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## 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.

Mandelbrot applet instructions

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)

(Your browser does not support Java. Go to www.java.com to get the Java Runtime Environment for your browser. Try the Manual Download if the automatic installation does not work.)

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 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 "Mass of star relative to Sun" against "Radius of orbit relative to Earth's":

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 maths. This is a clear departure from traditional Platonism which merely imagines mathematical objects as existing in some separate plane of existence: "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.

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 eventually 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-independence (for more on substrate-independence, see page 2 of this paper by Nick Bostrom).

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." 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 Mathematical Universe Hypothesis (this idea of a Platonic realm will be consider in the next section).

Max Tegmark continues this idea of hardware independence in his mathematical universe paper: "If the computer need only describe and not compute the history, then the complete description would probably fit on a single memory stick, and no CPU power would be required. It would appear absurd that the existence of this memory stick would have any impact whatsoever on whether the multiverse it describes exists for real". However, on the Is the Universe a Computer? page we see that it is possible to consider the universe to be a computer in which the laws of nature play the role of the software, and the elementary particles play the role of the hardware. If you try removing the hardware from that computer (as Barrow and Tegmark suggest) you won't be left with anything at all!

### 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:

1. A straight line can be drawn between any two points.
2. A finite line can be extended infinitely in both directions.
3. A circle can be drawn with any centre and any radius.
4. All right angles are equal to each other.
5. 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:

If a chalk triangle is drawn onto a ball then every angle of the triangle can be ninety degrees! This is an example of non-Euclidean geometry applicable to real physical reality.

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 arithmetic 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 some use in our 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."

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."

### 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 infinitesimally 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:

A circle can be approximated by an increasing number of straight-line segments (diagram taken from Wikipedia's entry for Archimedes).

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:

Russell's Paradox. Should S have itself as a member?

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:

Physical reality. A box 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:

An insight into the "mind of God"?
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.

Back to the main page

I have to say I don't agree that Platonism is somehow opposed to physical interpretation - or at the very least I don't understand what philosophy you propose would make these connections to physics clearer. If it is true that physics somehow trails mathematics, then we obviously require some other standard than physicality to judge what mathematics is "good". And I think you are underselling the relevance of even very esoteric mathematics to physical reality. Note that it might be better to say that quantum mechanics is based not on a 2-valued logical system, but on one with infinitely many possible states, since it predicts only probabilities. There is also the interesting sum of positive integers:

1+2+3+4+...

Clearly this series diverges, but it turns out that there are several ways of defining sums (based here on the Riemann zeta function - pretty darn "abstract" mathematics) which give the result:

1+2+3+4+...=-1/12

which actually turns out to have physical significance in certain experiments.

Perhaps by defining your philosophy more clearly you can sharpen your criticism of other mathematical philosophies.

Kevin Saff http://kevin.saff.net/ - Kevin Saff, 9th January 2007
Hi Kevin, thanks for your interesting comment.

You make a good point about quantum mechanics in that the Hilbert space can have a basis composed of infinitely many vectors. However - and this is key - we do not see those infinitely many possible states in physical reality. Once an observation is made we only see one eigenstate in physical reality - after the "quantum jump". So we don't encounter the infinity in physical reality. See http://www.ipod.org.uk/reality/reality_wavefunction.asp

Similarly with your other good point about the Riemann zeta function. Yes, this is an infinite series but, as you point out, it converges to a finite value. And it is this converging property which makes it useful for regularization in quantum field theory, producing a finite value which has applicability in physical reality. Yes, infinity can exist in mathematics, but I don't see it in physical reality. You can't have objects which are infinitely large, you can't travel at infinite speed, etc. Maybe the universe is infinite in extent, and who knows what goes on in black holes, but we don't seem to have infinity in physical reality. Regularization eliminates the infinities so that mathematics can become real. - Andrew Thomas, 10th January 2007
After reading this article and comparing the idea with one of my own theories, I think I found a unique relationship which is profound.

I think of the universe as being self-similar. I was trying to figure out the relationship between different scales (bubbles or dimensions of space) when I noticed a pattern. If the size of dimensions are exponential in nature (as appears to be the case in http://www.amherst.edu/~rloldershaw/NOF.HTM via a scaling factor) and you map the center and edge of a bubble to 0 and 1 on a logarithmic scale, then life seems to occur at the coordinates 0.5, 0.5, 0.5, 0.5... etc. In other words, when you map the distances, sizes, and complexities of objects from 0 to 1, we are at the 0.5 zone, what you call the "interesting bit". The 0.5 zone for atoms is Iron, which rests on the middle group of the middle period of the periodic table. Fission and Fusion push atoms toward this element, confirming its importance. Our galaxy, when structurally compared to an atom, has S (the core), P (the bars), and D (the spiral arms) orbital equivalents; it is the 0.5 of galaxies. The sun is of midrange size and distance from the center of the galaxy, a 0.5 star. Our planet is perhaps also of midrange size and distance from the sun, the 0.5 of planets, and is composed mainly of Iron, the 0.5 of atoms. Coincidence? I think not.

You can read some more of my ideas and related things on my page at http://www.bmfusion.com/

Chris - Chris Winn, 9th April 2007
In my view, mathematics is an activity we engage in in order to understand and cope with our world. The structures and processes of the physical world are not the same as their mathematical formulation; the map is not the territory. Other activities such as music, art, technology etc also contain valid descriptions of the world. In the Rosen diagram above, the mathematical system is depicted as being a sort of one-to-one image of the natural system. However, one could exchange the Theorems and Inferences on the Mathematical side of the model for any human activity, say, Notes and Scores for a Musical encoding and decoding of the universe. Of course each modelling system has its own terms, functions and relations. What sets mathematics apart, perhaps, is its predictive power, which is why it so revered as a model for reality. But once again, any model, however powerful, can only reflect an aspect of the universe, not the full picture, which, in all likelihood, can never be known in its entirety. - Jeremy Becker, 20th April 2007
Very nicely put. Thanks. - Andrew Thomas, 20th April 2007
I think Gödel's Theorem has in fact a profound link with physical reality: the lack of an absolute system of reference as proven by Relativity leads ultimately to the same paradoxes involving circular reasoning.
What about The Numbers being the ultimate (and only) reality, like Pythagoras said? - Marian, 29th September 2007
Hi Marian. Since you posted your comment I've added a section on "Meta Maths" to the main text which explains why I believe Gödel's Theorem is over-hyped as applied to physical reality - in fact, I believe it has no relevance at all to physical reality and the "real world" (see the "Meta Maths" section for an explanation). Somedays I wonder if the main function of Gödel's Theorem is just to produce endless popular science books about it!

(Gödel's Theorem may have far-reaching consequences for the future of pure **mathematics**, though. But that's not really the focus of this website. This website is concerned about the physical world, not the mathematical world.) - Andrew Thomas, 29th September 2007
Hi Andrew,
With your permission, let's try to detail some difficult stuff:

1. What is the status of Maths?
Let Gregory Chaitin speak: “Gödel's incompleteness theorem tells us that within mathematics there are statements that are unknowable, or undecidable. Omega –Chaitin's own discovery, an irreducible, infinitely complex number- tells us that there are in fact infinitely many such statements… something we cannot deduce from any mathematical theory.” And more: “To put it bluntly, if the incompleteness phenomenon discovered by Gödel in 1931 is really serious — and I believe that Turing's work and my own work suggest that incompleteness is much more serious than people think — then perhaps mathematics should be pursued somewhat more in the spirit of experimental science…” (Omega and why maths has no TOEs: http://plus.maths.org/issue37/features/omega/ ).
What we must hold is that Mathematics cannot be complete, cannot be closed in a final theory, that all theories, not only strictly mathematical ones, must have a limited power of prediction given by the quantity of information contained in their sets of axioms and rules of inference (Chaitin, Gödel's Theorem and information; http://www.cs.auckland.ac.nz/CDMTCS/chaitin/georgia.html ), that Maths must be pursued more as an experimental science, like Physics, and that mathematical objects must not be seen as similar to Platonic 'noeton', ideal things, but to mundane, physical things. And from the incompleteness of Mathematics goes that of the knowledge, most generally speaking, because Maths are the 'sine qua non' condition of all systematic knowledge.

2. What is the status of Physics?
The status of Theoretical Physics cannot be, by no means, better than the status of Maths; one can attain only a precision lesser or equal with that of the tool he uses. The consistent theories of Physics cannot be complete, they have limited power of prediction. (It will be very odd physical theories be logically better than mathematical ones!)
For example, from Descartes to Einstein, one basic issue in physics is the reference frame (system), which enables the mathematical approach to physical problems. The Theory of Relativity stipulates no absolute frame of reference (the absolute space, 'ether', and absolute time from the previous theories are proof by Einstein to be pure idealizations, they simply don't exist). Being related only one to each other, not to an absolute referential, the logic of the set of all the physical frames of reference lacks consistence and ends in a kind of circular reasoning. The frames of reference relate/'support' one another, but, the whole edifice is, metaphorical speaking, 'in air'.

(continued ...) - Marian Radulescu, arches_saccas@yahoo.com, 4th October 2007
(continued ...)

What about the physical reality itself? The physical things, obvious, don't have absolute existence. As expressed in Nagarjuna's philosophy (and one must not necessary be a Buddhist to agree with this statements of good sense): “all phenomena are without any 'svabhava', literally ‘own-nature' or ‘self-nature', and thus without any underlying essence; they are 'empty' of being independent” http://en.wikipedia.org/wiki/Nagarjuna ).

3. What is the relation between Maths and Physics?
We saw there is strong analogy between the two. It is like both, the things of nature and the concepts of knowledge, suffer by a syndrome of imperfection (incompleteness).
The common (and in my opinion false) idea is that math, along with the whole knowledge, ‘reflects' the physical world. (Almost all common idea prove, sooner or later, to be false!)
What about an reverse, Pythagorean, relation: the mathematical objects are the invisible building ‘material' for the manifest natures of the physical world and consciousness? All is information… (maybe, we'll talk about that with another occasion).

4. Why is that?
The primary cause for this state of logical and ontological illness (circular reasoning, infinities, incompleteness, imperfection, etc.) is the absence of the absolute natures, or, at least, of an absolute reference in the manifested world. Nor the physical existence, nor the knowledge can't be completed, closed, ‘perfect', in the absence of the absolute manifested as absolute. There is not such a ‘thing'.
But, we wonder, could physical nature and consciousness exist, as we experience them, just like that ‘in air', without the absolute prop, without an ‘urprinzip'?
I think, not. The existence of the Absolute is necessary. But, being ‘absolute', must be unique, more, must be the only existence (if there is another, then both are not absolute), and this seem to be in total contradiction with our experience.

5. The solution?
The Absolute is the unmanifested substrate, permanent, unchanging, complete ‘real reality'. Substrate for all manifested reality, which is changing, limited, incomplete, with an inferior degree of existence. Absolute is also ‘ineffable', 'aporreton' (Damascius, the last of the Neoplatonists), that is, nothing can be known and/or expressed in words about (if other, that is not the really absolute – "The Tao that can be told is not the permanent Tao; the name that can be named is not the permanent name." — the Dao De Jing).
This is the Meta-Physics. Maybe, with another occasion, about how this ideas might apply to the Quantum Mechanics. - Marian Radulescu, arches_saccas@yahoo.com, 4th October 2007
Thanks for that super contribution, Marian.

1) First I'll deal with your point number one about the status of maths. I agree with Chaitin's idea that mathematics is going to have to proceed more like an experimental science, like physics, rather than relying on small, elegant proofs. When you look at something like the distribution of the prime numbers, the positions appear to be **fundamentally random ** - there's no simple formula can calculate that. You can't compress that randomness down into a simple formula or computer program to produce those random prime number positions (this is the principle behind Chaitin's Algorithmic Information Theory: http://en.wikipedia.org/wiki/Algorithmic_information_theory ). So if you want to find the positions of the prime numbers you just have to get a powerful computer and crunch the numbers.

It could be said that the development of mathematics has so far been rather deceptive in that it has concentrated on a small number of elegant proofs, but - as Chaitin suggests - the rest of the mathematical universe is not neat and tidy like that: it is chaotic and random. And because you can't compress it to simple, elegant theorems you need to rely on number-crunching computers, and take an approach more like experimental physics.

For example, in his book, "Pi in the Sky", John Barrow considers an extraterrestrial civilisation with a very different approach to maths: "They saw mathematics as another branch of science in which all the facts were established by observation or experiment. They had used their fastest computers to check that every even number was equal to the sum of two prime numbers case by case through the first trillion examples. Needless to say, terrestrial mathematicians were shell-shocked; expecting the deepest logical insights, they found nothing but results established by empirical methods or generalisation from special cases. But what, in retrospect, is so fascinating about the development of mathematics in this culture, and which seems to have depressed our own mathematicians so much, is the speed and confidence with which it progressed. It had no worries about logical paradoxes or concerns about Gödel's theorems but had access to a vast area of truth which Gödel taught us must be out of the reach of deduction. If we took this approach to mathematics it would allow us to find possible truths that lie beyond the boundary of provability discovered by Gödel."

So there could be a form of maths without the restrictions of Gödel's theorem, and all those unattainable proofs would then be available to us.

But this talk of pure mathematics and the future of mathematics is rather outside the scope of this website which concentrates on the use of mathematics for modelling physical reality. As I said in the main text, anything else (with no physical applicability) I basically consider to be noodling!

To move on to your second point about the status of physics:

(continued ... - Andrew Thomas, 4th October 2007
(continued ...)

Your second point on the status of physics:

2) You say "The status of Theoretical Physics cannot be, by no means, better than the status of Maths; one can attain only a precision lesser or equal with that of the tool he uses." You seem to be implying that physics is subject to any restrictions imposed on it by the limitations of mathematics. For example, Gödel's Theorem of incompleteness leads to Turing's idea of uncomputable functions (again, this is considered in the section on "Meta Maths"). And if there are uncomputable functions in physical reality then it puts paid to the notion of the universe behaving like a digital computer (see the page "Is the Universe a Computer?" http://www.ipod.org.uk/reality/reality_universe_computer.asp ).

Your idea of circular reasoning in the Theory of Relativity is very interesting - perhaps you might like to look at the examples of circular feedback I have added to the "Meta Maths" section.

Once, again, Marian, thank you for a super contribution.
- Andrew Thomas, 4th October 2007
Thank you for your initiative, Andrew. - Marian, 4th October 2007
You keep saying that Math (thinking about and studying mathematical concepts) was "designed" to describe reality. Math (and our ability to practice it) wasn't "designed" to do anything (unless it was designed by God, in which case you still don't know how God came to practice Math). It simply arose in the minds of intelligent human beings and other species, no doubt shaped by evolutionary effects, or by the will of God, whichever you prefer. Experiments have shown that newborn babies are born able to count (the ability improves with age, but newborn babies can count), and also that monkeys, birds, and all sorts of animals can count up to small numbers. Saying we developed Math as an abstraction of the physical environment is inaccurate; it makes it sound like we conciously developed the basics of Math (I really mean the basics) to suit our needs, i.e. we could have chosen not to develop any Math at all if we had not wanted to. But this is false; one has no choice but to be born with the capacity to count.

In fact, for all we know, Math ability (i.e. the ability to count and that which may follow from that ability) arises simultaneously with or before conciousness.

Physical problems have motivated much of the developement of Math, concious developement by humans in order to solve physical problems, but they did not motivate the *start* of Math. They were motivation *near* the beginning, but they were not there *at* the beginning. The ability to count is developed before birth, and does not have a concious motive.

The all-singing all-dancing Math we know today has been motivated largely though not entirely by physical considerations. However the most basic Math, arithmetic, arises/arose at birth with no human motivation whatsoever, and in principle all mathematics (or certainly all physically applicable mathematics) could be developed from it.

Properly considered, the question "Why did Math develop?" does not have the answer "because we decided so, in order to suit our needs". We develop our basic Math capacity before we do or decide anything, and the deliberately poor answer given above does not take account of this. The answer is about the physical universe and how evolution has shaped us or how God has created us, whichever you prefer. Our minds are intrinsically mathematical, and our minds are part of this universe, making it mathematical to some degree.

You could then say "We merely evolved Math ability due to our situation in the physical world". But that then leaves the unanswered question of why the physical world would force such a chain of events. So you still have an unanswered question.

As for disappearing up it's own backside, let's just say if it were not for much of the crap coming out of it, much of modern science and most of modern physics (relativity, QM, etc.) would not have arisen. And Godelian considerations helped give rise to computers and modern computer science.

(continued ...) - Dan, 3rd December 2007
(continued ...)

As for Godel, his work is not overhyped - it is misunderstood. It does not say there are statements that we cannot prove and it does not say that math must become doubtful (that would be a question of certainty and uncertainty, which are emotions, no mathematical theorem will compel one to feel either one emotion or the other).

It says that for any formal system extending minimal arithmetic there is an arithmetical statement which can be expressed but not proven by that formal system. That is not to say that there are not other axioms we can recognise as true, add to the system, and obtain a proof of the statement.

Godel's work was brilliant because he revealed that provability of this or that theorem based upon these or those axioms was literally a problem of the theory of numbers. Similarly a statement of the theory of numbers can be recognised as a statement of the kind I just mentioned in mathematical logic. This is a profound insight and one that Godel developed to a high degree.

As for Chaitin, he was not the first to show that infinitely many undecidable statements exist for each formal system, that followed from Godel's work. Chaitin constructed, for a theory T, an infinite class of arithmetical statements which corresponded to each bit/digit of a certain real number omega, such that each bit was a 0 or a 1, and for each bit it could not be proved in T what that bit was. He then showed that if one took n bits and wanted to decide what they were, adding some of the information about the bits, i.e. less than n bits, for example what n-1 of the bits were, to the system T as axioms would not enable you to prove in the new system T' what all n bits were. You had to add at least n bits. In more wordy terms, this means that each bit of omega is proof-theoretically independent of all the others. Add it to a theory which cannot prove what any of the bits of omega are and you could still only prove what one bit of omega was.

Note that Matiyasevich's solution of the Hilbert's tenth problem was similar, the only difference being that not every set of n bits of his real number were independent, only some.

This is interesting mathematics but it's hardly great mathematics, and Chaitin wants to be great, so he uses fantastical language which is downright inaccurate at times. The bits of omega are not random (omega is), and Chaitin's theorem for an individual bit amounts to nothing other than "if T cannot prove what the bit is, then T cannot prove what the bit is".

Chaitin's approach to research is stupid. His approach is "if you can't solve a problem after working on it for a year, give up and guess the answer, then move on to apply that answer to new problems".

This philosophy has the following property. There is no positive integer N such that, following Chaitin's philosophy, you will never lose more than N years worth of mathematics due to that work being made based on a falsity.

(continued ...) - Dan, 3rd December 2007
(continued ...)

Before this becomes spam, a quick point about different possible universes. First of all, It may be the case that every true mathematical statement is realised in this universe.

Also,
"Hence, the different physical conditions in each universe would result in different mathematical systems."

*If* there are different possible universes then yes. But they are all mathematical models that can be interpreted within mathematics. Mathematics shifts seamlessly from 3 to 4 to n dimensions, where n is arbitrary.

And to say a true mathematical statement is true for any possible universe is surely not controversial, as we define the term "possible" in terms of what is mathematically or logically possible.

"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 investigation."

if only Platonists denied that n-valued logic had no application to our universe. Even if you don't consider it as a logic, one could consider it as a formal algebra that is bound to be of use somewhere. As far as I know, Platonists only claim that 2-valued classical logic applies to mathematical statements.

Not that I agree with Platonism. Talk of a mathematical object existing is nonsensical. But it is also nonsensical to disagree with nonsense. To ask whether 3/4 exists is nonsensical. To ask whether it exists in the real numbers does make sense. That is my take on the Platonism/antiplatonism debate. It is meaningless.

It is sometimes associated with Platonism the idea that mathematical statements are either true or false. I think some, but not all, mathematical statements are either true or false (I won't go into that), for example a positive integer square can not be both a sum of two squares and a difference of two squares. I verified that this was true myself, I didn't "make" or "create" that it was the case, I merely verified that it was so. - Dan, 3rd December 2007
Thanks a lot for those three comments Dan. Very interesting. There must be something about this subject makes people want to write essays!

1) Firstly I'll consider you first comment on human mathematical ability:

You said: "You keep saying that Math (thinking about and studying mathematical concepts) was designed to describe reality". Yeah, I think I misled you there - that's not really the idea I wanted to convey. I'm sure we didn't "design" maths in the sense that we had some choice as to how maths works. I have now changed the main text to say that maths was "developed" to match physics (rather than "designed"). As I said in the main text, physics and maths are both axiomatic systems, so the human-led development of maths will inevitably match the development of physics - no "design" involved. But whether or not we develop those tools, physical reality will carry on the same regardless. So physical reality seems more fundamental to my mind.

As you suggest, human mathematical ability would be "no doubt shaped by evolutionary effects" as it would give a survival advantage (counting food supplies, for example). And because evolution is shaped by physical processes this would explain why natural, inborn human mathematical abilities match physical reality.

So, yeah, I'm sorry I misled you on that one. I think we're actually in agreement on the fundamental issues.

On a different subject, I'm not saying Godel's work does not have important implications for mathematics, I'm just saying it appears to have no match or application in the physical world.

2) I'll now consider your second comment on Godel's Theorem:

You talk about Gregory Chaitin's "Omega" number in some depth (Chaitin describes Omega very clearly here: http://plus.maths.org/issue37/features/omega/ ) For those who aren't aware of it, Omega is a real number with a fractional part composed of an infinitely long string of binary digits (rather than decimal places) after the decimal point. Each of those binary digits (1/0) represents the "yes"/"no" answer to whether a particular computer algorithm will halt or not (this is Turing's Halting Problem, considered in detail on the page "Is the Universe a Computer?": http://www.ipod.org.uk/reality/reality_universe_computer.asp )

The Halting Problem is an example of an uncomputable problem, so each bit of the Omega number is uncomputable. Chaitin feels that this shows the limitations of the current mathematical methods based on elegant proofs, and suggests a more experimental method of mathematics, more like physics (I considered that in an earlier comment).

(continued ...) - Andrew Thomas, 4th December 2007
(continued ...)

You're not happy with Gregory Chaitin, are you! I know a lot of people have a problem with Chaitin's egotistical style (just see the Amazon reviews of his book "MetaMaths"), and he does his fair share of ego-stroking with that other "great genius" Stephen Wolfram. But I actually loved his book, and I think his work on Algorithmic Information Theory, complexity, compressibility, and randomness is important stuff. I do have some reservations of my own about his Omega number, though.

As I just explained, each bit of Omega is an answer to the halting problem for one particular computer program. And as Omega considers all possible computer programs, Omega has a fractional part composed of an infinite number of binary digits. So basically what we end up with is the halting problem times infinity: "No single computer program can solve ALL the halting problems" - yeah, but surely we knew that already? Why not just say "The answer to the halting problem is fundamentally random for each computer program" - that is enough to reveal that fundamental randomness which Chaitin feels is so important. I'm not convinced that combining all the results into a real number (Omega) gives us anything new. To my mind, Omega seems more of a marketing brand image on which he can hang his mathematical philosophy.

It can certainly be extremely difficult to calculate the bits of Omega: you might end up having to solve something like Fermat's Last Theorem just to find a single bit! But a small number of bits of Omega have been calculated. As Chaitin himself admits: "What is clear is that Omega can never be known in its entirety, but if the growth of our mathematical knowledge continues unabated, each individual bit of Omega can eventually be known." - see http://www.cs.auckland.ac.nz/~chaitin/prog.html

Chaitin says the important result is that Omega is fundamentally random in that it cannot be generated by any computer program with a complexity smaller than Omega itself - the bits are "true for no reason" (no reason simpler than they are), so there is a fundamental randomness at the heart of mathematics. And I do agree with him on that point, and I think that's important. But I would probably have guessed that anyway from the apparently random distribution of the prime numbers.

So, according to Chaitin, this randomness at the heart of maths means the world of mathematical truth has infinite complexity, and any mathematical axiomatic system only has finite complexity. So it's never going to be able to produce all the theorems: hence, incompleteness. And, according to Chaitin, we need to move to a more experimental style of maths to generate theorems which we will never be able to generate with our limited, finite, axiomatic reasoning.

Chaitin compares this suggested experimental approach with debugging computer software: "Proving correctness of software using formal methods is hopeless. Debugging is done experimentally, by trial and error. Experimentation is the only way to "prove" that software is correct. Traditional mathematical proofs are only possible in toy worlds". (For more detail on all of this, read Gregory Chaitin's book "MetaMaths").

(But this is all a bit off-topic for this website, which is concerned with physical reality, not mathematics). - Andrew Thomas, 4th December 2007
"From the very start, mathematics was designed to match physics, to be an effective tool."

That's the quote I noticed and took exception to, I equivocated "match physics" and "describe reality". Whereas, I would agree with something like "The situation of living organisms in the physical world and the laws of evolution caused the developement of innate mathematical intelligence in these organisms. These organisms then evolved into humans, and humans have since practised what we call Math and Physics, and the study of each has informed the other."

We do develop formal systems, but remember formal axiomatic systems (FAS) were first conceived in 1900 or so, and Math (and Physics) were around long before that. Also Math is not a FAS, the whole point of Godel's theorem is that there is no FAS which can express in it's formal language, and then prove, every true statement about the nonnegative integers. In a sense it says that no formal approach can tie down exactly what a nonnegative integers is. It would be presumptuous to assume that the laws of physics can be accounted for by a FAS, given that the laws of mathematics cannot.

As for what would happen without us, there would be no Math or Physics. There would still be physics, where physics is what we study in Physics, the laws of physics etc. A slight linguistic distinction to make things clear.

And math, where math is what we study in Math, equations, statements and numbers and stuff, it would be math. One cannot say it would change, but also one cannot say it would remain the same. Notions of changing and remaining the same do not apply to math, because the things of math are not physical. Change and remaining the same requires a notion of time, and time only applies to the physical. So all one can say of math is that it would be math. One can equally not say it would exist or would not exist, for that also would not make sense as I may have explained in a previous post (the bit about 3/4).

As for Godel's theorem, Math is a vast subject, and one would not expect to see a given theorem in a high percentage of articles. It can be used in Math to prove that there is no general algorithm for computing whether a Diophantine equation has a solution or not. Although it is a statement about nonnegative integers, it is more of interest philosophically than it is in the theory of numbers.

Godel's theorem and it's proof did have physical applications in the development of computers, which are quite useful. An application to Physics would have to be philosophical. One could say that if any question of whether some Diophantine equation has a solution is of physical relevance, then the laws of physics cannot be accounted for algorithmically (i.e. are infinitely complex). Personally I believe that the laws of physics cannot be so accounted for, as it is much more exciting, and nature never ceases to be exciting, or so I believe. - Dan, 6th December 2007
I agree, Dan. You might enjoy my page on "Is the Universe a Computer?" http://www.ipod.org.uk/reality/reality_universe_computer.asp which considers Diophantine equations and whether the "laws of physics cannot be accounted for algorithmically". - Andrew Thomas, 6th December 2007
I'll try to be brief... These excellent pages get a little off the path on this one. Quick comments:
1) Everything is discrete (in maths and physics). Physicists are moving this way if they're not already here.
2) The only way you can have something that appears to be infinite (or continuous) in maths or elsewhere is via a self-referring expression such as a continued fraction, or a recursive equation like Mandelbrot, or the ellipsis used to define the repeating pattern of an 'infinite' series. The simplest self-ref equation is the paradox.
3) Paradoxes happen all the time in real life, and are useful. For example every clock contains a paradox in its mechanism. Paradoxes cause trouble in reasoning *within* a system, but can always be resolved (held safely) in a larger system (in which there may be higher-level paradoxes, of course). The sky is not falling and maths is not impotent because of this.
4) Chaitin's stuff is certainly clever and interesting, imho, but he is just plain wrong in his interpretation of some things, including formal verification. - Richard Shoup, 26th September 2008
Thanks again, Richard. What clock mechanism paradox do you mean? Do you do this stuff professionally, or just passing interest? - Andrew Thomas, 26th September 2008
By 'clock', I don't mean analog time-*measuring* devices like a draining water vessel with marked time increments on the side. Rather, I mean time *counting* devices, like tick-tock-tick-tock etc. Every clock is based on some sort of oscillator, whether mechanical or electronic, and an oscillator is a paradox. "If true, then false; If false, then true" => "If left, go right; If right, go left" for a pendulum. Electronic clocks have a literal X => notX circuit at their core.
I've been doing research in this area and related subjects for many years (see http://www.boundary.org ), but funding for the really deep and tangled cross-disciplinary stuff like this is always problemmatic. - Richard Shoup, 26th September 2008
Wow, Richard, you've won an Emmy and an Academy Award!! How cool! You've got a very interesting site there and I'll have to read it properly. - Andrew Thomas, 26th September 2008
Yes I have, but that doesn't seem to matter much here. After all, it is rumored that you are an excellent web designer, but you seem to have put together more interesting stuff about reality here than most physicists understand... - Richard Shoup, 1st October 2008
Thanks a lot, Richard. Good luck with your venture. - Andrew Thomas, 1st October 2008
Found a slight "counterexample" to the idea that self-referencing systems do not appear in nature, specifically, in thinking about mobius strips.

A mobius strip can be made by taking a thin strip of paper, twisting it, and taping the two ends together.
Therefore if you put a marker on this strip at point p, and draw it along the length of the strip, you
eventually arrive back at point p, without ever lifting the marker off the strip of paper. Then, if you cut
the strip where you had taped it, you will find two lines, one on either side, which had been drawn in one
continuous action. Cool, eh?

Set S1 = set of all (longitudinal) lines in a mobius strip (MS)

[lines running across the length of MS, not sideways]

LineA = line on side A of MS

LineB = line on side B of MS

LineC = continuous line covering both sides of MS

[where LineA and LineB are what you get if you draw LineC on MS, then cut MS in half]

So S1 = {LineA, LineB, and LineC}

LineA and LineB do not contain any of each other's points.

LineC contains all points in LineA and LineB.

If we shrunk the width of the mobius strip to some arbitrary minimum, then it would only have lines A, B, C.

Since C = A + B, and there are only three lines, then S1 = LineC.

Therefore, S1, in containing LineC, therefore contains itself.

In other words:

S1 = {LineA, LineB, and LineC}

is the same as the statement:

S1 = {LineA, LineB, and S1}

If we added a condition that S1 was the set of longitudinal lines that could be drawn by marker M, and the width
of M was the same as the width of MS, then that would satisfy the condition of "shrinking MS" to an arbitrary minimum.

Since this can be accomplished with paper, scotch tape, sissors, and a marker, i.e., with nothing more than materials
in a kindergarten class, it follows then that a set can contain itself, even in the "physical" world, since it is
quite easy to make a mobius strip and draw a line round about it.

Thus, it is not true that sets containing themselves do not exist in the "physical/real" world, and therefore Russell's
paradox of self-referencing sets has to be taken seriously, and not just dismissed as a mathematical "trick".

I am unclear on the consequences of this for Tegmark's Mathematical Universe Hypothesis, but it would appear that the
mechanism for "selecting" viable universes (ones which can evolve life) out of the plentitude of mathematical objects
is not as simple as "constructivism", i.e., only including objects that can be made in a finite number of steps using
natural numbers. It would seem that something more is needed, something that can allow things like Russell's paradox
into the system, but not to such an extent as would undermine all of its axioms, roughly speaking.

Frank Erdman
Software QA Engineer
Austin, Texas
FrankErdman2000@yahoo.com
December 14, 2008 - Frank Erdman, 14th December 2008
Hi Frank, thanks for that, very interesting, not quite sure I understand it! I've had the scissors and the sellotape out trying to make sense of it.

Surely a Mobius strip only has one edge, so Line A is the same as Line B, the one edge? How can it be that "LineA and LineB do not contain any of each other's points", as you suggest?

And then you say "S1 = LineC", so I'd say S1 doesn't **contain** Line C, instead S1 **is** Line C. That's my take on it.

I think the three-dimensional Klein bottle is perhaps more relevant than the Mobius strip: http://en.wikipedia.org/wiki/Klein_bottle It only has one surface which is both "inner" and "outer". But does it actually "contain itself"? I'd say not. I can't think of any physical object that could contain itself in its entirety.

Very interesting - thanks, Frank. - Andrew Thomas, 15th December 2008
With interest I have read the article. Regarding the Mathematical Universe, I think the hypothesis of Max Tegmark are provable (so really are theorems). If we define the Universe as a state-space then we can prove our physical state-space is completely indepedent of the mathematical statespace.

Proof:

[1]: For a randomly chosen n-bit hash of the mathematical statespace, the probability that there is no string that hashes to itself is (1 - 1/2^n)^(2^n), which approaches 1/e for large n

Explanation: There are 2^n possible n-bit hash values of the mathematical statespace. Assuming the hash is a randomly chosen function from the set of those values to itself, a given single value will thus map to itself with probability 1/2^n. Thus, a given single value will not map to itself with probability 1 - 1/2^n. But there are 2^n such values, so the probability of _none_of them mapping to itself is (1 - 1/2^n)^(2^n). The
rest follows from the observation that, for x close to zero, ln(1 + x) is approximately equal to x. Thus, (1 - 1/2^n)^(2^n) = 1/e for large n.

So if the physical statespace emerges from this mathematical statespace and we prove that the physical statespace has less states then or equal to the number of states of the mathematical state space divided by e, we have proven the emergent physical statespace is completely independent of the mathematical statespace.

[2]: The number of States of a 3-dimensional emergent quantum physical Statespace from a mathematical statespace with n mathematical states is given with W = n.SQRT(pi)/2eGamma(3/2) = n/e. For a proof of this see eg https://www.wuala.com/FreemoveQuantumExchange/Documents/FreeMove+Quantum+Exchange+Physical+Law.pdf?lang=en (which is a presentation of our provable secure quantum information exchange system)

So given [1] and [2] the External Reality Hypothesis can be concluded, because there is no mapping of the mathematical statespace on the physical statespace. - Ronald Heinen, 15th January 2009
Thanks for that, Ronald. Sounds very interesting. I can't honestly say I understand it, though! Surely if "our physical state-space is completely indepedent of the mathematical statespace" then it's NOT a mathematical universe, is it? I don't get it. Feel free to explain a bit more about your state-space theories in a slightly-less mathematical way. Thanks a lot. - Andrew, 16th January 2009
You can see The Mathematical Universe as organized in layers, which are "separated" by hash functions. You can find a brief explanation on the link http://renewenergy.wordpress.com/transition/earth-system-analysis/

On this page you will find a brief explanation of the mathematical framework which I have presented last year on the 23rd International Conference on Statistical Physics, see http://www.statphys23.org/

In this mathematical framework the efficiency of prediction of a System is defined as the ratio between its excess entropy and its statistical complexity. A System S derives from another System Q if and only if S = f(Q) for some measurable function f. A derived System is emergent if it has a greater predictive efficiency than the System it is derived from. A dynamically autonomous System self-organized between time t and time t+T if and only if its statistical complexity increased between t and t+T.

In this mathematical model the physical world emerges from the mathematical world. If we constrain a mathematical statespace only with the postulates of quantum physics, we arrive at the physical law W=n/e, where W is the number of physical states and n is the number of mathematical states.

So we see that W=f(M) as explained in the previous definitions (S=f(Q)), where W=n/e. If we regard this function f as a one-way hash-function. We see that for a general hash function that (1-1/2^n)^2^n = 1/e. So the emergent function W=n/e is a perfect hash function because the mathematical state-space is perfectly hashed, because the mutual information between the mathematical state-space and physical state-space is 0. This can also be expressed with the Information-Theoretic equation H(n|W)=H(n). So although the physical state-space emerges from the mathematical state-space, the emergence function W=n/e is perfectly one-way and given the physical state-space W we know nothing about the mathematical state-space n. So these state-spaces are completely independent and the ERH is proven. - Ronald Heinen, 16th January 2009
Thanks very much for your contribution, Ronald. - Andrew Thomas, 16th January 2009
Thanks. Note that in the presented mathematical framework on Statphys23 for analyzing a complex system it is important to realize that particular levels don't have to exist in actuality. Levels are discerned from hierarchical analysis, aimed at constructing (discovering) Nature's “joints”.

About the existence of a "mathematical world" following references could be interesting:
http://en.wikipedia.org/wiki/philosophy_of_mathematics;
M.Balaguer, Platonism and Non-Platonism in Mathematics;
D.W.Hart, The Philosophy of Mathematics;
M.Panza,J.M.Salanskis, L'Objectivite Mathematique; Platonisme et structures formelles;
H.Field, Science Wothout Numbers;
H.Field, Realism, Mathematics and Modality; - Ronald Heinen, 16th January 2009

But I thought that Kurt Godel (and others) have shown that no mathematical system can be both complete and self-consistent?

That only abstractions like geometry -- which doesn't describe anything in the actual world where, for instance, there are no right angled triangles -- can be exempt from this state of affairs?

With good wishes,

Martin ( http://www.martin-woodhouse.co.uk ) whose brain and mind are apparently so constructed as to be unable to grasp anything more than elementary algebra and whose cavil here may therefore be disregarded as mere envy. - Martin Woodhouse, 6th August 2009
That's a wonderful website you have, Martin. You have had a very interesting life. I am so impressed you wrote six episodes of The Avengers. - Andrew Thomas, 6th August 2009
"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?"
1° Can this, or has this, been applied to Relativity and Special Relativity?
2° This does imply that there can be *NO* infinite anything.
3° Are particles or quanta, in contact with their neighbours before, during and after an eigenstate "jump"? - Denis-André Desjardins., 4th September 2009
Hi Denis-Andre,
1) The only connection to general relativity is that constructivism imagines spacetime to be discrete (as described in the main article).
2) Yes, if constructivism is correct then nothing in the mathematical or physical world could be infinite.
3) That's rather a peculiar question. I doubt there is such a concept as "contact" when we are dealing with point particles.
Thanks, Andrew. - Andrew Thomas, 4th September 2009
Read Emmanuel Kant. Mathematical truths are known a priori through experience, i.e., mathematics is NECESSARY to human cognition itself. We could not understand the world around us without the ability to process experience mathematically, e.g., distinguishing one object from another (for without this ability all experience would be seen as ONE thing as opposed to the various objects of experience). Essentially, mathematics also explains our ability to reason (logic is mathematical in nature, but deals with particular objects of experience instead of variables). Basically, I'm supporting your position. - Julio, 21st September 2009
Thanks for that, Julio. - Andrew Thomas, 21st September 2009
I am not a mathematician. Fibonacci numbers are intriguing. Counting forward after 1, 1, 2, 3, 5, 8 etc. appears to reach infinity. Counting backward, 2, 1, 1, 0, 1, -1, -2, etc. attains a "zero" Inversion at infinity and counting "Fib" rules in reverse, returns infinity where the initial zero was.. In short, "infinity" bounds infinity. And doing this below "zero" results in a positive OR a negative "infinity". So the world must consist of positive/negative real numbers AND infinities. (whatever that could mean!) but to me suggests positive/negative true AND false, i.e. as Bucky Fuller says, "unity is plural and a minimum of two". "Occupied space" (i.e. mass, a "reality"?) requires 4 collinear points for its definition, my reasoning always requires "4" points. - Ted Erikson, 22nd September 2009