Copernicus was wrong (maybe)


The Hubble Deep Field South looks very similar...
The Hubble Deep Field South looks very similar to the original HDF, demonstrating the cosmological principle. (Photo credit: Wikipedia)

No, I haven’t taken leave of my senses and decided the Sun moves around the Earth (but here is a pop quiz for all of you laughing at that idea – can you think of a simple experiment that would prove to a 10-year-old that the Earth moves around the Sun?).

In fact the issue here – the so-called Copernican Principle, or in its grander form the Cosmological Principle – was almost certainly not Copernicus’s view at all. But his paper – De Revolutionibus – opened the door to it, and indeed to the positivist idea of science in general.

The Copernican Principle states that there is nothing special about the position of Earth, the grander Cosmological Principle states that, at a sufficiently large scale, the universe looks the same in all directions – including (and this is important for what is coming) from where we are. In other words not just Earth is nothing special, but nowhere is anywhere special.

But what is that is all wrong? A fascinating article in this week’s New Scientist looks at just this.

That there are limits to the Cosmological Principle is obvious – the world is not smooth, even at some very large scales (look at the night sky – most of it is dark).

The standard scientific response to this is to state that at a sufficiently large scale – around 400 million light years – the matter density between galaxies and the inter galactic void evens out. But that assumption is based on our observations of our locality: yet what if, actually, we were in an atypical part of the universe? The atypicality could even be meta-typical (in other words we could have a super-Cosmological Principle but accepted that the universe was lumpy.)

This matters because our cosmological models are based on the assumption that the Cosmological Principle is correct and that therefore we are typical observers of the universe: hence the phenomena we see are ones that would be seen by any observer and are therefore artefacts of the universe’s physical nature and not our observational position.

So, for instance, we have data that appear to show that the expansion of the Universe is speeding up. We do not know why this is, so we call it “Dark Energy“. But what if the apparent speed up was because, actually, the universe was not isotropic (did not look the same in all directions) and the additional mass in one direction was impacting on the perceived rate of expansion of the universe?

The beauty of this question is that asking it does not mean challenging Einstein’s General Relativity – it’s not an exercise in metaphysical speculation but an argument firmly within the positivist realm bequeathed to us by Copernicus in the first place.

And finally…It is actually quite tough to devise a simple experiment to show that the Earth revolves round the Sun – but the orbits of Venus and Mercury are probably the best examples: these planets are never in opposition to the Sun. Though binoculars to observe the planets’ phases are probably needed to fully escape any Ptolemaic theories’ grasp.

Finally, finally… this book – Can You Speak Venusian? A Trip Through the Mysteries of the Cosmos – was very funny when I read it about 35 years ago, whether it has stood the test of time I am not sure.

Racey code can damage your (mental) health


Running the Hackney half marathonI’ve had a tough week. Ran a half marathon (quite badly, but I finished – see the picture) on Sunday, hobbled around York University (blister on my foot) on Monday before returning to London and coping with the added stress of a new job.

All the while the failure of my latest piece of code to work was annoying me too – eating away at me and making me wonder if the whole PhD thing was a bit of Quixotic nonsense or even Panzan stupidity.

Anyone who has chased down a bug for days will know the feeling – the root cause must be out there somewhere, even if you have been through your code dozens of times and can see nothing wrong.

Sitting on a long commute to my place of (temporary) work this morning I realised that my problem could only be down to one thing – a race condition.

I am emulating an LRU/2 type page replacement policy – with two page queues – a “high” and a “low”: pages are initially assigned to low, but can be pushed into high on a second reference. Pages only leave via low (they can be pushed out of high into low if we run out of space in high and so on).

With one thread running there was no problem, but when a second thread came on board the application fell over – and I realised it was because the manipulation of the LRU queues was not atomic.

And, a few dozen code changes later (done on the reverse journey back to London) – and some accounting for C++ template strangeness (it seems that an iterator across a map, means the map is not const – that took me a while to work out) and the problem has gone away and suddenly academic endeavour feels worth it again.

A vision in yellow


Some locations in the London Borough of Hackney.
Some locations in the London Borough of Hackney. (Photo credit: Wikipedia)

Yesterday I ran in the Hackney Half Marathon – my first ever race at that distance. It was a very hot day and I ran a poor race – I had aimed to keep a 5’30” pace per kilometre but in fact ran the first 7km faster than that and ran the first 10km three seconds faster than the 10km I ran last month: not very clever.

I ended up paying the price – going into meltdown at about 11 miles, though I did finish and that’s something to be proud of.

But it is fair to say I was in something of a state by the time I crossed the line, in 2:15:45 (my hope had been for something around 1:55).

Desperate for some sort of energy restoring drink (I missed the energy drink handout on the way round, possibly because by that point – at 8 miles – I was already approaching zombie runner state) I flopped to the ground and drank the “energy” drink I’d been given (actually I think it had little calorific value and the energy it claimed verged on new age mysticism).

Only feeling slightly better I decided to head off to the bag collection point, knowing I had two genuinely sugary Lucozade bottles in my bag.

As I stood up I seemed to get cramp in every muscle in both legs and then – strangely – my vision was washed a sort of sepia yellow.

I didn’t feel faint – particularly (though it’s fair to say I wasn’t fully compos mentis either) – but I did think, “oh here we go”. But the moment passed quite quickly – especially as I knew I had to complete standing up to end the pain in the legs – and I staggered off.

Since then I have read that this is quite likely to be caused by low blood pressure – or, less likelier, by hypoglycaemia. Low blood pressure might seem an odd cause in that I’d just undertaken some extremely strenuous exercise – but I dare say blood pressure in my brain was quite low, with blood diverted to my limbs, and suddenly lowered further by the effort of standing.

It was a stark reminder, though, of the way that our vision of the world is completely internally created. “Colours” are purely perceptional, not real.

The intelligent computer has arrived


The Turing Test, version 2, as described by Al...
The Turing Test, version 2, as described by Alan Turing in “Computing Machinery and Intelligence” (Photo credit: Wikipedia)

At least, that is the claim being made by the University of Reading and it seems to have some credibility – as a computer entered into their annual “Turing Test” appears to have passed – convincing a third of the judges that it was a human and not a machine.

This definition of intelligence relies on Turing’s own – in his famous 1950 paper “Computing Machinery and Intelligence”  (well worth reading, and no particular knowledge of computing is required) – a definition I like to think of as being summarised in the idea that “if something looks intelligent it is intelligent”: hence if you can make a computer fool you into thinking it is as intelligent as a 13-year-old boy (as in the Reading University case), then it is as intelligent as a 13 year old boy.

Of course, that is not to say it has self-awareness in the same way as a 13-year-old. But given that we are struggling to come up with an agreed scientific consensus on what such self-awareness consists of, that question is, to at least a degree, moot.

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When Apple made mainframes


English: The logo for Apple Computer, now Appl...
English: The logo for Apple Computer, now Apple Inc.. The design of the logo started in 1977 designed by Rob Janoff with the rainbow color theme used until 1999 when Apple stopped using the rainbow color theme and used a few different color themes for the same design. (Photo credit: Wikipedia)

These days we think of “mainframe computers” as lumps of “big iron” computing power that are typically designed to handle lots (millions) of simultaneous pieces of data and record manipulation.

But the term did not originate in that way – a “mainframe” computer was simply one where all the key components of a “Von Neumann machine” were inside one box – a “main frame”.

So, it is interesting to see Apple advertising from 1978 (see page 16 here) describe the Apple II computer – oh how much I would have loved to have had one of those – in these terms:

Apple is a fully tested and assembled mainframe computer.

If that sort of approach from a company associated with miniaturisation is long gone, one approach has very much remained.

The main body of the advert states:

Just take an Apple home, plug it in, hook up your color TV and any cassette tape deck – and the fun begins.

But in smaller type at the very end of the ad we are told (emphasis added):

Apple II plugs into any standard TV using an inexpensive modulator (not included).

Still, if you could afford one of these devices (Apple seems to have always charged about £1000 for its computers) then you could probably have afforded the extra cost of the modulator. And they were lovely machines.

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A better demonstration of the product rule


Inspired by The Theoretical Minimum: What You Need to Know to Start Doing Physics: here’s a better proof/justification for the product rule in differential calculus than the one I set out here last month.

We will start with what we will treat as an axiomatic definition of the differential of the function y=f(x):

\frac{dy}{dx} = \frac{df(x)}{dx} = \frac{f(x+\Delta x) - f(x)}{\Delta x} as \Delta x \rightarrow 0

In this case we have y=f(x)g(x), so \frac{dy}{dx} = \frac{f(x + \Delta x)g(x +\Delta x) - f(x)g(x)}{\Delta x}

From our definition we can substitute for f(x+\Delta x) and g(x + \Delta x) and simplifying our notation for presentational reasons so that \frac{df(x)}{dx} = f^{\prime} etc:

f(x+\Delta x) = f^{\prime}\Delta x + f(x)

g(x+\Delta x) = g^{\prime}\Delta x + g(x)

Giving (after dividing through by \Delta x ):

y^{\prime} =f^{\prime}g^{\prime}\Delta x + g(x)f^{\prime} + \frac{f(x)g(x)}{\Delta x} + g^{\prime}f(x) - \frac{f(x)g(x)}{\Delta x}

=f^{\prime}g^{\prime}\Delta x + g(x)f^{\prime} +g^{\prime}f(x)

As \Delta x \rightarrow 0 the first term falls to zero and so we are left with:

y^{\prime}=f^{\prime}g(x) + g^{\prime}f(x)

Which, of course, is the product rule.

Update: See this most excellent comment from Professor Rubin.

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My start up idea


I was travelling on a train today when someone sat opposite me wearing a tee-shirt that purported to show the cover of a 1950s motoring magazine and for some reason – perhaps the typeface used – I immediately thought of the much loved “Practical Computing” magazine.

I then entertained the fantasy that – with programming about to roar back into the English school curriculum – this was the moment to think about once again launching a magazine aimed at those with a mathematical/scientific interest in computing – people who thought programming could be fun and who wanted to know how to model four dimensions in three-dimensional space and so on.

Just an idea.

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