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.