the Rathouse
Series of Very Abbreviated Versions of Classical Philosophical Works for Very Busy People.
Chapter 2: The Nature of Philosophical Problems and their Roots in Science
"We must beware of mistaking the well-nigh senseless and pointless subtleties of the imitators for the pressing and genuine problems of the pioneer."

This chapter is based on a 1952 paper which Popper delivered to a meeting of the Philosophy of Science Group of the British Society for the History of Science.

He commenced by stating that he was not happy to talk about the present position of English philosophy because he thought it was more important to solve scientific or philosophical problems rather than talk about what he or other philosophers are doing or might do. Some years later it has turned out that this might have been a mistake because critical rationalism has been marginalised in the philosophy schools and it is possible that more attention to the state of play in the profession at large (and the games that are played) might have helped people who admired Popper's ideas to compete more effectively for an audience.

He moved on to talk about the nature of philosophical problems, after some comments about the description and classification of subjects.

"Disciplines are distinguished partly for historical reasons and reasons of administrative convenience (such as the organisation of teaching and appointments), and partly because the theories which we construct to solve our problems have a tendency to grow into unified systems. But all this classification and distinction is a comparatively unimportant and superficial affair. We are not students of some subject matter but students of problems. And problems may cut right across the borders of any subject matter or discipline."

His two main theses:

1. Every philosophy and every philosophical school is likely to degenerate under the influence of philosophical inbreeding to the point where its problems become practically indistinguishable from pseudo-problems and its talk becomes almost meaningless babble.

2. This is likely to happen as a result of what Popper called the "prima facie" method of teaching philosophy, that is, starting off by reading the works of the great philosophers without reference to the problem situation in science or mathematics or politics which concerned them.

It is likely that the situation has improved since that time due to the rise of the history and philosophy of science which has stimulated a great deal of research on ancient problem situations and the social and political context of the times, but the benefits have been dissipated by some of the ideological and sociological postures that are adopted in the reseach. It is interesting to note as an aside that Popper anticipated some of these tendencies notably the strong program in the sociology of science, in his chapter on the sociology of knowledge in The Open Society and its Enemies.

"What I mean by the prima facie method of teaching that of giving the beginner the works of the great philosophers to read; the works, say, of Plato and Aristotle, Descartes and Liebniz, Locke, Berkeley, Hume, Kant and Mill. What is the effect of such a course of reading? A world of astonishingly subtle and vast abstractions opens itself before the reader; abstractions of an extremely high and difficult level. Thoughts and arguments are put before his mind which sometimes are not only hard to understand, but which seem to him irrelevant because he cannot find out what they might be relevant to. Yet the student knows that these are the great philosophers, that this is the way of philosophy. Thus he will make an effort to adjust his mind to what he believes (mistekenly, as we shall see) to be their way of thinking. He will attempt to speak their queer language, to match the tortuous spiral of their argumentation, and perhaps even tie himself up in their curious knots."

He went on to argue that it is necessary to understand the real problems that concerned the great philosophers, not just the philosophical problems that they wrote about, and the real problems are actually problems in science and mathemtics, or in politics and law. The student needs to understand these problems, which means understanding the problem situation at the time, which means understanding some of the history of ideas.

He pursued his case with two examples, the first concerning Plato's theory of Forms and the crisis in Greek science following the discovery of the irrationality of the square root of two. The second is Kant's Critique of Pure Reason.

Kant's Problem

Taking the second example first, the setting for Kant's work was twofold (1) the apparently overwhelming triumph of Newtonian mechanics in the Principia and (2) Hume's critique of the empirical base of knowledge which woke Kant from "dogmatic slumber". The problem was the conflict between the apparent success of Newton in gaining true and unassailable knowledge of the cosmos, and Hume's scepticism about the logic of induction and hence the observational basis of our knowledge of the cosmos.

"Kant's proposed solution to this insoluble problem consisted of what he proudly called his 'Copernican Revolution' of the problem of knoweledge. Knowledge was possible because we are not passive receptors of sense data but their active digestors. By digesting and assimilating them we form and organise them into a Cosmos, the Universe of Nature...Thus our intellect does not discover laws in nature, but prescribes its own laws and imposes them upon nature. This theory is a strange mixure of absurdity and truth."

Popper's solution was of course to dissolve part (1) of Kant's dilemma, to insist that scientific knowledge, even that which appears to be as unassailable as Newton's theory at the time, is inevitably conjectural, so there is no need to speculate how it is that we can obtain certain and unassailable truths. Then if the best that we can manage is conjectural knowledge, Hume's scepticism about induction ceases to be a serious problem, thus taking the sting out of part (2) of Kant's problem situation.

The Geometric Program and the Theory of Forms

Turning to Popper's first example, I will draw on a previous account of the problem which Popper write up in some of the notes to The Open Society and its Enemies and in an Addendum to the 1962 edition.

"My thesis here is that Plato's central philosophical doctrine, the so-called Theory ofForms or Ideas, cannot be properly understood except in an extra-philosophical context; more especially in the context of the critical problem situation in Greek science which developed as a result of the discovery of the irrationality of the square root of two."

"It seems likely that Plato's Theory of Forms is both in its origin and in its content closely connected with the Pythagorean theory that all things are in essence numbers. The detail of this connection and the connection between Atomism and Pythagoreanism are perhaps not so well known."

From Addendum 1 to volume 2 of The Open Society.

I"n the second edition of this book, I made a lengthy addition to note 9 to chapter 6 (pp. 248 to 253). The historical hypothesis propounded in this note was later amplified in my paper 'The Nature of Philosophical Problems and Their Roots in Science' (British Journal for the Philosophy of Science, 3, 1952, pp. 1241 ff.; now also in my Conjectures and Refutations). It may be restated as follows: (1) the discovery of the irrationality of the square root of two which led to the breakdown of the Pythagorean programme of reducing geometry and cosmology (and presumably all knowledge) to arithmetic, produced a crisis in Greek mathematics; (2) Euclid's Elements are not a textbook of geometry, but rather the final attempt of the Platonic School to resolve this crisis by reconstructing the whole of mathematics and cosmology on a geometrical basis, in order to deal with the problem of irrationality systematically rather than ad hoc, thus inverting the Pythagorean programme of arithmetization; (3) it was Plato who first conceived the programme later carried out by Euclid: it was Plato who first recognized the need for a reconstruction; who chose geometry as the new basis, and the geometrical method of proportion as the new method; who drew up the programme for a geometrization of mathematics, including arithmetic, astronomy, and cosmology; and who became the founder of the geometrical picture of the world, and thereby also the founder of modern science-of the science of Copernicus, Galileo, Kepler, and Newton."

From the notes to chapter 6 of The Open Society.

"(2) According to tradition an inscription over the door of Plato's academy said: 'Nobody untrained in geometry may enter my house.' I suspect that the meaning of this is not merely an emphasis upon the importance of mathematical studies, but that it means: 'Arithmetic (i.e. more precisely, Pythagorean number theory) is not enough; you must know geometry!' And I shall attempt to sketch the reasons which make me believe that the latter phrase adequately sums up one of Plato's most important contributions to science. See also Addendum, p. 319."

"As is now generally believed, the earlier Pythagorean treatment of geometry adopted a method somewhat similar to the one nowadays called 'arithmetization'. Geometry was treated as part of the theory of integers (or 'natural' numbers, i.e. of numbers composed of monads or 'indivisible units'; cp. Republic, 525e) and of their logoi', i.e. their 'rational' proportions. For example, the Pythagorean rectangular triangles were those with sides in such rational proportions. (Examples are 3:4 15; or 5: 12: 13.) A general formula ascribed to Pythagoras is this: 2n + 1: 2n(n + 1): 2n(n + 1) + 1. But this formula, derived from the 'gnomon', is not general enough, as the example 8: 15: 17 shows. A general formula, from which the Pythagorean can be obtained by putting m = n + 1, is this: m2-n2: 2mn: m2 + n2 (where m > n). Since this formula is a close consequence of the so-called 'Theorem of Pythagoras' (if taken together with that kind of Algebra which seems to have been known to the early Pythagoreans), and since this formula was, apparently, not only unknown to Pythagoras but even to Plato (who proposed, according to Proclus, another non-general formula), it seems that the 'Theorem of Pythagoras' was not known, in its general form, to either Pythagoras or even to Plato. (See for a less radical view on this matter T. Heath, A History of Greek Mathematics, 1921, vol. I, pp. 80-2. The formula described by me as 'general' is essentially that of Euclid; it can be obtained from Heath's unnecessarily complicated formula on p. 82 by first obtaining the three sides of the triangle and by multiplying them by 2/mn, and then by substituting in the result m and n and p and q.)."

"The discovery of the irrationality of the square root of two (alluded to by Plato in the Greater Hippias and in the Meno; cp. note 10 to chapter 8; see also Aristotle, Anal. Priora, 413a6 f.) destroyed the Pythagorean programme of 'arithmetizing' geometry, and with it, it appears, the vitality of the Pythagorean Order itself. The tradition that this discovery was at first kept secret is, it seems, supported by the fact that Plato still calls the irrational at first 'arrhetos', i.e. the secret, the unmentionable mystery; cp. the Greater Hippias, 303b/c; Republic, 546c. (A later term is 'the non-commensurable'; cp. Theaetetus, 147c, and Laws, 820c. The term 'alogos' seems to occur first in Democritus, who wrote two books On Illogical Lines and Atoms (or and Full Bodies) which are lost; Plato knew the term, as proved by his somewhat disrespectful allusion to Democritus' title in the Republic, 534d, but never used it himself as a synonym for 'arrhetos'. The first extant and indubitable use in this sense is in Aristotle's Anal. Post., 76b9. See also T. Heath, op. cit., vol. I, pp. 84 f., 156 f. and my first Addendum on p. 319, below.)"

"It appears that the breakdown of the Pythagorean programme, i.e. of the arithmetical method of geometry, led to the development of the axiomatic method of Euclid, that is to say, of a new method which was on the one side designed to rescue, from the breakdown, what could be rescued (including the method of rational proof), and on the other side to accept the irreducibility of geometry to arithmetic. Assuming all this, it would seem highly probable that Plato's role in the transition from the older Pythagorean method to that of Euclid was an exceedingly important one-in fact, that Plato was one of the first to develop a specifically geometrical method aiming at rescuing what could be rescued from, and at cutting the losses of, the breakdown of Pythagoreanism. Much of this must be considered as a highly uncertain historical hypothesis, but some confirmation may be found in Aristotle, Anal. Post., 76b9 (mentioned above), especially if this passage is compared with the Laws, 818c, 895e (even and odd), and 819e/820a, 820c (incommensurable). The passage reads: 'Arithmetic assumes the meaning of "odd" and "even", geometry that of 'irrational"..' (Or 'incommensurable'; cp. Anal. Priora, 41a26 f., 503a7. See also Metaphysics, 983a20, 1061b1-3, where the problem of irrationality is treated as if it were the proprium of geometry, and 1089a, where, as in Anal. Post., 76b40, there is an allusion to the 'square foot' method of the Theaetetus, 147d.) Plato's great interest in the problem of irrationality is shown especially in two of the passages mentioned above, the Theaetetus, 147c-148a, and Laws, 819d-822d, where Plato declares that he is ashamed of the Greeks for not being alive to the great problem of incommensurable Magnitudes."

"Now I suggest that the 'Theory of the Primary Bodies' (in the Timaeus, 53c to 62c, and perhaps even down to 64a; see also Republic, 528b-d) was part of Plato's answer to the challenge. It preserves, on the one hand, the atomistic character of Pythagoreanism-the indivisible units ('monads') which also play a role in the school of the Atomists-and it introduces, on the other hand, the irrationalities (of the square roots of two and three) whose admission into the world had become unavoidable. It does so by taking two of the offending rectangular triangles-the one which is half of a square and incorporates the square root of two, and the one which is half of an equilateral triangle and incorporates the square root of three-as the units of which everything else is composed. Indeed, the doctrine that these two irrational triangles are the limits (peras; cp. Meno, 75d-76a) or Forms of all elementary physical bodies may be said to be one of the central physical doctrines of the Timaeus."

"All this would suggest that the warning against those untrained in geometry (an allusion to it may perhaps be found in the Timaeus, 54a) might have had the more pointed significance mentioned above, and that it may have been connected with the belief that geometry is something of higher importance than is arithmetic. (Cp. Timaeus, 31c.) And this, in turn, would explain why Plato's 'proportionate equality', said by him to be something more aristocratic than the democratic arithmetical or numerical equality, was later identified with the 'geometrical equality', mentioned by Plato in the Gorgias, 508a (cp. note 48 to this chapter), and why (for example by Plutarch, loc. cit.) arithmetic and geometry were associated with democracy and Spartan aristocracy respectively-in spite of the fact, then apparently forgotten, that the Pythagoreans had been as aristocratically minded as Plato himself; that their programme had stressed arithmetic; and that 'geometrical', in their language, is the name of a certain kind of numerical (i.e. arithmetical) proportion."

"It may be suggested that all this exegesis of Plato and cognate texts is a long way from the issues in epistemology that concern us at the present time. That may be true, and Popper's theory needs to be tested by more examples, by exploring whether contemporary debates in epistemology can be understood in isolation from problems in science or elsewhere. In defence of Popper's interest in the Platonic Theory of Forms, that is a central part of his exegesis of Plato's political thought in the first volume of The Open Society. It is my contention that the ideas in that book have huge contemporary relevance as we try to control the destructive force of defective ideas about the proper function of government."


These are some of my (RC) thoughts on the way that philosphy could be introduced. From the end of the article Cracking the Dogmatic Framework of Thought.

Philosophy could be introduced as a study of critical thinking and Bartley has proposed four types of criticism or tests that may be applied to arguments. These are the test of experience; the test of comparison with other theories; the check on the problem; and the test of logical consistency. None of these tests or checks are unproblematical and Bartley refers to them as 'non-justificationist criticism'.

The study of critical thinking that is proposed here could be taught at school, it could be used for an introduction to university courses in philosophy, it could be a core subject for all tertiary students. Its content could be adjusted for the interests and capacities of the class and it offers an alternative to the debacle of general studies where students of marketing and organic chemistry have to shuffle and fidget for a certain number of hours in lectures on Introductory Psychology or Medieval Drama. The course would consist of exploration and applications of the four methods of criticism to any theories or beliefs which interest the class.
The test of evidence and experience could lead to the philosophy of science, to a study of rules of evidence in law, to the use of diagnostic tests by doctors, motor mechanics or plumbers, and to the use of clues by detectives and archeologists.

The test of comparison with other theories would raise questions about the weight and authority to be assigned to assumptions imported into arguments, more or less uncritically, from other domains. For example the psychological theories assumed by literary critics, the physical theories assumed by geologists, the sociological theories assumed by engineers, the economic theories assumed by politicians. This part of the course should open student's eyes to the inter-dependence of the so-called disciplines and with any luck the artificial nature of boundaries between subjects would become apparent. At the same time students may learn how to use readily available resources, including other students and staff, to pursue problems from one discipline to another (for example by walking from the Philosophy Department to Physics or Life Sciences).

The check on the problem is in some ways the most fundamental criticism of all. This part of the course would indicate how a revised formulation of a problem may be decisive, how background theories can unconsciously direct how problems are identified and formulated, how fashions and fads (and funding) can dictate the directions of intellectual effort. It would lead to a study of the history of ideas, showing that problems have histories, that philosophical problems usually have their roots elsewhere, in science, or religion or in social and moral dilemmas, that powerful themes can leak from one discipline to another and preoccupations often run in parallel in more than one field.

The section on logic would call for study of both the formal and informal methods of argument. Formal logic concerns rules of inference and the way that logical steps can be used to draw out the consequences of an argument or of a scientific theory, perhaps for testing or for technological application. Informal logic encompasses the tricks of debate that may be used to cover up logical and factual defects in a position. Discourse by politicians, theologians, creation scientists and advertisers would furnish material for critical study.

All of this could lead to exploratory reading of the Great Philosophers, though preferably not until the students have a firm sense of their own interests and problems. In this mood they might be less deferential to the greats, more critical and at the same time more willing to learn. This would contrast with the traditional situation where the young student is confronted with soaring abstractions and profound arguments utterly unconnected with the historical background or the problem situations which agitated the titans of the past. The novice is completely overwhelmed (who am I to criticise the great?) or else clings to a critique provided by the teacher. The usual result is either a student who is indoctrinated into a system of thought, or else a person who is skilled in certain methods and techniques without any sense of purpose or perspective. No course can be rendered failsafe against authoritarian teachers, or against complete lack of interest on the part of students but the approach sketched above would provide interested people with a chance to avoid the more obvious dead ends of contemporary philosophy, and to apply imaginative criticism to their own professional and personal concerns.

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Conjectures and Refutations
Karl Popper