Mathematics and Certainty
Topic 1- Mathematics and Certainty
Having said something about the nature of formal systems, we must now look in more detail at the nature of mathematical certainty. To do this, let us begin by making two distinctions. The first concerns the nature of propositions. An analytic proposition is one that is true by definition. A synthetic proposition is any proposition that is not analytic. So we can say that every proposition is either analytic or synthetic.
The second distinction concerns how we come to know that a proposition is true.
A proposition is said to be knowable a priori if it can be known to be true independent of experience.
It is said to be knowable a posteriori if it cannot be known to be true independent of experience.
As with the analytic—synthetic distinction, we can say that every true proposition can be known either a priori or a posteriori. Combining the two pairs of distinctions, we can generate the following matrix:
Nature of Proposition
Analytic
Synthetic
How is it knowable?
A Priori
1
4 ?
A Posteriori
2 x
3
Now let us try to explain what might fit into each of the four boxes:
Box 1: This concerns propositions that are true by definition and can be known independent of experience.
Does anything go in this box? Yes!
We can put all definitions in this box because they can all be known to be true independent of experience.
For example take Bolivian bachelors. I have never been to Bolivia and I know nothing about the profile of the average Bolivian bachelor, but I can say with complete confidence that every Bolivian bachelor is unmarried.
Apart from knowledge of the English language, I do not need any experience of the world to verify the truth of this proposition.
It might, however, be described as a trivial truth. If you are told that all Bolivian bachelors are unmarried, you have learned nothing new about the world.
Box 2: For a proposition to go in box 2, it would
Having said something about the nature of formal systems, we must now look in more detail at the nature of mathematical certainty. To do this, let us begin by making two distinctions. The first concerns the nature of propositions. An analytic proposition is one that is true by definition. A synthetic proposition is any proposition that is not analytic. So we can say that every proposition is either analytic or synthetic.
The second distinction concerns how we come to know that a proposition is true.
A proposition is said to be knowable a priori if it can be known to be true independent of experience.
It is said to be knowable a posteriori if it cannot be known to be true independent of experience.
As with the analytic—synthetic distinction, we can say that every true proposition can be known either a priori or a posteriori. Combining the two pairs of distinctions, we can generate the following matrix:
Nature of Proposition
Analytic
Synthetic
How is it knowable?
A Priori
1
4 ?
A Posteriori
2 x
3
Now let us try to explain what might fit into each of the four boxes:
Box 1: This concerns propositions that are true by definition and can be known independent of experience.
Does anything go in this box? Yes!
We can put all definitions in this box because they can all be known to be true independent of experience.
For example take Bolivian bachelors. I have never been to Bolivia and I know nothing about the profile of the average Bolivian bachelor, but I can say with complete confidence that every Bolivian bachelor is unmarried.
Apart from knowledge of the English language, I do not need any experience of the world to verify the truth of this proposition.
It might, however, be described as a trivial truth. If you are told that all Bolivian bachelors are unmarried, you have learned nothing new about the world.
Box 2: For a proposition to go in box 2, it would