# Dorothy Wrinch's early papers

We give below a list of some of the papers which Dorothy Wrinch published between 1917 and 1927. For each of those we list we give a short extract, usually the introductory paragraphs, but sometimes, when it seemed more appropriate, a short extract from the conclusions. Let us note that we list only 23 of around 50 papers Wrinch published between 1917 and 1927.

1. Dorothy Maud Wrinch, Mr Russell's Lowell Lecture, Mind 26 (104) (1917), 448-452.

In this paper I propose to consider a few of the criticisms of Mr Russell's Lowell Lectures brought forward by Prof Saunders in the issue of Mind for January, 1917. Prof Saunders attempts to show that on purely general grounds Mr Russell's results in this book are of little or no philosophical value. It will be my object to prove that Prof Saunders has not been successful in his attempt.

It appears to be a fact that, on reflexion, it is much more difficult to doubt some kinds of propositions than others, and to doubt the existence of some kinds of things than others. Thus it seems to be more difficult to doubt the existence of our own sense-data than to doubt the existence of other people's sense-data or the existence of unperceived sensibilia. Again on reflexion, the existence of sense-data experienced by us in the immediate past seems less open to question than the existence of points and instants and physical objects which are never given in experience. Then it seems to be much more difficult to doubt propositions asserting that certain spatial or temporal relations hold between sense-data than to doubt propositions about other people's minds and mental states. Further the very hardest propositions to doubt seem to be the Laws of Logic. Thus there is an obvious sense in which the collection of data that Mr Russell has specified can be called "hard" data - "data which resist the solvent influence of critical reflexion," and they can quite properly, I think, be called comparatively certain. It seems too, as far as one person can judge, that our own sense-data and the Laws of Logic are the hardest of these hard data and that it is a fact that "the more we reflect upon these, the more we realise exactly what they are, and exactly what a doubt concerning them really means, the more luminously certain do they become". There is therefore a fairly well-defined body of data which appear to have a comparatively high degree of certainty. In view of the fact that the distinction between "hard" and "soft" data is one of degree it would be impossible to give a precise criterion of whether or not a certain datum is "hard". In his criticism of Mr Russell's premisses Prof Saunders has. not, I think, adduced any important considerations which cast any doubts on the hardness of the data in question: but just as one can only put forward arguments of a psychological nature. in support of Mr Russell's position on this point, so one can only use the same kind of weapons against it.

2. Dorothy Maud Wrinch, Bernard Bolzano (1781-1848), The Monist 27 (1) (1917), 83-104.

In Bolzano we find the virtues of human sympathy and insight coupled with the austerer virtues of the metaphysician and logician. He was a man of action as well as a man of ideas. He was well known for his kindly disposition and his broadmindedness. He possessed not only the sympathy with the poor necessary for a social reformer, but the ability to develop his ideas of social re construction on practical lines. Not only did he elaborate a theory of an ideal state, but he also introduced numerous reforms in the actual state of which he was a member. He studied theology very earnestly as a young man and later wrote a great deal on the subject. Even though his liberal views brought him into collision with those on whom his livelihood depended, yet he courageously continued his teaching and writing, always making it his aim to seek for truth. He was a metaphysician of some importance and his treatises on metaphysics are valuable, not only for the original thought which they contain, but also for his important criticisms of Kant. In aesthetics his work is by no means without interest, and to the psychology and ethics of his day he made very valuable contributions. But pre-eminently he was a mathematician and logician. In his work on mathematical analysis and mathematical logic, he stood out from all the other thinkers of his day. He was a man of many ideas and his intellectual equipment made him able to indicate to his followers the most fruitful lines of inquiry. All through his life he worked for the good of mankind, helping it on in its search for truth.

3. Dorothy Maud Wrinch, On the Summation of Pleasures, Proceedings of the Aristotelian Society, New Series 18 (1917-1918), 589-594.

Any discussion of the question of pleasure seems to be beset with difficulties. But there is one very interesting point which I hope I may be allowed to touch on, without entering into the intricate and pressing problems ordinarily discussed. I refer to the summation of pleasures. It is possible that the considerations to be urged in this paper apply to other kinds of value; but it seemed best to take pleasure only. I will assume that pleasures can be arranged in serial order, and I will pass over entirely the complicated questions which are involved in any attempt to justify such an assumption. This assumption is, of course, not so large as the assumption that pleasure is quantitative. Pleasures can be arranged in serial order if there is a relation "less than" which subsists between different pleasures, such that (1) a pleasure cannot be less than itself, (2) any pleasure less than a pleasure, itself less than a third pleasure, is less than the third pleasure, (3) of any two unequal pleasures, one is less than the other.

Now, the question I wish to discuss is this: Can the pleasure of several experiences together be expressed in all cases in terms of the pleasure of the experiences separately?

4. Dorothy Maud Wrinch, On the Nature of Judgment, Mind 28 (111) (1919), 319-329.

In putting forward this theory of judgment, my aim is not to offer criticism of Mr Russell's theory of judgment, nor yet to estimate its plausibility; I rather wish to offer suggestions as to the ways in which his idea for dealing with judgments of the form "aRb" can be extended so as to enable us to deal with more complicated judgments. Although I shall not be able to claim that I have dealt exhaustively with the various developments of which the idea that judgment is a multiple relation is capable - I shall try, at any rate, to refer to the various classes of possibilities which suggest themselves. I shall not attempt in this paper to give any answer to the question as to the truth of the theory: I am only going to try to show how it might be made to work. Whether or not the theory can be made to work (quite apart from whether or not the theory is true), depends, I hope to show, on various rather obscure questions. I shall content myself with showing that the answers given to these questions do determine the workableness. of the theory, and I shall not attempt at present to investigate the answers to them in any serious spirit.

But, in case, some may feel that the propositional theory of judgment as a dual relation is fairly satisfactory, and that any other theory is. so far unnecessary and without interest, may I suggest that in making up a theory to fit certain facts, if all the relevant facts are included, then there are none left by means of which one can judge between different theories, each of which fits in with all the given facts. There is no reason, I think, to believe that there is only one theory which can satisfactorily account for a certain group of facts. In view of this, it seems to me of interest to investigate how far this theory of judgment could be made satisfactory even if one is satisfied to some extent with some other theory, though one's unsatisfied desire if no suitable theory of judgment has been found would doubtless lend a stronger interest to this inquiry.

5. Dorothy Maud Wrinch, Existence, The Monist 29 (1) (1919), 141-145.

I am venturing to put forward for your consideration a few remarks on "Existence," because it appears to me that many questions of no little importance have been unduly complicated by too crude an idea of "existence."

In discussing different kinds of existence, it will be convenient first to discuss that kind of existence which is significantly predicable of what are called "sense-particulars." I will call this first kind of existence "primary existence." Primary existence is always implied in the naming of anything. Thus one cannot give a proper name to anything not having primary existence. But this kind of primary existence which can, I think, significantly be predicated of sense-particulars must be carefully distinguished from reality and non-reality, which are sometimes asserted of the same kind of thing. To call one's own sense-data real in waking life and unreal in dreams is not to be identified with asserting primary existence of some of one's sense-data, and not of others. All our sense-data have equally primary existence. Why we call some real and some unreal is because some give the usual kind of correlation with other sense-data, and some do not. Ghosts which cannot be touched are "unreal." Macbeth's dagger is "unreal" because the correlation with sense-data of touch which one has learned to expect, does not in fact occur in these cases. It may be, on the other hand, that the matter of correlation is merely unusual, as for example if one dreamt night after night of a certain object being at a certain place: i.e., if one's sense-data were correlated in a certain way. Then if one went to the place at which the sense-data ordinarily would have led one to expect to see certain sense-data correlated with the others in a particular way, one might not experience such a sense-datum. Again, if in what is called "waking life" one saw sense-data at different times which one could correlate by saying that they all belong to the sun, one would probably see certain sense-data as expected at other times; but if the sense-data had only been given in dreams, any reference of that kind would very probably be fallacious. Thus sense-data are said to be unreal when inferences usually true turn out to be false. Thus while it appears essential to predicate primary existence of all sense-data with which we are acquainted, sense-data can be said to be real or unreal in a definite sense.

6. Dorothy Maud Wrinch and Harold Jeffreys, On Some Aspects of the Theory of Probability, Philosophical Magazine 38 (1919)715-731.

The theory of probability suffers at the present time from the existence of several different points of view, whose relations to one another have apparently never been adequately discussed. On the one hand some authorities follow de Morgan and Jevons in regarding probability as a concept comprehensible without any definition, and perhaps indefinable, satisfying certain definite laws the logical basis of which is not yet clear. On the other hand, attempts have been made to give definitions of probability in terms of frequency of occurrence; of these one is due to Laplace, who was largely followed by Boole, and another to Venn. Frequency of occurrence being a well-understood mathematical concept, such a definition would be important if it could be carried out; for then the undefined notion of probability would be expressed in terms of others that are better understood, and its laws, if true, would become demonstrable theorems in pure mathematics instead of postulates. Thus the subject would acquire the certainty of any other portion of pure mathematics and it would be unnecessary to investigate its foundations independently. It appears, however, as we hope to show in the first part of the present paper, that the definitions offered either implicitly involve the very notion they are meant to avoid, or else make assumptions which are actually erroneous. We therefore consider it best to regard probability as a primitive notion not requiring definition.

Laplace defines probability as the ratio of the number of favourable cases to that of all possible cases, and then goes on to say "but that supposes the various cases equally possible," so that to understand this definition it is necessary to examine what Laplace meant by equally possible. The expression is meaningless as it stands, for a proposition relative to a set of data is always either possible or impossible; there can be no degrees of possibility. He indicates later that if a coin is unsymmetrical the probability of throwing a head may be greater than that of throwing a tail, though the difference may be small; yet, both are possible. In fact it seems that by equally possible he meant equally probable. Thus, as Poincaré has pointed out, it seems useless to attempt to make this definition satisfactory; it defines the probability of one proposition in terms of those of a set of others and not in terms of frequency alone, so that the notion Laplace set out to define reappears in the undefined concept of equally possible. The statement is, in fact, not a definition, but a simple and important principle of probability inference. Nor does it appear that there is any prospect of making any modification of it into a definition of probability; for there will always be the difficulty of deciding what are to be considered as unit alternatives. It is clear that even if it were possible to avoid introducing the notion of equally probable alternatives, some other way of distinguishing between sets of mutually exclusive and exhaustive alternatives would have to be found, and the immense variety of the circumstances to which it would have to apply seems to indicate that its scope must be at least as wide as that of truth; and it is very unlikely that a notion so general is capable of definition.

The view of Venn is much more complex. He considers that the notion presupposes a series, the terms of which are indefinitely numerous and represent the cases of an attribute $\phi$. From these one can pick out a smaller class, the members of which possess the further attribute $\psi$. If, then, we have chosen $n$ members in all and $m$ of them belong to the smaller class, the probability of $\psi$ given $\phi$ is defined as the limit of $\large\frac{m}{n}\normalsize$ when $n$ becomes indefinitely great. The form of this definition restricts the field of probability very seriously. In the first place it seems impossible to apply it to any case where the number of members of the first series is finite; one could attach no meaning to a statement that it is probable that the solar system was formed by the disruptive approach of a star larger than the sun, or that it is improbable that the stellar universe is symmetrical, for the indefinite repetition of entities of such large dimensions is utterly fantastic. Yet such cases as these are the very ones where the notion of probability is particularly valuable in science, and any definition that will not cover them is not satisfactory.

It may be urged, however, that this theory gives an adequate treatment of probability as applied to the class of cases with which it deals. Serious difficulties nevertheless present themselves. The existence of a probability on this theory requires that a limit shall exist to which a certain ratio tends in the long run; and one is led to ask what the evidence is for the existence of such a limit.

7. Dorothy Maud Wrinch, On the Nature of Memory, Mind, New Series 29 (113) (1920), 46-61.

In beginning a study of the phenomena of memory, it is expedient first to point out an ambiguity in the words "memory" and "remembering" as ordinarily used. Suppose I say "I remember the face of the girl I saw yesterday". I may mean one or other of two things. I may mean that a definite phenomenon is occurring which may be called "a memory of the face of the girl I saw yesterday". On the other hand, I may mean that I could produce a phenomenon of this kind. With the second meaning of the word, the fact of my remembering the so-and-so is a fact of the form: under certain circumstances, I shall have an act which is a remembering of the so-and-so in the first sense. A memory of the second kind can be called a dispositional memory and one of the first kind a memory act. A dispositional memory, then, can be said to be a possibility of memory acts. This same ambiguity occurs in the case of knowing and a differentiation of knowings into dispositional knowings, and acts of knowing is a necessary prelude to any investigation of the nature of knowings in general.
...
Since the question of the relation between acts and their corresponding dispositions is one relevant not only to the case of memory, but also to very many other groups of psychological phenomena, it is best to leave the discussion of its nature and to confine ourselves in this enquiry into the nature of memory, to a discussion of memory acts.

Two artificial restrictions on the field of memory acts are to be made in this paper. First, I wish to discuss only those memory acts in which images occur: and in the second place, I wish to limit the class of memory acts to be discussed to those memory acts, which are memories of physical objects or of events of the same status. One object of the first restriction is to exclude at once those acts in which sense-data occur. I mean cases where, e.g., one remembers a picture on seeing it again. These seem to me best called recognitions: though they share to some extent the properties of other memory acts, it is more convenient for the sake of the adequate discussion of properties they do not share, to discuss them separately. The other object of the first restriction is to exclude those cases of memory acts (if there are any such) in which there is no image element - the kind of occurrence that "remembering an idea" might possibly be. It does not seem clear that in every memory act an image occurs: it was therefore deemed best to make a restriction on the field of our enquiry which would exclude such imageless memory acts if they existed.

8. Dorothy Maud Wrinch, On the Theory of Probabilities, The Monist 30 (4) (1920), 618-623.

A study of philosophical literature and periodicals of the last thirty years leads to the conclusion that the theory of probability has made little progress since the time of De Morgan and Jevons. Coupled with the lack of progress there has been an enormous development in the use of statistics, and there is other evidence of a growth of interest in the practical problems of probability. A large extension in the applications of a science, without a corresponding growth of interest in its principles, is sometimes dangerous. In the particular case of the theory of probability it is clear that a serious re-examination of its principles is urgently needed. A hint of a breeze in this direction is to be seen in Mind, October, 1918, where Prof C D Broad restates the question of the relation of probability and induction. It is, I think, worth while to make a few general remarks in the lull before Professor Broad's lead is taken up and the whole question put on a more satisfactory basis.

Probability propositions are to be distinguished by the fact that they involve certain special notions. It is necessary first of all to discuss their form in order to see the nature of the entities to which the notions of probability can significantly be applied, and then to discuss what primitive ideas the system needs. It has been recognised that in all estimates of probability there is an implicit or explicit reference to a state of knowledge relative to which the probability of some proposition is to be assessed. Probability in general has reference to two propositions; the fundamental notion is then the probability of one proposition being true assuming another is true, or (as it may be more conveniently expressed) the probability of one proposition on another.

The propositions consist in a relation 'less than', 'greater than' or 'equal to' between probabilities, or in the evaluation of the probability of a certain proposition on a certain ground. It will be important to discuss the relation between these notions. We may, e.g., tentatively suggest that it may be necessary to take only one of these notions, 'the probability of', 'more probable than', 'less probable than', 'as probable as', as a primitive idea in the system. For we may use the relation 'more probable than' as primitive, defining 'less probable than' as its converse and 'as probable as' as the negation of the sum of these relations. (This is only satisfactory if to say that one entity is more probable than another, if it is neither more nor less probable than the other entity, is to give a permissible interpretation of the notions involved.) Then the probability of an entity will be defined as the class of equally probable entities. The problems of the nature of entities to which these notions can significantly be applied, and of the relation between them, complete the first division of the theory.

9. Dorothy Maud Wrinch, On the Structure of Scientific Inquiry, Proceedings of the Aristotelian Society, New Series 21 (1920-1921), 181-210.

The problems of scientific methodology have always claimed the serious attention of our Society, so that though I have no observations of a metaphysical nature to present to-night, I offer no apology.

The object of this paper is the study, in some of its aspects, of the structure of scientific inquiry. The subject is a large one and covers the major part of the field of scientific methodology. We shall confine our attention more particularly to the consideration of those problems of scientific structure which occur in the case of more advanced sciences and merely outline the corresponding problems which occur primarily in sciences in the more elementary stage.

The data of science as presented in experiment and observation have many interesting characteristics. These have been discussed by many writers. Whatever else may be true of these results of experiment and observation, it is quite evident that they are discrete and particular. We get some characteristic predicated of some term, some other of another term. We assert that this particular object which is under consideration has a certain property. But no light whatever can be thrown on scientific problems by statements of this kind alone. The fact that the deflection of light from a great distance, caused by passage through the sun's gravitational field is 1.74", is not in itself a fact significant for science. It is only when we begin to deal with general propositions regarding the behaviour of objects that we are entering the field of science. The proposition that bodies fall to the ground can be a starting point for science equally with a proposition as to the dates at which birds begin to sing in the spring or a proposition as to the behaviour of any particular substance in water. An observation of one body falling to the ground or of one bird singing, or of one case of a substance diffusing in water, are not in themselves of scientific significance. In science we seek to order the phenomena of the world into classes and subsequently to order these classes among themselves. The elementary stage of science deals with the problem of collecting phenomena together and ordering them in classes as, for example, in the general proposition as to the dates at which birds begin to sing in the spring. The central topic to be discussed in a study of the structure of science at this stage is the nature of the relations between the particular propositions which our experience yields and the general propositions with which science opens.

10. Dorothy Maud Wrinch and Harold Jeffreys, On Some Aspects of the Theory of Probability, Philosophical Magazine 42 (1921)369-390.

In order that a scientific method may be of any value, it must satisfy two conditions. In the first place, it must be possible to apply it in the actual cases to which it is meant to be relevant, in the second, its arguments must be sound. The main object of science is to increase knowledge of the world, and if a method is not applicable to anything in the world it obviously cannot lead to any knowledge. This principle is very elementary, and it is probably for that very reason that it is habitually overlooked in theories of scientific knowledge.

Any theory, whether scientific or purely logical, must rest on a set of primitive propositions, called postulates. In each case other propositions are deduced from these, one by one, by a purely logical process. The difference between pure logic and science lies in the nature of the primitive propositions. In all cases these include the postulates of pure logic; but in scientific investigation they also include two other kinds of proposition. The first of these consists of the facts of sensory experience, which do not form a part of logic; the second type are general propositions, involving the non-logical concept of probability. This is necessary in order to deal with the essential process of generalisation, and accordingly we think that any attempt to construct a theory of scientific knowledge without it is foredoomed to failure. The attempt has nevertheless been made several times in different ways. In this paper we hope to indicate the points at which these various attempts break down; in all cases it is found that they fail to satisfy the criterion of applicability in practice.

The second criterion is as necessary as the first, but it is usually satisfied in scientific theories. Nevertheless, the true nature of scientific argument is very imperfectly understood. It is not difficult to suggest a reason for this. The analysis of processes of reasoning has always been regarded by philosophers as their special province, and they have habitually regarded the scientific practice of proceeding from the particular to the general as formally fallacious: as indeed it is, if no postulate be introduced other than those of pure logic and the bare facts of sensation. Scientific writers, on the other hand, start with a firm conviction that their methods are valid; accordingly, if philosophical argument is opposed to science, they regard the fact merely as a good reason for condemning philosophy. The result has been that they look with disfavour on any analysis of the fundamental assumptions of science, since such discussions have proved almost entirely fruitless in the past. Thus any serious attempt at such analysis has come to be described as metaphysical and largely ignored, with a consequent loss of clarity in scientific discussion. A certain amount of attention has, however, at last been drawn to this need, partly by the slashing attacks of Karl Pearson and Ernst Mach on certain prevalent scientific concepts, and partly by the important physical results predicted by Einstein, largely based on the views of these earlier writers. Physical concepts have consequently been subjected to some discussion in recent books, but, we think, quite unsatisfactorily. Instead of investigating the actual nature and method of application of the fundamental postulates, these writers have surrendered completely to the philosophic criticisms and tried to treat scientific knowledge without using any form of generalisation, but with the introduction of certain new postulates which, if they occur in ordinary scientific use at all, are not primitive propositions. The result is that beautifully coherent deductive systems are obtained which would be perfectly satisfactory if their fundamental postulates were admitted; but when these are examined it is found in all cases that they are not directly known to be true, and that they can be verified only by the confirmation of the predictions based on them. But this does not prove that they are true: if two propositions $p$ and $q$ are so related that $p$ implies $q$, the fact that $q$ is true does not entitle us to say that $p$ is true, unless some further assumption is introduced; and this assumption is not a part of pure logic. Thus the truth of the alleged primitive postulates in these eases is only inferred by using a principle which the purely logical method was expressly designed to avoid.

11. Dorothy Maud Wrinch, The Idealistic Interpretation of Einstein's Theory, Proceedings of the Aristotelian Society, New Series 22 (1921-1922), 134-138.

The theory of relativity is a part of physics and shares with other theories which make up modern physics certain well-understood and well-established assumptions. The theory is of outstanding interest in physics mainly because of the intricacy of the deductions which are involved, but in no respect whatever is the theory "idealistic" or "realistic" in any sense in which any other branch of physics is not. The question before us is then, in fact: Is physics in accord with the idealistic interpretation of the external world?

Legitimacy of Analysis: High up on the list of assumptions which form the foundations of physics and of the theory of relativity is the assumption of the legitimacy of analysis. In science we believe that the facts with which we deal are capable of analysis; we believe that the facts have constituents in the sense that identical terms can form part of different facts. This identification of terms, occurring in different facts which come to our notice, is the main stimulus to scientific thinking. Unless the same term arrested our attention in several different complexes of fact, we could not attempt to build up science. For in science we collect together the facts at our disposal with a view to discovering general propositions. A large part of scientific thought consists in the building up of probability inferences of various kinds.

12. Dorothy Maud Wrinch, On the Lateral Vibrations of Bars of Conical Type, Proceedings of the Royal Society of London. Series A 101 (713) (1922), 493-508.

This paper contains a discussion of the lateral vibrations of a thin conical bar of circular section, which has its tip free. By means of a discussion of the roots of certain equations containing Bessel functions of the second and third orders, both of real and of imaginary argument, the frequencies and nodal arrangement associated with the first three tones are investigated, in the case when the base of the bar is clamped.

The lateral vibrations of conical bars of circular section were first treated by Kirchhoff, in 1879. In his investigations, Kirchhoff was concerned with the case of a bar with its tip free and its base clamped, and he limited himself to a discussion of the period associated with the gravest tone, and considered neither the higher periods nor the positions of the nodes associated with them. J W Nicholson, in the course of an investigation of the lateral vibrations of certain types of bars of variable section, discussed the case of a double cone (consisting of two equal cones placed base to base,) vibrating with both tips free, and discussed the periods and nodal ratios associated with the first three symmetrical tones. His results, however, throw no light on the question of the vibrations of a conical bar with a clamped base, owing, to the peculiar nature of the conditions at the centre.

The present paper, in addition, contains an investigation of the periods of the higher tones, and of the arrangement of the nodes associated with these tones. The discussion gives for the case of a clamped-free bar of conical type the results which correspond to those given by Lord Rayleigh, and Seebeck, and Donkin, for a bar of uniform cross section. General characteristics are worked out of the nodal arrangement in the higher tones when the tip is free, and nothing is known about the conditions at the base.

13. Dorothy Maud Wrinch, On Certain Methodological Aspects of the Theory of Relativity, Mind, New Series 31 (122) (1922), 200-204.

There are many aspects of the Theory of Relativity which involve problems until lately the exclusive property of Philosophy. Chief among these is the problem of Space and Time. In the theory these conceptions are given a definite status. There are now various different views of the characteristics of Space, held by different writers on Relativity; and it is a very satisfactory sign of the vigour of modern physics that such different systems as those of Einstein, Weyl, and Eddington should be elaborated. The only way of deciding between them will be by means of experimental tests; but at present no experimental test appears to be available to test the theory of Weyl though it is not impossible that some deduction might be made as to the shift of the lines in the spectrum of the sun or perhaps as to the size of the universe. But in spite of the fact that the views of these writers differ and there seems to be no way of deciding between them at present, if we eliminate all the parts of the theory about which there is not agreement, there is still something of fundamental importance in their treatment of the notion of Space. In the language of modern logic, they all alike use "Space" as a description. Space in the theory of Relativity is a constructed entity.

To say that space is a description involves many consequences. No description can ever be used as a proper name. We see two particular spots of colour and we say "This is darker than that". There is in some sense, a direct relation between the symbol "This" and one of the spots of colour. Whatever this characteristic may be in virtue of which "This" is in direct relation to the thing in the External World to which it refers, it is absent in the case of the symbol for space. There is nothing in the External World to which we can point as being represented by the symbol. And this property of the concept which makes it a description involves a further consequence. Any proposition in which the term occurs is not in its logically simplest terms. It can be analysed further. And we may easily see that the analysis of such a proposition will disclose some propositional function.

14. Dorothy Maud Wrinch and Harold Jeffreys, On Some Aspects of the Theory of Probability, Philosophical Magazine 45 (1923)368-374.

It is a universal belief among physicists that when a sufficient number of inferences from a quantitative law have been verified, the probability of the correctness of the next inference from it may be made to approach indefinitely near to unity. It was shown in a former paper that this proposition is not easily reconcilable with the other proposition, also believed by some physicists, that all laws of some infinite class are equally probable à priori; but that it is a necessary consequence of the contradictory proposition, which appears not implausible, that the prior probabilities of admissible laws are not all equal, the simpler in fact having the higher probabilities. It was pointed out further that, if the admissible laws are all arranged in a series according to their probabilities, the current process of taking the simplest law that fits the facts can never enable one to proceed more than a finite number of steps along this series, and accordingly the laws capable of being discovered must form an enumerable aggregate. In particular, since it appears that the constants in physical equations are often capable of continuous variation, or of discontinuous variation through more values than can be examined, we must suppose that laws containing such parameters are not in forms suitable for the comparison of their probabilities. They must have these parameters removed, by differentiation or otherwise, before such a comparison can be carried out. In particular, where a law can be expressed as a differential equation or as its integral, the latter involving one or more arbitrary constants, the latter cannot be the fundamental form of the law in our knowledge, while the differential equation may be. One consequence of the inadmissibility of quantities capable of discontinuous variation through more than a certain number of values is that the fundamental form of the law of gravitation cannot involve the mass of the attracting body explicitly. Thus the Newtonian form of it cannot be the fundamental one; nor can the Poisson form nor the equation in Einstein's system that corresponds to it, since the density occurs as a parameter in these and is capable of many values.

15. Dorothy Maud Wrinch, On Mediate Cardinals, American Journal of Mathematics 45 (2) (1923), 87-92.

A "mediate cardinal" is defined in "Principia Mathematica" as a cardinal which is neither inductive nor reflexive and it is established that the multiplicative axiom implies the non-existence of mediate cardinals. The converse implication is not established, and there seems to be no reason to suppose it is true. The relation of the existence of mediate cardinals to the multiplicative axiom is therefore one-sided and offers a contrast to the mutual implications of the comparability of cardinals, the well-orderability of classes and the multiplicative axiom. In this paper it is proposed to investigate other classes of cardinals which are not Alephs, beyond the mediate cardinals of "Principia Mathematica," and instead of the one-sided implication between the multiplicative axiom and the non-existence of mediate cardinals to establish an equivalence between the axiom and the non-existence of certain cardinals which are not Alephs.

16. Dorothy Maud Wrinch, Some approximations to hypergeometric functions, Philosophical Magazine (6) 45 (269) (1923), 818-827.

The importance of the Bessel functions ... in mathematical physics has long been recognised. These functions are, however, particular cases of the hypergeometric function with $s$ denominators ...

In the present paper, we are concerned with the hypergeometric function with four denominators ... and, further, with the hypergeometric function with five denominators and one numerator ...

17. Hugh E H Wrinch and Dorothy Maud Wrinch, Tables of the Bessel function $I_{n}(x)$, Philosophical Magazine (6) 45 (269) (1923), 846-849.

While Tables of the Bessel Functions of both kinds, $J_{n}(x)$ and $Y_{n}(x)$, are now fairly complete, much work remains to be done in regard to the corresponding tables for the so-called "Functions of Imaginary Argument," $I_{n}(x)$ and $K_{n}(x)$, which are of considerable importance in physics.

In the present paper we give a table of the values of $I_{n}(x)$, where $n = 0, 1, ... 6$, at intervals of a unit, and $x = 5, 6, 7, ... 15$, at intervals of a unit.
...
Previous tables of this function have been almost entirely limited to the functions $I_{0}(x), I_{1}(x)$ of orders unity and zero. The British Association report for 1896 gives $I_{0}(x)$ to nine places from $x$ = 0 to 5.100 at intervals of 0.001, following on a table of $I_{0}(x)$ of the same character in 1893. Aldis (1899) gives $I_{0}(x)$ and $I_{1}(x)$ to twenty-one places at intervals of 0.1 from $x = 0$ to$x = 6$, and a few extra values.

Anding (Leipzig, 1911) published tables of the logarithms of these functions,$I_{0}(x), I_{1}(x)$, from $x$ = 0 to 10, at intervals of 0.01. The British Association Committee in 1889 published some tables of the more general function $I_{n}(x)$ to twelve figures for $n = 0, 1, 2 , ... 11, x = 0$ to 6, with intervals 0.2 in $x$.

18. Dorothy Maud Wrinch, On Certain Aspects of Scientific Thought, Proceedings of the Aristotelian Society, New Series 24 (1923-1924), 37-54.

The "present heroic adventures and discoveries of Physics," to use our President's happy phrase, give much material to the student of scientific thought. Stupendous progress has been made during the last fifty years. On this occasion, we attempt to deal with certain aspects of scientific thought, which have proved to be of fundamental importance in Physics at the present time.

Science at the outset is concerned with the discovery of facts about the external world by means of experiment and observation. These facts are subsequently arranged in groups and by means of probability inference general propositions are suggested for consideration. The second stage in science opens with the statement of these general propositions about physical concepts. A physical concept, refined so as to be significant in science, is a short-hand way of referring to a class of properties. Our field, therefore, at this stage of science, is the body of the general propositions which cover the facts of experience and the general problem is the problem of the relations between properties. On this occasion, we leave on one side any discussion of the characteristics of scientific thought associated with the earlier stage of science and attempt only to draw attention to certain important principles of procedure characteristic of scientific thought in its conceptional stage.

19. Dorothy Maud Wrinch, The Quantum Theory in Relation to the Logical Concept of Continuity, Proceedings of the Aristotelian Society, Supplementary Volumes 4, Concepts of Continuity (1924), 27-33.

The Quantum Theory, in common with the Theory of Relativity, is of great interest and of vast importance to the student of Scientific Methodology. The relation of logical concepts to scientific theory is a study as yet in its infancy. Yet, it is precisely the nature of this relation which is the critical issue in the Quantum Theory regarded as a part of the fabric of modern science.

Mathematics, or Logic - if we may use this latter term to denote the general theory of the structure of concepts and so to include the first - is essentially concerned with possibilities and not with actualities. When it is introduced into scientific theorising it provides information as to what may happen, and never any information as to what does happen in fact. Given the relevant mathematical theorems, it is for the Natural Sciences to discuss the actual phenomena of the external world.

20. Dorothy Maud Wrinch, The Concept of Energy, Proceedings of the Aristotelian Society, Supplementary Volumes 5, Philosophy and Metaphysics (1925), 55-63.

Classical mechanics depended on the assumption of the Laws of Motion. Relativity Mechanics depends on Invariance Postulates. There is all the difference in the world between the tone of these postulates. The Invariance Postulates are colourless and abstract and skeletal in character. It is difficult to deny them with any enthusiasm. They are themselves, so structural in nature that the hot enthusiasms and violent preferences for one view rather than another which are sometimes to be seen in politics, feminism, art, literature, motoring, dress, education, seem out of place. Who can get warm about a four-dimensional space rather than a ten-dimensional space? If science requires the one when the consequences of the assumption are compared with facts of observation and experiment, we are content to assume it until it is contradicted by experience.

It is not that the Postulates are uninteresting or dull, or unworthy of attention from a mathematical or scientific point of view. The most entrancing worlds can no doubt be constructed from the assumption of fifteen dimensions. But the cold dignity of the postulates makes us judge them not in themselves but only in their consequences. If their consequences fit the facts we take them; if they do not, we reject them; and, if one assumption in its consequences fits the facts better than another, we prefer it and retain the other assumption, possibly for further service later on.

Thus it is not a progress to the "simple natures" that is going on. It is rather a progress to more austere postulates which is a characteristic of modem science. The modern concepts of Relativity, in their icy dignity, allow the deduction of physics not in a simpler fashion but in a more logical fashion. It is the aim of science to provide postulates, which by the application of logic alone, give the whole of observed phenomena.

But, at the end of all discussions, however enthusiastic, the grim fact emerges. Science depends on assumptions which we must accept or reject without logical reasons. And the Conservation of Energy, being a part of the structure of science, must be treated in the same way. We can deduce it from it other postulates. But how far these are to be accepted only the successful fitting together of the facts of experience and its logical consequences can decide. The stupendous successes of the Principle of the Conservation of Energy are almost sufficient to go to the head of the theoriser on science, for they have been very remarkable.

But we must hold fast to our general methodological principles. All is grounded on assumptions for which no logical reasons can possibly be given. We have no reply to the professional sceptic. No scientist ever has had a reply and no philosopher either.

But having made this frank confession of the state of affairs in science, we should be inclined to pass on to point out that philosophical doctrines are all in the same strange, embarrassing position as scientific doctrines. For philosophers and scientists must stand together. There is nothing to decide between them in this respect. Scientists and philosophers alike stand for ever on sand.

21. Dorothy Maud Wrinch, Scientific Methodology with Special Reference to Electron Theory, Proceedings of the Aristotelian Society, New Series 27 (1926-1927), 41-60.

It is very interesting to notice the fact that during the last years Electron theory has come to be the focus of interest for Scientific Methodology. Before this the theory of Relativity dominated our attention. There had been no theory previously which provided comparable material, either for the purpose of illustrating the principles of scientific method or for actually pointing the way to a further development of scientific ideas. However, recent advances in Electron theory have been so considerable and of so far-reaching a character that the material it offers is very nearly, if not quite, as important as the material offered by the Relativity theory. An examination of the present position in the subject certainly offers great scope to the logician and the philosopher.

A great deal of the material to which I shall refer has been available for discussion for some few years, and, indeed, on at least one occasion the Society has directed attention to certain aspects of it. There is, however, a special reason which has led me to this investigation at this particular time, for during the last six months a development has occurred which it is reasonable to think may prove to be of importance to Electron theory. Indeed, whatever may be its ultimate fate, it certainly provides interesting material for the illustration of the principles of scientific method. In this paper we shall not concern ourselves exclusively or even mainly with this new theory, which is as yet awaiting its thorough examination and discussion by professional physicists, but in the course of our investigation it will be worth while to direct the attention of the Society to its existence and to introduce those parts of it, which, whether they prove acceptable to the world of physics or not - and I cannot too strongly emphasise that the position with regard to the theory is at present entirely open - enable us to obtain some rough estimate of its value as a contribution to scientific theorising.

22. Dorothy Maud Wrinch, The Relations of Science and Philosophy, Journal of Philosophical Studies 2 (6) (1927), 153-166.

It is, I think, one of the outstanding characteristics of our age that during a short spell of thirty or forty years fundamental advances have been made in a large number of different sciences. These developments have altered almost every aspect of material life - they have certainly had great influence upon modern education, and upon modern ideas of politics, as well as upon a host of less important things. But chief of all we notice the effect of this Golden Age of Science in the birth of new ideas. The revolution in ideas has only just started. Where it will end no one can see.

Here we are concerned specially with the revolution in ideas as it effects Philosophy and Logic. For side by side with these tremendous changes in scientific outlook there has been a change in those parts of Philosophy which are concerned with the principles of Scientific Method - with those parts, in fact, of which the aim and object is the critical assessment of the progress of theoretical science. And this change has not been wholly dictated by the discoveries of Science. There has also been a vitally important development from the side of technical philosophy. It is to the happy synchronisation of these two apparently independent developments that we owe the fact that such unparalleled progress has lately been made, both in the domain of scientific practice and in the domain of theoretical science.

The domain of philosophy is so vast and broad that little of value can be said about it as a whole. In the past it has suffered too much from the tendency of philosophers to treat, on broad and comprehensive lines, topics and problems which would appear to need careful analysis into a host of subsidiary problems, each requiring the application of methods of inquiry which, owing to their successes in other fields, have come to be regarded as the methods most likely to be successful in dealing with the problem of knowledge. There have been too many attempts at theories designed to cover in their sweep the heterogeneous collection of questions which form the subject-matter of philosophy. In this article, therefore, I wish at the outset to delimit the parts of philosophy to be dealt with. We are concerned with that part of philosophy which has for its focus of interest the constructive and destructive criticism of scientific ideas.

23. Dorothy Maud Wrinch, Is the "Fallacy of Simple Location" a Fallacy?, Proceedings of the Aristotelian Society, Supplementary Volumes 7, Mind, Objectivity and Fact (1927), 237-243.

Is the "Fallacy of Simple Location" a Fallacy? If we treat the question strictly literally, it appears to me clear that events cannot have simple locations. However, taking the question more broadly as having reference to the nature of the relation between events and space time, I am of the opinion that this relation is such as to require very complicated logical analysis before its nature becomes plain, and that no conclusion can be arrived at except after an intensive consideration of the space time in question. Such an enquiry could, of course, be undertaken with respect to the space time of Professor Whitehead. But the important point, so far as I can see, is that there is no guarantee that the relation would be of the same nature in all the various space times which can be constructed and which very well may be constructed in the future. All that we can say quite definitely is that in all space times, propositions as to location will certainly contain terms which are abstractions from experience. The precise type of relation entailed by any one type of space time must, therefore, be investigated subsequently. The fact that in the types at present current the relation of events to space time has a certain logical structure and a high degree of complication which make it quite unfeasible to ascribe to events simple locations is no answer to the question before us in this symposium, and is indeed scarcely relevant to our enquiry.

Last Updated December 2021