What is Knowledge?

Knowledge is Based on the Resemblance, Contrariety, Quality, and Quantity of Ideas to Each Other

There are 7 kinds of philosophical relation:

1. Resemblance
2. Identity
3. Relations of time and place
4. Proportion in quantity or number
5. Degrees in any quality
6. Contrariety
7. Causation

These relations are divided into 2 classes:

1. Relations that depend entirely on the ideas that we compare
2. Relations that can change without any change in the ideas

From the idea of a triangle, we discover the relation of equality which its three angles bear to two right ones.

This relation does not change as long as our idea [the triangle] does not change.

On the contrary, the relations of contiguity and distance between two objects can change merely by their change of place, without any change on the objects themselves or on their ideas.

The place depends on 100 different accidents which cannot be foreseen by the mind.

It is the same case with identity and causation.

Two objects, which perfectly resemble each other and appeare in the same place at different times, may be numerically different. The power by which one object produces another is never discoverable merely from their idea.

Therefore, cause and effect are relations that arise from experience, and not from any abstract reasoning or reflection.

Quantity is not Intuitive Knowledge

No single phenomenon can be accounted for from the qualities of the objects as they appear to us, without the help of our memory and experience.

Therefore, only 4 of these 7 relations can be the objects of knowledge and certainty:

1. Resemblance
2. Contrariety
3. Degrees in quality
4. Proportions in quantity or number.

Three of these relations, resemblance, contrariety, and quality:

• are discoverable at first sight, and
• fall more properly under intuition than demonstration.

When any objects resemble each other, the resemblance will at first strike the eye or rather the mind. It seldom requires a second examination.

The case is the same with contrariety and with quality.

Existence and non-existence:

• destroy each other, and
• are perfectly incompatible and contrary.

It is impossible to judge exactly of the degrees of any quality, such as colour, taste, heat, cold, when the difference between them is very small.

But it is easy to decide that any of them is superior or inferior to another, when their difference is considerable. This decision we always pronounce at first sight, without any enquiry or reasoning.

We might:

• proceed in the same way in fixing the proportions of amount, and
• at one view observe a superiority or inferiority between any numbers or figures especially where the difference is very big.

We can only guess at equality or exactness from a single consideration.

Except in very short numbers, or very limited portions of space which are comprehended in an instant, and where we perceive an impossibility of falling into any considerable error.

In all other cases we must:

• settle the proportions with some liberty, or
• proceed in a more artificial way.

Geometry is the art by which we fix the proportions of figures.

It excels both in:

• universality and exactness, and
• the loose judgments of the senses and imagination.

Yet it never attains a perfect precision and exactness.

Geometry’s first principles are drawn from the general appearance of the objects.

That appearance can never afford us any security, when we examine the minuteness of nature.

Our ideas seem to give a perfect assurance that no two right lines can have a common segment.

But if we consider these ideas, we shall find that: they always suppose a sensible inclination of the two lines, and where the angle they form is extremely small, we have no standard of a right line so precise as to assure us of the truth of this proposition. It is the same case with most of the primary decisions of the mathematics.

Only the sciences of algebra and arithmetic can we:

• carry on a chain of reasoning to any degree of intricacy, and
• yet preserve a perfect exactness and certainty.

We are have a precise standard to judge of the equality and proportion of numbers.

We determine their relations according as they correspond to that standard, without any possibility of error. When two numbers are so combined, as that the one has always an unite answering to every unite of the other, we pronounce them equal.

Geometry can never be a perfect and infallible science for want of such a standard of equality in extension. Geometry and Mathematics Create Philosophical Absurdities by Insisting on Exactness or Obscurity Geometry falls short of that perfect precision and certainty which are peculiar to arithmetic and algebra.

But it excels at the imperfect judgments of our senses and imagination. I impute geometry’s defect from its original and fundamental principles being derived merely from appearances.

This defect must always:

• attend geometry, and
• keep geometry from ever reaching a greater exactness in the comparison of objects or ideas, than what our eye or imagination alone is able to attain.

I own that this defect so far attends it, as to keep it from ever aspiring to a full certainty:

But since these fundamental principles depend on the easiest and least deceitful appearances, they bestow on their consequences a degree of exactness, of which these consequences are singly incapable.

It is impossible for the eye to determine the angles of a polygon with 1,000 sides to be equal to 1,996 right angles or to make any conjecture that approaches this proportion.

But when it determines, that right lines cannot concur; that we cannot draw more than one right line between two given points.

Its mistakes can never be of any consequence.

The nature and use of geometry is to run us up to such appearances which, because of their simplicity, cannot lead us into any considerable error.

My second observation on our demonstrative reasonings is suggested by mathematics.

Mathematicians pretend that their object-ideas are of so refined and spiritual a nature, that they:

• do not fall under the conception of the imagination, and
• must be comprehended by a pure and intellectual view which only the soul can do.

The same notion:

• runs through most parts of philosophy, and
• is principally used to:
  • explain our abstract ideas, and
  • show how we can create for example, an idea of a triangle which shall:
    • neither be an isoceles nor scalenum, nor
    • be confined to any length and proportion of sides.

It is easy to see why philosophers are so fond of this notion of some spiritual and refined perceptions since by that means, they:

• cover many of their absurdities, and
• may refuse to submit to the decisions of clear ideas, by appealing to obscure ones.

To destroy this artifice, we only need to reflect on that principle so often insisted on, that all our ideas are copied from our impressions.

From this, we may conclude that the ideas copied from our impressions:

• must be of the same nature, and
• can never contain anything so dark and intricate, but from our fault.

An idea is by its very nature weaker and fainter than an impression.

But since it is the same in every other respect, it cannot imply a great mystery.

If its weakness renders it obscure, it is our business to remedy that defect as much as possible by keeping the idea steady and precise.

Until we have done so, it is in vain to pretend to reasoning and philosophy.