I followed a lead today that came at the end of Clifford Pickover’s The Math Book.
The last of Pickover’s 250 milestones in mathematics is Max Tegmark’s Mathematical Universe Hypothesis, which Tegmark published in 2007 in both scientific and popular articles. The hypothesis is that “our universe is not just described by mathematics – it is mathematics. Tegmark is a cosmologist at MIT and scientific director of the Foundational Questions Institute. His NewScientist article from that time, What the Universe if Really Made Of, can be found on his website.
There is also a link on his website to a roundtable discussion that Tegmark joined. The discussion was hosted in 2009 by the Philoctetes Center, a New York City based organization devoted to ‘The Multidisciplinary Study of Imagination.’ It seems the organization lost funding and shut down in 2010. This is unfortunate. I was intrigued by the topic – Mathematics and Religion – and listened for an hour.
The diversity of perspectives was interesting but, notwithstanding the unique focus of the discussion, the group found it difficult to get past the usual questions, about Platonism and the existence of God. And they unexpectedly, spent a fair amount of time arguing about what Spinoza said. This was despite the fact that novelist, philosopher and Harvard Research Associate, Rebecca Newberger Goldstein did a nice job of starting things off. I thought these words of hers opened the door to a number of paths that were never pursued:
Also, in mathematics, although it perceives by proofs, it happens that proofs are often the afterthought. It’s after you’ve already seen it, you have a heuristic grasp of it. Again, if this is a transcendent world, what are we seeing when we’re seeing these things, and how are we making contact with this transcendent world? So – although mathematics itself seems very, very certain, mathematics raises all sorts of questions about human knowledge and the limits of human knowledge, and how we can do what we do when we’re doing mathematics.
Max Tegmark made his argument:
…my guess is that the explanation is that reality is so well described by mathematics because it is ultimately purely mathematical, and not only is our physical world described by a mathematical structure, but it is a mathematical structure. So I think we’re all living in a gigantic mathematical object – not one of the simple ones that we learn about in high school math. We’re not living inside of a cube or a dodecahedron or in the set of integers, but there’s some more complicated mathematical object, maybe M-theory, maybe some – more likely something we haven’t discovered yet which somehow is our reality.
That’s his view in a nutshell – interesting and provocative. What might that imply about mathematics and religion? I don’t think anyone actually took this up. Later in the discussion he made what I thought was a nice observation about time and creation (in mathematics).
Like you asked, for example, a very interesting question of the integers, were they created or have they always existed. That whole question presupposes the existence of time, because you need time for something to first not exist and then exist. A creation event requires time. But we know, and Einstein taught us that there are two ways in which we can think of time. We can think of either a reality being this three-dimensional space where stuff happens over time, or we can think of living in this four-dimensional space-time, a four dimensional space where the fourth dimension has a minus sign in it, which makes it feel like time. And the space-time of Einstein of course, there’s nothing happening in there. If life is a movie then the space-time is like the entire DVD. It’s all in there. The DVD isn’t changing even though there’s all sorts of drama unfolding in the play.
So if you think of our reality as a mathematical object which contains the space-time, then time exists within this mathematical structure, rather than the mathematical structure existing in time.
Earlier in the discussion, Princeton mathematician Edward Nelson made a different observation of time that I liked very much.
Another interesting feature of mathematics is the tremendous time frame. Six is called a perfect number because it’s the sum of its divisors other than itself; one and two and three divide six, one plus two plus three equals six. And Euclid not only did geometry but did number theory in his book and he proved that numbers of a certain form connected with prime numbers were also perfect. And so, yes it was 2000 years later, in the eighteenth century Euler proved that every even perfect number was of Euclid’s form. That’s a tremendous timeframe to be working on the same problem, and it’s a major open problem today, do there exist odd perfect numbers. No one knows.
I did enjoy what I finally pulled out of the Spinoza discussion – the idea that God could be imagined as the complete image of what nature is.
But I was surprised to find that there was no opportunity to connect these thoughts to questions about us – how we work, how we’re built, what is the thing we call intuition, what is an image, how we may be motivated, or what it is that we seek. Why is it that mathematics has been seen as pointing to a transcendent reality? There is, after all, a perspective in biology that says that living is cognition – as in Maturana and Varela in The Tree of Knowledge where “cognition is an ongoing bringing forth of a world through the process of living itself.” How are the products of human cognition ‘the process of living itself,’ and what might this mean about religious images? The title of this work in biology certainly suggests that the ideas they present may be related to religious ones.
Despite words like ‘the multidisciplinary study of imagination,’ or Goldstein’s questions about what is it that we’re seeing when we see the idea for which we seek proof, in this discussion of mathematics and religion there was no talk about the organism, only about some of the images that the organism creates. I don’t think it’s possible to have an effective discussion of mathematics and religion without including talk about us.