Independently of any eighteenth century work on the geometry of parallels, Thomas Reid discovered the non-euclidean "geometry of visibles" in 1764. Reid's construction uses an idealized eye, incapable of making distance discriminations, to specify operationally a two dimensional visible space and a set of objects, the visibles. Reid offers sample theorems for his doubly elliptical geometry and proposes a natural model, the surface of the sphere. His construction draws on eighteenth century theory of vision for some of its technical features (...) and is motivated by Reid's desire to defend realism against Berkeley's idealist treatment of visual space. (shrink)

„My sole purpose in this paper is to try and correct what I take to be a common misinterpretation of Hume’s opinions on mathematics. I shall not enquire whether he was right or wrong in holding these opinions. Nor shall I offer opinions of my own.“.

In lieu of an abstract, here is a brief excerpt of the content:HUME ON SPACE AND GEOMETRY Hume's discussion of our ideas of space, time and mathematics in Book One of the Treatise is referred to by one recent commentator as 'the least admired part' of this work, while another finds it to be 'one of the least satis2 factory Parts'. Hume himself, it would appear, was not far from endorsing such opinions. The omission of any detailed comment on these (...) subjects from the first Enquiry can be taken as indicating his dissatisfaction with the earlier exposition as well as a felt incompetence to offer a suitable revision. For there was certainly no loss of interest on Hume's part in the philosophical problems of mathematics. In a letter to William Strahan in 1772 he speaks of an essay prepared for publication around 1755 entitled: On the Metaphysical Principles of Geometry, which he withdrew upon the advice of his friend the mathematician Lord Stanhope, who convinced me that either there was some defeat in the argument or in its perspicuity; I forget which... Defects of both kinds are present in the Treatise discussion of geometry and the infinite divisibility of space, which taken together with certain remarks in the Abstract of the Treatise and in the Enquiry, suggest that the problem defying satisfactory solution was the relation between these. Hume was of the opinion that the theory of infinite divisibility constituted an affront to human reason by virtue of the paradoxes it involved. Such paradoxes, he thought, must open the door to scepticism about the certainty of mathematical knowledge, since in the field of abstract reasoning reason is thrown into a kind of amazement and suspence, which, without the suggestions of any sceptic, gives her a diffidence of herself, and of the ground on which she treads. (E157) Hume's attempted solution involved the rejection of those abstract ideas employed in mathematics and most fundamentally, that of infinite divisibility, but he never succeeded in working out a theory of geometry 2. which preserved its desired status as a body of certain knowledge and yet was at the same time consistent with his empiricism. In this paper I set out to examine the relation between Hume's view of space and his theory of geometry. Firstly I take a detailed look at Hume's arguments against infinite divisibility in an attempt to discover why they take the form they do, and to understand what Hume has in mind when he speaks of that 'minimum' idea which is an 'adequate representation" of the most minute possible part of extension. The second section deals with Hume's account of the genesis of our idea of space. I propose, contrary to some commentators, that Hume does not treat the idea of space on an exact parallel with that of time. Although both are said to be abstract ideas they differ in the relation in which they first stand to impressions. My comments on time are confined to drawing attention to this difference since fuller discussion of passages dealing exclusively with time would require a separate paper. In the final section Hume's remarks about geometry are examined. Here I argue that Hume entertains two views of geometry in the Treatise, one of which is abandoned in the Enquiry while the other remains largely unaltered. Hume opens his account of our ideas of space and time with an attack on the thesis of infinite divisibility which he regards as belonging to that body of doctrines which are greedily embrac'd by philosophers, as shewing the superiority of their science, which cou' d discover opinions so remote from vulgar conception. (T26) Two principles, boti taken to be evident, form the basis of his argument. I shal] refer to them hereafter as principles A and B. Principle A: the capacity of the mind is limited, and can never attain a full and adequate conception of infinity. 3. Principle B: whatever is capable of being divided in infinitum, must consist of an infinite number of parts, and ( ) 'tis impossible to set any bounds to the number of parts, without setting bounds at the same time to the division. (T26, 27) From these two principles Hume draws the conclusion that... (shrink)

The essence of Hume’s eighteenth-century philosophy was that all the sciences were ‘dependent on the science of man’, and that the foundations of any such science need to rest on experience and observation. This title, first published in 1932, examines in detail how Hume interpreted ‘the science of man’ and how he applied his experimental methodology to humankind’s understanding, passions, social duties, economic activities, religious beliefs and secular history throughout his career. Particular attention is paid to the English, French and (...) Latin sources that shaped Hume’s theories. This is a full and fascinating title, of particular relevance to students with an interest in the philosophy of Hume specifically, as well as the philosophy of human nature and the methodologies applied to its study more generally_. _. (shrink)

There is a widespread belief, more often implied than explicitly asserted, that Hume considered all necessary propositions to be analytic.Of course Hume did not use the analytic-synthetic distinction explicitly. This only come to the forefront with Kant; and it is Kant who is probably the main source of the above-mentioned belief. Kant ascribed to Hume the view that mathematical propositions are, in his terminology, analytic. If this is correct, then since mathematics was for Hume the paradigm of a body of (...) necessary truths, it is plausible to infer that he considered necessary propositions to be one and all analytic. The absence in Hume's writings of the analytic-synthetic terminology is by no means decisive: though he did not have the terms he may well have possessed the concepts, as evidenced by his use of them. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:62. 'HUME ON SPACE AND GEOMETRY': ONE RESERVATION In so far as Rosemary Newman disagrees with any2 thing said in my 'Infinite Divisibility in Hume's Treatise ' - which seems, happily, not to be so very far - I hasten to report that I am now persuaded. Thus my suggested reason for refusing to allow that an impression of blackness could give rise to the idea of extension was (...) not, after all, 3 Hume's. And no doubt his conviction that whatever ^s capable of being divided in_ infinitum, must consist of an infinite number of parts (T26) was in part sustained by "the atomistic type of phenomenalism advanced in the first Part 4 of Book I of the Treatise". For to any objection that this nowadays so plainly unacceptable conviction had been accepted as obviously correct by such formidable predecessors as Berkeley and Hobbes, it could be replied that they too had their commitments to their own brands of atomism. In her further comments upon "Hume's phenomenalist atomism" Newman notices, as "a very curious claim", his statement that when - like the true philosopher he was! - he locked at a table:- My senses convey to me only the impressions of colour 'd points, dispos 'd in a certain manner (T34). So I was sorry that she seems to have missed the bold priority claim which, referring to the same passage, I made on Hume's behalf in that article: "Anyone familiar with the theories and the paintings of Seurat might also mischievously characterize the Hume of this Section as 'the Father of Pointillisme' "! But at the end of her paper, although I think only at the end, Newman begins to go very wrong indeed. She says: "Turning to the Enquiry there is little to indicate any shift in Hume's opinion of geometry as a body of synthetic but necessary knowledge." After remarking that Hume does not repeat "any of his earlier arguments" against infinite divisibility "nor those in support of a system of mathematical 63. points", she notices that, in the paragraph characterizing propositions stating only the relations of ideas, geometry "is classed as a science alongside arithmetic and algebra (E25) ".7 Next she quotes the end of that paragraph, and at once comments: "Flew thinks that in the Enquiry Hume is to be seen as 'restoring pure geometry to its place alongside Q the other two elements in the trinity',..." Newman does not, so far as I can see, offer any reason for dismissing this suggestion - apart, that is, from asserting that the paragraph from which she has just quoted "can be interpreted so as to remove any apparent inconsistency with the claim that the impossibility of conceiving the negation of a geometrical axiom has the same ground in the Enquiry as in the Treatise, namely, the invariant character of sensible space." She just tells us that "Where geometry is concerned I follow Atkinson's opinion in finding no reason to suppose that 9 Hume abandoned the Treatise stand." This I find, to say the least, surprising since the very sentence of Hume's Philosophy of Belief from which she quoted my contrary opinion does provide what I still consider to be a quite decisive reason. Since she does not discuss either my reason or the Hume passage to which I was referring, I can here do nothing else or better than repeat myself; adding only and by the way that Professor Atkinson's paper 'Hume on Mathematics' was published a year before that book:Besides restoring pure geometry to its place alongside the other two elements of the trinity, the Inquiry also sketches an account of applied mathematics. This is something the earlier book did not provide at all. 'Every part of mixed mathematics proceeds on the supposition that certain laws are established by nature in her operations, and abstract reasonings are employed, either to assist experience in the discovery of these laws or to determine their influence in particular instances... Thus it is a law of motion, discovered by experience, that the moment or force of any body in motion is in 64. the compound ratio or proportion of its solid contents and its velocity, and, consequently... (shrink)

In lieu of an abstract, here is a brief excerpt of the content: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... (shrink)