Author Archives: xentwtww

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:

axesmove

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

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.

Further maths

We said said we want to generalise the set of complex numbers, but we derived only the minimum  set of numbers necessary for our generating function, namely multidimensional numbers with integer coordinates. So our M4 set are only the Gaussian numbers.

In order to represent complex numbers we need fractions and real numbers. We can easily extend our scheme to define these and other useful sets. Even though not necessary for our purposes, I think it is interesting.

I’ll outline how to do that in a way that is hopefully easy to follow, but not very rigorous. For this post to make sense you must have read and understood the maths section.

Chains and fractions

Let us consider any two consecutive numbers on any chain.

f1

Let us say, for simplicity that the blue arrow corresponds to the open function +1, although of course the same applies to any other chain.

First we introduce an intermediate number.

f2

The green arrow would then correspond to the chain +1/2. It is defined by d22(x)=inc(x).

We can also go via two intermediate numbers:

f3

In this case the brown arrow corresponds to +1/3, which is the chain defined by d33(x)=inc(x).

With three intermediates:

f4

The red arrow corresponds to +1/4; it is defined by the chaind44(x)=inc(x).

Since d44(x)= d42(d42(x)), it holds that d42(x)=d2(x).

Continuing this way we can make fractions with arbitrary denominators, but this is unsatisfactory because as we increase the number of intermediates we lose the ability to express some numbers we could do previously. For example, in the diagram above with two intermediates we can express the number 1/3. Increasing the resolution to three intermediates loses this ability again.

Instead we now add intermediate chains and keep the previous ones.

We start as before:

f5

Now we add two intermediates, but keep the previous intermediate order:

f6

For the next step with three intermediates you might expect three new numbers, but you only get two, because we have one of them already; two applications of the +1/4 chain is equivalent to one of the +1/2.

f7

This diagram shows the fractions ¼, 1/3, ½, 2/3, and ¾, but we can combine the chains to express other fractions. With a finite depth of nested intermediates we have generated the set Qn of fractions, where Qnis the set generated by repeated application of chain d to 0 and where dnn(x)=inc(x).

Hierarchies of Infinities

If we allow infinite applications of functions then we can generate new sets: hierarchies of infinities and infinitesimals as well as real numbers.

Infinities

Any number that can be reached by a finite number of applications of an inc function to zero is finite. Conversely, numbers for which this is not true, we call “infinite”.

“Finite” and “Infinite” are properties of the numbers, like “even” or “odd”.

We have denoted repeated applications of an open function to an object with a superscript. For open function f and all objects p:

f0(p) = p
fn+1(p) = f(fn(p))

This defines the positive integers n as a finite applications of f. Remember that we use the shorthand notation  f3(p) for f(f(f(p))).

We pick any infinite n and call it ∞.

Hilbert’s Hotel

All the usual chain and loop axioms apply to infinite numbers.

loopinf

“Infinite” is a property of a number. The number ∞ – 1 is less than ∞, but still infinite.

The number ∞ + 5 is a number 5 applications of the inc function away from infinity the same way that 1000 + 5 = 1005.

This is contrary to the usual treatment of infinity that treats a set of infinite numbers as a number and takes ∞ + 5 = ∞.

In Hilbert’s Hotel an infinite number of rooms are all occupied. A new guest can be accommodated by moving all guests up one room. An infinite number of new guests can also be accommodated by moving all the guests from room n to room 2n, freeing all the odd numbered rooms.

This only works when you confuse the property “infinite” with the number “infinity”.

Hilbert’s hotel is as much a paradox as saying that my two-seater sports car has an even number of seats ergo it can accommodate 4 people.

If the hotel has ∞ number rooms and a new guest arrives then we have ∞ + 1 guests. If an infinite number of guests arrive then we end up with 2∞ guests. We can’t just conjure up new rooms and all the usual rules of arithmetic still apply.

Real numbers

We can express any fraction with a finite depth of intermediate numbers.

Transcendental numbers, such as π or√2, cannot be expressed as a finite fraction. We can approximate them to an arbitrary accuracy using fractions with bigger and bigger denominators, meaning more and more intermediate numbers, but to express a transcendental number accurately we need an infinite number of intermediates.

ff1

The blue arrow depicts an inc function again.

The grey arrow is the function d(x) , where Δ = d(0)  is an infinitesimally small number, such that it takes infinity applications of the grey functions to advance the distance of a blue function.

This way we can use our chain functions to also generate the real numbers.

Automorphisms

Consider the structure of the functions +1 and +1/2.

ff2

The +1 chain function is shown in blue and the +1/2 chain function is shown in green.

Now consider the functions +1 and +2:

ff3

The +2 function is shown in orange.

You will notice that these two structures are isomorphic, i.e. there is a one-to-one mapping between them.

In other words the blue function is to the green function like the orange function is to the blue function.

Hyperreals and Infinities

Here is the diagram from above again:

ff1

Looking at the grey d function (adding an infinitesimally small value to a number), we see an automorphism similar to the one shown in the previous section. The grey function is isomorphic to the blue function.

If we consider the grey function to be +1 then the numbers on the blue function chain will be a series of infinities. These have been studied by Cantor. If we look upon the chain function d(x) as inc(x) then the numbers 1,2,3,… become ∞, 2∞, 3∞, … After an infinite number of infinities at ∞∞ we arrive at a number that Cantor says has the cardinality of aleph1.

We can keep going scaling up infinities. There is never a final biggest number that cannot be exceeded.

We can of course apply this mapping the other way and consider the blue function to be in which case the numbers generated by the grey function might be related to surreal and hyperreal numbers. Maybe a mathematician reading this could confirm or refute?

More sets

We can apply infinitesimals and infinities to loop functions, too. I don’t know whether these sets have already been explored.

A loop function with an infinite order is not the same as a chain function. A loop function returns to the first value after infinite applications, while chain functions continue to make new elements even after infinite applications or more.

For x ≠ 0
rot(x) = x
inc(x) ≠ x

 

Rotational numbers

In this post I’d like to talk about the mathematics section. In particular I would like to address some common objections.

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. In fact it is telling that the professional mathematicians I discussed this with often seemed to prefer to derive this as a special case of a higher level concept, like a module over a ring, rather than accept the elegant derivation from first principles.

Other common objections are (1) concern about the absence of multiplication, and (2) discomfort about using the output of an open function to define new sets because it is contrary to the requirement that functions must have a well defined domain and range.

Multiplication is essential to many areas of mathematics and we learn it at primary school so it maybe helpful to take a step back and review what it is in essence.

We use unary inc open functions to generate the natural numbers. We can create a new binary function that repeats the application of inc. Clearly the second argument, the loop counter that determines the number of repetitions, must be a positive integer. We arrive at the binary addition function. We find that the function is commutative, meaning that we can swap the loop counter and the first argument and get the same result. The inverse function to addition is subtraction. This is not closed in the natural numbers. We extend the range set for subtraction and arrive at the set of integers. We can retroactively extend the domain for addition to include negative numbers. We don’t want to break commutativity, so we invent a rule to allow negative numbers as the second argument (the loop counter).

Now we repeat the process and repeat adding the same number to itself and arrive at multiplication. We find that this is also commutative. The inverse function is division which again is not closed under the integers, so we need to extend the range set to the fractions. Again we extend the domains for addition and multiplication to include fractions. Because we want to keep commutativity we allow fractions as the loop counter argument.

We repeat the process again and arrive at potentiation. This is no longer commutative, so we get two inverse functions: roots and logarithms. The inverse functions again require a range extension to real numbers. Retrofitting fractions and real numbers to addition, multiplication and potentiation results in some rules that don’t appear intuitive at first sight, but that work to maintain the expected laws.

The next level is tetration or the power tower. Again, there are two inverses. Finding sensible domain extensions becomes quite difficult from this point and is an active area of research.

The point of all this is to demonstrate that defining new sets as the range of functions is perfectly natural and that multiplication is merely the hyper2 function that has many applications and regularities.

The conventional insistence of all unity roots being complex numbers rather than defining them to have independent dimensions (where the complex numbers are the order 4 special case) stems from the desire to maintain general multiplication.

In the mathematics section we don’t strictly generalise complex numbers, but Gaussian numbers, because we only require integer coordinates. The same methods can be used to derive real numbers and some other interesting sets. I’ll do that in the next post.


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.

bounce0

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.

bounce1

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.

bounce2

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.

bounce3

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.