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Particles and Tortoises

Some primitive peoples believed that the world was created from a tortoise, or maybe from the shattering of a giant rainbow. Where was the flaw in their theories?

The mistake that the primitive peoples made when they created their mythical theories about the origins of the world was that they considered tortoises or rainbows as ‘atomic’ entities which exist apart from their world, rather than being products created by the world. An easy enough mistake; so easy in fact, that modern physicists make the same mistake when they consider particles as the basic building blocks of the world.

The classical idea was that particles exist (this probably arose because our concept of the world is based upon experiencing ‘objects’), and that particles cause and react with ‘fields’. The modern interpretation got rid of the fields and replaced them by special particles on their own. To explain the existence of particles such as protons or omega-particles, the theories postulate that they are made up of other particles still.

The notion of particles is not very satisfactory however because these particles all behave in a most ‘unparticle-like’ manner! They spontaneously appear and disappear and, furthermore, colliding them at high energies transmutes them into other particles still.

It is even stranger than that. Most of the fundamental particles are not stable in isolation, but very swiftly disintegrate into a swarm of other particles.

Particles do not even stay in the same place; they get smeared out into things like clouds. It is fundamentally impossible to know the position and velocity of a particle at the same time.

So counterintuitive is all this behaviour that scientists have speculated about the limits of human understanding of how the world works.

Some physicists seem to have resigned themselves to throwing common sense out of the window, which is a mistake, because it is entirely applicable to particles, which are a construct of common sense.

In our heads we make up a model of the world as we experience it through our senses. This model is, of course, tailored to the size scales and time scales that we evolved to deal with in our lifetimes.

Relativity effects and quantum effects are not normally encountered so we have no appropriate model for them in our heads. This is similar to being more scared of heights on a tower than in a plane, because the human brain has evolved to fear falling off trees and cliffs, but not to deal with the heights that planes fly at.

A particle is a model created for human scales. If we want to understand how the world works, we need to look at concepts that exist outside our world; we need to start with pure mathematics.

The dress

Sorry for the long quiet period, but I have been busy in the world of work.

Recently there was some fuss about the picture of the dress that different people perceive in different colours. I am sure you have seen it. Admittedly this has nothing to do with the connection of mathematics and physics, the topic of this blog. Still I thought it was pretty cool.

Here is how you can see both interpretations. The optical illusion is due to your brain performing a white balance operation by guessing at the lighting conditions. In one case your brain thinks the picture has been taken in yellow light, in the other case in blue light. If you want to switch the colour you see, you need to give your brain some more information about the surrounding. See the two pictures below:

yellow light

blue light

The picture of the dress has another interesting property. The pictures of the stripes are inverses of each other. I saw a yellow and white dress (at first). Inverting the picture to its colour negative gives a white and yellow dress.

Can amateurs still make discoveries in mathematics?

In a maths forum someone posted a link to a Spiegel article (in German) about crackpots. This elicited a predictable stream of comments with one group ridiculing the crackpots and another complaining about the arrogance of the academics.

The interesting point of view was that all the low hanging fruit have been discovered. Amateurs have no chance of success because there are more professional mathematicians in the world than ever before and all the progress is at the outer fringes of knowledge which require many years of study to get to.

How does that square with my rotational numbers?

To reiterate, I don’t see them as a radical new discovery, but a reinterpretation of existing knowledge.

Rotational numbers are not an advanced concept at the cutting edge of mathematical research, but a basic number set that, in my view, was passed over in the relentless drive towards abstraction.

The rules and laws of rotational numbers have already been discovered: as cyclotomic fields. The difference is that in my system the unity roots are not complex numbers, but fundamental constants,like 1,-1 or i. The number i is the order 4 rotational constant, which makes complex numbers a subset of rotational numbers rather than the one underlying set. This is much more general and elegant, but at the price not having general multiplication between rotational numbers.

You could argue that someone would have pointed this out before, if it was really a better view. Are there precedents where mathematicians are attached to a viewpoint for historical reasons that turns out to be sub-optimal?

It turns out that there is at least one.

You can read it up here.

All the circle formulae use radii, but pi is defines as circumference by diameter. A full 360 in radians is 2 Pi. It makes a lot more sense using tau, where:

\tau = 2 \pi

So it turns out that all these centuries we have used the wrong constant. In Carl Sagan’s “Contact” aliens send the digits of Pi as the header to their messages, since this is a recognisable sequence that could not have arisen by natural processes. But they would have sent tau not pi.

Just like rotational numbers, the tau constant does not break existing laws or adds new ones. It is just a better and more natural interpretation of existing knowledge.

Max Tegmark

Someone commented “… as you can see, the multidimensional ideas have been thoroughly explored before, and found unworkable,” and pointed me to the article “On the dimensionality of spacetime” by Max Tegmark.

On many points I seem to agree with Tegmark, judging by “The Mathematical Universe”, where he argues that out physical world is an abstract mathematical structure.

Tegmark analyses the characteristics of partial differential equations with n space + m time dimensions. His very clever anthropic* argument states that all higher dimensional combinations of n+m result in universes that are unpredictable. Any observer with a consciousness similar to ours, i.e. characterised by a linear sequence of thoughts, would require the ability to extrapolate from memories and to make predictions about the future.

* “Anthropic” means that if all the the universes exist but in all but one there is no life to experience them, then us observers would forcibly find ourselves in that one.

Firstly I’d like to point out that in my theory we don’t have flat n+m dimensions but a hierarchical structure and that the world can be approximated in the lowest dimensionality that exhibits space and time properties, which is 3 space and 3 time dimensions. According to Tegmark’s paper this should result in an unpredictable world . If I understand it correctly then the problem with that reasoning is that it assumes a traversal at an arbitrary angle through the time dimensions. In my theory the subjective path that observers take through the time dimensions is down the path of the steepest complexity gradient. (The chapters under the heading  “Multi-dimensional time” in the menu on the right explain this in detail.)

To paraphrase this part of my theory, a traversal too far off the line of steepest complexity gradient would not allow observers to make sense of cause and effect. We should remember that the mathematical patterns that exhibit these complexity gradients are made by a simple generating function, and that any partial differential equations that describe the behaviour are emergent effects.

In the function that generates a structure with 3 space dimensions (x,y,z) and 3 time dimensions (tx, ty, tz) the pairs (x,tx), (y,ty) and (z,tz) are interchangeable (the interchangeability of the space dimensions is consistent with our experience of the world).  The steepest complexity gradient (for stationary patterns) is therefore the diagonal between tx,ty,tz as shown by the red arrow t in this diagram:


There is only one diagonal that passes exactly through the origin, but if we allow some sideways drift then far far away from the origin we have lots of these red time lines that are to all intents and purposes parallel.

I have made an intuitive argument with a lot of hand waving that we experience time as one dimensional as macroscopic observers, while at the microscopic level the effects of neighbouring time lines becomes important.

Max Tegmark’s paper provides a much more scientific explanation as to why a consciousness with the ability to make predictions cannot travel across time-lines or through multiple time dimensions simultaneously. In my opinion his argument supports rather than contradicts my thesis.

Crackpots: Humanity’s unappreciated resource

Crackpot taxonomy

In my view, there are two types of crackpots. There are those that have wild ideas stemming from ignorance. Conspiracy theorists often fall into this category. Crackpots of the second type have wild ideas because they reject commonly accepted wisdom.

Yes, the two types also have features in common. They tend to make grandiose claims in all caps: I HAVE DISCOVERED THE WAY THE WORLD WORKS. They are often outsiders not conversant with the accepted style, standard notation and terminology of the field.

The first type is always wrong. The second type is almost always wrong.

Perhaps presumptuously, if I have to be a crackpot, I fancy myself being the second type. I freely admit that my ideas may well turn out to be wrong, but I am convinced that they are worth investigating. The problem is that the academic world seems to be entirely geared towards gradual improvements to human knowledge. Crackpots of category 2 are facilitators of forward leaps. After all, all progress depends on new ideas and in my experience in just about any field the bottleneck is ideas. Should not all fresh ideas be welcome?


Examples always help to make a point. The theory of continental drift, which is of the same monumental importance to geology as evolution is to biology, was proposed by Alfred Wegener in 1912, but was belittled and ignored for decades because the most eminent scientist of the field attacked it. This is an example of a theory that was considered crackpot science, but turned out to be correct, in the light of evidence that was discovered much later.

Here is an example of an idea that is most likely incorrect, from another field: history.

Another German, Heribert Illig, came up with the rather stunning theory that a medieval emperor bumped the calendar for 300 years in order to reign in the year 1000, and invented a history for the missing years. According to Illig, Charlemagne is a fabricated character and the correct date now is 17something. This theory would explain some apparent historical inconsistencies, but has trouble in other areas and is most probably wrong.

We all agree that correct ideas are good, but I would also argue that both the ideas above have merit irrespective of their eventual correctness, which is not known at the time they were proposed.

In my opinion in a perfect world, instead of being attacked and ridiculed, Wegener and Illig should both have been thanked for their inspired and original ideas. In both cases evidence would eventually refute or confirm the ideas, and this ought to be received with the same good grace by the idea’s originators regardless of the outcome. Without shame if they turn out to be wrong, without gloating if they struck it lucky.

Unhealthy attachment to ideas

This blog entry has been prompted by a paper by Brian Martin: “Strategies for Dissenting Scientists”.

One of the points that the paper makes is how scientists become attached to viewpoints in which they have heavily invested and become unreceptive to ideas that would undermine their status and would make their lifelong commitment be apparently wasted.

In my view the same applies to crackpots. At some point some become so attached to their theories that they become impervious to counterproof, and so category 2 crackpots morph into category 1.

Not me though. I would welcome it if someone comprehensively disproved my ideas in a way that I can accept.

It sounds like a suspicious caveat to insist that I myself accept the counter proof, but several people have read drafts of my paper and pointed out perceived fatal flaws that are not valid in my view. While I realise that it is classic crackpot behaviour to reject critical peer reviews, I do think that a minimum of open mindedness is required.

Personal perspective

If in 1900 an inventor came to a patent office with a mechanical perpetual motion machine, the patent clerk could save the trouble of having to understand the machine by the shortcut of simply noting that the law of conservation of energy and the existence of friction makes it impossible. However the discovery of superconductivity provides a counter example to this strategy.

Similarly, there is a shortcut to dismissing my idea out of hand since it has a deterministic mechanism at its core. In my view though, the proofs that preclude determinism make the unspoken assumption of having a single time dimension. With only one time dimension the path of an electron can only be calculated as a probability and by taking all the possible paths it could take into account. This is an observer-centric way of looking at the world. I made the analogy of a geocentric world-view in my introduction. (It made me smile when I discovered that I get 40 points on my crackpot index for Copernican comparisons.)

With multiple time-dimensions there are many parallel and interacting copies of observers and electrons and the electrons actually do take all the possible paths. I am sorry, this still makes much more sense to me.

I have come across a paper by Max Tegmark that makes an anthropic argument for the existence of a single time dimension, but I object to his reasoning. I will deal with that paper in a future blog post.

One would have thought that the mathematical part of my idea, rotational numbers, would be easy to prove or disprove, but I am having the same trouble there. All the criticism levied against it (apart from that due to fixable mistakes that I made) boils down to a rejection of an unfamiliar approach. So far at least.

Everybody has a world-view

So far this all sounds like an attack on the narrow-minded establishment that refuses to read my theory; but in reality everybody, me included, behaves the same way. Every person, and especially a scientist, has a world-view; some intuition that tells us how the world works. It is the same mechanism under which religious people operate. Some scientists are attached to religion in an unscientific way. Some others are attached to scientific theories in a quasi-religious way. If they encounter something that does not fit their world view, then it is rejected or ignored.

For me anything to do with superstition, UFOs or any religious arguments hit the instant-off button, because they have no legitimacy in my world-view. Can I really complain if physicists who worship relativity theory with religious fervor refuse to take seriously any theory where relativity is not built in at the base level, but is an emergent and approximate phenomenon?

The problem

The first half of the problem is the ratio of category 1 to category 2 crackpots. I have come across many crackpot theories on the internet. While there are a few where I get the sense that the originator has some intuitive feel for something or a novel insight, most seem to be very clearly crazy. The second half of the problem is that it is very hard to tell one from the other. I experimentally posted on an “against the mainstream” forum recently and was surprised how hard it was not to sound like another raving loony when forced to describe my idea in short forum posts.

If you look at the history of just about every single big idea you find the same pattern. Overwhelming hostility towards the originator of the idea, who then becomes engaged in a monumental struggle for acceptance which often only comes posthumously. This makes you wonder how many potentially world-changing ideas fall by the wayside because the inventors lack the stamina to pursue them.

There is a freakonomic perspective to this also.

Imagine a group of people testing thousands of rocks to find one gold nugget. If it is laborious to test each rock and costly to declare a false positive then the economically winning strategy for every member of the group is to reject all their rocks without testing, since in all likelihood they will all be duffers. The downside is that the one nugget will not be found. This analogy is not limited to the scientific world of objective measures. For example I think it also occurs in the book publishing industry.

So who owns the problem of sifting new ideas? Academics appear to think that this is not part of their job description. I wonder, if not them, then who? Academics are busy with their own projects, and there are strong disincentives for academics to even interact with crackpots. Furthermore, since the chances are against them encountering the one nugget, dealing with category 1 crackpots can be exasperating and offensive.

For example, Gerard ‘t Hooft, who already has a Nobel prize and so does not need to fear association with the wrong sort has been inundated with crackpot theories and has a big section of website devoted to them. Some physicists have extremely scathing sections on their webpages designed to repel purveyors of crackpot theories. This blog post is trying to come to the defence of at least category 2 crackpots.

Maybe someone should found a crackpot idea analysis institute. This would be a nice systemic solution, but what about me? What can I do right now? I am still hoping for some help from the academic community, hard as it is to come by.

To be clear, I only make a claim to category 2 status and to join the crackpots that enrich the world with new ideas. I can’t be sure that my idea contains a nugget, I merely believe that it is worth investigating.

Balls bouncing in a box – in two time dimensions

Let’s kick off this blog with a demonstration of the effects of multiple time dimensions.

We simulate 12 conventional objects bouncing around in a box. By “conventional” I mean that each object is at a well defined position in space and time.

We will start with one dimension of time and then extend it to two and plot the paths of one of the objects. We see that even though the each particle has a definite position at each time coordinate (t1,t2), there is an uncertainty about the direction in which it is currently moving. Each particle also has multiple histories as to how it got to this point.

Let’s first look at the one dimensional time case.


Here we have the 12 objects in a box, 6 blue, 6 red. Similar colours repel, different colours attract. There is no collision detection nor friction. We show the state of the box for the first 10 times, and we are tracing the path of one of the objects.

We are starting from a very highly ordered state, at later times the state in the box is less well ordered. We can see that time flows from left to right. This simulation with only one dimensional time shows the familiar common-sense, Newtonian picture of the objects. At each time each object is in a well defined place. The black line traces one unique history for the object that we are monitoring.

Now let us have a look at the same system with 2 time dimensions.


If we start from the highest ordered state in the top left corner and we want to move along to successively less ordered states, we could move from left to right in the top row, or we could move down the left column, or we could follow a path in between, for example diagonally across or at an angle.

The initial conditions for the vertical and horizontal time directions are very slightly different. (If we made the initial vertical and horizontal components the same, we would get a perfectly symmetric pattern.)

As you can see on the following diagram, which shows one particular space at some time (t1,t2), there is a very definite position for the black ball, but there are several ways it could have got there, and in fact, it did. There is an inherent uncertainty about its momentum. This is a natural consequence of having more than one time dimension and not a designed-in ‘feature’ of our simulation. There are multiple histories associated with each ball.


If we don’t know which (t1,t2) time we are at, but only on which generation we are (meaning distance from the origin, i.e. t1+t2) then it gets even more interesting. At generation 12 we could be on any of the 12 times on the diagonal of the 2-dimensional diagram. If we plot the location of the black particle in each of these boxes on top of each other, using an opacity of 1/12 we get the following picture.


The location of the object is smeared into a cloud. If we test for the position of the object we get the situation of the previous diagram, where we know where the object is, but can’t say by which path it got there. The probability of finding it at a particular location is determined by the blackness level of the cloud picture.

These are natural consequences of having more than one time dimension.

The code that produced these pictures is available on the downloads page.

First post!

Welcome to my blog.

Hopefully this is relevant to anyone with an interest in Physics, but the Mathematics section stands alone and I would be very grateful for feedback from Mathematician, even when negative. Actually especially when negative so I can try to address the criticisms.