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Archive for November, 2007

In music, there is a chord known as the “Jimi Hendrix chord”. For those who know about these things, it’s a dominant 7 #9 chord – e.g. C7#9 contains the notes C E G Bb D#. The dissonance between the E and the D# (the major and minor third in C respectively) creates the gritty, edgy, crunchy rock sound that Hendrix uses in Foxy Lady and Purple Haze.

There is a story, possibly an urban legend, that Purple Haze is so named because the Jimi Hendrix chord in its introduction made Hendrix see a purple haze. Other theories invoke copious amounts of LSD and marijuana, but it is the triggering of a purple haze that I want to focus on.

This phenomenon, of one sensory experience involuntarily triggering a second, usually unrelated sensory experience, is known as synesthesia. It is a neurological condition, and appears in a variety of forms. For example, some synesthetes (as they are called) associate letters and numbers with colours – for example, a black 5 written on a page is seen to be green; a 2 seen to be red.

At first glance, the condition doesn’t seem very interesting. Most people would connect the word ‘sunset’ with an orange-red colour for the following reason:

  • First, the word “sunset” connects with the concept of a sunset.
  • Next, the concept of a sunset connects with a mental picture of a sunset.
  • Finally, the mental picture of the sunset fills the mind with an orange-red glow.

The mind does all this in an instant, so that the word “sunset” and the colour “orange-red” link seamlessly.

We might postulate that synesthesia involves the same sort of connections, albeit a bit less obvious. For example, the number 2 could trigger a childhood memory of a refrigerator magnet ‘2’ that happened to be red. As time goes by, the connection between the number 2 and the colour red remains even when the fridge magnet is forgotten.

But synesthesia is more than simply association – the number 2 doesn’t just remind them of the colour red. When synesthetes see a black 2, they will tell you that it “really is red”. But is there any way to test how real this mental response is?

In 2001, Ramachandran and Hubbard performed the following ingenious experiment. (See the Wikipedia article on synesthesia for more details.) They presented synesthetes and non-synesthetes with displays composed of a number of 5s, with some 2s embedded among the 5s. These 2s could make up one of four shapes; square, diamond, rectangle or triangle – see the diagram below:

Subjects were asked to identify the hidden shape. If recognising the number triggered a concept that triggered a colour, then the colours wouldn’t appear until after the number was recognised. Thus, if synesthesia is just a subjective, mental connection, then it won’t help a synesthete to find the hidden 2’s.

The results were astounding. Non-synesthetes took about 20 seconds to find the shape; synesthetes took about a second.

How do you explain that? Seeing something that isn’t there is one thing, but having it improve your ability to discern shapes is something else. Its like having an imaginary friend who actually helps with the laundry. (This is why synesthesia isn’t usually classified as a neurological condition, because it is often advantageous to the “sufferer”.) Most explanations involve rejecting the linear processing of sense data we invoked previously. Some researchers have suggested that increased cross-talk between different regions of the brain that are specialized for different functions could explain it.

What if you could train your mind to use synesthesia? What if a piano student who struggles to read music could be taught to see each note on the page as a different colour? Could children be taught to see harmful objects as red and harmless ones as blue? What if you saw interesting blog posts as red, and boring ones as blue?

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At the beginning of this month, New Scientist’s cover story detailed the work of Oxford physicist Joy Christian. He recently released a paper on quant-ph (apparently its a cheap copy of of astro-ph) with the provocative title:

Disproof of Bell’s Theorem by Clifford Algebra Valued Local Variables.

Here’s the idea. Quantum mechanics doesn’t predict deterministic quantities – it only gives the probability of measuring various outcomes. If quantum mechanics is the final word in microscopic physics, then it suggests that, at its most fundamental, the universe is really quite weird – it has a ghostly, fuzzy existence, only making up its mind when observed.

But what if quantum mechanics is just an approximation to a better theory? What if the reason that quantum mechanics only gives us probabilities is that it is ignoring hidden variables?

Enter Bell’s theorem, which states that “no physical theory which is realistic as well as local in a specified sense can reproduce all of the statistical predictions of quantum mechanics.” Roughly, realistic means that reality exists independently of being observed; local means that no physical influence can travel faster than the speed of light. In other words, we either get spooky particles that pop into existence when they are observed, or we mess with causality via ‘spooky action at a distance’, as Einstein said.

Enter Christian. He claims that Bell’s reasoning is correct, but relies on an unjustified assumption. Bell assumed that the hidden variables would commute under multiplication: like ordinary numbers, a x b = b x a. Christian claims that if we instead consider non-commutative variables then Bell’s theorem fails. Such variables are provided by Clifford algebras. (Consult Wikipedia and references therein. I know practically nothing about Clifford algebras, other than that they are particularly useful for dealing with rotations).

There have been critics, replies and further work so the debate is not over. This is either the first step in a major revision and extension of quantum mechanics, or the mathematical equivalent of cold fusion. The whole thing thing almost makes me want to go learn Clifford algebras.

Almost.

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The Surprise Quiz Paradox

I love a good paradox – especially ones I can’t see a resolution to. I’ve run into this one a few times and wanted to look into it further.

The story is as follows. A teacher is worried that her class isn’t working consistently through the term, choosing instead to “cram” on the night before an exam. So she announces to the class that there will be a quiz on the work they will cover this week. It will be a surprise quiz, sometime next week. The students, not knowing which day the quiz is on, will not have the option of staying up the night before. They must work consistently so that they are always prepared for the quiz.

One enterprising student, however, quickly realises that the test cannot be on Friday. His reasoning is sound – suppose that a student hadn’t worked consistently through the week. Suppose it is Thursday night, and the quiz hasn’t happened yet. Then the student knows that they should stay up all night and cram, which defeats the purpose of the surprise quiz. Thus, the quiz won’t happen on Friday.

So far, so good. But suppose that it is Wednesday night, and the quiz hasn’t happened yet. The quiz must be on Thursday or Friday. But we just saw that it won’t be on Friday. Thus it must be on Thursday. So the student should stay up all night cramming. Which defeats the purpose of the surprise quiz. Thus, the quiz won’t happen on Thursday.

You can see guess what happens next. On Tuesday night, the student would know that the quiz was on Wednesday. On Monday night, the student would know the quiz was on Tuesday. Thus, the student knows that the exam is on Monday, so it is obviously not a surprise quiz. The student concludes that there is no such thing as a surprise quiz.

On Tuesday, the teacher hands out the surprise quiz. The student is, frankly, surprised.

Any mathematicians out there will recognise an induction. Mathematical induction is a form of proof that works like this:

We are attempting to prove a set of statements U(n), where n = 1,2,3 … For example, U(n) could be the mathematical formula:

U(n): 1 + 2 + 3 + … + n = (n2 + n)/2

Since we have an infinite number of statements, we can’t check them one-by-one. Instead, we line them up like dominoes, and then push the first one over. More precisely, we prove the following

Step 1: If U(n) is true for any particular value of n (say, n = k), then it is true for the next value of n (i.e. n = k + 1).
Step 2: U(n) is true for n = 1 i.e. U(1) is true.

Step 2 says U(1) is true. Thus, by Step 1, U(2) must be true, because U(1) “knocks it over”. But then, by Step 1, U(3) must be true, because U(2) knocks it over. And so on.

Returning to the surprise quiz paradox, a number of the discussions of the paradox on the net (e.g. here and here ) suggest that any attempt to put the paradox in the form of an induction will fail because the term “surprise” cannot be given a precise, mathematical meaning.

However, such a formulation has been given here, in one of the comments. The problem is set out as follows:

Premise 1: On exactly one day out of the next n days, there will be a quiz.
Premise 2 (the “surprise” requirement): On the evening of day k (given that the quiz didn’t happen on days 1,2,…, k), there does not exist a proof that the quiz will be on day k+1.

The claim, then, is that these premises are inconsistent. This is shown as follows.

If the quiz was to occur on day n, then on the evening of day n – 1, there would be a simple proof that the quiz would occur on day n:
Proof:

  • The quiz must occur on one of the days 1,2,3 … n (premise 1)
  • The quiz did not occur on days 1,2,3 … n-1 (by assumption)
  • Thus, the quiz must occur on day n.

Since premise 2 forbids such a proof, the quiz cannot occur on day n.

Now the induction shows itself. If premise 1 holds, but premise 2 requires that the quiz cannot happen on day n, then we can formulate a new premise:

Premise 1a: On exactly one day out of the next n – 1 days, there will be a quiz.

But Premise 1a, along with premise 2, can be used to formulate this premise:

Premise 1b: On exactly one day out of the next n – 2 days, there will be a quiz.

And so on. Thus, we reach the conclusion that the announcement of a surprise quiz to the class is self-defeating.

(At this point I am reminded of a passage in John Barrow’s book, “Impossibility”, where he presents a similar argument along the lines that you can only predict the future if you keep the prediction to yourself. That is a topic for another post.)

At this point, I will not attempt to resolve the paradox, because I’m not sure what the resolution is. I’m told that many philosophers have written papers on this very paradox in its various forms (other versions include a man sentenced to hang). I’ll simply leave you with a list of references to further investigation, and invite you to provide your solutions.

Wikipedia: Unexpected hanging paradox

Stanford Encyclopedia of Philosophy: Epistemic Paradoxes

PrawfsBlawg – especially the discussion in the comments.

Cornell course notes: Discrete Mathematics in Computer Science – Induction

“What is a proof?” – Robin Cockett, University of Calgary

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UK Young Astronomer’s Meeting

It’s on again! Despite the name, this meeting is targetted at early-career astronomers (students and post-docs) of all ages and will be held on Friday the 7th of December at the Royal Observatory, Edinburgh. In fact, it’s been advertised for a while and registration has officially closed, but word on the street is that there are spaces for anyone keen enough to email the organisers.

Info at http://www.roe.ac.uk/ifa/YAM.

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