Objections

The Indivisibility of the Quantum of Space is Rejected by Mathematicians

Our system on space and time has 2 parts intimately connected.

Part 1 depends on this chain of reasoning:

• The mind’s capacity is not infinite.
• Consequently, no idea of space or time consists of an infinite number of parts or inferior ideas.
• Consequently, space or time consists of a finite number which are simple and indivisible.
• Therefore, it is possible for space and time to exist conformable to this idea.

If it were possible, then they certainly actually exist since their infinite divisibility is utterly impossible and contradictory.

Part 2 is a consequence of Part 1.

The ideas of space and time finally resolve themselves into indivisible parts, which are nothing in themselves.

• They are inconceivable when not filled with something real.

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 space without matter, or
• a time when there was no succession or change in any real existence.

The intimate connection between these parts of our system is why we shall examine the objections against them.

First are the objections against the finite divisibility of space.

Objection 1: It is more proper to prove this connection and dependence of the one part on the other, than to destroy either of them.

According to schools, space is divisible to infinity because the system of mathematical points is absurd.

• The system of mathematical points is absurd because a mathematical point is a non-entity.
• Consequently, it can never create a real existence.

This would be true if there were no medium between:

• the infinite divisibility of matter, and
• the non-entity of mathematical points.

But there are obviously media:

• the colour or solidity on these points, and
  • Colour and solidity cannot be extended infinitely. This is why this medium proves the finite divisibility of space.
• the system of physical points.

A physical point is real space.

• A real space can never exist without parts different from each other.

Wherever objects are different, they are distinguishable and separable by the imagination.

Objection 2: This is derived from the need for penetration.

If space consisted of mathematical points, then a simple and indivisible atom that touches another, must necessarily penetrate it.

It is impossible for a point to touch another point by its external parts. This is because a point has no parts.

The point must therefore touch the other point:

• intimately, and
• in its whole essence

This is the very definition of penetration.

• But penetration is impossible.
• Consequently, mathematical points are equally impossible.

I answer this objection by giving a more proper idea of penetration.

Penetration is 2 bodies uniting completely to create one body.

But this penetration is just:

• the annihilation of one of these bodies, and
• the preservation of the other, without our being able to distinguish which is preserved and which is annihilated.

Before the union, we had the idea of 2 bodies.

• After the union, we have the idea only of 1 body.

It is impossible for the mind to preserve the difference between 2 bodies of the same nature existing in the same place and time.

I define ‘penetration’ as the annihilation of one body on its approach to another.

Does anyone see a need 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, and
• may be distinguished into two parts, each preserving its separate existence, 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 will happen to them?
• Shall the red or the blue be annihilated?
• If these colours unite into one, what new colour will they produce?

Our imagination and senses have a natural infirmity and unsteadiness when employed on such small objects.

This creates these objections.

Put a spot of ink on paper and walk away until the spot becomes invisible.

When you walk back to the paper, the spot gradually becomes visible until you see it clearly.

Even then, the imagination still finds it difficult to break it into its component parts, because of its uneasiness in thinking of such a minute object.

This infirmity makes it almost impossible to answer the questions on minute objects.

Objection 3: Mathematics objects against the indivisibility of space.

Mathematics initially seems:

• favourable to the the indivisibility of space, and
• perfectly conformable in its definitions, if mathematics were contrary in its demonstrations.

My present task then must be to:

• defend the definitions, and
• refute the demonstrations.

A surface is defined as length and width without height.

• A line is length without width or height.
• A point has neither length, width or height.

These definitions only apply to things existing in space by indivisible points or atoms.

How else could anything exist without length, width or height?

I find 2 answers to this argument which are both not satisfactory.

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.

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.

Anyone 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 because 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, and
• denying their idea.

The second answer to the foregoing argument is founded on this latter principle.

It has been pretended in L’Art de penser that though it is impossible to conceive a length without any width, yet by an abstraction without a separation, we can consider the one without regarding the other.

In the same way, we think of the distance between two towns and overlook its width.

The length is inseparable from the width both in:

• nature, and
• 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 space.

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.

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 win.

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

• hides himself, and
• fairly delivers his arms.

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 enough 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, and
• maxims which are not precisely true.

Mathematics and Geometry are Countered by the Subjectiveness of ‘Equality’

Geometry creates shapes without the utmost precision and exactness.

It takes the dimensions and proportions of shapes roughly and with some liberty.

Its errors are never considerable.

It would not have errors if it aimed for an absolute perfection.

I ask mathematicians, what do they mean when they say one line or surface is equal to, or greater or less than another?

This question will embarrass them whether they maintain space by:

• indivisible points, or
• quantities divisible to infinity.