Internal Vs External Space: Numbers are Imaginary

8. Quantity and number differ only in our minds

Conversely, the number ten can be understood, as well as the continuous quantity of ten feet, even if we do not attend to that determinate substance: because the concept of the number ten is plainly the same, whether it is referred to this measure of ten feet or to anything else.

The continuous quantity of ten feet, although it cannot be understood without some extended substance of which it is the quantity, can nevertheless be understood without this determinate substance.

But in reality, it cannot happen that the smallest part of this quantity or extension be removed without an equal part also being taken away from the substance; nor conversely, that the smallest part of the substance be removed without an equal part of quantity and extension also being taken away.

The following are different only in our minds, but not in physical reality:

• Actual quantity-thought versus metaphysical quantity-idea
• Actual numbered-thought versus metaphysical number-idea

For example, we see a log that we estimate to measure 10 feet.*

• The 10-feet-estimate is our thought of its occupied space which is different from our ideal [substance] of 10 feet of space.
• Yet we do not make a distinction between both because the 10-foot-idea is understood to represent the 10-foot-estimate

Conversely, the idea of 10 feet* can be understood without looking at the log. This is even if the 10 feet needs a representation of something so that the idea of 10 feet can be understood [in discussion?].

But in reality, the following cannot happen:

• the smallest part of this quantity-thought is only removed when an equal part is also removed from the quantity-idea
• the smallest part of the quantity-idea is only removed when an equal part is also removed from the quantity-thought

• conversely, that the smallest part of the substance be removed without

and extension also being taken away.

without thinking of this determinate substance, we can conceive both*:

• the number 10
• 10 feet (as a continuous quantity)

This is because the concept of the number 10 is the same in 10 feet, 10 things, etc.

We can conceive a continuous quantity of 10 feet without thinking of this or that determinate substance,

Although we cannot conceive 10 feet without some extended substance of which it is the quantity.

It is in reality, however, impossible that any, even the least part, of such quantity or extension, can be taken away, without the retrenchment at the same time of as much of the substance, nor, on the other hand, can we lessen the substance, without at the same time taking as much from the quantity or extension.

9. Physical matter, when distinguished from metaphysical quantity, is confusedly conceived as something metaphysical.

When people distinguish physical matter from metaphysical space or number, they either:

• mean nothing by the word “physical matter”, or
• form in their minds merely a confused idea of metaphysical substance, which they falsely think to be physical.

Then they leave to metaphysical space the true idea of this physical matter.

• This metaphysical space they call an accident. But this means that their words are not in harmony with their thoughts.

10. What is space or internal place?

Space or internal place, and the physical matter in it, are not different in reality.

They are merely different in the mode conceived by us.

The same extension in length, width, and height, which constitutes space, constitutes body.

Their only difference is that we see the space occupied by the body as different from the body. Whereas we see space to be united to extension.

• After a body has occupied a space, we know that the extension of that space was not removed at the same time. The same extension remains there so long as it is of the same size and shape, and preserves the same location relative to certain bodies around it, by means of which we determine this space.

Physical Space Versus Metaphysical Space

11. Space, in reality, is not different from corporeal substance

It is the same extension which constitutes the nature of body as of space.

These 2 things are mutually diverse only as the nature of the genus and species differs from that of the individual, provided we reflect on the idea we have of any body.

For example, we think of a stone.

• We reject all that is not essential to the nature of the body of a stone.

We can reject:

• hardness
  • This is because if the stone were liquefied or reduced to powder, it would no longer have hardness. Yet it would still be a body.
• colour
  • This is because we have frequently seen stones so transparent as to have no colour.
• weight
  • This is because we have the case of fire. Fire is very light. But it is still a body.
• temperature
  • They are not considered as in the stone
  • A change of temperature does not cause the stone to lose its nature as a body.

After this examination, we find that nothing remains in the idea of body, except that it is something extended in length, width, and depth.

• This something is comprised in our idea of space, not only of that which is full of body, but even of ‘void space’.

12. Space differs from body in our mode of conceiving it.

There is, however, some difference between them in the mode of conception.

If we remove a stone from the place where it was, we see that its extension [of stone] is also taken away. This is because we regard its extension as particular and inseparable from the stone itself.

But the same extension of place where this stone was remains, even if that place is now occupied by air or even if it is vacant.

This makes us consider extension in general.

We think that the same extension is common to stones, wood, water, air, and other bodies, even to a vacuum itself, provided it:

• be of the same magnitude and shape as before, and
• preserve the same situation among the external bodies which determine this space.