2+2=5

Another day, some SJW-professor said something like "treating 2+2=4 as an absolute truth is a form of cultural oppression". The equation would only be true from a white western point of view. (I guess Muhammad al-Khwarizmi wanted to raise from his grave.) Some SJW minions tried to defend her on Twitter, using very weak arguments e.g. non-decimal numeral systems.

Others even mentioned cultural appropriation, being the Arabs the legit owners of the number symbols, and we westerners "robbed" them. (Actually, the decimal positional system was created by the Indians; al-Khwarizmi made it known in the West.)

This elementary equation remembers me of something I have been saying frequently, due to the increasingly polarized political scene here in Brazil (and also in the USA). If a communist says 2+2=4, and a conservative says 2+2=5, I *will* agree with the former. We should not embrace anti-intellectualism just because intellectuals are often SJWs or commies. Replace "2+2=5" by "Sun gravitates around the Earth", "Earth is flat", "Cloroquine cures COVID"... you get the idea.

A benevolent interpretation of what that SJW-professor said, is that our western math culture cherishes more some branches of mathematics, while other branches are overlooked.

For example, calculus (and the whole "math of continuum") gets much more airtime at schools than discrete math. Schools care more about measuring than about counting. Integers and fractions are taught as "numbers for children", simple subsets of the real numbers. At least here in Brazil, almost every college degree must include a calculus course. I graduated in accounting, had calculus classes, and I don't understand why. Even in software engineering, we hardly ever use calculus. A computer is a machine of discrete states, after all.

I must admit calculus was a big achievement for the humankind. Among other things, it has an enormous capacity of modeling physical phenomena. Another important fact is, Newton and Leibniz were famous and admired in their lifetimes, just like we admire Jeff Bezos and Steve Jobs. These guys projected a long shadow.

Perhaps because of that, calculus gets disproportionate classroom airtime today — because the founding fathers of calculus lived and died respectably, while discrete math was built up by troubled people. Galois dueled and died at 20; Abel died poor at 27; Cantor went to the asylum for a while due to depression; Gödel went nuts and starved himself to death; and so on.

Teaching calculus fits well in a top-down, hierarchical education system, since it is a matter of memorizing and applying integration and differentiation rules that were invented centuries ago. Discrete math is a challenge for professors and students alike, since learning it is ultimately to learn to think; the axioms to memorize are few and self-evident.

Having said all that, let's accept the provocation and analyze why we say that 2+2 equals 4, and under which circumstances 2+2 could be equal to 5. Because you need to demonstrate knowledge of why 2+2=4 is such an universally accepted truth, and only then you can start suggesting alterative universes where 2+2=5. Smoking a joint and speaking nonsense is not enough to become a scholar :) and we all know true-blue mathematicians do prefer stimulant drugs (Erdös was addicted to coffee and benzedrine.)

To start with, we must define what is 2 or "two". 2 is just a symbol, and "two" is just a name. In the context of Peano axioms, two is simply the second successor of the number zero. The definition of "four" is analogous. We could use any other symbols to represent 2 and 4, but this doesn't change their essence.

By now, let's define "zero" as the first (or zeroth?) natural number, that does not succeed any other, without any other assumptions about what "zero" means. The actual value of zero will be defined later.

Actually, we should have started by defining what is a number. Accordingly to Peano, a (natural) number is any object that has a well-defined successor, which is also a number. Is the alphabet a number system? No, because Z has no successor, so it is not a number. Not being a number, Z can't be a successor for Y, so Y is also not a number, and this defect propagates back to the letter A.

Then, we turn our attention to the symbol =, that we colloquially call "equal". We use the term "equality" too loosely in our lives. When we say "equal", we often mean "equivalent". There is a whole family of equivalence relations, and many of them do fulfill our common-sense definition of "equal".

For some relation to be an equivalence relation, it must be

reflexive (a=a is always true)
symmetric (if a=b, then b=a)
transitive (if a=b and b=c, then a=c)

There are quite relaxed equivalence relations, e.g. making a=b if both are even or both are odd. It doesn't look too useful, but it does match the common-sense of equality in modulo-2 arithmetic.

Peano says natural numbers are equal when they have the same successor. This is a pretty airtight definition, for integers. But there are many other situations where equality and equivalence will diverge, either by practical or by theoretical reasons.

For example, 500 cents are equivalent to $5, but they are not equal to $5. In many countries, a merchant can refuse inconvenient means of payment e.g. purchasing a car with coins. If coins can't purchase a car, they cannot be equal to money in a bank account, right? This doesn't break the equivalence relation; you can always go to the bank with your coins and get a cashier's check (and dirty looks from the cashier).

In the realm of fractions, 2/3 is equivalent but not equal to 4/6. The equivalence relation for fractions is

a/b=c/d iff a.d = b.c (for a,b,c,d integers)

Like any other equivalence relation, this one partitions the field of fractional numbers in subsets called equivalence classes, each set containing an infinite collection of fractions (e.g. 2/3, 4/6, 6/9, 8/12...). In each class, the "irreducible" fraction with relatively prime parts (in the example, 2/3) is the one that represents the whole set — also called leader, representative form or canonical form.

How we can be sure the equivalance relation is tight enough to match our common-sense definition of equality? Leibniz said that, if x=y, then P(x)=P(y) is true for every conceivable predicate or function P. This is quite good at least for infinite fields, it rules out some funky equivalence relations like "x=y if both odd or both even", and accepts reasonable ones like "2/3=4/6".

In finite fields, this is not enough; it is possible that two different polymomials evaluate to the same value when the x's are replaced by numbers. So they are equivalent from a certain POW, but certainly not equal.

Not even for real and complex numbers the symbol = can be blindly believed as absolute equality. For example, the complex number 4 can be represented in polar form as 4∠0°. Or 4∠360°, or 4∠720°, or 4∠1080°... all these representations yield the same result when converted to the cartesian form a+bi. Can we say then they are absolutely equal?

Well, is seems so, until you extract their square roots. The root of 4∠0° is 2∠0° = +2. But the root of 4∠360° is 2∠180° = −2. Numbers that have different square roots cannot be equal, even though they may be equivalent in other instances. (*)

So the complex number "4" is not just a number; it is the representative form of an equivalence class: an infinite set of numbers in the form 4∠n.360° (integer n). When we say the root of +4 is "+2 or −2", we mean that half of the elements of the set +4 has root +2, and the other half has root −2.

(And yes, the square root of the real number +4 is just +2. When we say "the root of +4 is +2 or −2", like in the Bhaskara's Formula, we are implicitly saying that we are operating in the field of complex numbers, not (only) the real numbers. All in all, the fundamental theorem of algebra depends on complex numbers to be true.)

Finally, we can turn our attention to the symbol +, and to define what it means to "sum". The first Peano axiom for addition is

a + 0 = a

This means that zero does not append anything; it is the neutral element for addition. This axiom finally allows us to define "zero" as a number that is worth nil. The second axiom is

a + S(b) = S(a + b)

where S(x) is the successor function e.g. S(0)=1. This is profound and perhaps not obvious how it works:

2 + 2 = ?
if 2=S(1), then
2 + S(1) = S(2 + 1)
if 1=S(0), then
2 + S(S(0)) = S(2 + S(0))
2 + S(S(0)) = S(S(2 + 0))
Using the first axiom,
2 + S(S(0)) = S(S(2))
2 + S(S(0)) = 4

Using only the axioms, we can also prove that addition is associative and commutative.

Of course, we could define the binary operation "add" in any other way. But before we do that, we should ponder whether the Peano's definition of addition is valid from a teleological point of view. Teleology is the study of purpose and utility of the things. The motto can be found at Luke 6:44: "Each tree is recognized by its own fruit."

Peano is teologically valid, since it accurately reflects our material reality. If I have an empty cabinet (and we define "empty" as "zero", since there is no emptier cabinet than an empty cabinet), I put 2 cans of food in there, close it, open it, put 3 more cans, close, open again, I shall find 5 cans in there. If I repeat the experience, putting 3 cans, than 2, again I end up with 5.

Even if you try to argue the result could be something other than 5 cans, your belly knows full well that 2+3 cans are 5 meals. (In this case, smoking pot could help you think more clearly, since it increases the appetite. You won't get away with 2+2 cans for 5 meals.)

One thing that does not work is to add a new and conflicting axiom to Peano e.g. define that 2+2=5. This would allow to "prove" that S⁴(0)=S⁵(0) and finally 0=1, x=S(x) and therefore S() is an innocuous function and all numbers are absolutely equal.

But yes, we can create a whole new set of axioms, with a valid and consistent binary operation that does not work like addition at all. For example, if we imagine a transparent triangle with marked edges

   / \
  / 1 \
 /     \
/ 3   2 \
---------

and two basic movements: rotate it (symbol "r") or flip it by the top edge (symbol "f"), we can build an "arithmetic" using the "numbers" r and f to express all configurations of the triangle reachable by rotating or flipping it. BTW the grand total is six configurations.

The composition of operations "r" and "f" is not commutative. Making f+r (flip, then rotate) reaches a different configuration than r+f (rotate, then flip).

The point here is, the "algebra" of triangle manipulation is weird, is completely alien from number arithmetic, and yet it posesses an order, it respects an aesthetics. And it is also teleologically valid (triangles are real-world objects, and pretty useful too).

So yes, the arithmetic we learn since childhood is not the only possibility, and it is not even the best fit, depending on what real-world process we are trying to model.

Modular arithmetic is also useful and weird. Is this perhaps the way to define 2+2=5 and get away with it?

Let's try to create a finite group modulo 10 in which 2+2=5, while keeping addition commutative, zero as neutral. Given that 5+5=0 (mod 10), 5 is an element with order 2, and 5 is its own inverse. But if 5=2+2, 2+2+2+2=0, and 2 would be order 4, which is impossible — the order of every element must divide the group order (10). You may insist, but you will never wrap up this group.

Note that we have not said a thing about the value of 2 or 5; we just considered their relatioship with the group size and with the neutral element. We didn't even demand a definition for the operation +. And it seized up already.

In a group modulo 11, every non-neutral element has order 11. Then we can make 2+2=5, since 5x11=2x11=0 (mod 11). But, if 5+6=0, we can no longer accept that 2+2+2=6, otherwise (2+2)+(2+2+2)=0, making "2" an element of order 5, which is forbidden. In a group where 2+2=5, it must be true that 6=2x9, so 5+6=0 is explained by (2x2)+(2x9)=0, since 2x11=0.

This adjustment creates a rabbit hole, and the result is simply a shuffling of the arithmetic symbols. The "progressive" arithmetic will be isomorphic to the "conservative" arithmetic, there will be an 1:1 relationship between elements of the former and of the latter. The progressive-5 has the same role of the conservative-4, the sjw-6 does the same as the western-7, and so on.

Notes

(*) Remember that, for real and complex numbers, powers and roots are not defined by repeated multiplication. They are defined in terms of exponentials and natural logarithms:

a^b = e^{b.ln a}

This is true even when b=1, so we can say every number has an equivalent exponential representation. In particular,

(−2)¹ = e^{ln −2}

But the logarithm of a negative number is not unique, it is any number in the form

ln x = ln |x| + i(π+2πn) (n integer)

That is, behind a seemingly boring negative number, we have a whole set of numbers in the form

−2 = { e^{ln 2 + i(π + 2πn)} }

Since we can express positive numbers as a product of negative numbers, we can extend this reasoning to the positive numbers as well:

+2 = −1.−2 = e^{ln 2 + i(π + π + 2πm + 2πn)}
+2 = { e^{ln 2 + i(2πn)} }

Moreover, representing a complex number in exponential form is the same as representing it in the polar form, the only requirement is to express the angle as radians:

n∠θ = e^{ln n + i.θ}

For example,

+1 = 1∠0º = e^{ln 1 + i.0} = e^{0 + 0} = e⁰
−1 = 1∠180º = e^i.π
i = 1∠90º = e^i.π/2
√2+√2i = 2∠45º = e^{ln 2 + i.π/4}