The Infinite Divisibility of Space and Time

The Limitation of the Mind Leads to Indivisible Units Space and Time

Wherever ideas represent objects, the relations, contradictions and agreements of the ideas are all applicable to those objects.

• This is generally the foundation of all human knowledge.

But our ideas represent the smallest parts of space.

Whatever divisions and subdivisions these parts may lead to, they can never become inferior to some ideas that we create.

This means that whatever appears impossible and contradictory after comparing these ideas must be really impossible and contradictory.

Anything that can be 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.

It follows that if any finite space is infinitely divisible, then a finite space* has an infinite number of parts.

• Conversely, if a finite space cannot have an infinite number of parts, then no finite space can be infinitely divisible.

But this latter supposition is absurd.

I clear this up by thinking of the smallest idea that I can about space.

This smallest idea is a real quality of space. I repeat this idea to create a compound idea of space that becomes bigger.

If I did this infinitely, then the space would akso be infinite.

Therefore:

• an infinite number of parts is the same as the infinite space*, and
• no finite space can have an infinite number of parts.

Consequently, no finite space is infinitely divisible [Footnote 3].

How to Clear Up Spacetime

I first take the smallest idea that I can form of a part of space.

• Since this is the smallest idea, it can be used to discover the real qualities of space.

Time and Space is an Illusion

Nicolas de Malézieu is a noted author who proposed that existence in itself:

• belongs only to unity, and
• is never applicable to number, but the units which make up that number.

This argument is very strong and beautiful.

20 men exist only because 1, 2, 3, etc. exist.

If you deny the existence of those 1, 2, 3, etc, then the existence of the 20 falls.

It is therefore absurd to suppose any number to exist, and yet deny the existence of its units.

Metaphysicians feel that space:

• is always a number, and
• never resolves itself into any unit or indivisible quantity.

It follows that space can never exist at all.

It is in vain to reply that any specific amount of space is a kind of unit that:

• allows an infinite number of fractions, and
• can be subdivided infinitely.

Two Kinds of Unity

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.

This kind of unity is merely a fictitious denomination.

The mind may apply it to any quantity of objects that it collects together.

Such a unity can no more exist alone than a number can, as being a true number in reality. 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, and
• perfectly 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, and
• 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, and
• 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 obvious from the nature of motion.

If the infinite divisibility of time is impossible, then the infinite divisibility of space must also be impossible. The most obstinate defender of the doctrine of infinite divisibility will regard these arguments are difficulties. It is impossible to give them any clear answer.

But it is absurd to call this demonstration as a ‘difficulty’ just to elude its evidence.

It is in probabilities, not in demonstrations, that:

• difficulties can take place, and
• one argument counter-balances another and reduces 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 the 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, and
• 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.

• 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 space, for otherwise, why do we talk and reason about it?

This idea of space, 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 space which consists of parts or inferior ideas that are perfectly indivisible.

Consequently:

• this idea implies no contradiction,
• it is possible for space really to exist conformable to it, and
• all the arguments against the possibility of mathematical points are:
  • mere scholastich quibbles, and
  • 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 space are equally sophistical since these demonstrations cannot be just without proving the impossibility of mathematical points, which is absurd.