Notes on Hume's Treatise
by G. J. Mattey
Book I, Part II OF THE IDEAS OF SPACE AND TIME
§ IV Objections answer'd
1. The capacity of the mind is not infinite, so the ideas of extension and duration consist of a finite number of simple and indivisible parts.
2. (A consequence of 1), something existent is always required. So there is no idea or conception of vacuum.
Objection I. There is no mean between infinite divisibility and mathematical points. But the latter are non-entities. But, we can bestow a color or solidity on the points (which are yet said not to be physical).
Objection II. Simple and indivisible atoms would interpenetrate, since one would touch the other "intimately, and in its whole essence."
But we can substitute a better idea of penetration: union of body produces no more extension. But this would rather be annihilation, since we then have the idea of only one. But then, there is no reason to think they are annihilated. Rather, there are two objects side-by-side. This is hard to see because of the infirmity of our senses.
Objection III. Mathematical definitions in geometry.
Hume undertakes to defend the definitions of geometry but reject its demonstrations, which conflict with his system.
The definition of a plane (length and breadth without depth), a line (a length without breadth or depth), and a point (having neither length nor breadth nor depth) can only understood on the assumption of indivisible points. This alone can give a clear idea of something defined negatively.
There are two responses given. 1) the objects of geometry are ideal only, and "never can exist in nature."
But this is "absurd and contradictory," since we can conceive of them by a "clear and distinct idea," which implies the possibility of its existence. "'Tis vain to search for a contradiction in any thing that is distinctly conceiv'd by the mind."
2) The "Port Royal Logic" of Nicole and Arnauld (L'Art de penser) allows that we can make a distinction of reason , an "abstraction without a separation" to account for the lack of a dimension. "The length is inseparable from the breadth both in nature and in our minds; but this excludes not a partial consideration."
Hume could fall back on the argument from the finitude of the mind and the resultant impossibility of its comprehension of an infinite number of parts which extension would contain. He looks for new absurdities in this reasoning.
We can think of the dimensions as terminations, (a point terminates a line), but if the ideas are infinitely divisible, then the mind cannot fix on the terminating part. "It immediately finds this idea to break into parts, and upon seizing the last of these parts, it loses its hold by a new division, and so on ad infinitum , without any possibility of its arriving at a concluding idea." The real terminating idea, which there must be, does not consist of parts.
The schoomen reacted to this strong argument by mixing mathematical points (terminators) in with matter, which itself is infinitely divisible. Others tried to dodge it by "a heap of unintelligible cavils and distinctions." But this is just to concede the argument.
So the definitions of mathematics destroy the demonstrations it claims to make.
The conclusion is that the demonstrations rest on ideas which are not exact and principles which are not precisely true. "When geometry decides any thing concerning the proportions of quantity, we ought not to look for the utmost precision and exactness. None of its proofs extend so far. It takes the dimensions and proportions of figures justly; but roughly, and with some liberty. Its errors are never considerable, nor would it err at all, did it not aspire to such an absolute perfection."
Hume asks the mathematicians: what does it mean to say that one line is equal to another? Any answer, either based on indivisible points or infinitely divisible quantities, will be an embarrassment.
Although there are few mathematicians who defend the doctrine of indivisible points, they would answe by claiming that lines are equal when the number of points is equal. Though the answer is correct, it is useless due to grossness of the senses.
On the other hand, on the assumption of infinite divisibility, there is no answer to the question. Infinite numbers cannot be equal or unequal with respect to each other.
Perhaps congruity, i.e., the touching of the end of overlaid lines, is the required relation. As a relation, it requires a comparison made by the mind. Again, the grossness of the senses will prevent our determining this. Besides, because "the contact of large parts wou'd never render the figures equal, any comparison will require minima, so the notion of congruity is no advance over the doctrine of mathematical points.
[Appendix] Many philosophers refuse to define equality, opting instead for the perception of "the whole united appearance and the comparison of particular objects." Hume endorses this view.
The mind often at one view can determine the proportions of bodies, often with certainty and infallibility. This holds for cases of gross inequalities.
But judgements are often subject to "doubt and error" as much as judgments on other subjects. So we correct them "by a review and reflection." We also juxtapose objects to correct our judgment in another way. But this correction is subject to further correction with finer instruments.
So we form a "mix'd notion of equality" derived from comparing appearances, which is a looser method and using a common measure, which is a stricter. We should stop here, but we don't. We go on to reason that there are much smaller bodies which we cannot measure, and of whose proportions therefore we must remain uncertain. So we suppose an imaginary standard. But it is "plainly imaginary," and the correction we would be accomplishing by it is "a mere fiction of the mind." This is most obvious in the case of time, but it also applies to musical notes, judgment of colors, etc. They are imagined to be subject to a strict standard of equality.
The same reasoning holds for the distinction between a CURVED line and a STRAIGHT ("right") line. We know what they are by sight, but we cannot define them. All we see is a "united appearance," so even Hume's system of indivisible points cannot be applied, and infinite divisibility is even in a worse condition. Too, we are able to perform corrections and thus form "a loose idea of a perfect standard to these figures, without being able to explain or comprehend it."
That the straight line is the shortest distance between two points is "one of the properties of a right line" rather than "a just definition of it." [Cf. Kant, who considers the claim to be synthetic.] A straight line can be comprehended alone, while that of the shortest distance requires comparison with other lines.
Moreover, to use the alleged definition, we would need a precise idea of what is a shortest line, and we have none. "An exact idea can never be built on such as are loose and indeterminate."
Nor can we provide a precise standard to the notion of a flat ("plane") surface. It cannot be defined as the "flowing" of a straight line. First, we understand the notion of a flat surface independently of how the surface is generated. Second, we have no precise standard of a straight line. Third, a straight line may flow irregularly, so we would have to define a plane as flowing between to parallel straight lines on the same plane, which results in a circular argument.
So the most important ideas of geometry, equality and inequality, straightness and flatness, are not exact and determinate. We cannot tell the difference in dubious cases, and we cannot form an idea of them "which is firm and invariable." We can only appeal to "the weak and fallible judgment, which we make from the appearance of the objects, and correct by a compass or common measure." Any supposition of a further correction is useless or imaginary. And we cannot suppose that a god could form a perfect geometrical figure. "As the ultimate standard of these figures is deriv'd from nothing but the senses and imagination, 'tis absurd to talk of any perfection beyond what these faculties can judge of; since the true perfection of any thing consists in its conformity to its standard."
This casts doubt on geometrical demonstations, such as that two straight lines cannot have a common segment, or that only one straight line can be drawn between two points. This may work for sensible lines, but consider two lines that form a very acute angle to each other over a long distance. "I perceive no absurdity in asserting, that upon their contact they become one." To object to this, you need a definition of a straight line and to show that there would be a deviation in the case of the two straight lines merging. But this can get the mathematician back to the notion of indivisible points (which would prohibit the merging), and in any case, this is not the standard of the straight line, or if it were, it would be of no use. "The original standard of a right line is in reality nothing but a certain general appearance; and 'tis evident right lines may be made to concur with each other, and yet correspond to this standard, tho' corrected by all the means either practicable or imaginable.
[Appendix] Mathematicians always have a dilemma. If they appeal to an exact standard, they reinstate indivisible points, which want to avoid doing, and which furnish no practicable standard, or the accept the gross judgments based on appearances, which do not support their conclusions.
Alleged geometical demonstrations of infinite divisibility do not have the force of other geometrical arguments. The reason "why geometry fails of evidence in this single point, while all its other reasonings command our fullest assent and approbation" is that no idea of quantity is infinitely divisible. Any attempt to show that the quantity itself is infinitely divisible is bound to land in absurdity.
Hume tries to illustrate this point with the geometrical proposition that there can be only one point of contact between a straight line and a circle tangent to it. There is more than one in any diagram that is drawn, so the case is pushed off to "certain ideas, which are the true foundation of all our reasoning." If you form the idea of them, you find either that they touch in a mathematical point or that they overlap for some space. If the former, then mathematical points are possible in idea, and therefore are possible in fact, which is an undesirable outcome. On the other hand, if they must concur, then the fallacy of the geometrical demonstration of their non-concurrence is exposed.