Here are some advantages to philosophy as a secondary focus which are relevant to success or failure at philosophy:
Less pressure to make philosophy progress. Less scary to get stuck. Got a good alternative to work on so philosophy issues can be on the back burner until you’re ready to try again, without this seeming bad.
Helps avoid problems like ivory tower philosopher, armchair philosopher, rationalist, theory disconnected from reality.
Enables problems from primary focus to drive/motivate philosophy interest, helps you care and be interested. (This works particularly well if you have strong interest in some other focus already. It would make less sense if someone was unclear on their interests or motivations, or already favored philosophy.)
Pursuing philosophy problems relevant to primary interest can help focus on good philosophy issues that actually matter.
Less resources for philosophy (time, mental energy, etc.).
More temptation to evade philosophy difficulties and retreat to your primary field.
Split-focus, multi-tasking, task switching (which you might also do with philosophy as primary focus and something else as secondary, but you could do just philosophy, or a more uneven split like 80%+ philosophy which would reduce switching. Whereas if philosophy is a secondary focus but you’re taking it seriously, then you have at least two significant focuses to go back and forth between.).
Your primary field is designed for single focus, not split focus. There will be ongoing friction.
The people in your primary field will be bad at philosophy, and the more you learn philosophy the more it’ll make you different than them and possibly lead to conflict. You’ll have a big incentive to get along with them, stay compatible with them, think about things in ways that are adequately similar to how they think, etc.
This is the kind of post that I think bystanders often kinda ignore and don’t learn much from, and also don’t say anything about. But you could take it as an example of how to look into an error and try to brainstorm things in a kinda similar way for some other type of error that you want to investigate.
Here was my reasoning for saying that arithmetic is not a bottleneck issue for me:
For exams, I basically decide what grade I’m going to get by increasing or decreasing the amount of material I study. I don’t do this quantitatively, but I meet all my goals with respect to grades. Doesn’t this show that my bottleneck is my willingness to study?
For everything else in my career (right now at least), the main issue I seem to be facing is that I don’t know wtf to work on, and it’s very hard to see how arithmetic errors could possibly be the root cause of something like that.
A context where it sounds like arithmetic errors could potentially be a bottleneck issue for me is if I get into another situation where I need to do a whole bunch of long 30 page calculations by hand, since the main problem here is the amount of time it takes me to do them. But that’s not an issue I’m facing right now.
I wasn’t thinking about bottlenecks for getting good grades or test scores (or choosing what to work on). I was thinking about the issue of understanding math and physics well. I think arithmetic errors (or their underlying causes) could be important for that.
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”?
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.