Part 2, Sec 1: Space and Time

Part 2: THE IDEAS OF SPACE AND TIME

SEC 1: THE INFINITE DIVISIBILITY OF OUR IDEAS OF SPACE AND TIME

• Philosophers often greedily embrace whatever:
  • has the air of a paradox
  • is contrary to the first and most unprejudiced notions of mankind, as showing the superiority of their science.
    • Their science could discover opinions so remote from vulgar conception.
• On the other hand, anything proposed to us which causes surprise and admiration, satisfies the mind that it:
  • indulges itself in surprise and admiration
  • will never be persuaded that its pleasure is entirely without foundation.
• From these dispositions in philosophers and their disciples, arises that mutual complaisance between them.
  • While the philosophers furnish plenty of strange and unaccountable opinions, their disciples so readily believe them.
• The most obvious example of this mutual complaisance is in the doctrine of infinite divisibility.
  • I shall examine this beginning with the ideas of space and time.

• The capacity of the mind:
  • is limited
  • can never attain a full and adequate conception of infinity.
• This is obvious from the plainest observation and experience.
  • It is also obvious that:
    • whatever is capable of being divided to infinity, must consist of an infinite number of parts
    • it is impossible to set any bounds to the number of parts, without setting bounds at the same time to the division.
• The idea we form of any finite quality is not infinitely divisible.
  • By proper distinctions and separations, we may run up this idea to inferior ones which will be perfectly simple and indivisible.
• In rejecting the mind’s infinite capacity, we suppose it may arrive at an end in the division of its ideas.
  • The evidence of this conclusion cannot be evaded.

• Therefore, the imagination may reach a minimum idea which it cannot further subdivide or reduce without annihilating it.
• When you tell me of the 1/1,000th and 1/10,000th part of a grain of sand, I have a distinct idea of:
  • these numbers
  • their different proportions.
• But the images I form in my mind to represent the sand themselves, are:
  • not different from each other
  • not inferior to that image of a grain of sand itself.
• What consists of parts is distinguishable into those parts.
  • What is distinguishable is separable.
• But the idea of a grain of sand is:
  • not distinguishable
  • not separable into 20, much less into 1,000, 10,000, or an infinite number of different ideas.

• The same is true for the impressions of the senses and the imagination.
• Put a spot of ink on paper, fix your eye on that spot, and go to such a distance that finally you lose sight of it.
  • The moment before it vanished, the image or impression was perfectly indivisible.
• The minute parts of distant bodies do not convey any sensible impression, not because of the lack of light striking our eyes.
  • It is because they are removed beyond that distance, at which their impressions were:
    • reduced to a minimum
    • incapable of any further reduction.
• A microscope or telescope, which renders them visible, does not produce any new rays of light.
  • It only spreads the rays which always flowed from them.
  • By that means, it:
    • gives parts to impressions, which to the naked eye appear simple and uncompounded
    • advances to a minimum, what was formerly imperceptible.

• We may hence discover the error of the common opinion, that:
  • the mind’s capacity is limited on both sides
  • it is impossible for the imagination to form an adequate idea of what goes beyond a certain degree of minuteness and greatness.
• Nothing can be more minute, than some:
  • ideas which we form in the fancy
  • images which appear to the senses.
    • Since there are ideas and images perfectly simple and indivisible.
• The only defect of our senses is that they:
  • give us disproportioned images of things
  • represent as minute and uncompounded what is really great and composed of a vast number of parts.
• We are not sensible of this mistake.
  • We take the impressions of those minute objects to be equal to the objects.
  • We find by reason that there are other objects vastly more minute.
  • We too hastily conclude that these more minute objects are inferior to any idea of our imagination or impression of our senses.
• We can form ideas no greater than the smallest atom of the animal spirits of an insect 1,000 times smaller than a mite.
  • We should rather conclude, that the difficulty lies in enlarging our conceptions to form a just notion of a mite or an insect 1,000 times smaller than a mite.
• To form a just notion of these animals, we must have a distinct idea representing every part of them.
  • According to the system of infinite divisibility, this idea is:
    • utterly impossible
    • extremely difficult because of the vast number and multiplicity of these parts.

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SEC 2: THE INFINITE DIVISIBILITY OF SPACE AND TIME

• Wherever ideas are adequate representations of objects, the relations, contradictions and agreements of the ideas are all applicable to the objects.
  • This is generally the foundation of all human knowledge.
• But our ideas are adequate representations of the most minute parts of extension.
  • Whatever divisions and subdivisions these parts may be arrived at, they can never become inferior to some ideas which we form.
• The plain consequence is that whatever appears impossible and contradictory on the comparison of these ideas, must be really impossible and contradictory.

• Everything capable of being infinitely divided contains an infinite number of parts.
  • Otherwise the division would be stopped short by the indivisible parts, which we should immediately arrive at.
• If therefore any finite extension is infinitely divisible, it is not contradictory that a finite extension contains an infinite number of parts.
  • And vice versa, if it is contradictory to suppose that a finite extension contains an infinite number of parts, no finite extension can be infinitely divisible.
    • But this supposition is absurd.
    • I clearly see why:
      • I first take the least idea I can form of a part of extension.
      • Being certain that there is nothing smaller than this idea, I conclude that whatever I discover by its means must be a real quality of extension.
      • I then repeat this idea many times and find the compound idea of extension.
        • This compound idea arises from its repetition.
        • It always adds and becomes double, triple, quadruple, etc. until at last it swells up to a considerable bulk as I repeat the same idea.
        • When I stop adding the parts, the idea of extension ceases to add.
        • If I carried this addition to infinity, I clearly perceive that the idea of extension must also become infinite.
• On the whole, I conclude that:
  • the idea of all infinite number of parts is individually the same idea with that of an infinite extension
  • no finite extension is capable of containing an infinite number of parts
    • consequently, that no finite extension is infinitely divisible [Footnote 3].

Footnote 3:

• It has been objected to me that:
  • infinite divisibility supposes only an infinite number of proportional, not of divided parts
  • an infinite number of proportional parts does not form an infinite extension.
• But this distinction is entirely frivolous.
  • Whether these parts be called divided or proportional, they cannot:
    • be inferior to those minute parts we conceive
    • form a less extension by their conjunction.

Nicolas de Malézieu

• I subjoin another argument proposed by a noted author, Nicolas de Malézieu, which seems very strong and beautiful to me, that existence in itself:
  • belongs only to unity
  • is never applicable to number, but on account of the unites which the number is composed of.
• Twenty men exist only because one, two, three, etc. exist.
  • If you deny the existence of those one, two, three, etc, then the existence of the 20 falls.
• It is therefore absurd to suppose any number to exist, and yet deny the existence of unites.
• According to metaphysicians, extension:
  • is always a number
  • never resolves itself into any unite or indivisible quantity.
• It follows that extension can never exist at all.
• It is in vain to reply that any determinate quantity of extension is a unite, but such-a-one that:
  • admits of an infinite number of fractions
  • is inexhaustible in its subdivisions.
    • For by the same rule these 20 men may be considered as a unit.
    • The whole earth or the whole universe may be considered as a unit.
• That term of unity is merely a fictitious denomination.
  • The mind may apply it to any quantity of objects it collects together.
  • Such a unity can no more exist alone than a number can, as being in reality a true number.
• But the unity which can exist alone and whose existence is necessary to the existence of all numbers, is of another kind.
  • That unity must be perfectly:
    • indivisible
    • incapable of being resolved into any lesser unity.

• All this reasoning takes place with regard to time, along with an additional argument.
  • Time’s inseparable property and essence is that:
    • each of its parts succeeds another
    • none of its parts, however contiguous, can ever be co-existent.
• The year 1737 cannot concur with the present year 1738.
  • Every moment must be distinct from and posterior or antecedent to another.
• Thus, time must be composed of indivisible moments.
  • Because there would be an infinite number of co-existent moments or parts of time if:
    • the division of time could never be ended
    • each moment, as it succeeds another, were not perfectly single and indivisible.
      • This will be an utter contradiction.

• The infinite divisibility of space implies the infinite divisibility of time, as is evident from the nature of motion.
  • If the latter is impossible, the former must be equally so.

• The most obstinate defender of the doctrine of infinite divisibility will regard these arguments are difficulties.
  • It is impossible to give them any clear and satisfactory answer.
• But nothing can be more absurd than this custom of calling a difficulty what pretends to be a demonstration, and then trying to elude its force and evidence.
  • It is not in demonstrations as in probabilities, that:
    • difficulties can take place
    • one argument counter-balance another and reduce its authority.
• A demonstration, if just, admits of no opposite difficulty.
  • If it is not just, it is a mere sophism.
    • It consequently can never be a difficulty.
  • It is either irresistible, or has no force.
• To talk therefore of objections and replies, and balancing of arguments in such a question as this, is to confess that:
  • human reason is nothing but a play of words, or
  • the person himself, who talks so, has not a capacity equal to such subjects.
• Demonstrations may be difficult to be comprehended, because of abstractedness of the subject.
  • But demonstrations can never have such difficulties as will weaken their authority after they are comprehended.

• Mathematicians are used to saying that:
  • here there are equally strong arguments on the other side of the question
  • the doctrine of indivisible points is also liable to unanswerable objections.
• I will take these arguments in a body and prove at once that it is impossible they can have any just foundation.

Nothing We Imagine Is Absolutely Impossible

• It is an established maxim in metaphysics, that whatever the mind clearly conceives, includes the idea of possible existence, or in other words, nothing we imagine is absolutely impossible.
  • We can form the idea of a golden mountain.
    • We can conclude that such a mountain may actually exist.
  • We can form no idea of a mountain without a valley.
    • We therefore regard it as impossible.

• We must have an idea of extension, for otherwise, why do we talk and reason about it?
  • This idea of extension, as conceived by the imagination, is divisible into parts or inferior ideas.
  • But it is not infinitely divisible.
  • It does not consist of an infinite number of parts.
    • For that exceeds the comprehension of our limited capacities.
• Here is an idea of extension which consists of parts or inferior ideas that are perfectly indivisible.
  • Consequently:
    • this idea implies no contradiction
    • it is possible for extension really to exist conformable to it
    • all the arguments against the possibility of mathematical points are:
      • mere scholastich quibbles
      • unworthy of our attention.

• We may carry these consequences one step further.
  • We may conclude that all the pretended demonstrations for the infinite divisibility of extension are equally sophistical.
  • Since these demonstrations cannot be just without proving the impossibility of mathematical points, which is absurd.

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