Friday, April 22, 2016

Prelude to Mathematical Neo-Platonism

"Is not being, then, unified number,
and beings number unfolded,
and Intellect number moving in itself,
and the living creature number
embracing everything?" (Plotinus)

This post argues for a Mathematical Neo-Platonism (MNP), where a transcendent source – analogous to the One in historical Neo-Platonism (NP) – is seen as generating the Platonic reality of mathematics, which in turn generates the physical universe in which we find ourselves. First I will discuss some interesting parallels between NP and Zermelo-Fraenkel set theory (including the axiom of Choice, together abbreviated as "ZFC"). Subsequently I will discuss the consequences of Mathematical Monism (MM) in light of the reduction of mathematics to ZFC. MM is the claim that all of reality – including physical reality – consists entirely of mathematical structures. According to physicists like Max Tegmark, MM follows from the success of modern mathematical physics, since the latter describes physical reality entirely in mathematical terms. I will argue that MM leads to MNP when we take into account the reduction of math to ZFC, where the existence of the empty set, designated by "", and a small number of other axioms are sufficient to generate the entire universe of pure sets in which the bulk of mathematics fits. Thus, given MM and the reduction of math to ZFC, Leibniz's famous question "Why is there something rather than nothing?" reduces to: Why does exist? And why do the axioms of ZFC hold? As I will argue in my next post, it is only from a Neo-Platonic perspective that these questions become fully answerable.

The analogy between set theory and Neo-Platonism

Is there any substance to MNP? Or is it no more than a pretentious sounding but ultimately empty combination of words, a mere flatus vocis? Very interesting in this regard is the remarkable role played by axiomatic set theory in contemporary mathematics. As most mathematicians nowadays recognize, axiomatic set theory functions as the foundation for virtually all of mathematics – and some mathematicians would go even further than this, e.g. John Mayberry: "set theory is not really, or not just, a foundation for mathematics. It simply is modern mathematics." (1988: 353) This privileged role played by axiomatic set theory holds in particular for ZFC, which is standardly used in mathematics and mathematical logic. As I will show in the following, ZFC reproduces surprisingly many of the conceptual structures characteristic of NP, notably its hierarchical universe deriving from a single and indeterminate source. Now, suppose that this analogy between ZFC and NP holds up under closer analysis. Wouldn't we then be justified in concluding that ZFC = MNP, since ZFC reproduces NP in the context of mathematics? Let's see how far this analogy goes.

As already noted, ZFC functions as the standardly used axiomatic foundation for virtually all of mathematics. It is important to keep in mind just how remarkable this intellectual achievement is! Especially when you realize that ZFC requires only one existential axiom, namely the existence
, and a small number of other axioms stating which operations may be performed on sets in general, in order to generate an endless "cumulative hierarchy" of pure sets (i.e. sets containing nothing but sets), starting from as the hierarchy's sole existential basis. Amazingly, virtually all of mathematics can be located somewhere in this hierarchy (commonly called V). As the mathematician Enderton explains: "[M]athematical objects (such as numbers and differentiable functions) can be defined to be certain sets. And the theorems of mathematics (such as the fundamental theorem of calculus) then can be viewed as statements about sets. Furthermore, these theorems will be provable from our axioms [i.e. the axioms of ZFC, PS]. Hence our axioms provide a sufficient collection of assumptions for the development of the whole of mathematics – a remarkable fact." (1977: 11) Remarkable indeed!

The hierarchical universes of NP and ZFC
To what extent does ZFC reproduce the core ideas of NP? As already noted, both NP and ZFC present a hierarchically structured universe. Whereas ZFC gives us the set-theoretical hierarchy V deriving from
, NP gives us the metaphysical hierarchy One Intellect Soul Nature (where "" stands for emanation). And this analogy is all the more apt insofar as the Neo-Platonic hierarchy is, like the set-theoretic hierarchy, a graded unfolding of increasing multiplicity. For Plotinus, the One is an utterly undifferentiated unity, which generates the plurality-in-unity of the Intellect, which in turn produces the still more complex multitudes of Soul and Nature, finally terminating in the utter chaos of unordered Matter. Likewise in ZFC, where the hierarchy V starts on the 0th level (called "V0") with the utterly simple unity of , from there on generating ever higher levels of complexity, such that V1={}, V2={, {}}, V3={, {}, {{}}, {, {}}}, and so on ad infinitum. Thus "in set theory one is always climbing upward" (Devlin 1993: 47). And the higher one goes in this endless hierarchy, the more complexity one encounters, until finally sets are generated which are so mind-bogglingly huge that they escape mathematical understanding altogether. True, the limit of what mathematicians do understand about V is constantly being raised higher, thanks to continuing mathematical research. However, since V is literally endless, it is clear that some limit will always remain for us finite human beings: the limit may be eternally moving upwards, but beyond it will always remain utterly incomprehensible multiplicity. It is tempting to see this as analogous to the way in which the Plotinian hierarchy ultimately dissipates in the incomprehensibility of unordered Matter.

The 'empty unity' at the beginning
It is also at their starting points, however, that the set-theoretic hierarchy V and the metaphysical hierarchy of NP are surprisingly similar. For NP, this starting point is the utterly undifferentiated One. For V, this starting point is
, the empty set. The analogy between them is obvious: both and the Plotinian One are, in a sense, 'empty unities'. Let's take a closer look at this. Exactly why does Plotinus say that the One is an undifferentiated and therefore 'empty' unity? This is actually a very complicated question, but for now the simple answer must do. Plotinus reasons roughly as follows: if the One were a definite something, i.e. if it had well-defined properties, then its properties would in a sense limit its nature (since by having these properties it would not have the contrary properties), and thereby its power to produce would be limited as well, so it would no longer be omnipotent. Thus, in order to be the cause of Everything, the One itself must be (or rather: contain) Nothing. As Plotinus writes about the One: "It is because there is nothing in it that all things come from it." (Enneads, V.2.1.1-5) But, as we have seen, in ZFC the starting point of V is likewise a kind of 'unified nothing', namely, , the set which collects nothing. In a sense, then, could be seen as the set-theoretic 'equivalent' of the Plotinian One. The crucial question is obviously what this 'equivalence' is worth. Is it no more than a coincidental analogy? Or does it rather reveal a substantial, doctrinal agreement between NP and ZFC?

Before dealing with this question, however, it is interesting to note that this analogy between
and the Plotinian One has been noted before, notably by Rudy Rucker in his widely read book Infinity and the Mind. Rucker (1995: 40) clearly refers to this analogy when he writes: " is the One obtained by taking together... nothing". Although Rucker does not mention Plotinus in this context, the Neo-Platonic overtones of his mention of "the One" are loud and clear. Also because Rucker (in the same section) explicitly states that he inclines towards a Platonic interpretation of set theory (i.e. as describing an ideal reality existing outside of space and time). So if the set-theoretic universe (i.e. hierarchy V) constitutes a Platonic realm of ideal objects, then Rucker's description of as "the One" is clearly suggestive of a Neo-Platonic view, where is seen as a kind of transcendent, metaphysical source of V, analogous to the Plotinian One. It would seem, then, that our notion of MNP has already been anticipated by Rucker! Unfortunately, he does not develop this suggestion any further, so at best Rucker remains a 'closet Mathematical Neo-Platonist', hiding under the official cover of MP.

The Pythagorean element in NP
Let's return to the question how far the analogy between
and the Plotinian One extends. Is this indeed a substantial equivalence? One way to discuss this issue is by reversing the approach taken so far. Up till now we have mainly focused on the question to what extent ZFC repeats the core ideas of NP – but we can also turn this around and ask: to what extent did NP anticipate ZFC? There has, after all, always been a mathematical side to NP, partly because mathematics was already dear to Plato himself (who saw in geometry a privileged way to understand the ideal reality of the Forms), but mainly because the development of NP in the first centuries CE coincided with a revival of interest in the mathematical philosophy of Pythagoreanism. The philosophical and religious doctrines of the historical Pythagoras are largely unknown, but the claim that "All is number" was and is widely attributed to him. Thus Pythagoreanism has come to be understood as a form of Mathematical Monism (MM), the claim that all of reality consists of mathematics. However, together with this surprisingly modern doctrine, Pythagoreanism always went hand in hand with ascetic and magical practices, the latter based on a numerological belief in the magical properties of certain numbers. This numerological side obviously conflicts with the scientific aspect of Pythagoreanism. And this also holds for the Neopythagorean revival of the first and second centuries which strongly influenced the development of NP.

For the Neopythagoreans, reality consisted of a hierarchy flowing from God into something they called the original Monad and then into the original Dyad and finally into the numbers that shape physical reality. This scheme clearly anticipated the Plotinian hierarchy of the One unfolding into the multiplicities of Intellect, Soul and Nature (cf. Remes 2008: 15). In fact, Plotinus was seen by some of his contemporaries (notably Longinus) as the principal expositor of Pythagoras's doctrines (cf. Gatti 1996: 12-13). This Pythagorean aspect of NP comes clearly to the fore in the very title of Plotinus's master piece, the Enneads (the 'Nines'), although it should be remembered that this title came not from Plotinus himself but from his editor Porphyry. When Plotinus died in 270, Porphyry took it upon himself to prepare the master's manuscripts for wider circulation and by shifting around Plotinus's original texts (sometimes even breaking them up into smaller sections), Porphyry obtained 54 treatises which he divided into six groups of nine, "a combination of mystical numbers that delighted him" (Wallis 1995: 46).

Given this Pythagorean side of NP (which grew even stronger in later Neo-Platonists, notably Iamblichus), one could say that NP already was a form of MNP right from the start. One could even say, with some goodwill, that the Neo-Platonic hierarchy in its mathematical aspect – i.e. as an unfolding of increasing multiplicity out of an original but empty unity – anticipated the set-theoretic hierarchy of ZFC and other axiomatic set theories. In a sense, given NP's commitment to the Pythagorean claim that "All is number", NP can even be said to have anticipated MM – thus Plotinus, for example, writes: "Is not being, then, unified number, and beings number unfolded, and Intellect number moving in itself, and the living creature [i.e. the World Soul, PS] number embracing everything?" (Enneads, VI.6.4.29-31) But obviously the claim that NP anticipated ZFC and MM ultimately runs afoul of the lack of scientific rigor and the intrusion of numerology into the mathematical aspect of NP, which really did not go beyond vague claims about the One generating the Dyad which in turn generates all numbers that somehow "embrace everything". The how and why of this mathematical unfolding of the One remained shrouded in mystery.

All in all, Plotinus and later Neo-Platonists really lacked the mathematical theories that would have allowed them to precisify the Neopythagorean claim that reality is a mathematical outflow from a single transcendent source. But can't we now say that this situation has changed with the development of axiomatic set theory in combination with the MM of modern, i.e. mathematical physics? Doesn't ZFC+MM allow us to revive the Neoplatonic claim that reality is a mathematical outflow of the One? If MM is true, then the reduction of mathematics to ZFC implies that
stands not only at the origin of V but also at the origin of physical reality, because the latter would then be a substructure of V. In short: MNP=ZFC+MM?

Modern physics and Mathematical Monism
So let's examine MM more closely. What exactly does it say? And is it acceptable? To repeat, MM is the claim that all of reality – including physical reality – is quite literally composed of mathematical objects. And, as already said, although MM is of quite ancient origin, tracing back to the Pythagorean claim that "All is number", it is nowadays making a comeback due to the success of modern, mathematical physics. The basic point behind this comeback is simple enough: since physics describes and explains the whole of physical reality in essentially mathematical terms, the huge experimental success of physics shows that the 'substance' of this reality is ultimately nothing but what these mathematical terms refer to, namely, mathematical objects such as numbers, vectors, functions, fields, topological spaces, etc. One contemporary and very outspoken advocate of MM, the cosmologist Max Tegmark, summarizing the main results from relativity theory and quantum mechanics, describes this thoroughly mathematical nature of physical reality as follows:

"[W]e saw that the very fabric of our physical world, space itself, is a purely mathematical object in the sense that its only intrinsic properties are mathematical properties – numbers such as dimensionality, curvature and topology. [W]e saw that all the "stuff" in our physical world is made of elementary particles, which in turn are purely mathematical objects in the sense that their only intrinsic properties are mathematical properties [...] such as charge, spin and lepton number. [W]e saw that there's something that's arguably even more fundamental than our three-dimensional space and the particles within it: the wave function and the infinite-dimensional Hilbert space where it lives [...] and the wave function and Hilbert space are purely mathematical objects." (Tegmark 2014: 253-254)

It is on the basis of these considerations that Tegmark puts forward his own version of MM, which he calls the "Mathematical Universe Hypothesis" (MUH): "our external physical reality is a mathematical structure" (idem: 319). Similar arguments for MM can be found in scientists and philosophers like John Wheeler, Frank Tipler, Roger Penrose, W.V.O. Quine, James Ladyman, Steve French, Don Ross, Vlatko Vedral, and many, many others. I am inclined to agree with them that modern physics shows MM to be true. However, I can't really argue for that claim here (other than referring to the authority of said scientists and philosophers), simply because of lack of space. So in the following I will simply assume that MM is true, as shown by physics, and then develop the joint consequences of MM and ZFC, to see if this leads anywhere in the direction of MNP.

Three remarks on Mathematical Monism
Before I continue, however, there are three remarks about MM that I would like to make in order to put things in proper perspective. Firstly, it is important to keep in mind that MM does not equate physical reality with the mathematical realm in its entirety. That would simply be incorrect, because the vast majority of topics studied by mathematicians have no connection to physics whatsoever or to any other science apart from pure mathematics. Thus we have to assume that if physical reality is a mathematical structure, then it is a relatively small substructure of a much, much (perhaps infinitely) larger realm of mathematical objects. True, it always remains possible that particular results from pure mathematics find unexpected applications in physics (as happened, for example, with non-Euclidean geometries which were first developed by pure mathematicians but which turned out to find a surprising application in relativity theory). But even if more and more of pure mathematics is 'sucked into' physics, it seems extremely likely that pure mathematics will always dwarf the mathematics needed for physics. This means that MM faces a particular obligation, namely, to explain why certain substructures of mathematics and not others are singled out as constituting physical reality, and exactly what this 'singling out' amounts to.

Secondly, note that MM should be understood as a radicalized form of Mathematical Platonism (MP). Both original Platonism and MP are dualistic in that they make principled distinctions between the spatiotemporal realm of physical reality on the one hand and the ideal realm of Forms / mathematical truths on the other. MM, on the other hand, is monistic in that it reduces all of reality, including physical reality, to mathematics. In that way MM can be seen as a radicalization of MP insofar as it takes the ideal realm of mathematics to 'engulf' the spatiotemporal realm. Tegmark puts this very clearly: "Mathematical structures are eternal and unchanging: they don't exist in space and time – rather space and time exist in (some of) them." (Tegmark 2014: 318) This conception is also known in the context of special relativity as the "block universe", where time is the fourth dimension of the geometrical structure of spacetime, such that the entire universe, from past to future, exists 'at once' as a single, mathematically structured 'block'.

My third remark is a caveat. Although I agree with the argument that the success of modern physics shows that MM is true, I also think it is crucial to note that this argument limits the truth of MM to physical reality, i.e. reality as described by physics. Thus there might be non-physical realities that escape mathematical treatment. Indeed, I think this follows from the famous Hard Problem of Consciousness, which shows that the reduction of reality to mathematics stops short of how we experience reality, insofar as the qualia of that experience refuse direct reduction to non-conscious building blocks, be it the physical structures that form the human brain or the mathematical structures that model the functional organization of the brain (see Chalmers 1996). So here, in the Hard Problem of Consciousness, MM reaches its limit. However, as I will argue in my next post, this fact – that consciousness falls outside of mathematics – is precisely what will make a Neo-Platonic approach to mathematics possible. For now, however, I will abstract from the Hard Problem and simply assume that MM is true tout court. Why? Because this puts in very sharp relief the most fundamental question of ontology...

Why does
exist? Mathematical Nihilism averted
If MM were true tout court, i.e. if all of reality reduces to mathematics, then – given the reduction of mathematics to ZFC – Leibniz's famous question "Why is there something rather than nothing?" would in turn reduce to: Why does
exist? And why do the axioms of ZFC hold? Now it might be thought that this turns Leibniz's question into a 'no-brainer' because – as is sometimes said – " is nothing". So to explain why physical reality exists, we simply have to assume that 'in the beginning there was nothing' et voilà we have , the sole existential assumption needed to derive the whole of mathematics! But, obviously, this approach is nonsensical. First of all, is not nothing, it is rather {nothing}, i.e. a set with nothing in it. And a set, even if it is empty, is something, not nothing: "It is not the same thing as nothing because it has whatever kind of existence a set has, although it is unlike all other sets." (Gardner 1977: 15). This difference between and nothing is clarified by a nice example from Enderton': "a man with an empty container is better off than a man with nothing – at least he has the container". (Enderton 1977: 3) Secondly, even if we were warranted in simply assuming the existence of , we would still need to explain why the other axioms of ZFC are valid, because it is only in combination with them that yields V. Clearly, then, we have to reject the idea that ZFC provides an easy or otherwise attractive solution to Leibniz's question because 'it starts from nothing'.

Nevertheless, this confusion of
with some kind of 'primordial nothingness' explains a lot of the current 'sexiness' of axiomatic set theory, as if the latter were a sort of 'Mathematical Nihilism' giving new content and credibility to the idea of an absolute creatio ex nihilo. Such a view can be found, for example, in Jim Holt's popular book Why does the world exist?. Commenting on the set-theoretic construction of V on the basis of , he writes: "Out of sheer nothingness, a remarkable profusion of entities has come into being." Then, bringing in the hypothesis of MM, i.e. that all of reality reduces to mathematics and thereby to set theory, he writes: "The whole show of reality can be generated out of the empty set – out of Nothing." (Holt 2013: 40) Admittedly, for Holt this is merely one of the many possible solutions to Leibniz's question he examines in his book, so we should perhaps not pin him down on it. The situation is different, however, with the popular French philosopher Alain Badiou, who has developed a wide-ranging set-theoretic ontology (including a political theory) based on a conception of as a kind of "primordial void" on which all existence is somehow founded, "the nothing from which everything proceeds" (Badiou 2005: 59). Thus he writes: "In Set Theory, the primitive name of Being is the void, the empty set. The whole hierarchy takes root in it. In a certain sense, it alone "is"." (Badiou 2006: 98) As the scare quotes around "is" indicate, Badiou takes (the sole basis of existence, according to him) to be itself inexistent, i.e. nothing. If this is the foundation of Badiou's ontology, then clearly that ontology is based on a mistake.

Confusion of with 'nothing' widespread
This confusion is not just confined to foggy philosophers, however. It can also be found in otherwise
respectable set theorists and mathematicians, people who really should know better. For example, Keith Devlin (1993: 36) in The Joy of Sets writes about the construction of V: "we commence with nothing, that is to say, the empty set". Mary Tiles (1989: 124) in The Philosophy of Set Theory writes that the universe of sets "is a wholly abstract universe generated, as it were, out of nothing" (admittedly, she qualifies her claim with the phrase "as it were", but still, the damage has been done). John D. Barrow (2000: 167), a prominent theoretical physicist and mathematician, writes about set theory that "it has enabled us to create all of the numbers from literally nothing, the set with no members". But the most curious case of this confusion can be found in Enderton, whose remark about "a man with an empty container" we quoted earlier precisely to dispel the confusion! For if we look at the context of that remark, we see that Enderton himself is confused as well. Here is what he writes:

"Note that {
} , because {} but . The fact that {} is reflected in the fact that a man with an empty container is better off than a man with nothing – at least he has the container." (Enderton 1977: 3)

Clearly, the notion of a container here represents the notion of a set, since a set is a container of sorts (it 'contains' its members). So when Enderton speaks of "an empty container" one naturally supposes he is referring to
. But a closer look reveals that this is not the case. Enderton in fact uses the difference between a man with an empty container and a man with no container to clarify the difference between {} and . So, in this analogy, {} is represented by the empty container, whereas is represented by no container at all! This clearly indicates that Enderton makes the mistake of identifying with nothing. If {} is represented by the empty container, a container with nothing in it, then this means that is nothing. Similarly, if is represented by a man with no container at all ("a man with nothing" as Enderton writes), then this also means that is nothing. So even if Enderton has in his hands the germ of the insight that ≠ nothing given his example of the difference between an empty container and no container, he fails to see the true significance of this example. Apparently, then, thinking about the empty set is a tricky affair and its confusion with nothing is always lurking, even for the ablest expositors of set theory.   

Transition to MNP
≠ nothing. This means that Leibniz's question, even in its set-theoretic form ("Why does exist? And why do the ZFC axioms hold?"), still presents us with a genuine problem, i.e. a problem not solved by simply assuming the 'existence of nothing' (whatever that is supposed to mean). As I will argue in my next post, it is especially in the light of Leibniz's question that the importance of NP will become apparent, because Plotinus was actually the very first philosopher who grappled with this problem in a systematic way and who developed a systematic answer in the form of his notion of the One as causa sui. So, if we return to our analogy between and the Plotinian One, and we allow ourselves to speak rather loosely, we could say that Plotinus supplies us with a way to understand how has brought itself into existence! More about this in my next post.

-Badiou, Alain (2005), Being and Event. New York: Continuum.
-Badiou, Alain (2006), Briefings on Existence: A Short Treatise on Transitory Ontology. Albany: SUNY Press.
-Barrow, John D. (2000), The Book of Nothing. London: Jonathan Cape.
-Chalmers, David J. (1996), The Conscious Mind: In Search of a Fundamental Theory. Oxford University Press: New York and Oxford.  
-Devlin, Keith (1993), The Joy of Sets: Fundamentals of Contemporary Set Theory. New York: Springer.
-Enderton, Herbert B. (1977), Elements of Set Theory. New York: Academic Press.
-Gardner, Martin (1977), Mathematical Magic Show. London: Penguin.
-Gatti, M.L. (1996), "Plotinus: The Platonic Tradition and the foundation of Neoplatonism", in: Lloyd P. Gerson (ed.), The Cambridge Companion to Plotinus. Cambridge: Cambridge University Press.

-Holt, Jim (2013), Why Does The World Exist? One Man's Quest for the Big Answer. Profile Books: London. -Mayberry, John (1988), "What are numbers?", in: Philosophical Studies, 54 (3), 317-354.
-Remes, Pauliina (2008), Neoplatonism. Stocksfield: Acumen.
-Rucker, Rudy (1995), Infinity and the Mind: The Science and Philosophy of the Infinite. London: Penguin Books.
-Tegmark, Max (2014), Our Mathematical Universe. New York: Alfred A. Knopf.
-Tiles, Mary (1989), The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise. Mineola: Dover.
-Wallis, R.T. (1995), Neoplatonism. London: Gerald Duckworth & Co.

1 comment:

  1. This statement is a gross exaggeration : "ZFC functions as the standardly used axiomatic foundation for virtually all of mathematics". A more correct statement would be "ZFC is the standard reference of a possible axiomatic foundation for virtually all of mathematics, as mentioned by usual mathematics textbooks passively copying each other". But over 99% of mathematical works make no explicit use of this particular formulation of set theory. In my site on the foundations of mathematics I present another axiomatic formulation of set theory which is much closer to the practical use of set theory in mathematics.