Anathem Metaphysics

(“Metatheorics”, I should say)

Most Neal Stephenson fans probably read his book Anathem last fall, but I just finished. There are many reviews available, so I won’t add one here, but I got a big charge out of some of the philosophical ideas Stephenson put in the book.

In the course of the novel, Stephenson proposes an interesting relationship between platonic truths, the multiverse, and human rationality/consciousness, which I have enjoyed comparing to my own ideas (discussion below the fold).

The main idea is that the platonic realm is not a single transcendent world out there, but rather the source of platonic ideas is to be found in the multiverse (called the “polycosm” in the book; the overall theory is called “complex protism” where protism=Platonism and “simple protism” would be the positing of a single platonic realm).

The core of this idea is one I endorse here in the non-fictional realm, and it was very cool to see it in the novel. Modal judgments about what is possible and what is necessary play a crucial role in our reasoning. At the same time, both philosophers and theoretical physicists have proposed that our world is one member of a set of many possible worlds. If we propose that the logical possibilities we explore with our minds are identical with real possibilities present in the multiverse, then the multiverse can be seen as the source of our rationality, and of our knowledge of abstract truths. (The philosophical term for the match-up between logically possible worlds and metaphysically possible worlds is modal rationalism).

But how does a human mind access the multiverse? First, let me note that the type of multiverse being proposed in the book is inspired by the many-worlds interpretation (MWI) of quantum mechanics (QM). So the solution is to have the mind exploit the quantum realm. There’s a dialogue between the characters Orolo and Erasmus (pp. 543-548) where they discuss how the presence of the same person’s brain in multiple adjoining and interfering worlds gives the brain access to possibilities. Stephenson’s characters later make the point (which I liked) that it isn’t that the human brain is the only thing which is in contact with possibilities; one should assume everything in the world experiences some contact, but it is in our own brains that we can best see the evidence manifested (pp. 690-92).

I’ve come at this in similar spirit, with one apparent difference when it comes to interpreting QM. I believe that our world is comprised of quantum measurement “collapse” events, which each embody the actualization of one of many possibilities. The fact that we humans are ultimately grounded in quantum-level events gives us a kind of direct acquaintance with possibilia in everything we do. That makes our ability to have modal knowledge intelligible, even though we don’t yet know the manner in which our brain/body system leverages this presumably micro-level contact into macro-level knowledge.

It appears Stephenson’s idea is to try to preserve the MWI (which attempts to do away with measurement events), while allowing contact between parallel worlds to be exploited by the mind. MWI would not typically be seen as allowing any contact. Also, usually in MWI, worlds branch (what we think of as a measurement is a splitting of worlds), whereas Stephenson wants to keep worlds in parallel. Of course, positing contacts between parallel worlds is very helpful for the creating the exciting parts of the novel which involve characters and spaceships moving between worlds.

The other aspect of Stephenson’s multiverse which is interesting is that the informational contact between worlds has a flow, where some worlds are upstream or downstream from one’s own world. It seems he places a single purely platonic world at the source of the information flow (whereas I would follow the usual philosophical tradition of identifying logical and mathmatically necessary truths to be those things true in all worlds). I’m not sure this makes sense or was philosophically motivated, but it was a neat twist.

There are allusions to many other philosophical and scientific ideas in the book, and Stephenson discusses many of the sources which inspired these in this acknowledgments page on his website. There’s more to follow up on there – one philosopher he discusses who I have not read is Edward N. Zalta.

[I have a number of old posts on related topics, including
Making Abstract Truths Intelligible
Modal Realism, Modal Rationalism
Multiverses -- Physical and Metaphysical]

Making Abstract Truths Intelligible

The problem with abstract objects is that their existence seems required to provide truthmakers for our propositions about them. Nominalist and other deflationary accounts can’t adequately meet this requirement. Yet how could abstract objects be real if they aren’t part of our concrete world? In modern discussions, abstract objects are causally inert by definition. If they somehow exist in some platonic realm, how could we know them in the absence of any causal connection? (A previous post on abstract objects is here).

Well, we have been considering here a model of causality that incorporates abstract modal realism. The concrete world is a causal network of events which are actualized possibilities. The set of possibilities available to be actualized in an event is constrained by preceding or adjacent events but the outcome isn’t fully determined prior to a new actualization. Unactualized possibilities may be considered “abstract” in that they are non-concrete yet real, and abstract seems as good a term as any for this mode of being (see note on terminology at end of post).

So a certain kind of abstract entity does enter into causal connections. Specifically, the concrete events of the world make contact with abstract possible events. Then the question is can we use this theory to make sense of our seeming knowledge of abstract truths, such as the prototypical logical and mathematical ones?

Well, I don’t have a developed model of how our macroscopic brain/body system would accomplish this. But given the centrality of modality to our reasoning, and given an independently motivated theory that we, as natural systems, exist in a web of actualized possibilities, we can try to connect the dots. The idea would go something like this: we have a direct primitive acquaintance with possibility, which we leverage into knowledge of idealized abstract truths through a process of counterfactual analysis. It is this notion of “primitive” acquaintance which makes this process not just a matter of conceptual or psychological construction, but a matter of reaching toward metaphysical truths.


Terminological note: it is easy to get misled by terminology here and I have probably been sloppy at times. Often in discussions of modal realism (See SEP article on Actualism), a distinction is drawn between the “actual” and “mere” possibilia. Unless one subscribes to David Lewis’ model of modal realism, where possible worlds are all concrete and the term actual is an indexical, I suggest using actual and concrete interchangeably and ask the reader’s forbearance to not misread the fact that possibilities are “non-actual” as saying they don’t exist. They exist, but are non-concrete, hence abstract.

Gödel’s Platonism

I got around to reading Rebecca Goldstein’s brief and engaging book on Gödel -- Incompleteness: The Proof and Paradox of Kurt Gödel. The book centered on the irony that Gödel’s own philosophical interpretation of his work (which indeed may have driven his efforts to begin with) was in complete opposition to how it was most commonly interpreted by others.

Gödel was a Platonist, believing that the mind was able to make contact with absolute mathematical reality. Given that he was an attending member of the Vienna circle in the 1920’s, which was the locus of logical positivism, many assumed he was of like mind, believing there was no truth beyond what man could empirically discover. Gödel’s extreme reluctance to speak or write on his views helped make this misunderstanding possible. Indeed, the incompleteness theorems have often been co-opted by sloppy post-modernists (along with relativity theory and the uncertainty principle) in making the case for truth relativism. They would focus on the conclusion that we can’t construct formal systems (large enough to at least encompass arithmetic) that are both complete and provably consistent and treat this fact as revealing a limitation in our ability to reach absolute truth. Gödel believed the actual lesson was that the human mind can and does perceive truth beyond the capability of formal systems (equivalently, algorithmic computing machines).


[UPDATE 12 January 2012:  For a review quite critical of Goldstein see Solomon Feferman's here.  He says she has no basis for taking Godel's later platonism and suggesting it motivated his earlier seminal work.]

To digress a moment, I just about forgot I have an old post on Gödel. This blog is ably serving one of its functions -- an external memory module. In that post I noted the consensus of experts that while the incompleteness theorems may point toward philosophical conclusions (such as thinking the mind surpasses a computer), they don’t provide any proofs after you depart their formal setting. However, one philosophical stance I said they did appear to support was the notion of an ultimate limit on “objective” knowledge. (Note I also maintained that such an observation need not lead to thorough-going relativism.) Now, in reading more about Gödel’s own views, I’m not feeling confidant that assertion quite captures things.

In one of those happy coincidences, the recent update to the Online Papers in Philosophy blog maintained by Jonathan Ichikawa [UPDATE: this blog not longer exists] had a paper by eminent mathematical logician Solomon Feferman which examined one of Gödel’s rare talks (the 1951 “Gibbs” lecture). In the talk, Gödel presented the philosophical implications in terms of a disjunction thus: “Either…the human mind…infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable Diophantine problems.” By the most reliable accounts, Gödel did indeed believe that the mind surpassed finite systems and therefore there were no (ideally) unsolvable problems, but he expressed a bit of caution in this talk by presenting the disjunction.

Feferman, in keeping with the usual deflationary mode of papers by experts on this topic, respectfully shows how the imprecision and complexity of these issues prevent one from reaching a logical proof of Gödel’s claim (or even ruling out that the disjuncts could both be true). Once again, the broad statements about the mind and its capabilities can’t be derived from the mathematical arguments which inspire them. Still, many interesting facets of these issues are illuminated in the discussion.

The Feferman paper included one (unpublished) Gödel quote which struck me as very insightful. Gödel (responding to something Turing had written) says: …”mind, in its use, is not static, but constantly developing, i.e., we understand abstract terms more and more precisely as we go on using them…though at each stage the number and precision of the abstract terms at our disposal may be finite, both…may converge toward infinity…” We are clearly finite and contingent creatures, and so the idea that we are in direct contact with Platonic truths is hard to support; but we do appear to have a gift of rationality which allows us to converge toward absolute truths. I see a connection here to the idea of modal rationalism, which I’ll explore in a future post.

One more note: that OPP update had another paper which touches on this topic, Philip Ebert’s “What Mathematical Knowledge could not be”. This is a nice survey of positions on the reality of mathematical objects; it doesn’t itself advance an argument.

Platonism on Tap at Maverick Philosopher

Posts on the subject are underway at Bill Vallicella's active and interesting blog (see here and here). Good group of commenters as well.

Driven to Abstraction

Plato just won’t go away.

I’ve been thinking about whether the truthmakers for modal truths could involve abstract possible worlds, but this requires backing up a bit in order to consider the status of abstract objects in general.

Defining what it is to be "abstract" is not trivial. Lewis made this point in On the Plurality of Worlds. His breakdown of several ways to approach the question is followed by Gideon Rosen in this brief SEP article on Abstract Objects. The most often used methods are to define abstract objects in terms of what they are not, and then work out the idea using examples. Abstract objects are neither physical nor mental – usually they are thought of as unchanging and causally inert (the potential role of abstract objects in a theory of causation is something to come back to, however). Numbers and universals (“redness”, “roundness”) are paradigm examples.

This SEP article on Platonism in Metaphysics, by Mark Balaguer, summarizes the state of abstract objects in modern metaphysics. He surveys the landscape by seeing how the main candidates for abstract object status (mathematical objects, properties, propositions, possible worlds) fare under Platonism and its main rivals (nominalism, immanent realism, conceptualism). It’s an exceptionally reader friendly article, and was a helpful review for me.

One item I thought was interesting is that, according to Balaguer, the strongest argument for Platonism is a truthmaker argument. The things denoted in literally true statements (“3 is prime”) must exist as abstract objects in order to make the statement true. Alternative accounts of how these statements work or attempts to deny their truth all have problems and objections.

Of course, Platonism has strong objections. Balaguer singles out the epistemological issue as the biggest problem: if abstract objects are non-physical (and exist outside of space-time) how can we have knowledge about them? Different accounts have been proposed regarding how this can work; objections have been lodged, and the debate continues.

There is a running “meta-theme” here which I’ve been thinking about as I’ve tried to survey different topics in metaphysics (ontology, causality, modality). These metaphysical questions are difficult, and simple solutions obviously don’t work or the debates would have ended long ago. What this means to me is that the common presumption that something like physicalist monism should be the “default” metaphysical position is unfounded. More “extravagant” metaphysical systems need to be weighed in the quest to find a better mousetrap for explaining how the world works.

Speculative Thought on the Platonic Realm

One of the challenges of naturalism is that, taken strictly, it seems to rule out the existence of ideas or concepts as transcendent entities. Even for scientifically minded individuals, some concepts seem to exist in the sense of our discovering them, even though they are not part of nature. The prototypical examples are in the arenas of logic and mathematics.

I have often been critical of the disproportionate focus in academic philosophy on concepts and language as things of primary analytical interest, as opposed to their being derivatives of our (very complex) interaction with the rest of the natural world. But there are persuasive arguments that some of these concepts seem stubbornly non-reducible to nature.

It occurred to me that a way to naturalize these seemingly platonic concepts would be to picture them as gaining traction from a larger multi-verse. Science increasingly points us to the idea that our observable universe is a part of a larger complex of universes, each with potentially different characteristics along certain dimensions. As we and our world co-evolve, perhaps we reach toward ideas that do find a natural incarnation somewhere in this much larger meta-world.