Propositions Of Geometry Research Articles

'Hume on Space and Geometry': A Rejoinder to Flew's 'One Reservation'

66. ' HUME ON SPACE AND GEOMETRY * : A REJOINDER TO FLEW ' S 'ONE RESERVATION '. ? Flew' s reservation about my assertion that the Enquiry contains no significant revision of the Treatise conception of geometry as a body of necessary and synthetic knowledge, appears to involve two charges. Firstly, he alleges that I dismiss but offer no substantial argument against his own view that the Enquiry restores pure geometry "to its place alongside the other two elements of the trinity ". Secondly, he thinks that I have committed a serious sin of omission in failing to take into consideration a certain passage in the Enquiry (page 31, section 27) which, in his opinion, constitutes "a quite decisive reason" for supposing that Hume abandoned the Treatise position, in that it contains an account of applied mathematics, a subject on which the Treatise is silent. To the charge of omission I plead guilty, and I shall presently seek to remedy this defect and to state why I do not share Flew' s view of the significance of the said passage. To the former charge I am not so ready to plead guilty, and I shall counter-charge that Flew has failed not only to do justice to my proposed reading of Enquiry page 25, section 20, but also to provide clear and convincing support for his own. One difficulty encountered in answering Flew lies in understanding whether or not his claim about the status of geometry as a pure science in the Enquiry is put forward independently of his interpretation of E31. As originally presented in Hume's Philosophy of Belief this appears to be the case, and if so, on what evidence does it rest? 2 In his book , Flew' s statement about Hume replacing pure geometry alongside arithmetic and algebra follows directly upon a paragraph in which he comments that Hume's references to geometrical propositions at E25 signify "an important advance from the position of the Treatise" . For whereas the status of perfect precision and certainty was 67. there withheld from geometry (T71) , Hume now speaks of the certainty and evidence of the propositions of Euclidean geometry , and declares such propositions to be discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe . (E25) Now, I argued at some length in my paper (pp. 21-24) that Hume is prepared to grant scientific status to geometry in the Enquiry because he has come to regard its certainty as that proper to a mathematics of space. He has abandoned the Treatise contrast between an ideal or theoretical standard of equality, viewed therein as necessary to geometry considered as a science proper, and the merely sensible standard of equality upon which geometry as practised depends . It should not be forgotten that Hume recognises two kinds of certainty where "Relations of Ideas" are concerned, intuitive and demonstrative. Since he fails to provide any detail of the place of each of these in mathematics, the door is left open for speculation along the lines followed in pp. 28-29 of my paper. For there is nothing in ?25 which implies that the certainty of all geometrical propositions is of the same kind, or that the process of geometrical demonstration is of the same logical nature as demonstrative reasoning in arithmetic and algebra. In his comment on my position Flew makes no reference to my proposed reason why Hume classes geometry as a science in the Enquiry.· Yet, even if this reclassification does mark an advance of a kind, the fact remains that E25 does not furnish incontrovertible evidence for that advance which Flew thinks he finds there. Nor, therefore, is it conclusive against my assertion that Hume in both works regards the propositions of geometry as necessary and synthetic truths. Given that Flew does not wish to deny the inclusion of a synthetic element even in the Enquiry account of mathematics (see H. P. B. pp. 64-66), I find it difficult to see why he supposes that the logical status of the propositions of geometry is different in the Enquiry from that in the Treatise, since in the latter too they are held 68. to be truths which depend...

Kant on Outer and Inner Intuition

According to Kant, the primary propositions of geometry and arithmetic express synthetic apriori judgments. Though necessary, such judgments, being synthetic, must have a basis-a very special basis-in experience. Kant's theory of sensibility was his attempt to exhibit this special basis and thereby safeguard his characterization of the basis truths of mathematics. His theory's primary theses include the following: Sensibility, in contrast to understanding, is the capacity to have intuitions. Intuitions are either sensory (outer) intuitions or introspective (inner) intuitions. The former are representations of extended objects standing in spatial relationships; the latter are representations of temporal sequences of states of mind such as experiences, thoughts and feelings.' Space is the form of outer intuition; time is the form of inner intuition. Kant also asserted that neither the objects nor the forms of any intuitions have transcendental reality; they merely have phenomenal (empirical) reality.2 However, although neither outer nor inner intuition is apprehension of independent (transcendental) reality, both are related to bodies of synthetic apriori knowledge. Geometry Kant straightforwardly associated with outer intuition. In contrast, nowhere in the Transcendental Aesthetic, indeed nowhere in the Critique of Pure Reason, did Kant shed much light on the basis of arithmetic or the nature of the truths associated with inner intuition.3 In fact, considering its importance for his theory of

Rationalists and Current Concepts

THE RATIONALIST ANNUAL for 1941 (London: The Rationalist Press Association, 1940, pp. 86, 1s. net) includes, as usual, a number of articles which are calculated to stimulate controversy, even though some of them may appear to have been written with a sub-intention pour épater le bourgeois. Dr. Gilbert Murray in “Why Despair of Reason?” discusses the thesis that the modern psychologists, in diminishing the authority of reason, have played an important part in turning “a decent and humane world into an ill-managed lunatic asylum”; while Sir Peter Chalmers Mitchell in “Religion in Trouble”, in citing the Union of Soviet Republics as an example of the inefficacy of prayer, expresses profound admiration for the achievement of that “new force of civilization” in education and public health. From the former as a “capacity-catcher” he looks forward to “an almost colossal contribution to the progress of mankind”. In a contribution on the laws of Nature, Prof. J. B. S. Haldane expounds the modern concept of their nature and authority, noting in particular such modification as has been introduced by the recent work of Jeffreys and Milne, summing up in the conclusion that “so far from being laid down by the arbitrary word of a creator, they [natural laws] may prove to be a system as intimately and rationally knit together as the propositions of geometry, and yet changing and evolving with time like the forms of plants and animals.”

Propositions and Corollaries in Petroleum Production

All important industries have certain basic principles in common whichgovern them and give them a family resemblance. In addition, each industry hasa certain individuality due to some principle or principles which govern, orare accepted as governing, that industry and which is not effective in others.The customs and practices of an industry depend on these basic principles muchas the corollaries follow a proposition in geometry. After the proposition isproved or is accepted as proved, the corollaries are seen to be true almostwithout additional evidence or proof. In many cases, the corollaries are moreimportant in practical affairs than the proposition on which they depend. Bycontinued application the corollaries may come to be accepted as law and thefact that their truth depends on the correctness of the fundamental propositionis forgotten or ignored. If it is discovered that an error was made indemonstrating the proposition and that the proposition is untrue, thecorollaries are, in most cases at least, equally untrue. Considerable time maybe required, however, to realize the fact that the corollaries have failed, particularly if they have been embodied in custom and law. For example, in the business of government the proposition that women areintellectually inferior to men was formerly considered, by most men at least, to be duly proved if not axiomatic. This proposition was never embodied in law, but its corollaries-that the rights of the wife in property and even in her ownchildren were inferior to the rights of the husband, and that women had noplace in governmental affairs-were duly incorporated in law or so completelyapproved by custom that they had the effect of law. Developments of the past few decades have proved, to the satisfaction of themajority, that woman's intellectual inferiority to man had not beensatisfactorily demonstrated, and that the proposition that she was inferior, although accepted for many generations, was untrue. It was not so great a shockto men to discover that this proposition was wrong as it was to find out thatits corollaries were also untrue and that the laws embodying them were not goodlaws and were subject to repeal.

Postulates and Sequence in Euclid

Almost any proposition in geometry is implied by another, already accepted; that may have been implied by still others, but in this regression we finally reach statements that appear not to be dependent upon implication. They are the postulates or definitions upon which all the science of geometry is based— the “first causes.” They are originally the result of analyzing the world outside our minds, to clarify the ideas of the simplest objects in it and of their simplest relations. Some such primary statements can by taking thought be shown to be implied by others. In framing a geometrical system, as Euclid did, it is necessary to choose which of the results of our analysis shall be our postulates or definitions. The choice usually made for elementary geometry to-day is in good part the choice made by Euclid.1 We can, however, be more explicit than he, particularly in regard to the idea of what we call number.

News and Notes

Dr. John H. Minnick was elected President of the National Council of Teachers of Mathematics at the annual meeting at Atlantic City. Dr. Minnick has been unusually active in secondary school mathematics. He is now engaged in training high school teachers in mathematics in the University of Pennsylvania. Many readers will recall Dr. Minnick’s Tests of Abilities in Geometry, a scientific monograph on the nature of the abilities which are involved in proving a proposition in geometry.

The Fundamental Definitions and Propositions of Geometry, With Especial Reference to the Syllabus of the Association for the Improvement of Geometrical Teaching

I DESIRE to offer some suggestions respecting the form and arrangement of the elementary definitions and propositions of the Euclidian geometry. It has appeared to me that the recent German textbooks upon the subject have made a great improvement upon the older system, as developed in the works of Euclid and Legendre. I have but recently obtained the “Syllabus of the Association for the Improvement of Geometrical Teaching” and compared it with the corresponding parts of a summary of my own, the latter still in an inchoate state.