Strong proofs, weak proofs, and very weak proofs: Strong proofs can be seen as adding a burdock to a cluster of burdocks ... in mathematics in particular but (in a somehow softer meaning) also in physics:
The term "proof" is often used in science - but in what sense?
- Some proofs are so complicated that they are very difficult to comprehend even for the best specialists (and to be proven as proofs),
- more and more proofs are by computers (and mainly by brute computational force): These proofs check e.g. thousands or even million of cases which are all components of the subject lemma (e.g. as has been the case with the four color problem).
In any case, a mathematical proof is interwoven in the system of mathematics (which is for a physicist (see Einstein) as a system and in its perfectness a magic secret):
In this sense, a proof in mathematics is always “strong”.
On the other side, refutation is unavoidable if there has been found at least one case violating the lemma – then the lemma is not in the system of mathematics.
In science we have to be even more careful: If the term “it has been proved” is used, it can only mean:
1. One (or a small number of) experiment(s) with measurements of some precision deliver(ed) in their limits the claimed property.
2. Then we claim that the experiment could be repeated as often one liked to do (and where you would like to do) with the same results in the experimental limits.
Disproof (falsification) means that we have a problem with at least one failing experiment and we have to go “back to the drawing boards”).
Two examples from physics:
- Newton used the solar system and Kepler’s (i.e. Brahe’s) data, and the Jovian satellite system, as the experiments to derive celestial mechanics – and he had, as we know, the magic feeling to have found a law valid for the entire universe, earth and heaven.
- Cockroft and Walton initiated a nuclear reaction by bombarding Lithium with protons and observed experimentally (“proved”) the validity of the equivalence of matter and energy according to the famous formula
E = m c² .
But there are invisible interconnections within many proofs:
The statements and the experiments are often not isolated but networked and parts of “the system”: We have in science therefore marginal statements which are loosely coupled with the system, and substantial statements that are tightly coupled:
- Loosely coupled statements can be falsified without general impact,
- Tightly coupled statements (as E = m c² ) hold the system together. Falsifying means a revolution (probably a Nobel prize).
Through this integration and the securing through the system by many bolts clicking in, we can call this second class a “strong” scientific proof – the hooks to the network give strength (and the looser proof is just a “weak” one).
In another Blogpost, we have introduced the notion of “scientific hardness” - this is just another view of the hardness of a scientific area.
But we have also “very weak” proofs through a practical issue:
Often, experiments and statements use statistical data – and this makes it very difficult to achieve “proofs” even for a singular experiment:
- “hard” (and reliable) are statistical experiments with very large numbers, e.g. in classical physics with 10**23 or more objects, or experiments in the Internet, e.g. by Amazon, with millions or hundreds of millions of clicks,
- “weak” (or just invalid) are experiments with small numbers and by depending on small differences for the proof, as, e.g.,
o “leukemia in the proximity of high tension lines
(or mobile ground stations)”,
o “cancer in the proximity of nuclear reactors”.
It is very easy to make wrong statistical experiments and to draw wrong causal conclusions, and very hard to conduct and analyze correctly. There are many cases where claimed pro-results (e.g. “magnetic fields increase cancer risk”) after a professional analysis on the same data proved clearly the null hypothesis!
Never believe statistical medical studies where only one or two cases more or less make the difference: Try to make experiments under trivial conditions where possible.
The probably most striking wrong and even counter-intuitive “proofs” are examples of Simpson’s paradox which are true ("proven"):
“The mortality in Sweden is higher than in Costa Rica” (in spite of the extraordinary Swedish health care system) – is it really healthier to live in Costa Rica?
”The mortality of babies with low birth weight of smoking mothers is lower than the mortality of low-weight-babies of non-smoking” – is it healthier for a baby to have a smoking mother?
The explanations are:
- In all age groups in the population, Swedish mortalities are lower than the Costa Rican' - but the Costa Rican population in total is so much younger that the total mortality is higher,
- Smoking mothers are from all parts of the population, mothers with low-birth-weight babies have probably some serious health problem. The mother's smoking affects the birth weight more neutral – together, this gives the observed crazy statistics.
If there are underlying unknown degrees of freedom, the statistical result can be completely nonsense – again: make things simple as possible (but not simpler), a quote loaned from Albert Einstein.