The most important philosopher of science since Francis Bacon (1561-1626), Sir Karl Popper finally solved the puzzle of scientific method, which in practice had never seemed to conform to the principles or logic described by Bacon -- see The Great Devonian Controversy, by Martin J. S. Rudwick, for a case study of Baconian rhetoric and expectations being contradicted by actual practice and results. Instead of scientific knowledge being discovered and verified by way of inductive generalizations, leaping from perceptual data into blank minds, in terms that go back to Aristotle, Popper realized that science advances instead by deductive falsification through a process of "conjectures and refutations." It is imagination and creativity, not induction, that generates real scientific theories, which is how Einstein could study the universe with no more than a piece of chalk. Experiment and observation test theories, not produce them. This was not, in retrospect, so hard to understand; and some philosophers, like Kant, had come close to recognizing it. It is still subject to some dispute, though mainly from those who misunderstand the rejection of induction or who demand positive epistemic reasons for crediting theories that are derived negatively, by falsification (see "Criticism of Karl Popper in Anthony O'Hear's An Introduction to the Philosophy of Science"). That is a reasonable enough demand, but the answer can only come from deeper philosophical epistemology, not from science or philosophy of science. That deeper epistemology is, in fact, Friesian.
The most important question about Popper for The Proceedings of the Friesian School is why he says that, rather than a Positivist, he is more a Kantian in the Friesian tradition:
It seems to me that the view here upheld is closer to that of the 'critical' (Kantian) school of philosophy (perhaps in the form represented by Fries) than to positivism. [The Logic of Scientific Discovery, Hutchinson of London Ltd., 1959, 1977, p.105 note]
He undoubtedly says this because Fries had held that synthetic propositions a priori, or the First Principles of Demonstration, do not need to be proven. This follows from Hume and Kant's definition of "synthetic" (can be denied without contradiction) and from Aristotle's definition of "First Principles" (are not justified by derivation from other propositions). Since Popper thought that justification works through falsification, and never through verification, he obviously agreed that such propositions do not need to be proven in the sense of logical derivation. It is now common in science to use falsifiability as a criterion for dismissing theories or claims as parts of science. Popper's own critique of Marx and Freud as unfalsifiable was a classic study, and the salutary influence of the principle in discussion of psychics or astrology is occasionally seen.
Popper, however, misunderstands the rest of Fries's theory, accusing him of "psychologism" in the sense that Fries supposedly relies on a psychological or subjective sense of certainty to justify instances of immediate knowledge. This is not true. On the very page cited above, Popper says:
I admit, again, that the decision to accept a basic statement, and to be satisfied with it, is causally connected with our experiences -- especially with our perceptual experiences. But we do not attempt to justify basic statements by these experiences. [ibid. p.105, boldface added]
Whether or not there is a causal relation between perceptions and statements or beliefs is actually irrelevant, and Popper commits a grave error by dwelling on it. We justify statements about experience by reference to the objects of experience. There is in fact no other way to justify them except by memory, hearsay, or inference. Reference is not a causal relationship but a fundamental logical property of concepts and propositions, as Popper well understands. The problem with reference to the objects of experience, as Descartes discovered, was the question of access to those objects. Popper's mistake, in criticizing the Positivists, was to accept a Positivist, and Empiricist, premise, that we only have access to perceptions, to contents of the mind, not to the objects themselves. Popper misses the Kantian aspect of Friesian theory that immediate knowledge consists of phenomenal objects, which as objects, are not merely psychological or subjective. One's psychological attitude, or its origin, is therefore irrelevant; and the cognitive force of immediate knowledge lies in the intersubjective availability of empirical objects, our direct acquaintance with them, and the possibility of their being shown to others by way of justification. (These issues are discussed in the essay "Ontological Undecidability.") Furthermore, Popper himself realized that the test of falsification cannot be applied to everything, for it is not clear how the principle of falsification itself could be subject to a falsifying test. If the principle can then be known to be true, there must be some means of verification for certain things after all. That must return us to Fries' original considerations.
Popper's connection with Nelson's Neo-Friesian School is recounted in his Unended Quest, an Intellectual Autobiography [Open Court, 1985]:
I had met Julius Kraft (of Hanover, a distant relation of mine, and a pupil of Leonard Nelson [note]), who later became a teacher of philosophy and sociology at Frankfurt; my friendship with him lasted until his death in 1960.
Julius Kraft, like Leonard Nelson, was a non-Marxist socialist, and about half our discussions, often lasting into the small hours of the morning, were centered on my criticism of Marx. The other half were about the theory of knowledge: mainly Kant's so-called "transcendental deduction" (which I regarded as question-begging), his solution to the antinomies, and Nelson's "Impossibility of the Theory of Knowledge". Over these we fought a hard battle, which went on from 1926 to 1956, and we did not reach anything approaching agreement until a few years before his untimely death in 1960. On Marxism we reached agreement fairly soon. [pp. 74-75.]
Nelson, as it happens, regarded the "transcendental deduction" as question begging also, where the idea of "transcendental" knowledge dangerously obscured the question whether such knowledge would be synthetic a priori or a posteriori. If the former, an argument to prove the existence of synthetic a priori knowledge would be circular; but if the latter, it would then seem that necessary a priori knowledge would somehow be certified by contingent and a posteriori knowledge. The Friesian theory of Deduction addresses that dilemma.
The solution to the antinomies was the view that phenomenal objects were not things in themselves, which Popper evidently did not want to accept. And the argument of Nelson's "Impossibility of the Theory of Knowledge" was simply that a theory of knowledge cannot prove that knowledge exists, for it would have to presuppose the possibility that the existence or non-existence of knowledge could be known, which would beg the question.
What Popper finally got Kraft to agree to is not stated, though the implication is that Popper carried the argument. Such an outcome, sadly, seems to have spelled the end of Nelson's own students taking Friesian epistemological principles seriously, which left very little for which to recommend Fries and Nelson to the philosophical community or the general public.
Finally, in Nelson's tradition of political activity, Popper stands as one of the great opponents of totalitarianism, and its philosophical roots, in the century. The Open Society and Its Enemies , like the nearly contemporaneous Road to Serfdom  of F.A. Hayek (who used Popper's ideas and helped him professionally), were seminal works in the history of liberty and the counterattack against the intellectual and political assaults, from both political right and left, on freedom and classical liberal principles in the Twentieth Century. As with Nelson's work, Popper's exposure of the irrationality and danger of the doctrines of philosophers like Hegel and Marx has yet to have its influence properly felt. Unfortunately, Popper did not accept Hayek's own defense of free market capitalism, and both he and most of his recent followers have accepted the muddled welfare-statism whose principles are more or less assumed as truisms in most political debate in Europe and the United States. Below is a link to an exchange with an important associate and supporter of Popper, funded by philanthropist George Soros, which began hopefully but then turned acrimonious as the differences between Popper and Hayek emerged. Indeed, it was surprising and disturbing just how acrimonious the exchange quickly became -- reflecting, perhaps, the reenergized hostility to capitalism found in the recent left (and thoughtlessly aided by some successful capitalists, like Mr. Soros). How Karl Popper, one of the greatest critics of totalitarianism, could be now used in the defense of statism and serfdom (as Hayek put it), is a very tragic development.
Letter from Sir Karl Popper to K.L. Ross, 12 December 1992
The Friesian Trilemma
Criticism of Karl Popper in Anthony O'Hear's An Introduction to the Philosophy of Science
Criticism of Karl Popper in Martin Gardner's Are Universes Thicker Than Blackberries?
Exchange with Mark Notturo on Popper and Hayek
History of Philosophy
Philosophy of Science
Popper on Home Page
The Karl Popper Web
Leonard Nelson was an outstanding personality, one of the small band of Kantians in Germany who had opposed the First World War, and who upheld the Kantian tradition of rationality.
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