When using the method of reduction to absurdity, we must keep the setting as specific as possible so that we may figure out the reason why our assumption is physically impossible. For example, only within the setting of microscopic diffraction [Eis, p.67, Fig. 3-6] we are able to figure out the physical reason why the simultaneous measurement of position and momentum is physically impossible [Eis, p.68, l.13-l.22]. We can also reach a contradiction in the general setting of Fourier integrals [Coh, p.287, (10) & p.1464, (44)] and gain a broader view. However, in the latter case, although we reach a contradiction we still cannot pinpoint a physical reason because the general setting is far from reality and we lose track of the Compton effect and the diffraction setting [Eis, p.68, l.24-l.25].
Reduction to absurdity can be used to identify the main cause that leads to the conclusion.
Let us prove that heat cannot be completely converted into work. If our emphasis were the validity of the conclusion, then a straightforward argument [Kit, p.230, (7)] is better than the one using reduction to absurdity [Kit, p.228, l.−6-p.229, l.−4]. However, in the former argument, the real cause is somehow buried in the complicated equation [Kit, p.230, (7)]. In the latter argument, we start with a simpler equation dQ/dσ =τ, and adopt the viewpoint that the entropy must be removed. If heat were completely converted into work, we would quickly reach a contradiction. Therefore, that the entropy cannot be permitted to pile up is an important cause that leads to the conclusion. In my opinion, the viewpoint that leads to a contradiction through the shortest argument is the most important clause.
[Kit, p.230, the first equality of (7)] gives the natural definition of the Carnot efficiency. It is good practice to check the consistency of this natural definition before we further develop the definition's mathematical expression [Kit, p.230, the second equality of (7)].
Reduction to absurdity can often reveal the impacts of a concept on various aspects. The feedback of these impacts implicitly defines the meaning of the concept so that its mathematical expression cannot be otherwise. For example, Reduction can show the consistency of the Carnot efficiency's natural definition [Kit, p.240, l.13-l.21]. It would be a terrible waste if one goes through a lot of argument without understanding its purpose.
How physicists solve the problems of contradiction.
(Ampere's circuit law) Add an imaginary term
Add a mathematical term [Sad, p.381, (9.20)] to solve the problem of the contradiction between [Sad, p.381, (9.18)] and [Sad, p.381, (9.19)].
Add a physical term (displacement current) [Sad, p.382, Fig. 9.10(b)] to solve the problem
of the contradiction between [Sad, p.382, (9.25)] and [Sad, p.382, (9.26)].
(The zero-point energy) [Lev2, p.71, l.3] proves that the zero-point energy cannot be zero, while [Coh, p.357, l.12-l.23]
proves that the zero-point
energy cannot be small. Consequently, the latter argument is more general than the former one.
[Wangs, p.65, l.10] says that the derivation of Gauss' law in [Wangs,
pp.65-66, §4.3] is more direct than that in [Wangs,
pp.58-60, §4.1]. In fact, the important
difference between these proofs is that the proof in [Wangs,
pp.65-66, §4.3] is more effective
[Wan3, pp.107-110] than that
in [Wangs, pp.58-60, §4.1]. The derivation
of [Wangs, p.60, (4-10)] is based on the the following method of reduction to absurdity: If there exists an r0 such that
(Ñ×E)(r0)¹r(r0)/e0,
we will have a contradiction. In contrast,
in [Wangs, pp.65-66, §4.3], we prove [Wangs,
p.66, (4-26)] for a given r without using the method of reduction to absurdity.
The argument using reduction to absurdity in [Hall, p.465, l.-10-l.-6]
can be replaced by a straightforward argument [Sad, p.144, (4.83)].
Maxwell uses contradictions to find another source term, displacement
current, for the magnetic field. Without displacement, [Wangs, (21-3)] contradicts [Wangs,
(1-49)] and the boundary condition [Wangs, (20-31)] will fail [Wangs, p.352,
l.9-l.11; p.353, l.5-l.7]. In contrast, if we assume the existence of
displacement current, [Wangs, (1-49)] will be satisfied [Wangs, p.349, (21-5)],
as will the boundary condition [Wangs, (20-31)] (see [Wangs, p.352, l.4-l.5;
p.353, l.4].
Exhaustive considerations can make an argument more complete. If the energy gap between the impurity ground state and lowest excited state is large compared with kBT,
then the impurity scattering is elastic [Ashc, p.321, l.8-l.10]. Because kBT
is an electron's average energy, it is impossible for an impurity to
absorb energy from most of the electrons. The few electrons with exceptionally high
energy also cannot lose energy because the low energy levels lack spaces [Rei,
p.389, Fig. 9.16.1 (the key point); Ashc, p.321, l.11-l.13 (the way that this more
precise and detailed statement is formulated actually blurs the key point.)] for
them to land (Pauli's exclusion principle).
The proof of qi1
= qt2 given in [Hec, p.188, l.c., l.-5]
is straightforward, while the proof using reduction to absurdity given in [Hec,
p.188, l.-13-l.-8] is
based on a single aspect: the minimum deviation is unique [Hec,
p.188, Fig. 5.57].