Paul P. Mealing

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Saturday, 19 February 2011

Metaphysics in mathematics revisited

I recently wrote a post on E. Brian Davies’ book, Why Beliefs Matter (Metaphysics in mathematics, science and religion). Davies is Professor of Mathematics at Kings College London, so his knowledge and erudition of the subject far outweighs mine. I feel that that imbalance was not represented in that post, so this is an attempt to redress it.

Davies’ book is structured in 5 parts: The Scientific Revolution; The Human Condition; The Nature of Mathematics; Sense and Nonsense; and Science and Religion.

Davies addresses mathematical Platonism in 2 parts: The Human Condition and The Nature of Mathematics. Due to the nature of my essay, I believe I gave him short thrift and, for the sake of fairness as well as completeness, I seek to make amends.

For a start, Davies discusses Platonism in its wider context, not just in relation to mathematics, but in its influence on Western thought, regarding religion as well as science. Many people have argued that Aquinas and Augustine were both influenced by Platonism, to the extent that Earth is an imperfect replica of Heaven where the perfect ‘forms’ of all earthly entities exist. There is a parallel view expressed in some interpretations of Taoism as well. Note that one doesn’t need a belief in ‘God’ to embrace this viewpoint, but one can see how it readily marries into such a belief.

Davies discusses at length Popper’s 3 worlds: World 1 (physical); World 2 (mental); and World 3 (cultural). Under a subsection: 2.7 Plato, Popper, Penrose; he compares Popper’s 3 worlds with Penrose’s that I expounded on in my previous post: Physical, Mental and Mathematical (Platonic). In fact, Davies concludes that they are the same. I’m sure Penrose would disagree and so do I.

There is a relationship between mathematics and the physical world that doesn’t exist with other cultural ideas. Even non-Platonists, like Paul Davies and Albert Einstein, acknowledge that the correlation between mathematical relationships and physical phenomena (like relativity and quantum mechanics for example) is a unique manifestation of human intelligence. In his book, The Mind of God (a reference to Hawking’s famous phrase) Paul Davies devotes an entire chapter to this topic, entitled The Mathematical Secret.

On the other hand, Brian Davies produces compelling arguments that mathematics is cultural rather than Platonic. He compares it to other cultural entities like language, music, art and stories, all of which are products of the human brain. In one of his terse statements in bold type he says: Mathematics is an aspect of human culture, just as are language, law, music and architecture.

But, as I’ve argued in one of my previous posts (Is mathematics evidence of a transcendental realm? Jan. 08) there is a fundamental difference. No one else could have written Hamlet other than Shakespeare and no one else could have composed Beethoven’s Ninth except Beethoven, but someone else could have discovered Schrodinger’s equations and someone else could have discovered Riemann’s geometry. These mathematical entities have an objectivity that great works of art don’t.

Likewise I think that comparisons with language are misleading. No one has mathematics as their first language, unless you want to include computers. Deaf people can have sign language as a first language, but mathematics is not a communicative language in the same way that first languages are. In fact, one might argue that mathematics is an explanatory language or an analytic language; it has no nouns or verbs, subjects and predicates. Instead it has equalities and inequalities, propositions, proofs, conjectures and deductions. Even music is more communicative than mathematics which leads to another analogy.

Is music the score on the page, the sounds that you hear or the emotion it creates in your head? Music only becomes manifest when it is played on a musical instrument, even if that musical instrument is the human voice. Likewise mathematics only becomes manifest when it is expressed by a human intelligence (and possibly a machine intelligence). But the difference is that mathematical concepts have been expressed by various cultures independently of each other. Mathematical concepts like quadratic equations, Pascal’s triangle and logarithms have been discovered (or invented) more than once.

Davies makes the point that invention is a necessary part of mathematics, and I wouldn’t disagree. But he goes further, and argues that the distinction between invention and discovery cannot be readily drawn, by comparing mathematics to material inventions. He argues that a stone axe may have been the result of an accidental discovery, and Galileo’s pendulum clock was as much a discovery as an invention. I would argue that Galileo discovered a principle of nature that he could exploit and people might say the same about mathematical discoveries, so the analogy can actually work against Davies’ own argument if one rewords it slightly.

In my previous post, I did Davies an injustice when I referred to his conclusion about mathematical Platonism being irrelevant. In section 3.2 The Irrelevance of Platonism, Davies explains how some constructivist theories (like Jordan algebras) don’t fit into Platonism by definition. I don’t know anything about Jordan algebras so I can’t comment. But the constructivist position, as best I understand it, says that the only mathematics we know is what we’ve created. A Platonist will argue that the one zillionth integer of pi exists even if no one has calculated it yet, whereas the constructivist says we’ll only know what it is when we have calculated it. Both positions are correct, but when it comes to proofs, there is merit in taking the constructivist approach, because a proof is only true when someone has taken the effort to prove it. This is why, if I haven’t misconstrued him, Davies calls himself a mathematical ‘pluralist’ because he can adjust his position from a classicalist to a formalist to a constructivist depending on the mathematics he’s examining. A classicalist would be a Platonist if I understand him correctly.

I still haven’t done Davies justice, which is why I recommend you read his book. Even though I disagree with him on certain philosophical points, his knowledge is far greater than mine, and the book, in its entirety, is a worthy contribution to philosophical discourse on mathematics, science and religion, and there aren’t a lot of books that merit that combined accolade.

The forgotten man

This is excellent journalism, whatever your view is on the story. It makes me angry, because the person being punished is allegedly the person who brought us the famous video footage showing ‘collateral damage’ in Iraq, which Assange called ‘collateral murder’. Is he any different to the guy who attempted to stop the tanks going to Tiananmen Square? In both cases they have effectively disappeared and become enemies of the state in their own countries.

As the title of the programme says, Private Bradley Manning has become ‘the forgotten man’, as all news coverage focuses on the indictment of Julian Assange for an alleged double rape in Sweden.

I won’t make any character or personality judgements concerning Assange because they are irrelevant to the issue. Assange may be narcissistic and he may be a delusional crusader, but it doesn’t change the case against him or the arguments concerning his journalistic rights to make public, information that may embarrass heads of government. Because, as far as I can tell, that’s exactly what he’s done.

When this first came to a head, i.e. information was leaked, our (Australian) government toed the American party line and told us that what Assange had done was dangerous, jeopardised national security and put lives at risk in the field of combat. But, after examining the evidence, the Attorney General’s Department issued a statement saying Assange had done nothing illegal under Australian law.

It should be stated that, in Australia, Assange has a lot of support, especially from journalists. All journalists know that if they had obtained the same information they would have done the same thing. Whistleblowers are always persecuted by the body that they’ve betrayed, because you can’t whistleblow without betraying the hand that feeds you. Democracies like to think that they are fairer than other countries but if you whistleblow on your government, then, even in a democracy, you won’t escape the full force of the law they can bring to bear upon you. This is true of Australia just as it is of America.

It is evident from the 4 Corners programme (refer link) that they are attempting to break Manning through torture (solitary confinement 23 hrs a day is torture) so that he will turn evidence against Assange for espionage.

Assange’s barrister, Geoffrey Robertson QC, argues that Assange won’t get a fair trial in Sweden and it will be a closed court. Assange believes that the case in Sweden is really a ploy to get him to America so they can put him on trial for espionage. Robertson (another ex-pat Aussie) is a well known human rights lawyer and famously took on Salman Rushdie’s case when he was issued a death sentence fatwah by Iran’s Ayatollah Khomeini in 1989.

What’s most alarming in the entire programme, is footage from FOX News showing right wing political commentators recommending, on American national television, that Assange should be ‘taken out’ by CIA operatives.

The solution to unwanted news in America is apparently to shoot the messenger, literally.

Sunday, 6 February 2011

Metaphysics in mathematics, science and religion

Why Beliefs Matter; Reflections on the Nature of Science, by E. Brian Davies, is one of the best books I’ve read on science, philosophy and religion, and I’ve read lots of books in all those fields. Davies is Professor of Mathematics at King’s College London and a fellow of the Royal Society. He gives one of the best arguments I’ve encountered against mathematical Platonism, which is high praise indeed from a self-confessed mathematical Platonist like myself.

There is much in this book to be commended, not least his conscientiousness in separating philosophy from science and of pointing out that ‘beliefs’ like the anthropic principle are, in fact, metaphysical considerations rather than truly scientific (it can’t be tested). He outlines the significant difference between the philosophical and scientific ramifications of quantum mechanics, which I’ve expressed myself in a post on Science, Philosophy, Religion (November 2009).

More than anything else, he reinforces the intellectual reality that philosophy often deals with questions for which there may well be no definite answers. And whilst science can provide answers in the form of empirical evidence as well as mathematically based laws to explicate them, the bigger questions, concerning our existence, the origin of the universe and a potential higher purpose, remain elusive.

The scope of Davies' book includes the history of science, the mind-body problem, induction, determinism, artificial intelligence and the modern day ‘warfare’ between science and religion, especially in America (this is not an exhaustive list). I’ll only cover 2 apparently unrelated topics: mathematical Platonism and religion and science.

Davies has no particular barrow to push, and is candid in his disagreement with his fellows on all topics, expressing bewilderment, bordering on amusement, at the hostility one often encounters concerning questions for which there are no definitive answers. One such topic is the philosophy of mathematics and its various ‘schools of thought’ that borders on religious zeal. He calls himself a mathematical ‘pluralist’ because he can see merit in alternative views. As far as mathematical Platonism goes, he expresses appreciation of its appeal to both mathematicians and physicists without necessarily agreeing with them. In his conclusion he calls it ‘irrelevant’, but only because it doesn’t really provide any theoretical benefit. In other words, being a Platonist won’t give you an advantage in understanding mathematics – it’s purely a philosophical position, with no real practical ramifications in executing formulae or even searching for new ones.

He points out that mathematical Platonism has quasi-religious overtones, which I don’t shy away from. I’ve written at least 3 posts previously on this topic, so I won’t labour the point here. It’s a very good example of a philosophical position based more on a ‘feeling’ or ‘sense’ of abstract reality, which its proponents (like myself) then support with rational argument. One of Davies’ strongest arguments is that we are the only species (that we know of) in the entire universe who can not only appreciate mathematics but make it manifest. Without an intelligence like ours, it remains completely hidden which makes its apparent essentiality questionable.

I have 2 not-unrelated responses to this argument. Firstly, all the laws of the universe, that we have discovered, from quantum mechanics to relativity to thermodynamics to the DNA code, would remain complete secrets in the exact same way, yet the universe, that we observe and exist within, is completely dependent on all these things. Secondly, mathematics is the only way we can quantify and interpret these very same laws, which leads me to contend that the mathematics is just as essential as the laws themselves. DNA is a 4 letter code, by the way, that is completely analogous to computer code, so life entails mathematics at a fundamental level.

The alternative view to this is that mathematics is an intellectual construct, purely of human origin, that has allowed us to unravel some of nature’s deepest mysteries. Roger Penrose, whose Platonist philosophy is discussed in some detail by Davies, manages to incorporate both views in a non-contradictory though paradoxical manner, which is what sold me on mathematical Platonism eventually (see below). In other words, I am a convert who came to this position via Pythagoras. In my early years of studying science, I saw mathematics as a tool that physics had seconded, but even then I struggled to reconcile natural laws with their apparent and deeply enigmatic mathematical precision (more on this below).

Davies postulates a hypothetical that there may be a species somewhere in the universe who can fathom nature’s secrets heuristically without mathematics. I can remember, when I was much younger, contemplating the same scenario and even entertained writing a sci-fi story that incorporated such a species. However, I gave up on the enterprise, when I realised that, philosophically, my world-view had changed. Physics, especially quantum mechanics, is so fundamentally dependent on mathematics for its interpretation, that any other methodology appears impossible, which is not to say that it is. Whilst quantum mechanics remains a conundrum in terms of envisaging the ultimate reality of the universe (or universes), it remains, mathematically, a completely consistent and eminently reliable metatheory.

Of course Davies’ discussion on this topic is much more comprehensive than what I’ve presented. I’ve just re-read my post on Schrodinger’s book, What is Life? and his quote concerning mathematics, “…whose truth is not only unassailable, but is obviously there forever; the relations held and will hold irrespective of our inquiry into them. A mathematical truth is timeless, it does not come into being when we discover it.” Davies also quotes Einstein, who wasn’t a Platonist: “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality.” This neatly encapsulates the paradoxical nature of mathematics that I described above.

Davies disputes the ‘independence’ claim because geometry and arithmetic didn’t evolve independently of the physical world that we’ve investigated since we first learnt to count. He also argues that not all ‘objects of reality’ can be defined or described mathematically, but he’s talking about ‘...the contents of our conversations and the products of our culture, in which mathematics is completely useless.’ No one would argue with that, but it’s obvious that Einstein’s ‘objects of reality’ are physical objects rather than cultural artefacts and mental constructs specific to the human race.

I disagree with Davies on his first point too, because mathematical investigations, at least since Newton, have evolved independently of the physical world only to be married back into it when our science has caught up with our mathematics. The most famous examples would include Riemann’s geometry being married to Einstein’s general theory of relativity and complex algebra being essential to Schrodinger’s equations for quantum mechanics. It’s as if mathematical discoveries precede physical discoveries, and, in fact, necessarily so.

There is a sense that mathematics entails a world (at least in the abstract realm) greater than our world, with multiple dimensions that can extend to infinity, of which string theory represents a potential multitude. In other words, there is more mathematics than we need to describe our physical theories, which is why Max Tegmark argues that all mathematically possible universes could exist in a multiverse. If one takes this at face value, then the mathematical world extends beyond the physical world (as Penrose points out) in the same way that the physical world extends beyond our mental world. Hence the paradox that the mind is an effective subset of the physical world, and even if the physical world is not a subset of the mathematical world (as per Tegmark), it appears, at least, to follow mathematical rules, yet mathematics is a product of the human mind. Penrose represents this relationship between the physical, mental and mathematical (Platonic) worlds as a closed circuit, one being a subset of the one before, just as I’ve described them above. Davies also addresses this aspect of Penrose’s philosophical discourse in his book, but I’ll leave that for the reader to pursue if they’re interested.

In a section called The Human Condition, Davies introduces the subject of ‘induction’ by referring to Aristotle’s ‘four types’ of causes that he recorded and discussed in the 4th Century BC. He did this in reference to a clay pot. The ‘material cause’ is the materials that the pot is made from which is the clay. The ‘formal cause’ is its shape or form which is a pot or vase. The ‘efficient cause’ is the process involving the Potter who made it. And the ‘final cause’ is the whole reason it was made which is to store something.

I have to admit I’d never come across this before, despite having read and studied Aristotle at Uni, and Davies makes particular reference to the 4th ‘final cause’ which has disappeared in the philosophy of science, and is arguably the principal source of friction that lies between science and religion. Davies rightly points out that since Descartes, and even more so after Darwin, final cause has no place in science. This is a particular issue of contention I've had with many fundamentalists, like William Lane Craig (refer The God Hypothesis, December 2008). Even if there is a final cause for the universe, science can't tell us anything about it – it’s purely a metaphysical question.

Aristotle’s final cause refers to a human artefact, and it’s not difficult to see how God became an anthropomorphic equivalent who created the universe, life and us, which means we are the final cause. I really don’t have a problem with this, purely from a philosophical viewpoint, because it makes God dependent on us rather than the other way round. If we are the final cause then, without us, there is no reason for God to exist. Few people appreciate the reverse logic that this argument entails: it doesn’t make sense for God to exist without a purpose, and the only purpose we can come up with is us.

In a recent post (Cycles of Time, last month) I gave considerable space to the exposition of entropy, aka the 2nd law of thermodynamics. A corollary to the 2nd law is that the universe is not teleological and by inclusion neither is evolution. I would suggest that this, and not the Book of Genesis, is the main philosophical difference between science and religion. Religion infers that the universe has a purpose and science infers that it doesn’t.

Davies expounds at length on the indeterminism inherent in chaos theory as well as quantum mechanics. Another Davies (Paul Davies), when he still resided in Australia, wrote an excellent book on chaos theory called The Cosmic Blueprint. The significance of chaos theory, and its particular relationship with entropy, is that very small changes can lead to huge differential consequences. In a not-so-recent issue of New Scientist (16 October 2010) their feature article described how chaos theory appears to rule evolution. In particular, evolution is fractal in the same way that branching blood vessels are in the human body. Fractal relationships appear everywhere in nature; the best example being a coastline (Davies’ example in Cosmic Blueprint). The Mandelbrot set is fractal and so are Pollock’s paintings (like Blue Poles hanging in the Sydney Art Gallery). Fractals demonstrate the same relationship at all scales, which means, in evolutionary terms, that speciation branches appear in similar ratios at all levels. The article explains how, over 65 million years, major climatic events, major tectonic events and major evolutionary events all follow the same ‘chaotic’ patterns, though ‘...connections between them are hard to discern.’

Brian Davies, like Schrodinger (What is Life?), explains how radioactivity is statistically highly predictable whilst individualistically it is impossible to predict. In fact, Schrodinger begins his book with an exposition on how almost everything in physics is statistically determined: from magnetism to the photo-electric effect to the behaviour of gases and fluids. It’s only at a macro scale that physics appears predictable. The point is that between chaos theory, the 2nd law of thermodynamics and quantum phenomena, the universe is a lottery. As Stephen Jay Gould famously said, if you were to rerun the universe you’d get a completely different result. This flies in the face of all religious philosophy.

The last 60 pages of Davies’ 240 page book (so 25%) is devoted to a section titled, Science and Religion. He starts off with a philosophical aphorism: “We must learn to live with the fact that some disagreements cannot be resolved.” Throughout his book he places terse statements in bold type like the following:

Christian theologians ignore the fundamentalist challenge at their own peril. It is the greatest threat to rational thought and toleration at the present time.

To outsiders like myself, America appears to be one of the most polarised societies in the Western world: politically, intellectually and religiously. The all-consuming debate between evolutionary science and fundamentalist religion really doesn’t exist anywhere else in the world, certainly not to the same degree of hostility and, dare-I-say-it, desperation. It’s only taken on a global perspective because American culture is so pervasive, especially on the internet.

Davies points out that humanist philosophy goes back even further than Christianity, citing Socrates, Aristotle, Plato and even Confucius. Confucius is the earliest known philosopher (500 BC) to evoke a fundamentally empathetic approach to ethics: ‘Don’t do to others what you wouldn’t want done to yourself.’ He acknowledged the importance of trust between rulers and their subjects, arguing that trust was the last commodity a ruler could afford to lose. (Someone should point this out to Egypt’s Mubarak.) Davies argues that Mill's utilitarian philosophy has probably been the biggest influence on Western democracy, because it’s inherent in civil rights and feminist movements witnessed in the last half of the last century. Even though no one invoked Mill as the model to follow; utilitarianism is concerned with the greatest benefit to the greatest number.

At the end of his book, Davies discusses the religious views of famous scientific figures, both historical and contemporary. He is not afraid to criticise Dawkins’ The God Delusion, even though he obviously is not completely at odds with Dawkins’ philosophy. Dawkins polarises people almost like no one I know, yet he’s neither a villain nor a hero. He has a demeanour not unlike an Australian politician: provocative, rhetorically aggressive, disputatious, uncompromising and unapologetic. On the blogosphere, if you criticise Dawkins, as I have done a few times, you suddenly become a Christian apologist to his supporters. It’s a sign of insecurity that people can’t deal with criticism without adopting an extreme position. Davies, like myself, takes Dawkins to task for treating all religions and all religious followers the same. It doesn’t help his cause to alienate people who would otherwise support him. ‘The worst feature of Dawkins’ book is its failure to get to grips with the variety of religious belief. Dawkins’ real enemy is fundamentalism, but he attacks religion indiscriminately.’ I agree completely.

Davies ends with a poem by William Cecil Dampier, from which I’ll quote the last verse:

And Nature smiles – still unconfessed
The secret thought she thinks –
Inscrutable she guards unguessed
The Riddle of the Sphinx


Davies follows with these words:

The riddle of our place in the universe may never be solved, and I am content that this should be so. The struggle to divine the meaning of life is a part of being human.

Science can’t solve this riddle either; in fact it tells us that our existence is a completely arbitrary phenomenon built upon an accumulation of arbitrary phenomena. The end result (so far) is mind and mind seeks its own purpose because that’s its nature.

Addendum 1: I need to point out, in all fairness to Davies, that his discourse on mathematics is far more erudite than mine, which is not apparent from my presentation above. I attempted to address this in a later post, Metaphysics in mathematics revisited.

Addendum 2: In March 2012, I give a more definitive response to the question of teleology after re-reading Paul Davies' Cosmic Blueprint and blogging about it.