Source
Nicholson (2026), LIT-1821, §5, reporting Weiss (1963, 1969a, 1971c, 1973, 1977, 1978); read in NOTE-1340 from the Europe PMC full text.
What was actually shown
The inequality as given. For each part a of a system S, measure “all the fluctuations from the mean for a range of physical and chemical parameters over a given period of time”, call the cumulative record va, and do the same for S, giving VS. Then “What we would invariably find, if we studied any system in this way, is that the sum of the individual variances of the parts is far greater than the total variance of the whole”:
VS ≪ ∑ (va + vb + vc + vd + … + vn)
Nicholson says it is “the only mathematical expression we find in Weiss’s theoretical writings about systems”, used “sometimes as a way of illustrating how one might go about empirically identifying systems in nature” (§5).
What it can show. Read it with one scalar quantity xᵢ per part, and the whole’s quantity either their sum or their average. For any random variables,
Var(∑ᵢ xᵢ) = ∑ᵢ Var(xᵢ) + 2 ∑ᵢ<ⱼ Cov(xᵢ, xⱼ).
- Average. Var((1/n)∑xᵢ) = (1/n²)[∑Var(xᵢ) + 2∑Cov]. With independent parts it is ∑v/n², far below ∑v for any n of more than a few. Every aggregate meets the inequality, so it cannot distinguish a system.
- Sum. With independent parts Var(∑xᵢ) = ∑v, and the inequality fails. It holds exactly when 2∑ᵢ<ⱼ Cov is close to −∑v, so that the parts’ deviations cancel in the total. That is compensation, and it is a mark of regulation that an aggregate does not show.
A numerical check (numpy, 100 parts, 20,000 samples, seed 0; my script): independent parts gave VS = 0.010 against ∑v = 100 for the average and VS = 101 against ∑v = 100 for the sum. Parts held to a fixed total, plus a small common fluctuation, gave VS ≈ 25 against ∑v ≈ 99 for the sum, and the identity matched to three decimals.
What could have failed. If the inequality as stated separated systems from aggregates, independent parts would have violated it under both readings. They violate it only under the sum.
What this does not say
- That Weiss’s systems lack compensation. Cells and embryos may well show strongly negative covariance among parts in an extensive quantity. The point is that the inequality as reported does not require it, so finding VS ≪ ∑v is no evidence of it unless the quantity is extensive.
- That Weiss meant the average. The paper fixes neither the parameters nor how the whole’s value is formed from the parts’, and it sums variances across “a range of physical and chemical parameters”, whose units differ. The two readings above are the simplest that make the formula a statement at all.
- That variance reduction is the whole of Weiss’s concept. His definitions also require restoration “after non-destructive disturbances” (Weiss 1971c, quoted in §3), a dynamical property no single-time variance measures.
- That any measurement was made. Neither Nicholson nor the passages he quotes report a value of VS or of ∑v for any system.