Mathematics: The Modern God

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 F signify core memes outlined in my blog on the workings of the mind.