Inductive reasoning is the process where a small observation is used to infer a larger theory, without necessarily proving it.
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Most scientists use this method to generate theories about how the universe works and discover the laws governing our very existence.
Many ancient philosophers used induction for making observations and constructing theories.
For example, the Ancient Greek philosophers believed that theories could be proved by logic alone and did not need experiments. They thought that mathematically strict laws, deduced from smaller observations, governed the universe.
Generally speaking, inductive reasoning and deductive reasoning are a circular process and inextricably linked.
J. J. Thompson's Cathode Ray-Experiment was an excellent example of this process, where he had ideas about how electrons behaved and generated theories about their nature.
Inductive reasoning works in the opposite direction, where an initial observation leads to the discovery of a certain pattern. This allows a tentative prediction to be made which leads to a general theory about how things work.
An excellent example of this process in action is the discoveries and works of the great Charles Darwin. He made some observations about how the Darwin Finches vary from each other across the Galapagos archipelago.
After some thought and reasoning, he saw that these populations were geographically isolated from each other and that the variation between the sub-species varied over distance.
He therefore proposed that the finches all shared a common ancestor, and evolved and adapted, by natural selection, to exploit vacant ecological niches. This resulted in evolutionary divergence and the creation of new species, the basis of his 'Origin of Species'.
This was an example inductive reasoning, as he started with a specific piece of information and expanded it to a broad hypothesis. Science then used deductive reasoning to generate testable hypotheses and test his ideas.
Of course, very few examples of induction are this clear cut. Usually, more information is gathered and assimilated over a longer period. Darwin incorporated other information and spent many years gathering more information before publishing his work.
These two methods should be seen as a cycle, with inductive reasoning generating a theory, with deduction and experimentation validating or falsifying the theory. This, in turn, leads to inductive enhancements of the theory and more testing.
For example, Bohr used Thompson's experiments as observations about atomic structure. Inductive reasoning led to the creation of theories and hypotheses about the structure and nature of atoms. This is an example of induction taking initial observations and expanding them into full-blown theories.
Deductive reasoning was used by Bohr to create individual hypotheses, isolating parts of the theory and testing them, to prove that his ideas were scientific truth. This is an example of the inductive/deductive cycle in action.
The process is not without critics and Hempel's famous 'Raven Paradox' highlighted flaws in the logic. Despite this, for the vast majority of scientific disciplines, induction is a powerful and essential foundation.