Reminds me of how Ben Franklin taught himself to write by reading famous passages and then attempting to write them from memory, comparing his version to the original version afterward and seeing what he did differently and why.
Like, I have expert knowledge in law, and I'm working toward at least more advanced (though far far from expert) knowledge in coding-things, and it very nicely matches how I interact with legal problems and can dimly see glimmers off at the end of a long tunnel in coding problems.
But at the same time, it's inspiring as all hell, because mathematics from the outside doesn't seem like the kind of thing that could ever produce the same satisfying and effective experiences of cognition.
You talk about the dungeons, monsters, level-bosses, etc but nobody can relate.
Edit: In a later part, the answer says "Terence Tao is very eloquent about this here: ..." so likely not (based on factors like typical personality of a mathematician, but I could be wrong since I don't know him personally).
Mathematics is a language. Most people only learn arithmetic, and then only by rote. Sometimes, it is disheartening. I've had people who refused to accept that Ph.D. is Doctor of Philosophy.
I like to point out that nobody has successfully brought me a bucket of negative apples.
Err... I'm a mathematician. I am an applied mathematician. I am not smart enough to be a philosopher of mathematics, though I have my Ph.D.
Anyhow, math is a language that is capable of telling stories, truths, and even falsehoods.
Are symmetry groups not "physical" enough? I can rotate a piece of paper in the plane and then undo this operation just fine.
Algebraic and topological structures seem to manifest themselves readily within nature. I should think infinities provide greater concern.
So basically what you are saying is that, we can't be so certain that we humans invented math. Math is there, structures were there to be discovered ? Why is there such a thing as monster group ? We named those things, but if we didn't exists, it would still be there in the abstract world ? This is what is bugging me lately, I can't decide if math was invented or not. Some parts may be invented, but some parts may be discovered.
Accordingly "the working philosophy of most mathematicians is a mostly unexamined Platonist-Formalist hybrid ... Math is a product of human minds but not bendable to human will. Exploring it is like exploring a new tract of country ... but the mathematical countryside does not come into existence until you explore it."
As for infinities... I'm not sure if they drove Cantor insane or if he was already insane. Some infinities are bigger than others, and I have strong opinions about infinity.
0.999... = 1
To say "I have three apples" is merely a way of modeling a configuration of the world. The apples may be in my pocket, or in my hand, or in a bag I'm holding, or even on my desk in front of me. They may be in the barn out back, but in all cases we can say I have 3 apples. Saying "i have 3 apples" is a simplification of a ton of configurations of the state of the world where I have ownership of apples. Similarly I have -3 apples is a summary of all the configurations of the world where I owe three apples.
Neither is any more real than the other. And yes it is philosophical.
Your enterprise can have an obligation to deliver apples to someone, which has all the same effects as negative apples and should be booked as such. If you acquired an enterprise with such an obligation, they gave you negative apples.
It's important to account for them that way in e.g. seeing the implications of the Put-Call Parity Theroem.
The post may not have been clear enough, but the point was to show that it's still very much a philosophy.
But when they did I am going to take the square root of the number of them and break the universe.
I just disagree with your example and I dislike it because I've literally had an argument with an engineer (an engineer!!!) about whether negative numbers were real who insisted on sticking to this sort of example. Negative indicates direction, this isn't really abstract. If it's increasing your balance of apples, it's positive. If it's decreasing your balance of apples, it's negative. Many people understand this quite well in the sense of monetary debt. I "possess" negative $xxxx, because that's the amount I owe someone else, in turn the holder of the debt "possesses" positive $xxxx because they expect to receive it at some point in the future.
If this is philosophy it's the barest levels, and it's philosophy that any culture with a concept of ownership and debt would have no difficulty with.
Bucket of negative apples, hmm. I feel slightly sheepish to say this but I immediately visualised it to be "a bucket that you intend to put n apples in, but didn't." Thank god I'm in a semi-technical field now :D
In electrical engineering, imaginary/complex numbers are critical in circuit analysis once you move into the AC domain. You can try doing it with sines and cosines, but that's infeasible as soon as you move beyond anything basic.  But when you transform them into complex numbers using Euler's formula (one of the most beautiful things in maths, IMHO, especially Euler's identity) , it becomes "easy".
Then there's also the Fourier Transform  which transforms time-domain signals to frequency-domain.
So my visualisation of imaginary numbers is something involving circles and spirals being twisted and shifted all over the place. Hard to explain, but I've reached the point where I feel I have a good grasp on them. :)
Oh yes no doubt j is very useful - but that's what I meant by it being part of the Toolbox. To visualise it as an entity is something else though, and I know that you're not really supposed to do that, especially when it was 'invented' to be useful in the first place.
Perhaps with more experience and perseverance, I might have reached an understanding like yours. I do find myself regretting for giving up on engineering so soon, but hey, I can still take my time to understand what I couldn't before. Thanks for the links :)
Developing an intuition about what rotation a particular complex number will give you is more complex, admittedly. But this is enough to visualize the effects, if not precisely mentally predict the result.
In hindsight, I think that I approached it wrong, and it was far too simplistic anyway. I've been reading popular maths books lately and it's made me realise that there are more ways than just visualisation to understand something. Ironically (a happy one!) this is making me excited about maths for the first time in my life.
The complex number notation describes an object in space or an action to be performed on objects in space. Other than it being more compact, it's not substantively different than a statement describing an action like "travel 50m northeast, then turn left, then travel 40m".
A key distinction here is that one has a defined implementation. Making that distinction is philosophy. Talking within the framework of implementation is mathematics.
And it's not really my fault that nobody can bring me negative apples. It's not like I made the rules! Sheesh! ;-)
I just don't see the value in conflating both terms (philos and math) when there's a distinction pointed out above.
You can't bring me a bucket of negative apples. You can bring representations of future apples, or something like that. However, negative apples do not actually exist. In all these years, nobody has brought me a basket of negative apples.
In the moment where you have an empty bucket and a debt of apples, I'd call that a bucket of negative apples.
You can call it negative apples, if you want. However, it's just an empty bucket. ;-)
In other words, it's all in your head. A real, physical, "negative apple" does not exist in this world. Only the human philosophical concept of "owing someone apples" or similar such things. That's the guys' point. It doesn't matter whether you can bring representations of negative apples because those all depend on a philosophical bedrock.
Philosophy today is pretty useless.
Studying philosophy teaches you how to think. If you consider that useless...
Program slow speed limits in dangerous areas, install sensor augmentation in areas where vehicle located sensors don't work well, rebuild roads to work better, etc.
I think Philosophy studied with other disciplines is a particularly potent combination. Of my friends who have done this (they are very few) they are perhaps the most inspiring and trenchant thinkers I know.
Math is fascinating. It is also just a tool.