Kant and The importance Of Mathematics

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Kant wants to know what kind of knowledge can validate experience and what do we need in order to validate experience?

To do so he distinguishes between different kinds of propositions and sentences.

  1. Propositions where the truth or falsity of which DOES NOT depend on experience (a-priori propositions)
  2. Propositions where truth or falsity of which DOES depend on experience (a-posteriori propositions) ex: mathematics a-priori.

The natural sciences clearly use a-posteriori propositions. Where we must experiment in order to know how the world works. EX: gravity. Or the sentence: My friends blue bag has color: if you understand the meaning of blue we know it has color.

There is also a difference between analytic and synthetic propositions:

The truth or falsity of analytic propositions depends only on the meaning of the words composing them, and synthetic propositions are all non-analytic ones.

Ex: a bachelor is an unmarried man – This is true by definition (analytic)

Meaning it depends on the words and understanding the language.

With synthetic propositions we cannot just analyze the concept or words.

EX: Bachelors are happy.

Maybe its true, maybe its false, but this sentence does not just analyze concepts or words.

A true analytic proposition is one that you cannot conceive of its opposite.

EX: can I imagine a bachelor who is a married man? No because it contradicts itself.

Before Kant it was assumed that there were only two kinds of propositions, Analytic-a-priori-necessary and Synthetic-a-posteriori-contingent. Whatever is not necessarily true or necessarily false are contingent.

According to Kant, each of the two modern philosophical traditions innate-ism and empiricism, attempt to found knowledge of one of these two kinds.

Innate-ism attempts to found knowledge on analytic a-priori propositions as we saw with Descartes arguments for the existence of G-d. Descartes rejects the argument from design, because he does not know the world exists before he proves that g-d exists. He proves the existence of G-d by reason only by using the definition of g-d as g-d is perfection. So, if we define G-d as perfection we can prove that g-d exists. And G-d necessarily exists because necessity is by definition part of perfection.

He uses an a-priori analytic proposition, but for Kant those say nothing about reality!

All analytic sentences can do is analyze words. You cannot make something exist by playing with words, by defining it etc.. So, when Descartes says think about the concept of G-d and you will understand that G-d necessarily exists, he is according to Kant just analyzing words and this is not enough to know that something exists outside of the world.

Kant argues that Descartes thought he could prove something exists by simply playing with words but analyzing words or concepts can only teach us about language and not about the world. So, we cannot rely on analytic a-priori propositions in order to know things. This will not give us knowledge because they do not teach us anything new, it can only teach us what we already think.

Empiricism attempts to found knowledge on synthetic a-posteriori propositions only, but Kant argues that it inevitably leads to skepticism, because empirical knowledge cannot be founded on empirical knowledge. So, this leaves us needing a third kind of proposition to validate our knowledge. We need both a-priori and synthetic propositions. Things we can know before we experience the world and based on some a-priori knowledge we can validate our a-posteriori knowledge.

The answer: mathematics is synthetic a-priori knowledge.

Kant argues that we actually synthesize when we calculate.

For example: 7+5 = 12 is an arithmetical proposition.

I know what 5 is I know what 7 is.

I have to then count, and when I add 5 to 7 I actually imagine 7 points, fingers, balls etc.. and I then add 5 to those and now I see ,12.

This is easily misunderstood because beforehand all philosophers agreed that mathematics is valid a-priori experience. Meaning that points, fingers or balls don’t make mathematics true or false they should be true or false independently of experience. So, the question here is why do we need to count?

The key word is intuition. Kant does not mean we need to count in reality such as with fingers. I can simply imagine fingers and I can do it inside my mind. Meaning that it is not a-posteriori knowledge. It is something that comes from within and not actual experience but rather something I do. This action is something that needs to be taken inside. We need to do something inside our mind and this thing is not analysis. We must synthesize the 5 and 7, thus combining them to bring them together and when we do that we use intuition. They can be pure and independent of experience. We see things with our minds eye. We don’t just imagine, we can just understand things without either appealing to reason or depending one experience. Like when we calculate. We see that 5 plus 7 equals 12. When we analyze we use reason but not when we count because we just see the way how when we have 7 points and add 5 points to the 7 points we get 12 points.

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