Introduction

One of the topics that seems to come up from time to time on Lesswrong is the supernatural vs a naturalistic explanation for the universe. 

I felt that I could make a contribution to this discussion as a theoretical physicist (now moved into a different area), which actually came about from a discussion I had with a particular esteemed professor from the University of Melbourne, in 2012. Prior to this moment, I had slowly (perhaps nascently) been developing a metaphysical philosophy that was a little closer to Functionalism than Physicalism (these two metaphysics are compatible with one another), but since Functionalism was shown by Prof. Searle to be 'incomplete' in a certain sense that I won't go into right now, I continued my search as a side project, and I found that after much training both during Ph.D and in post-doctoral research in theoretical physics and pure mathematics, I found a 'bug' (or a 'feature') worth expounding that continued on from my initial thoughts on metaphysics many years prior. What I arrived at was a kind of reverse-Epiphenomalism, which I will explore below. 

The 'bug' (or 'feature') is that while I found many scientists will typically attest to a kind of physicalism or completely naturalistic philosophy by lip-service, when in conversation and in daily work in their profession within science (or mathematics) I found that there was a tacit working-ethos of something more akin to Mathematical Realism. I want to defend this philosophy a little bit, as I would currently consider myself to fall into that category (would love to hear others' view on this). What I mean is, when queried, many physicists and mathematicians would frame mathematical truths as actually true, 'real', that is, existing entities, and that there is a kind of Platonic 'math world', and in physics apply the process of Ansatz, where we apply a proposed mathematical object or principle and propose a theory, before beginning to query it and conduct experiments to see if we can falsify it. I found that a) this belief reminded me strongly of a talk by Richard Feynman where he explains the scientific enterprise as not really being so much a following of the scientific method per se (many great scientists have intuited great ideas prior to the experiment being conducted), but that it begins often with a kind of 'special guess' (Feynman's terminology). This special guess seems to come from some deep place, some mental insight arrived at through an unclear process, some inspiration, and then the guess is applied (as an Ansatz) and the theory constructed from that point. Furthermore, I was reminded of b) my uncle's former supervisor Roger Penrose a few years ago wrote Road to Reality: A Complete Guide to the Law of the Universe, and in it depicts a diagram (not reproduced here) that proposes a Platonic 'math world' and a 'real world' which follows a subset of these mathematical principles. Now I am aware of Penrose's position that he doesn't take this metaphysics as literally true, but I wondered why, and I decided to delve into it a little more, bear with me. 

Mathematical Realism

I did a more thorough (and better cited) defence of Mathematical Realism in the manuscript I will refer to (which is here: https://arxiv.org/abs/1306.2266 - also, I know the current arXiv version has a typo in the first line - it is fixed in Peer-Review) but the point I wish to make (which I might not convince you all on) is that, if you think of any mathematical object: a straight line, the number '3' (or any number) or any mathematical theory, there is a sense that these objects have a distinct existence, are 'provable' in a stricter sense than anything in science, specifically as the word 'proof' really has the format of an abuse notation in science - in science we infer, and collect evidence for or against things, but the word 'proof' really needs to be reserved for a mathematical procedure that is carried out (I know it has other definitions such as in copy-editing, or courts of law, but I am meaning only in the interface between science and mathematics in the context of Natural Philosophy). So, for example, if you were to have a map with a straight line marked on it, is it really there? It's notional of course. but, is the line on the page really straight? Not really, it's being compared to an ideal line that isn't in the universe per se. If you rubbed out, or altered the line, would you destroy the actual concept of the straight line for everyone? Of course not, and furthermore, the concept of the straight line has a kind of objective existence, and doesn't have the same properties as other objects in the universe - it doesn't have a 'location' for a start, and seems eternal, immutable, static, but also provable in principle from any future race or alien, from anywhere in the universe, if they decided to replicate a definition of a concept of a straight line. 

Now, I am keenly aware it can be argued (and often is) that these ideals can be thought of as a kind of 'fiction' agreed upon by humanity for a time, and this harmonises it with Naturalism. Because then, all these mathematical theorems are simply derivable from brain function. This has some troubling aspects as it has assumed structure being followed by brain function to start with, and assuming the thing to query doesn't lead to good explanations. Furthermore, one is then in difficulty if asked whether molecules and atoms in the primordial universe moved, according to what equation, and how they were organised (in mathematical groups, for instance)? So we actually get a clear solipsism if we go too far with that approach. If atoms in the primordial universe can't have followed any mathematical principles in their motion, then it's all back-generated from our minds somehow. It's certainly an irrefutable stance (I can't disprove it), but most practicing physicists would kind of believe in a more straight forward 'there is absolutely truth' and 'the universe existed before we did' approach. That is, the atoms and particles in the early universe did undergo processes and fall into groups, and so forth, following mathematical principles. This is the stance that I am taking in this work but would love to hear feedback on missed alternatives. (For the Jordan Peterson followers out there, this view could be seen as an extension of Judeo-Christian ethos growing out of monastic tradition that learning, books, education and universities are important institutions built on a philosophical disposition that the universe is inherently reasonable and queryable. That also, is quite an assumption and we have no ability to prove it to be the case - but it has worked so far, as Dawkins has historically pointed out - though I differ on Dawkins that the philosophical underpinnings of science are scientific statements, they are in fact philosophical statements, so we need to assume this ethos to begin with prior to conducting science). 

In that sense, the view I am taking is that mathematical principles are prior (they don't have a constraint in time or location and are timeless), and that any physical interaction between two objects in the universe are governed by some principle - the 'interaction' itself is an abstraction, and so, the universe can't really flow, evolve or function without the intersection of abstractions obtruding, and so phenomena are only linked via abstractions (and abstractions with themselves) but no two physical objects can be linked or related in any way without first defining an abstract entity called a 'relationship' - hence, a 'reverse' version of Epiphenomenalism.

Developing a formalism to be able to explore the process of Ansatz

Flash forward to my 2012 meeting with a particular professor (of String Theory, for reference), and we were discussing, lo-and-beyond, multiple universes. We were at a workshop for particle accelerator experimentation - the discussion was whether there would be any practical measurement in principle to show if there was anything outside of the universe. The typical position of many would be to dismiss it as impossible in-principle, because it could be assumed you could 'draw a box' around anything found, and define it as being in the universe. (In this post, I will refer to this work I put together afterwards that disproves this mathematically). But actually, this professor related an interesting anecdote. 

The anecdote: let us imagine that life developed on earth, not when it did, but many millions of years later or more. One observation would be there would be no starlight. The universe would have expanded to such a degree that no stars would be visible except the sun, in fact, we would not (until telescopes were invented) even have a concept of a star, and we would only know the sun (moon) and an empty sky. A natural question for an astronomer in this fictional world, would be, that the sun seems fundamental to the universe and we would like to calculate the distance from the earth to the sun. Unfortunately, no matter what we do, we cannot derive from first principles this distance. The reason? Well, it isn't fundamental, it's kind of an arbitrary value, which works for us, and other earth creatures, to live at this comfortable temperature range, etc. So we are left with a situation where a person desires to find a fundamental physical answer to a physical phenomenon but cannot from first principles derive it from some mathematical principle. Now, consider the situation where the subatomic particles in our universe have certain masses, electric charges, and other properties. After half a century or more of trying, we haven't got a method for deriving these values from scratch, but they need to be assumed from hand, as being values arrived at stochastically from some dynamical process (dynamic chiral symmetry breaking, or some other symmetry of the Standard Model, for this model, which is electroweak theory + QCD, is, as it has been known since Weinberg pointed it out, an Effective Field Theory). String Theorists (and other proposed methods of unifying General Relativity and the Standard Model) might propose a solution for generating these values mathematically - but, this is still speculative, as there is no firm evidence if (and which) of these theories are correct. (String Theory is not a single theory but covers an extremely broad range of parameters, and string phenomenologists will generally approach their art through methods of some kind of proposed compactification of dimensions to try to "get" an Ansatz for the universe but this as so far not been that successful). 

After thinking over the professor's thoughts, I felt inspired to try to put some philosophical thinking down on paper to try to see if we could 'prove' multiple universes as arising from some stochastic process of abstract objects with a large configuration space - noting as a guess, the symmetry between the large and small scale Pareto distributions: of all the sperm, only a few lead to a foetus, of all the planets only a few seem habitable, of all the wealth, only a few have most of it, of all the space in the universe, only a small portion has a high concentration of mass (galaxies) - is it so much of an extension to imagine the same could apply to universes? 

So, I set out to do a couple of things: 

• Write down a mathematical formalism for the process of an Ansatz so we can explore it, probe it, and under that formalism (at least!) make some conjectures about it;
• Explore some simple properties of the large parameter space of mathematical objects and the constraint that only a few of them are 'followed' as mathematical principles that guide nature (ie how particles move, etc. not all theories are true!);
• Make it as general as possible so as not to add superfluous structure that could add artefacts that would distort the discoveries that could be made;
• As a key first step, I noted that any mathematical formalism of non-mathematical objects (physical objects) will inevitably lead to a 'nesting' structure (I called it the Labelling Principle), where a reference of an object is already a 'pointer' to 'what is meant' by the pointer, and what is meant by 'what is meant by' a pointer is also a pointer to the same object, etc. Of course, any physical object, in order to be referred to, needs to have a mathematical representation from within the mathematical framework being defined and there's no way around that. 
• I then had to separate these two notions of existence. Mathematical objects that 'exist' in a real, or provable sense (I prove a theorem, for example) aren't exhibiting the same kind of existence as 'somewhere in the world a black swan exists'. It's similar to how the word 'causality' can mean two distinct things: a physical property to do with the Minkowskian geometry of the universe ('cause-and-effect'), and the 'ground-consequence' structure of a logical argument. To distinguish these two concepts existence, I call the mathematical one 'exist' and the physical one in the universe extant.
• Then, I equated existence (in a mathematical sense) of objects in the formalism, to only occur for objects if they are extant, that is, I could query whether observed, extant phenomena constituted evidence for the extantness of a thing but setting up the formalism so that the mathematical theorem would be inconsistent (invalidated, and disproved by contradiction) so I could kind of model the process of an Ansatz and see it happen on paper. This was the main major constraint I added to the theory which otherwise isn't too much more structured than good ol' fashion Zermelo–Fraenkel theory with an axiom of choice (ZFC). 
• One of the first interesting discoveries was that this led to a very natural explanation of Eugene Wigner's 'Unreasonable Effectiveness'. It simply falls out of the theory from a simple cardinality argument. 
• Speaking of cardinality, I drew on Georg Cantor's universality paradox that gave sudden, crucial insight on how there might be multiple universes. 
• First, I indeed needed to add some (reasonable) very general definitions of what a universe 'is' (in the Tegmark classification sense), and adopted a pose that the universe should at least be definable and practically usable - I am a physicist afterall - and also I needed to supply some clarification of  what is 'evidence' in the context of this formalism.
• Then, what I found is something very interesting, one is able to prove, using the theory and using a Cantor argument, that inclusion of mathematical principles leads to a Universal Set which can't be consistently defined (a real problem for practising physicists) and so it is a proof by contradiction. In other words, If you want to define a universe in a consistent manner, it has to leave out certain objects. And these objects are the mathematical principles. So it's fairly neat. 

Putting all this together, as we talk of religion and spirituality, and taking the stance of a practising science where we are drawing on a philosophy that truth is important, we need to seek truth and that the universe is reasonable and queryable so that science can be done on it (a Judeo-Christian-derived view) I thought it worthwhile to show how this is kind of incompatible (or at least, not straightforwardly compatible) with Naturalism. And once the door is open, many other things outside the universe could be considered. 

But I've been thinking of this for about 10 years, and then spent another 10 years developing it and thinking about it to myself (the actual writing down of this article was 1 year only), and I have struggled to get some real feedback on this proposal so I need to get some Lesswrong peer review happening.

Why now?

With recent works such as The Master and His Emissary, and the more ground-breaking works of Prof. Iain McGilchrist, together with the views on what is real, (what is meta-real) in deep narratives espoused by Jordan Peterson, it feels like people are coming to this particular same thought from different directions, this Mathematical Realism view that I had already held for a while, and can argue quite effectively for it. Now, when 2-3 fields (neurology/psychiatry, psychology, physics) converge on a similar thinking pattern, I think it's worth taking notice as that doesn't usually happen and it might indicate that we are onto something here. Therefore, I wanted to place here the work (it is currently going through Peer-Review) but is up as a pre-print: 

On the existence of other universes https://arxiv.org/abs/1306.2266.

I wished to print some of the main theorems here in this post, but without knowing how to get a tex plugin, the typesetting doesn't come across properly, so I will leave it here, with some concluding summaries regarding it:

Abstract

Natural philosophy necessarily combines the process of scientific observation with an abstract (and usually symbolic) framework, which provides a logical structure to the practice and development of a scientific theory. The metaphysical underpinning of science includes statements about the process of science itself, and the nature of both the philosophical and material objects involved in a scientific investigation. By developing a formalism for an abstract mathematical description of inherently non-mathematical (physical) objects, an attempt is made to clarify the mechanisms and implications of the philosophical tool of Ansatz. Outcomes of this style of analysis include a possible explanation for the philosophical issue of the ‘unreasonable effectiveness’ of mathematics as raised by Wigner, and an investigation into formal definitions of the terms: ‘principles’, ‘evidence’, ‘existence’ and ‘universes’, that are consistent with the conventions used in physics.

It is found that the formalism places restrictions on the mathematical properties of objects that represent the tools and terms mentioned above. This allows one to make testable predictions regarding physics itself (where the nature of the tools of investigation is now entirely abstract) just as scientific theories make predictions about the universe at hand. That is, the mathematical structure of objects defined within the new formalism has philosophical consequences (via logical arguments) that lead to profound insights into the nature of the universe, which may serve to guide the course of future investigations in science and philosophy, and precipitate inspiring new avenues of integrated research.

Summary

To summarise and encapsulate the final thesis and import of this work that will follow from the presentation of the formalism, we will find there is no logically consistent way to define a universe that includes all abstractions (such as mathematical objects), and that such a claim need only rely on logic to prove it. That is, objects outside the present universe can be proved to exist using a self-consistency definition, and that constitutes the only evidence that would be allowed in this case.

An attempt was made to classify other universes in a general fashion, and to clarify the characteristics and role of evidence for theories that provide at least a partial description of a universe. The connection between phenomena that constitute evidence and the theory itself was established in a proposed Duality Theorem. Instead of focusing on attempting an ad- hoc identification of extra-universal phenomena from experiment, the formalism was used to derive basic properties of objects that do not align with our universe. As a first example toward such a goal, a fundamental object was identified, which satisfies the necessary properties for evidence, and whose extantness does not coincide with our universe. This paves the way for future investigations into more precise details of the properties of objects and methods amenable to this type of formal inquiry.

Note that this proof demonstrates that, in principle, there are objects that exist in a universe different from our own, due to the constraints of logical consistency. The nature of the test-case object is quite rudimentary, but it serves as an initial example. The formalism as a whole, though, has been especially fruitful in producing the unanticipated results of immediate relevance to the scientific community; in particular, a possible explanation for the ‘unreasonable effectiveness’ of mathematics is put forward. This represents an important development in the philosophical understanding of physics. Furthermore, the clarification of Ansatz: the process of applying an hypothesis to the physical world, and then testing it against experiment, represents the main achievement of the work. The introduction of a framework within which one can interrogate the nature of how hypotheses are applied represents a long-term ambition, seemingly missing from the current literature, which future research can develop and refine.