Suppose I am standing on a train platform and the train is travelling at 20km/h from my reference frame. If it goes 10km/h faster (from my reference frame), then it will be travelling at 20km/h + 10km/h = 30km/h from my reference frame. Standard addition. — Andrew M

If you are standing on a train platform and the train is travelling at 20km/h according to your reference frame, and the driver switches the headlamp on and emits light at the speed c, then in your reference frame you will measure the light as travelling at c, and not at c + 20km/hr. That was the puzzling observed phenomenon in the Michelson-Morley experiment that led to Special Relativity in the first place. If something is travelling at the speed of light and it emits light, everyone will measure that light as travelling at exactly the speed of light, whatever their reference frame. No standard addition ever works with light. c + c = c, and that's the observation that led to the equation. I don't see how and why it can be dismissed as only a superficial feature of different reference frames. Please enlighten me! (Pun intended...)

As for physics, aren't conservation laws just the same sort of thing? This much stuff is always this much stuff. It can't be not equal to itself. No matter what you do, you can't make it not equal to itself. — petrichor

Thank you for your interesting and intelligent discussion Petrichor. The problem I have with it is that the whole point of coming up with the Einstein-Poincaré equation in the first place was to explain the strange but true observed fact that c + c = c. By all of your discussion it would be absurd for that to be true, but it is, and it is precisely that absurdity that led them to come up with the solution. When you add c and c together using that equation and not c + c = 2c, you get the required answer, that c + c = c, which is true. And that's the observed fact that underlies my whole discussion.
I've been thinking about this all day and I think the answer might be that it is not the reference frames that matter, but that no actual 'thing' is being added when you add velocities. Taking two apples and physically putting them with two apples and counting the resultant number of apples is qualitatively different from a moving object accelerating and doubling its previous speed, or something travelling at the speed of light emitting light which is observed by someone at rest relative to the light as travelling at the speed of light. In these instances, nothing is actually being added to anything in reality, so that saying that going from 2 mph to 4 mph is 'adding' 2 mph to the original 2 is just a human interpretive parsing of a phenomenon that didn't actually involve any addition. So that far from finding a genuine counter-example, I have confused two things that are different, made a category mistake perhaps by saying that adding velocities is the same type of thing physically as adding apples, when it isn't. I don't know, I'm not sure about this at all, I'll have to think about this one a bit longer...

This veneration of 'the Greeks' is very cultural, very 'Western Europe classical education'. Vast amounts of paintings, plays and poems have been derived from Greek and Greco-Roman myths, and they have long been held up in the schools (particularly British public schools) as the Golden Age of art and philosophy - even the idea of a Golden Age being Greek. But in fact the corpus of Greek philosophy doesn't make anything like as big a contribution as the venerators would have you believe. Aristotle's Ethics are all about the social acceptance or non-acceptance of the man-about-town; Kant's Categorical Imperative would have bewildered him. Plato's Republic is the first great Fascist tract. The 'Academy' was a garden in Athens where small numbers of people used to meet. Only a few sentences of Democritus have survived, and his arguments about Atom and Void are interesting, but they are hardly nuclear physics or cosmology.
As SophistiCat has correctly pointed out, the Babylonians were much better at maths, they knew more than Euclid did, and in fact human civilisation first went past the lost mathematics of the Babylonians in the Enlightenment. 'Pythagoras's theorem' of geometry isn't by Pythagoras at all, it's Babylonian, as was all of the foundational material that the Pythagoreans contributed to the development of Greek geometry, and they got it from the Persians.
Furthermore, the lost books of 'the Greeks' were mainly transmitted to modern Europe via the great Muslim civilisation of the early medieval era, via such melting pots as the Sicilian Norman kingdom of Roger II. The great Caliphate of Baghdad under Haroun al-Rashid and his successors produced some of the finest works of science and philosophy ever written before modern times, far outstripping their Greek influences - at a time when the Europeans were a pack of Christian savages.
Don't believe the propaganda. 'The Greeks' were a small group living in Athens during a period of about 50 to 100 years; Greek culture, language and belief spans a huge timescale from the Mycenaean conquest of the Agean from the Cretans, through the Alexandrian Asian Greco-Buddhist kingdoms of Bactria and Gandhara, important for Buddhist philosophy, to the thousand years of Eastern Orthodox Christian Byzantium, which contributed nothing to philosophy at all. It is certainly true that the Late Bronze Age Babylonians and the Islamic Golden Age scholars were more important than 'the Greeks', for reasons of astronomy as well as maths and philosophy. I can't think of anything at all that I find important in modern science and philosophy that was said by a Greek; contrariwise, where do you think 'algebra' comes from? Just because the word 'metaphysics' comes from Greece doesn't mean that Greek metaphysics is important, while there is a very good reason for so many of the stars in the night sky having Arabic names like Betelgeuse.
I went to a British public school that venerated 'the Greeks', and today I have as much contempt for that veneration as I have for so much of the British public school system. Americans have inherited that tradition. It is a false one. I recommend you drop it.

Popper's excellent discussion is related to the concept of discrimination. We should of course discriminate at all times between right and wrong, good and evil, abuse and decency, and we should only make sure that we are not applying false discrimination, such as between skin colours as if they were good and bad. Voltaire's famous quote, "I disagree with what you say but I will fight for your right to say it", should be infamous. In England at the moment huge numbers of newspaper editors, politicians and other public figures are getting away with serious abuse crime, incitement to hatred and driving desperate people to suicide, then posturing about claiming free speech and the freedom of the press. The freedom of the press has only ever meant freedom from control by abusive power, it doesn't mean freedom to incite racist violence, which should be prosecuted. If you are saying foul things, then my quote is "I disagree with what you say, and I will fight to have you put in a cage where you can no longer hurt people".

In ‘The Open Society and its Enemies’ Popper has a very specific thing to say about Freud. He presents an example, or rather two versions of one scenario. In one, a man is walking along a riverbank when he hears the cry of a child, swept into the river by the swift current and drowning. The man reacts instantly, flinging himself into the dangerous flow and saving the child at great personal risk. In the second scenario, a man is walking along a riverbank when he hears the cry of a child, swept into the river by the swift current and drowning. He stops his ears to the suffering cries, and, as the child sinks beneath the water, walks grimly on.

In the first scenario, the Freudian explanation is that the man is suffering from unresolved conflict issues with his father, and needs to prove himself and his masculinity. To do this, he risks life and limb to do a heroic deed. In the second scenario, the Freudian explanation is that the man is suffering from unresolved conflict issues with his father, and needs to prove himself and his masculinity. To do this, he forces himself to be aggressive and uncaring, and deliberately rejects the child to show how strong-willed and manly he is.

Popper’s point is that Freudian analysis and psychological interpretation is never predictive, and consequently has nothing whatsoever to tell us. It is a belief system, one that is used to interpret whatever happens according to its own rules. Popper goes on to mention the case of the Revolutionary Socialist, who sees in every single line of the newspaper, from the front page headline to the adverts at the back, clear proof of the class struggle and the malignant effects of capitalism. We could add to this discussion the case of the American christian fundamentalist, who reads exactly the same newspaper and sees clear proof of the war between angels and demons in every line.

So it isn’t that Freud’s claims are unfalsifiable. The point is that they are meaningless. They are a belief system though which you see the world and interpret it, just as the revolutionary Socialist and the christian fundamentalist do. It doesn’t matter what people do, how they behave, what they say their motivations and desires are, because whatever that is there will be a Freudian interpretation that can be imposed later and believed by Freudians.

Freudian psychology never predicts, which would make it falsifiable, it just retrospectively imposes an interpretative belief system. One person says that someone did something because they were unconsciously struggling against their father for the love of their mother, another person says that they did it because the devil made them do it, yet another says it is the driving historical force of capitalism that made them do it. Popper’s point is that this is all just meaningless, and has nothing to offer at all.
That’s his take on Freud anyway. His take on Marx occupies most of a long volume and is much more complex. And that’s because he dismisses Freud with a sneer, but he considers Marx to be one of the Open Society’s enemies, and consequently a lot more important.