Section 4


by David Hume Icon

The Indivisibility of the Quantum of Space is Rejected by Mathematicians

Our system on space and time has two 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 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 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, and 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 it can touch it by its external parts, from the very supposition of its perfect simplicity that excludes all parts.

It must therefore touch it:

  • 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 two 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 two bodies.

After the union, we have the idea only of one body.

It is impossible for the mind to preserve the difference between two 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 two 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.

Equality is Different from Infinity

A few mathematicians defend the hypothesis of indivisible points.

Yet these have the readiest and most proper answer.

They only need to reply that:

  • lines or surfaces are equal when the numbers of points in each are equal, and
  • as the proportion of the numbers varies, the proportion of the lines and surfaces also varies.

This answer is proper 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.

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 with exact numbers that:

  • an inch has fewer points than a foot, or
  • a foot fewer than a yard.

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

Those who imagine that space is divisible to infinity will not be able to:

  • use this answer, nor
  • fix the equality of any line or surface by counting 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 shapes have an infinite number of parts, and
  • infinite numbers can neither be equal nor unequal to each other.

It is true that:

  • the inequality of a mile and a yard is in the number of the feet that make them, and
  • the inequality of a foot and a yard is in the number of their inches.

But 1 inch in a foot is the same 1 inch in a yard.

It would be impossible to infinitely list all the decimals of the inches which make a foot or a yard.

We must fix a standard of equality different from an enumeration of the parts.

Some people (see Dr. Barrow’s mathematical lectures) pretend that:

  • equality is best defined by congruity, and
  • any two shapes are equal when all their parts correspond to and touch each other, after placing one on the other.

But equality is a relation.

It is not a property in the shapes themselves.

It arises merely from the comparison made by the mind between them.

If it consists in this comparison and mutual contact of parts, we must:

  • at least have a distinct notion of these parts, and
  • conceive their contact.

This would make us think of these parts in the smallest forms that can be thought of, since the contact of the shapes would never render the shapes equal.

But the smallest parts that 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 fair but useless standard.

We must therefore look to some other solution to this.

Many philosophers refuse to assign any standard of equality.

They assert that it is enough 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, and
  • 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, and
  • calls as ‘greater’, ’less’ and ’equal’.

The mind’s decisions on 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.

Even this correction is susceptible of:

  • a new correction, and
  • different degrees of exactness, according to the:
    • nature of the measuring instruments, and
    • 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 used to these judgments and their corrections, and
  • finds that the same proportion, which makes two figures appear equal, makes them also correspond to:
    • each other, and
    • any common measure that they are compared with.

Sound Reason convinces us that there are bodies vastly smaller than those that we can see.

But a false reason would persuade us that there are bodies infinitely smaller.

We clearly know that we have no instrument or any way to prove this.

We are sensible, that the addition or removal of one of these smallest parts is not discernable.

But we still imagine that two shapes that 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, and
  • 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.

We cannot make any correction beyond what our instruments can make.

Even if this standard is only imaginary, the fiction is very natural.

The mind 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 way to determine proportions in space.

Yet the various corrections of our measures and their different precisions, have given a similarly obscure notion of a perfect equality.

The case is the same in many other subjects:

  • A musician finds his ear becoming everyday more delicate.
  • He corrects himself and 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.

He imagines that light and shade are exact comparisons.

A mechanic forms the same fiction with regard to motion.

He imagines that swift and slow are comparisons.

We may apply the same reasoning to curve and straight lines.

It is easy to see the difference between a curve and a straight line.

But it is impossible to define them in order to fix their precise boundaries.

When we draw lines on paper, the lines run along from one point to another in a certain order.

They can produce a curve or a straight line.

But this order is perfectly unknown.

Only the united appearance is observed.

Thus, even on the system of indivisible points, we can only create a distant notion of some unknown standard.

We cannot go even this length on that standard of infinite divisibility.

We are reduced merely to the general appearance and to see them as curved or straight lines.

We cannot:

  • give a perfect definition of these lines, nor
  • produce any very exact way of distinguishing one from the other.

But this does not hinder us from correcting the first appearance by:

  • a more accurate consideration, and
  • a comparison with some rule to give us a greater assurance.

We create 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 straight line when they say that it is the shortest way between 2 points.

This is more the discovery of one of the properties of a straight line.

A person who things of a straight line immediately sees it as the shortest way between two points.

He does not see this shortest way being an accident.

A straight line can be comprehended alone.

But this definition is unintelligible without a comparison with other lines.

I object to this definition with 2 arguments.

  1. 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 straight line was the same as the shortest way between two points.

  1. We have no precise idea of equality and inequality, shorter and longer, more than of a straight line or a curve.

Consequently, the one can never give 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 flat surface also cannot have a precise standard.

We can only distinguish such a surface through its general appearance. Mathematicians say that a flat surface is created by the flowing of a straight line. We can immediately object that straight lines constrained into a plane necessarily creates a flat surface. This description explains a thing by itself, and is circular reasoning.

The ideas most essential to geometry are those of:

  • equality and inequality, and
  • a straight 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, and
  • such a surface is a plain one.

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, and 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, and
  • employ the supposition of a deity whose omnipotence may enable him to:
    • create a perfect geometrical figure, and
    • describe a straight line without any curve or inflexion.

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, and
  • the most vulgar and obvious principles?

How can he prove:

  • that 2 straight lines cannot have one common segment?
  • that it is impossible to draw more than one straight 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 straight lines incline on each other with a sensible angle.

But it is absurd to imagine them to have a common segment.

But if these two lines approach at the rate of an inch per 100 kilometers, they will eventually meet.

By what rule do you use to assert that the line cannot make the same straight line with those two, that form so small an angle between them?

You must surely have some idea of a straight 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 straight line?

If so, I must inform you that:

  • besides that in judging after this manner, that space 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 straight line.
    • if it were, is there any such firmness in our senses or imagination to determine when such an order is violated or preserved?

The original standard of a straight line is in reality nothing but a certain general appearance.

Straight lines may be made to concur with each other, and yet correspond to this standard, though corrected by other means.

This dilemma meets mathematicians whatever side they turn to.

If they judge of equality by the exact standard of the enumeration of the minute indivisible parts, they:

  • employ a standard that is useless, and
  • actually establish the indivisibility of space.

If they employ, as is usual, the inaccurate standard derived by comparing objects on their general appearance, corrected by measuring and juxtaposition, their infallible first principles are too coarse to afford any subtle inferences.

The first principles are founded on the imagination and senses.

The conclusion, therefore, can never go beyond, much less contradict the imagination.

Thus, no geometrical demonstration for the infinite divisibility of space can be supported.

This is also 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, and
  • to prove this through directly opposite ideas.

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

For example, all mathematicians are judged by the diagrams they write on paper. They tell us these diagrams are loose drafts which only convey certain ideas which are the true foundation of all our reasoning. This is satisfactory, if they just refer to them as ideas. A circle touching a line

I ask our mathematician to think of a circle touching a straight line.

Do they intersect or touch at a precise point?

If they intersect at a precise point, then it means a precise point exists.

Or does their intersection and touch occupy the same space?

If he says that they occupy the same space, then it means:

  • that shapes cannot be analyzed beyond a certain degree of minuteness, and
  • that the circle and straight line creates a new shape.