In all Theoretical Sciences of Reason, Synthetical Judgements 'a priori' are contained as Principles

5. All Theoretical Sciences of the Abstract Mind Have Principles Based on Active-thinking Within-the-mind

1. Mathematical judgements are always active-connective-thinking.

Mathematical conclusions all proceed according to the principle of contradiction.

An active-connective-thinking proposition can be discerned through contradiction.

But this is possible only when it is preceded by another active-thinking proposition, from which the other is deduced, but never of itself.

Proper mathematical propositions are always within-the-mind judgements.

They are not sense-based because they carry with them the idea of cause-effect which cannot be given by experience.

7 + 5 = 12 might be a merely passive proposition.

But if we regard it more narrowly, we find that our conception of the sum of 7 + 5 contains only the uniting of both sums into one.

whereby it cannot at all be cogitated what this single number is which embraces both.

The idea of 12 is not obtained by merely thinking of the union of 7 and 5.

We must go beyond these ideas, and imagine our 5 fingers* and then, by one by one, add the idea of 7.

Finally, I see see the number 12 arise.

Arithmetical propositions are therefore always active-connective-thinking. This is more easily proven by large numbers.*

“A straight line between 2 points is the shortest” is an active-thinking proposition.

The idea of straight is qualitative, not quantitative.

The idea of the shortest is therefore wholly an addition. Passive-thinking will not lead to an idea of a straight line from a shortest line.

Passive-knowing must here help for us to combine the idea that the shortest line is straight.

Some few principles preposited by geometricians:

• are really active-thinking
• depend on the principle of contradiction.

They serve like identical propositions linking in the chain of method, not as principles—for example, a = a, the whole is equal to itself, or (a+b) —> a, the whole is greater than its part.

These principles derive their validity from pure ideas. Yet they are only admitted in mathematics because they can be presented passively.

What causes us here commonly to believe that

We tend to think that:

• the predicate of such apodeictic judgements is already in our minds.
• therefore, the judgement is passive.*

But this wrong idea is merely the equivocal nature of the expression.

We must join in thought a predicate to a given conception.

• This necessity cleaves already to the conception.

We really actively-think about this [idea that comes passively] obscurely.

Then it becomes manifest that the predicate necessarily is actively-connected to these ideas by passive knowing.

2. Physics contains in itself within-the-mind active-thinking judgements as principles.

Two examples are:

• The Conservation of Matter
• Newton’s Third Law

Both of these:

• are active-thinking propositions yet conceived within-the-mind
• have the relation, as cause and effect, clear within-the-mind

The idea of matter does not include its permanence, but merely its presence in the space which it fills.

I therefore really go beyond the idea of matter, in order to think on to it something a priori, which I did not think in it.

3. Metaphysics has active-thinking propositions within-the-mind

Metaphysics dissects and thereby passively illustrates the ideas which we form of things within-the-mind.

But we seek to widen the range of our within-the-mind knowledge.

This is why must use principles that add something to the original idea through active-connective-thinking within-the-mind beyond the limits of experience.

For example, in the proposition, “the world must have a beginning” and such like.

Thus metaphysics only has active-connective-thinking within-the-mind propositions.