### SEC 4: OBJECTIONS ANSWERED

- Our system on space and time has two parts intimately connected.
- The first depends on this chain of reasoning.
- The mind’s capacity is not infinite.
- Consequently, no idea of extension or duration consists of an infinite number of parts or inferior ideas.
- It consists of a finite number which are simple and indivisible.

- It is therefore possible for space and time to exist conformable to this idea.
- If it were possible, it is certain they actually do exist.
- Since their infinite divisibility is utterly impossible and contradictory.

- If it were possible, it is certain they actually do exist.

- Consequently, no idea of extension or duration consists of an infinite number of parts or inferior ideas.

- The mind’s capacity is not infinite.

- The second part of our system is a consequence of this.
- The ideas of space and time finally resolve themselves into indivisible parts.
- These indivisible parts are nothing in themselves.
- They are inconceivable when not filled with something real and existent.

- The ideas of space and time are therefore not separate nor distinct ideas.
- They are merely ideas of the manner or order, in which objects exist.
- In other words, it is impossible to conceive:
- a vacuum and extension without matter, or
- a time when there was no succession or change in any real existence.

- The ideas of space and time finally resolve themselves into indivisible parts.
- The intimate connection between these parts of our system is why we shall examine the objections against them.
- We begin with the objections against the finite divisibility of extension.

- The first objection is that it is more proper to prove this connection and dependence of the one part on the other, than to destroy either of them.
- It has often been maintained in the schools, that extension must be divisible to infinity because the system of mathematical points is absurd.
- Such a system is absurd because a mathematical point is a non-entity.
- Consequently, it can never form a real existence.

- Such a system is absurd because a mathematical point is a non-entity.
- This would be perfectly decisive, if there were no medium between:
- the infinite divisibility of matter
- the non-entity of mathematical points.

- But there is evidently a medium: the bestowing a colour or solidity on these points.
- The absurdity of both the extremes is a demonstration of the truth and reality of this medium.

- The system of physical points is another medium.
- It is too absurd to need to refute.

- A real extension, such as a physical point, can never exist without parts different from each other.
- Wherever objects are different, they are distinguishable and separable by the imagination.

- It has often been maintained in the schools, that extension must be divisible to infinity because the system of mathematical points is absurd.

- The second objection is derived from the necessity of PENETRATION, if extension consisted of mathematical points.
- A simple and indivisible atom that touches another, must necessarily penetrate it.
- It is impossible it can touch it by its external parts, from the very supposition of its perfect simplicity which excludes all parts.

- It must therefore touch it:
- intimately, and
- in its whole essence
*secundum se, tota, et totaliter*- This is the very definition of penetration.
- But penetration is impossible.
- Consequently, mathematical points are equally impossible.

- A simple and indivisible atom that touches another, must necessarily penetrate it.

- I answer this objection by substituting a juster idea of penetration.
- Suppose two bodies, containing no void within their circumference,:
- approach each other
- unite so that the resulting body is no more extended than either of them.

- This is what we mean when we talk of penetration.
- But this penetration is just the:
- annihilation of one of these bodies
- preservation of the other, without our being able to distinguish which is preserved and which is annihilated.

- Before the approach, we have the idea of two bodies.
- After it, we have the idea only of one.

- It is impossible for the mind to preserve any notion of difference between two bodies of the same nature existing in the same place at the same time.

- Suppose two bodies, containing no void within their circumference,:

- I use ‘penetration’ to mean the annihilation of one body on its approach to another.
- Does anyone see a necessity in a coloured or tangible point being annihilated on the approach of another coloured or tangible point?
- On the contrary, does anyone not perceive that from the union of these points, an object results which:
- is compounded and divisible
- may be distinguished into two parts, each preserving its existence distinct and separate, despite its contiguity to the other?

- Let him conceive these points in different colours, to better prevent their coalition and confusion.
- A blue and a red point may surely lie contiguous without any penetration or annihilation.
- If they cannot, what possibly can become of them?
- Shall the red or the blue be annihilated?
- If these colours unite into one, what new colour will they produce?

- What chiefly gives rise to these objections and renders it so difficult to answer is the natural infirmity and unsteadiness of our imagination and senses when employed on such minute objects.
- Put a spot of ink on paper and go to such a distance that the spot becomes invisible.
- You will find that on your return and nearer approach, the spot:
- first becomes visible by short intervals
- afterwards becomes always visible
- afterwards acquires only a new force in its colouring without adding its bulk
- afterwards, when it has increased to be really extended, it is still difficult for the imagination to break it into its component parts, because of its uneasiness in the conception of such a minute object as a single point.
- This infirmity affects most of our reasonings on the present subject.
- It makes the imagination almost impossible to answer intelligibly the many questions about it.

- There have been many objections from the mathematics against the indivisibility of the parts of extension.
- Though at first sight, mathematics seems:
- favourable to the present doctrine
- perfectly conformable in its definitions, if mathematics were contrary in its
*demonstrations*.

- My present business then must be to:
- defend the definitions
- refute the demonstrations.

- Though at first sight, mathematics seems:

- A surface is defined to be length and breadth without depth.
- A line to be length without breadth or depth.
- A point to be what has neither length, breadth nor depth.
- All this is perfectly unintelligible on any other supposition than that of the composition of extension by indivisible points or atoms.
- How else could anything exist without length, breadth, or depth?

- I find two answers to this argument.
- I think neither answer is satisfactory.

- The first is, that the objects of geometry are mere ideas in the mind.
- It never did and never can exist in nature.
- No one will pretend to draw a line or make a surface entirely conformable to the definition.
- We may produce demonstrations from these very ideas to prove that they are impossible.

- It never did and never can exist in nature.

- Can anything be more absurd and contradictory than this reasoning?
- Whatever can be conceived by a clear and distinct idea necessarily implies the possibility of existence.
- He who pretends to prove the impossibility of its existence by any argument derived from the clear idea, in reality asserts that we have no clear idea of it, because we have a clear idea.
- It is in vain to search for a contradiction in anything that is distinctly conceived by the mind.
- If it implied any contradiction, the idea would have never been conceived.

- There is therefore no medium between:
- allowing at least the possibility of indivisible points
- denying their idea.

- The second answer to the foregoing argument is founded on this latter principle.
- It has been pretended (L’Art de penser), that though it is impossible to conceive a length without any breadth, yet by an abstraction without a separation, we can consider the one without regarding the other.
- In the same way as we think of the distance between two towns and overlook its width.

- The length is inseparable from the width both in:
- nature
- our minds.

- But this does not exclude a partial consideration and a distinction of reason.

- In refuting this answer, I shall not insist on the argument that:
- if it is impossible for the mind to arrive at a minimum in its ideas, its capacity must be infinite to be able to comprehend the infinite number of parts making up its idea of any extension.

- I shall try to find some new absurdities in this reasoning.

- A surface terminates a solid.
- A line terminates a surface.
- A point terminates a line.
- If the ideas of a point, line, or surface were not indivisible, we could never conceive these terminations.
- For let these ideas be supposed infinitely divisible.
- Let the fancy try to fix itself on the idea of the last surface, line or point.
- It immediately finds this idea to break into parts.

- On its seizing the last of these parts, it loses its hold by a new division, and so on to infinity, without any possibility of its arriving at a concluding idea.
- The number of fractions bring it no nearer the last division, than the first idea it formed.
- Every particle eludes the grasp by a new fraction like quicksilver, when we try to seize it.

- But in fact, there must be something which terminates the idea of every finite quantity.
- This terminating idea cannot consist of parts or inferior ideas.
- Otherwise it would be the last of its parts, which finished the idea, and so on.

- This is a clear proof that the ideas of surfaces, lines and points admit not of any division.
- Those ideas of:
- surfaces cannot have divisions in depth
- lines cannot have divisions in breadth and depth
- points cannot have divisions any dimension.

- Those ideas of:

- The school was so sensible of the force of this argument.
- Some of them maintained that nature mixed among those particles of matter divisible to infinity, a number of mathematical points to terminate bodies.
- Others eluded the force of this reasoning by a heap of unintelligible cavils and distinctions.

- Both these adversaries equally yield the victory.
- A man confesses to the superiority of his enemy if he:
- hides himself
- fairly delivers his arms.

- A man confesses to the superiority of his enemy if he:

- The definitions of mathematics appear to destroy the pretended demonstrations.
- If we have the idea of indivisible points, lines and surfaces conformable to the definition, their existence is certainly possible.
- But if we have no such idea, then we can never conceive the termination of any figure.
- Without the termination, there can be no geometrical demonstration.

- None of these demonstrations can have sufficient weight to establish this principle of infinite divisibility.
- Because with regard to such minute objects, they are not demonstrations as they are built on:
- inexact ideas
- maxims which are not precisely true.

- Because with regard to such minute objects, they are not demonstrations as they are built on:
- When geometry decides anything about the proportions of quantity, we should not 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.
- It would not err at all if it did not aspire to such an absolute perfection.

- I ask mathematicians, what they mean when they say one line or surface is
*equal*to, or*greater*or*less*than another? - Let any of them answer.
- This question will embarrass them whether they maintain the composition of extension by:
- indivisible points, or
- quantities divisible to infinity.

- This question will embarrass them whether they maintain the composition of extension by:

- There are few mathematicians who defend the hypothesis of indivisible points.
- Yet these have the readiest and justest answer to the present question.

- They only need to reply that:
- lines or surfaces are equal when the numbers of points in each are equal
- as the proportion of the numbers varies, the proportion of the lines and surfaces is also varied.

- This answer is just and obvious.
- But this standard of equality is entirely useless.
- We never determine objects to be equal or unequal from such a comparison.

- The points of any line or surface are so minute and confounded with each other, that it is impossible for the mind to compute their number.
- Such a computation will never afford us a standard to judge their proportions with.

- No one will ever be able to determine by an exact numeration, that:
- an inch has fewer points than a foot, or
- a foot fewer than an ell or any greater measure

- This is why we seldom or never consider this as the standard of equality or inequality.

- Those who imagine that extension is divisible to infinity will be unable to:
- use this answer or
- fix the equality of any line or surface by a numeration of its component parts.

- The equality or inequality of any portions of space can never depend on any proportion in the number of their parts, since:
- according to their hypothesis, the smallest and biggest figures have an infinite number of parts
- infinite numbers can neither be equal nor unequal to each other.

- The inequality of:
- an ell and a yard consists in the different numbers of the feet which make them.
- a foot and a yard consists in the number of their inches.

- We must fix some standard of equality different from an enumeration of the parts, as:
- that quantity we call an inch in the one is equal to what we call an inch in the other
- it is impossible for the mind to find this equality by proceeding to infinity with these references to inferior quantities.

- There are some (see Dr. Barrow’s mathematical lectures) who pretend that:
- equality is best defined by congruity
- any two figures are equal when all their parts correspond to and touch each other, after placing one on the other.

- To judge of this definition, let us consider that since equality is a relation, it is not a property in the figures themselves.
- It arises merely from the comparison which the mind makes between them.

- If it consists in this imaginary application and mutual contact of parts, we must:
- at least have a distinct notion of these parts
- conceive their contact.

- In this conception, we would run up these parts to the greatest minuteness which can possibly be conceived.
- Since the contact of large parts would never render the figures equal.

- But the minutest parts we can conceive are mathematical points.
- Consequently, this standard of equality is the same with the standard derived from the equality of the number of points.
- We have already determined this to be a just but useless standard.

- Consequently, this standard of equality is the same with the standard derived from the equality of the number of points.
- We must therefore look to some other quarter for a solution of the present difficulty.

- There are many philosophers who refuse to assign any standard of equality.
- But they assert that it is sufficient to present two equal objects to give us a just notion of equality.

- They say all definitions are fruitless without the perception of such objects.
- If we perceive such objects, we no longer need of any definition.

- I entirely agree to this reasoning.
- I assert that the only useful notion of equality or inequality is derived from the whole united appearance and the comparison of particular objects.

- The eye or rather the mind, at one view, is often able to:
- determine the proportions of bodies
- pronounce them equal to or greater or less than each other, without examining or comparing the number of their minute parts.

- Such judgments are common and, in many cases, certain and infallible.
- When the measure of a yard and a foot are presented, the mind cannot question that the first is longer than the second, than it can doubt of those obvious principles.

- There are therefore three proportions, which the mind:
- distinguishes in the general appearance of its objects
- calls as ‘greater’, ‘less’ and ‘equal’.

- Its decisions about these proportions are sometimes infallible, but not always.
- Our judgments of this kind are not more exempt from doubt and error than our judgments on any other subject.

- We frequently correct our first opinion by a review and reflection.
- We pronounce those objects to be equal, which at first we esteemed unequal.
- We regard an object as less, though before it appeared greater than another.

- This is not the only correction which these judgments of our senses undergo.
- We often discover our error:
- by a juxtaposition of the objects, or
- if that is impractical, by using some common and invariable measure which informs us of their different proportions after being successively applied to each.

- We often discover our error:
- Even this correction is susceptible of:
- a new correction
- different degrees of exactness, according to the:
- nature of the measuring instruments
- care which we employ in the comparison.

- We form a mixed notion of equality derived from both the looser and stricter methods of comparison when the mind:
- is accustomed to these judgments and their corrections
- finds that the same proportion, which makes two figures appear equal to the eye, makes them also correspond to:
- each other
- any common measure they they are compared with.

- But we are not content with this.
- Sound reason convinces us that there are bodies vastly more minute than those which appear to the senses.
- A false reason would persuade us that there are bodies infinitely more minute.
- We clearly perceive that we do not have any instrument or art of measuring which can secure us from ill error and uncertainty.
- We are sensible, that the addition or removal of one of these minute parts, is not discernible either in the appearance or measuring.
- We imagine that two figures which were equal before, cannot be equal after this removal or addition.
- We therefore suppose some imaginary standard of equality which:
- exactly corrects the appearances and measuring
- reduces the figures entirely to that proportion.

- This standard is plainly imaginary.
- Equality is the idea of an appearance corrected by juxtaposition or a common measure.
- The notion of any correction beyond what we have instruments and art to make, is a mere fiction of the mind.
- It is useless and incomprehensible.

- Even if this standard is only imaginary, the fiction however is very natural.
- The mind most usually proceeds this way with any action, even after the reason which started it has ceased.
- This appears very conspicuously with regard to time.

- We have no exact method of determining the proportions of parts, not even so exact as in extension.
- Yet the various corrections of our measures and their different degrees of exactness, have given as an obscure and implicit notion of a perfect and entire equality.

- The case is the same in many other subjects.
- A musician finds his ear becoming everyday more delicate.
- He corrects himself entertains a notion of a complete
*tierce*or*octave*, by reflection and attention, without being able to tell where he derives his standard. - A painter forms the same fiction with regard to colours.
- A mechanic with regard to motion.

- To painter, the one light and shade; to the mechanic, swift and slow are imagined to be capable of an exact comparison and equality beyond the judgments of the senses.

- We may apply the same reasoning to
*curve*and*right*lines.- Nothing is more apparent to the senses than the distinction between a curve and a right line.
- We form these ideas most easily.

- But however easily we may form these ideas, it is impossible to define them to fix their precise boundaries.
- When we draw lines on paper or any continued surface, there is a certain order which the lines run along from one point to another, that they may produce the entire impression of a curve or right line.
- But this order is perfectly unknown.
- Only the united appearance is observed.

- Thus, even on the system of indivisible points, we can only form a distant notion of some unknown standard to these objects.
- We cannot go even this length on that standard of infinite divisibility.
- We are reduced merely to the general appearance, as the rule by which we determine lines to be curve or right ones.

- We cannot:
- give a perfect definition of these lines
- produce any very exact method of distinguishing one from the other.

- But this does not hinder us from correcting the first appearance by:
- a more accurate consideration
- a comparison with some rule, of whose rectitude from repeated trials we have a greater assurance.

- We form the loose idea of a perfect standard to these figures, without being able to explain or comprehend it:
- from these corrections
- by carrying on the same action of the mind, even when its reason fails us.

- Mathematicians pretend to give an exact definition of a right line when they say that it is the shortest way between two points.
- This is more the discovery of one of the properties of a right line, than a just deflation of it.

- If upon mention of a right line, does a person:
- not think immediately on an appearance of a right line?
- not consider this property only by accident?

- A right line can be comprehended alone.
- But this definition is unintelligible without a comparison with other lines which we conceive to be more extended.

- In common life, it is established as a maxim, that the straightest way is always the shortest.
- It would be as absurd as to say that the shortest way is always the shortest, if our idea of a right line was the same as the shortest way between two points.

- Secondly, we have no precise idea of equality and inequality, shorter and longer, more than of a right line or a curve.
- Consequently, the one can never afford us a perfect standard for the other.
- An exact idea can never be built on such as are loose and undetermined.

- The idea of a plain surface is as little susceptible of a precise standard as that of a right line.
- We do not have any other means of distinguishing such a surface, than its general appearance.

- It is in vain, that mathematicians represent a plain surface as produced by the flowing of a right line.
- It will immediately be objected that:
- our idea of a surface is as independent of this method of forming a surface, as our idea of an ellipse is of that of a cone.
- the idea of a right line is no more precise than the idea of a plain surface.
- a right line may flow irregularly and form a figure different from a plane
- we must therefore suppose a right line to flow along two right lines, parallel to each other, and on the same plane.
- This description explains a thing by itself, and returns in a circle.

- The ideas most essential to geometry are those of:
- equality and inequality
- a right line and a plain surface

- These are far from being exact and determinate, according to our common method of conceiving them.
- We are incapable of telling if the case is doubtful in any degree, when:
- such figures are equal
- such a line is a right one
- such a surface is a plain one.

- We are incapable of telling if the case is doubtful in any degree, when:
- But we cannot form an idea of that proportion or these figures which is firm and invariable.
- Our appeal is still to the weak and fallible judgment, which we:
- make from the appearance of the objects
- correct by a compass or common measure.

- If we join the supposition of any further correction, it is of such-a-one as is useless or imaginary.
- We would:
- have recourse to the common topic in vain
- employ the supposition of a deity whose omnipotence may enable him to:
- form a perfect geometrical figure
- describe a right line without any curve or inflexion.

- We would:
- The ultimate standard of these figures is only derived from the senses and imagination.
- We cannot talk of any perfection beyond what these faculties can judge of.
- Since the true perfection of anything is in its conformity to its standard.

- Since these ideas are so loose and uncertain, I ask any mathematician: what infallible assurance he has of:
- the more intricate and obscure propositions of mathematics
- the most vulgar and obvious principles?

- How can he prove to me that two right lines cannot have one common segment?
- Or that it is impossible to draw more than one right line between any two points?

- If he tells me that these opinions are absurd and repugnant to our clear ideas, I would answer that I do not deny, where two right lines incline on each other with a sensible angle.
- But it is absurd to imagine them to have a common segment.

- But supposing these two lines approach at the rate of an inch in 20 leagues, it is not absurd to assert that they become one on their contact.
- By what rule or standard do you judge, when you assert that the line, in which I have supposed them to concur, cannot make the same right line with those two, that form so small an angle between them?
- You must surely have some idea of a right line, to which this line does not agree.

- Do you therefore mean that it does not take the points in the same order and by the same rule, as is peculiar and essential to a right line?
- If so, I must inform you that:
- bbesides that in judging after this manner, that extension is composed of indivisible points.
- Perhaps this is more than you intend.

- this is not the standard from which we form the idea of a right line.
- if it were, is there any such firmness in our senses or imagination to determine when such an order is violated or preserved?

- bbesides that in judging after this manner, that extension is composed of indivisible points.

- If so, I must inform you that:
- The original standard of a right line is in reality nothing but a certain general appearance.
- Right lines may be made to concur with each other, and yet correspond to this standard, though corrected by all the practical or imaginable means.

- This dilemma meets mathematicians whatever side they turn to.
- If they judge of equality or any proportion by the accurate and exact standard, namely the enumeration of the minute indivisible parts, they:
- employ a standard useless in practice
- actually establish the indivisibility of extension, which they endeavour to explode.

- If they employ, as is usual, the inaccurate standard derived from a comparison of objects on their general appearance, corrected by measuring and juxtaposition, their infallible first principles are too coarse to afford any subtle inferences which they commonly draw.
- The first principles are founded on the imagination and senses.
- The conclusion, therefore, can never go beyond, much less contradict these faculties.

- If they judge of equality or any proportion by the accurate and exact standard, namely the enumeration of the minute indivisible parts, they:

- This may let us see that no geometrical demonstration for the infinite divisibility of extension can have so much force as what we naturally attribute to every argument supported by such magnificent pretensions.
- We may also learn why geometry falls of evidence in this single point, while all its other reasonings command our fullest assent and approbation.

- It seems more requisite to give the reason of this exception, than to show that we really must:
- make such an exception
- regard all the mathematical arguments for infinite divisibility as sophistical.

- Since no idea of quantity is infinitely divisible, it is most absurd to try:
- to prove that quantity itself admits of such a division
- to prove this by means of directly opposite ideas.

- All arguments founded on this absurdity will have a new absurdity.

- I might give as instances those arguments for infinite divisibility derived from the point of contact.
- All mathematicians will agree to be judged by the diagrams they describe on paper.
- They tell us these diagrams are loose drafts which only convey certain ideas which are the true foundation of all our reasoning.
- I am satisfied with this.
- I am willing to rest the controversy merely on these ideas.

- I desire therefore our mathematician to form, as accurately as possible, the ideas of a circle and a right line.
- I then ask, if on the conception of their contact:
- he can conceive them as touching in a mathematical point, or
- if he must necessarily imagine them to concur for some space.

- I then ask, if on the conception of their contact:
- Whichever he chooses, he runs himself into equal difficulties.
- If he can imagine those figures to touch only in a point, he allows the possibility of that idea and consequently of the thing.
- If he says that in conceiving those lines’ contact, he must make them concur, he acknowledges the fallacy of geometrical demonstrations when carried beyond a certain degree of minuteness.
- Since he can prove an idea of concurrence to be
*incompatible*with the ideas of a circle and right line, though simultaneously acknowledging these ideas to be inseparable.

- Since he can prove an idea of concurrence to be

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