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In mathematics, the standard method of proving something is to start with the axioms and then apply the rules of logic to arrive at a theorem. In science, the parallel exercise is to start with a basic theory that consists of a set of fundamental entities and the laws or principles that are assumed to apply to them (all of which serve as the scientific analogues of axioms) and then apply the rules of logic and the techniques of mathematics to arrive at conclusions. For example, in physics one might start with the Schrodinger equation and the laws of electrodynamics and a system consisting of a proton and electron having specific properties (mass, electric charge, and so on) and use mathematics to arrive at properties of the hydrogen atom, such as its energy levels, emission and absorption spectra, chemical properties, etc. In biology, one might start with the theory of evolution by natural selection and see how it applies to a given set of entities such as genes, cells, or larger organisms.< [Read more…]

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In the previous post in this series, I argued that in the case of an existence claim, the burden of proof is upon the person making the assertion. In the absence of a preponderance of evidence in its favor, the claim can be dismissed. As has often been said, “What can be asserted without proof can be dismissed without proof”. The basis for this stance is the practical one that proving the non-existence of an entity (except in very limited circumstances) is impossible. Hence if we do NOT have a preponderance of evidence in favor of the existence of an entity, we conclude that it is not there.

In the case of a universal claim, however, the situation is reversed and the default position is that the claim is assumed to be true unless evidence is provided that refutes it. So in this case, the burden of proof is on the person disputing the assertion, again for eminently practical reasons.
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The logic used in arriving at scientific conclusions closely tracks the legal maxim that ‘the burden of proof rests on who asserts’. It should be noted that the word proof used here does not correspond the way it is used in mathematics, but more along the lines used in law. As commenter Eric pointed out in response to the previous post in this series, in the legal arena there are two standards for proof. In criminal cases, there is the higher bar of proving beyond a reasonable doubt, but in civil cases the standard is one based on the preponderance of evidence. So if the preponderance of evidence is in favor of one position, it is assumed to be true even if it has not been proven beyond a reasonable doubt. Scientific propositions are judged to be true not because they have been proven to be logically and incontrovertibly true (which is impossible to do) or because they have been established by knowledgeable judges to be beyond a reasonable doubt (which is not impossible but is too high a bar to result in productive science), but because the preponderance of evidence favors them. Evidence plays a crucial role here as it does in legal cases.
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For a long time, religion claimed to reveal eternal truths. No one except true believers seriously says that anymore because science has become the source of reliable knowledge while religion is increasingly seen as being based on evidence-free assertions. So some believers tend to try and devalue the insights science provides by elevating what we can call truth to only those statements that reach the level of mathematical proof, because such a high bar can rarely be attained and thus everything else becomes a matter of opinion. They can then claim that scientific statements and religious statements merely reflect the speaker’s opinion, nothing more.
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As I discussed in the previous post in this series, our inability to show that an axiomatic system is consistent (i.e., free of contradictions as would be evidenced by the ability to prove two theorems each of which contradicted the other) is not the only problem. Godel also showed that such systems are also necessarily incomplete. In other words, for all systems of interest, there will always be some truths of that system that cannot be proven as theorems using only the axioms and rules of that system. So the tantalizing goal that one day we might be able to develop a system in which every true statement can be proven to be true also turns out to be a mirage. Neither completeness nor consistency is attainable.
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Within mathematics, Euclidean geometry is the prototypical system that demonstrates the power of proof and serves as a model for all axiomatic systems of logic. In such systems, we start with a set of axioms (i.e., basic assumptions) and a set of logical rules, both of which seem to be self-evidently true. By applying the rules of logic to the axioms, we arrive at certain conclusions. i.e., we prove what are called theorems. Using those theorems we can prove yet more theorems, creating a hierarchy of theorems, all ultimately resting on the underlying axioms and the rules of logic. Do these theorems correspond to true statements? Yes, but only if the axioms with which we started out are true and the rules of logic that we used are valid. Those two necessary conditions have to be established independently.
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The idea of scientific infallibility, that the knowledge generated by science should be true and unchanging, suffered a series of blows in the late 19th and early 20th centuries that saw the repeated overthrow of seemingly well-established scientific theories with new ones. Even the venerable Newtonian mechanics, long thought to be unchallengeable, was a casualty of this progress. Aristotle’s idea that scientific truths were infallible, universal, and timeless, fell by the wayside, to be replaced with the idea that they were provisional truths, the best we had at the current time, and assumed to be true only until something better came along.
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