Sorry but I don't have time to listen to the whole video and in the first 15 minutes Peikoff didn't explain very much. But it seems to me that with "certainty" he doesn't mean 100% certainty. He talks about inductive reasonings so he is right in the field of probabilities, right? Is there any argument against that later in the video?

@Dietl Yes, the video is long... He argues that there is a need for the concept "certainty" or "absolute" and that the modern definition of "absolute" as a principle independent of any other fact or cognition; i.e., as something unaffected by anything else in reality or in human knowledge; is mistaken. In other words, if you define "certainty" as something impossible for humans, then it, as a concept, is useless and somewhat mystical. As humans, our knowledge is contextual and so are concepts, such as "certainty." Certainty, as a useful concept, would mean something like, "with all the available evidence, using logic, there is no reason for doubt."

I see. Thanks for clarifying.

@BeerAndWine
I change my mind when evidence is able to do so but not when a claim is independant of evidence. You have to be able to identify which are which.

@BeerAndWine Yes, if new knowledge is obtained it doesn't mean we were necessarily wrong, only that we are not omniscient. Take Isaac Newton's physics as an example. There have been new discoveries in physics, but nothing he discovered was wrong in the context and on the level he was working.

@BeerAndWine
Circularity doesn't make it not knowledge. Something being circular is only an issue if you want to justify an argument. Consider the rules of chess. It is 100% true (within the rules of chess) that a pawn can only move one step forward. This is something you can know therefore knowledge. You might say there could be a world where the rules of chess are different, but that wouldn't be the same chess we mean. So you can't justify that the way we play it is the only true form of the game but within the rules of chess you can make true and false statements which are not probabilistic.

@BeerAndWine
I disagree concerning math. Math does not depend on a world. It is a construct just like chess in my previous post. A world where "2+2=3" is true is not possible because what would that even mean. There is no evidence you can provide which would make this statement false and in the same sense there can't be a world where it is not true. If no new knowledge can change the truth of a statement it must be true (or false for that matter).

It's okay to not call self referential knowledge knowledge, but that would mean that there is no knowledge in maths, knowing rules in not knowledge and creating a definition for a word in also not knowledge. I think what we need to consider is your definition of knowledge. Shouldn't "knowing X" also imply "having knowledge of X"? For instance, shouldn't "knowing the rules of chess" imply "having knowledge of the rules of chess"?

@BeerAndWine
I think you are confusing a few things when you bring quantum mechanics to the table as a refutation of mathematical truths. The math is independant of anything in our world. No matter what happens. There can be things that can't be described by a certain mathematical system. But what you are doing here is trying to find out how math applies(!) to this world.
What you descibe here with your example with the sets is not really relevant for maths, because you describe a physical event where at the start you have 2 sets of 2 things. If you counted those things you would get 4 things. But you go on to "bring them together" and then something happens and the number changes. But what we would learn hear would be that bringing things together changes the amount you have. Similarily you can look at quantum mechanics. There is a certain weirness going on were you need unintuitive principles to descibe what is going on but math is still true even if it doesn't apply.

@BeerAndWine
"math working different"
Which means it applies differently. We created math so we can apply it to the world and to reflect reality but that doesn't mean that it is wrong when it doesn't apply. When you count you compare the world with the set of numbers and assign each thing you want to count a certain number. When you count differently in a weird world you just make a different assigning of the numbers. The numbers don't change.

@BeerAndWine
The definitions of what we mean with '2', '3', '+' and '=' make the statement "2+2=3" false. I think the error in your reasoning becomes abit more apparent when you consider the statement "all bachelors are unmarries". What you are proposing is a world in which there is a married bachelor. This is not possible.
I do think the best we can say is that abstract knowledge is 100% certain but we can not be certain about how it applies to the world.

I'm not 100% certain in my ability to do maths and I might make mistakes but there must still be statements which can't be wrong even if I am not able to identify them. I know the following:
"2+2=4 is true or I made a mistake."
I see your point and I think it is very healthy to never stop doubting but there is a line that must be drawn with logical truths else you might end in an asylum

@BeerAndWine I also appreciated our little philosophical discourse
As for math I can only repeat my previous point is a slightly different form,so let's leave it at that. I totally agree that outside of our minds nothing is 100% sure. But there are other things that we can be 100% sure namely our direct experiences. Not what they represent or the source of them, but the sensory input itself. If you have the imperssion of a red apple you can be 100% sure that at that moment you have that impression. You can't be sure if it's an illusion but the experience itself is there. Again, that's not much because memories of those impressions can again be false but that's another certainty.

@BeerAndWine it gets closer than religion lol

g