ALL SCIENCE IS FOUNDED IN THE POSTULATE OF EXISTENCE.
47. We have said that the idea of being is not the sole form perceived, but that it is a form necessary to all perception. We do not mean by this to say that we cannot perceive without the actually existing; but that existence enters in some degree as a condition of every thing perceived. We will explain ourselves. When we simply perceive an object, and affirm nothing of it, it is always offered to us as a reality. Our idea certainly expresses something, but it has nothing excepting reality. Even the perception of the essential relations of things involves the condition that they exist. Thus, when we say that in the same circle or in equal circles equal arcs are subtended by equal chords, we suppose impliedly this condition, "if a circle exists."
48. Since this manner of explaining the cognition of the essential relations of things may seem far-fetched, we will endeavor to present it under the clearest possible point of view. When we affirm or deny an essential relation of two things, do we affirm or deny it of our own ideas or of the things? Clearly of the things, not of our ideas. If we say, "the ellipse is a curve," we do not say this of our idea, but of the object of our idea. We are well aware that our ideas are not ellipses, that there are none in our head, and that when we reflect, for example, upon the orbit of the earth, that this orbit is not within us. Of what, then, do we speak? Not of the idea, but of its object; not of what is in us, but of what is without us.
49. Nor do we mean that weseeit thus, but that itisthus; when we say the circumference is greater than the diameter, we do not mean that we see it thus, but that it is thus. So far are we from speaking of our idea, that we should assert it to be true although we did not see it, and even although it were not to exist. We speak of our idea only when we doubt of its correspondence with the object; then we do not speak of reality, but of appearance, and in such cases our language is admirably exact, for we do not say,it is, but,it seems to us.
50. Our affirmations and negations, therefore, refer to their objects. Now, we argue thus: what does not exist is pure nothing, and nothing can either be affirmed or denied of nothing, since it has no property or relation of any kind, but is a pure negation of every thing; therefore, nothing can be affirmed or denied; there can be no combination, no comparison, no perception, except on condition of existence.
We sayon condition, because we know the properties and relations of many things which do not exist; but in all that we do know of them, this condition always enters: if they exist.
51. Hence it follows that our science rests always on a postulate; and we purposely use this mathematical expression in order to show that those sciences which are called exact by antonomasy do not disdain this condition which we exact from all science. The greater part of them commence with this postulate: "Let a line be drawn, &c.," "Suppose B to be a right angle, &c.," "Take a quantity A greater than B, &c." This is the way the mathematician, with all his rigor, always supposes the condition of existence.
52. It is necessary to suppose this existence, otherwise nothing could be explained. Common sense teaches us what has escaped some metaphysicians. To prove it, letus see how a mathematician, who never dipped into metaphysics, would talk. We will suppose the interlocutor to set out to demonstrate to us that in a rectangular triangle the square of the hypothenuse is equal to the sum of the squares of the base and perpendicular; and that we, in order to exercise his intelligence, or rather to make him show us, without himself being aware of it, what is passing in his own mind with respect to the perception of its object, put various questions to him, in reality searching, although apparently asked out of ignorance. We will adopt the form of a dialogue for the sake of greater clearness, and will suppose the demonstration to be given from memory, without the aid of figures.
Demonstration.Drop a perpendicular from the right angle to the hypothenuse.
Where?
Why, in the triangle of which we speak, of course.
But, sir, if there be no such triangle——
Why then, what are we talking of?
We are talking of a rectangular triangle, and the case supposed is that there is none.
Is not, but can be. Take paper, a pencil, and ruler, and we will have one right away.
That is to say, you speak of the triangle we may make?
Yes, sir.
Ah, I understand; but then we should have it; now, we have not got it.
All in good time. But if we had drawn it, could we not drop the perpendicular?
Certainly.
That is all I meant to say.
But you were saying drop——
No doubt we cannot drop a perpendicular in a triangle unless the triangle exists, since then there is neither vertexof a right angle, hypothenuse, nor any thing else; but when I say, drop a perpendicular, I always suppose a triangle; and as it is evident that the triangle may exist, I do not express the supposition, but understand it.
I comprehend this; but then we should drop the perpendicular only in this triangle, but you spoke as if we might drop it in all triangles.
I only took this triangle for an example; we can clearly do with all others what we can do with this one.
With all?
Certainly. Can you not see how, in every rectangular triangle, a perpendicular may be drawn from the right angle to the hypothenuse?
Yes, in your figure; but since what is in my head is not a triangle, for I imagine some with sides a thousand miles long, and there is not in my head room enough—
There is no question of what is in your head, but of triangles themselves—
But these triangles do not exist; therefore, we can say nothing of them.
Yes; but may they not exist?
Who doubts it?
Well then, if they do exist, be they large or small, in one position or another, here or there, is it not true that a perpendicular may be drawn from the vertex of the right angle to the hypothenuse?
Evidently.
I have then only to say that, in every rectangular triangle, this perpendicular may be drawn.
Then you do not speak of those which do not exist? Is it not so?
I speak of all, whether they do or do not exist.
But a perpendicular cannot be drawn in a triangle which does not exist. What does not exist is nothing.
But perhaps that which does not exist may exist; and I see with perfect clearness how every thing said would be verified,supposing it to exist. Thus we can and do speak of all existences and non-existences without any exception.
We leave it to the reader to judge if we have not, while thus rudely troubling our good mathematician with our importunate questions, made him reply as would have replied every one not at all acquainted with metaphysics. It is evident that these replies ought to be accepted as reasonable, as satisfactory, and as the only ones in this case that all the mathematicians in the world could give.
This being so, all that we have advanced is found in these replies and explications. All science is founded on the postulate of existence; every argument, to demonstrate even the most essential properties and relations of things, must start with the supposition of their existence.