Dear Friends,
Past the link if you don't receive the images/can't access the links.
http://www.ipod.org.uk/reality/reality_mathematical_universe.asp
Love and Light.
David
| Home | Quantum Introduction | The Quantum Casino | Quantum Entanglement | Quantum Decoherence | Quantum Reality | It's A Small World |
| The Cosmic Universe | The Anthropic Principle | The Arrow of Time | The Mathematical Universe | Is the Universe a Computer? | Living in the Matrix | The Intelligent Universe |
The Mathematical Universe
On this page we will consider the rather uncanny central role that mathematics appears to play in the structure of the universe. We will also consider the theory that certain mathematical structures have a form of "reality" all of their own.
It could be said that the role mathematicians is to discover truths which are already "out there". These truths are no inventions of clever men - it does not matter who invented the mathematical structure of complex numbers, for example. Such structures have been there since the beginning of time - an eternal truth - waiting to be uncovered.
This view that certain mathematical concepts are eternal truths which apparently have an independent reality of their own was proposed in ancient times by the great Greek philosopher Plato (c. 360 BC). Consequently, the mathematical structures with this apparent reality are called Platonic.
And perhaps the most stunning example of a mathematical structure which has recently been discovered is the Mandelbrot Set.
The Mandelbrot Set
It was Benoit Mandelbrot who first introduced us to the beauty of the Mandelbrot set in 1980. However, when Mandelbrot first saw the strange patterns printed by his computer he suspected his computer was broken!
The Mandelbrot Set is produced by a remarkably simple mathematical formula - a few lines of code describing a recursive feedback loop - but can be used to produce beautiful, coloured, computer plots. What makes it so extraordinary is that it is possibly to endlessly zoom in to the set revealing ever more beautiful structures which never seem to repeat themselves. It's almost as if it is a mathematical object with an independent existence of its own, and we are the explorers investigating this uncharted mathematical world.
Referring back to our earlier discussion, it might be thought that the Mandelbrot set has a Platonic reality all of its own - it doesn't matter who creates the diagram, or which computer is used, the structure will always appear the same. To quote Roger Penrose from The Emperor's New Mind: "The Mandelbrot Set is not an invention of the human mind: it was a discovery. Like Mount Everest, the Mandelbrot Set is just there!"
The applet below allows you to explore the Mandelbrot Set for yourself.
You can zoom into the image below by creating a window around the area you want to zoom to. To do this, just click on the image and drag a box (Tip: try zooming in to the very narrow region between the red and the black areas).
Just click once to reset to the original image.
(You need to have Java installed in your browser to run the applet, so if you cannot see the Mandelbrot set immediately below, go to www.java.com to get the Java Runtime Environment for your browser).
(Applet created by Andreea Francu)
I tried playing around with the applet above and produced the images below. See if you can do better:
Using the applet I discovered the image on the left below. Don't you think it looks like a piece of coastline?
The reason for this is that the border of the Mandelbrot Set has a shape like a fractal (a fractal coastline actually has infinite length, revealing endless detail as you zoom in. For more information see the section "The Tower of Turtles" on the It's a Small World page). Fractals are geometric shapes found throughout nature which are characterised by having many similar branches. Fractals are, in fact, self-similar because no matter how far you zoom into them they still resemble the original object (a branch of a tree, for example, resembles the whole tree):
So here we can see an example of how mathematics is an underlying force in the design of nature. Simple mathematical rules are responsible for the beautiful complexity of nature. Maverick genius Stephen Wolfram stresses the importance of mathematics in designing nature in his controversial book A New Kind of Science (see this Forbes article). We have a tendency to assume that evolution is the sole factor in designing nature. However, Wolfram reveals the undoubted role of mathematics alone in designing not only trees but also objects such as sea shells:
The fractal "Pascal's Triangle" creates the pattern on sea shells
(picture from Stephen Wolfram's A New Kind of Science)
Just as a sidethought, I sometimes wonder if the complex structure of the Mandelbrot Set can provide valuable insights into the nature of our universe (after all, the universe appears to follow mathematical principles). Whenever we talk about the Mandelbrot Set we are used to seeing beautiful, colourful pictures of swirling complex patterns. But what is not generally realised is that the wonderful complexity can only be found in the very thin border region of the set. The iterative equations used to produce the set can be used to assign a colour to every pixel in the entire (infinite) 2-dimensional Argand plane (i.e., treating the Cartesian coordinates to represent complex numbers), but the black area which stretches outside the colourful area is completely blank (the "Big Boring Area" in the diagram below). Similarly, the area inside the Mandelbrot border is completely blank (the "Small Boring Area" in the diagram below). Only the very thin border between these two regions contains the fractal complexity which has made the Mandelbrot Set so famous (the "Interesting Bit" between the two red arrows below).
Maybe we could imagine the complex structure of the Mandelbrot Set as a "mini universe". Then it might be possible to find an analogy between these three regions of the Mandelbrot Set and our universe. Let's try equating distance from the centre of the Mandelbrot Set with increasing scale in our universe. The vast, unbounded region on the outside of the set would equate to the largest scales (e.g., galaxies), whereas the very smallest scales toward the centre of the set would equate to elementary particles. The thing is, it could be said that the most interesting thing in the universe (intelligent life) occurs in the very narrow fractal border region (human scale) between these two (rather predictable and boring) extremes of scale.
Copernicus discovered that the earth does not hold a special position in the universe - but maybe he was considering the wrong "space" (John D. Barrow from The Infinite Book: "While Copernicus's idea that our position in the universe should not be special in every sense is sound, it is not true that it cannot be special in any sense"). If we move away from considering our positioning in physical space and instead consider the positioning and conditions of Earth in an abstracted"Mass of star relative to Sun""Radius of orbit relative to Earth's": mathematical space then we would find it to be very special indeed. Consider a number of variables describing the conditions and positioning of Earth, such as the scale of the planet, its distance from the sun, its surface conditions, the positioning of the neighbouring planets, and then consider each of these variables in an abstracted mathematical space. For example, the diagram below plots against
In that case, if we stood back and considered our findings for all our variables we would find that Earth's position and conditions are very special indeed: it's in a very narrow region called the Habitable Zone (indicated as a curvy grey band on the diagram above). Two red arrows are superimposed on the diagram to indicate that the Habitable Zone is analogous to the Interesting Bit shown on the Mandelbrot Set diagram considered above. In fact, you could say the Earth is right in that interesting fractal border region of the Mandelbrot Set where interesting things happen (see Wikipedia on planetary habitability hypothesis). The most interesting thing in the universe (intelligent life) occurs in this very narrow fractal border region, just as if it was a galactic Mandelbrot Set.
If I was an observer of the universe, that's the region I'd zoom into.
Max Tegmark's Mathematical Universe
Let's return to the discussion on mathematical Platonism from the top of this page, the idea that mathematical concepts are eternal truths which have an independent reality existing outside of space and time. What sort of reality are we talking about for these mathematical concepts? Are we talking about a hard, physical reality which we can touch? Well, not according to Roger Penrose in The Emperor's New Mind (which firmly advocates some form of mathematical Platonism): "It is the Mandelbrot Set's 'mathematician-independence' that gives it it's Platonic existence". So this traditional Platonism is definitely intended to be different from the more usual concept of "hard" physical reality.
However, one physicist has taken the idea a step further ...
Max Tegmark is a physicist at MIT who likes to explore speculative theories about reality in his spare time. One of his recent theories - which has generated a lot of publicity (and ire!) - is the Mathematical Universe Hypothesis or MUH (see his arXiv paper The Mathematical Universe, or this considerably shorter arXiv article). According to the MUH, mathematical things actually exist, and they are actually physical reality! So the world around us actually is"The Mathematical Universe is in this sense very contrary to traditional Platonism: it does not say that existence is a mere shadow, an imperfect copy of some eternal object 'out there' in some inaccessible realm, but in fact the Platonic relations are all there is" - see here. maths. This is a clear departure from traditional Platonism which merely imagines mathematical objects as existing in some separate plane of existence:
So how did Tegmark come to such a remarkable conclusion? Well, as a first step he considered the definition of what we mean by "physical reality". It is certainly not easy to obtain any such definition of "reality", but a simple definition would be something we could touch, a tangible reality composed of solid objects (though light is also "real" even though we cannot touch it). Most physicists seem to agree that "physical reality" should be external to any observer, an observer-independent, objective reality, i.e., it does not matter who is looking - two observers would get the same results when they analyse the world around them (Tegmark called this the External Reality Hypothesis). Tegmark realized that any such reality - external and independent of any human observer - would have to be free of any human "baggage": "If we assume that reality exists independently of humans, then for a description to be complete, it must also be well-defined according to non-human entities - aliens or supercomputers, say - that lack any understanding of human concepts. Put differently, such a description must be expressible in a form that is devoid of any human baggage like 'particle', 'observation' or other English words." - see here. And so a universe of pure mathematics (and therefore completely free of human baggage) would appear to be a likely contender for the basis of our reality: "A description of objects in this external reality and the relations between them would have to be completely abstract, forcing any words or symbols to be mere labels with no preconceived meanings whatsoever."
In an interview with Max Tegmark in Discover magazine, the interviewer finds it hard to believe that reality is mathematics: "I don’t feel like a bunch of equations. My breakfast seemed pretty solid". Such an attitude is understandable, but we have to imagine how the universe would appear to an inhabitant of the mathematical universe. As Tegmark explains: "To understand the concept, you have to distinguish two ways of viewing reality. The first is from the outside, like the overview of a physicist studying its mathematical structure. The second way is the inside view of an observer living in the structure." Tegmark then goes on to refer to the outside view as the "bird" view, and the inside view as the "frog" view.
With this in mind, we can realize that a "frog-like" occupant of a mathematical universe would be surrounded by mathematical objects defined in terms of their relationships with other "real" mathematical objects. They would define their "real" objects in just the same, circular way we define our "real", tangible objects - in terms of other objects which we already consider real. For example, we might say: "I know the apple is real because I can hold it with my hand". As Brian Whitworth says in his paper The Physical World as a Virtual Reality: "Reality is relative to the observer, so by analogy, a table is 'solid' because our hands are made of the same atoms as the table". So any occupant of a mathematical universe would experience that same circular definition of reality which we experience. In other words, they would not be able to tell their reality was composed of purely mathematical objects! Yes, their breakfast would indeed seem "pretty solid" to them.
I have a couple of reservations about the mathematical universe:
Firstly, I fail to see how such a structure - supposedly composed of all possible mathematical structures - could be logically consistent. Physical reality is known to be logically consistent, but this mathematical reality would be riven with destructive paradoxes and inconsistencies. For example, Russell's paradox (which is considered in details in the Meta Maths section below): "S is the set of all sets which do not have themselves as a member. Is S a member of itself?". Such mathematical paradoxes do not exist in physical reality. Also, if physical reality really is mathematics then this would suggest that objects which could contain themselves could actually exist (which of course, they do not - again, see the Meta Maths section below). And as Tegmark admits, Gödel's Theorem implies that we can never be certain that mathematics is consistent: it leaves open the possibility that a proof exists demonstrating that 0 = 1.
Mathematics on its own does not lead to a sensible universe: we seem to need to apply constraints in order to obtain consistent physical reality from mathematics (in order to remove the destructive paradoxes and inconsistencies: unrestricted axioms lead to Russell's paradox - see here). Max Tegmark relies on quoting David Hilbert: "Mathematical existence is merely freedom from contradiction", implying that mathematical structures simply do not exist unless they are consistent. But implicit in Hilbert's quote is some unexplained way of avoiding the previously-discussed paradoxes. What mechanism is selecting that consistent subset of maths from the whole of maths? It's easy for mathematicians such as Max Tegmark or David Hilbert to apply restrictions (maybe applying something like Zermelo-Frankel set theory which avoids Russell's paradox), but that's clearly a manual intervention in the natural process. How does physical reality achieve that? In my view, Tegmark seeks to avoid paradoxes and inconsistencies using methods containing implicit, unstated restrictions on mathematical structures.
In order to explain away these problems, Tegmark eventally proposes an extreme form of mathematical constructivism: "A mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps" (we will consider constructivism in a later section of this page).
Secondly, the mathematical universe idea implies that mathematics must have some kind of independent "reality" of some form: "mathematical structures in Plato's realm of ideas ... exist 'out there' in a physical sense". It is suggested that all mathematical structures are actually real, our universe being one of those structures. So we're dealing with a Platonic viewpoint here, a very strong Platonic viewpoint. But does the Platonic viewpoint really survive close scrutiny? That is what we will discuss in the next section (well, the section after the next blue box bit).
Simulated Universes and "Substrate-Independence"
In his book Pi in the Sky, John Barrow attempts to gain insights into the working of a possible "Platonic realm". He achieves this by considering a universe which has been created by being simulated on a computer - a so-called "simulated reality". Eventually, intelligent beings might evolve within the simulation. John Barrow says this about these emergent intelligent beings: "They will not, of course, be able to determine that they are part of someone else's simulation, but it is very likely that some self-conscious parts of the simulation would indulge in 'theological' speculations about the origin of the 'world' about them, the nature of its initial state and what lay before it, and whether these considerations point to some Initiator of Everything". Essentially, these beings would be living in The Matrix, though they would be completely unaware of it (for more about simulated reality, see the Living in the Matrix page).
Barrow then makes the point that the type of computer hardware which would run this simulation is completely irrelevant (it could be either a Mac, or a Windows PC, or any other type of computer, for example - though it would have to be far more powerful than any of today's computers!). This proposition seems fairly sensible, and is called substrate-independencepaper by Nick Bostrom). (for more on substrate-independence, see page 2 of this
Barrow then builds on this idea of substrate-independence: "When we reach this stage we see that we really have no need for the computer hardware we started with; indeed, its particular identity is really irrelevant. We could have run our program on all manner of different types of computer architecture. But surely, if we are of the Platonic viewpoint, we need not have run the program on any hardware at all. This means that we can think of the mathematical formalism as containing self-conscious states - 'minds' - within it."Mathematical Universe. Essentially Barrow is making the point that if we are of the Platonic persuasion (i.e., we believe in mathematical structures having some kind of existence in a separate Platonic realm) then we would be forced to accept that our universe itself could be a construct in that mathematical realm - the idea behind Max Tegmark's
Is there really a "Platonic realm"?
From my earlier discussion about Max Tegmark's "Mathematical Universe" it was probably clear that I have strong reservations about this idea that mathematics has some sort of reality all of its own. So let's consider the claims of the Platonists - the view that certain mathematical concepts are eternal truths which apparently have an independent reality of their own. The Platonists would consider these mathematical truths to be "obviously correct" and thus occupy a special position in a vaulted mathematical realm. In order to examine their claim, let's consider the foundations of mathematics.
Mathematical proofs are built-up from simple axioms: self-evident truths. These simple axioms are combined via the laws of logic (the basic rules of reasoning which we use in everyday life, perhaps without realising it) to create more complex theorems. The Platonists would consider these simple axioms to be so obviously correct that they could not possibly be any other way. They were considered to be eternal truths occupying a special position, lying in waiting to be discovered, and certainly not the invention of men.
One of the earliest sets of "obviously correct" axioms came from Euclidean geometry. Euclid's five postulates (axioms) were:
- A straight line can be drawn between any two points.
- A finite line can be extended infinitely in both directions.
- A circle can be drawn with any centre and any radius.
- All right angles are equal to each other.
- Parallel lines which are extended indefinitely never meet.
The certainty and absolute truth of Euclid's axioms underpinned the whole structure of mathematics for 2000 years, from navigation to architecture. So it came as a tremendous shock when, in the early 19th century, other forms of non-Euclidean geometry were discovered. Non-Euclidean geometries apply to curved surfaces, and they contradict Euclid's 5th postulate because parallel lines can meet when they are drawn on a curved surface.
This discovery of alternative geometries shook the certainties of mathematics, and this saw an abandonment of the ideal of absolute truth. And not only was absolute truth in axioms undermined: the ideal of absolute truth in the laws of logic themself was undermined (remember how axioms can be combined using the laws of logic to generate theorems - the structure of mathematics). Conventional bivalent logic is based on the principle that every proposition takes exactly one of two truth values (i.e., "true" or "false"). This principle was based on the classic laws of thought devised by Aristotle. However, in the early 20th century it was discovered that entirely consistent families of logic could be more than two truth values. For example, ternary logic has three truth values indicating "true", "false", and "unknown".
So the question now arises as to why the particular axioms (Euclidean geometry) and laws of logic (bivalent logic) were selected for our universe. For example, why isn't ternary logic dominant in our universe instead of true/false bivalent logic? This notion of specific mathematical and logical structures apparently being "selected" to play a dominant role in our universe appears to give weight to the Platonists' argument of there being a select realm populated by "perfect" mathematical objects, with certain axiomatic and logical systems apparently occupying exalted positions.
However, on closer inspection we find this idea that only a limited subset of axioms and logic apply to our physical reality really does not stand up to scrutiny. For example, non-Euclidean geometry can apply to physical reality if one is considering the geometry of curved objects:
The example above shows how a chalk triangle drawn onto a ball will obey the laws of non-Euclidean geometry - even in our physical reality. So those who would say that non-Euclidean geometry is not applicable to our universe are very mistaken. (Though, of course, Euclidean geometry has proved to be more useful over the years as we tend to draw our triangles on flat surfaces. And planet Earth is locally flat, so our geometry on the surface of the Earth does not reveal its non-Euclidean reality.) The truth is that certain types of geometry are emphasised or preferred in different universes, but this does not preclude the fact that other types of geometry would still be useful for some applications.
And not only are non-Euclidean axioms applicable to our physical reality, but so is multi-valued logic (i.e., not just binary true/false logic). We could think of propositions with the ternary truth values of "true", "false", and "unknown". For example, it is not yet possible to determine if the statement "It's going to be a white Christmas" is true or false - at this moment in time we have to regard the statement as "undecided". No doubt we could imagine situations in which four-or-more valued logic would be appropriate. So maybe two-valued logic does not hold such an exalted position. Two-valued logic dominates only because the vast majority of propositions can be determined with certainty (i.e., true or false), but this does not mean there cannot be roles for other mathematical structures, axioms, and logic systems in our physical reality - they're just not so common.
As John D. Barrow explains in his comprehensive study of mathematical Platonism, Pi in the Sky: "The attempt to create a heavenly realm of universal blueprints that are truly different from the particulars founders under the weight of another simple consideration. Plato wants to relate the universal abstract blueprint of a perfect circle to the approximate circles that we see in the world. But why should we regard 'approximate' circles, or 'almost parallel lines', or 'nearly triangles' as imperfect examples of perfect blueprints. Why not regard them as perfect exhibits of universals of 'approximate circles', 'almost parallel lines' and 'nearly triangles'? When viewed in this light the distinction between universals and particulars seems to be eroded". No mathematical structure can claim superiority over any other structure.
To sum up, the Platonists' claim - that certain mathematical structures which play dominant roles in our physical reality have an exalted position in some Platonic realm - does not really stand up to scrutiny. It's easy to get fooled by beautiful, complex mathematical structures. And when "real" objects (such as humans and computers, or any physical process) come along they can endow an illusory, deceptive reality to those structures (such as the Mandelbrot Set). It's easy to get taken in and entranced by this "grand illusion" (read Stanislas Dehaene on the cognitive illusion of mathematical Platonism here).
Arithmetic in Different Universes
In the last section we saw how geometry in a different universe might be very different from geometry in our universe (if that other universe had non-Euclidean space, or if it had more than three spatial dimensions). Euclid's five postulates - the axioms of geometry in that universe - would have been different. So we can see how geometry (a branch of mathematics) could be different in another universe which had different physics. But could all of mathematics be different in another universe?
OK, you might see how geometry could be different in another universe, but surely, you might think, arithmetic (another branch of mathematics) could never be different in another universe. Arithmetic, you might protest is just too fundamental, too "obviously correct" - it could surely never take any other form? I know it can be very hard to imagine how arithemetic could be different in a different universe with different physics. However, I do believe that could be the case. Let me explain ...
Let's say in our universe I have a bag. And I put a ball in my bag. And then I put another ball into my bag. I will then have two balls in my bag, and I would use this fact to generate one of my axioms of arithmetic in my universe: 1 + 1 = 2
But now let's say I'm in a different universe with different physics (e.g., let's imagine it is a simulated universe of the type described in the Living in the Matrix page, and it has been programmed so that physical objects behave differently in that universe). Again, I have a bag. And I put a ball in my bag. And then I put another ball into my bag. But in this universe physical objects behave differently so that when two objects touch they merge to become one object (it would be easy to program objects so they behaved in that manner). So I now only have one ball in my bag. In which case the mathematics developed by the simulated beings would be based on the axiom: 1 + 1 = 1
(Let's ignore the fact that life could surely never evolve in such a universe!)
The whole system of mathematics they would develop would be different to the system of mathematics we have developed in our universe because it would be based on different fundamental axioms. Those axioms would be based on the physical axioms of their universe, but those physical axioms are different from our universe. But those different axioms would seem completely normal and "obviously correct" to the simulated beings, just as our axioms seem "obviously correct" to us!
Note, however, that even this extremely strange "1+1=1" system of arithmetic would have someour universe, if we were performing arithmetic on objects which naturally merge together - such as jellies. As was explained in the previous section: "The truth is that certain types of mathematics are emphasised or preferred in different universes, but this does not preclude the fact that other types of mathematics would still be useful for some applications." use in
But why should the preferred system of mathematics follow the underlying physics of that universe? This is the question we will consider in the next section.
Why does Physics follow Mathematics?
The principle of axioms might go some way to explain the almost uncanny match between mathematics and physics. Mathematics has been almost uncannily useful in explaining the natural sciences. It is almost weird the way that developments in mathematics, and the discovery of new mathematical structures, has been later matched by discoveries in physics which involve the similar structures in the physical world (see The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner, and the entertaining short story Unreasonable Effectiveness by Alex Kasman). For example, in 1931 Paul Dirac predicted the existence of the positron purely by considering mathematics - the first time a particle had been predicted from purely mathematical considerations. For this reason, mathematics has been called the "language of nature".
In order to get a better understanding of this apparently uncanny match, we should consider the origin of mathematics. Mathematics tools were originally developed for analysing the physical world. Geometry, for example, was developed as a tool for measuring plots of land and constructing buildings, while counting and arithmetic were developed for commerce and trading of goods. So mathematics was developed as an abstraction from physical reality. We would do well to remember that at the start it was physical reality that provided the motive for developments in mathematics, as the axiomatic principles behind mathematics were established in those early days. I believe that provides a clue as to why we now find developments in mathematics are later being shown to have equivalent counterparts in physics.
So why is mathematics such a remarkable tool for describing physical reality? Well, it is important to realise that at its most fundamental level, physical reality has to be based on physical axioms. As a result, physical structures and processes are axiomatic systems themselves. For example, the presence or absence of a physical object would be modelled as "true" or "false", so clearly this is a situation in which bivalent logic (true/false logic, considered in the previous section) is applicable. As a result, the principle that an object cannot be both present ("true") and absent ("false") at the same time is modelled by the logical law of noncontradiction. So if we start with simple, physical axioms we can build a full physical system, and that will be modelled by a mathematical system built from mathematical axioms.
So physical axioms can be combined to describe the behaviour of larger, more complex physical systems: macroscopic behaviour results from microscopic behaviour (read about the building of a ladder of effective theories on the It's a Small World page). Indeed, David Hilbert wanted to see a full mathematical treatment of the axioms of physics (see here). In particular, he considered the kinetic theory of gases in which the pressure, temperature, or volume of gases could be found by considering the statistical mathematical behaviour of smaller constituent molecules.
If mathematics was developed as a model of the behaviour of physical reality, and if physical reality is an axiomatic system, then it should be no surprise that the resultant mathematics turned out to be an axiomatic system. From the very start, mathematics was developed to match physics, to be an effective tool. Developments in physics provided the motive and inspiration for developments in mathematics. As developments in physics are now stalling (with the requirement for ever-larger particle accelerators) it's little wonder that developments in mathematics have forged ahead and are later found to mirror developments in physics. The use of symmetry in discovering new elementary particles has been especially remarkable, but it should be remembered that the initial inspiration for exploring mathematical symmetry came from exploring the natural beauty of the macroscopic physical world.
So we should not forget that the strength of mathematics lies in its ability as an abstracted tool for describing physical reality, which was its initial function. Mathematics is now appearing as nothing more than an abstracted language for describing (and building upon) the physical axioms (for example, the presence or absence of physical objects). So the role of mathematics in regard to its apparently mystical power to reveal hidden truths becomes less surprising and less amazing: we have two axiomatic systems matching each other.
"The integers were made by God; all else is the work of man."
- Leopold Kronecker
Mathematical Constructivism
(or ... "Why there is no such thing as infinity")
So in the discussion so far I have considered mathematics as first and foremost an abstraction from physical reality. However, the goal of pure (as opposed to applied) mathematics is to consider structures which do not necessarily have roles in physical reality. So this inevitably means that a distinction must be drawn between those structures which have relevance to physical reality and those which do not. But where do we draw the line? How do we decide which are the structures with applicability to the "real world"? To answer this question, let us consider some of the more exotic mathematical constructs, starting with the concept of "infinity". Is there really anything "infinite" in physical reality?
Many have asked this question. For example, "nothing in the physical world (outside mathematics) corresponds to the notion of infinity" (quoted from here), and the presence of infinity in physics theories is generally taken to represent a flaw in the theory (the elimination of troublesome infinities is a major reason behind the popularity of string theory).
Richard Feynman raised doubts about the relevance of infinitessimally small scales: "It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space?" Recent developments in the quantum gravity fields of string theory (which has a minimum distance scale) and loop quantum gravity (see Atoms of Space and Time) suggest that space is composed of incredibly small discrete elements, rather than being continuous. Indeed, Gregory Chaitin devotes a chapter of his book Meta Maths to convince us that space is discrete, not infinitely divisible (Chaitin's motive is to gain support for the idea that the universe behaves like a digital computer - see the page Is the Universe a Computer?).
Paul Davies considers this idea that certain mathematical structures have no relevance to reality in his paper Emergent Biological Principles and the Computational Properties of the Universe: "For example, the use of differential equations assumes the continuity of spacetime on arbitrarily small scales, the frequent appearance of numbers like 
implies that their numerical values may be computed to arbitrary precision by an infinite sequence of operations. Many physicists tacitly accept these mathematical idealizations and treat the laws of physics as implementable in some abstract and perfect Platonic realm. Another school of thought, represented most notably by Wheeler and Landauer, stresses that real calculations involve physical objects, such as computers, and take place in the real physical universe, with its specific available resources. In short, information is physical. That being so, it follows that there will be fundamental physical limitations to what may be calculated in the real world."
Perhaps an answer as to how we might achieve this division of mathematical structures into "physically relevant" and "physically irrelevant" groups comes from an approach to mathematics called constructivism. Constructivism says that mathematics should only include statements which can be deduced by a finite sequence of step-by-step constructions, starting from the "natural" numbers (1, 2, 3, etc.). This is a major departure from conventional maths because many of the more exotic mathematical structures (such as infinity, and irrational numbers) would no be eligible as part of maths under the strict rules of constructivism. To quote Gregory Chaitin from his book Meta Maths (in an attack on the existence of irrational numbers which are generated from an infinite series): "Some mathematicians have what is called a 'constructive' attitude. This means that they only believe in mathematical objects that can be constructed, that, given enough time, in theory one could actually calculate. They think that there ought to be some way to calculate a real number, to calculate it digit by digit, otherwise in what sense can it be said to have some kind of mathematical existence?"
So Gregory Chaitin believes that irrational numbers (which have infinitely many fractional digits) have no basis in reality. Let's examine this claim by considering the irrational number, . There is clearly a relationship between and 
, most obviously because has an infinite length of fractional digits, but more interestingly because of one way we can calculate . Archimedes realised that you could approximate a circle by drawing an increasing number of straight-line segments:
As the number of straight-line segments increases toward infinity, the approximation gets closer and closer to the accurate value of , so this ties in with the idea that an infinite series can generate a perfectly accurate value for :
However, if spacetime really is discrete, composed of extremely small straight lines at the smallest scales, then there will be a limit on this subdivision into straight lines. There will always be that minimum distance scale (discussed earlier), the smallest straight line possible in the real world. This would mean that the irrational number (and the associated ) would have no relevance to the real, physical world. We end up with a mathematics which juggles physically irrelevant symbols and which are circularly-defined, defined in terms of each other. However, mathematical constructivism, which insists that only a finite number of steps could be employed to produce a mathematical structure, now appears to be a perfect match for physical reality, unable to generate irrational numbers such as , and no place for .
(This idea that mathematical constructivism is the best tool for modelling our universe - and speculation as to why that might be the case - is considered further on the page Is the Universe a Computer?).
Meta Maths: Mathematics about mathematics
We now move on to consider the implications of the theories of metamathematics: mathematics about mathematics. Metamathematical theorems impose fundamental limitations on mathematics, which in turn impose limitations on computation. Would these have an impact on our model of the physical world? For example, the theory of uncomputable functions asserts that there are some problems which we cannot solve by using a computer, and this would restrict any model of the physical world based on the idea of the universe behaving like a computer.
At the root of all of these metamathematical theorems lies a form of self-referential circular paradox which has its origins in the granddaddy of all circular paradoxes: Russell's paradox. Russell's paradox is stated in everyday terms in the story of the Barber of Seville: "A man of Seville is shaved by the Barber of Seville if and only if the man does not shave himself. Does the Barber shave himself?" If he does then he doesn't, but if he doesn't then he does!
In mathematical terms, Russell's paradox is expressed in terms of sets: "S is the set of all sets which do not have themselves as a member. Is S a member of itself?" The paradox is that if S does not have itself as a member then it should be in S, and if S does have itself as a member then it shouldn't be in S:
This circular style of Russell's paradox was influential in the formation of other metamathematical theorems:
- Gödel's Incompleteness Theorem was formed by formalizing Russell's paradox. Essentially, Gödel considered the statement "This statement cannot be proved to be true". If the statement is false, then we can prove it to be true - a contradiction. But if the statement is true, then we have a true statement which we cannot prove! In essence this is the same self-referential circular paradox as in Russell's theorem. (Gregory Chaitin presents a fascinating discussion about the controversy in 20th century mathematics which arose as a result of Russell's paradox and Gödel's Theorem here).
- In computational theory, the halting problem proves there are some problems which cannot be solved by a computer. The proof involves feeding a computer program round to operate on itself, in the same circular fashion. The proof is considered in detail on the page Is the Universe a Computer?
So all of these metamathematical theorems involve a form of circular feedback whereby an object can operate on itself. But does this approach have any relevance to physical reality? Can physical objects behave in this manner? I believe not. For example, Russell's paradox involves an object (the set S) somehow containing itself. It is no problem to do this in abstract non-physical mathematics, but no physical object can contain itself. A box, for example, cannot contain itself:
(Compare with the diagram for Russell's paradox immediately above)
In the real world, the Barber of Seville's situation is simple: the barber can shave another man, or the barber can shave himself. That's it. There is no paradox. There can never be a paradox with simple, physical objects such as barbers and customers. Russell's paradox introduces the abstract mathematical concept of the set which has no direct equivalent in the physical world, but it is this idea of the set (an object which can include itself) which causes all the troubles in the abstract mathematical world. We seem to have the equivalent of Zermelo-Frankel set theory at work in the physical world, which contains the Axiom of Separation which states that objects (sets) can only contain other objects (sets) as members but cannot contain themselves - thus avoiding Russell's paradox:
The Axiom of Separation in Zermelo-Frankel set theory.
"Objects can only be composed of other objects" or, in other words,
"Objects shalt not contain themselves".
It is my belief that these metamathematical theorems have no basis in physical reality and can safely be ignored in our quest for the nature of reality. I consider them to be nothing more than interesting intellectual mathematical diversions - much like solving the logic problems in Puzzler magazine: quite fascinating, but of no practical use. More harshly, I consider all of these obscure mathematical structures which have no role in physical reality as a case of maths "disappearing up its own backside" (no doubt in a self-referential, circular manner!): mathematics for mathematics' sake. Indeed, Max Tegmark considers that the reason his Mathematical Universe must be composed of only a computable subset of all possible mathematical structures is "because the rest of the mathematical landscape is a mere illusion, fundamentally undefined and simply not existing in any meaningful sense". And according to Gregory Chaitin, the physicist Karl Svozil referred to pure maths as an "unreal mental mindscape fantasy world". Again, according to Chaitin, the real world is composed of only a subset of a larger mathematical world: "math deals with the world of ideas, which transcends the real world".
By definition, metamathematics is mathematics about mathematics. It is not mathematics about reality.
No comments:
Post a Comment