The Scientific Method/Components of the Method

Principles

The laws of nature as we understand them are the bases for all empirical sciences. They are the result of postulates (specific laws) that have passed experimental verification resulting in principles that are widely accepted and can be re-verified (using observation or experiments).

World View (Axioms, Postulates)

The word "axiom" comes from the Greek word αξιωμα (axioma), which means that which is deemed worthy or fit or that which is considered self-evident.

To "postulate" is to assume a theory valid due to be based on an a given set of axioms, resulting on the creation of a new axiom, this is so due to be self evident, "axiom," "postulate," and "assumption" are used interchangeably.

Generally speaking, axioms are all the laws that are generally considered true but largely accepted on faith, they cannot be derived by principles of deduction nor demonstrable by formal proofs—simply because they are starting assumptions, other examples include personal beliefs, political views, and cultural values. An axiom is the basic precondition underlying a theory.

Another thing one should be aware is that some fields of science predate the scientific method, for instance alchemy is now part of chemistry and physics and math was created even before we had numbers, one should have particular attention that in some fields the definitions or nomenclature may be out dated or be so for historical reasons, due to their use since before the definition of scientific method, and that mathematics uses not only the scientific method but also logical deductions, that result in theorems.

Take for instance the use of the word "axiom" in mathematics, this particular field has gone by several changes, especially in the 19th century, but due to historic reasons an "axiom" in mathematics does have a particular meaning.

Euclid's geometry, is based on a system of axioms that look self-evident. So, in physics, Euclid's geometry was used as natural (and the only) choice, the complete theory could be drawn from the axioms, resulting in the whole geometry to be considered to be true and self evident. This changed in the early 19th century, Gauss, Johann Bolyai, and Lobatchewsky, each independently, took a different approach. Beginning to suspect that it was impossible to prove the Parallel Postulate, they set out to develop a self-consistent geometry in which that postulate was false. In this they were successful, thus creating the first non-Euclidean geometry. By 1854, Bernhard Riemann, a student of Gauss, had applied methods of calculus in a ground-breaking study of the intrinsic (self-contained) geometry of all smooth surfaces, and thereby found a different non-Euclidean geometry.

It remained to be proved mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by Beltrami in 1868. With this, non-Euclidean geometry (both Lobatchewsky and Lobatchewsky) was established on an equal mathematical footing with Euclidean geometry. But this raised issues, "what geometry is true?". Even more, "does the latest question make sense?". All three geometries are based on different system of axioms, all are consistent.

Physics has helped answer these questions. While Euclid's geometry is used on Newtons' mechanics (normal distance), Riemann's geometry became fundamental for Einstein's theory of relativity. Moreover, Lobatchewsky's geometry was used later in quantum mechanics. So, question "Which one of these theories is correct for our physical space?" was answered in the surprising way: "All geometries represents physical space, but on a different scale".

All this influences what to think about axioms. From end of 19th century - beginning 20th century, math didn't appeal to the "self-evidence" of the axioms. It takes the freedom to freely choice axioms. What does, math say, that if the axioms are true, then the theory is followed from the axioms. Correspondence to the real world should be established separately. Axioms doesn't provide any guarantee.

Example of conflict of mathematics/theoretical physics and the scientific method

The best example lies with quantum mechanics. Many facets of quantum mechanics are merely mathematical models explaining the behavior and interaction between subatomic particles. One of the major stumbling blocks in quantum mechanics lies within one its' fundamental theories: quantum superposition (all particles exist in all states at the same time). There are many interpretations of this, the standard being the Copenhagen interpretation. This states, basically, that the act of measuring (or observing) the state of a particle collapses the superposition effect, altering it's state to the value defined by the measurement. This shows that the superposition effect, while being one of the most widely accepted and fundamental principles of quantum mechanics, can never actually be directly observed, even if it is possible for experiments to be devised to corroborate the theory.

Theory

Consists in a set of statements or principles devised to provide an explanation to a group of facts or phenomena. For instance a mathematical theorem; in the mathematical field we have to be careful on how we apply the definition since a theorem may be considered an axiom in itself, they can be accepted as valid until proved false (due to the infinite nature of numbers, it is common to propose limits to sets to provide validation), and other mathematical theorems may depend or be created over each others assumption of validity.

Hypothesis

The hypothesis, or the model is a way for us to make sense of the data. We try to fit the data into some kind of model, and that model is our hypothesis.

Predictions

A key component to the scientific method is the ability to predict. We can make predictions about something, and then test those predictions to see if they are correct. If the predictions are true, it's likely that the hypothesis is correct.

Theorems

Are not part of the scientific method but may be a cause of some confusion. Most theorems have two components, called the hypotheses and the conclusions. The proof of the theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions. The concept of a theorem is therefore fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.

Verification

The fundamental step of turning an hypothetical relation into a principle, by validating it with real world data. Any verified hypothesis becomes a principle.

Observations

an act or instance of viewing or noting a fact or occurrence for some scientific or other special purpose.

Experiments

Experiments are key to the scientific method. Without experiments, any conclusions are just conjecture. We need to test our observations (to ensure the observations are unbiased and reproducible), we need to test our hypothesis, and then we need to test the predictions we make with our hypothesis.

Setting up of a proper experiment is important, and we will discuss it at length in section 2.