Addition Theorems and Composition

Occasionally one sees the claim (even from professional physicists who should know better) that “special relativity implies 1+1=1”. This bizarre claim is based on conflating composition with addition as discussed below. A similar confusion arises in other contexts in which arithmetical addition is conflated with other concepts such as the logical union.

Perhaps the simplest case is cardinality. For a set S of distinct objects, let C(S) denote the cardinality of S, i.e., the number of objects in the set. The cardinality of the union of two sets equals the arithmetic sum of the cardinalities of the individual sets. Symbolically we can write this as

However, a given function of the union of two sets need not be the sum of that function of the individual sets. For example, let A(S) denote the average age of a set S of people. In this case we have

where n₁ and n₂ are the numbers of people in the respective sets. Similarly we could consider the temperatures of two glasses of water, and the temperature of the combined water if we pour both glasses into a single container. With simplistic assumptions about specific heats, etc., the result would be the same as the expression for average ages, except with the masses of water replacing numbers of people.

Another example comes from the field of probability theory, in which the probability of the union of two sets (e.g., subsets of some sample space) is

In the common notation of Boolean logic the union or “logical OR” is sometimes denoted with the plus (+) symbol, because it is associated with “adding” two sets together, although it isn’t actually strictly additive unless the sets are mutually exclusive, in which case the subtractive term is zero. If the sets are independent the subtractive terms is the product P(S₁)P(S₂).

Examples in which we are actually evaluating a function of a literal sum of two quantities can be found in trigonometry, such as the expression for the hyperbolic tangent of the sum of two quantities in terms of the hyperbolic tangents of the two individual quantities

In the limit as x and y go to infinity the values of tanh(x) and tanh(y) approach 1, as does the value of tanh(x+y). Thus in this limit the above expression approaches 1 = 1, which is not a surprising equality. Nevertheless, some people enjoy saying that the above trigonometric identity expresses the counter-factual relation 1+1 = 1. The motivation for this strange claim can perhaps be inferred from considering the case of very small values of x and y (magnitudes much smaller than 1). For very small x the value of tanh(x) approaches x, so each side of the above identity is asymptotically tanh(x)+tanh(y), and so those people reason that the expression always formally represents that arithmetical sum, even when it doesn’t even approximate that arithmetical sum.

The example involving the hyperbolic tangent has a well-known application in special relativity. Every pair of inertia-based coordinate systems S_i and S_j is related by a Lorentz transformation L_ij with relative velocity parameter v_ij, and given any three systems S₁, S₂, S₃ we have L₁₃ = L₁₂ L₂₃ where the right side denotes matrix multiplication, which signifies the composition of the two transformations.  Letting V(L_ij) denote the velocity parameter v_ij of the Lorentz transformation L_ij, we can say V(L₁₃) = V(L₁₂ L₂₃).  For co-linear systems this can be expanded explicitly as

We can define a quantity called rapidity as ϕ_ij = atanh(v_ij), and note the additive property

This can be written as

Taking the hyperbolic tangent of both sides and making use of the trigonometric identity, this gives

Each of the above expressions could be called an “addition theorem” (and there are others for elliptic functions, etc.), and the last in particular is closely related to the claim about special relativity and the expression 1+1 = 1. For small values of velocities we have the approximate relation v₁₃ = v₁₂ + v₂₃, and if we pretend to think that this continues to apply as the speeds approach 1, then it would imply 1 = 1 + 1, but of course that is not valid reasoning, because as the speeds approach 1 the denominator of the above expression approaches 2, so we actually have 1 = (1 + 1)/2, which, again, is not a surprising fact.

One difficulty of explaining this to students is that many authors, including Einstein himself at times, refer to the velocity composition formula as “velocity addition”. From the standpoint of a mathematically literate person this is understandable, because there are many well-known “addition theorems”, including not only trigonometric identities but also involving the “addition” of points on elliptic curves, and so on. It is understood that these don’t refer to arithmetical addition. However, to beginning students the word “addition” seems to unequivocally mean arithmetical addition, which leads them into cognitive dissonance when they assert things like x+y = (x+y)/(1+xy), which obviously is not true unless xy=0. The mistake is equating actual arithmetical addition with an “addition formula” which actually represents the composition, as explained above.

To be fair, in his 1905 paper Einstein explicitly refers to “the composition of velocities” in section 5 of the paper. However, in his 1907 survey article he already calls it the addition theorem, as he does in his popular booklet on relativity written in 1916, and in his Princeton lecture of 1922. As a result, it became standard to refer to the composition formula as the addition formula, which confuses some students. Contributing further to the confusion is the tendency on the part of some teachers to strive to make relativity sound as bizarre and unintuitive as possible, so they delight in telling students things like “In relativity, 1 + 1 = 1!”, as if they are intentionally trying to undermine the student’s confidence in sound reason and logic. Obviously velocities add in the usual way in special relativity, provided everything is expressed in terms of a single system of coordinates, and arithmetic still works the same as it always did. Special relativity does not introduce a new kind of arithmetic, it introduces a new relationship between relatively moving systems of inertial coordinates, due to the inertia of energy. There is nothing weird or counter-intuitive about this.

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