Alice's Adventures in Wonderland - the Math

Jack Dikian

November 2010

Alice's Adventures in Wonderland (commonly shortened to Alice in Wonderland) is an 1865 novel written by English author, mathematician, logician, Anglican deacon and photographer Charles Lutwidge Dodgson under the pseudonym Lewis Carroll.

Much has been said of the mathematical undercurrent in Alice’s adventures in wonderland. Before reflecting upon the math it’s useful to look at the times, and backdrop of mathematics in the latter half of the nineteenth century.

How Dodgson came to write his novel has been much documented – in the way of a very brief summary, the story goes that in 1862, Dodgson, together with the Reverend Robinson Duckworth, rowed in a boat up the River Thames with three young girls, Lorina Charlotte Liddell, aged 13, Alice Pleasance Liddell, aged 10, and Edith Mary Liddell, aged 8, the daughters of Henry George Liddell, the Vice-Chancellor of Oxford University and Dean of Christ Church College, as well as headmaster of the nearby, private, Westminster School.

As they rowed, Dodgson made up and told the girls a story about a bored little girl named Alice who goes looking for an adventure. The three girls loved it, and Alice Liddell asked Dodgson to write it down for her. Two years later he did just that, and in November 1864 he gave Alice the handwritten manuscript of what he then called "Alice's Adventures Under Ground," illustrated by his own drawings.

The math

In the latter half of the nineteenth century, it was a turbulent period for mathematicians, with the subject becoming more and more abstract. Non-Euclidean geometry, Symbolic algebra and the growing use of imaginary numbers were just some of the developments that challenged classical mathematics.

Martin Gardner, whose The Annotated Alice was published in 1960, Helena Pycior of the University of Wisconsin-Milwaukee, and more recently Melanie Bayley, of the University of Oxford in England, have examined linkages between mathematics and the experiences of the characters in the novel

According to Bayley, Dodgson was dismayed by what he saw as the declining standards of mathematical rigor. She suggests Dodgson additions for publication was a wicked satire on those new developments.

What of the links with mathematics - Here are just a few examples

In the chapter "Advice from a caterpillar." Alice has fallen down the rabbit hole and eaten a cake that has shrunk her to a height of just 3 inches. The Caterpillar enters, smoking a hookah pipe, and shows Alice a mushroom that can restore her to her proper size. But one side of the mushroom stretches her neck, while another shrinks her torso, so she must eat exactly the right balance to regain her proper size and proportions. Bayley believes this expresses Dodgson's view of the absurdity of symbolic algebra.

Bayley suggests that the overall madness of Wonderland reflects Dodgson's views on the dangers of this new symbolic algebra. Alice has moved from a rational world to a land where even numbers behave erratically.

In another scene, the hallway, Alice tries to remember her multiplication tables, but they have slipped out of the base-10 number system she is used to. Yet another, perhaps, unsubtle but pointed remark.

In the caterpillar scene, Alice's height fluctuates between 9 feet and 3 inches. Alice, bound by conventional arithmetic where a quantity such as size should be constant, finds this troubling: "Being so many different sizes in a day is very confusing," she complains.

To survive in Wonderland, Alice must act like a Euclidean geometer, keeping her ratios constant, even if her size changes.

The baby's discomfort with the whole process, and the Duchess's unconcealed violence, signpost Dodgson's virulent mistrust of "modern" projective geometry, Bayley says. Everyone in the pig and pepper scene is bad at doing their job.

EPR Paradox & Bell's Inequality rule

Jack Dikian

July 2006

Albert Einstein and two colleagues, Boris Podolsky and Nathan Rosen (EPR) intended to show that Quantum Mechanics could not be a complete theory of nature and that some other theory would have to be invoked in order to fully describe nature.

Key words: EPR, Existence of hidden variables, Bell's Inequality rule and local theories of quantum mechanics.

To be uploaded...

Making Meaning of Time

Jack Dikian

August 2001

Introduction

When thinking about a world where the linearity of time is no longer maintained – that is, a world where the 24 hour measure of time is no longer divided in equal intervals; the time that elapses between 3pm and 3.15pm is different from the time that elapses between say, 4pm and 4.15pm.

It become very clear that the word “time” is used as a means of measuring the passing of time. So a clock measures intervals of time, but not of time itself. The immediate question, therefore, is what is it that we are measuring? We can use all sorts of instruments and fall back on common descriptors such as;

“the years are going by so quickly”,

“Time is like a river”,

“Time flies by when we are having fun”

These don’t, however, tell us what it is that we are measuring. We are still left without an understanding of what exactly is an interval of time.

The Psychology

Despite the preciseness of our instruments we are still measuring physical time in contrast to a more subjective personal time. Psychologically, time seems to be over before we know it. The past only exists as memory and we can’t remember the future. So, therefore we are always in the moment. Interestingly enough, unlike other spatial dimensions of perception, time is not directly available to any sensory system but can only be sensed through the unfolding of events in the world we live in.

Before examining the mathematics and/or the physics of what we call time it may be useful to briefly look at the psychological construction of time that we use to make meaning of our world.

The orderly nature of our world is born in part from the fact that events’ spatial characteristics are structured in and over time. Speech, body movements and walking gaits in which the sequence of words or actions unfold with a characteristic rhythm and tempo over a given time span. This spatio and temporal structure influences how an event is perceived and remembered, as well as providing some guidance over the accuracy with which the event’s velocity and total duration are subsequently judged. We routinely decide (or to be more précis estimate) whether there is sufficient time to cross a street ahead of an approaching car.

The activities we undertake in everyday life also vary in cyclic patterns over the course of a day or a course of a seasion and provides a scheme that serves to coordinate those individuals in any particular society. Different cultures have different conceptualizations and attitudes toward time which can be reflected in the overall pace of life. In any individual, the temporal perspective and their relative orientation toward the past, present, and future is central to one’s mental well-being and the degree of ego strength displayed in coping with life’s difficulties. As well, in the clinical analysis of timing disorders mechanisms mediating circadian-driven behaviors are quite distinguished from those governing the internal clock and sensitivity to an event’s duration and overall velocity.

The Physics

Firstly, we all tend to perceive time as a smooth flowing and continuous motion. We in the usual course of life never experience time in chunks, slices, or something characterized by broken gaps. This is somewhat consistent with our perception of other natural phenomena such as energy, and the space around us. However, at the quantum level of matter energy is not released continuously - there is a limit to how small a change in energy an atom can experience - it is released in discrete quanta by the emission of photons. This lends itself to the question;- are individual frames of time so small that it only gives the appearance of being smooth and continuous.

In experiments using instruments that can 'slice' moments in time to a small enough granularity in order to capture a chemical reaction and dealing with time at the Femtosecond level, that is one thousandth of one trillionth of a second, (1/1,000,000,000,000,000th) there seems to be no indication of time passing in discreet steps. Time still appears to flow smoothly. It is off course entirely possible that if it does move in discrete steps that we have not yet reached a level small enough to observe it. The smallest measurement of time that can have any meaning in contemporary physics is a 'Planck Time', and is equal to 10-43 seconds.

That is a decimal point followed by 43 zeros and then 1 (0.00000000000000000000000000000000000000000001)

Interestingly, there is nothing in the laws of physics to suggest that time actually flows from the past through the present and into the future. At the subatomic level there is no distinction between the past and the future. By studying these particles it would be impossible to determine the order in which the events took place. At this level there is no way to distinguish the past from the future by simply looking at each particle pairs.

However, we are clearly able to discern the arrow of time or the direction of time at the macrosporic level. We would never see for example, a bottle hitting the floor and breaking before it actually falls off the table. So it seems at the macro level (the level we experience reality) there is a natural inbuilt arrow of time pointing from the past to the future. The distinction between the past and future is expressed mathematically by thermodynamics.

The second law of thermodynamics states that the entropy of a closed system always increases, entropy being the measure of disorder. In other words disorder will always increase and if left to its own devices, a system will run to disorder, and not order, giving us an arrow of time.

One of the most peculiar qualities of time is the fact that it is measured by motion and may possibly therefore emerge through motion. Our perception of “time” may therefore be mostly an illusion. Our memory creates the illusion of the past. Our conscious perception of events gives us the feeling of the present. The future is a mental construct patterned on the memory of experience. The concept of time emerges as our mind tries to make sense of the world we are in.

In general relativity matter produces curvature in space-time, and this concept can be extended to moving objects showing that moving objects can also curve space-time. In an expanding universe, time is related to the expansion of space. Slower time is associated with slower expansion. This extra dimension is not a time dimension in which we can travel. Time is just thought of as the presence of motion and forces and is caused by the expansion of space. Expansion of space, here, is the prime mover behind all the action, it imparts time as the presence of motion and forces to matter in an area of space.

The relationship of the clock and speed of light is fixed at a basic level in a given region of space. Clocks may measure a different speed of light if located in a region of space which is different from where light is being observed. For example a clock on earth measuring speed of light in a region next to a distant neutron star where time is slower will measure a slower speed. While a clock located on the surface of neutron star will measure a faster speed of light in distant space.

Moving objects produce moving wave like curvature in space. Once within the vicinity of the moving object mass-less particles like photons are picked up and carried forward by the time differential which is present in these waves. Also the velocity of light is decreased by the slower time within the moving wave. Within the moving object like earth light is pulled along in the direction of motion and it also slows down to conform to the slower time around the earth.

The faster we travel through the dimensions of space, the slower we travel through the dimension of time, and vice versa. Thus an astronaut zooming along at light speed has used up all their speed 'allocation' in the space dimensions, and as a consequence does not travel through time. This would seem to suggest that the speed of light really is the limiting speed within the universe, and if we had no motion at all through space then we would be travelling at light speed through time. The more movement we make across space the less we make in the direction of time.