Of Mereology and Math

[M]atter is ultimately particulate. I assume that every material thing is composed of things that have no proper parts: “elementary particles” or “mereological atoms” or “metaphysical simples.” (Peter van Inwagen [12], p.

“Fundamental” refers to the foundations of something, or the basis on which other things rest (fundare = ‘to found’). Hence it often implies that something is being generated (built) from it, or being made to rest on it (i.e. reduced to it). There exists a dependence relation between less and more fundamental things that define ‘levels’ of reality. Fundamentalism is simply the view that there is a terminus: a unique final layer to the cake. As mentioned above, this is usually taken to be a domain of undecomposables, something like ultimate lego pieces, and so the relevant domain is that of mereology (concerned with the part-whole relation and composition).

What can be reduced (what has parts) is not fundamental according to this mindset. Hence, we can simply insert a variety of things into the schema ‘Can x be reduced?’ to tick off what is and isn’t fundamental (where x can be ‘water,’ ‘wardrobes,’ ‘waiters,’ and so on). If something can be reduced, then it is often asserted that that thing does not really exist (mere appearance versus reality)—less derogatory is to say that it is emergent, or scale-dependent. John Kemeny and Paul Oppenheim’s mid-century eliminativist-reduction account [7] would have us depose the reduced theory, in favour of the deeper, reducing theory (much as the atomists and Parmenides supposed the illusory, or conventional nature of what was derived from their fundamental ontologies)—thus, we might say: ‘I believe that Max Tegmark is really a bunch of excitations of quantum fields’; or, if we have read Tegmark’s book, ‘I believe that Max Tegmark is really a mathematical sub-structure in a multiverse of such structures.’ It is rare these days to find people espousing this radical eliminitivism.

The mereological account, of reduction to simples, is already in trouble in standard quantum field theory in which there is, strictly speaking, no “basic, elementary, eternally persisting, concrete, physical stuff” as such. As Rolf Hagedorn points out, Dirac’s discovery of anti-matter was the most decisive in understanding the nature of elementary particles. Before this, atoms were more or less Democritean: immutable and untransmutable. As he says, the fact that quantum field theory makes any particle a complex dynamical system (of virtual particles which comprise the ‘physical’ particle) implies that “A-TOMs are dead” ([6], p. 106). However, the main challenges come from complexity science.

Nobel prizes are routinely awarded for finding the smaller, simpler constituents of complex systems. Going deeper is tantamount to going smaller. This assump­tion (more fundamental = smaller = more basic = more important) will guide the expenditure of billions of dollars, and countless physicist-hours. Of course, there is a famous precedent here: the ill-fated superconducting supercollider, cancelled in 1987 (after 2 billion dollars had already be spent). A debate about ‘fundamentality' occurred between elementary particle physicist Steven Weinberg (on the necessity of reducing to the smallest to get to the fundamentals) and condensed matter physi­cist Philip Anderson (on the side of complexity as no less fundamental). As Max Dresden rightly notes, “most physicists would agree that among the sciences physics is surely the most fundamental discipline... [b]ut this unanimity disappears rapidly when different areas within physics are considered” ([5], p. 133). In his Dreams of a Final Theory, Steven Weinberg argues that the fact that the arrows of explanation seem to repeatedly converge on deeper more fundamental theories points to some final theory: the ultimate attractor for all explanatory arrows. But Anderson finds examples that violate this.

Anderson had already presented the case against what we might call ‘micro­imperialist fundamentalism' in 1972, in his paper “More is Different”—Anderson was explicitly arguing against that idea that “if everything obeys the same funda­mental laws, then the only scientists who are studying anything really fundamental are those working on those laws” ([1], p.

His examples were based on broken symmetry scenarios, and show how in certain limits one can only understand the systems through the emergent laws and through the emergent degrees of freedom obeying them. While not denying that their is some underlying basis in individual components, these are not (and cannot be) employed to do the work. Not even a perfect theory of the elementary constituents would enable us to deduce the goings on in these limits. Hence, such systems are irreducible in the sense that one cannot find a unique micro-grounding which would imply the properties and laws of the macro-level; yet there is no denial that the micro-level exists, nor that if it did not the macro-level would not exist. In this sense we often speak of the ‘autonomy of levels.' This situation is well established in the area of ‘effective field theories,' in which the idea of a ‘final theory' is dispensed with in favour of a more pragmatic vision of a tower of theories, with their own level­dependent ontologies and laws no less fundamental than any other—this amounts to a kind of theory-based version of the pluralism versus monism debate mentioned above.

The distinction between these two approaches to fundamentality (level-based ver­sus ultimate) can itself be couched in a further distinction based on the mathematical representations employed.

What Dirac called “the mathematical quality in Nature” has of course been recog­nized for millennia. In his Metaphysics Aristotle referred to the Pythagoreans' belief that the principles of mathematics are “the principles of everything there is” (Meta­physics, 1.5, 985b23-986a1). Of course, the mathematical quality tends to break down as we consider more everyday systems. It is well known that as complexity goes up, so must the likelihood of using numerical methods: the more complex a sys­tem is, the harder it is to describe through mathematical laws. Whether this pushes us to speak of mathematics itself as fundamental (since it is involved in both the complex cases and the elementary cases, though in very different ways) is a matter for further investigation (we briefly discuss it below). This leads to a (more sociolog­ical) speculation that the split in fundamentalisms (unique level versus autonomous levels) might be due to this difference in mathematical modelling employed, and in some deeper view of mathematics (Platonism versus anti-realism) that the members of the camps hold.

However, the fact that the ‘same' mathematics can be transferred from one system to another in cases of universality should give us pause for thought.

It is easiest to consider an example here. Consider heating iron to its critical temperature, so it demagnetizes, with its spins pointing any which way. At this phase, the correlations between its atoms (whether their spins are pointing in the same direction or not) are given by identical critical exponents as water at its critical point (where water's phases meet). This indicates that the critical exponents are independent of the microscopic details of the matter, so that the systems occupy the same universality class. Systems at critical points obey conformal symmetry: one can rescale in various ways and the system looks identical (i.e. it is a fractal). One can adopt the view that it is such symmetry that is doing the work in generating the properties of critical systems, just as it is the symmetries (e.g. U(1), SU(2), and SU(3)) that generate the physics of elementary particles. In this case, one can treat the Weinberg versus Anderson debate as a mistake, since the truly fundamental layer is the physical symmetries rather than whatever systems obey those symmetries.

Of course, we can, if we are that way inclined, push for further explanation, and demand to know ‘why these exponents?’ and ‘why these laws?’ But one can ask the same of so-called fundamental, elementary particles: why these properties and laws.

Philosopher David Lewis saw it as “a task of physics to provide an inventory of all the fundamental properties and relations that occur” ([8], p. 292). If this is reasonable, and it sure seems to be, then the fact that certain phenomena would not appear in that inventory if we based it purely on the most elementary level indicates that we need to expand our inventory, lest physics be incomplete—we might call this “constitutive incompleteness.” This should not be taken as meaning that a ‘theory of everything’ is an impossibility: it simply means that by looking only at reality’s lego we are probably not going to find it. Moreover, what this complexity/critical phenomena work showed is that order is just as crucial as the basic elements and, in many cases, is more important such that, contra Weinberg, the arrows of explanation must point to order not elements (or micro-details more generally).

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