Philosophical knowledge is the knowledge gained by reason from concepts; mathematical knowledge is the knowledge gained by reason from the construction of concepts.

Philosophical knowledge is knowledge which reason gains from concepts; mathematical knowledge is knowledge which reason gains from the construction of concepts.

In so far as theories of mathematics speak about reality, they are not certain, and in so far as they are certain, they do not speak about reality.

For the things of this world cannot be made known without a knowledge of mathematics.

Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

There is no certainty in sciences where one of the mathematical sciences cannot be applied, or which are not in relation with these mathematics.

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics...

Knowledge is invariably a matter of degree : you cannot put your finger upon even the simplest datum and say 'this we know'.

No human investigation can be called real knowledge if it does not pass through mathematical demonstrations; and if you say that the kinds of knowledge that begin and end in the mind have any value as truth, this cannot be conceded, but rather must be denied for many reasons, and first of all because in such mental discussions there is no experimentation, without which nothing provides certainty of itself.

In the early 1900s, a great mathematician was expected to comprehend the whole of known mathematics. Mathematics was a shallow pool. Today the mathematical waters have grown so deep that a great mathematician can know only about 5% of the entire corpus. What will the future of mathematics be as specialized mathematicians know more and more about less and less until they know everything about nothing?

I surmise that mathematical knowledge amounts to the crystallization of officially endorsed delusions in an intellectual quicksand

[K]nowledge that is not Infallible is not certain knowledge.

If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.