Time and physics

Einstein's rejection of the notion of time as we know and experience it was squarely based in classical physics and classical mathematics. One problem with such a view is that it assumes the existence of infinite information (e.g. infinite decimal expansions).

Nicolas Gisin, a physicist at the University of Geneva, wants to reformulate standard physics in terms of intuitionistic mathematics. This approach holds the promise of resolving some of the paradoxes and confusions which have bedevilled theoretical physics for over a century.

Information is physical. We now know that there are strict limits on how much information can exist within any specific volume of space.

Nathalie Wolchover writes: "The universe’s initial conditions would, Gisin realized, require far too much information crammed into too little space. “A real number with infinite digits can’t be physically relevant,” he said. The block universe, which implicitly assumes the existence of infinite information, must fall apart."

Wolchover's non-technical article on Gisin's ideas and reactions to them by fellow physicists is well worth reading. This is how it begins:

Strangely, although we feel as if we sweep through time on the knife-edge between the fixed past and the open future, that edge — the present — appears nowhere in the existing laws of physics.

In Albert Einstein’s theory of relativity, for example, time is woven together with the three dimensions of space, forming a bendy, four-dimensional space-time continuum — a “block universe” encompassing the entire past, present and future. Einstein’s equations portray everything in the block universe as decided from the beginning; the initial conditions of the cosmos determine what comes later, and surprises do not occur — they only seem to. “For us believing physicists,” Einstein wrote in 1955, weeks before his death, “the distinction between past, present and future is only a stubbornly persistent illusion.”

The timeless, pre-determined view of reality held by Einstein remains popular today. “The majority of physicists believe in the block-universe view, because it is predicted by general relativity,” said Marina Cortês, a cosmologist at the University of Lisbon.

However, she said, “if somebody is called on to reflect a bit more deeply about what the block universe means, they start to question and waver on the implications.”

Physicists who think carefully about time point to troubles posed by quantum mechanics, the laws describing the probabilistic behavior of particles. At the quantum scale, irreversible changes occur that distinguish the past from the future: A particle maintains simultaneous quantum states until you measure it, at which point the particle adopts one of the states. Mysteriously, individual measurement outcomes are random and unpredictable, even as particle behavior collectively follows statistical patterns. This apparent inconsistency between the nature of time in quantum mechanics and the way it functions in relativity has created uncertainty and confusion.

Over the past year [...] Nicolas Gisin, has published four papers that attempt to dispel the fog surrounding time in physics. As Gisin sees it, the problem all along has been mathematical. Gisin argues that time in general and the time we call the present are easily expressed in a century-old mathematical language called intuitionist mathematics, which rejects the existence of numbers with infinitely many digits. When intuitionist math is used to describe the evolution of physical systems, it makes clear, according to Gisin, that “time really passes and new information is created.” Moreover, with this formalism, the strict determinism implied by Einstein’s equations gives way to a quantum-like unpredictability. If numbers are finite and limited in their precision, then nature itself is inherently imprecise, and thus unpredictable. [...]

On this view the future is open (rather than closed or predetermined), and time is closer to how we experience it – and so intuitively envisage it to be – than most physicists have supposed.

Anti-metaphysical musings

I have been looking recently at some material relating to "the metaphysics wars", and thought it worthwhile to jot down a few notes.

No doubt, my general position would be characterized by those with other views as scientistic. It is also anti-metaphysical in that I don't see the traditional philosophical discipline of metaphysics as having much point these days.

I don't deny that there are very interesting questions in the philosophy of physics, the philosophy of mathematics and the philosophy of logic which may be characterized as metaphysical. The meta-thinking that goes on at the margins of physics, other sciences and mathematics, etc. is necessary and valuable.

But somehow, when such thinking moves away from the discipline in question and becomes more generally philosophical, problems arise.

Timothy Williamson is perhaps the most powerful and impressive advocate for this broader kind of metaphysics (and analytic philosophy generally). As an avowedly non-religious person, he can't be dismissed as having ulterior motives of a religious nature; and, being at home with formal – and specifically modal – logic, he can't be dismissed as natural language-bound or as being daunted in any way by technical rigor.

Some of the points he makes in this interview are good ones – such as noting the light that modal logic can undoubtedly throw on the workings and nature of natural language (via Montague grammar, for example), and perhaps also on the foundations of set theory – but I have to say that I am strongly inclined to reject the basic thrust of his argument in defense of metaphysics, and, by extension, philosophy (as he understands it).

Essentially the questions he seems most interested in are reminiscent of medieval scholasticism. I too have great respect for thinkers such as Avicenna (to whom he refers approvingly) and respect also for more recent – and more mathematically sophisticated – exponents of that general tradition of thought (such as Bolzano, to whom he also refers), but it seems to me that it is now incumbent upon any thinkers who aspire to deal with questions of what there is in a fundamental sense to base their accounts – at least in large part – on contemporary physics; or on mathematics if they are restricting their focus to mathematical realities.

Williamson seeks to defend the relative independence of his core preoccupations from science by invoking the old shibboleths, scientism and reductionism, and rejecting naturalism as a confused and inadequate concept.

I grant that mathematics does pose problems for advocates of strong forms of naturalism and empiricism, and there are real unresolved issues in the philosophy of mathematics. But my preference is to address these issues in a broadly scientific and mathematical context rather than in a purely logical or philosophical one, or – worse – not to address them at all and instead merely to use them as a kind of justification or license for logical excess and metaphysical self-indulgence.

Williamson cites Quine as an example of scientistic naturalism.

"Quine privileged natural science, and in particular physics, over all other forms of inquiry, to the point of not taking very seriously any theory that couldn't be reduced to part of natural science."

Williamson's view, by contrast, more or less allows the analytic metaphysician carte blanche, and Williamson's own approach to analytic metaphysics is clearly – in my view at any rate – insufficiently constrained and guided by science.

Here, for example, is an extract from an old interview in which he explains his developing views:

"My work on vagueness and ontology doesn’t really concern ontology. Probably my most distinctive ontological commitment comes from my defence of a controversial principle in logic known as the Barcan formula, named after the American logician Ruth Barcan Marcus, who first stated it. An application of this principle is that since Marilyn Monroe and John F. Kennedy could have had a child (although they actually didn’t), there is something that could have been a child of Marilyn Monroe and John F. Kennedy. On my view, it is neither a child nor a collection of atoms, but rather something that merely could have been a child, made out of atoms, but actually has no location in space and time. The argument can be multiplied, so there actually are infinitely many things that could have been located in space and time but aren’t. It takes quite a bit of work to show that the Barcan formula is better than the alternatives! That’s what my next book will be on. The working title is Ontological Rigidity."

The book was actually called Modal Logic as Metaphysics, and this is how he recently stated its main point:

"I am ... saying that it is necessary what there is. Necessarily everything is necessarily something. There could not have been more or fewer things than there actually are, and which particular things there are could not have been different. What is contingent is only what properties those things have, and what relations they have to each other. I call that view necessitism. Its denial is contingentism. Who knows how far back necessitism goes? Maybe Parmenides was some sort of necessitist..."

On the face of it, talking about (apparently countable) things (minus their properties and relations!) as given strikes me as breathtakingly naïve in the context of a physics-based understanding of reality. I can only imagine that Williamson is – like the medieval scholastics – implicitly asserting a privileged role for logic.

Quine's assertion of a privileged role for physics makes a lot more sense to me.

Admittedly I haven't looked at Williamson's ideas in any depth, but what I have seen so far – and what he says in this latest interview – really makes me question whether it would be worth the effort. I am intrigued, however, by what is driving such thinkers.

Strangely, Williamson appears not to be quite sure whether his latest work is meaningful or not – or at least seems unwilling to commit himself on the matter. There is (don't you think?) just a touch of arrogance in this passage (from Chapter One of Modal Logic as Metaphysics)?

"This book compares necessitism and contingentism. Which is true? Of course the question has a false presupposition if the definitions of 'necessitism' and 'contingentism' lack meaning or content. But if every enquiry must first establish its own meaningfulness we are on an infinite regress, since the enquiry into the meaningfulness of the previous enquiry must first enquire into its own meaningfulness, and so on. Better to act on the assumption of intelligibility: readers can decide for themselves whether they understand the book as they go along, and recycle it if they don't."

This passage is a combination of facile reasoning and rhetorical sleight of hand. By using the word 'understand' in the final sentence, he subtly shifts the focus to the reader's possible inadequacy and away from the original question concerning the work's meaningfulness.

In fact, I am tempted to see Williamson's work as emblematic of a broader trend. On the basis of my (admittedly limited) knowledge of the history of the relevant intellectual cultures, I discern, since the middle years of the 20th century, a disturbing falling off in intellectual seriousness in secular circles accompanied by an equally disturbing rise in anti-scientific name-calling and credulity amongst those thinkers who remain favorably disposed towards religion.

I'll finish here with a few comments about Paul Horwich, Williamson's great philosophical antagonist, whose deflationary views on truth I have referred to favorably in the past.

Horwich is opposed to the sort of traditional theoretical philosophy ('T-philosophy') which Williamson defends. I have made the point that, though I broadly accepted Horwich's account of truth, I doubted that his Wittgensteinian view of philosophy was compatible with a continuation of philosophy as an academic discipline. And, interestingly, Williamson makes a similar point in the recent interview.

"...Horwich didn’t explicitly call for T-philosophy not to be funded. I pointed out that if the picture of philosophy in his book were accurate, philosophy should be abolished. The reader encounters just two sorts of philosophy: irrational T-philosophy, and level-headed Wittgensteinian debunkers of T-philosophers. Philosophy is presented as an activity in which some people make a mess and others clear it up. Why on earth should taxpayers fund that? It looks as though we’d be better off simply abolishing the activity altogether."

Finally, I was surprised (and a bit disappointed) to learn recently that Horwich rejects naturalism, and even more unequivocally than Williamson does. He cites not only mathematical but also moral claims as a basis for his view.

Horwich is more thoroughly Wittgensteinian than I had previously thought.

Necessary freedom

The mathematician G.H. Hardy – most famous amongst the general public for his having 'discovered' the self-taught prodigy Ramanujan – said that the only other career that might have suited him was journalism.

When I first read this it surprised me, even bearing in mind the fact that journalism in early 20th-century England was very different from journalism today.

Clearly Hardy could write – his short book, A Mathematician's Apology, is a minor classic. But it's very clear from that essay that his identity was inextricably bound up with being a mathematician, and nothing else.

Late in life he attempted suicide, not just because of the general effects of failing health but also – and perhaps mainly – because his mathematical powers had deserted him.

Rather depressingly, he claimed (in his Apology) that most people don't have any significant talent for anything. But "[i]f a man has any genuine talent he should be ready to make almost any sacrifice in order to cultivate it to the full." Anyone, he asserted, who sets out to justify his existence and his activities has only one real defense. And that is to say, “I do what I do because it is the one and only thing that I can do at all well."

Why did he mention journalism, I wonder? It's particularly puzzling because journalism is so utterly different from mathematics generally – and especially from Hardy's style of doing and thinking about mathematics with its focus on timeless beauty.

This is in addition to the fact that mathematics is normally associated with the sciences. So, naïvely, I would expect a mathematician to say that, had he not pursued mathematics as a career, he might have become a scientist or engineer of some kind, for example.

But Hardy, though he was attracted to biology in his youth, exhibited in his adult life no great interest in or high regard for science, and he had a quite negative attitude to applied science. He prided himself on the fact (as he saw it) that his work had no practical applications.

And he disliked new technologies. He had a telephone installed in his house which he ostentatiously avoided using: it was for the use of any guests who fancied that kind of thing.


By journalism Hardy certainly didn't mean writing about scientific (or mathematical) subjects for a general audience. He meant, presumably, mainstream journalism. And my guess is that he was attracted to it for three basic reasons.

Firstly, he recognized that he had a second talent, a gift for writing – and writing with style and wit and conciseness. (He was famous amongst his friends for his postcards.)

Secondly, though scornful of politicians, he did have an interest in politics and was active in a pacifist organization, the Union of Democratic Control, during World War 1. Significantly, one of the leading and most impressive figures involved in this organization was the French-born journalist E.D. Morel.

And last but not least, I suspect that Hardy saw in the lifestyle associated with journalism (as in the academic lifestyle of the time) a kind of freedom which for a certain kind of person is not just desirable but necessary.

Does mathematics pose problems for physicalism?

For those who want to see the world as being comprised at a fundamental level entirely of such processes as physics and the other sciences might model or describe, mathematics seems to pose problems.

Some say that our moral sense, or love and emotion, or our perception of beauty somehow undermine a physicalist viewpoint. But they don't really. All these things can be understood as complex products of simpler physical processes (evolution, biology, social interaction, etc.).

And the realm of the mystic may be timeless, but is subjective – or at least cannot be shown to have an objective existence.

But mathematics seems to take us into a non-empirical but demonstrably objective realm.

Mathematics works in many ways like a branch of science (and of course is an inextricable part of science), but is essentially concerned with abstract patterns and relations rather than with the empirical world directly.

Nonetheless, as a human activity even pure mathematics is clearly a part of the empirical world.

In fact, as digital computers become more fully integrated into mathematical research and practice, and discrete mathematics continues (as I suspect it will) to replace continuous as the basis for our most plausible and accurate descriptions of reality (since physical reality at its most fundamental levels appears to be discontinuous), the view of mathematics as timeless and Platonistic will most likely fade.

Be that as it may, for various reasons many still adhere to a full-fledged mathematical platonism, and see the existence of mathematics and our access to mathematical truths (which on some views gives rise to what has become known as the 'access problem') as evidence of the inadequacy of physicalism; even sometimes as evidence for a religious or spiritual view of reality.

There are, of course, many competing philosophies of mathematics, some of them (like platonism) realist (in the sense of accepting the real existence of abstract mathematical objects), others anti-realist. In general, the former approaches seem more or less incompatible with physicalism, and the latter compatible.

As there is no scientific way of deciding between these approaches, the physicalist is really under no pressure. He or she can just point to one or other of those ways of seeing mathematics which do not entail accepting independently-existing abstract objects, etc.

I have alluded in the past to the anti-realist views of the mathematician Timothy Gowers. And I have just come across someone else whose views appeal to me.

Sharon Berry, who has recently completed work on her Ph.D. at Harvard*, is not an anti-realist or anti-platonist like Gowers, but her approach is basically empirical, and the platonism she countenances is sufficiently weak not to put me off too much.

Her dissertation is on the so-called access problem which she addresses in what seems like a refreshingly straightforward and down-to-earth way. She argues that mathematical knowledge can be reduced to 'knowledge of a kind of broadly logical possibility, that is possibility with regard to the most general principles about how any objects can be related by any relations.' She calls this notion 'combinatorial possibility', and claims we can account for our knowledge of combinatorial possibility 'by appealing to general constraints on relationships between concrete physical objects.'

What is particularly interesting about the notion of combinatorial possibility is that it is tightly tied to the empirical world: 'one can infer possibility from actuality.'

Our access to good (but incomplete) methods of reasoning about combinatorial possibility is explained by our experiences with concrete objects, and so – if indeed mathematical knowledge can be reduced to a knowledge of combinatorial possibility – our (partial) access to mathematical truth is also explained in terms of these ordinary experiences.

Berry believes that her approach to the access problem meshes neatly with a relatively robust approach to claims about mathematical objects. 'The key idea is that quantifiers can take on different senses in different contexts. These senses correspond to different standards that we might apply when assessing questions of existence.'

Her view is that lower standards operate in everyday contexts than are required in discussions of 'fundamental ontology'. And lower standards apply also in mathematical discussions.

So long as these higher and lower standards are seen in pragmatic terms (and not in terms of different kinds of objects actually having different degrees of being), this approach seems doubly attractive. It does justice to the subtleties of human communication as well as avoiding the implicit dogmatism of standard realist and anti-realist stances.

Berry herself, taking ontological and metaphysical discourse in general rather more seriously than I am inclined to, may not be entirely happy with my pragmatic interpretation. But her views do certainly reflect empirical and pragmatic tendencies.

As I said, there is probably no way to decide which, if any, of the available positions in the philosophy of mathematics are on the right track and which are not. Some look more plausible than others, it must be said, but such judgments are always going to be affected by prior metaphysical (or anti-metaphysical) tendencies and such like.

Which is not to say that all work in this area lacks significance. Arguably, both Gowers's and Berry's perspectives have significance and value.

As I suggested above, so long as there are plausible positions available which do not entail fully-fledged realism or platonism, the physicalist need not feel that his or her physicalist stance is under threat.

And, so, though I don't feel obliged to make (or capable of making) an unequivocal assessment either of Gowers's or of Berry's approach, I do value them both as possible (and, on the face of it at least, plausible) alternatives to full-blown mathematical platonism.


* Both a short and a long dissertation abstract are included in her CV which is available via her website.

Possible worlds and possible worlds

I have trouble seeing philosophy as an intellectual discipline. The gist of my thinking is that 'philosophy' is a word which has changed its meaning quite dramatically over the centuries as various sciences have split off from it and I'm not sure that it has much meaning left.

If one has a theological view of the world, philosophy's position will not be threatened as it can resume (or continue) its traditional role as a secular complement to theological discourse. But if one denies that there are truths we can intuit or know by non-empirical, non-deductive means, then, arguably, there is no place for a non-scientific intellectual discipline (unless it be seen as an art form or as a kind of game).

Of course, the study of formal deductive systems, logical or mathematical, is non-empirical, but it is continuous with science.

All that is left of philosophy for someone who rejects claims to substantive intuitive knowledge of a religious or moral kind are reflections on the various intellectual disciplines (physical, social and historical sciences, mathematics, logic, etc.). Such meta-thinking is best carried out (I would presume) by the practitioners of the various intellectual disciplines rather than by outsiders (whether or not they are designated as 'philosophers').

I do recognize, however, that much pure and applied work in certain disciplines (logic, mathematics, psychology and linguistics come to mind) draws strongly on philosophical traditions of thought, and raises issues which previously have been addressed by philosophers. An example of such work is the attempt (drawing on theoretical work in logic and mathematics as well as linguistics) to model the processes of natural language.

Computational linguistics clearly has great practical and commercial importance at the moment, but it can also be seen as a project the relative success [or failure!] of which has implications for the way we see human language – and ourselves.

My reading of the current state of play is that the formal approaches which followed in the wake of Chomsky's early attempts to give an explicit analysis of the syntax of ordinary language have not delivered as expected, just as early work in the field of artificial intelligence produced very disappointing results. Both of these research projects underestimated the importance of contextual factors and real world knowledge which is inevitably a part of intelligent human functioning and communication. Formal systems need in some way to be integrated into this real world context, but, even if they are, it is still possible that many important aspects of language and communication will remain out of reach. I am thinking in particular of aspects of language use which depend on social awareness, a sense of the sorts of things that people with autism spectrum disorders have trouble dealing with, including subtleties of tone and style.

There is a huge body of theoretical work in the syntax and semantics of natural language which shows, if nothing else, that there are countless ways of conceptualizing and formalizing (at least aspects of) natural language. In the light of this profusion, the key question – it seems to me – is not which theoretical approaches are true (whatever that might mean) but which are useful.*

We may want to postulate possible worlds and use set theory to model the semantics of natural language, including complex noun phrases and verb tenses and auxiliaries. But sets and 'possible worlds' are only one way (albeit a possibly enlightening one) of representing the way, for example, words like 'must' or 'could' or 'should' work. No claim need be made that such possible worlds exist. They are merely useful fictions.

Physicists, of course, also talk about other possible worlds, parallel universes and so on, but they are making ontological claims. Their concern is primarily with how the world (or the multiverse) is rather than with formal systems, though they use formal systems to model the operations of nature (as linguists may use formal systems to model the operations of natural language).

But the possible worlds of logicians and linguists are – notwithstanding some outlandish claims by certain logicians – merely formal constructs, to be judged entirely by their usefulness. The other worlds of the physicist may well prove in fact to be 'out there' – to exist in the normal sense of the word, though they may be inaccessible to us.

I am aware that the question of what existence consists in is a traditional philosophical one, but is it a serious or potentially productive question? I think not. Most of the confusions can be resolved simply by accepting that we use words like 'exist' in various ways.

The one area which does seem to raise important issues is mathematics. Just as there are possible worlds and possible worlds (the 'worlds' of the logician and the worlds of the physicist), so there are formal systems and formal systems, and, as we move from, say, the first-order predicate calculus to formal systems which can encompass arithmetic we cross a kind of threshold. Mathematics needs to be clearly distinguished from logic. But this is a topic for another time.



* This is not to say that the exercise of trying to create formal representations of natural languages may not reveal interesting things about natural language and provide new, more concise, more explanatory ways of understanding aspects of the grammar of those languages than traditional grammars provided. But, although such rarefied goals are not pointless, nor are they the sorts of goals for which society is likely to provide support. Traditional grammarians were, after all, essentially pedagogues and their grammars were pedagogical aids.

Scientism

'Scientism' is a scare word. Its chief purpose is not so much to describe a position as to describe-and-attack. It is generally used by those who are ill-disposed to what they see as a narrowly scientific view of the world to suggest that there are other sources of knowledge than ordinary human perceptions and judgements and the empirical and deductive methods of science and mathematics.

Many thinkers take human consciousness (which of course incorporates value judgements, etc.) as a window into something like a spiritual or moral realm, but I would have thought that our perceptions and judgements are entirely the result of processes associated with individual organisms interacting with each other and the wider organic and inorganic environment in which we find ourselves.

The fact that mathematics seems to subsist in a world of its own - accessible to human reason and not empirical in the normal sense - poses problems for radical empiricism, but computational approaches to mathematics (incorporating the concept of information as something physical) may lead to a natural way of sustaining a physicalist, anti-Platonic outlook.

The very fact that we have so many alternative philosophies of mathematics and no clear way to decide which of them (if any) is 'true' suggests that the traditional categories and concepts in which these competing approaches have been framed might be the real source of confusion, and that radically new ways of addressing the questions are needed.

The trend certainly seems to be towards seeing mathematics as being rather closer to physics than has previously been thought to be the case.

Our knowledge of the world may conveniently be divided between ordinary, everyday knowledge (incorporating skills, commonsense, values and so-called intuitions) and theoretical knowledge. The latter is valuable in my view only to the extent that it is scientific (broadly interpreted).

If this is scientism, I can live with it.