KANT: HOW IS PURE MATHEMATICS? SECTION 7

"All mathematical cognition has this peculiarity: it must first exhibit its concept in intuition, and do so a priori, in an intution that it is not empirical but pure. Without this mathematics cannot take a single step, hence its judgements are always intuitive; whereas philosophy must be satisfied with discursive judgements from mere concepts, and though it may illustrate its apodeictic doctrines through intuitions, can never derive them from it"

I think that here Kant is saying that math shows it concepts as an instinct and something that you should already know. It is not something that can be proven it is something that is pure and comes naturally. He also compares philosophy to math saying that philosophy has many ideas and proceeds to a conclusion through reason instead of intuition.

I don't understand because math really doesn't come naturally unless you learn it. For example, when you learn 2+2=4. You may know this now and when someone asks you this the answer comes naturally to know because you know it but at first you did not know it and someone had to teach it to you so it really isn't natural.

Kant: HOW IS METAPHYSICS IN GENERAL POSSIBLE?

"Without solving this problem reason never is justified." (section 40)

Here Kant is talking about how problems within the metaphysics will never be reasoned as true if they are not solved. This applies to everything in life. Nothing is proven true unless it is solved. To figure out that 2+2=4 one must solve the problem 2+2 to know that it equals 4.

Kant: How is Metaphysics in General Possible?

"Physics cannot altogether refuse and dispense with the testimony of the latter; because with all its certainty, it can never, as philosophy, rival mathematics." (section 40)

I am not sure what the "latter" is, but I am assuming that it is referring to later things in the future. I do not understand what the testimony is though. Also, why can philosophy not rival with mathematics? I do not get it. Does anyone know what Kant is talking about here??

Kant: How is Pure Mathematics Possible?

"If two things are quite equal in all respects as much as can be ascertained by all means possible, quantitatively and qualitatively, it must follow, that the one can in all cases and under all circumstances replace the other, and this substitution would not occasion the least perceptible difference." (section 13)

Here Kant is saying that if two things are equal in every aspect that one could substitute one for the other and nothing would change. It is very interesting and slightly confusing that Kant can put such an obvious thing into such ridiculous phrases. Although this is correct in our minds I do think that there is a way around everything. And even though something appears to be equal it may not be.

Kant: How is Pure Mathematics Possible?

"How then is it possible for human reason to produce a cognition of this nature entirely a priori?" (section 6)

A priori is a person's ability to have the idea of something without learning it from experience previously. In reponse to Kant's question I think that it is not possible for a person to know mathematics without first learning or experiencing it. People are born with natural instincts, yes, but I do not think that the knowledge of mathematics is one of them.

Kant: How is the Science of Nature Possible?

"...nature is the complex of all the objects of experience." (section 16)

Immediately after reading this I thought that not everything in nature comes from experience. Many things are either a figment of our imagination or grown naturally. Then Kant went on to say this:

"And with this only are we now concerned, for besides, things which can never be objects of experience, if they must be known as to their nature, would oblige us to have recourse to concepts whose meaning could never be given in concreto (by any example of possible experience)." (section 16)

Then I knew that Kant was saying exactly what I was thinking. Although many things are from experience there are even more things that are not from experience. Mant creatures, plants, and organisms are not from experience.

Kant: How is the Science of Nature Possible?

"But experience teaches us what exists and how it exists, but never that it must necessarily exist so and not otherwise. Experience therefore can never teach us the nature of things in themselves." (section 14)

Here Kant is saying that experience teaches us everything. It teaches us not only about things, but also where they come from, but in doing this we do not learn from experience is if something absolutely must exist. He says then that experience will never be able to teach us the real origin of things alone. I kind of see where Kant is coming from when he says this. I agree with him that experience does teach us, come the phrase, "The old and the wise". I was a little thrown off when he says that we do not learn if things do really exist because clearly if we learn of them than they at one point had existed. As far as his last statement in this section I do not think that anyone can learn the original origin of anything or anyone unless they are it.