by Jan Westerhoff
02 October 2012
from Livasperiklis Website



Jan Westerhoff is a philosopher at the University of Durham and the University of London’s School of Oriental and African Studies, both in the UK, and author of  Reality: A very short introduction 

(Oxford University Press, 2011)




It’s relatively easy to demonstrate what physical reality isn’t.
It is much harder to work out what it is.


How do electrons know how to make an Airy pattern?

(Image: GI PhotoStock/Science Photo Library)


NOTHING seems more real than the world of everyday objects, but things are not as they seem.


A set of relatively simple experiments reveals enormous holes is our intuitive understanding of physical reality. Trying to explain what goes on leads to some very peculiar and often highly surprising theories of the world around us.


Here is a simple example.


Take an ordinary desk lamp, a few pieces of cardboard with holes of decreasing sizes, and some sort of projection screen such as a white wall. If you put a piece of cardboard between the lamp and the wall, you will see a bright patch where the light passes through the hole in the cardboard. If you now replace the cardboard with pieces containing smaller and smaller holes, the patch too will diminish in size.


Once we get below a certain size, however, the pattern on the wall changes from a small dot to a series of concentric dark and light rings, rather like an archery target. This is the “Airy pattern” - a characteristic sign of a wave being forced through a hole (see image).


In itself, this is not very surprising. After all, we know that light is a wave, so it should display wave-like behavior.


But now consider what happens if we change the set-up of the experiment a bit. Instead of a lamp, we use a device that shoots out electrons, like that found in old-fashioned TV sets; instead of the wall, we use a plate of glass coated with a phosphor that lights up when an electron strikes it. We can therefore use this screen to track the places where the electrons hit.


The results are similar: with sufficiently small holes we get an Airy pattern.


This now seems peculiar:

electrons are particles located at precise points and cannot be split. Yet they are behaving like waves that can smear out across space, are divisible, and merge into one another when they meet.

Perhaps it is not that strange after all.


Water consists of molecules, yet it behaves like a wave. The Airy pattern may just emerge when enough particles come together, whether they are water molecules or electrons.


A simple variant of the experiments shows, however, that this cannot be right. Suppose we reduce the output of the electron gun to one particle each minute. The Airy pattern is gone, and all we see is a small flash every minute. Let’s leave this set-up to run for a while, recording each small flash as it occurs.


Afterwards, we map the locations of all the thousands of flashes.


Surprisingly, we do not end up with a random arrangement of dots, but with the Airy pattern again. This result is extremely strange. No individual electron can know where all the earlier and later electrons are going to hit, so they cannot communicate with each other to create the bulls-eye pattern.


Rather, each electron must have travelled like a wave through the hole to produce the characteristic pattern, then changed back into a particle to produce the point on the screen.


This, of course, is the famous wave-particle duality of quantum mechanics.


This strange behavior is shared by any sufficiently small piece of matter, including electrons, neutrons, photons and other elementary particles, but not just by these. Similar effects have been observed for objects that are large enough in principle to be seen under a microscope, such as buckyballs.


In order to explain the peculiar behavior of such objects, physicists associate a wave function with each of them.


Despite the fact that these waves have the usual properties of more familiar waves such as sound or water waves, including amplitude (how far up or down it deviates from the rest state), phase (at what point in a cycle the wave is), and interference (so that “up” and “down” phases of waves meeting each other cancel out), what they are waves in is not at all transparent. Einstein aptly spoke of a “phantom field” as their medium.


For a wave in an ordinary medium such as water, we can calculate its energy at any one point by taking the square of its amplitude.


Wave functions, however, carry no energy. Instead, the square of their amplitude at any given point gives us the probability of observing the particle if a detector such as the phosphor-coated screen is placed there.


Clearly, the point where an object switches from being a probability wave, with its potential existence smeared out across space, and becomes an actual, spatially localized object is crucially important to understanding whether matter is real.


What exactly happens when the wave function collapses - when among the countless possibilities where the particle could be at any moment, one is chosen, while all the others are rejected?


First of all, we have to ask ourselves when this choice is made. In the example described above, it seems to happen just before the flash on the phosphor screen.


At this moment, a measurement of the electron’s position was made by a piece of phosphor glowing as the particle struck it, so there must have been an electron there, and not just a probability wave.


But assume we cannot be in the lab to observe the experiment, so we point a camera at the phosphor screen and have the result sent via a satellite link to a computer on our desktop. In this case, the flash of light emitted from the phosphor screen has to travel to the camera recording it, and the process is repeated: like the electrons, light also travels as a wave and arrives as a particle.


What reason is there to believe that the switch from probability wave to particle actually occurred on the phosphor screen, and not in the camera?


At first, it seemed as if the phosphor screen was the measuring instrument, and the electron was the thing being measured. But now the measuring device is the camera and the phosphor screen is part of what is measured.


Given that any physical object transmitting the measurement we can add on to this sequence - the camera, the computer, our eyes, our brain - is made up of particles with the same properties as the electron, how can we determine any particular step at which to place the cut between what is measured and what is doing the measuring?


This ever-expanding chain is called the von Neumann chain, after the physicist and mathematician John von Neumann.


One of his Princeton University colleagues, Eugene Wigner, made a suggestion as to where to make the cut. As we follow the von Neumann chain upwards, the first entity we encounter that is not made up in any straightforward fashion out of pieces of matter is the consciousness of the observer. We might therefore want to say that when consciousness enters the picture, the wave function collapses and the probability wave turns into a particle.


The idea that consciousness brings everyday reality into existence is, of course, deeply strange; perhaps it is little wonder that it is a minority viewpoint.


There is another way of interpreting the measurement problem that does not involve consciousness - though it has peculiar ramifications of its own. But for now let’s explore Wigner’s idea in more depth.


If a conscious observer does not collapse the wave function, curious consequences follow. As more and more objects get sucked into the vortex of von Neumann’s chain by changing from being a measuring instrument to being part of what is measured, the “spread-out” structure of the probability wave becomes a property of these objects too.


The “superposed” nature of the electron - its ability to be at various places at once - now also affects the measuring instruments.


It has been verified experimentally that not just the unobservable small, but objects large enough to be seen under a microscope, such as a 60-micrometre-long metal strip, can exhibit such superposition behavior.


Of course, we can’t look through a microscope and see the metal strip being at two places at once, as this would immediately collapse the wave function. Yet it is clear that the indeterminacy we found at the atomic level can spread to the macro level.


Yet if we accept that the wave function must collapse as soon as consciousness enters the measurement, the consequences are even more curious.


If we decide to break off the chain at this point, it follows that, according to one of our definitions of reality, matter cannot be regarded as real. If consciousness is required to turn ghostly probability waves into things that are more or less like the objects we meet in everyday life, how can we say that matter is what would be there anyway, whether or not human minds were around?


But perhaps this is a bit too hasty. Even if we agree with the idea that consciousness is required to break the chain, all that follows is that the dynamic attributes of matter such as position, momentum and spin orientation are mind-dependent. It does not follow that its static attributes, including mass and charge, are dependent on in this.


The static attributes are there whether we look or not.


Nevertheless, we have to ask ourselves whether redefining matter as “a set of static attributes” preserves enough of its content to allow us to regard matter as real. In a world without minds, there would still be attributes such as mass and charge, but things would not be at any particular location or travel in any particular direction.


Such a world has virtually nothing in common with the world as it appears to us.


Werner Heisenberg observed that:

“the ontology of materialism rested upon the illusion that the kind of existence, the direct ‘actuality’ of the world around us, can be extrapolated into the atomic range. This extrapolation, however, is impossible… Atoms are not things.”

It seems that the best we are going to get at this point is the claim that some things are there independent of whether we, as human observers, are there, even though they might have very little to do with our ordinary understanding of matter.


Does our understanding of the reality of matter change if we choose the other strong definition of reality - not by what is there anyway, but by what provides the foundation for everything else (see “Reality - The definition“)?


In order to answer this question, we have to look at the key scientific notion of a reductive explanation.


Much of the power of scientific theories derives from the insight that we can use a theory that applies to a certain set of objects to explain the behavior of a quite different set of objects. We therefore don’t need a separate set of laws and principles to explain the second set.


A good example is the way in which theories from physics and chemistry, dealing with inanimate matter, can be used to explain biological processes. There is no need to postulate a special physics or a special chemistry to explain an organism’s metabolism, how it procreates, how its genetic information is passed on, or how it ages and dies.


The behavior of the cells that make up the organism can be accounted for in terms of the nucleus, mitochondria and other subcellular entities, which can in turn be explained in terms of chemical reactions based on the behavior of molecules and the atoms that compose them.


For this reason, explanations of biological processes can be said to be reducible to chemical and ultimately to physical ones.


If we pursue a reductive explanation for the phenomena around us, a first step is to reduce statements about the medium-sized goods that surround us - bricks, brains, bees, bills and bacteria - to statements about fundamental material objects, such as molecules. We then realize everything about these things can be explained in terms of their constituents, namely their atoms.


Atoms, of course, have parts as well, and we are now well on our way through the realm of ever smaller subatomic particles, perhaps (if string theory is correct) all the way down to vibrating strings of pure energy.


So far we have not reached the most fundamental objects. In fact, there is not even an agreement that there are any such objects.


Yet this is no reason to stop our reductionist explanation here, since we can always understand the most basic physical objects in terms of where they are in space and time. Instead of talking about a certain particle that exists at such-and-such a place for such-and-such a period of time, we can simply reduce this to talk about a certain region in space that is occupied between two different times.


We can go even more fundamental.


If we take an arbitrary fixed point in space, and a stable unit of spatial distance, we can specify any other point in space by three coordinates. These simply tell us to go so many units up or down, so many units left or right, and so many units back or forth. We can do the same with points in time.


We now have a way of expressing points in space-time as sets of four numbers, x, y, z and t, where x, y, and z represent the three spatial dimensions and t the time dimension. In this way, reality can be boiled down to numbers.


And this opens the door to something yet more fundamental. Mathematicians have found a way of reducing numbers to something even more basic: sets.


To do this, they replace the number 0 with the empty set, the number 1 with the set that contains just the empty set, and so on (see “Reality - Is everything made of numbers?“ - below insert).






-   Reality   -

Is Everything Made of Numbers?

by Amanda Gefter

extracted from New Scientist Magazine - 29 September 2012


Amanda Gefter is a writer and New Scientist consultant

 based in Boston, Massachusetts.


The fact that the natural world can be described so precisely

by mathematics is telling us something profound,

says Amanda Gefter.

WHEN Albert Einstein finally completed his general theory of relativity in 1916, he looked down at the equations and discovered an unexpected message: the universe is expanding.

Einstein didn't believe the physical universe could shrink or grow, so he ignored what the equations were telling him, Thirteen years later, Edwin Hubble found clear evidence of the universe's expansion. Einstein had missed the opportunity to make the most dramatic scientific prediction in history.

How did Einstein's equations "know" that the universe was expanding when he did not?


If mathematics is nothing more than a language we use to describe the world, an invention of the human brain, how can it possibly churn out anything beyond what we put in?

"It is difficult to avoid the impression that a miracle confronts us here," wrote physicist Eugene Wigner in his classic 1960 paper "The unreasonable effectiveness of mathematics in the natural sciences" (Communications on Pure and Applied Mathematics, vol 13, p 1).

The prescience of mathematics seems no less miraculous today.


At the Large Hadron Collider at CERN, near Geneva, Switzerland, physicists recently observed the fingerprints of a particle that was arguably discovered 48 years ago lurking in the equations of particle physics.

How is it possible that mathematics "knows" about Higgs particles or any other feature of physical reality?

"Maybe it's because math is reality," says physicist Brian Greene of Columbia University, New York.

Perhaps if we dig deep enough, we would find that physical objects like tables and chairs are ultimately not made of particles or strings, but of numbers.

"These are very difficult issues," says philosopher of science James Ladyman of the University of Bristol, UK, "but it might be less misleading to say that the universe is made of maths than to say it is made of matter."

Difficult indeed.


What does it mean to say that the universe is "made of mathematics"?


An obvious starting point is to ask what mathematics is made of. The late physicist John Wheeler said that the "basis of all mathematics is 0 = 0". All mathematical structures can be derived from something called "the empty set", the set that contains no elements.


Say this set corresponds to zero; you can then define the number 1 as the set that contains only the empty set, Z as the set containing the sets corresponding to 0 and 1, and so on, Keep nesting the nothingness like invisible Russian dolls and eventually all of mathematics appears.


Mathematician Ian Stewart of the University of Warwick, UK, calls this,

"the dreadful secret of mathematics: it's all based on nothing" (New Scientist, 19 November 2011, p 44).

Reality may come down to mathematics, but mathematics comes down to nothing at all.

That may be the ultimate clue to existence - after all, a universe made of nothing to require a physical origin at all.

"A dodecahedron was never created," says Max Tegmark of the Massachusetts Institute of Technology. "To be created, something first has to not exist in space or time and then exist."

A dodecahedron doesn't exist in space or time at all, he says - it exists independently of them.

"Space and time themselves are contained within larger mathematical structures," he adds.

These structures just exist; they can't be created or destroyed.

That raises a big question: why is the universe only made of some of the available mathematics?

"There's a lot of math out there," Greene says.


"Today only a tiny sliver of it has a realization in the physical world. Pull any math book off the shelf and most of the equations in it don't correspond to any physical object or physical process."

It is true that seemingly arcane and unphysical mathematics does, sometimes, turn out to correspond to the real world.


Imaginary numbers, for instance, were once considered totally deserving of their name, but are now used to describe the behavior of elementary particles; non-Euclidean geometry eventually showed up as gravity.


Even so, these phenomena represent a tiny slice of all the mathematics out there.

Not so fast, says Max Tegmark.

"I believe that physical existence and mathematical existence are the same, so any structure that exists mathematically is also real," he says.

So what about the mathematics our universe doesn't use?

"Other mathematical structures correspond to other universes," Tegmark says.

He calls this the "level 4 multiverse", and it is far stranger than the multiverses that cosmologists often discuss.


Their common-or-garden multiverses are governed by the same basic mathematical rules as our universe, but Tegmark's level 4 multiverse operates with completely different mathematics.

All of this sounds bizarre, but the hypothesis that physical reality is fundamentally mathematical has passed every test.

"If physics hits a roadblock at which point it turns out that it's impossible to proceed, we might find that nature can't be captured mathematically," Tegmark says.


"But it's really remarkable that that hasn't happened, Galileo said that the book of nature was written in the language of mathematics - and that was 400 years ago."

If reality isn't, at bottom, mathematics, what is it?

"Maybe someday we'll encounter an alien civilization and we'll show them what we've discovered about the universe," Greene says.


"They'll say, 'Ah, math. We tried that. It only takes you so far, Here's the real thing,' What would that be? It's hard to imagine. Our understanding of fundamental reality is at an early stage."





All the properties of numbers also hold for all these ersatz numbers made from sets.


It seems as if we have now reduced all of the material world around us to an array of sets. For this reason, it is important to know what these mathematical objects called sets really are.


There are two views of mathematical objects that are important in this context.

  • First, there is the view of them as “Platonic” objects. This means that mathematical objects are unlike all other objects we encounter.


    They are not made of matter, they do not exist in space or time, do not change, cannot be created or destroyed, and could not have failed to exist.


    According to the Platonic understanding, mathematical objects exist in a “third realm”, distinct from the world of matter, on the one hand, and the world of mental entities, such as perceptions, thoughts and feelings, on the other.


  • Second, we can understand mathematical objects as fundamentally mental in nature. They are of the same kind as the other things that pass through our mind: thoughts and plans, concepts and ideas.


    They are not wholly subjective; other people can have the very same mathematical object in their minds as we have in ours, so that when we both talk about the Pythagorean theorem, we are talking about the same thing.


    Still, they do not exist except in the minds in which they occur.

Either of these understandings leads to a curious result.


If the bottom level of the world consists of sets, and if sets are not material but are instead some Platonic entities, material objects have completely disappeared from view and cannot be real in the sense of constituting a fundamental basis of all existence.


If we follow scientific reductionism all the way down, we end up with stuff that certainly does not look like tiny pebbles or billiard balls, not even like strings vibrating in a multidimensional space, but more like what pure mathematics deals with.


Of course, the Platonistic view of mathematical objects is hardly uncontroversial, and many people find it hard to get any clear idea of how objects could exist outside of space and time.


But if we take mathematical objects to be mental in nature, we end up with an even stranger scenario.


The scientific reductionist sets out to reduce the human mind to the activity of the brain, the brain to an assembly of interacting cells, the cells to molecules, the molecules to atoms, the atoms to subatomic particles, the subatomic particles to collections of space-time points, the collections of space-time points to sets of numbers, and the sets of numbers to pure sets.


But at the very end of this reduction, we now seem to loop right back to where we came from: to the mental entities.


We encounter a similar curious loop in the most influential way of understanding quantum mechanics, the Copenhagen interpretation. Unlike Wigner’s consciousness-based interpretation, this does not assume the wave function collapses when a conscious mind observes the outcome of some experiment.


Instead, it happens when the system to be measured (the electron) interacts with the measuring device (the phosphor screen). For this reason, it has to be assumed that the phosphor screen will not itself exhibit the peculiar quantum behavior shown by the electron.


In the Copenhagen interpretation, then, things and processes describable in terms of familiar classical concepts are the foundation of any physical interpretation. And this is where the circularity comes in.


We analyze the everyday world of medium-sized material things in terms of smaller and smaller constituents until we deal with parts that are so small that quantum effects become relevant for describing them.


But when it comes to spelling out what is really going on when a wave function collapses into an electron hitting a phosphor screen, we don’t ground our explanation in some yet more minute micro-level structures; we ground it in terms of readings made by non-quantum material things.


What this means is that instead of going further down, we instead jump right back up to the level of concrete phenomena of sensory perception, namely measuring devices such as phosphor screens and cameras.


Once more, we are in a situation where we cannot say that the world of quantum objects is fundamental. Nor can we say that the world of measuring devices is fundamental since these devices are themselves nothing but large conglomerations of quantum objects.


We therefore have a circle of things depending on each other, even though, unlike in the previous case, mental objects are no longer part of this circle.


As a result, neither the phosphor screen nor the minute electron can be regarded as real in any fundamental sense, since neither constitutes a class of objects that everything depends on. What we thought we should take to be the most fundamental turns out to involve essentially what we regarded as the least fundamental.


In our search for foundations, we have gone round in a circle, from the mind, via various components of matter, back to the mind - or, in the case of the Copenhagen interpretation, from the macroscopic to the microscopic, and then back to the macroscopic.


But this just means that nothing is fundamental, in the same way there is no first or last stop on London Underground’s Circle Line.


The moral to draw from the reductionist scenario seems to be that either what is fundamental is not material, or that nothing at all is fundamental.