Archive for String Theory

With Strings Attached?

Posted in Biographical, The Universe and Stuff with tags , on May 6, 2019 by telescoper

Image Credit: Flickr user Trailfan.

I was flicking through various posts on the interwebs this morning while I was having my breakfast and came across one that nearly made me choke on my muesli. What it’s like to be a theoretical physicist is a piece in Stanford University news. In it I found the following quote:

String theory feels like a little superpower that I have, this physical intuition that enables me to make connections and have insights into things that by rights I should not be able to say anything interesting about.

I’ve tried many times that in a way that doesn’t come across as arrogant, but I’m afraid I’ve failed – especially because (speaking as a physicist) I don’t think string theory has so far given us any profound insights into physics at all.

Now I’m mindful of the fact that many mathematicians thing string theory is great. I’ve had it pointed out to me that it has a really big influence on for example geometry, especially non-commutative geometry, and even some number theory research in the past 30 years. It has even inspired work that has led to Fields medals. That’s all very well and good, but it’s not physics. It’s mathematics.

Of course physicists have long relied on mathematics for the formulation of theoretical ideas. Riemannian geometry was `just’ mathematics before its ideas began to be used in the formulation of the general theory of relativity, a theory that has since been subjected to numerous experimental tests. It may be the case that string theory will at some point provide us with predictions that enable it to be tested in the way that general relativity did. But it hasn’t done that yet and until it does it is not a scientifically valid physical theory.

I remember a quote from Alfred North Whitehead that I put in my PhD DPhil thesis many years ago. I wasn’t thinking of string theory at the time, but it seems relevant:

There is no more common error that to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain.

My problem is not with string theory itself but with the fact that so many string theorists have become so attached to it that it has become a universe in its own right, with very little to do with the natural universe which is – or at least used to be – the subject of theoretical physics. I find it quite alarming, actually, that in the world outside academia you will find many people who think theoretical physics and string theory are more-or-less synonymous.

The most disturbing manifestation of this tendency is the lack of interest shown by some exponents of string theory in the issue of whether or not it is testable. By this I don’t mean whether we have the technology at the moment to test it (which we clearly don’t). Many predictions of the standard model of particle physics had to wait decades before accelerators got big enough to reach the required energies. The question is whether string theory can be testable in principle, and surely this is something any physicist worthy of the name should consider to be of fundamental importance?

 

P.S. This rant reminded me of the time I got severely told off by a very senior British physicist (who shall remain nameless) when I was quoted in Physics World as saying that I thought that in a hundred years time string theory would be of more interest to sociologists than physicists…

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Counting String Theory Standard Models

Posted in The Universe and Stuff with tags , on October 22, 2018 by telescoper

I saw a paper on the arXiv and couldn’t resist a (snarky) comment. Here is the abstract:

We derive an approximate analytic relation between the number of consistent heterotic Calabi-Yau compactifications of string theory with the exact charged matter content of the standard model of particle physics and the topological data of the internal manifold: the former scaling exponentially with the number of Kahler parameters. This is done by an estimate of the number of solutions to a set of Diophantine equations representing constraints satisfied by any consistent heterotic string vacuum with three chiral massless families, and has been computationally checked to hold for complete intersection Calabi-Yau threefolds (CICYs) with up to seven Kahler parameters. When extrapolated to the entire CICY list, the relation gives about 1023 string theory standard models; for the class of Calabi-Yau hypersurfaces in toric varieties, it gives about 10723 standard models.

Isn’t  10723 also the number of angels that can dance on the head of a pin? That number of models for the price of one theory looks like a bargain to me!

But, seriously, people often complain that string theory isn’t really scientific because it isn’t predictive. That clearly isn’t true. String theory is the most predictive theory ever: it can predict anything you want!

 

MaxEnt 2016: Some thoughts on the infinite

Posted in The Universe and Stuff with tags , , , on July 10, 2016 by telescoper

I thought I might do a few posts about matters arising from talks at this workshop I’m at. Today is devoted to tutorial talks, and the second one was given by John Skilling and in the course of it, he made some comments about the concept of infinity in science. These remarks weren’t really central to his talk, but struck me as an interesting subject for a few tangential remarks of my own.

Most of us – whether scientists or not – have an uncomfortable time coping with the concept of infinity. Physicists have had a particularly difficult relationship with the notion of boundlessness, as various kinds of pesky infinities keep cropping up in calculations. In most cases this this symptomatic of deficiencies in the theoretical foundations of the subject. Think of the ‘ultraviolet catastrophe‘ of classical statistical mechanics, in which the electromagnetic radiation produced by a black body at a finite temperature is calculated to be infinitely intense at infinitely short wavelengths; this signalled the failure of classical statistical mechanics and ushered in the era of quantum mechanics about a hundred years ago. Quantum field theories have other forms of pathological behaviour, with mathematical components of the theory tending to run out of control to infinity unless they are healed using the technique of renormalization. The general theory of relativity predicts that singularities in which physical properties become infinite occur in the centre of black holes and in the Big Bang that kicked our Universe into existence. But even these are regarded as indications that we are missing a piece of the puzzle, rather than implying that somehow infinity is a part of nature itself.

One exception to this rule is the field of cosmology. Somehow it seems natural at least to consider the possibility that our cosmos might be infinite, either in extent or duration, or both, or perhaps even be a multiverse comprising an infinite collection of sub-universes. If the Universe is defined as everything that exists, why should it necessarily be finite? Why should there be some underlying principle that restricts it to a size our human brains can cope with?

On the other hand, there are cosmologists who won’t allow infinity into their view of the Universe. A prominent example is George Ellis, a strong critic of the multiverse idea in particular, who frequently quotes David Hilbert

The final result then is: nowhere is the infinite realized; it is neither present in nature nor admissible as a foundation in our rational thinking—a remarkable harmony between being and thought.

This comment is quoted from a famous essay which seems to echo earlier remarks by Carl Friedrich Gauss which can be paraphrased:

Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn’t belong in mathematics.

This summarises Gauss’s reaction to Cantor’s Theory of Ininite Sets. But to every Gauss there’s an equal and opposite Leibniz

I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author.

You see that it’s an argument with quite a long pedigree!

When I was at the National Astronomy Meeting in Llandudno a few years ago, I attended an excellent plenary session that featured a Gerald Whitrow Lecture, by Alex Vilenkin, entitled The Principle of Mediocrity. This was a talk based on some ideas from his book Many Worlds in One: The Search for Other Universese, in which he discusses some of the consequences of the so-called eternal inflation scenario, which leads to a variation of the multiverse idea in which the universe comprises an infinite collection of causally-disconnected “bubbles” with different laws of low-energy physics applying in each. Indeed, in Vilenkin’s vision, all possible configurations of all possible things are realised somewhere in this ensemble of mini-universes. An infinite number of National Astronomy Meetings, each with the same or different programmes, an infinite number of Vilenkins, etc etc.

One of the features of this scenario is that it brings the anthropic principle into play as a potential “explanation” for the apparent fine-tuning of our Universe that enables life to be sustained within it. We can only live in a domain wherein the laws of physics are compatible with life so it should be no surprise that’s what we find. There is an infinity of dead universes, but we don’t live there.

I’m not going to go on about the anthropic principle here, although it’s a subject that’s quite fun to write or, better still, give a talk about, especially if you enjoy winding people up! What I did want to say mention, though, is that Vilenkin correctly pointed out that three ingredients are needed to make this work:

  1. An infinite ensemble of realizations
  2. A discretizer
  3. A randomizer

Item 2 involves some sort of principle that ensures that the number of possible states of the system we’re talking about  is not infinite. A very simple example from  quantum physics might be the two spin states of an electron, up (↑) or down(↓). No “in-between” states are allowed, according to our tried-and-tested theories of quantum physics, so the state space is discrete.  In the more general context required for cosmology, the states are the allowed “laws of physics” ( i.e. possible  false vacuum configurations). The space of possible states is very much larger here, of course, and the theory that makes it discrete much less secure. In string theory, the number of false vacua is estimated at 10500. That’s certainly a very big number, but it’s not infinite so will do the job needed.

Item 3 requires a process that realizes every possible configuration across the ensemble in a “random” fashion. The word “random” is a bit problematic for me because I don’t really know what it’s supposed to mean. It’s a word that far too many scientists are content to hide behind, in my opinion. In this context, however, “random” really means that the assigning of states to elements in the ensemble must be ergodic, meaning that it must visit the entire state space with some probability. This is the kind of process that’s needed if an infinite collection of monkeys is indeed to type the (large but finite) complete works of shakespeare. It’s not enough that there be an infinite number and that the works of shakespeare be finite. The process of typing must also be ergodic.

Now it’s by no means obvious that monkeys would type ergodically. If, for example, they always hit two adjoining keys at the same time then the process would not be ergodic. Likewise it is by no means clear to me that the process of realizing the ensemble is ergodic. In fact I’m not even sure that there’s any process at all that “realizes” the string landscape. There’s a long and dangerous road from the (hypothetical) ensembles that exist even in standard quantum field theory to an actually existing “random” collection of observed things…

More generally, the mere fact that a mathematical solution of an equation can be derived does not mean that that equation describes anything that actually exists in nature. In this respect I agree with Alfred North Whitehead:

There is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain.

It’s a quote I think some string theorists might benefit from reading!

Items 1, 2 and 3 are all needed to ensure that each particular configuration of the system is actually realized in nature. If we had an infinite number of realizations but with either infinite number of possible configurations or a non-ergodic selection mechanism then there’s no guarantee each possibility would actually happen. The success of this explanation consequently rests on quite stringent assumptions.

I’m a sceptic about this whole scheme for many reasons. First, I’m uncomfortable with infinity – that’s what you get for working with George Ellis, I guess. Second, and more importantly, I don’t understand string theory and am in any case unsure of the ontological status of the string landscape. Finally, although a large number of prominent cosmologists have waved their hands with commendable vigour, I have never seen anything even approaching a rigorous proof that eternal inflation does lead to realized infinity of  false vacua. If such a thing exists, I’d really like to hear about!

The Map is not the Territory

Posted in History, The Universe and Stuff with tags , , , , , , , , on January 27, 2015 by telescoper

I came across this charming historical map while following one of my favourite Twitter feeds “@Libroantiguo” which publishes fascinating material about books of all kinds, especially old ones. It shows the location of London coffee houses and is itself constructed in the shape of a coffee pot:

Coffee
Although this one is obviously just a bit of fun, maps like this are quite fascinating, not only as practical guides to navigating a transport system but also because they often stand up very well as works of art. It’s also interesting how they evolve with time  because of changes to the network and also changing ideas about stylistic matters.

A familiar example is the London Underground or Tube map. There is a fascinating website depicting the evolutionary history of this famous piece of graphic design. Early versions simply portrayed the railway lines inset into a normal geographical map which made them rather complicated, as the real layout of the lines is far from regular. A geographically accurate depiction of the modern tube network is shown here which makes the point:

tubegeo

A revolution occurred in 1933 when Harry Beck compiled the first “modern” version of the map. His great idea was to simplify the representation of the network around a single unifying feature. To this end he turned the Central Line (in red) into a straight line travelling left to right across the centre of the page, only changing direction at the extremities. All other lines were also distorted to run basically either North-South or East-West and produce a regular pattern, abandoning any attempt to represent the “real” geometry of the system but preserving its topology (i.e. its connectivity).  Here is an early version of his beautiful construction:

Note that although this a “modern” map in terms of how it represents the layout, it does look rather dated in terms of other design elements such as the border and typefaces used. We tend not to notice how much we surround the essential things, which tend to last, with embellishments that date very quickly.

More modern versions of this map that you can get at tube stations and the like rather spoil the idea by introducing a kink in the central line to accommodate the complexity of the interchange between Bank and Monument stations as well as generally buggering about with the predominantly  rectilinear arrangement of the previous design:

I quite often use this map when I’m giving popular talks about physics. I think it illustrates quite nicely some of the philosophical issues related with theoretical representations of nature. I think of theories as being like maps, i.e. as attempts to make a useful representation of some  aspects of external reality. By useful, I mean the things we can use to make tests. However, there is a persistent tendency for some scientists to confuse the theory and the reality it is supposed to describe, especially a tendency to assert there is a one-to-one relationship between all elements of reality and the corresponding elements in the theoretical picture. This confusion was stated most succintly by the Polish scientist Alfred Korzybski in his memorable aphorism :

The map is not the territory.

I see this problem written particularly large with those physicists who persistently identify the landscape of string-theoretical possibilities with a multiverse of physically existing domains in which all these are realised. Of course, the Universe might be like that but it’s by no means clear to me that it has to be. I think we just don’t know what we’re doing well enough to know as much as we like to think we do.

A theory is also surrounded by a penumbra of non-testable elements, including those concepts that we use to translate the mathematical language of physics into everday words. We shouldn’t forget that many equations of physics have survived for a long time, but their interpretation has changed radically over the years.

The inevitable gap that lies between theory and reality does not mean that physics is a useless waste of time, it just means that its scope is limited. The Tube  map is not complete or accurate in all respects, but it’s excellent for what it was made for. Physics goes down the tubes when it loses sight of its key requirement: to be testable.

In any case, an attempt to make a grand unified theory of the London Underground system would no doubt produce a monstrous thing that would be so unwieldly that it would be useless in practice. I think there’s a lesson there for string theorists too…

Now, anyone for a game of Mornington Crescent?

 

All models are wrong

Posted in The Universe and Stuff with tags , , , , , , , , , on May 17, 2013 by telescoper

I’m back in Cardiff for the day, mainly for the purpose of attending presentations by a group of final-year project students (two of them under my supervision, albeit now remotely).  One of the talks featured a famous quote by the statistician George E.P. Box:

Essentially, all models are wrong, but some are useful.

I agree with this, actually, but only if it’s not interpreted in a way that suggests that there’s no such thing as reality and/or that science is just a game.  We may never achieve a perfect understanding of how the Universe works, but that’s not the same as not knowing anything at all. 

A familiar example that nicely illustrates my point  is the London Underground or Tube map. There is a fascinating website depicting the evolutionary history of this famous piece of graphic design. Early versions simply portrayed the railway lines inset into a normal geographical map which made them rather complicated, as the real layout of the lines is far from regular. A geographically accurate depiction of the modern tube network is shown here which makes the point:

geo_tubemap

A revolution occurred in 1933 when Harry Beck compiled the first “modern” version of the map. His great idea was to simplify the representation of the network around a single unifying feature. To this end he turned the Central Line (in red) into a straight line travelling left to right across the centre of the page, only changing direction at the extremities. All other lines were also distorted to run basically either North-South or East-West and produce a much more regular pattern, abandoning any attempt to represent the “real” geometry of the system but preserving its topology (i.e. its connectivity).  Here is an early version of his beautiful construction:

Note that although this a “modern” map in terms of how it represents the layout, it does look rather dated in terms of other design elements such as the border and typefaces used. We tend not to notice how much we surround the essential things with embellishments that date very quickly.

More modern versions of this map that you can get at tube stations and the like rather spoil the idea by introducing a kink in the central line to accommodate the complexity of the interchange between Bank and Monument stations as well as generally buggering about with the predominantly  rectilinear arrangement of the previous design:

I quite often use this map when I’m giving popular talks about physics. I think it illustrates quite nicely some of the philosophical issues related with theoretical representations of nature. I think of theories or models as being like maps, i.e. as attempts to make a useful representation of some  aspects of external reality. By useful, I mean the things we can use to make tests. However, there is a persistent tendency for some scientists to confuse the theory and the reality it is supposed to describe, especially a tendency to assert there is a one-to-one relationship between all elements of reality and the corresponding elements in the theoretical picture. This confusion was stated most succintly by the Polish scientist Alfred Korzybski in his memorable aphorism :

The map is not the territory.

I see this problem written particularly large with those physicists who persistently identify the landscape of string-theoretical possibilities with a multiverse of physically existing domains in which all these are realised. Of course, the Universe might be like that but it’s by no means clear to me that it has to be. I think we just don’t know what we’re doing well enough to know as much as we like to think we do.

A theory is also surrounded by a penumbra of non-testable elements, including those concepts that we use to translate the mathematical language of physics into everday words. We shouldn’t forget that many equations of physics have survived for a long time, but their interpretation has changed radically over the years.

The inevitable gap that lies between theory and reality does not mean that physics is a useless waste of time, it just means that its scope is limited. The Tube  map is not complete or accurate in all respects, but it’s excellent for what it was made for. Physics goes down the tubes when it loses sight of its key requirement, i.e. to be testable, and in order to be testable it has to be simple enough to calculate things to be compared with observations. In many cases that means a simplified model is perfectly adequete.

Another quote by George Box expands upon this point:

Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful.

In any case, an attempt to make a grand unified theory of the London Underground system would no doubt produce a monstrous thing so unwieldly that it would be useless in practice. I think there’s a lesson there for string theorists too…

Many modern-day physicists are obsessed with the idea of a “Theory of Everything” (or TOE). Such a theory would entail the unification of all physical theories – all laws of Nature, if you like – into a single principle. An equally accurate description would then be available, in a single formula, of phenomena that are currently described by distinct theories with separate sets of parameters. Instead of textbooks on mechanics, quantum theory, gravity, electromagnetism, and so on, physics students would need just one book. But would such a theory somehow be  physical reality, as some physicists assert? I don’t think so. In fact it’s by no means clear to me that it would even be useful..

Infinity or not Infinity?

Posted in The Universe and Stuff with tags , , , , , , on May 14, 2011 by telescoper

Most of us – whether scientists or not – have an uncomfortable time coping with the concept of infinity. Physicists have had a particularly difficult relationship with the notion of boundlessness, as various kinds of pesky infinities keep cropping up in calculations. In most cases this this symptomatic of deficiencies in the theoretical foundations of the subject. Think of the ‘ultraviolet catastrophe‘ of classical statistical mechanics, in which the electromagnetic radiation produced by a black body at a finite temperature is calculated to be infinitely intense at infinitely short wavelengths; this signalled the failure of classical statistical mechanics and ushered in the era of quantum mechanics about a hundred years ago. Quantum field theories have other forms of pathological behaviour, with mathematical components of the theory tending to run out of control to infinity unless they are healed using the technique of renormalization. The general theory of relativity predicts that singularities in which physical properties become infinite occur in the centre of black holes and in the Big Bang that kicked our Universe into existence. But even these are regarded as indications that we are missing a piece of the puzzle, rather than implying that somehow infinity is a part of nature itself.

The exception to this rule is the field of cosmology. Somehow it seems natural at least to consider the possibility that our cosmos might be infinite, either in extent or duration, or both, or perhaps even be a multiverse comprising an infinite collection of sub-universes. If the Universe is defined as everything that exists, why should it necessarily be finite? Why should there be some underlying principle that restricts it to a size our human brains can cope with?

On the other hand, there are cosmologists who won’t allow infinity into their view of the Universe. A prominent example is George Ellis, a strong critic of the multiverse idea in particular, who frequently quotes David Hilbert

The final result then is: nowhere is the infinite realized; it is neither present in nature nor admissible as a foundation in our rational thinking—a remarkable harmony between being and thought

But to every Hilbert there’s an equal and opposite Leibniz

I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author.

You see that it’s an argument with quite a long pedigree!

When I was at the National Astronomy Meeting in Llandudno a few weeks ago, I attended an excellent plenary session that featured this year’s Gerald Whitrow Lecture, by Alex Vilenkin, entitled The Principle of Mediocrity. This was a talk based on some ideas from his book Many Worlds in One: The Search for Other Universese, in which he discusses some of the consequences of the so-called eternal inflation scenario, which leads to a variation of the multiverse idea in which the universe comprises an infinite collection of causally-disconnected “bubbles” with different laws of low-energy physics applying in each. Indeed, in Vilenkin’s vision, all possible configurations of all possible things are realised somewhere in this ensemble of mini-universes. An infinite number of National Astronomy Meetings, each with the same or different programmes, an infinite number of Vilenkins, etc etc.

One of the features of this scenario is that it brings the anthropic principle into play as a potential “explanation” for the apparent fine-tuning of our Universe that enables life to be sustained within it. We can only live in a domain wherein the laws of physics are compatible with life so it should be no surprise that’s what we find. There is an infinity of dead universes, but we don’t live there.

I’m not going to go on about the anthropic principle here, although it’s a subject that’s quite fun to write or, better still, give a talk about, especially if you enjoy winding people up! What I did want to say mention, though, is that Vilenkin correctly pointed out that three ingredients are needed to make this work:

  1. An infinite ensemble of realizations
  2. A discretizer
  3. A randomizer

Item 2 involves some sort of principle that ensures that the number of possible states of the system we’re talking about  is not infinite. A very simple example from  quantum physics might be the two spin states of an electron, up (↑) or down(↓). No “in-between” states are allowed, according to our tried-and-tested theories of quantum physics, so the state space is discrete.  In the more general context required for cosmology, the states are the allowed “laws of physics” ( i.e. possible  false vacuum configurations). The space of possible states is very much larger here, of course, and the theory that makes it discrete much less secure. In string theory, the number of false vacua is estimated at 10500. That’s certainly a very big number, but it’s not infinite so will do the job needed.

Item 3 requires a process that realizes every possible configuration across the ensemble in a “random” fashion. The word “random” is a bit problematic for me because I don’t really know what it’s supposed to mean. It’s a word that far too many scientists are content to hide behind, in my opinion. In this context, however, “random” really means that the assigning of states to elements in the ensemble must be ergodic, meaning that it must visit the entire state space with some probability. This is the kind of process that’s needed if an infinite collection of monkeys is indeed to type the (large but finite) complete works of shakespeare. It’s not enough that there be an infinite number and that the works of shakespeare be finite. The process of typing must also be ergodic.

Now it’s by no means obvious that monkeys would type ergodically. If, for example, they always hit two adjoining keys at the same time then the process would not be ergodic. Likewise it is by no means clear to me that the process of realizing the ensemble is ergodic. In fact I’m not even sure that there’s any process at all that “realizes” the string landscape. There’s a long and dangerous road from the (hypothetical) ensembles that exist even in standard quantum field theory to an actually existing “random” collection of observed things…

More generally, the mere fact that a mathematical solution of an equation can be derived does not mean that that equation describes anything that actually exists in nature. In this respect I agree with Alfred North Whitehead:

There is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain.

It’s a quote I think some string theorists might benefit from reading!

Items 1, 2 and 3 are all needed to ensure that each particular configuration of the system is actually realized in nature. If we had an infinite number of realizations but with either infinite number of possible configurations or a non-ergodic selection mechanism then there’s no guarantee each possibility would actually happen. The success of this explanation consequently rests on quite stringent assumptions.

I’m a sceptic about this whole scheme for many reasons. First, I’m uncomfortable with infinity – that’s what you get for working with George Ellis, I guess. Second, and more importantly, I don’t understand string theory and am in any case unsure of the ontological status of the string landscape. Finally, although a large number of prominent cosmologists have waved their hands with commendable vigour, I have never seen anything even approaching a rigorous proof that eternal inflation does lead to realized infinity of  false vacua. If such a thing exists, I’d really like to hear about!

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Maps, Territories and Landscapes

Posted in The Universe and Stuff with tags , , , , , , , , on January 10, 2009 by telescoper

I was looking through recent posts on cosmic variance and came across an interesting item featuring a map from another blog (run by Samuel Arbesman) which portrays the Milky Way in the style of  a public transport map:

mwta

This is just a bit of fun, of course, but I think maps like this are quite fascinating, not just as practical guides to navigating a transport system but also because they often stand up very well as works of art. It’s also interesting how they evolve with time  because of changes to the network and also changing ideas about stylistic matters.

A familiar example is the London Underground or Tube map. There is a fascinating website depicting the evolutionary history of this famous piece of graphic design. Early versions simply portrayed the railway lines inset into a normal geographical map which made them rather complicated, as the real layout of the lines is far from regular. A geographically accurate depiction of the modern tube network is shown here which makes the point:

tubegeo

A revolution occurred in 1933 when Harry Beck compiled the first “modern” version of the map. His great idea was to simplify the representation of the network around a single unifying feature. To this end he turned the Central Line (in red) into a straight line travelling left to right across the centre of the page, only changing direction at the extremities. All other lines were also distorted to run basically either North-South or East-West and produce a much more regular pattern, abandoning any attempt to represent the “real” geometry of the system but preserving its topology (i.e. its connectivity).  Here is an early version of his beautiful construction:

Note that although this a “modern” map in terms of how it represents the layout, it does look rather dated in terms of other design elements such as the border and typefaces used. We tend not to notice how much we surround the essential things with embellishments that date very quickly.

More modern versions of this map that you can get at tube stations and the like rather spoil the idea by introducing a kink in the central line to accommodate the complexity of the interchange between Bank and Monument stations as well as generally buggering about with the predominantly  rectilinear arrangement of the previous design:

I quite often use this map when I’m giving popular talks about physics. I think it illustrates quite nicely some of the philosophical issues related with theoretical representations of nature. I think of theories as being like maps, i.e. as attempts to make a useful representation of some  aspects of external reality. By useful, I mean the things we can use to make tests. However, there is a persistent tendency for some scientists to confuse the theory and the reality it is supposed to describe, especially a tendency to assert there is a one-to-one relationship between all elements of reality and the corresponding elements in the theoretical picture. This confusion was stated most succintly by the Polish scientist Alfred Korzybski in his memorable aphorism :

The map is not the territory.

I see this problem written particularly large with those physicists who persistently identify the landscape of string-theoretical possibilities with a multiverse of physically existing domains in which all these are realised. Of course, the Universe might be like that but it’s by no means clear to me that it has to be. I think we just don’t know what we’re doing well enough to know as much as we like to think we do.

A theory is also surrounded by a penumbra of non-testable elements, including those concepts that we use to translate the mathematical language of physics into everday words. We shouldn’t forget that many equations of physics have survived for a long time, but their interpretation has changed radically over the years.

The inevitable gap that lies between theory and reality does not mean that physics is a useless waste of time, it just means that its scope is limited. The Tube  map is not complete or accurate in all respects, but it’s excellent for what it was made for. Physics goes down the tubes when it loses sight of its key requirement: to be testable.

In any case, an attempt to make a grand unified theory of the London Underground system would no doubt produce a monstrous thing so unwieldly that it would be useless in practice. I think there’s a lesson there for string theorists too…

Now, anyone for a game of Mornington Crescent?