I suspect you still haven't worked out my position if you think I've suggested that. I'm growing tired with this whole thread. I'm only writing one more response to adx and then I'm done. You and adx can have the last word.
Fair enough, anything I write still remains open to criticism from others.
[...] I meant natural, philosophically, in the sense of being of or directly or closely related to natural things, say, quantities of countable things or something perceivable to humans[...]
This is an incredibly peculiar thing to say given what you're suggesting about only 'natural numbers' (a squishy philosophical definition you're making up) being okay and not the complex numbers.
It's not a squishy definition that I've made up, it is quite an established concept... the concept of being a "natural entity" at least is, I just stretched the definition to numbers, there is a good treatment of that in Mill's Ratiocinative and Inductive Logic (much of the work isn't unique to Mill, but compiled into a cohesive system written natively in English avoids some of the poorer translations of others'). A reference I totally forgot about was
this site of Peter Smith which does go quite some way to demonstrate how maths is simply a branch of philosophy and that philosophy isn't as squishy as you may think (i.e. not just about drinking wine and pondering).
Why are irrationals a whole other discussion? [...] we can't count it on our fingers... we're not even in the 17th century anymore. Welcome to ancient Babylon apparently... 
Irrationals are a whole other discussion because there is not necessarily a perfect and infinitely precise
process for representing them physically, the emphasis there on "process", whilst a rational number would be as difficult to represent to some arbitrary precision, an irrational would require an infinite precision no matter how precise the process was... which is a whole other discussion, because that is possible with some, maybe not all.
Again, you are missing my point, modern maths does away with the dependency on physical representation by abstracting it beyond that necessity of representing numbers physically. In the more modern maths and natural philosophy, the "on-paper" representation of maths does not necessitate that the numbers are physically representable (i.e. avoiding the problems of geometric proofs) - yet, what you are doing by teaching complex numbers as immediate physical concepts is incredibly 17th century, whereas what I am suggesting is that the complex numbers could just be taught for what they are as just one possible representation of a vector.
like sqrt(-1), is super mysterious and mystifying and has also been suddenly branded as icky in this conversation because we can't count it on our fingers
I think you are slightly biased from your teaching experiences, that is certainly not what I am suggesting. I am still suggesting that there is a difference between "maths as an abstract language" and the physical processes it describes. The power gained by modern maths through that abstraction is in the fact we can work with totally realisable numbers and separately bridge between the number on paper to the physical quantity through isomorphism and metrics... that is especially implicit in engineering, it is something we often do without thinking, i.e. the rms of a 1V pk sine-wave, maybe 1/sqrt(2) on paper, but could be 1.707 with some uncertainty as far as we can measure. The complex j is not immediately there on the scope, we just add that on when representing it on paper.
The more axiomatic and abstract our mathematical system has gotten, the more useful it has become. Thank goodness we don't just count on our fingers and toes anymore...
So why insist on breaking that nice abstraction by teaching complex numbers as non-abstract things?
You can go round and round chasing your tail about whether math is 'physical' unless you're counting sheep or whatever. I'm not worried about that. Math is logic and the universe is, evidently, logical.
Again, (modern) maths is not itself physical, I have no problem with that, only when somebody says it is. But... the universe isn't necessarily logical, science is logical and the behaviours and patterns arrived at through scientific study are logical, but only within science... that's the squishier end of philosophy, I mean, we don't know with complete certainty that the bible is wrong, only that is doesn't agree with the concepts arrived at through science.
But sure, some people would rather huddle around and dismiss it all as philosophical mumbo jumbo ickiness because they can't find sqrt(-1) between their thumb and forefinger.
At this point... whatever.
I'm sorry you feel that way.
But, your reference to l'Hopital sums it up so nicely, as you lead into it with "As another example, what is 0 times infinity? 0 divided by 0? Infinity divided by 0?", suggests you havn't quite understood the question yourself, l'Hoptital it would give the value to a function that contains terms that individually tend to those values... not of the pure numbers themselves necesarily.
You haven't understood the example. I'm not motivated enough to explain it further given what else I'm reading here.
How many other interpretations could there be? l'Hopital doesn't relate to the question as you wrote it.
The complexities of nature are kinda irrelevent to the maths, the maths describes only our observations and patterns amongst them, it all exists within the artificial construct of logic that is related to human reasoning, nature just does its own thing.
Yet there are some here who want to reduce both nature, and our math, to nothing more compelling than counting on fingers and toes.
Exactly, but I'm hypothesizing that where a lot of the philosophy and relationship between mathematical and the physical world get mixed up into a "the maths works out, therefore it must be physical", so yes, to most of the world, without being taught the more formal logic and constructs behind maths, what else do they have to go on? Just the word of a teacher?
[...] I suspect you still havn't worked out that my gripe is not with maths itself, but with how people are so quick to ignore the fact it is only describing links between the observations etc, [...]
I suspect you still haven't worked out my position if you think I've suggested that.[...]
I suspect you think I may be attacking you personally if you suspected that I suspected that of you, because I didn't and I'm not.