Maybe it’s established context that 25 is “lying on the real number line,” and so the deduction of 3 from 1 could be valid, but this is a really stupid example because why would anyone define what the real number line is before you define what a real number is?

Okay, I agree in theory that such things could be bottlenecks for understanding mathematics and physics well.

But I only see it as being relevant for like, people for whom learning physics is too hard and they give up. And maybe one of the reasons why it’s too hard for them is that they make too many errors. (edit: too many arithmetic errors*)

In my own case, I don’t think there aren’t any fundamental obstacles to me learning a new piece of physics if I want to learn it. I don’t think I’ve failed to learn things that I’ve wanted to learn in physics. Maybe arithmetic could be a bottleneck if my goal was like, “learn X in one day”?

I think you’re looking for errors in my understanding of very basic stuff, because you are trying to prove to me that my education has failed me. Am I correct?

Not exactly. I wasn’t looking for or expecting an error re the real number syllogism. I had a followup question in mind but didn’t get to it yet. (BTW your trig answer was problematic but I didn’t get to that yet either, and you didn’t volunteer information about mathematical induction or recursion so those are still pending to potentially follow up on.) And “failed [you]” is something like a value judgment, which wasn’t my point.

I was trying to help with identifying things that I thought you wanted to identify:

My broader purpose was to help discuss/clarify/explain the broad societal situation in which you’re making career choices, which could help you understand your options better, make more informed decisions, and have realistic expectations/plans/goals.

Okay, got it. Yes – I’m still interested in this conversation.

Regarding the syllogism thing, I think you interpreted it as more snarky than I actually meant it to be.

Basically, what happened is that the first change I thought of would make it valid but depended on some context (the context of knowing that 25 is an integer). Rather than asking for technical details about what you actually wanted me to do, I decided to make an edit such that it’s trivially true. I said Lol because I thought my edit was different from what you expected and therefore funny.

Your edit isn’t correct. From a contradiction, anything follows by a series of steps that you could do, but you didn’t include those steps, so it isn’t a valid syllogism.

I know induction as well as I possibly could because I have used it a lot for math proofs.

I understand recursion because I’ve programmed with functional programming languages before. I actually started out wanting to do theoretical CS at the beginning of my career, and know (or did know) e.g. the stuff in CLRS.

I expect that someone who just did 100 induction practice problems involving arithmetic sequences could solve his 101st faster than I could, so maybe this is a sense in which I don’t know it “as well as I possibly could,” but does that really mean he understands mathematical induction better than I do? I think this is what you have in mind, but I generally wouldn’t describe this as someone knowing induction better than me, or a lack of knowledge about induction, etc.

Maybe you have an instrumentalist premise? Your sentence reads as absurd to me because conceptual understanding exists and is not maximized by using something a lot for calculations/proofs/problems (or any use).

Being able to do a skill and understanding a concept related to the skill are different things.