Paul P. Mealing

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Monday, 2 March 2009

The Unreasonable Effectiveness of Mathematics

This is a quote from Physics Nobel Laureate Eugene Wigner, and it appears in the first sentence on the front fly leaf cover of Mario Livio’s book, Is God a Mathematician? On the back fly leaf cover we learn that ‘Mario Livio is a senior astrophysicist and head of the Office of Public Outreach at the Hubble Space Telescope Science Institute in Baltimore, Maryland.’ He’s also written a few other books on mathematics, including The Golden Ratio and The Equation That Couldn’t be Solved (neither of which I’ve read).

I use Wigner’s quote as the title of this post, rather than the title of Livio’s book, because it’s the quote that he keeps returning to and attempting to address, rather than any real attempt to address the question on the cover of his book.

It’s a very good read: erudite, thought-provoking and as balanced as one could expect from someone who has their own philosophical standpoint. I’ve addressed this issue on 2 previous posts: Is mathematics evidence of a transcendental realm? (Jan.08) and Where does mathematics come from? (Sep.07). The Jan.08 post is effectively a review of Gregory Chaitin’s book, Thinking about Godel and Turing, and the Sep.07 post is more of a critique, than a review, of George Lakoff’s and Rafael Nunez’s book, Where Mathematics Comes From. Livio gives a good account of Lakoff’s and Nunez’s views, but he doesn’t mention Chaitin (even in the bibliography).

I don’t wish to reiterate ideas I’ve already explored in those posts, but I do wish to say that Livio demonstrates why scholars like him publish books, and amateurs like me only publish on a blog.

Livio’s book covers topics as varied as statistics and probability theory, Euclidean and non-Euclidian geometry, and logic - all in revelatory detail, yet easy to read. The most interesting topic as far as I was concerned was on knots, and their application to biology and the most recent theoretical investigations in cosmology (string theory).

He also gives brief biographies on, what he considered to be, the giants in mathematics: Archimedes, Galileo, Descartes and Newton.

But it is an extended quote from Wigner that establishes the tone, if not the intent, of Livio’s treatise:

‘The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.’

Livio effectively encapsulates his position when he states that there are 2 aspects of mathematics which he calls: ‘active’ and ‘passive’. By ‘active’ he means the deliberate use of mathematics as a tool to investigate and understand natural phenomena. This aspect supports the view that mathematics is ‘invented’ rather than ‘discovered’. We humans, with our preternatural intellectual abilities, create mathematical models that provide accurate facsimiles of nature’s laws, and can even formulate predictions and forecasts by employing the ‘scientific method’.

To appreciate what Livio means by ‘passive’ it is best to quote his own words:

‘But there is a “passive” side to the mysterious effectiveness of mathematics, and it is so surprising that the “active” aspect pales by comparison. Concepts and relations explored by mathematicians only for pure reasons – with absolutely no application in mind – turn out decades (or sometimes centuries) later to be the unexpected solutions to problems grounded in physical reality!’

(I make a similar point, though not as eloquently, in my Mar.08 post, The Laws of Nature.)

Livio gives a number of examples of this ‘passive’ aspect of mathematics, but amongst the most commonly known are: Maxwell’s equations predicting electromagnetic waves traveling at the speed of light (an example of a pure mathematical construction predicting a yet-to-be discovered physical phenomenon); and Einstein using Riemann’s geometry to postulate his General Theory of Relativity (an example of pure mathematics employed for a completely unexpected natural phenomenon). One must remember that, no one, during Riemann’s time, thought the universe was other than Euclidean, which meant Riemann’s geometry was considered to be purely intellectual play with no possible application in reality.

This leads to a view held by Roger Penrose, which is expounded upon in a couple of his books, The Emperor’s New Mind and Shadows of the Mind, and is also referenced by Livio. Penrose makes the observation that there are 3 aspects of the world of mathematics, and they all interrelate. He refers to them as the ‘Platonic world’, the ‘Mental world’ and the ‘Physical world’. He depicts their interrelationship pictorially, a bit like the rock, paper, scissors game. I will try and explain.

Human thought can grasp some of the Platonic world, but not all of it, which we apply to the Physical world, even though some of it seems beyond our abilities of comprehension. Likewise the Physical world seems to incorporate some of the Platonic world but not all of it. So far so consistent. The enigma is turned into the rock, scissor, paper analogy when one realises that the human mind is a product of the physical world, which can then comprehend the Platonic world, of which the Physical world is a ‘shadow’. This is effectively the way Livio explains it as well, but without the analogy.

This leads me to what I consider to be the greatest mystery of the Universe: that it created the means to comprehend itself. As Einstein famously said: ‘The most incomprehensible thing about the universe is that it is comprehensible.’ Now I don’t know if Einstein was making a metaphysical statement, as he certainly wasn’t a Platonist like Penrose or his good friend and Princeton colleague, Kurt Godel. But without mathematics, it is clear that our comprehension of the universe would be severely limited indeed, and whether Einstein was referring specifically to that or not, I would suggest that his statement only makes sense in that context.

This leads to another point that Livio makes, almost in passing, but I consider to be highly relevant and it is to do with the most abused of concepts: ‘truth’.

I recently had an argument with a blogger (Armageddon Thru To You), who referred to
‘absolutes of truth’ which override all so-called scientific truths. I contended that the only truths I can be sure of are mathematical truths, and he responded: ‘There are absolutes of truth that humanists cannot understand because they don't acknowledge the existence of a perfect being (God)’.

This is relevant to Livio’s book, because he spends 6 pages on the subject of Galileo’s famous confrontation with the Catholic Church in 1633. He ends this section with the following commentary:

‘Still, at a time when there are attempts to introduce biblical creationism as an alternative “scientific” theory (under the thinly veiled title of “intelligent design”), it is good to remember that Galileo already fought this battle almost four hundred years ago – and won!’

 I make a similar reference to Galileo and compare it to the current ID/Creationism debate in my Nov.07 post, Is evolution fact? Is creationism myth?

These days, no one wants to get into an argument about whether the earth goes round the sun or not, so they think it’s irrelevant, but, as Livio points out, in 1600, the Catholic Church saw it as a direct intervention on its intellectual turf. It contradicted the Bible, and as far as they were concerned, the Bible was sacrosanct: the Word of God or ‘absolute truth’, to quote my aforementioned interlocutor. I’ve no doubt that there will come a day when evidence of evolution will be so monumental that only the most die-hard fundamentalists will question it – in other words, it will be no more contentious than Galileo’s position is today.

My point being that one can’t read a text like Livio’s and not be struck by how much we have learnt from the study of science, none of which appears in the Bible or any other religious text. An appeal to ‘absolute truth’ not only rings hollow in the face of thousands of years of accumulated knowledge, it shouts credulity. Mathematics has provided us with more truths, both in abstraction and in physical reality, than any other endeavour. To attempt to trump it with ‘absolute truth’ is to make a mockery of the human intellect.

And this brings me, to what I consider, to be the most revelatory portion of Livio’s book. The relevance of the study of knots to our understanding of DNA. Livio explains how a mathematical ‘discovery’ by John Horton Conway in the 1960s also describes the way enzymes ‘unknot’ DNA to allow for ‘replication or transcription’ (Livio’s description).

DNA is one of the greatest mysteries of the universe – a code for life itself. It raises the metaphysical question of why does the universe exist? Most scientists and philosophers would say it is an accident. Paul Davies attempts to address the question in his book, The Goldilocks Enigma, by expanding on an original idea proffered by his mentor, John Wheeler, that there is a causal loop between conscious intelligence and the universe itself. In a way, it attempts to address the mystery I alluded to in Einstein’s famous quote: that the universe created the means to comprehend itself.

The answers to this question are neither biblical nor scientific – they are philosophical. And perhaps it’s best to quote Bertrand Russell from The Problems of Philosophy, as Livio does in his closing paragraph of his book:

‘Thus, to sum up our discussion of the value of philosophy; Philosophy is to be studied, not for the sake of any definite answers to its question, since no definite answers can, as a rule, be known to be true, but rather for the sake of the questions themselves; because these questions enlarge our conception of what is possible, enrich our intellectual imagination and diminish the dogmatic assurance which closes the mind against speculation; but above all because, through the greatness of the universe, which philosophy contemplates, the mind is also rendered great, and becomes capable of that union with the universe which constitutes its highest good.’ (Emphasis is mine)

P.S. I sent this post as a link to Mario Livio, and I've posted his response in a comment below.

Addendum: I came across this very relevant quote in It Must be Beautiful; Great Equations of Modern Science, edited by Graham Farmelo (Granta books, 2002).

'One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.' (Heinrich Hertz, on Maxwell's equations of electromagnetism.)


Mario Livio said...

Dear Mr Mealing,

Thank you very much for your note.

I have read your blog with great interest, and I greatly appreciate your detailed and thoughtful comments.

I can only hope that many people will read your very interesting discussion.

All the best,

Mario Livio

PK said...

To me the greatest mystery in the universe (one which Hofstadter comes closest to managing, howsoever obliquely, to apprehend) is not comprehension, nor even the existence of physical systems or beings that commit ratiocination, but just sentience itself. There's a tautology here, as you yourself observed, since absent the coadunate faculties of awareness and intelligence, neither I nor anyone else could make that particular observation (or any other observation). You've said it: once somebody utters the word, "I," the rest of Descartes' proclamation is implicit, and can safely be omitted for the sake of verbal economy.

I *am* still intrigued, though, by the phenomenon of sentience, whether manifested in conjunction with the level of intelligence of a human, or that of a Galapagos turtle.

Mathematical systems are just sets of rules for the manipulation of symbols, and those symbols can be runic icons scratched onto scraps of parchment, electronic representations of ones and zeros resident in the RAM of this or some other computer, physical objects manipulated by mechanical systems, constellations of thought-objects in the form of dendritic connectivities, or anything else imaginable that may or may not exhibit the property of isomorphism to some aspect of the observable universe, even when that property is discovered ex post facto, the system of symbol manipulation having been created strictly for its own sake as an intellectual diversion.

The universe is a system of objects subject to rules of interaction that is isomorphic to itself, so if all we're looking for is the discovery of a "math" that's isomorphic to reality in the sense that it reveals hitherto undiscerned facts about the universe, we've already got one: which is the universe, itself.

I've always been fascinated by the idea of semantics. To my mind, there are only isomorphisms linking one syntax to the next, representing, perhaps, "turtles all the way down,' but never reaching some numinous, non-syntactic thing (what would such a thing look like?) called a "semantics." I did a lot of work in the early days of MT, and it always involved the tripartite model of syntactic source-->intermediate semantic representation-->syntactic target, but what was the "semantic representation," really? It was just another syntax, differentiated only by its ostensibly lesser proximity to any particular human language.

Anyway, I believe that sentience and comprehension do exist, and that they're not (at least the former isn't) strictly syntactic (i.e., systems of symbolic or physical object manipulation), but they might be epiphenomena of those. The most intractably rationalistic part of me, though, sees in mathematics only machines that reformulate strings of symbols, and though I wouldn't ever endorse Searle's reductio ad absurdum, I do have trouble finding an "effectiveness" that is unreasonable in any preternatural sense.

That having been said, I really very much enjoyed the post.

Paul P. Mealing said...

Thanks PK,

You are nothing if not a deep thinker.

A couple of points you raise, which I've considered elsewhere.

Mathematics being isomorphic to reality is something I've alluded to myself. In a previous post, written Sep.07 Is mathematics invented or discovered? I took issue with
George Lakoff's use of the term metaphor to explicate all mathematical discoveries or inventions. I asked the rhetorical question: metaphors for what? My rhetorical answer: metaphors for reality.

In regard to sentience, I asked the question on Stephen Law's blog, in a forum discussing Dawkin's The God Delusion, 'What's the point?' in the context that the God question was the wrong question. (I'm not averse to the God concept per se, by the way, only to the way people tend to wield it.)

In other words, what's the point of the universe's existence?

There are only 2 answers to that question: there is no point; or the point is consciousness, because that's the end result. By consciousness I mean sentience - I don't make a distinction.

As I quoted Russell (which I stole from Livio), in philosophy there are not necessarily right or wrong answers, there are only points of view which we attempt to support via our intellectual exertions.

Glad you enjoyed the post. I hope I can continue to entertain.

Regards, Paul.