/sci/ - Science & Math

>>6901522

The idea is that in order for "not (not P) is true" to entail "P is true" then you need the law of the excluded middle which says that every sentence must be either true or false. So, in order to get rid of the excluded middle and allow "not (not P)" to (possibly) have a different truth value from "P" you need to get rid of "Double Negation Introduction". This is one of the big fundamental differences between the proof system for classical/normal logic and the proof system for intuitionist logic.

So yea, to in the first sentence
>"Suppose p is true. Then ... contradiciton. Thus p is false"
you're just proving a negation (applying the rule of negation introduction) and in the second sentence
>"Suppose p is false. Then ... contradiction. Thus p is true"
you're actually having to use either the law of excluded middle or double negation.

You can think of the intuitionist thought process to be kind of analogous to the scientific thought process. In particular you can't prove a negative.

I hope I've made this clearer instead of more confusing.

>>6901483
By the way, here's another crappy picture.

>>6515904
You can construct all the integers by using a circle of radius 1 over and over again along a straight line. Addition can then be done with unit circles on a straight line and subtraction works the same way just in the opposite direction.

Then you can do division and multiplication in the same way. Here's a picture for multiplication. Construct two straight lines with integers marked (as described above) so that they intersect only at 0. Then choose two points, p and q, from each line respectively. Draw the line from 1 to q, then draw another line parallel at p. It will intersect at pq, refer to the diagram.

(cont.)

>>6451473
I was posting from my phone, this should say.

>Draw a line from 1 to q. Then draw another line parallel to this one and crossing through the point p,

The picture is self explanatory though.