A priori and a posteriori

The Latin phrases a priori and a posteriori are philosophical terms of art popularized by Immanuel Kant's Critique of Pure Reason. They are used with respect to reasoning (epistemology) to distinguish necessary conclusions from first premises (i.e., what must come before sense observation) from conclusions based on sense observation (which must follow it). Thus, the two kinds of knowledge, justification, or argument may be glossed:
 * A priori knowledge or justification is independent of experience, as with mathematics (2+2=4), tautologies ("All bachelors are unmarried"), and deduction from pure reason.
 * A posteriori knowledge or justification is dependent on experience or empirical evidence, as with most aspects of science (evolution) and personal knowledge.

Quotes

 * It is much more high and philosophical to discover things a priori than a posteriori. And therefore the Peripatetics have not been very solicitous to gather experiments to prove their doctrines, contenting themselves with a few only, to satisfy those that are not capable of a nobler conviction. And indeed they employ experiments rather to illustrate than to demonstrate their doctrines.
 * Robert Boyle, The Sceptical Chymist, Introductory Preface

These views on space as professed by the greatest scientists are in large measure to be attributed to the discoveries of non-Euclidean geometry supplemented by the investigations of the psychophysicists. ...By the time men are of an age to philosophise, they have been subjected for so many years to beliefs based on inferences from experience, that the beliefs have remained, whereas the inferences, owing to the monotony of their repetition, have become second nature and appear intuitive. ... Were three-dimensional Euclidean space an a priori condition of the understanding, it would have been quite impossible for mathematicians to wend their way through the non-Euclidean hyperspaces of relativity. Neither can three-dimensional space be considered to be imposed by sensibility, since, as Poincaré tells us, after a certain amount of perseverance, he was aided to a considerable degree by sensibility when investigating the problems of Analysis Situs of four dimensions.
 * Up to this point we have restricted our attention to space as a mere extension. But space, as understood in common practice, implies considerably more: it represents a three-dimensional Euclidean continuum. When thus particularised, Kant's arguments as to its a priori character are no longer tenable in the light of modern discovery; and we must assume that this special form we credit to space arises entirely from our co-ordination of sense impressions conducted in the simplest way possible. On no account may we consider three-dimensional Euclidean space to be imposed a priori either by sensibility or by the understanding. ...it is generally conceded by scientists that the a priori doctrine of three-dimensional Euclidean space is one of the most pernicious teachings that philosophy has ever attempted to impose upon science. ...
 * A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein (1927)


 * The only justification for our concepts and system of concepts is that they serve to represent the complex of our experiences; beyond this they have no legitimacy. I am convinced that the philosophers have had a harmful effect upon the progress of scientific thinking in removing certain fundamental concepts from the domain of empiricism, where they are under our control, to the intangible heights of the a priori. For even if it should appear that the universe of ideas cannot be deduced from experience by logical means, but is, in a sense, a creation of the human mind, without which no science is possible, nevertheless this universe of ideas is just as little independent of the nature of our experiences as clothes are of the form of the human body. This is particularly true of our concepts of time and space, which physicists have been obliged by the facts to bring down from the Olympus of the a priori in order to adjust them and put them in a serviceable condition.
 * Albert Einstein, Four Lectures Delivered at Princeton University (May, 1921) as quoted in The Meaning of Relativity (1922)


 * As an interpreter of nature... Leibnitz stands in no comparison with Newton. His general views in physics were vague and unsatisfactory; he had no great value for inductive reasoning; it was not the way of arriving at truth which he was accustomed to take; and hence, to the greatest physical discovery of that age, and that which was established by the most ample induction, the existence of gravity as a fact in which all bodies agree, he was always incredulous, because no proof of it, a priori could be given.
 * John Playfair, Dissertations on the History of Metaphysical and Ethical: And of Mathematical and Physical Science, Vol. 1. Dissertation Third, p. 572.


 * There is nothing physical to be learned a priori. We have no right whatever to ascertain a single physical truth without seeking for it physically, unless it be a necessary consequence of other truths already acquired by experiment, in which case mathematical reasoning is alone requisite.
 * Peter Guthrie Tait, Lectures on Some Recent Advances in Physical Science, With a Special Lecture on Force, (3rd edition) Lecture I (p. 6)


 * Behold the mighty Dinosaur, Famous in prehistoric lore, Not only for his weight and strength But for his intellectual length. You will observe by these remains The creature had two sets of brains— One in his head (the usual place), The other at his spinal base. Thus he could reason a priori As well as a posteriori.
 * Bert Leston Taylor, A Line-o’-verse Or Two, The Dinosaur