Proponents of the Beijing Olympics assured the world seven years ago that the human rights situation was improving in China, and that everything would be all right by 2008. Awarding the games to the communist regime was even supposed to “help” this to happen. This was an irresponsible lie even back in 2001, propagated by people who should have known better. Of course the communists will always continue to disrespect human rights, simply because at the very moment that they are removed from power via a democratic election, they will be held responsible for their past actions. And when you know you are a bandit, this is the last thing you want to happen.

But the rest of the world can still make a difference. Athletes: don’t go to Beijing. Make a stand, even if it’s the only time you ever make a political stand in your entire life. People everywhere: make a note of the companies that are official sponsors, and stop buying their products. Sell any shares that you own in these companies. You can always buy them back later at a lower price

And don’t forget to tell everyone why you are doing this.

And a personal note to Mr. Dalai Lama the 14th: I very much appreciate and agree with your total commitment to a non-violence policy. But you are doing a disservice to your people by not demanding full sovereignty for your country. And please do whatever you can to stop all this superstitious reincarnation bullshit. This is truly opium for the people, and an obstacle for anyone interested in spiritual liberation.

Time is what prevents everything from happening at once.
Space is what prevents everything from happening to me.

I won’t address the dimensionality of time here, I’ll save that for a later post maybe. Let me just briefly remark that I think the observation that time is 1-dimensional is a consequence of the fact that only a 1-dimensional time makes it possible for different observers to share an objective reality where causality is possible. So let’s stick with one single time dimension, and consider space and how it’s possible to see things at a distance.

In order to see things clearly at a distance, it must be possible to transport light signals across space with little or no distortion. Since light is a type of electromagnetic wave, the propagation of waves must allow such transporting of signals. The wave equation in n-dimensional space looks like this:

where c is the propagation speed of the waves, and represents the stuff that is “waving”, whatever it is. More precisely, is a function of space and time coordinates, and the value of that function can be any mathematical object for which partial derivatives are defined: for example a simple scalar, a complex number or quaternion, a vector, a matrix, and a lot of other things.

Now imagine a wave that propagates from a point-like source, symmetrically in all directions. The symmetry allows us to write the wave equation in terms of only r and t, where r is the distance from the source. This spherically symmetric wave equation looks like this:

In order to transport an arbitrary signal, the wave function must hold for all solutions of the form

where f is an arbitrary function that represents the shape of the source signal, and is some fixed function that is independent of f. The factor attenuates the signal as it spreads outward from the source. Since the wave equation must hold for any f, it must in particular hold for the case , which gives the condition:

which, ignoring irrelevant constants and the pathological case n=1, has the solution

which gives us for the general case, since is independent of f.

Inserting this into the spherically symmetric wave equation produces, after simplifications,

where and are the first and second derivates of the function with respect to its parameter. After further simplification the final result is simply n=3. Voila! So faithful propagation of arbitrary signals via spherically symmetric waves is only possible in 3 space dimensions.

Notes:

• I only bothered with so-called “retarded” waves. Since the wave equation only uses the second derivative with respect to t, any linear combination with a solution that uses -c instead of c is also a solution. Solutions of the form f(r+ct) are called “advanced” waves and are the time-inverse of the normal retarded solutions.

• Spherical symmetry is an idealization for electromagnetic waves, since the electric and magnetic fields (or the Faraday tensor if you prefer) have directions in space which are transverse to the direction of wave propagation. See The Hairy Ball Theorem. This doesn’t affect our conclusion, however.

• There are some speculative exotic theories (string theory etc) that postulate several extra “compact” space dimensions in addition to the familiar ones that we already know about. This does not affect our conclusion either, since such hypothetical dimensions do not contribute to any degrees of freedom in a macroscopical wave equation.

The perpetrator Anton Nilson emigrated to the Soviet Union after he was released from prison, and spent a decade participating in Lenin’s Red Terror against liberals, social democrats, anarchists, and anyone else who didn’t submit to Lenin’s totalitarian rule.

All this happened a 100 years ago, but some people still refuse to learn the lesson that communism is pure evil. Two influential swedish social democrats Olof Palme and Ingvar Carlsson, both who served as prime ministers of Sweden, actually celebrated the terrorist Anton Nilson on various occasions. Maybe they didn’t know who Anton Nilson was, and the crimes he committed. Or maybe they did? Either way it’s very very scary to know that such blood-thirsty people have actually led governments in my country.

Sedan har det dykt upp ett antal återkommande begrepp i debatten som jag tror det är nyttigt att skärskåda:

• “Teknikneutral” betyder att man istället för att bara spana på radiosignaler som avsändaren har sänt ut för spridning överallt i etern, vill kunna spana på alla meddelanden som skickas överhuvudtaget, oavsett teknik. Det tycker jag är helt OK så länge samma grundförutsättningar gäller som tidigare: FRA bör få spana på allt som sänds via kabel och som sprids överallt, d.v.s. all annonserad multicast (RFC1112 m.fl.), allt som skrivs på publika websajter, och även allt som är allmänt åtkomligt via FTP och Bittorrent naturligtvis. För detta krävs ingen massiv avlyssningsapparatur. Det räcker med en vanlig internetanslutning. Med FRA:s budget så skulle dom kunna köpa sig en miljon vanliga internetanslutningar. Om jag var FRA-chef så är det precis vad jag skulle göra.
• “Kontrollstation” betyder i praktiken att regeringen och FRA själva ska utvärdera i efterhand huruvida regeringen och FRA har missbrukat sina befogenheter. Hallå?
• “Integritet” betyder på FRA-lobbyisternas nyspråk att FRA först öppnar ditt brev, tar en kopia på det, granskar kopian maskinellt, sparar information om innehåll och metadata, och till sist “förstör” den sparade informationen efter att ha läst och tolkat den och konstaterat att den för närvarande inte är intressant för FRA:s kunder.

Jag kanske ska påpeka att jag inte är principiellt mot avlyssning av kabeltrafik. Det finns situationer där detta är befogat och önskvärt. Men det är en stor skillnad mot FRA-lagens massavlyssning av alla. Och det kom som en kalldusch för mig att hela den borgerliga alliansen (med undantag för riksdagsledamot Camilla Lindberg) accepterade massavlyssningen. Jag har röstat anti-socialistiskt i alla val jag har deltagit i, men från och med FRA-lagen kan jag inte längre rösta på något av allianspartierna. Att rösta för ett visst parti innebär att man som individ legitimerar dess politik, och därför är det helt uteslutet för mig att någonsin mer rösta för något av de partier som klubbade igenom denna lag.

That is not what this post is about, though. Instead I want to point out the difference between sums and sums, or rather the confusions that occur due to the different interpretations that can be applied to the word “sum”.

An arithmetic expression like 1+23+99+42 is typically called a “sum”. But “the sum” of the numbers in that expression is 165 which is a number, not an expression. The expression 1+23+99+42 has distinct parts: these parts are the numbers 1, 23, 42, 99, and the operators that connect those numbers. The sum 165 on the other hand is just one single part all by itself. Or perhaps it consists of 165 parts which are all copies of the number “1″. Or perhaps it consists of the prime factors 3, 5, and 11 multiplied together. The possibilities are endless, and it doesn’t really matter since all these “perhapses” refer to implicit structure, not to anything explicit in the original expression.

The resulting confusion is entirely semantic. When we say “2+2 = 4″ we only mean that 2+2 and 4 have values that are identical, not that the expressions themselves are identical. They can’t be identical since the expressions have different structure. Things that are identical can’t have any properties that are non-identical (cf. Leibniz’ law).

The same confusion reappears in many other contexts. One example is “the will of the people”, which is a straight-forward concept when it refers to a collection of varying individual preferences, but it’s something completely different when it refers to a single collectively held preference. The latter is not even well-defined in the general case, due to a mathematical result in social choice theory known as Arrow’s theorem. This has far-reaching consequences for politics and ethics.

Having built-in Latex support is nice!