Paul P. Mealing

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Sunday, 4 October 2009

Quantum Tunneling

In some respects this logically follows on from a post I wrote in July this year, Quantum Mechanical Philosophy, which is one of the more esoteric essays I’ve written on this blog. Hopefully, this essay will be less so, as the source material is well written and aimed at the uninitiated.

But I need to recount the gist of that post to make the relevant connection: specifically, the enigmatic Bell’s Theorem or Bell Inequality. To summarise, Bell’s Theorem arose from a thought experiment created by Einstein in an attempt to prove Bohr’s interpretation of quantum mechanics (the famous ‘Copenhagen Interpretation’) wrong.

The thought experiment was elaborated upon by Podolsky and Rosen, so it became known as the Einstein-Podolsky-Rosen or EPR experiment. It examines the purported ‘action-at-a-distance’ phenomenon predicted by quantum physics for certain traits of particles or photons, which Einstein described, quite accurately, as ‘spooky’. If you have 2 particles with a common origin (could be photons with opposite polarisation or subatomic particles like electrons with opposite ‘spins’), then separate them over any distance whatsoever, you will not know what the spin or polarity, or whatever quantum mechanical trait you are measuring, is, until you take the actual measurement. The ‘spooky’ bit is that as soon as you make the measurement the ‘twin’ particle will instantaneously become the opposite. Before the measurement or observation is made the particles are in, what’s called, a ‘superposition’ of states – it can be either one or the other.

Einstein realised that this conjecture contradicted his special theory of relativity, which states that no signal or means of communication between particles of any kind can travel faster than the speed of light, which had already been confirmed by experiment. John Bell developed a mathematical equation that analysed correlations of hypothetical results from the thought experiment that would categorically prove either Einstein or Bohr wrong.

Alain Aspect developed a real experiment to test Bell’s Inequality (made the thought experiment actually happen) and proved Einstein wrong (long after Einstein had died, by the way).

As I point out in that previous post, the upshot of this is that either faster-than-light actions are possible (called non-locality) or there is no objective reality. Non-locality is self-explanatory (you can’t communicate faster than the speed of light) but no objective reality means that the thing doesn’t exist until someone measures it or takes an observation. I discuss this in more detail (lots of detail) in my previous post, but that’s effectively the Copenhagen interpretation of quantum mechanics: at the subatomic scale, particles don’t exist until they are measured or observed. A less extreme and more popular interpretation is that they remain in a superposition of states until they interact with something else. If you want to delve deeper, read my previous post, but you may be none the wiser. The philosophical implications of this have never been truly resolved.

My conclusion was to accept non-locality (faster-than-light connections) in order to keep objective reality, and I made specific reference to David Bohm’s unpopular interpretation, known as the ‘hidden variables theory’. Bohm believed that there was a hidden set of parameters that govern the particles which we can’t see or detect.

To quote David Deutsch (who doesn’t agree with Bohm at all): ‘A non-local hidden variable theory means, in ordinary language, a theory in which influences propagate across space and time without passing through the space in between.’

And this leads me to quantum tunneling, because that’s exactly what quantum tunneling does, only it happens over short distances, not the distances used in the EPR experiment, which could theoretically include the other side of the universe.

I’ve just read an excellent book on this subject, Zero Time Space subtitled, How Quantum Tunneling Broke the Light Speed Barrier, authored by Gunter Nimtz and Astrid Haibel. Originally published in German in 2004, it was published in English in 2008. This book could be read by people with only a rudimentary knowledge of physics, as it contains only a few simple equations, among them Planck’s equation: E = hf where E is energy, f is frequency of a ‘wave’ and h is Planck’s constant, 6.6 x 10-34 Js (Joules seconds). The authors also include Snell’s law of refraction and the universal wave equation of wavelength times frequency equals velocity (I can’t find the symbol, lambda, for wavelength, in my arsenal of fonts). One of the annoyances is that there is a type-setting error in this particular equation (in the book). If someone is going to include equations, especially for people unfamiliar with them, I wish they could at least get them checked during type-setting. The same applies to Richard Feynman’s excellent book on relativity theory, Six Not-So-Easy Pieces where I found 3 type-setting errors amongst the equations scattered throughout the book. In both cases the books are aimed at people who are not familiar with the material, which means they won’t know the errors are there.

Putting that one (some may say petty) criticism aside, it’s a very good book on quantum mechanics for people who know very little about physics. It includes a short history of physics leading up to Einstein’s theories of relativity (with particular reference to the Special Theory) as well as quantum mechanics. They do this because the whole point of the book is to highlight how quantum tunneling breaks Einstein’s special theory of relativity, and therefore reinforces non-locality, as I described in my previous post. So the authors go to some pains to give the reader an overview of both Einstein’s theory and quantum mechanics, in conjunction with the historical context. It’s very well done.

Nimtz and Haibel, by the way, make no reference to Bell’s Theorem, as it would probably confuse readers who are unfamiliar with it – I hope I haven’t put people off by referencing it here. Having said that, they do discuss ‘entanglement’ towards the end of their book, which is the state I described above concerning ‘twin’ particles interacting at a distance. In particular, they raise this phenomenon in their lengthy discussion on causality, as they are at pains to explain that ‘tunneling’ does not affect causality as some people might be led to believe. Even so, it still manages to confound common sense, as I explicate below.

In the forward to the book, they briefly discuss the ‘myth… about the half-life of knowledge… It suggests that our knowledge is being declared invalid every five years by new knowledge.’ They then go on to dispel the most common representation of that myth: Newton’s theory of gravitation is still valid, even in the light of the theory of relativity… Einstein’s theory has extended theory rather than disproved Newton’s theory.’

I made the same point in my essay on The Laws of Nature (March 08), explaining that Einstein’s equations reduce to Newton’s when certain parameters become negligible. The authors raise this point, because, whilst quantum tunneling appears to contradict Einstein’s special theory of relativity, in their own words: ‘Einstein deals with free space, whereas tunneling is not free space.’ In other words there are constraints on relativity theory in the same way that there are constraints on Newton’s theory, but there are various aspects of nature where one is more significant than the other. It’s one of the reasons that I’m a bit sceptical about a grand unified theory (GUT), a meta-theory of everything. Many people would love to prove me wrong, and a part of me would like to see that, but another part wouldn’t because I don’t believe there will ever be an end to physics.

During this discussion they make another statement, relevant to the stability of scientific knowledge: ‘Mathematical proof has been regarded since Pythagoras and Plato as eternal, metaphysical truth.’ A statement I would agree with. For example, Reimann geometry hasn’t displaced Euclidean geometry, it has just extended our knowledge, both of the mathematical world and the physical world (through Einstein’s theory of General Relativity).

I’ve discussed on other posts, the relationship between mathematics and the natural world (refer The unreasonable effectiveness of mathematics, March 09), but no where is that more significant than in quantum mechanics. QED (Quantum Electrodynamics), for which Richard Feynman, Julian Schwinger and Sin-Itoro Tomonaga jointly won the Nobel Prize, is the most successful theory of all time. Without mathematics, quantum mechanics would be indecipherable, quite literally. Intriguingly, there are imaginary numbers in quantum theory that are completely relevant to quantum tunneling. Without imaginary numbers (created by the square route of -1, called i) quantum mechanics would never have been articulated as a meaningful theory at all.

As Nimtz and Haibel point out, it is the imaginary component of the equation that does the tunneling. When this was first derived, people just assumed that these imaginary components were unnecessary remnants of the mathematics, but that’s not the case. When tunneling occurs there is an interface where part of the signal is reflected and part is transmitted through ‘the tunnel’. The part that is reflected is mathematically ‘real’ and the part that is transmitted is mathematically ‘imaginary’. (I've since been informed this is not correct - refer Addendum 2 below.) A tunnel, by the way, is a barrier, where the particle or wave theoretically can’t travel, because it doesn’t have enough energy. The authors point out that it even occurs in the sun, otherwise the fusion, which gives us sunlight, would never occur. I should add that quantum tunneling is a feature of all transistor devices. In fact, it's the very feature that makes transistors work (called 'tunnel diode' by Nimtz and Haibel).

Both of the authors have performed experiments, to not only detect quantum tunneling, but to also measure the time elapsed. As predicted by Thomas Hartman in 1962, there is a time elapse at the ‘entrance’ to the tunnel, or the ‘interface’, between the medium and ‘the tunnel’, but the actual time spent in the tunnel is zero. This is called the Hartman effect. To quote the authors: ‘So the wave packet spreads in the tunnel in zero time and is everywhere from the entrance to the exit. This non-local phenomenon makes one feel eery.’ An understatement, if I’ve ever read one.

One of the authors, Gunter Nimtz, participated in an experiment that tunneled Mozart’s symphony in g-minor through a waveguide at superluminal speed: 4.7 times the speed of light. The elapsed time occurred at the entrance to the tunnel, as predicted by Hartman, not in the tunnel itself. In an exposition, that I will not try to repeat here, the authors explain how this quirk of nature (the elapsed time at the entrance to the tunnel) allows superluminal communication without impacting causality. The speed in the tunnel is infinite – as the Americans like to say: go figure. The title of the book, Zero Time Space, is therefore entirely appropriate.

They end the book with a brief description of wormholes and hypothetical warp drives, beloved of Sci-Fi writers, like me, that require exotic negative gravity amongst other improbabilities.

Of all the incredible manifestations of the universe, only consciousness is arguably more inexplicable or more mysterious (but no more weird) than quantum phenomena. If we didn’t observe it, no one would believe it. And if we didn’t have the mathematics to describe it, no one would be able to fathom it, even remotely.

Addendum 1: I came across this - it's very entertaining as well as informative.

Addendum 2: I would like to acknowledge Timmo (refer comments thread below) who has valiantly tried to correct all my mistakes. In particular, that the imaginary component of Schrodinger's equation plays no greater role in tunneling than the real component, if I understand Timmo correctly. Also he points out that tunneling and non-locality are independent phenomena, and possibly I misled people on that point.

He also corrects some faux pas I made concerning the Lorenz transformation and Godel's Incompleteness Theorem in response to comments I've made since the post was posted.

I confess I don't know as much as I appear to, and I wish I understood more than I actually do.

And I would like to thank Timmo for reminding me of how much I don't know.


larryniven said...

Wait, so does stuff that undergoes quantum tunneling travel across some other dimension than time? I guess it has to alter its spatial coordinates, or else it wouldn't be much of a tunnel, but doesn't some other variable have to change also? Apparently that variable isn't time, but still.

Also, as a side note, the no-objective-reality interpretation would directly falsify George Berkeley's slogan that "to be is to be perceived," or at least his hypothesis that God sees everything in the universe all the time. If that turns out to be the wrong interpretation, though, I guess it doesn't much matter - I'm not sure whether the other one affects him or not.

Paul P. Mealing said...

That's a very good question. I don't know if you read my previous post on this subject, but I did speculate that maybe it does infer a 'hidden' dimension.

In that post I also discuss the implications of taking a literal reading of the 'Copenhagen Interpretation', which I suggest borders on solipsism.

I am one of those who believes that eventually we will have an explanation of quantum mechanics that everyone will understand, but that explanation may include extra dimensions or time temporarily running backwards.

In regards to what variables change - well there are only 3 that govern velocity: the other 2 being distance and time. So if distance remains constant, zero time produces infinite velocity.

Even using the Lorenz transformation (as Einstein did for relativity theory) you still only have the same 3 variables, because 'c' is constant.

Having said that, the Lorenz transformation obviously doesn't apply to quantum tunneling, which is one of the reasons that Einstein bridled at Bohr's interpretation (aka the Copenhagen interpretation).

Regards, Paul.

Paul P. Mealing said...

I just realised that I spelt Lorentz wrong.


larryniven said...

"I am one of those who believes that eventually we will have an explanation of quantum mechanics that everyone will understand, but that explanation may include extra dimensions or time temporarily running backwards."

Then you, sir, have a much higher opinion of humans than I do! I'm willing to say that we'll develop a system that is in principle understandable, but we'd be extremely lucky (in my opinion) if even just the scientists involved understood it.

Paul P. Mealing said...

John Wheeler once said that until you can explain it to the person on the street (or words to that effect) you don't really understand it.

Yes, I am a heretic about many things. I think we will have an understanding of quantum mechanics that makes sense, and yet I believe we will never understand the subjective nature of consciousness.

On both counts, I know I'm in the minority, philosophically.

Regards, Paul.

Andrew Louis said...

Why is it such a horrible thing to consider that there is no Objective Reality? Nothing is lost in the consideration, i.e. the way we negotiate our sorroundings remains the same. Not to mention the fact that we can finally ditch all the dogma Platonism.

Andrew Louis said...

I'm generally of the school of thought which would say, truth does not exist in the world, it exists in lanuage. This isn't to say that the world doesn't exist out there, only that truth doesn't.

So to say that there isn't an objective reality (and that perhaps it's supported by science to boot) is actually quite nice, and pushes us away (again) past the old dogma of the appearance/reality distiction.

At this point I'm not familiar enough with you to say one way or another why this is a bad thing for you and your general philosophical position

Paul P. Mealing said...

Hi Andrew,

I discuss the various quantum mechanical interptretations in an earlier post Quantum mechanical philosophy, based on a series of interviews Paul Davies conducted with some of the major figures in quantum physics, and some of whom are no longer with us.

The problem with the 'no objective reality' position is that it's hard to distinguish from solipsism. To reference an anecdote, that John Searle credits to Bertrand Russell, about a woman who wrote to him (Russell) claiming she was a solipsist and couldn't understand why she never met any others.

As for Platonism, I do call myself a Platonist in a mathematical sense, though I don't believe in Plato's forms per se. I think Roger Penrose gives the best interpretation of this, which I describe in another post.

As for only finding truth in language, we discussed this on Stephen Law's blog a little while ago, and I discuss it in reference to Don Cupitt's book, Above Us Only Sky, on my post, The Existential God, which you've already visited once.

Well talking about Platonist mathematics, I've argued previously that the only 'truths' you can be certain of are mathematical. Distinguishing truth from reality is the tricky bit. Does truth only exist when a conscious entity 'senses' it? In other words, is truth purely subjective? Well I actually think that mathematical truths are objective truths in that they exist independently of us, which is why I am a Platonist.

Regards, Paul.

Andrew Louis said...

Mathematical truths exist independently of us? How so?

What about propositional truth?

Paul P. Mealing said...

Hi Andrew,

I thought that might get a reaction,and, to be honest, I would have been disappointed if it didn't, considering your previous comment.

I've written 3 essays on this subject over the last 2 years: a review of Mario Livio's Is God a Mathematician?; review of Gregory Chaitin's Thinking about Godel and Turing; and George Lakoff's and Rafael Nunez's Where Mathematics Comes From.

The first one is the most recent and probably the most accessible.

Mathematics is more to do with the relationships between the numbers than the numbers themselves. These relationships are universal and are discovered not invented - in that respect they exist independently of us. It is true that without a conscious intelligence like ours we would not know they exist but that applies to all things epistemological.

Nature's laws also exist independently of us in exactly the same way, and it is no surprise that many of nature's laws, in fact most that we know of, are only cognisant to us via mathematics.

I concede that my view is not popular amongst philosophers or ordinary people, but the more I learn about mathematics the more I think it's the correct view. Pythagoras was the first in Western culure to appreciate the inherent relationship between mathematics and the natural world, but, I expect, even he would be surprised at how far-reaching his insight turned out to be.

And don't forget that mathematical truths are the closest we will ever get to undeniable objective truths in either the abstract or the physical world.

Regards, Paul.

Paul P. Mealing said...

I came across this quote (just today) in American Scientist (Sep-Oct 2009) from Solomon Feferman, who is Professor of mathematics and philosophy, emeritus, at Stanford University. He also holds a professorship in humanities and sciences, emeritus, at the same institution, and is author of a book, Light and Logic and co-author of another, Life and Logic.

In American Scientist, he is reviewing a book by Jeremy May, Plato's Ghost: The Modernist Transformation of Mathematics, though I don't think it has much to do with Plato at all.

The quote, however, is very relevant to our discussion, and to put it into context, he's discussing Ernst Zermelo's work in the first decade of the 20th Century, who "...isolated the controversial axiom of choice to establish... Cantor's transfinite numbers, and showed that it had been used implicitly by mathematicians for many arguments else where..."

Feferman then explains: "Philosophically, the justification for the axiom system... that Zermelo introduced requires a Platonist account... of mathematics... [whereby] mathematical truths hold whether or not they can be established by human beings."

He then elaborates: "As a philosophy of mathematics, Platonism is deeply problematic, but no effort thus far to replace it with one brand or another of logicism, formalism, constructivism, nominalism or fictionalism... has succeeded in accounting for the accepted body of mathematics while being philosophically convincing."

This is a lengthy and fragmented extract, but the point is that Zermelo introduced an axiom that mathematicians were already using implicitly, which was inherently Platonist by nature, according to Feferman. In other words, mathematicians find it hard to avoid Platonism even when they don't like its philosophical implications.

Regards, Paul.

Andrew Louis said...

All you’ve really done is expounded upon your original assertion, you haven’t shown anything to be necessarily so… e.g. “These relationships are universal and are discovered not invented”. This is a belief statement, not proof.

The “made/found’ distinction is a pretty classic Platonist move as well.

You said,
“…it is no surprise that many of nature's laws, in fact most that we know of, are only cognisant to us via mathematics.”

I can say something quite similar and produce the same effect.
“…Its no surprise that the hippopotamus, and in fact all animals, are only cognizant to us via language…”

As a pragmatist, there is no difference between your statement and mine. However surely you’d suggest otherwise, and I’m betting you’ll make an appearance/reality distinction of some sort…..


Paul P. Mealing said...

Hi Andrew,

You are right: "This is a belief statement, not a proof."

There are no proofs in philosophy, which is a common misunderstanding, I might add, so I'm not offering one.

There are only points of view supported by rational arguments. The major difference between philosophy and science is that science often deals in right and wrong answers and philosophy often does not. I find that many people who argue philosophy in the blogosphere fail to appreciate this salient difference.

You seem to follow a philosophical viewpoint that Cupitt expresses specifically in his book: Above Us Only Sky.

To quote: “You can have more-or-less anything, provided only that you understand and accept that you can have it only language-wrapped – that is, mediated by language’s secondary, symbolic and always-ambiguous quality.”

You may want to read my arguments in response to Cupitt's position, where I specifcally discuss mathematical 'truths', unless you've already read it.

You say: “…Its no surprise that the hippopotamus, and in fact all animals, are only cognizant to us via language…”; with the implication that this is no different to my statement concerning mathematics.

However, mathematics is not like other languages at all, which is a point I make in my argument against Cupitt. No one has mathematics as their first language, except, arguably, a computer. But more significantly, mathematics is a universal language even though it can be represented in any base arithmetic you want to envisage. It is a mistake to think of mathematics as a language in the same way as human verbal languages.

Mathematics is unique, epistemologically, in that it is not only universal, but it contains 'truths'. In fact, it is the only medium that arguably contains standalone truths. It is this feature that gives it its Platonic status.

This is a minority view, as I've said before, but it's hard to avoid if you study mathematics in any depth.

I'm not a mathematician by the way, but I've read a lot of books by mathematicians, the most recent being Marcus du Sautoy's Finding Moonshine; Mathematics, Monsters and the Mysteries of Symmetry. The book expounds on the history of symmetry, from Pythagoras to the 20th Century, culminating in an 'Atlas' of all the basic symmetrical objects discovered in all possible dimensions. The largest of which is a 'Monster' existing in 196,883 dimensional space with 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 symmetries.

There is a world of mathematics that exists, of which we will never fully know. The number pi and Euler's constant, e, otherwise known as the base of the natural logarithmical scale, are numbers that can never be computed yet they have significance well beyond our initial expectations.

The mathematical realm is close to magical, especially when one considers its relationship to nature. It is the bridge between the natural world and our intellect, without which we would know very little about the universe at all.

I cannot 'prove' a philosophical viewpoint but let me state the obvious: no mathematical rule, relationship, proof or object, has ever been invented. We invent the symbols, even what base arithmetic we will use, therefore we arguably invent the numbers, but that's it. But no one invented pi or e - they, like all other mathematical relationships, objects and proofs, were discovered. Arguably, 0,1,pi, i and e are the only numbers we need to do all our mathematics.

Regards, Paul.

Andrew Louis said...

You said:
“There are no proofs in philosophy, which is a common misunderstanding, I might add, so I'm not offering one….. There are only points of view supported by rational arguments.”

so lets hear that rational argument ;)

You said mathematical language is universal across cultures. In what way is that so?

Surely we can both agree that language about, lets say, COLOR, is universal. Surely every culture has different names and symbols for them, etc. etc. but every culture has a language surrounding colors.

1.) So your first argument is “UNIVERSIALITY”. However, as other languages and forms of language have universality, I’m not sure you pass the test here.

What is a “stand alone” truth? And in what way is that standalone truth not contingent?

You never stated how my statement was different then yours?

You said,
“…it is no surprise that many of nature's laws, in fact most that we know of, are only cognisant to us via mathematics.”

I can say something quite similar and produce the same effect.
“…Its no surprise that the hippopotamus, and in fact all animals, are only cognizant to us via language…”

Andrew Louis said...

I've never read Cupitt, however it seems as though he draws heavily upon Richard Rorty, whom I'm a huge fan of. Maybe I'll have to give it a look see.

Paul P. Mealing said...

Hi Andrew,

If I can’t explain the difference between mathematics and ordinary human languages in a way that you can comprehend it, I’m not sure I can enlighten you any further. However, I will try, using your own analogy.

Mathematics is similar to ordinary language in as much as we invent a number system in which to write it down and communicate it. But the mathematical objects it describes are as objective as the colour that you use as an analogy. So the number system doesn’t determine the mathematics any more than the language, be it English, French or whatever, determines what the colour actually is that we experience. (Colours, by the way, only exist in our minds, not out there, but that’s another discussion – irrelevant to this argument.)

So, in the same way that there are objects in the physical world that are universally comprehended and described, albeit using different languages, there are also objects in the abstract world of mathematics that are equally universal, albeit described and communicated using different number systems.

This should explain exactly what I mean - there is a language that we use to communicate mathematics but that language is not mathematics anymore than the language you speak to describe the physical world is the physical world.

If you don’t appreciate this subtle point then I’m not sure we can discuss this in a way that is going to be enlightening for either of us.

If that’s not a rational argument, by the way, then you have me stumped.

Regards, Paul.

Andrew Louis said...

Again, you're not giving me a rational argument, you're just telling me what you think.

For example, I used the idea of colors being universal, just as numbers are. But you didn't argue for your point at all, you just said, "Colors are in your head". Which isn't an argument at all.

What you've said amounts to, "Numbers are out there, colors are in your head."

Paul P. Mealing said...

Sorry Andrew,

If you can't follow my arguments then I don't think I can help you. The fact that colours only exist in our heads is irrelevant to this argument, I admit, but I made the comment (as an aside) because it's true, and I also made it in my post on Cupitt, so I needed to be consistent in case you read it.

You asked why the laws of nature being mathematically cognisant is different to descriptions of animals being cognisant in a human language.

The laws of nature in physics, like gravity or electromagnetic phenomena or quantum phenomena, all obey mathematical relationships, and they do it universally within certain parameters, so different laws dominate at different scales. This is unique, epistemologically.

We apply names to physical objects, like hippopotami for example, as you mentioned, but this is not the same thing at all. Physical laws actually follow mathematical rules - it's a completely different concept.

You are obviously frustrated, but the more I try to explain my case, it seems the more I confuse you.

"numbers are out there, colours are in your head."

Colours are a psychological phenomenon it is true. Einstein argued that the integers are invented by us, so he would have agreed with you. My argument is that we may invent the numbers but not the relationships between the numbers.

Numbers are a concept is the best way to define them, but mathematics is a lot more than numbers. Actually the 'atoms' of mathematics are the primes, because all the other rational numbers can be derived from them. These primes can exist in any number system we envisage, so the base arithmetic we use is unimportant. The whole point is that mathematics follows its own rules that we can't make up - that's what we can't get away from - that's why it's objective, universal and independent of us. And then nature also follows these rules (often in ways we don't predict) which makes it epistemologically unique.

This is a philosophical point of view, and not everyone agrees with it. I'm sorry if I can't argue my case adequately, but it's the best I can do.

Physics has been a lifetime passion for me, so it's hard to impart that to someone else.

Regards, Paul.

Andrew Louis said...

Paul, you have it all wrong, I’m not confused at all, I understand completely what your position is, you’re simply not making a case for it. i.e. you’re arm waving, and just restating your opinion.

The only argument you made was relative to “UNIVERSIALITY”. i.e. your argument is that universiality supports your point (if it’s universal, it exists independently). I pointed out that the language of color is universal, therefore (by your reasoning) color should also exist independently of us for the same reason. However you respond by simply saying, “Color is in our heads”. But that’s not a refutation at all.

You said:
“The whole point is that mathematics follows its own rules that we can't make up - that's what we can't get away from - that's why it's objective, universal and independent of us.”

Ok, show me an example of that… How do you know? Do you understand that you haven’t shown anything, you’ve just made a statement?

If mathematics exists independently of us, demonstrate that. e.g. you say that gravity follows universal mathematic relationships. i.e., you are essentially saying that the law of gravity is universal and absolute.

You also appealed to the appearance/reality distinction, which I stated in the beginning.
“My argument is that we may invent the numbers but not the relationships between the numbers.”

In effect you are saying that the numbers represent some universal/absolute concept, i.e. an underlying reality behind the appearance. But again, you haven’t shown that, or even justified it.

I hope I don’t sound overly contentious, or like I’m being a bastard, I’m really interested in your justification.

Paul P. Mealing said...

“In effect you are saying that the numbers represent some universal/absolute concept, i.e. an underlying reality behind the appearance. But again, you haven’t shown that, or even justified it.”

In regard to colour, it’s purely subjective and its universality comes from consensus. When I see something that I call green, we agree on that and we both agree that it’s different to red, say. But I have no idea really that what I see as green is the same as what you see, because it’s a subjective experience. The only thing we know objectively is that green is reflected light within a certain frequency range. Many animals don’t see the colours that we see, and some see colours that we don’t. So the experience of colour occurs in the mind. I discuss this in my essay on Cupitt, giving an example of some monkeys that can’t see colours that we can see, but can be made to see them, if we genetically engineer them (it’s been done).

In regard to mathematics, I’ve written 3 essays on this subject that are chock-full of rational arguments, if you’d care to read them, with references to people who are far more knowledgeable than me, but in particular, Roger Penrose, whose mathematical philosophy has been my biggest influence.

Penrose is Emeritus Rouse Ball Professor of Mathematics at Oxford University and won the 1988 Wolf Prize in physics, jointly with Stephen Hawking. This is his philosophical comment on Godel’s Incompleteness Theorem from The Emperor’s New Mind:

‘The notion of mathematical truth goes beyond the whole concept of formalism. There is something absolute and “God-given” about mathematical truth. This is what mathematical Platonism… is. Any particular formal system has a provisional and “man-made” quality about it… Real mathematical truth goes beyond mere man-made constructions.’

You want justification; well, how about examples; any mathematical relationship will do. 2+2=4 is a universal relationship independent of any number system you want to choose. If this relationship is not universally true then accountants, engineers, economists, physicists, climatologists – in fact, anyone who deals in numbers would be in huge trouble.The multidimensional ‘Monster’, expounded upon by Marcus du Sautoy, that exists at the other extreme of mathematics, that I referenced in an earlier comment, is also an example.

If you can’t appreciate that these are universal mathematical truths then, quite frankly, I am wasting my time. I admit I am a dilettante, not an academic, and I can’t explain to you what you’re unable to comprehend. The above examples are fundamentally true for me, apparently false for you. I refer you to Penrose’s quote above if you want someone with more authority.

Paul P. Mealing said...

Further justification: the Chinese discovered Pythagoras’s famous theorem 500 years before Pythagoras, though they didn’t prove it, and they discovered Pascal’s triangle 300 years before Pascal, though I’m unsure if they appreciated its significance to the binomial theorem. And yet, like the West, they had to learn what zero was from India.

To quote from the earliest of the essays that I referenced in my earlier comment, and wrote over 2 years ago, Is mathematics invented or discovered?

Mathematics is essentially a problem-solving endeavour, and this begs the question: if one is looking for a solution to a puzzle, does the solution already exist before someone finds it, or only after it’s found? Think: Fermat's Last Theorem; solved by Andrew Wiles 357 years after it was proposed. Think: Poincare's Conjecture; proposed 1904 and solved in 2002 by Grigory Perelman (read Donal O'Shea's account). Think: The Reimann Hypothesis; proposed 1859, still unsolved. To answer that question, I suggest, is to answer the philosophical conundrum: is mathematics invented or discovered?

The point is that anyone could make these discoveries, anyone with the necessary intelligence – no one invented them - otherwise they would all be unique to the person who found them, the way musical symphonies or other works of art are to those who created them, like Beethoven’s symphonies or Shakespeare’s plays. Do you understand the point I'm making here?

Surely you must understand that there are mathematical ‘truths’ that we may never know, which is the crux of Godel’s Incompleteness Theorem, by the way. Reimann’s hypothesis has so far not been solved, which means there exists a truth ‘out there’ that we currently don’t know. For a start, mathematics is infinite on many different levels, so it is impossible for us to know its entire scope and magnitude in exactitude. For example, the numbers pi and e run on forever, effectively incomputable, so we will never know their full extent – it’s impossible - yet we know that all those digits exist if someone or a computer wants to compute them to whatever length they can, but, of course, they will always continue to exist at limits that can’t be computed.

You ask me to justify my philosophical position, which I’ve covered in 3 essays written over the last 2 years, yet you haven’t even read them, apparently, and you don’t even have an alternative position to oppose mine. I have been arguing a philosophical position over days now, and you don’t even have one of your own. To be perfectly honest: that's a bit rich.

Regards, Paul.

scott roberts said...

Well, I was going to make a comment on your post (which I will in a moment), but now I also want to make a comment on this argument with Andrew. On the latter: the difference between mathematics and other languages is that a mathematical concept, once learned, is independent of our sense perceptions, while concepts of hippopotami are not. This is what makes them certain, in a way that concepts of sense perceptible objects cannot quite be. So, ironically, in the connotative use of the words 'subjective' and 'objective', it is that which is independent of the senses that is truly 'objective' (which just means we shouldn't use those words the way we usually do.)

Now on to the original post. There is another option to the choice between 'non-locality' and 'rejecting objective existence' when confronted with the Bell/Aspect business. It is the rejection of spatiotemporal objective existence. That is, assume that space and time are, like color, added in the act of perception, rather than being inherent in the "things themselves". If interested, Samuel Avery has written a short book exploring this option: The Dimensional Structure of Consciousness: A Physical Basis For Immaterialism.

Timmo said...

As Nimtz and Haibel point out, it is the imaginary component of the equation that does the tunneling. When this was first derived, people just assumed that these imaginary components were unnecessary remnants of the mathematics, but that’s not the case.

What do you mean that the 'imaginary components were unnecessary remnants of the mathematics'? A particle moving freely through space with a definite momentum, not interacting with anything, is described by a wavefunction:

ψ(x) = N*exp[2πipx/h]

which has both real and imaginary parts. (N is some normalization constant.) The imaginary part is not some extra residue that is essential to describing that particle. For example, without the imaginary part, ψ would not be an eigenfunction of the momentum operator:

-ih/2π d/dx ψ = pψ

Also, such a particle is equally likely to be found over any given interval, so the probability of finding it between a and b is:

∫ dx |ψ(x)|^2 = N(b-a).

This result cannot be got by assuming only the 'real' part of the wavefunction matters

Re[ψ] = cos[2πipx/h]

Also, I should add that both the real and imaginary parts of the wavefunction are, in general, part of tunneling. That is, it is not the case that the real part of the wavefunction vanishes inside the barrier while the imaginary part remains finite. So, I don't see in what sense it is true that 'the imaginary part does the tunneling'.

And this leads me to quantum tunneling, because that’s exactly what quantum tunneling does, only it happens over short distances.

In what way is tunneling a manifestation of non-locality? I suppose you are arguing that because the tunneling particle cannot, classically speaking, smash through the box that there is an action-at-a-distance between the confined particle and the instrument which later detects it on the other side of the barrier. The problem with this argument is that one cannot, in advance, say that the particle is in the box, only that the probability of finding it there is much greater than the probability of finding it outside.

The reason I say this is that the Hamiltonian operator H never commutes with the position operator X.

[H, X] ≠ 0

So, if we say that the particle has a definite energy E, it would be hasty to go on to say that it is also in the box, because the position of the particle is not also simultaneously well-defined. One can say that the probability to find the particle inside the box is greater than the probability to find it outside the box, but that does not imply that the particle was once inside the box and then later went outside the box.

Timmo said...


I should add that there is a nice internet applet showing the wavefunctions of a particle encountering a barrier here:

Timmo said...

...the Lorenz transformation obviously doesn't apply to quantum tunneling...

*gape* What do you mean by this!?

It is a basic principle that the laws of physics are invariant under Lorentz transformations. So, if one takes a problem that involves tunneling and then boosts to another reference frame, then everything is still OK. Tunneling is just as much a consequence of the Dirac equation as it is of the Schroedinger equation, and the Dirac equation is relativistically covariant.

Timmo said...

Surely you must understand that there are mathematical ‘truths’ that we may never know, which is the crux of Godel’s Incompleteness Theorem, by the way.

No. This is a sloppy gloss.

Godel's First Incompleteness Theorem states that any recursively axiomatizable, omega-consistent theory that can represent all recursive functions and express all recursive relations has at least one undecidable sentence. That is, a formal system capable of capturing a certain amount of arithmetic cannot be simultaneously consistent and decide every sentence. But, this does not show one does not 'know' whether a particular undecidable sentence is true.

Using the Fixed-Point theorem, one can construct a sentence G which effectively 'says' that no natural number is the Gödel number of a proof of G (i.e., G asserts that it is not provable). The curious thing is that we can explicitly construct G and know that G is true. That is, I might write down some formal theory T which satisfies the conditions for the Godel theorem. There is a rote procedure for constructing the Godel sentence for T. That sentence can't be proved in T (assuming it is consistent), but I know that the Godel sentence is true (because I know that T is consistent) and what that Godel sentence is (because there is a rote procedure for constructing it).

The Godel theorem is a profound theorem about recursion and the relative power of a wide class of formal systems. The claim that there are things about mathematics we will never know is trivial. Surely one didn't need to invent mathematical logic to learn that!

Paul P. Mealing said...

I've never had so many comments on a post before, especially in one go.

Firstly, to address Scott Roberts.

Your point about space and time being "added in the act of perception" is very Kantian, as you know by referring to the "things themselves", but, personally, I've always rejected that aspect of Kant's philosophy, even though I agree in principle that the 'thing-in-itself' is something we may never know, and quantum mechanics is arguably the best example of that.

The reference you make to Samuel Avery sounds like something I'd be interested in.

Regards, Paul.

Paul P. Mealing said...

Hi Timmo,

You have me outclassed, but I welcome someone who knows more about physics than I do.

In regard to your first comment I was quoting from Nimtz and Haibel's book, specifically in tunneling through an air gap between prisms:

'The spreading of evanescent modes corresponds to the tunneling process of particles in quantum mechanics.

The evanescent modes, as they occur with total reflection are imaginary mathematical solutions of the law of refraction. Formerly they were rejected by scientists as non-physical because these solutions describe field modes with no real wavelength.'

These 3 sentences follow each other exactly like this in the book, so the inference is that the 'evanescent mode' does the tunneling and is also the imaginary component, and it is the authors that claim it was originally rejected. I'm happy to be corrected (by someone with more knowledge) if this inference is incorrect.

As you've probably worked out, I'm not that familiar with Schrodinger's equations. The similarity with non-locality (I know it's not the same thing) is that the actual 'tunneling' occurs with no time differential, effectively simultaneously.

I wasn't aware of this before I read their book.

I'll respond to your other comments with other comments.

Regards, Paul.

Paul P. Mealing said...

Hi Timmo, again.

I tried the link but I needed a plugin that didn't work or wouldn't download so I missed it.

Yes, I know the Dirac equation effectively 'marries' Schrodinger's equations with Einstein's special relativity (therefore the Lorenz transformation) but I admit I'm not familiar with the specifics.

I just assumed that, because tunneling is infinite velocity and zero time, the Lorenz transformation didn't apply, because it forbids velocity greater than c.

So how does Dirac's equation overcome this, if it can be explained in ordinary English?

I take your point: my reference to Godel's Incompleteness Theorem, in that context, was most likely "a sloppy gloss". I actually do know that: "this does not show one does not 'know' whether a particular undecidable sentence is true."

And I also know that: "The claim that there are things about mathematics we will never know is trivial. Surely one didn't need to invent mathematical logic to learn that!"

So why my statement, made in the heat of argument, infers that we need Godel's Theorem to 'prove' that there are things we may never know about mathematics, that is not true, and, yes, it's trivial. I just added it as a 'by the way', when, as you point out, it's not really relevant. Godel's Theorem, like Turing's, effectively says that we can't decide in advance whether a conjecture, like Reimann's or Goldbach's, can be proved or disproved by knowing if it has a limit.

I'm sure you'll tell me if I have that wrong.

Thanks for your comments. I don't mind being corrected by someone with a lot more knowledge than me.

Regards, Paul.

Paul P. Mealing said...

Hi Timmo, I acknowledge your contribution to this post as an addendum in the original post.

In regard to Godel's Theorem, it occurred to me, when I made that reference between the trivial and the profound, that they both deal with infinities. We can never know all mathematical truths because mathematics is infinite in scope on many different levels.

Likewise, Godel's theorem, or Turing's, because I find it easier to follow, also works because mathematics is infinite. A 'routine', trying to find a solution to an algorithm, could possibly run forever, and we can't tell in advance if that's true or not. If mathematics wasn't infinite then Turing's (and Godel's) Theorem would not apply.

I guess that's why I made that jump from the trivial to the profound.

Regards, Paul.

Timmo said...

Hi Paul,

Sorry to be so sluggish in replying.

These 3 sentences follow each other exactly like this in the book, so the inference is that the 'evanescent mode' does the tunneling and is also the imaginary component, and it is the authors that claim it was originally rejected

It looks like Nimtz and Haibel used some unfortunate language (I don't quite see what they are getting at), and I can see how you would draw that inference from what they said!

Complex numbers z = x + iy, as you know, have a real x and imaginary part y. Because they have two parts they are useful for describing wave phenomona: one wants to keep track of the both the amplitude and phase of the wave. Classically, when describing light waves it is handy to use complex numbers, and then at the end of the problem just take the real part. It turns out that the quantum-mechanical wavefunction ψ is a complex number and needs to be so.

I think what you are noticing here is that the probability to find the particle here does not just depend on what's happening in this neighborhood, but everywhere in the problem.

So how does Dirac's equation overcome this, if it can be explained in ordinary English?

Actually, this is a very profound question. Suppose I have a box with a shudder that contains a radioactive nucleus that emits electrons. Somewhere across the room you have a geiger counter. If I open the shudder, electrons can come out and your geiger counter can click. It turns out that there is a finite probability for your geiger counter to click as soon as I open the shudder. How can this be consistent with relativity if the electrons can't fly from the source to the detector in just an instant?

The problem is even more accute when we start comparing different reference frames. In one frame F, I open shudder (event A) and your geiger counter across the room click (event B) simultaneously. If we boost to another reference frame, then , according to the theory of relativity, those events will not necessarily be simultaneous. In F*, A happens before B. But, in some other frame, F**, B happens before A -- your geiger counter clicks before I even open the shudder!

So, how is is the finite probability for instantaneous detection compatibile with the requirement that there is no action at a distance? The solution is the introduction of anti-particles. Electrons have a sort of twin, the positron, which has the same spin, mass, and opposite electric charge. When electrons and positrons come together they can annihilate each other.

Here's what happens when the shudder opens and the counter goes off: I open the shudder and electrons come streaming out. Next to your detector, an electron-positron pair randomly appear from the vacuum. You detect the electron from that pair, and then some time later an electron coming from my detector annihilates the positron. All of the interactions in this picture occur locally.

What does it mean for the Dirac equation to be Lorentz invariant? Suppose you and I use different intertial reference frames. The principle of relativity states that whatever laws of physics I write down in my notebook using my co-ordinates should be the same laws of physics you write down in your notebook using your co-ordinates -- when you change from one reference frame to another the mathematical form of the laws should remain the same. The Lorentz transform equations are a set of relations connecting the space and time co-ordinates of my reference frame to your inertial frame. The Schrodinger equation changes form under this transformation so it can't be consistent with relativity. But, Dirac dreamed up another quantum-mechanical law of motion -- the Dirac equation -- which does not change form when you change from one reference frame to the other.

Timmo said...


If mathematics wasn't infinite then Turing's (and Godel's) Theorem would not apply.

Yes, one of the fascinating 'Godel results' is that if we have a sufficiently powerful formal mathematical theory T then there is no rote procedure or algorithm for deciding whether or not a given statement in the language of that theory T is a theorem of T.

But, this doesn't have anything in particular to do with the infinity of true statements of, say, number theory. Simple (classical!) propositional logic has infinitely many theorems (the tautologies). But, there is a rote procedure for determining whether or not any formula of propositional logic is a tautology: namely, truth-tables. No matter how big and horrible a thing you write down (with just p's and q's and &'s and ~'s and V's), I can work it out in a finite number of steps using truth-tables with no creativity at all. The incompleteness and recursive undecideability really all flows from recursive functions and their properties.

Timmo said...

Personally, I am uncomfortable with terms like 'quantum non-locality' because it is never used consistently with the typical definition of locality that you offer in terms of interactions and how they propagate through space. Strangely, both physicists and philosophers use 'non-locality' to refer to other things. The probability P to find an electron at point x depends up the entire experimental setup. P(x) does not just depend on x and what's right next to x. So, in that sense, quantum mechanics is non-local, and tunneling would be an example of that kind of 'non-locality.' But, notice this is different from the definition given in terms of interactions before.

There is a strong temptation to infer from 'non-locality' that there is genuine, true-blooded non-locality of the kind you see in Bohm's hidden variable theory. It is a logical jump, though a very tempting one because it makes 'non-locality' more understandable.

Bohmian mechanics is certainly compelling. It offers a very reasonable solution to the measurement problem, if nothing else. Bohm, like Bohr, clearly perceived and stressed the wholeness inherent in quantum-mechanical phenomena.

I, personally, have not assented to Bohmian mechanics because no one has yet devised a Bohmian-style interpretation of relativistic quantum field theory, the trouble being the requirement that it be consistent with the theory of relativity. Though, there are some people thinking hard about this and it will be interesting to see what they come up with!

Paul P. Mealing said...

Thanks Timmo, for your enlightening comments. I appreciate them very much, especially your explanation of Dirac's resolution of tunneling by the production of an anti-particle.

It's very easy to fall into the trap of thinking you know more than you do, so I don't mind being brought to earth.

Regards, Paul.

Paul P. Mealing said...

Hi Timmo,

Back in May I wrote a review of a book by Kerson Huang on gauge fields that led to a discussion of Peter Woit's book on String Theory called Not Even Wrong. Woit is critical of String Theory by the way.

You don't have to, but I wouldn't mind if you cast a critical eye over my post. In particular, I speculate that gauge fields seem to be scale-dependent, due to what Huang describes as the 'RG trajectoy'. Is this, of itself, correct? Or am I once again making inferences that aren't true?

Regards, Paul.

Timmo said...

Hi Paul,

I appreciate them very much, especially your explanation of Dirac's resolution of tunneling by the production of an anti-particle.

It's not specifically tunelling that is resolved by introducing anti-particles, but the problem of how to make quantum mechanics consistent with the theory of relativity. What's really interesting to me is that when you blend quantum mechanics with relativity you get new things you might not have otherwise, such as anti-particles. Another thing that comes out is spin and particle statistics. You can have spin in non-relativistic quantum mechanics and handle it ad hoc, but the spin of the electron comes out as a necessary consequence of the Dirac equation! Also, there is the 'spin-statistics' theorem, which says that you have to have two kinds of particles, bosons which like to be in the same state and fermions which are never in the same state, and they have half-integer or integer-spin respectively. It's a beautiful and deep thing, and I'd like to understand it better.

It's very easy to fall into the trap of thinking you know more than you do, so I don't mind being brought to earth.

I'm sorry if I've come off that way. The difficult thing about thinking through the philosophical implications or suggestions of modern physics is that it requires both significant philosophical and scientific acumen/sophistication. I've heard some physicists wax philosophical and it can be ugly. I wish my insight went deeper and farther than it does -- otherwise, I'm doomed to talk nonsense, too!

You don't have to, but I wouldn't mind if you cast a critical eye over my post.

I'll look it over. :-)

Timmo said...

Hi Paul,

It just occurred to me that if you are interested in the interpretation/philosophy of quantum mechanics, you should check out David Albert's book Quantum Mechanics and Experience. I am critical of Albert's cursory dismissal of Bohr's complementarity interpretation as well as the eventual many-minds position he takes up. At the same time, the book is very clear, accessible, and written in chatty prose. It's quite an achievement, actually, as he explains some very technical points (like how Bohm's hidden variables theory resolves the measurement problem). If you're interested in these topics, this book is required reading.

Paul P. Mealing said...

Thanks Timmo,

Definitely sounds like something I should read. I have 4 books in front of me right now, but I'll keep it in mind.

Thanks a lot, Paul.