Principle of induction
This article is about the term as it is used in the sciences. For mathematical induction, see Proof by induction
The principle of induction is perhaps most succinctly described as the reason that we believe that the Sun will rise tomorrow. It is a form of reasoning whereby general statements are derived from a collection of singular observations.
If something seems to happen repeatedly, such as an apple falling to the ground when it leaves the tree, one can use this prior behavior to predict what will happen next. The reason we believe that the Sun will rise tomorrow is that it has done so, without exception, countless times in the past.
David Foster Wallace stated it this way:
The Principle of Induction states that if something x has happened in certain particular circumstances n times in the past, we are justified in believing that the same circumstances will produce x on the (n + 1)th occasion. 
Induction, and Deduction, in Natural Science
Induction, also known as inductive reasoning, is central to scientific investigation. Pure deduction can be used in proving mathematical theorems, because the theorems are purely about abstract notions. But it can't be used to establish scientific theories, because we haven't been given fundamental axioms or postulates about how nature works.
The way scientific discoveries work is generally along these lines:
- Observations of natural phenomena are made, for example, the motions of the points of light that we see in the sky that we call "planets".
- Humans use their ingenuity and reasoning powers to formulate theories or models to explain these observations.
- These theories and models sometimes take on extremely elegant characteristics, that make them appear to be equivalent to mathematical postulates. But they aren't.
- Treating these theories as though they were mathematical postulates, people "prove theorems" from them, often predicting other phenomena.
- If the predicted phenomena are later observed to be true, confidence in the theory increases tremendously. For example, conclusions drawn from the equations in Newton's Theory of Gravitation led to the discovery of Neptune, General Relativity was found to explain the precession of the perihelion of Mercury, and Quantum Mechanics was found to explain atomic spectra in extreme detail.
- But these aren't really theorems. They are based on inductive reasoning from observations.
The most celebrated physical theories describe phenomena in terms of simple and elegant mathematical equations. Those equations then take on the status of mathematically established fact. For example, Newton's Theory of Gravitation can be described in terms of an inverse-square attraction, and his "laws of motion" can be described by the equation F = ma. Many "theorems" can be deduced from these, such as conservation of momentum.
But they are not actual theorems, and they could be "falsified" by observations that are inconsistent with the predictions. When such inconsistencies arise, they are generally fairly minor: If we were to observe apples falling upward tomorrow, it would upset our world view in a catastrophic manner. The universe really does appear to behave consistently; that's what make scientific induction work in the first place. But, in the end, it's a matter of human belief.
Observations that falsify well-established scientific theories are generally fairly minor, and often lead to extensions of the theories. Quantum mechanics has overthrown our interpretation of F = ma, but that formula is still accepted as extremely important and very nearly true.
An important aspect of a scientific theory is the notion of "falsifiability", made famous by philosopher Karl Popper. Simply put, a theory is falsifiable if one could imagine circumstances under which it wouldn't be true. If one can't, then the theory may not be scientifically acceptable. For example, gravity might have been an inverse-cube law. Observations were performed that led Isaac Newton to conclude that it was an inverse-square law. An inverse-cube law would have led to different observations. A theory that says, for example, "blue is a triangular color", is unfalsifiable. One couldn't conceive of an experiment to test it. Unfalsifiable "scientific theories" are generally met with great skepticism.
- “Everything and more, a compact history of infinity” by David Foster Wallace (Weidenfeld, 2003)