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The beautiful thing about philosophy is that it identifies the truly great problems, ones where arguments can be made for each side with equal validity. The website philpapers.org, an online repository of philosophy articles and books, recently conducted a very interesting survey about some of these grand debates.

It consisted of 30 questions, all current issues with well established alternative positions in philosophy under intense debate. They surveyed 1803 philosophy faculty members and/or PhDs and 829 philosophy graduate students, and then tabulated the results. I also went through and gave each question a “controversy” score – the lower the score, the lower the consensus in the answers (for the curious, via mean square error).

Is this the real life – or is it just fantasy?

The least controversial issue was Question 6, “External world: idealism, skepticism, or non-skeptical realism?“. This dealt with the structure of the external world, where and how it exists, and what we can know about it.

4.2% of respondents believed that the external world was best described by idealism, that reality is totally dependent on the mind. Plugged into the Matrix? Your “reality” could be described by an idealist perspective.

4.8% chose the viewpoint of skepticism, that the external world can never really be known in its true form.

9.2% reinforced some stereotypes of philosophers and gave an answer best described as “other”.

81.6% of respondents agreed in the clearest consensus of the survey that the external world was best described by a perspective of non-skeptical realism. This means that “reality” exists independent of the mind (the realist part, we aren’t making it all up in our heads) and that we can draw reasonable and consistent conclusions from it (the non-skeptical part).

Life may not be a waking dream after all. Now, onto controversy!

Kill one to save a thousand?

You are a superhero. The love of your life dangles from a fraying rope above a pit of spikes, while nearby a speeding train full of orphans rushes toward the edge of a cliff. You only have enough time to save one of the two – what do you do?

Normative ethics is the branch of philosophy that deals with questions like these – how “should” you act in a certain situation? Is there certain approach one should use? This was the subject of one of the most controversial issues, Question 20 – “Normative ethics: deontology, consequentialism, or virtue ethics?“.

18.1% chose the perspective of virtue ethics, first advocated for in a significant sense by Aristotle. Virtue ethics emphasizes the character of the person who is faced with a difficult decision. Do you save the children in the train or your dangling love? It doesn’t matter – what matters is your character and your intent in making the decision.

23.6% argued for consequentialism, the philosophy that spawned the approach of “the ends justify the means”. Whether an action is right or wrong depends on the outcome of the situation, not the specific actions you chose. There are a number of different variants of this which would “score” the final situations according to whether it maximized happiness, economic benefit, liberty, love, or any number of possible ideals to yourself or to others. If you were an egoist you would send the orphans over the cliff and save your love to maximize your own happiness. If you were a utilitarianist, you would save the orphans because it would benefit the largest number of people.

25.8% of respondents chose deontology, where it is the actions you take that are judged rather than the results of those actions. Deontologists are concerned with rules and duties, and an action following these duties can be considered morally correct even if it produces dire consequences. If you were a married superhero who had sworn a vow to protect his love – orphans be damned, there is a duty to perform.

32.3% of philosophers, presumably not wanting to commit to paper their rationale as to whether they’d kill orphans or their love, chose “Other”.

What about the rest?

Here’s a table ranking the “consensus” on each issue, from least to most controversial. The lower the mean square error, the less disparity there is in the magnitude of responses – and so less consensus is reached. I think this is a pretty decent metric, if you have any better suggestions, feel free to leave a comment.

Now you can go to fancy cocktail parties and hold your own in philisophical debates – or at least have the confidence that you’re stating a position that cannot be proven wrong.

My three dimensional unfolding of the quaternion Julia sets finally finished rendering. There are a fair bit of compression artifacts in the embedded version, click on the Vimeo button on the bottom right side of the video to watch it in full quality HD.

Since each quaternion can be described using four numbers, I unfolded these four dimensional quaternion Julia sets into three dimensional space, and animated the final coefficient.

But once I did that I noticed some radial symmetry along the y-z plane – it looks like something that’s been made on a lathe. This means that we can “index” all these shapes in a more sensible manner by collapsing things along this axis of symmetry. While previously we could index all of our shapes with four coefficients a, b, c, and d.

We can now index them with four coefficients a, r, theta, and d after this transformation. But there’s a nice side effect now that our coordinate system reflects our symmetry – if we vary theta, the appearance of the Julia set doesn’t change, the object just appears to rotate about the a axis.

So really we can index all possible shapes using only three coefficients – a, r, and d. This is awesome – it means we can use this symmetry to collapse a dimension and completely illustrate a discrete approximation of this four dimensional set in three dimensional space. The following images (click for 1080p full resolution images) illustrate the full set of these possible shapes – a is the horizontal axis, r is the vertical axis, and values iterate by 0.25. The grey sphere in the first image is the origin, and the images start at a d value of 0 and iterate upward by 0.25. We find that there exists additional symmetry with our d parameter – namely that d = -d, so we only need to illustrate the absolute value to see all shapes.

d = 0.00

d = 0.25

d = 0.50

d = 0.75

d = 1.00

When d = 1.25 there are only a few bits of unconnected dust loops visible. This analysis only covers a single “slice” – namely the plane normal to (0,0,0,1). I’d be very interested to see if there are any other symmetries…

Japan’s consumer goods lived up to their surreal billing during a recent stay at Narita Airport in Japan on my way to IAC 2009. A collaboration between Japanese designer Takada and Bandai’s Banpresto division, the Facebank combines limited practicality with pure awesomeness.

The eyes are optical sensors which start the chewing motion as soon as your hand goes near (and as soon as you turn out the lights in the room as I also found out). Standing about 10cm tall, it takes 4 AAA batteries and retails for about ¥2000.

I recently saw a very interesting photo of a sea shell on Flickr.

The patterns on the shell appear to be very similar to that of a mathematical structure called a Sierpinski triangle – and this is no coincidence.

A snail’s shell can grow only by adding on new material in a thin layer on the lip the shell. The pigmentation cells lie in a narrow band on this lip, and decide whether to switch on or off depending on the pigmentation of the area immediately around it. In short, the pigmentation patterns can be modelled as elementary cellular automata very accurately.

Several elementary cellular automata rule sets produce similar structures to that seen on the shell. Combine these basic rules with a little bit of noise due to nature, and you get these beautiful pattens with a bare minimum of computational effort.

The snail that grew the shell above is from the family Conidae. Other species have slightly different rules for pigmentation, but all produce their patterns by a method that can be modelled as cellular automata.

It is said that when the 20th century is long gone, it will be remembered for two revolutionary theories – those of relativity and quantum physics. While both have led to a deeper understanding of our world, quantum physics stands alone in its profound weirdness – the ability to accurately predict totally counter-intuitive aspects of the physical world. From the simple indisputable oddity of the double slit experiment to the philosophical implications of Schrodinger’s cat, it becomes clear that we still understand very little of the actual mechanics of our world.

When explanations are lacking, the mystical is often brought up to fill the void. This has degenerated today into complete pop-psychology crap such as The Secret or What the Bleep Do We Know, but the role that human consciousness plays as an “observer”, if any, was considered very early by the founders of these theories. These arguments brought forth by some of the finest thinkers of our time echo to this day.

Niels Bohr

Winner of the Nobel Prize in Physics in 1922, employed by the Manhattan Project, and father of the Bohr model familiar to every high school student, Niels Bohr was first accused by Einstein of introducing “mystic” elements in his explanation of quantum physics – mystic elements which in Einstein’s view had no place in science.

This was part of the famous Bohr-Einstein debates, and was perhaps not a fair criticism. Bohr appeared to not worry excessively about the “reality” underpinning the equations of quantum theory, and was simply more concerned about the equations of quantum theory rather than their implications. He rejected the hypothesis that the wave function collapse requires a conscious observer, insisting that “It still makes no difference whether the observer is a man, an animal, or a piece of apparatus”.

His view is perhaps best summarized in the following quote recalled by Heisenberg:

This argument looks highly convincing at first sight. We can admittedly find nothing in physics or chemistry that has even a remote bearing on consciousness. Yet all of us know that there is such a thing as consciousness, simply because we have it ourselves. Hence consciousness must be part of nature, or, more generally, of reality, which means that, quite apart from the laws of physics and chemistry, as laid down in quantum theory, we must also consider laws of quite a different kind. But even here I do not really know whether we need greater freedom than we already enjoy thanks to the concept of complementarity.

In short, if the numbers work out, don’t worry too much.

Wolfgang Pauli

But some did worry. Pauli was a skeptic’s skeptic – a man so dedicated to rationality it led him down a strange path. In 1927 the Solvay Conference was busy reaching consensus that Bohr’s approach was the best way to regard quantum physics (the Cophenhagen Interpretation), but Pauli was equally confident in a different interpretation. He tried to trace out just what part of consciousness it is that seems to prevent an in-depth, rational understanding. Deeply influenced by Schopenhauer’s The World as Will and Representation, Pauli appropriated his concept of a will “which breaks through space and time”.

He viewed that the acquisition of knowledge in mathematics or quantum physics “gives rise, however, to a situation transcending natural science” that can even acquire a “religious function” in human experience. This is not a belief in the religions of old, but as Pauli states “I do not believe in the possible future of mysticism in the old form. However, I do believe that the natural sciences will out of themselves bring forth a counter pole in their adherents, which connects to the old mystic elements.”

Perhaps the most interesting viewpoint on Pauli was that of Heisenberg, who viewed Pauli’s paradigm as even more rational than Bohr’s equation-focused approach because only he acknowledged that the scientific evidence pointed to something pre-rational or ‘mystical’. Pauli claimed that consciousness was presented philosophically by mystics and studied scientifically by psychologists. With the advent of quantum mechanics, physicists should then be able to unify both approaches. Unfortunately, we continue to wait.

Albert Einstein

Einstein was a scientific superstar, with fame not equalled to this day. One day, a quote was making the round in British newspapers that Einstein subscribed to the theory that “the outer world is a derivative of consciousness”. His response was swift and critical.

No physicist believes that. Otherwise he wouldn’t be a physicist. Neither do [Eddington and Jeans]. . . . These men are genuine scientists and their literary formulations must not be taken as expressive of their scientific convictions. Why should anybody go to the trouble of gazing at the stars if he did not believe that the stars were really there?

Einstein’s opposition to Bohr’s view or more “mystical” approaches is often cast as the great divide between the philisophies of idealism and those philosophies based on realism. Pauli often referred to Einstein’s “philosophical prejudice” assuming that reality is independent of any mind. In fact, his approach and objections were far more subtle. His major concern was the incredibly subjective aspect of consciousness introducing an unmeasurable “geist”, and this clash with the precise and well defined philosophical principles of physics such as locality or determinism.

This led to Einstein’s famous attempt at “breaking” quantum physics, the EPR paradox. At first a thought experiment which appeared to demonstrate quantum physics violating the seemingly well established principle of locality, later experiments showed that quantum physics instead proved locality false.

Violations of locality and determinism seemed to bother Einstein greatly, and this can be seen in his famous quote objecting to the randomness involved in wave function collapse under Bohr’s interpretation, that “God does not play dice”.

Bohr, summing up the debate perfectly, replied “Einstein, stop telling God what to do with his dice.”

John von Neumann

The “last of the great mathematicians”, von Neumann solved one of the great problems of quantum theory. While the theory itself was established and experimentally verified, it lacked a “deep” mathematical understanding based on an axiomatic approach. He treated the world as a Hilbert space, an infinite dimensional structure.

While classical mechanics approached the world as a collection of points with six different characteristics (position and momentum along the x, y, and z axis), von Neumann’s approach considers a quantum system as a point in infinite dimensional space, corresponding to the infinite amount of possible outcomes. This led to very interesting implications in terms of “measurement”. While the “measurement” of a classical system simply involved finding one or more of those six values, the “measurement” of a quantum system involved mathematical operators acting on an infinite amount of values to produce a finite measurement.

The interesting conclusion arises when we consider the “real” interpretation of these mathematical operators. While we may say that an scientific instrument has caused wave function collapse, we run into the problem that no physical system (and a scientific instrument is a physical system completely described by quantum mechanics) can cause wave function collapse. We can describe the entire ensemble perfectly as a Hilbert space. But we do not experience this Hilbert space – we measure and experience only finite values.

The conclusion von Neumann reached is that consciousness, whatever it is, appears to be the only thing in physics that can ultimately cause this collapse or observation. This does not mean that consciousness is “required” for the universe to work, but that wave function collapse appears to be caused by consciousness and we observe only a tiny slice. We are therefore an “abstract ego” acting as a measurement device on the infinite values of true reality.

Today

Today, the argument has largely died down, a combination of practicality and lack of any suitably shocking experimental results. The majority of physicists today take the approach of “it works”, namely that quantum physics produces accurate predictions of the real world and that the mathematical formalism is just that – a mathematical formalism that produces accurate results.

It may not reflect the true reality of the world (whatever it is), but it is suitably accurate to any level of precision that we are physically able to obtain. One may stay awake at night wondering “why”, but one will not get much work done with this approach. Perhaps more clarity lies in the future, but in the meantime – we will have to tolerate crap that tells us we can “will” our way to riches with quantum mechanics (and coincidentally make the authors rather rich, will indeed) instead of a rational approach dedicated to the pursuit of truth.