Saturday, March 01, 2014

Consistency arguments in theoretical physics

Originally written with a different audience in mind

Arguments involving the internal consistency of a theory or a system of ideas have assumed increased prominence in modern mathematical and theoretical physics as well as in other branches of human thought that were or are being inspired by physics.

It is possible to underestimate these arguments; and it is possible to overestimate them as well. And indeed, numerous thinkers err in both ways.

The goals of this essay are to explain the basic logic and assumptions behind these arguments, to present several examples, to show that these arguments are often equivalent to logical steps that may be described without the word “consistency”, to sketch a probabilistic argument suggesting that theories passing a consistency check are more likely, and to clarify both basic fallacies.

The first basic fallacy is to dismiss the arguments rooted in the consistency altogether; we will try to see that the reasoning based on consistency has often been useful, successful, and demonstrably valid. The second basic fallacy, one discussed at the very end of this essay, is the assumption that the internal consistency of a system of ideas proves (i.e. is sufficient for) the validity of such a system.

Consistency and proof by contradiction

At the level of mathematical logic, the character of arguments involving consistency is rather trivial. If proposition P implies proposition non(Q) i.e. if it can be proven that P → ¬ Q, we may say that P and Q are inconsistent.

Equivalently, the proposition “P and Q are inconsistent” may be expressed as Q → ¬ P i.e. it is possible to prove non(P) assuming Q. The equivalence of the two implications holds and is known as “transposition”; the equivalence is a valid rule to exchange the antecedent with the consequent of the conditional statement and to negate both propositions at the same moment. Both propositions are equivalent to ¬ (P ∧ Q), too.

All of mathematics and physics depends on mathematical logic which is why these elementary logical rules and principles are omnipresent in exact sciences. For example, every proof by contradiction in mathematics is nothing else than a proof exploiting the logical consistency (more precisely, inconsistency) of a set of assumptions.

To see a simple example, let us prove that there are infinitely many primes. The proof by contradiction is a proof showing the internal inconsistency of the usual assumptions about integers, the usual definition of primes, and the (false) assumption that there are finitely many primes. The proof proceeds as follows: If there were finitely many primes, one could calculate their product and add one. The resulting sum (“the sum”) would be greater than any of the (finitely many) primes so it couldn't be one of the entries on the original (finite) list of primes. However, according to the definition of a prime, the sum would be a prime as well because it is not divisible by any other prime i.e. because it is only divisible by one and by itself.

That's a contradiction: The sum is a prime and it is not a prime at the same moment. It follows that the original set of assumptions is logically inconsistent. At least one of the assumptions has to be false. By inspection, it is not hard to see that the false assumption is the assumption that the number of primes is finite.

May all mathematical descriptions fail?

It is perhaps necessary to briefly address two possible general objections against arguments using consistency – or against any argument involving the language of mathematics: 1) the world doesn't have to be consistent; 2) the world may admit no mathematical description at all which would apparently imply that every mathematical argument is invalid, too.

The first suggestion is impossible because a well-defined Yes/No question in Nature can't have both contradictory answers. Individual human beings may be confused and hold beliefs that are inconsistent. But it is always possible to verify and correct the system of propositions and to see that some assumptions are untrue or some steps in the proof of contradiction are invalid. In this way, we reach a system of proposition that avoids demonstrable contradictions. Such a “theory” is superior in comparison with an inconsistent one, to say the least.

As far as the general criticism of mathematics is concerned, we must realize that the language of mathematics (and mathematical logic in particular) is being used in physics and in natural sciences because it is the most accurate language that avoids confusions and contradictions. Aristotle has pioneered mathematical logic of the Western style and the subsequent improvements have simply eliminated all sources of confusion. But in principle, every language used to study Nature that is sufficiently refined and that is promoted to a sufficiently controllable form may be seen to be equivalent to mathematical logic.

On the other hand, the identification of the “reality around us” with mathematical objects similar to numbers and functions doesn't have to be counted as an independent, questionable assumption. Instead, it may be described as a general result of the scientific method. A priori, only propositions about our direct observations or perceptions (i.e. “we saw a full moon 28 days after the previous one”) may be considered “valid axioms”. Numbers and other mathematical objects have appeared in our theories describing Nature simply because all non-mathematical attempts to explain the whole body of our observations have been falsified or at least found significantly incomplete. In other words, numbers, functions, and other mathematical structures entered physics because they seem necessary for a viable explanation of the logical relationship between (propositions about) our direct perceptions or observations.

The extra assumptions commonly used in physics

Previously, we reviewed a proof of the infinite size of the set of primes. The same proof may be formulated in many ways, using various words. And we don't mean just the translation from one human language to another; or a translation from “plain English” to the language of mathematical symbols. After all, the phrase “proof by contradiction” doesn't contain the word “consistency” at all. It is not hard to modify the wording of the proof so that one avoids the word “contradiction”, too.

Given the ubiquity of proofs by contradiction in mathematics, one might be surprised that physicists' arguments that boil down to logical consistency may be questioned at all. After all, the dismissal of the requirement of logical consistency is tantamount to the denial of mathematical logic as a whole. And rational reasoning in mathematics, sciences, or other fields would be impossible without mathematical logic incorporated into the foundations.

However, there is something special about the consistency arguments in physics, something that goes beyond the generic proofs by contradiction in mathematics. What are the special new ingredients?

The first ingredient may be the empirical data. The compatibility (or consistency) of a hypothesis with the empirical data is the key criterion that decides about the fate of the hypothesis according to the scientific method. If the observed data disagree with predictions of a hypothesis, the hypothesis is falsified. But in this essay, we don't want to talk about the empirical data. Because the references to the internal consistency are characteristic steps used by theorists, not experimenters, we want to focus on the internal consistency i.e. consistency of assumptions that are parts of a theory. What are the extra ingredients we routinely encounter in physics?

One of them involves a special class of the physical assumptions that often belong to the “list of axioms” whose consistency or inconsistency is being analyzed. The other ingredient is the physical nature of the contradictions that are being arrived at in the process of the proof.

To be more specific, the important physical assumptions that may result in contradictions (inconsistency) with others are symmetries (or, almost equivalently, the independence of the predictions on some transformations or changes of conventions used to observe the system or to talk about it); the fact that probabilities must be numbers between 0 and 1; the fact that the sum of probabilities of all mutually excluding options equals 1. The typical last step of the contradiction in such physics proofs is a calculation of the value of a continuous physical quantity that should be equal to two different numbers according to two different arguments.

Examples of consistency arguments before string theory

Since the beginning of the 20th century, a common theme in theoretical physics was the “reconciliation of two pre-existing foundations” that resulted in the justification of a rather specific new theory. This sort of reasoning is an example of a consistency argument. Einstein's special theory of relativity; the uncertainty principle underlying quantum mechanics; Einstein's general theory of relativity; quantum field theory including its general predictions such as the existence of antimatter; and string theory may all be viewed as results of such “reconciliations”.

In each case and in many others, one may deduce a rather specific new theory out of some previous, less accurate or less complete theories or assumptions, because the older assumptions are compatible but just barely so. If they were completely incompatible, they couldn't simultaneously hold at all. On the other hand, if they were easily compatible, they wouldn't have much to do with each other and we wouldn't learn anything out of their combination (we don't learn much from the fact that the leaves are green and that the Sun is round). Modern physics often falls in the middle. It almost looks like we may derive that the assumptions are incompatible except that there is always a loophole. The assumptions are compatible but only if the loophole is exploited. Because we know that the assumptions hold, their consistency implies that all the propositions we summarized as the “loophole” – along with all of their implications – are true, too.

The special theory of relativity may be derived from two postulates: 1) the equal form of physical laws relevant for all inertial observers (those that are in a the state of uniform motion with respect to each other), and 2) the constancy of the speed of light (independently of the speed of the source of light as well as the observer). These two postulates look nearly inconsistent because the first postulate used to be believed to imply that the relative speed between two objects adds up i.e. is equal to \(u+v\). However, the second postulate requires that the relative speed of light shouldn't be \(c+v\); it should still be just \(c\). Einstein realized that these two postulates are compatible (and mechanics and electromagnetism+optics are consistent) if a particular loophole is exploited. The loophole involves the insight that the simultaneity of two events depends on the observer, too. Galilean transformations are replaced by the Lorentz transformations and the composition formula for two speeds becomes \((u+v) / (1 + uv / c^2)\). All the other consequences also follow. When the speed of objects increases, their length is being contracted, the time is slowing down, the masses are increasing, the speed of light is the maximum allowed speed of material objects or the information, and so on.

Similarly, the Heisenberg uncertainty principle of quantum mechanics is the necessary loophole that is needed to reconcile the fact that particles sometimes behave as pointlike objects; and sometimes they act as waves (e.g. in the double slit experiment). Mathematically, the uncertainty principle is translated to a nonzero value of the “commutator” \(xp – px = i\hbar\) and more or less all the exotic (as well as mundane) phenomena associated with quantum mechanics follow from this fact.

The general theory of relativity (found by Einstein in 1915) follows from the consistent reconciliation of the insights of the special theory of relativity (1905) and from the existence of the gravitational force (as previously understood by Newton). The gravitational influence seems to immediately change when the source of the gravitational field gets changed. However, the special theory of relativity prohibits such an instantaneous “action at a distance” (which would operate more quickly than by the speed of light). The two principles (gravity, special relativity) are only consistent because of a loophole: the gravitational signals also propagate (only) by the speed of light and it's possible because they are disturbance in the spacetime geometry itself. They cannot be disturbances or waves in any other field or medium because the “equivalence principle”, i.e. the fact that all objects accelerate in a gravitational field at the same rate (in the vacuum), would be contradicted.

There are several other very important examples and hundreds of less important examples. Quantum field theory results from the reconciliation of the special theory of relativity with quantum mechanics. Particles with wave-like properties have to be “created” and “annihilated” by local quantum fields for the rules of special relativity to be obeyed. However, one may see that such local quantum fields are able to create and annihilate antiparticles (particles of antimatter), too.

Finally, string theory seems to be the only known logically consistent reconciliation of the rules of quantum field theory (arrived to in the previous paragraph) and those of the general theory of relativity (Einstein's modern theory of gravity). In this case, we don't possess any “complete and direct proof” that string theory is an inevitable implication of the two assumptions. This failure partially boils down to the fact that we don't have any “truly universal” definition of string theory, something that would allow us to decide which laws of Nature are string theory and which laws of Nature aren't (research from the mid 1990s has demonstrated that “the theory” we called “string theory” is actually more than just a “theory of strings” and their splitting/joining interactions; it inevitably includes many other building blocks such as D-branes and other processes).

However, in practice, we are always able to decide whether a new proposed theory generalizing the general theory of relativity as well as quantum field theory is or isn't string theory. And whenever a candidate for a consistent theory of quantum gravity comes close enough to string theory, we may show that it is actually necessary for the constraints of string theory to be obeyed exactly and the resulting theory must be one of the numerous (seemingly inequivalent but at the end, exactly equivalent) descriptions of string theory.

Examples of consistency arguments in string theory

We have seen that “consistency arguments” exploded in theoretical physics about 100 years ago. They became even more omnipresent with the rise of string theory. Detailed string theory examples are arguably too technical for a philosophical essay. Nevertheless, I want to sketch and distinguish two ways in which references to consistency are used as arguments in the context of string theory.

First, consistency is often being exploited in a derivation of some particular technical result that apparently didn't have anything to do with consistency. Second, the apparent consistency of string theory and all of its insights and aspects is a rational reason to strengthen our belief that string theory as a whole is correct or at least unique and worth our (or physicists') time.

Concerning the first first point, it is interesting to note that even Einstein's equations of the general theory of relativity (which say that the Ricci tensor vanishes, if we restrict our analysis to the vacuum) may be derived from consistency within string theory. A priori, it is possible to study the vibrations and motion of one-dimensional strings on any spacetime geometry. The results should be independent of our parametrization of the curve spanned by the string in the spacetime. In general, however, we find out that the results depend on the parametrization. Such a dependence is sometimes referred to as a “Weyl anomaly” or “conformal anomaly”. (In general, “anomalies” are symmetry-breaking terms in the equations that may be made inevitable by the rules of quantum mechanics even though they would seem obviously absent according to the rules of classical physics.) If this dependence were there (and if the results for cross sections etc. included the corresponding term that would be nonzero), the physical theory would make ambiguous, mutually contradictory predictions. It would be an inconsistency. (More precisely, there would be new “degrees of freedom” that could create waves on the string and processes involving these new “degrees of freedom” would sometimes be predicted to occur with negative probabilities.)

It turns out that there are many potential terms in the “Weyl anomaly” and they are proportional to the values of the Ricci tensor at each spacetime point. The condition that the propagation of strings is logically consistent – i.e. that it produces unique, not self-contradictory, results – requires the Ricci tensor to vanish at each spacetime point. (More generally, we may derive all the right low-energy field equations including the source terms and other terms.) The field equations may be derived in several ways but the consistency of the propagation of other strings on a given background is one of them.

Surprisingly enough, one may also derive certain nearly “philosophical” conclusions that would look like “arbitrary assumptions” in all simpler theories. In particular, all approximate theories including quantum field theory may assume a certain number of spacetime dimensions. Theories with different numbers of spacetime dimensions describe “completely separate worlds” that cannot be related or compared. The number of spacetime dimensions seems to be one of the first assumptions we have to make and there is apparently no way (except for observations) to decide that one assumption is correct or wrong.

However, in superstring theory, the “Weyl anomaly” also contains terms proportional to (D – 10). Their vanishing implies that the spacetime has to have ten spacetime dimensions in total (it is not incompatible with our basic observations if six of them end up being compactified). It is just one of the prominent examples of the ability to string theory to pick the right values and choices that previously seemed to be “up to us”. String theory is a sufficiently complex theory whose “ill siblings” would have many potential or real inconsistencies. String theory manages to avoid all of these problems. A welcome side effect of this “surprising consistency” is the theory's ability to calculate quantities (such as the total number of dimensions) that previously looked “obviously incalculable”.

This leads me to the second aspect of consistency in string theory. The surprising consistency and cancellation of all conceivable inconsistencies and “anomalies” may be used as an argument in favor of the validity of the theory as a whole. Given the fact that the predictions of string theory only differ from the predictions of the approximate theories (quantum field theories) in extreme environments that will probably never be achieved by experiments, at least not in a foreseeable future and directly, we must ask: How may such reasoning be justified?

Even though one cannot “directly test” the new phenomena predicted by string theory, it is important to notice that even the “more mundane phenomena” that are predicted to occur similarly by quantum field theory and by string theory may increase the probability that string theory is valid. The reason is a difference between quantum field theory and string theory. In quantum field theory, the individual particle species and their interactions are more or less added to the theory one by one. So their qualitative and quantitative properties may be adjusted to achieve the internal consistency and the compatibility with the experiments.

In string theory, the diverse particle species (Higgs boson, leptons, quarks, gauge bosons, graviton) may be deduced as states of vibration of the same object, the fundamental string, and the same applies to different interactions. It means that there is a lot of potential for contradictions. String theory could have been incompatible with the existence of the Higgs boson or the graviton or the lepton or the gauge boson or the electromagnetic interaction or something else. Or it could have produced one of the infinitely many a priori possible inconsistencies or anomalies. But with a very small number of discrete choices, all the potential internal inconsistencies and inconsistencies between string theory and basic qualitative empirical observations are absent. These virtues are nothing else than “tests that string theory has passed” and by Bayes' theorem, the probability that a hypothesis is correct is increasing once it passes some tests that could have a priori failed. (We know how it works in the case of detailed empirical tests. But even tests of the internal mathematical rigidity or qualitative features of the list of particles and interactions are non-trivial tests that could have failed but they didn't. And that's why the probability goes up.)

Finally, string theory boasts two closely related general features: the resilience in extreme conditions and the dual descriptions of the same equations (the so-called “dualities”).

When we adjust the parameters of an incompletely defined theory to extreme values (a huge value of the coupling constant, a tiny size or a nearly singular shape of the compactification manifold, very high energies, a huge number of colors of quarks etc.), we expect to end up with a mysterious situation where predictions are impossible – i.e. either undetermined or inconsistent. But even though no complete definition of string theory is known, this doesn't happen. Instead, consistency considerations imply that there is always a unique way to predict what happens beyond a critical point or in extreme conditions. This property of string theory means that “a whole theory that is capable of answering arbitrarily extreme questions” almost certainly exists even if we don't know the most universal definition of the theory yet.

This virtue is sometimes described by the slogan that “physicists are discovering, not inventing, string theory”. Let me mention the following analogy. We call two people, a maritime explorer and a writer of fiction, and ask them to talk about a new continent they are just visiting. Is it possible to decide which of them is “making things up” and which of them is seeing a real new continent? A good strategy exists: We ask them to walk in various directions and describe what they are seeing. After some time, we will notice that the writer doesn't know what to say or her testimony will become internally inconsistent. String theory is analogous to the actual new continent being discovered by the explorer. He may have only seen parts of it but it is clear that wherever he goes, his testimonies continue to make sense. They are well-defined and internally consistent. Even if we decide that we never want to move to the new continent, it may be a good idea to listen to the explorer because there are not too many continents (and there is only one theory beyond quantum field theory with an analogous degree of consistency, string theory).

The other general virtue of string theory is the existence of “dualities”. A duality is an equivalence between two (or several) sets of equations i.e. laws of physics but it must be an equivalence that is “very hard to see”. For example, the renaming of all fields from \(F\) to \(2F\) wouldn't be counted as a duality. Since the 1990s, it was becoming clear that string theory offers us a huge number of such dualities. We may describe two physical systems that look very different, when it comes to some qualitative features and the words that seem most relevant for a description of the events in the (hypothetical or real) world. But when the physical implications of the two theories are analyzed in detail, we find out that for every measurement in one world, there is a measurement in the other world and the results always exactly agree. So despite the differing ways to think and talk about the events, there is actually no way to distinguish these two worlds; they are one world, they are one theory. One of the languages or descriptions may be more natural in a certain situation than the other but qualitatively speaking, both languages or descriptions are equally legitimate.

The existence of dualities increases the probability that “string theory is as real as a continent” because the different dual, equivalent descriptions of the same physics are analogous to photographs taken by completely different explorers from different perspectives. A pair (or collection) of two-dimensional photographs that are consistent with the hypothesis that they are actual photographs of the same three-dimensional object dramatically increases the probability that the three-dimensional object is real and the photographs were not Photoshopped.

Again, by itself, this feature doesn't guarantee that string theory is the right description of Nature around us (direct or indirect comparisons of the predictions with the empirical data are needed for that). However, what it does (nearly) guarantee is that string theory isn't just a collection of ideas that someone has made up, like the fictitious continents invented by the writers of fiction novels. Instead, it is a theory whose properties are objectively given, not determined by human choices or conventions, and because the theory seems capable of describing and predicting all phenomena in Nature, it deserves the physicists' attention. It deserves much more attention than the “theories” that are being invented by people who are adding one assumption on top of another – “theories” analogous to the fictitious continents – and the theories sometimes presented as “alternatives to string theory” belong to this disappointing camp.

Internally consistent theories may be false

Most of the essay has argued that the arguments referring to consistency are valid, important, and theories that have passed some (or many) consistency checks should be taken (more) seriously because they are probably teaching us something new, something that we previously didn't know.

However, it is also possible to overestimate the power of the consistency. The most fallacious abuse of the power of consistency (or the language that includes this word) is to pick a particular theory and to claim that it must be correct because every new insight or observation seems consistent with the theory while overlooking the existence of some competing theories that are equally (or more) consistent.

The arguments involving consistency are often used in a twisted and illogical manner, as a rationalization of an opinion, a theory, or an ideology that a given person decided to believe at the very beginning. That's why it is important to emphasize that the consistency checks that a theory passes for its probability to increase must be sufficiently non-trivial – in the sense that a (large) majority of competing, superficially similar but different theories (or every competing theory) fails the tests. There must exist a sufficient potential for (i.e. prior probability of) inconsistencies and the given promising theory must “surprisingly” manage to avoid all these inconsistencies. And whenever the original theory is modified (or made more complicated) in some way and whenever the modification is known to be logically unnecessary (i.e. whenever the modification is not just a correction of an error that people would previously make), some of the passed tests become trivial because some features of the theory were “fitted” in order to pass the test. Only when the same theory passes other, inequivalent tests (without additional modifications of the theory), the success may be considered an argument that increases the probability that the theory is correct.

Logically flawed excuses that use the word “consistency” are so widespread and numerous that some of the proponents of these invalid explanations could feel discriminated against if they were not mentioned. That is why I will mention no examples. However, when these logical traps, fallacies, and circular reasoning are avoided, arguments referring to the logical consistency are very powerful and important. And their power – especially in theoretical physics but perhaps in other disciplines, too – is likely to keep on increasing in the future.


  1. Hey, what do you think of paraconsistent mathematics? It has applications in QFT ! And I think it is very interesting.

  2. ''Serbia, Iran, Iraq, Libya, Syria'' - as for Serbia, the corrupt and autocratic rule of Milošević was toppled by his war-weary people for a reason. It really speaks for the democracy if former Serb Radical Party, the hardest of the hardline, can come to power as a reformed party and continue with moderate policies. The Islamic Republic of Iran is an extremely nasty regime even by the standards of the Islamic world and it's clamoring for nukes, If you argue that opposing this regime is bad per se, then I wonder where your loyalties lie in case of the North Korean monster.

    Now as for Syria, the chemical weapons deal was fair enough and the assistance has been suspended ( It's sad to say but the brutal Assad regime was lesser evil for its neighbours than parts of the opposition i.e. various Islamic nuts keen on fighting their sectarian all against all war. It is likely that the more secular parts of the opposition will have to find a compromise with the regime against the al-Qaida style jihadists.

  3. Forget about "rights" for a moment, consider (legitimate) interests instead. Imagine that instead of the neo-tsarist Putin, you had a modernizing, pro-civil society politician like Aleks Navalniy in power, or Garri Kasparov. Would they act any differently in the present crisis? Would they say, oh well who cares about the Russians and the Russian speakers in the Crimea and in eastern Ukraine, that's another country and none of our business? Would they shrug the shoulders about the scary threat to Russia's long-term security position with respect to the deep-water port, a vital element in Russian national defense? No, they would not. They would act roughly along the same lines as Putin. Heck, even Obama or McCain would act similarly to Putin, were they in his position.

    That was the other huge mistake the new Ukrainian government made, in addition to repealing the language law (the interim president has announced he will veto it -- too little too late). They should have rushed to assure Russia that its military self-defense interests would be respected, that the deep-sea port would remain Russia's to use through the entire 21st century, and that they would not seek NATO membership. None of which happened, of course. On the very day that Timoshenko was released from prison, Merkel called her and urged her in the strongest possible terms to work towards reconciliation immediately. The stupid broad (you know which one I mean) ignored her.

    And now here we are. It's a replay of Georgia 2008 when Saakashvili recklessly gambled on NATO coming to his aid, and promptly lost. It's deja-vu all over again. I did not think one could find enough stupid people in the world to repeat Saakashvikli's blunder, but I was wrong: there are enough of them in Kiev alone.

  4. Funny that one would need arguments like ''idiocy'', ''uneducated'', ''stupid'' and like for the guys who don't buy into one's pet theories.

    You may of course pretend as if the idea that Russian Federation had some ''special rights'' with regard to Ukraine while no-one has them with regard to the Czech Republic is some universally accepted truth as opposed to mere theory of yours, but in fact it isn't. It's merely your theory.

    Now, Soviet Union was de facto unitary state till Gorbachev. You may argue that all the republics are successors (and not the official successor Russian Federation), de facto they are, but I don't see the relevance for the idea that one of them has some special rights concerning others. Furthermore, I don't see why Russia should have special rights concerning Ukraine but Ukraine shouldn't have special rights concerning Russia.

    All this talk is degenerating into babble about some ''geopolitical rights to other countries'' that exist no-where in the international law or agreements. True, they do exist in the poor brains of certain ambitious leaders and honestly, Putin's Russia is no exception here.

  5. 1.''Russians and the Russian speakers in the Crimea and in eastern Ukraine, that's another country and none of our business?''

    Then how come the US is not speaking in the name of all Anglophones? Would you like Islamic countries to express their ''concerns'' with regard to Russia's Muslim populated areas?

    2. There may be common things that practically every Russian ruler would agree on, there are differences, too. Yeltsin for example used the ''Russophone interests'' rhetoric concerning Baltic states, however, de facto he let them be. As far as language laws were concerned, these were tougher than in Ukraine

    3. ''Would they shrug the shoulders about the scary threat to Russia's long-term security position with respect to the deep-water port, a vital element in Russian national defense?'' How realistic it is that someone is going to attack Russia militarily thru Ukraine? NATO? I thought such fantasies were the playground of Zyuganov and Zhirinovsky.

    As for the steps you suggested the new government in Ukraine should have done, then yes I agree, but not from the point of view that the Russia's possible fears are justified but from the pragmatic point of view, i.e. they should have offered Putin a kind of ''soft landing'' so that his hawks wouldn't get too angry.

  6. Hi, incredible idiot.

    Ukraine is not a successor state just de facto. Ukraine is a successor state to the USSR, much like eleven other countries, also de iure. It is a fact of the history. It is a fact of the political science. It is a fact of the international law. Your attempts to obscure and relativize this basic fact only show that you are either totally infamiliar with some totally basic facts about the politics of Europe and Asia or they show that you are an intentional hardcore demagogue.

    I have never said that no one had geopolitical rights in the Czech Republic. Quite on the contrary, I was explaining that since the beginning of the Czech statehood, someone had these rights and who had these rights was very clear and resulted from some previous big events, usually hard fought wars.

    The Roman Empire gave the statehood to the Czech kingdom, so it had these geopolitical rights for most of the second millennium. The rights were effectively transposed to the Austrian Empire because an extra level of hierarchy emerged. But nothing changed about the general fact that German speaking powers had these geopolitical rights in the Czech lands for nearly 1000 years.

    This only changed in 1918 when Austria and Germany lost the First World War. With our Czechoslovak legions etc., Czechoslovakia was created as a successor state and it was a de iure winner of the war. This First World War rearranged the alliances, so from a heartland of the German speaking empires, we became a remote buffer zone of the true Western powers, like France and Britain, effectively the winners of the First World War.

    The German occupation was really an attack not just on us, it was an attack on our Western allies as well, as they should have known and as they were taught the hardway less than two years later. They were obsessed with idiotic ideas about appeasement instead of basics of strategic thinking and political science. Others like Stalin knew better. At any rate, the Second World War reverted the German occupation, and because the USSR was the major power that defeated the Nazis on our territory, pro USSR forces, namely the communists, got very strong.

    Geopolitically, we switched from the Western bloc of allies to the Soviet bloc. This ended in 1989 when we became a part of the Western world and later NATO because the USSR lost the Cold War. At every moment of our history, the geopolitical layer of our country belonged to some larger power that knew that it could count with us in the case of greater games. It is complete nonsense that we have ever had some complete independence. The current independence is within the limits of being compatible with NATO.

    Ukraine and its parts have a similar history but with totally different sequences of the greater powers. Most of the time, Ukraine geopolitically belonged to Russia. It has never really been different although parts of Ukraine used to belong to other Eastern European countries.

  7. Demagogue or not, I really forgot that Ukraine and Belarus took over the seats in the UN. In fact, Stalin initially demanded 16-17 seats (Karelian-Finnish SSR definitely too, probably RSFSR as well), but it was considered as too much for him. It's pure formality. It has no bearing as to your preposterous claims.

    ''Your attempts to obscure and relativize this basic fact'' - how is this ''basic fact'' that Moscow had those two additional puppet seats in the UN validate your outlandish theory that Moscow has some special rights to Ukraine? I asked for a simple explanation - why does Russia still have those ''geopolitical rights'' with regard to Ukraine, whilst Austria/Germany lost them 1918 with regard to Czech lands? Instead of a straight answer to a polite question you answered with vulgar smearing. You were unable to offer a sensible explanation.

    So, as for ''incredible idiot'' - you can call all kinds of names, like there are ones who call anyone who disagrees with them an FSB agent or Nazi or commie, but it usually shows one's utter lack of arguments and confirms once more Goethe's judgement when he said ''Durch Heftigkeit ersetzt der Irrende, was ihm an Wahrheit und Kräften fehlt''

  8. We see a naked imperialism at work: a Smith & Wesson beats four aces. Disregard those idiotic 1994 guarantees - the world has changed (and they don't suit us any more). Mr. Yanukovich is a legally elected president - and that gives him a license to kill.

  9. May I remind you, there is such a thing known as the "Anglosphere". Formally speaking, it comprises the U.S., the UK, Canada, Australia, and New Zealand. These countries do not spy on each other, they spy on everyone else. And when push comes to shove, the stronger will come to the aid of the weaker, as in WW I and WW II. You may find it unfair that there is a class of "masters of the universe", but that is how it is. Unlike the U.S. and Australia, however, there are strong, direct historic ties of ancestry and culture between Russians in Russia and Russians in Ukraine. If anything, the Russians are entitled to come to the aid of their expat fellow Russians much more than the U.S. would be to come to the aid of Australia. Muslim powers are already interfering in places like Chechnya, see Saudi jihadists there.

    The Baltic states were never comparable to the Ukraine-Russia ties. Lithuania has stronger historic ties to Poland, Latvia to Sweden, and Estonia to Finland. I don't approve of the discrimination that these newly independent countries meted out against their Russians, but it did not threaten Russian identity in its very core.

    A national defense planner must consider every scenario, including worst-case scenarios. In some far distant future, a NATO invasion of Russia may seem so far-fetched as to be a laughable idea, but we are not yet living in that era.

  10. Russia is not acting against the will of a Ukrainian government because there's no credible government in Ukraine right now. Moreover, all other signatories of the 1994 Budapest memorandum have violated the memorandum as well.
    3. Refrain from using economic pressure on Ukraine in order to influence its politics.

    This was clearly violated by the U.K. - a signatory - as it was pumping money to reshape its pre-existing its ties to Russia. America is offering tens of billions now, too. The point has been violated by Russia, too.

    "Imperialism" may be used as an expletive among the Marxists but "imperialism" is a fact about the real world. Nations of different sizes always have and must have unequal relationships of some kind. We see it all the time. We see it in the EU, too. The idea that it could be otherwise is utterly silly.

  11. OK, you made an attempt at last. Fünf, sitzen. There was absolutely nothing in your diatribe that would justify your ''theory'' that one post-soviet country has after 20+ years still ''geopolitical rights'' over the other (to the extent of invading) while Hitler didn't have such ''justified rights'' against its Czech victims 20 years after WWII. It's pure scolasticism. Лень читать.

    And now throw a few additional bomzh style invectives around for the good impression!

  12. Did you even read his post? He clearly explained the difference -- why russia has those rights and germany didn't. Germany lost those rights because of the outcome of a major war.

  13. Russia has spent the past 100 years in constant risk of extermination.

    World War I -- Germany had a strong chance of conquering Russia. 20+ million Russians dead.
    Interwar period -- Stalin had to industrialise as fast as possible in order to survive the inevitable next war, which led to massive casualties.
    World War II -- Germany almost takes over Stalingrad. 20+ million Russians dead.
    Cold War -- At risk of nuclear extermination from the technologically and economically superior united states.
    Yeltsin Years -- Poor implementation of economic "shock therapy" leads to robber barons stealing a huge swath of Russia's wealth, plunging life expectancy and plunging economy.
    Bush-Obama Years -- Constant attempts to absorb the entire former USSR into NATO.

  14. "oh well who cares about the Russians and the Russian speakers in the Crimea and in eastern Ukraine,"

    They are not in danger. They are used as an excuse, just as Hitler did with his lebensraum expansion not too far from our blog host, before WW2.

    Yes, a bully may be "understandable". He's still a wrong.

  15. That is some of the finest apologia ever seen. Holodomor, the fault of the west. Amazing.

  16. Lokalkosmopolit, if they don't have rights then they have duties to protect.

  17. I find myself in substantial agreement with you.

  18. Russian deployments to Cuba? That is just stupid. One little naval vessel visiting Havana is not a deployment. How long do you think it would take for the US to sink that vessel?

  19. What was the original audience of this nice text?

  20. LOL, I don't actually quite know, Dilaton, I just know it was different.

  21. Figures: Obama misses National Security Meeting on Russian Invasion of Ukraine. As Instapundit (Glenn Reynolds commented), "couldn't he move his tee time?"

    The power of the U.S. president is vastly overestimated, anyway. The U.S. government and its policies are like a supertanker, slow to change direction, speed up or slow down. The "secret power" behind the elected officials is the accumulated institutional inertia of choices and decisions made by a continuously existing republic for the past 200+ years. When you join the government, you "buy into" those constraints and make them your own.

    However, image and perception still play an important role. The fecklessness of Obama, an empty suit whom no-one takes seriously, and his sidekick Kerry don't go unnoticed abroad. I can't prove it but I think Putin would have been more cautious with George W. Bush still in office. You just know that Putin has "wargamed" the current situation with his senior staff many times in the past weeks and months, while Obama was out on the golf course. First Syria, now Ukraine: how does Putin manage to show up the U.S. and EU despite being militarily and economically weaker than them?

    It's because he is better at timing and brinksmanship. He knows the other guy will blink first. However, I fear that one day Putin will miscalculate and then the consequences for us all will be dire.

  22. With our intervention in Kuwait we may have violated Saddam's "sphere of influence" in the first gulf war, but we had the rest of the world with us. Our cost was only 20% of the total and it cost very few American lives. The second gulf war accomplished absolutely nothing but had an enormous cost in both lives and dollars to absolutely no purpose.

    Like most rational people I would love to see China take over North Korea but it is not going to happen.

  23. You make a great deal of sense but could you not, at least, do credit to the English language? Your writing is very poor. I think you can do better.

  24. Again, Hitler was trying to *expand* his Lebensraum - Germany/Austria lost both Sudeten *and* the whole Czech lands and Slovakia in the wake of WW1.

    On the other hand, Russia is just preserving its "Lebensraum" that has always been a geopolitical Russian Lebensraum.

    Russians are clearly at danger. The main party that is friendly to them was violently removed from power in Kiev. Within a few days, the new government stripped the Russian language to be used as an official language anywhere in Ukraine, and so on. And the rallies in Kiev are known to be organized *primarily* by people who are as blinded anti-Russo-nationalists as some Nazis were anti-Semites.

    Which greater threat for the Russians do you want?

    The word "bully" is just a lame propagandist label. The rational term for what you want to say is a "large country". Every power of this magnitude acts and has to act in similar ways. Unlike smaller countries that are threatened from "distant enough" exterior, powers are threatened from nearby and from inside. So they have to counteract this process. This for the stability of the power - but also for the safety of all individuals who live there.

    Other powers would behave and did behave in analogous ways, whether it was the Roman Empire, Ottoman Empire, British Empire, Third Reich, USA, and so on, and so on. All of them had to be "bulies" at some point and one may scream that it's wrong. But it's a law of physics - geopolitics. Protesting such things in general is analogous to protesting the laws of gravity. The Earth is such a stealing bitch by attracting objects, isn't She?

  25. You owe it to the readers to exercise some care in composing your responses. We are all in a hurry; that's not an excuse.

  26. Although my experience of Ukrainians is somewhat limited they all seemed to me to be mad as hatters. You can see this is true by the fact that they want to join the EU. Only a truly insane people would want to do this. In my opinion the Ukrainians would perhaps be better off if Putin re-assimilated them and ruled the place with some sort of metallic fist.

  27. O.K., this analysis in the Kyiv Post covers all the important angles and lays out the likely scenarios.

    Going to switch off my Ukraine newsfeed now and let events unfold to their predicted conclusion.

  28. Hm, I dont know why, but for some reason I was thinking that maybe you have used that text in some kind of a nice speech to teach a bunch of overreaching pompous science philosophers some reason and correct way of thinking ... ;-)


  29. You're missing the point, which is how our actions are perceived by Putin.

    I am well aware of how nasty these regimes are, but the so-called responsibility to protect is a formula for endless war. Why for example, doesn't the US liberate Cuba? The communist regime there is quite nasty, and Cuba is only 90 miles away. The fact is these interventions have nothing to do with the actions of the various regimes and are driven by other considerations, generally economic, sometimes military advantage.

    The US press is fawning all over Obama and treating the coup perpetrators as freedom fighters.

  30. Despite our Noble Prize winning Presidents soaring words about freedom, and peace. His true aim is none of that. He is truly a corporate whore who will say and do anything to further the cause of the corporate Reich, in its drive to enslave the world. Even to the point of sending us to the brink of Nuclear war.

    What should be clear is that no one really knew who this man was, nor did we know his intent, his disguise, as a mild mannered man of conscience, was almost perfect. This is certainly not the man that this country elected president.

    He must look longingly at North Korea, and hope to shape this country into an American version of it at some point in the future, freeing our corporate dictators of the burden of our civil rights, and free speech.

  31. The article is nice and self-consistent, except by the phrase: "In particular, all approximate theories including quantum field theory may assume a certain number of spacetime dimensions". This is just partially correct when not assuming that physical theories have to explain experiments! Let's take QCD (for instance) and formulate the theory on the lattice. The gauge coupling on the lattice has to be adimensional (to allow perturbation theory to connects to DIS). Thus, in this very narrow context, such a regularizable and renormalizable nonabelian field theory also fixes (automatically and univocally) the dimensionality of space-time to be D=4.

  32. What 'costs' exactly does Obama think he can bring to bear against the
    Russians themselves, or those in the Crimea? NATO and the EU are
    outflanked, outgunned and ill prepared (even more so with a clearly
    CIA-inspired coup in the offing in Venezuela to deal with), Russia hold
    all the cards. The best we can hope for is an annexation of Crimea back
    to Russia, which would safeguard their interests and moorings for the
    Black Sea fleet (a level of alarm is being raised that this
    militarization may indeed be in part a move to secure seaborne nuclear
    weapons held by the Black Sea fleet, rather than having them fall into
    the hands of resurgent Ukrainian neo-fascists). The idea of a Russian
    Crimea isn't half as unsettling as the idea of those neo-Nazi and
    fascists at the vanguard of this 'revolution' controlling what arms
    Russia doesn't secure.

  33. Russia entering Ukraine pretty much settled this issue. Ukraine is going to split. Sorry, Russian military involvement won't stabilize Ukraine or at least not in the long term. The other side isn't just a small part of the country it's a huge chunk. It will perpetuate the separatist feelings even more and may even turn the other side to terrorism if Russians succeed (if their goal is united Ukraine under their influence).

  34. "Russians are clearly at danger. ... Within a few days, the new government stripped the Russian
    language from its status as an official language anywhere in Ukraine,
    and so on"

    If this is a standard of "clear danger" that justifies armed invasion, then I hope England never hears about Quebec language laws.

    "This is desirable for the stability of the power/empire - but also for the safety of all individuals who live there."

    ... all individuals, just not including those who until recently were free citizens of another country.

  35. The fun begins. Zbigniew Brzezsinski just suggested on CNN that NATO should offer some sort of military assistance to the new government in Ukraine and that NATO should reposition some of its forces so that they would be more useful. Hopefully this means only into Poland. But I think that would be wildly provocative.

    Dear sweet Jesus, the US/EU leadership is clowns all the way down. We can only pray that Zbig doesn't speak for the elites.

  36. Hi, thanks, but note that the QCD coupling constant is only dimensionless (the right word for "adimensionful") in the classical approximation. Quantum mechanically, the coupling runs which is equivalent to a dimension of "g" slightly different from zero.

  37. It's scary. Of course our Polish brothers are a major nation that hates Russians. But the truth is that the Ukrainian-Polish hatred exists, too -partly because many Poles haven't accepted the territorial losses to Ukraine.

    Why does someone want all these horrible negative sentiments to wake up? In Czechoslovakia, the Velvet Revolution helped to wake up the Slovak separatism or nationalism, but it was relatively peaceful so things could go fine. There is nothing peaceful about the true emotional relationships among the ethnic groups and people across Ukraine. This is not really just about the Russians and Ukrainians. Any conflict is a polarizing thing that opens too many cans of worms.

  38. Typical of O'Drama making more 'threats' to yet another country that
    does not accept that the US is the savior of the world. For gods sake
    just shut up and get your own house in order before dictating to others.
    After the absolute chaos and mess you have created in Syria, Libya,
    Egypt, Iraq, Afghanistan and now Ukraine, when will you learn to keep
    your big nose out of other countries affairs? Cameron also needs to take
    note and not commit the UK to any action in support of the warmonger

  39. Obama angers me with his dishonesty, incompetence, and arrogance, but
    now he's starting to frighten me with his ignorance of foreign policy. It's bad enough that he acts like a divisive bully here at home, but
    when he tries to play Mr. World Leader and issue warnings to people who
    can run circles around him regarding foreign policy, then I start to
    worry. Putin is not someone this little nit-wit should be ordering
    around. I'm not a Putin fan by any means, but Obama is not even in his

  40. If the US is interested in "liberating" the inhabitants of regions with naval bases they ought to check out Bahrain, where the Saudi army is protecting a local tyrant-and the US Navy-from a deeply wronged and courageous population.

    The hypocrisy of US foreign policy never ceases to amaze. As to Obama, the man is shameless, a prostitute completely devoid of principle or self awareness, an example of the ravages that racism (in America), comprador fascism (in Indonesia) and a mother's careerism (at the Ford Foundation) wreak upon the psyche. And, through it, the Universe.

  41. Putin is not a drunkard Yeltsin who would for vodka sign anything and
    everything. Russians are not to blame in this game around Ukraine, it
    seems they were too generous. Now, it's time for Crimea (and any part of
    Ukraine willing to do the same) to join home again. That's the only
    solution for this new problem, again, not caused by Russians. Ask
    yourself, who created this mess, who are these "serial killers" who do the
    same across the world? The pattern is the same all the time, only the
    victims are new.

  42. You are right. I already apologized in a post after that. I was in a hurry and didn't have time to make fixes.

  43. Please, Frank... First, it is questionable whether England still has any geopolitical power over Canada and whether it could even win a war against Canada.

    Second, French speakers are almost 80% in Quebec:

    Over 11 million of Ukranian citizens (more than the total population of the Czech Republic) speak Russian, 1/4 or so:

    It's surely not the same situation as in Quebec.

  44. A bit hostile in your comments, but I will defend my argument. America was an isolationist country until WWI with the exception of the Western Hemisphere. Europe did not have a major affect on us until World War One. It was our "intervention" that brought that war to a quicker end. After that we withdrew to isolationism again. Hitler came out of the ruble of post war Germany and racked up several nations, before the invasion of Poland. They were Chechoslavakia, Austria, and Spain. Spain was won in a bloody Civil war backed by the Nazis. He kept spreading after Britian and the US declared war. If we had done nothing the Nazi empire might have spread all the way to the Pacific. It was after WWII that we did "intervene" in world politics and won the cold war without a nuclear holocaust or a hot war with the Soviets. They stayed contained and eventually imploded on themselves do to the inefficient communist system. What would an unchecked Soviet Union have turned into?
    Where I think we went wrong was with are dealings in Central America, Iraq and Afghanistan. But no country and no one person is always correct. I'd rather stop or contain empire builders in an imperfect way then let them run wild. Has America made mistakes? Yes. Has America bullied other countries for Economic reasons? yes? America does need learn from the mistakes of the past on these issues. Has America defended other countries right to govern themselves? Yes. Has America spread the ideas of political freedom and the American economic dream? Yes. I think these last to things far out weigh the mistakes America has done.

  45. Right and both the US and EU are backing the fascists. Russia lost 20
    million people in WWII, they certainly do not want Nazis on their border
    and I don't blame them.

  46. Weren't you calling for Ukrainians to have nuclear weapons only a short while ago?

    Anyway, the "20 million people lost in WW II" is not the figure for Russia but for the Soviet Union. And a vastly disproportionate number of those victims were Ukrainians, not Russians -- this goes for civilians as well as troops. After all, Hitler's armies occupied only five percent of Russia at any time, but all of Ukraine.

    Makes one wonder how in a people so ravaged by the Nazis, there can today be anyone putting up posters of Hitler on the wall (see report by BMWA1's friend).

  47. I would say the Russia-US situation is more tense now than even during the war in Georgia. Honestly, I've never seen the situation between the two as bad.

    One thing is for sure Putin is a geo-political master, there is a reason Arab countries sometimes refer to him as "The Fox." Putin ordering a military drill around the Ukraine border region is a stroke of genius. Thousands of Russian soldiers doing maneuvers will confuse US intelligence and spy satellites masking raids and not being able to tell which is a real world mission and which is just part of the drill.

    He holds a lot of cards, mainly Russia's gas pipelines, powerbase in Crimea, rebels ranks divided among EU pick Klitschko, US pick Yatsenyuk, and Yulia Tymoshenko with none controlling the Neo-Nazi's. My best guess would be to never bet against Putin.

  48. I think that most Poles have rather warm feelings towards Ukraine. The loss of territories isn't a big deal for Poles - they can live without Lviv. The problem are some nationalists who deny existence of some basic historical facts:

    Anyway, polish Internet is 90% pro-Ukrainian in this conflict. Poles know their history well, and they know that strong Russia is the last thing they need. In 1939 Poland was attacked both by Nazis and Soviets. The real reason for soviet aggression was Ribbentrop-Molotov pact, but the official one was exactly the same as in the case of Georgia and Ukraine.
    Furthermore, though Poles are very anti-Islamic, they make some exceptions from this rule. They really like Tatars (they have very long history of friendship dating back to 13th century, and one has to notice that Tatars constituted an important part of polish-lithuanian nobility) and they like to some degree Turks, so if one asked a typical Pole if he would prefer Crimea to be Tatar or Russian, I believe that he would prefer it to be Tatar. In fact, a typical Pole would prefer Crimea to be under governance of China, Kazakhstan of Jamaica, because when it comes to Russia, they farther away the better.

  49. ''Czechoslovakia had lots of emigrants but they made just 300,000 in those 40 years of communism or so. Ukraine's emigration to Russia easily beats it - by a factor of more than two - in just two months'' - I can't bet on Czechoslovakia, however emigration from the USSR was small first and foremost because the border was closed. Similarly, there was a reason Berlin Wall was constructed, and it was not meant to protect ''West German fascists'' from invading.

    It is true, that Russian media, not just RT is reporting of mass emigration, but it has been disproved by Ukrainian media this may also be inaccurate, given that Ukrainian media have even more important things to follow at the moment, However, the Russian state media blunder of presenting Ukrainian queues at the border with Poland as ''people fleeing to Russia'' is true anyway.

  50. Afghanistan too? I mean, you suppose the 2001 invasion itself was a blunder, not only the fact that the American led coalition couldn't install a stable government?

  51. I have to say that probably for the first time I see your reasoning flawed ...

    By your reasoning Russia had the right to basically destroy the whole Europe because most of the countries are part of the NATO and there are several big cities that could have been attacked from e.g. submarines...
    Also Latvia and Estonia are direct neighbours, Turkey is (almost) neighbour - all NATO member states.

    Whatever is going on in Ukraine I can hardly see any direct threats to Russia, so I see no justification for their current actions.

  52. On The Nature And
    Origin Of The Universe...

    Science Replaced By 2013 Gravity Comprehension !!!

    נברא היקום יש מאין

    New Science 2013
    versus classical science

    Attn classical
    science hierarchy ( including Darwin and Einstein…)

    “I hope that now you
    understand what gravity is and why it is the monotheism of the universe…DH”


    Gravity is the natural selection of
    self-attraction by the elementary particles of an evolving system on their
    cyclic course towards the self-replication of the system. Period

    ( Gravitons are the
    elementary particles of the universe.
    RNA genes and serotonin are the elementary particles of Earth life)


    כח המשיכה
    הוא הבחירה הטבעית להיצמדות הדדית של חלקיקי היסוד של מערכת מתפתחת במהלך התפתחותה
    המחזורית לעבר שיכפולה. נקודה

    ( הגרוויטון הוא
    חלקיק היסוד של היקום. הגנים, הנוקלאוטידים של חומצה ריבונוקלאית והסרוטונין הם
    החלקיקים היסודיים של חיי כדור הארץ) Dov
    Henis (comments from 22nd century)

  53. This former US amabassador to theSoviet Union seems to mostly agree with you Lubos

  54. Excellent, I would endorse everything he wrote.

  55. Dear Lubos,
    I think you are right in analyzing this conflict in terms of empires and spheres of interest. Anybody not doing this is a fool. But you should not forget the Western ideas of democracy, nation and human rights in the analysis. The Austrian empire was doomed even without the first World War because the nations in that Empire were looking for independence and sooner or later they would have gotten it. The thing which can be changed by war is where the borders between nations are but one nation ruling another one is not really something working well in a world where the Western ideas have gotten tremendous success. The best argument for Putin to have a saying about the fate of the Ukraine is still the large number of Russians which live on the Crimea and in Eastern Ukraine. This is why the Czech Republic and the Baltic states may enjoy independence from Russia now while the Ukraine doesn't. For these reasons I think that the splitting of the Ukraine may be the best outcome.

  56. So would I. This pic shows Ukrainians holding up placards in front of the U.S. embassy in Kiev:

    "Save Ukraine," "Help Us," "Save Us". Now it dawns on them that they're in deep doo-doo, only it's rather late. No, ladies, Obama won't ride to your rescue in shining armor. You made your bed, now lie in it.

  57. There are 28 countries in the European Union that have vested economic and political interest in the stabilization of all of Europe. We need to keep our noses out of this. This continuous warring has bankrupted our country. We need to maintain a mighty military defense but back off of any further offensive wars. We don't seem to be able to resolve these conflicts and get out without looking like a loser or leaving the country in worse shape than when we invaded.

  58. Let's face it, Eugene, the creation of this crisis is fully the result of
    Obama's foreign policy, or more precisely, the result of his master's
    foreign policy since Obama is nothing more than the "house" servant. Recall,
    his neocon in residence in the State Department, Victoria Nuland (who
    is married to neocon Robert Kagan) said that the U.S. spent $5 billion
    dollars to create this "coup." While their aim may have been to put
    their own moderate faction in power, in fact western Ukraine's neo-nazis
    are assuming power.

    This is lunacy on the part of the warmongers pulling Obama's strings
    and the compliant MSM that parrots the lies of the government. For
    Obama to “warn” Russia not to interfere is a new low for Obama’s
    “Audacity of Hypocrisy.” Putin has every right to counter Obama's
    imperial over reach.

  59. We invade and blow up 2 countries, Iraq, and Afghanistan, spend 13 years in wars, waste thousands of soldiers lives, waste trillions of dollars, and we're telling a super power country like Russia, to watch their step? What goes on in Ukraine and Russia is none of our business. Keep this country out of it. Thanks to the stupidity of our last president, and the stupidity of our current president, we don't have the resources for any more wars.

  60. Nice post, thanks.

    Lubos, I wonder what do you think is the connection (if you think there is one, ofcourse) between consistency and replicability?

  61. Hi Lubosi. I am kind of late to reading this one, but, in your introductory prove of primes you have a mistake:" the sum would be a prime as well because it is not divisible by any other prime i.e. because it is only divisible by one and by itself." That is wrong. The sum would not necessarily have to be a prime. It only would not be divisible by any primes of the original list, so it would have to be divisible by some prime, not on the list. For example 2*3*5*7*11*13 + 1 = 30031 = 59*509.

  62. Honza,

    Luboš is not wrong. Such a thing is impossible! :)

    You forget to note the assumption (gambit) about there being a finite number of primes.

    Briefly, using your example, this assumption means that the primes stop at 13. So by that assumption, 59 CANNOT be prime because there aren't any beyond 13. After all, that's what it means to assume the assumption, as it were. :)

    The contradiction necessarily follows of course, proving the assumption false.

  63. Dear Honzo, as John said, you are missing the point - what you say is exactly the observation behind the proof that the set of primes cannot be finite because it is a contradiction.

    In your case, {2,3,5,7,11,13} is the hypothesized full list of primes, and your "example" is nothing else than a proof that the list cannot be a full list of primes because 59, 509 would have to be primes as well, thus contradicting the assumption that the primes never exceed 13.

  64. That was not my point. The inconsistency stands, of course. My beef was with the claim that, procedure of p(1)*p(2)*...*p(n) + 1 = X will generate a number X, which is a prime (divisible by 1 and itself only). It is mostly not true. The number X will be in most cases composed number. It will not be divisible by any of the generating primes, for obvious reasons, but by other primes, not on the list. Nevertheless, there is no reason to expect X to be prime itself. That's it.

  65. Dear Honzo, no, you are still *completely* missing the whole point of this proof.

    It may be rigorously proven, and I have given you (and I am going to give you again) the simple proof, that *if* {p(1), p(2), p(3), .... p(n)} is the full list of primes, *then* the sum

    p(1)*p(2)*...*p(n) + 1

    is a prime, too. That's because a prime is a number that is not divisible by any other prime except for itself.

    So we may prove that

    p(1)*p(2)*...*p(n) + 1

    is a prime as well and it is larger than any element of the assumed full list of primes, so it's a contradiction. One of the assumptions has to be wrong and the only assumption that may be wrong is the assumption that the set of primes is a finite set.

    So this proves that the set of primes is *not* a finite set.

    Please, if you have never completed any proof by contradiction, try to study what a "proof by contradiction" means at least for 30 minutes before you post another comment about this trivial thing, OK?

  66. It is not about not getting the proof, just about where the disconnection is. The way I have learned it at age of 7 was, that assuming that there is a finite set of primes, you can show that there must exist another one, outside that set (contradiction), without necessarily stating what it is.

  67. Dear Honzo, your new comment only appeared 29 minutes after mine which is less than 30 so you have the first strike.

    This latest comment of yours is irrational. The proof that you should have learned when you were 7 but you failed is *exactly* the same proof as mine except that you have misunderstood some critical portions of that proof and you are still misunderstanding them.

    When you write "you can show that there must exist another one," it's correct that you write "you" and not "I" because I can show it but you clearly can't.

    The way how I - and others - can show it is to actually construct a particular new prime given these assumptions, and the "product plus one" is by far the simplest one, and morally speaking, the unique one that obeys the required conditions.

    How else do you want to realize the step "I can show that there are other primes"?