, we could say that “In the Beginning” God establishes a bunch of facts about mathematical structures, heeding laws of logic and some further rules and inclinations we cannot possibly give an account of. In this way the mathematical facts, all of them, including the facts about the things about which there are no facts, are established. (I bring in God here not because I think we need to, but only to highlight the claim that facts about mathematical structure are fundamental.)

Mathematical structures run deep and are not always obvious on the surface. For example, consider a game called “Game of Fifteen” in which the integers 1-9 are laid out before us, and we take turns trying to be the first with a set of three numbers that add up to 15. You take 7; I take 8; you take 3, and I take 5, and so on. If we study this game, we will discover that there are in fact eight ways to get to 15 by adding together three numbers, such as 8, 1, and 6, or 4, 9, and 2, or …. We might even display the eight ways of reaching 15 with three numbers in the following table:

Every row, and every column, and each diagonal, sums to 15. So our “Game of Fifteen” turns out to be the same as Tic-Tac-Toe. In other words, Game of Fifteen and Tic-Tac-Toe have the very same mathematical structure; they are the same game, expressed in two different ways. Those two seemingly different games are different only because of the ways they are expressed; the underlying logic or structure is the same. Mathematics offers a neutral way of describing that structure, and that way describes the structure inherent to these seemingly different games.

If something like the Mathematical Universe Hypothesis is true, then the universe is ultimately one (or more) such mathematical structures, dressed up in various ways, according to our experience. The most fundamental job of science – and particularly, the job of physics – is to discover those structures, and recognize any non-mathematical differences as purely superficial.

2. Energy as matter
There is a difference between mathematics and physics. Mathematics is the study of pure form. The only constraint upon the mathematician is the constraint of what forms and structures are mathematically possible (whatever that means). The mathematician need not worry about how big or complicated they are, or whether they could ever be encountered in our reality. The physicist, by contrast, has further constraints. They are also interested in mathematical structures, but only the ones that “fit” in our universe, or the ones that best model what we encounter in our experience. For, it seems, not all mathematical structures are equally realized: not all of them can make it into what we experience as reality.

What is the difference between a structure as explored by a mathematician and the same structure as explored by the physicist? What is the difference between a pure triangle and one formed by three orbiting satellites? The difference is that the first is ideal, and the second is real. What does that mean? It means that there is no energy attached to the first: it cannot be detected or measured by the effects it has on anything. The second has some degree of energy: it has effects, it can be measured, and the measurements we make will always have only some degree of accuracy, and can never be infinitely precise.

This is, in a sense, the biggest difference there can possibly be between two things. It is the difference between existing as an idea and existing as a real thing. But it is nearly impossible to say much about this difference, or about what “energy” is or how it magically transforms an idea into reality. We can define energy as the capacity to do work, but we are thereby presuming we know what work is; and work presumes that there are things offering some form of resistance. I do not know what to say about energy, other than that it is what makes something real as opposed to ideal.

Is it possible to deny this distinction between reality and ideality? The most promising way of doing so, it seems to me, is to make all idealities realized; in other words, to insist that every possibility is an actuality in some possible universe or other. In this view, when mathematicians are busy thinking about structures too big or complicated to exist in our world, they are thinking of structures existing in other possible worlds. In this case, “ideal” just means: not in this world. But there are problems with this view, especially when I try to integrate it with my “everything is structure” thesis. If everything is ultimately the structure it has, then there is no way to have different objects in different possible universes sharing the same structure. An equilateral triangle in one world would be strictly identical with all equilateral triangles in all worlds, since so far as structure goes, there is only one equilateral triangle. But it is hard to see how one object can exist in more than one universe. We might try to argue that we can have multiple equilateral triangles in different universes in virtue of the other things in those universes: so one equilateral triangle shares a universe with Aristotle, another with Archimedes, etc., and that is what makes the equilateral triangles distinct from one another. But would that mean that, for example, as soon as Aristotle or Archimedes dies, the triangles become different entities? Can relations to other things make a thing what it is? But then how can that square with the “everything is structure” thesis?

So, in all, I would rather insist upon a distinction between reality and ideality, even if I can’t properly explain that distinction. Additionally, it just seems to me there is indeed a basic difference between ideality and reality. The difference is as real as every moment of my experience, which conveys to me an actual sensation rather than a merely possible one. To live in a world and have experience is to affirm the reality of energy, and the difference it makes. If there were no dividing line between the real and the ideal, it seems very difficult to explain why (what seems like) reality would seem so much more powerful than any possible alternative.

A further reason for distinguishing ideality from reality is the existence of time. It seems like time exists. If all were ideal, it is not clear that time would have to exist. It does not factor into mathematics, except possibly as just one further dimension to quantify. Truths about triangles do not depend on the year or the minute. But since time does (apparently) present itself in our reality, there must be some difference between ideality and reality that explains why time should be present (unless time is an illusion).

3. Different levels of interpretation
The claim I have been advocating so far is that reality is structure plus energy. One need say nothing more, if one is concentrating entirely on the world of atoms and molecules. But, over time, there have evolved very complicated systems, living beings, that somehow manage to represent the environment to themselves, and even themselves to themselves. Such representations are interpretations. Once there are interpretations, what we take for reality becomes much more complicated, because the terms, properties, entities, and quantifiers employed in interpretations need not reflect the structures of reality. All that matters, so far as the system itself is concerned, is that the representations manage to secure some sort of success for the system.

In this way nearly everything that we spend our time thinking about and acting upon comes into existence: ideas, feelings, people, relationships, money, food, power, beauty, truth, risk, promise, hope, and on and on and on, without end. At the ultimate level, there is nothing that happens that cannot be explained in principle in terms of energy and structure. But at various levels of interpretation, what we seek to explain cannot be explained at such a basic level. If I promise to give you something tomorrow, and feel the pressure to make good on my promise, that phenomenon is best explained at the level of people, promises, obligation, and expectation – none of which are themselves best explained in terms of mathematical structure. Higher levels of interpretation emerge somehow out of basic structures that do not contain their terms. The “somehow” here should not imply that the emergence is miraculous. It means only that the explanation is not likely to do justice to the phenomena, at some level of interpretation. This is a fact not about ontology, but about the limits of explanation.

But it would be wrong to take the higher levels of interpretation as purely illusory, even if they do not align clearly with what ultimately exists. We exist as highly-evolved systems in high levels of interpretation: everything we think we are, and all that we think we experience, is at a high level of interpretation. If we seek to understand anything at these levels, we will have to make use of other elements at those levels. It is analogous to understanding what is going on in a film. The film is nothing but a rapid succession of images, or a changing distribution of pixels, conjoined with sound waves. But the plot of a film cannot be understood in such terms. To understand the plot, we must talk about characters and events and motivations, none of which are encoded in pixels or soundtracks. This does not mean there are no facts about the plot; in explaining it, we could get the plot right or get it wrong. But the facts do not exist at the same level as the images, pixels, or soundtrack.

Although I am describing different “levels” of phenomena, it is not a well-organized structure. It is a mess. Interpretive languages are extremely flexible and filled with metaphors, analogies, and powers of representation that resist static structures. As stated earlier, what matters for the system employing this language is not “getting things right” in the sense of capturing an actual order of things, but “getting things done” in the sense of achieving the aims of the system. Interpretations exist so that systems can generate behavior that will be to some extent successful for them. There are many errors that can be made, but systems that manage to exist (and reproduce) are systems that manage to gain enough practical success toward those ends to be around to talk about it. For those of us who exist within these many levels of interpretation, we do not think about our lack of tethering to the ultimate foundation of reality; that is to say, we do not worry about worrying about the great many things other than structures and energy. We simply live our lives, as oblivious to the levels of interpretation as fish are oblivious to water.

Three examples may help to illustrate how much difference levels of interpretation make. First, consider Anselm as he decides whether to pick a flower. He weighs reasons for doing so (the pleasure of carrying it along with him, how pretty it would look in his room) against reasons for not picking it (it will remain undisturbed, show its beauty to others, and continue to adorn the footpath). In the end, he decides to leave it be. The weighing of reasons, the exploration of his range of actions and their effects on others, are all very real considerations, and if he acts on the basis of his reasons, we hold him accountable for what he does, and worthy of praise or blame. But not even a tiny segment of these considerations has any representation whatsoever in his neurons, let alone in the mathematical structures describing their operations, or the behavior of the electrons composing them. The considerations exist because of a larger social context in which Anselm operates, one which knows how to identify reasons and offer general guidance in weighing them against one another. It is in this social tradition that Anselm is raised and taught to think and speak and act. Notice that this social tradition developed without any attention to the metaphysics of structure and energy. It developed out of previous interpretations, cultures, traditions, and ideas – all of which are somehow generated from the ultimate metaphysics, without being wholly explicable by reality at that fundamental level.

Secondly, consider the flower under the threat of being picked by Anselm. What sort of flower is it? What are its parts? Does it have distinctive colors and smells? Is it a weed? Is it native to the landscape? Answering any of these questions requires complex interpretation. We shall have to identify the flower from its surrounding air and soil, describe the various functions of its various parts, the history of the landscape and the desires of local gardeners, and so on. In Tegmark’s terms, there is a lot of baggage that comes with these questions, and there is a wide opportunity for human ideals to shape what we think we are asking and how we are expected to answer. To put the matter very pointedly, none of the questions we raised can be asked in mathematics or physics. They can be asked and answered very sensibly in botany and ecology. But this does not mean that botany and ecology are somehow illusory or fictional. They simply operate at a different interpretive level of experience, and different levels of explanation.

As a third example, consider Anselm as he decides how he shall pray to God. Again, as in the case of the flower, he weighs reasons against one another, together perhaps with his own unnameable feelings of what is appropriate. These considerations are also rooted in broader contexts and traditions, which are shaped by larger interpretive influences that are not connected to any ultimate metaphysics. And if Gaunilo comes along and calls Anselm a fool for praying, Gaunilo is also acting within a rhetorical context, offering objections and arguments, and insisting that Anselm engage with him in a critical discourse. Some objections hit their mark, others are deftly answered, according to a robust set of interpretive traditions and ways of reasoning about human experience and the language we employ. It is, in other words, a giant complicated mess of interpretive levels, culture and traditions, language, history, and psychology.

An interesting consequence of this giant interpretive mess is that, from the standpoint of ultimate metaphysics (that is to say, structure and energy), all of our interpretive talk is equally real or well-grounded, so long as that talk is well-behaved in the appropriate context. Some people talk about physical forces, some about sprites and fairies, some about God and angels, some about healing waves and crystals, and so on. It is all equally rubbish, or all equally valid, so long as it operates within contextual parameters. Even when we engage in metaphysics, as this short essay does, we are laying down terms and views and beliefs, all at various levels of interpretation that are far removed from what ultimately exists. At all these various levels of interpretation – levels other than what I am calling “ultimate metaphysics” – we can, within those levels, identify what is plausible, what is possible, what is more likely, what is responsible, what is irresponsible, and so on. We can praise and criticize one another for what we say, do, and believe, and we can do so for better and worse reasons. But all of this exists only at high levels of interpretation. In other words, what we say matters only because we think it matters. Reality, as mere structure and energy, pays us no attention. Indeed, in its eyes, it is not even clear that we exist, since as conscious, social beings, we inhabit the upper rafters of our own interpretive cathedral.

Conclusion
This short essay has been an attempt to make good on a simple claim: that reality is structure plus energy. I have noted several questions along the way that have not been answered, and are worth thinking about further. The final section is only the briefest sketch of all of the phenomena that animate our conscious lives, so clearly there is much more to say about that as well. But in the end I believe this metaphysics squares best with what we understand our universe to be, and what we understand ourselves to be.