Where does induction collide?

The form of reasoning that moves from particular observations to a general conclusion, where the evidence makes the conclusion probable but never guarantees it: many observed white swans support the claim that all swans are white without proving it. David Hume's 1748 An Enquiry Concerning Human Understanding pressed the problem of induction, asking what justifies the leap from past cases to unobserved ones, and argued that no reasoning makes the conclusion certain. The referent is a pattern of uncertain, evidence-driven inference in logic and epistemology, and it is the exact opposite of the mathematical method that shares the word, which delivers certainty rather than probability; it has nothing to do with a magnetic field or a signaling embryonic cell.[1]

Unit

A single inductive inference from observed cases to a general claim

Recovery means

Not applicable

Measured by

Appraised by inductive logic and probability: how strongly the evidence raises the likelihood of the conclusion

A rigorous method of proof that establishes a statement for every natural number in two steps: showing it holds for a starting case, then showing that whenever it holds for one number it must hold for the next. Bussey's 1917 The Origin of Mathematical Induction traced the history of the method and set out its precise logical form. Despite sharing the word with the philosopher's induction, this is not probable reasoning at all: the referent is a deductive technique that yields complete certainty across infinitely many cases, whereas inductive reasoning leaves its conclusions merely likely, and it is unrelated to a changing magnetic field or a developing embryo.[2]

Unit

A single proof by starting case and inductive step, covering all natural numbers

Recovery means

Not applicable

Measured by

Checked by verifying the base case and the inductive step are each valid, which together entail the general result

The production of a voltage, called an electromotive force, across a conductor when the magnetic field passing through it changes: move a magnet near a coil of wire and a current flows in the coil. Faraday's 1832 Experimental Researches in Electricity reported the effect and the law relating the induced force to how fast the field changes, the principle behind electric generators and transformers. The referent is a physical effect in wires and magnets, measured in volts, and it has nothing to do with a form of argument, a proof technique, or one cell steering another; here the word names a cause and effect in the laboratory, not a step in reasoning.[3]

Unit

An induced electromotive force across a conductor, in volts

Recovery means

The induced force falls to zero once the magnetic field stops changing

Measured by

Measured as the voltage or current induced in a circuit, and predicted from the rate of change of magnetic flux through it

The process by which one group of embryonic cells directs the developmental fate of neighboring cells through chemical signals, so that a tissue forms only where it receives the right cue. Spemann and Mangold's 1924 Über Induktion von Embryonalanlagen durch Implantation artfremder Organisatoren showed that a small region, the organizer, when grafted into a second embryo could induce a whole extra body axis, founding the study of induction in development. The referent is a signaling relationship between living cells, and despite the shared word it is not an inference, a proof, or a voltage: it names how an embryo builds its body, not how anyone reasons and not how a magnet drives a current.[4]

Unit

One inductive interaction: a signaling tissue and the responding cells whose fate it changes

Recovery means

Not applicable

Measured by

Demonstrated by grafting or removing the signaling tissue and observing whether the neighboring cells change their fate