Roger Penrose is a guy who has a way of asking good questions and providing controversial answers. On page 20 of his monumental survey of applied mathematics, "The Road To Reality" he explains an interesting theory about three "worlds":

• R*:The "real", physical world of "stuff" and phenomena, which he (quite reasonably) takes to exist apart from the other two worlds;
• M:The world of ideas - what I call the "meme space". In principle, this is all the theories, observations, direct or indirect perceptions that could ever be made by the human mind;
• F:The world of forms, mathematics and logic.

While he contends is that each world "maps" completely to the others. In principle at least, each world can be explained in terms of the other.  But he allows us to imagine that the mapping is incomplete. It is the incompleteness that interests me:

• There may be aspects of reality that are "illogical" or cannot be described by mathematics. Such a possibility goes against the grain for Penrose, who is, after all, a mathematician.
• There may be ideas that correctly describe the world, but are beyond the ability of the human mind to grasp, even "in principle"
• There facts of math and logic that are beyond the ability of the human mind to comprehend. Questions that we cannot even think to ask, postulates - even "important ones" that we will never discover

"In principle" raises a red flag. It quite often means "This is obviously false, but lets pretend its true". It is a good habit to challenge every use of this term and ask precisely what is meant by it.

Penrose's claim amounts to believing that the world can be completely understood by the mind and the way to do it is through mathematics. Thinking like a mathematician, he feels that mathematics "maps" completely to the real world and therefore the two worlds are the same "for all intents an purposes" (another red flag expression). 

Mathematics is claimed to govern the workings of the universe, both seen and unseen. Denial of Penrose's optimism amounts to claiming that, although mathematics rules the universe, many of its "laws" are beyond the reach of human intellect. Even worse, things might happen that are fundamentally illogical. This debate sounds vaguely theological.

Mathematics does not share the personal attributes of the Old Testament God but this new god is all-knowing and, in a sense, separate from the universe governed by the laws of mathematics. This is because the world of mathematics is larger than the universe we inhabit. Other universes can easily be described by mathematical constructs such as the Shrodinger Equation (just fiddle with the constants in the equation). String Theory, a perfectly consistent model of something that looks like a universe, suffers from the problem that it describes too many universes. Ours is a special case.

True believers in a God are familiar with the idea that the "ways" of God's are ultimately unknowable to the human mind. The same is true of the "math god".  Godel's Completeness Theorem guarantees that there are some things that are "true" in mathematics that cannot be proven. In fact, there are an infinite number of such things and you never know if your favourite "obvious" postulate is one of them. Mathematics moves in mysterious ways.

Penrose's belief that the universe can, in principle, describe all phenomena we experience can be seen as faith: pure and simple. Penrose, whose scientific background is in subatomic physics, shares the faith of his peers. He regards subatomic physics as "fundamental" and believes (against all evidence) that all of the more complex phenomena of the world can be "explained" in terms of "fundamental" principles. This is the "reductionist" faith, modern fundamentalism. It is logically identical to the claim that everything you need to know about life can (in principle!) be learned from the Bible. Like religious fundamentalism, reductionism can only be supported if you spend most of your time among other true believers. In the early 21st century, this is not hard to do, since reductionism seems "obvious" to  almost everyone, just as the existence of God was obvious to Descartes.

Richard Dawkins has made his fame by attacking religion, challenging centuries-old "obvious" ideas that have long since lost their relevance to human experience. Religion is reductionist since it claims that everything comes down to God, even though we are not quite sure about the details. So religion is a special case of reductionism. Challenging reductionism is like a vaccine against a huge category of nonsense.

Laughin in "A Different Universe" destroys reductionism with an "emperor has no clothes" argument. He provides one example after another where reductionism fails. He provides an alternative theory as to why this failure is a fundamental aspect of reality.  

As things stand now, it is rare to find anyone who draws a parallel between the universe of mathematics and the heavenly realm. Most of us think we are immune from the kind of nonsense that people took for granted 200 years ago. But the mere fact that we take it as obvious that the mathematical world as real and, in some sense, superior to the world we live in marks us as "true believers" in the old tradition. At least we can certainly claim that mathematics is far more useful than theology when it comes to "explaining" the phenomena of the world, but we cannot defend our assumption that the universe "obeys" mathematical laws or that mathematics is somehow "fundamental".

Lets look at another aspect of Penrose's claim: that mathematics, in principle, completely describes the real world and, indirectly, the brain and the human mind. So, things like consciousness, compassion and creativity could be (in principle!) reduced to a branch of neuroscience. This idea applies reductionism to the link between the brain and the mind, even if the workings of the brain cannot be explained in mathematical terms.

The poster boy for this idea (that consciousness can be, in principle, be explained by workings of the brain) is Daniel Dennett. Typically, Dennett is his own best critic. He  identifies the weakness of his own approach as "Greedy Reductionism".

Like the theist view of the world, reductionism needs to be abandoned as one of those "obvious" ideas that simply doesn't fit experience. Mathematics and logic create a wonderful, fascinating world but we will probably never know if mathematics somehow underlies all of reality itself. It seems likely that there is a lot of things we'd like to know about the universe that we will never know. The worlds that Penrose talks about communicate in interesting ways, but they are not the same in any meaningful sense. 

*R, M and L are terms for "core memes", borrowed from my blog on the mind

The Real World - The "R and Q Memes"

The "R" meme is the most powerful and widely-shared meme: that there is a "real world out there" independent of our minds. This is the intuitive insight that is denied or at least challenged by "Idealism" [1].

Denial of R turns out to be a powerful "intuition pump". We imagining that the "real world" does not exist at all, and it's "all in our minds". This is a rather strong denial of Idealism, since an Idealist would, I think, admit that the real world exists -- just that we can't know anything about it other than what pops up in our minds.

At the risk of spoiling the plot, the view of this blog is that the brain can be thought of as a fantastically complex and specially-constructed knot that intimately ties the mind M to the world R. This connection is so intimate that it is, in practice, impossible to say where mind ends and world begins [2]. In effect, this frames "Idealism" as a discussion about a distinction without a difference -- not unusual for philosophical discussions in general.

One of the pillars of R is he "Standard Model of Physics" (I will call it Q) an idea that claims to be "isomorphic" with the "real world" [2] at almost all contexts of experience. That means that we have a complete explanation of things like atomic and sub-atomic particles and forces operating on us. The contexts where this model doesn't apply include the interior of black holes and the early time of the Universe (between the "big bang" and "inflation". Q includes a specific and elegant description of its own limitations, captured in the idea of an "effective field theory", which may be thought of as a formal definition of what's meant by "for all intents and purposes" when applying Q to specific phenomena.

Q is a powerful meme, underlying a sweeping range of ideas we have about reality, including things like photosynthesis, the lifetime of stars and the behavior of semi-conductors. Particle physicists and cosmologists tend to wax poetic about this idea, calling it "fundamental" and the "theory of everything". This over-states the matter. The mind, not R, is "fundamental". R is an idea, not reality itself, so therefore included in the world of memes (even it it turns out to be false!) In this sense, the theory of memes (especially as envisioned by Hofstadter) is more "fundamental". The idea that Q is "fundamental" is the bedrock of reductionism and fails for all the reasons reductionism fails. Q enthusiasts also tend to rely on the common confusion between a description and an explanation. The fact that all observed phenomena are consistent with Q does not imply that Q "explains" or predicts anything at all. Q fails to predict many "emergent" properties, such as the chemical properties of matter and becomes of marginal importance as the level of complexity increases. For example, Q doesn't help much when explaining how he mind works.

[1] "Idealism" The idea that we cannot know anything about the world except through the prism of our minds. This has many modern variants, such as the Copenhagen Interpretation of Quantum Mechanics (Q meme). Its shadow can be seen in Einstein's introduction of the "observer" into physics, banishing the all-knowing "god" view of the universe. Idealism can be contrasted with the idea we see in Eastern philosophies (Zen), which claim that the mind is an illusion.

[2] In Quantum Mechanics, the fact that the theory and the world are not the "same thing" can never be swept under the carpet. "Isomorphic" is an idea borrowed from mathematics, which says that there is a one-to-one mapping between one domain to another.  For. example, "isomorphism" means that all the properties of a "real" electron are exactly mirrored in the electron of Q meme (leaving nothing out). Conceptually, "isomorphism" can be taken to mean that the two domains are, "for all intents and purposes", the same thing, which almost amounts to claiming the universe itself is an idea. The R meme would deny this, claiming that there "must" be aspects of the universe that cannot be "isomorphic" to human ideas. Some things are forever beyond the grasp of the human mind (It is certainly true that Q itself is beyond the grasp of all but a tiny number of actual minds).

The concept of "effective field theory" (part of Q) is an open admission that isomorphism fails at some point, meaning that Q does not "map" to everything in the real universe. Even in its own terms, Q is not a "theory of everything".