### 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.

- It is also obvious that:
- 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.

- It is because they are removed beyond that distance, at which their impressions were:
- 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.

- According to the system of infinite divisibility, this idea is:

### 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.

- 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.
- 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.

- Whether these parts be called divided or proportional, they cannot:

- 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.

- Time’s inseparable property and essence is that:
- 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.

- Because there would be an infinite number of co-existent moments or parts of time if:

- 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.

- It is not in demonstrations as in probabilities, that:
- 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.**

- If it is not just, it is a mere sophism.
- 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 can form the idea of a golden mountain.

- 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.**

- Consequently:

- 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.

Words: 2041

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