## Tuesday, June 15, 2021 ... //

### Widespread defense of "1+1 isn't 2" means that the whole education system has collapsed

Would-be fancy apologists for the woke insanity are wrong about everything

Back in 2017, the following wonderful satirical minifilm was released:

Danny, a boy who struggled with math, wrote that $$2+2=22$$ into a test. His old-fashioned teacher, an older woman, did everything she could to professionally teach him that $$2+2=4$$ but it was hard. Instead, the spoiled brat's wrong answer was increasingly violently defended by Danny's dumb parents, the teacher's dumb principal, activists on the street, and an Orwellian legal committee of the city hall. The plot ended with a happy end. When the teacher was fired, she got a compensation of $2 thousand for the last month plus$2 thousand for the present one. She could use the new standards to claim that the total is $22 thousand. ;-) She would be even richer with$20002000 dollars.

I have really liked the video and created the Czech subtitles. Now, in 2021, this satirical video is no longer satirical, at least not in California where a whole movement of staggering, aggressive morons is openly and proudly embracing Danny's attitude (see the real fights about "math is racist" in the media). The value of the mathematical expressions may be anything and any constraint that "only one answer is right" is an oppression of some kind, they scream. And by "them", I mean numerous individuals or members of a mob who aren't safely confined in a padded cell where they belong.

Note that the 2017 video sort of deceptively suggested that the people defending the wrong answers may have been right-wing American patriots. It has always been nonsense but by now, it is spectacularly clear that these enemies of mathematics are overwhelmingly "new leftists" or, using a term suggested by the Urban Dictionary, "woketrash". And their leading justification to hate the "only correct answers in mathematics" is that such unique answers are said to be "racist".

Well, with their generalized definition of racism, everything that has any value is racist. For example, Czechia's soccer team has defeated the increasingly annoying and degenerated Scots 2-to-0, Patrick Schick scored both, a header and the goal of the tournament from the halfway line. And you easily find Scots who are so fudged up that they tell you that the cheering on the Czechs is racist. Of course, the only real racists are themselves because they're incapable of admitting that Czechs – a nation that may always be kicked into, according to their Nazi-PC hybrid ideology – showed some soccer skills that the Scots simply don't possess (a repetition of Slavia's victory over Glasgow Rangers at the national level).

In the past, values of expressions such as $$1+1$$ were rightfully considered the most unquestionable and neutral truths you may find which means that every individual, group, or movement that starts to deny that $$1+1=2$$ is just dishonest, degenerated, totally fudged up. The claim that someone actually denies $$1+1=2$$ was always an exaggeration or a straw man. Times have changed. Sadly, these individuals, groups, and a movement are omnipresent today because whole nations have failed to fight this cancer culture. The kids who were disputing that $$1+1=2$$ should have been beaten, whipped, punished etc. but their educators have failed. A whole insane ideology has developed that it was wrong to beat these spoiled brats; in his famous Cargo Cult Science commencement speech, Richard Feynman turned out to be a visionary who knew how wrong and far-reaching it would have been to allow self-described "experts" to prevent the educators from physically disciplining the kids. So we live in the world where these lying spoiled brats have gotten away with it and they grew into lying, aggressive, self-confident adults. How can you educate or eliminate them today? It seems too late, the corresponding nations (and it's almost all the Western nations) are already badly damaged.

The idea that $$1+1=2$$ is racist is utterly preposterous. Mathematics is a collection of truths and methods whose truthfulness is more independent of human cultures and races than anything else. Mathematics really works even in hypothetical universes that don't respect even the most general laws of string/M-theory that underlie our world (or our multiverse). The laws of physics are somewhat less universalist because they only work in some physical setups (string theory works in the whole multiverse while the Standard Model may only be OK for our Universe within the multiverse)... And you may continue to chemistry, biology, economics, sociology..., disciplines that are increasingly dependent on the environment where we live, the existence of life forms similar to ours, the existence of humans, or the existence of cultures and (increasingly specific and increasingly man-made) conventions that we are familiar with (but extraterrestrial aliens are not).

If you forget about politics and you try to be a serious mathematician or a serious physicist, is there a basis for questioning that the "right value of $$3+3$$" is $$6$$? The short answer is No. Let's look at this silly problem because you may read a lot of absolute nonsense about this issue which could only turn "controversial" in a mankind that became as degenerated as the contemporary one.

OK, so $$3+3$$ is a sum of two things which I chose to be the same object, namely "three" (twice). And there is a sign (plus) in between. The result will depend on some axiomatic framework that has objects, the identification of $$3$$ with an object in the scheme, and the identification of $$+$$ with an operation. Great. You may say that $$3$$ means Baden, $$+$$ is the merging of strings, and the result of $$3+3$$ is Baden-Baden, a town in Germany. This is a ludicrous answer, of course, because the interpretation of $$3$$ as "Baden" is unusual (or completely newly fabricated) and useless. Also, $$+$$ may be the union of strings but that is surely not the "default" meaning of the sign $$+$$ (it is the same meaning that was promoted by Danny in the video! It is also how the sign may be used in numerous programming languages). Also, if you wrote "Baden" as $$3$$, you should probably write "Baden-Baden" as $$33$$ which means that insisting on "Baden-Baden" is internally inconsistent to some extent.

Fine. Let's forget about "Baden" or any other totally fabricated redefinition of the digit $$3$$. The digit (or the word "three") is supposed to represent the integer $$3\in\ZZ$$ or a generalization of it. Also, $$+$$ is a well-known operation. Make no mistake about it, $$3+3$$ is equal to $$6$$ ("six"). The operation is true in integers $$\ZZ$$ with addition but it is true in all additive groups which contain $$\ZZ$$ as a subgroup. So it is true for the addition in $${\mathbb N}, {\mathbb Q},\RR,\CC$$, quaternions, and many other algebras. Even if $$3$$ were a symbol for a vector, like $$3\cdot \vec e$$ in some vector space, the result of $$3+3$$ would still be $$6\cdot \vec e$$ which we clearly chose to represent by the digit $$6$$.

The sign $$+$$ may be some operation such as the "union of strings" but it shouldn't be. As long as you use the mathematical symbols at least a little bit decently i.e. with some knowledge of or respect towards the subtleties that the proper mathematical or quantitative reasoning cares about (unlike a sloppy layperson's "thinking"), $$+$$ has a much more unique meaning than some "random similarly sounding words". It should be a symbol for an associative operation! And it's usually "times" or "multiplication" that is reserved for "possibly non-commutative" operations. The addition of three apples and three oranges in quantum mechanics (as objects) is represented by$\ket \psi = \ket{\text{3 apples}} \otimes \ket{\text{3 oranges}}$ where the symbol $$\otimes$$ in the middle is the tensor product, not a sum! A sum in between wave functions in quantum mechanics is "OR", not "AND"! The symbol $$+$$ should better be a commutative and associative operation, not only an associative operation.

So the only "freedom" that you have is to consider some other commutative, associative groups that contain objects that may be reasonably called $$3$$. What are they? The only other possibility is a cyclic group $$\ZZ_K$$. But $$K$$ must be at least four for an element of $$\ZZ_K$$ to be sensibly called $$3$$. On the other hand, when $$K\geq 7$$, the result is already called $$6$$ just like it is called $$6$$ in $$\ZZ$$, the regular integers. The only other possible values are those in $$\ZZ_4,\ZZ_5,\ZZ_6$$ and the values of $$3+3$$ are $$2,1,0$$, respectively. That's it!

But even this ambiguity is a matter of cheating. Why? Because the speech and mathematical symbols depend on some axioms, contexts, and conventions. While you may be adding $$3+3$$ within $$\ZZ_4,\ZZ_5,\ZZ_6$$, these are three vastly less important or canonical contexts than $$\ZZ$$ and the additive groups that contain $$\ZZ$$ as a subgroup. In fact, I find it legitimate to say that the symbol $$+$$ is only properly used for $$\ZZ,\RR,\CC\dots$$ and the usage of $$+$$ in $$\ZZ_4$$ is at least a partial abuse of the notation. Also, while the mathematics of cyclic groups is "legitimate mathematics", it is almost unavoidable that one learns these abstract things only after he learns the proper integers $$\ZZ$$. The "more exotic number systems" which includes $$\ZZ_K$$ are only extra derived, awkward floors built upon the foundations of the "proper" mathematics which is that of the integers $$\ZZ$$ and their strict extensions! I believe that there is no mathematician in the world who uses the term "cyclic group" for $$\ZZ_4$$ but isn't familiar with the addition in $$\ZZ$$. I would also say that even the decision to use the symbol $$3$$ for an element of $$\ZZ_4$$ is just a shortcut. It should never be claimed to be a "proper default meaning of $$3$$ unless the usage of the symbol is preceded by a $$\ZZ_4$$ disclaimer"!

In other words, even if $$\ZZ_K$$ is admitted to have the right to use the same symbol $$+$$ for the "addition" and $$3$$ for an element, it is still true that $$\ZZ_K$$ is less important a piece of mathematics than $$\ZZ_K$$ and its extensions. It is the equality that is completely untrue here. When we compare $$\ZZ_5$$ and $$\ZZ$$, there objectively exists the $$\ZZ$$ supremacy! It is this point that the woketrash wants to deny – because indeed, their whole mental defect is about claims that "everything is equal" even though most pairs of things are simply not equal. Almost all of them want this "equality" because they are inferior in the relevant respects and an equality would be a (perceived) improvement (or a professional promotion).

You may phrase the problem differently. A sensible person may say that it is legitimate to use $$+$$ as an addition in $$\ZZ_6$$ but there is still the question whether a schoolkid should assume that the objects and symbols are those in $$\ZZ_4$$ or $$\ZZ_5$$ or $$\ZZ_6$$ when he or she is asked to calculate $$3+3$$ (and whether it is right for teachers or the society to "encourage" the non-standard answer). And the answer "they should not" is clearly the only correct answer in an education system that produces better, smarter kids, a system that leads to a self-improving society. Why? Because the fabrication of clearly non-standard cyclic groups as a justification for "unusual" answers to $$3+3$$ is almost guaranteed to be a method for lousy schoolkids (or their parents) to cheat, to sell their failures in mathematics as a virtue. The kid simply must know the "right context" as well and the failure to know the "right context" is still a failure (which is almost always accompanied by other failures). Some kids may really be smart and confused about the terminology but a vast majority who will use the cyclic groups as an excuse are just cheating bastards who don't even understand the simple material but want to obfuscate the fact and get away with it.

A school that works decently simply mustn't allow such things. The schoolkids and teachers communicate and have to communicate so they must have some shared understanding of the language and the context and unless they specified the unusual context recently, the right context for $$3+3$$ is the addition within $$\ZZ$$ or its extensions. It is the most appropriate context in the whole world that uses these symbols (and yes, basically all nations in the world of 2021 have adopted the Arabic digits and the symbol $$+$$). This meaning of $$3+3$$ is so universal exactly because the beef, and the whole basics of the normal mathematics, are the most important "context" not only on Planet Earth but in the rest of the Universe and even in other multiverses, as I have mentioned. The importance of mathematics transcends not only nations and races; it transcends the planets, galaxies, universes, and multiverses, too!

Another approach of the woketrash to justify the wrong answers is to identify the sign $$+$$ with things that "vaguely sound" as addition. I have mentioned the wave functions in quantum mechanics – where they incorrectly interpret the sign $$+$$ in between the wave functions as "AND" (in the sense of adding objects such as a dead cat and an alive cat). But in between the wave functions, much like in between the probability distributions, $$+$$ means that each of the summands is possible, so "one OR the other" is possible. The $$+$$ symbol simply means something very close to "OR", not "AND". After all, this is why "OR" is known as the logical addition and "AND" is the logical multiplication!

One more physics-inspired incorrect justification of some "bizarre new meanings of $$+$$" could abuse the special theory of relativity where the speeds are "added" differently than in the Newtonian-Galilean space-and-time. When objects A,B move by the speeds $$u,v$$ to the left and to the right, respectively, their relative speed is $$u+v$$ in the Newtonian-Galilean physics but the total relative speed is$\frac{u+v}{1+uv/c^2}$ in the Einsteinian spacetime i.e. according to the special theory of relativity. In particular, when you substitute one of the speeds to be $$u=c$$, the result will be $$c$$, too. The relative speed of light and anything else is the speed of light again! Indeed, the statement that the light always has the same speed relatively to anything (in the vacuum) is one of the postulates of special relativity. You may rederive this postulate from a more general "result" of the theory for the "addition" of the speeds.

OK, so the "addition" of speeds is modified which could mean that $$3+3$$ may yield a different answer as well, can't it? No, it can't. This line of relativization of $$3+3$$ is absolutely indefensible and only a person who has completely failed to think quantatively (let alone scientifically) may agree with this totally wrong argumentation. First, the symbol $$3$$ doesn't normally refer to speeds because in the SI units, speeds are dimensionful. But even if you used some units, probably $$c=1$$, where the speeds are dimensionless, there is a deeper problem. The "addition" of speeds discussed above isn't a real addition (for tachyonic speeds $$3c$$ and $$3c$$, the rule above gives $$6c/(1+9)=0.6c$$). That is why I carefully wrote the word "addition" into quotation marks. The "addition" of speeds simply isn't a real addition and you couldn't write $$u+v=$$ as the left hand side of the displayed expression above. The "addition" is just some operation that "has some vague analogies" with the proper addition.

I will stop using "addition" and we will call it "composition of speeds" because the usage of the same word "addition" is clearly just a method to create misunderstandings and encourage a sloppy and mudding thinking (often deliberately). Well, the composition of speeds in special relativity just shouldn't be represented by a $$+$$ because this operation isn't even associative! A schoolkid who has heard some buzzwords about relativity and who would use this wrong argument to relativize the value of $$3+3$$ is still a bad schoolkid who hasn't learned the things properly because a good kid simply knows that $$+$$ should be associative while the composition of speeds is not.

Incidentally, the composition of speeds along one direction in special relativity is a commutative, associative group. And in fact, it is isomorphic to the normal group $$\RR$$ with the regular addition (which is, via exponentiation, equivalent to the real positive numbers with multiplication!). How does it work?

The composition of speeds is additive with the regular rule of "addition of real numbers" as long as you replace the speeds by the rapidities (which were already introduced by Varićak and Whittaker in 1910, building on Minkowski's realization that the boosts are analogous to rotations). The rapidity $$w$$ is a "hyperbolic" or "Minkowskian" counterpart of an angle. If the spacetime were Euclidean, the speed would be the ratio $$dx/dt$$ and – think about the trigonometric functions within a triangle – this ratio would simply be $$\tan\beta$$, the tangent of an angle. In relativity, the speed is instead (after we convert $$dt$$ to a distance by multiplying it by $$c$$) $\frac{v}{c} = \frac{dx}{c\cdot dt} = \tan w$ or, if you prefer the inverse relationship,$w = \text{arctanh}\, \frac{v}{c}.$ The rapidity $$w(v)$$ corresponding to the speed $$v$$ is the hyperbolic arc-tangent of the speed (expressed in the units of the speed of light). The cool property of the rapidities is that they really add – in the proper sense of addition of real numbers – when the corresponding speeds are composed. Let us explain the mathematical identity behind this assertion. First, let us convert the speeds to the units of the speed of light and use $$\beta$$ for $$v/c$$. The composition of speeds $$\beta_1,\beta_2$$ is given by the formula above$Comp(\beta_1,\beta_2) = \frac{\beta_1+\beta_2}{1+\beta_1\beta_2}.$ It is the same formula as before (the sum over one plus the product) but I could eliminate the division by $$c^2$$ in the $$uv$$ part. Now, let us write all the speeds on the right hand side as hyperbolic tangents of the rapidities $$w_1,w_2$$:$Comp(\beta_1,\beta_2) = \frac{\tanh w_1 + \tanh w_2}{ 1 + \tanh w_1 \cdot \tanh w_2} = \dots$ But the funny identity is that this complicated ratio is actually equal to the simple $$\tanh (w_1+w_2)$$. When the speeds are composed, the speeds are not added (in the proper mathematical addition) but the corresponding rapidities are added! Once again, the identity that allows this to work says$\tanh (w_1+w_2) = \frac{\tanh w_1 + \tanh w_2}{ 1 + \tanh w_1 \cdot \tanh w_2}$ This formula for "tanh" of a sum is totally analogous to the formula for the tangent of a sum, except for the opposite relative sign in the denominator! So yes, the composition of speeds (along a line) in special relativity is mathematically equivalent to the regular additive group $$(\RR,+)$$ with the normal addition but the additive real numbers can't be identified with the speeds. Instead, they must be identified with the corresponding rapidities!

Confusing the "addition" with "composition of speeds" (which is equivalent to the conflation of "speeds" with "rapidities") is just a proof that the speaker doesn't really understand these things. And when he or she doesn't understand, he or she just shouldn't use them!

Everything that he or she says is at least slightly wrong (usually brutally wrong) and it should still earn a failing grade. However, it's clear why this stuff is often being presented. It is just BS ideology to defend a wrong result; it is a tool for the lousy people to get good grades, degrees, and to contaminate numerous influential institutions where they couldn't get if the meritocratic checks and balances worked. A schoolkid shouldn't be allowed to brag about "knowing special relativity" if he or she actually misunderstands even the fact that $$3+3=6$$. It is clear that such a "relativity-based virtue signaling" is totally pretentious and the schoolkid just pretends to be extremely smart while it is totally dumb or ignorant or lazy (the kid could have been promising in the past, it could have been you, me, or someone else, but the habit to justify wrong results by a posteriori invented bullšiting involving much fancier topics is just wrong).

A decent teacher must never allow such stupid, aggressive spoiled brats to strengthen their arrogance. If you fail to beat a similar spolied brat in a similar situation, you are failing as a teacher. You don't deserve your salary, the salary is really a form of theft of the taxpayer's money and the taxpayer has all the rights to catch you on the street and demand the stolen money to be returned.

Three plus three is equal to six and all claims that this and similar assertions may be relativized are either 100% pure lies or at least highly demagogic and misleading argumentative farces that prove that the speaker is either totally ignorant, dumb, or deceptive (in most cases, he or she belongs at least to two of these groups). Any demagogy of this kind – whatever its purported degree of fanciness is – will find its (sufficiently moronic) target group that will swallow it. But I insist that everyone who has rightfully earned at least a STEM-related undergraduate degree knows that it is just wrong to deny that $$3+3=6$$.

And that's the memo.