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muckmagister · 2 years ago

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Your imaginary number explanation only works on paper and not in a physical space. That's kind of what pisses me off about math - so much of it just isn't physically real and most of it is just used for tracking wealth/money/other capitalist bullshit. Math should be about efficient ways to count and measure, not arbitrary ways to put people into debt or make rich people seem even richer so they can scam other rich people.

I also think this is why so much of our understanding of physics is flawed - our counting system has less to do with the real world and more to do with social structures, so attempting to apply it to physics doesn't work.

I just. Maybe I'm just out of the loop bc I genuinely cannot process simple math equations but I really feel the reason for that is bc it's not grounded in a sense of reality OR in a mindset I can understand. Idk I'm autisitc and also there's a likely chance I have discalculia or some other learning disability/disorder that impairs math comprehension but seeing talk abt these kinds of math problem really make me think that I'm not the problem here.

enter at your own risk, you've been warned

ok oouuh i- Well. I think you seem to have a pretty cynical outlook on maths. To say that it's all about putting people into debt and letting rich people scam each other is just. That baffles me, to be honest. It's like saying language is all about constructing the most brutal insults possible and telling people to kill themselves. Like yeah a lot of people use language to do just that but that's just one tiny part of it. Maths isn't just about efficient ways to count and measure either, by the way, it's its own language, its own art form, a vast mountainous landscape with peaks we've yet to climb and valleys we've yet to discover. Forgive me for getting poetic but maths just isn't that... small, y'know?

I also disagree that this is why our understanding of physics is flawed. To be clear: It's flawed because we're idiots. We're idiots figuring out how our new toy works by throwing it around and pulling and squishing it randomly, because it didn't come with a manual and we have no idea what we're doing. Also, mathematics isn't our way of understanding physics, it's our way of building models that try and predict what would happen in certain circumstances. That might just sounds like I'm being semantic, but I think it's important to point out that nobody actually, truly, has any idea how physics really works. In fact, no ones ever even seen physics, when we make observations about the physical world, we're just seeing what physics does. So we use maths to make models that predict what we expect to see in our observations, and then we use those models and our observations together to try and infer the nature of reality. But we're all just guessing at that point, really.

So with all that in mind, to say that our understanding of physics would improve if only we had a better suited system of mathematics is. Well, I disagree. I also think that there isn't just some "better" system of mathematics out there waiting for us to discover it. I'm reminded of Gödel's incompleteness theorem. I encourage you to look it up on your own if you want, but basically, in any reasonable mathematical system there will always be true statements that cannot be proved. So there certainly are other systems of maths and we have looked into them. They do have their own upsides and downsides depending on how they work, but to say one system is better than another is, well, if I may use the language analogy again, it's like saying one language is better than another, when really it's just different sets of words and rules for us to say the same sentences.

So when it comes back to our current system of mathematics, it's not better or worse than any other system except for how widespread it is, just like how most of the world uses metric, a standardised system of maths makes it easy for anyone to communicate the same mathematical ideas to one another. But there are other systems of mathematics that become useful tools in more niche areas of maths for people to talk and understand those difficult topics.

With all that being said now too, I don't think our current system of mathematics, or any other system really, inherently has any less to do with the real world than it does with our social structures, or anything else for that matter. In fact, I don't think any system of mathematics inherently has anything to do with anything except itself. People didn't invent maths, we discovered it and then we invented ways of talking about what we learned. But numbers are their own thing. Their own abstract metaphysical objects. And they'd exist whether there was anything alive to know about them or not. Saying that there's a special way to use them to understand physics to me implies that numbers are what physics is, in a way, but like I said we have no idea what physics is. It just so happens that numbers are an incredibly useful way of talking about what physics looks like. But it's also incredibly useful to talk about our social structures. Frankly we're incredibly lucky that we've been able to saddle these concepts from abstract space, that our minds are powerful enough to reach out beyond the real to give us these tools of ideation that we can use for so many things.

You said that maths difficult for you because "it's not grounded in a sense of reality OR in a mindset I can understand" and I think those two things are the same problem in a way. I know I just did some big talk about how maths is really this abstract thing that isn't inherently connected to the real world, but we're all still the idiots I said we were, so a lot of maths does need to get grounded in reality for it to be reasonably comprehensible. And being able to ground a mathematical concept is often about shifting your mindset until you can see it at an angle you understand.

Like, you also said that my explanation only works on paper and not in a physical space, and I actually disagree! But I do think I get where you're coming from. I suspect you're thinking about numbers like physical objects, as in, the number 1 is like having 1 apple. So well of course, negative numbers kinda make sense right? I mean you can't just have less than 1 apple but you can get an apple taken away. But when you introduce i, it's like. What does that even mean? And you'd be right that it really doesn't make sense when you think of numbers as objects like that. But numbers aren't physical objects, we're just taught like that as kids, 1 apple, 2 bananas, 3 oranges, because they want us to build the association between the symbols (1, 2, 3) and the idea of that amount of something, of that value, of 1 being 1 apple or 1 step. But that's it, 1 isn't the apple, or the step, it's the idea of "oneness" or "singleness" and 2 is the idea of "doubleness", 3 is "tripleness" etc. It's difficult to even describe the idea of "singleness" without using physical metaphors, which is why we rarely ever do think about numbers in their purely abstract forms. And it's of course why we teach kids with apples and bananas, and why we go on to teach addition and subtraction by giving them more and taking them away. Even though, likewise, addition and subtraction isn't really about giving and taking. Now, I'm not saying you can't or shouldn't think about numbers as physical objects, or addition and subtraction as giving and taking. I'm just saying those are the initial ways we ground maths in reality in order to teach the concepts in the first place. And for a lot of people, this understanding of numbers is all they need for the rest of their lives. But the truth is there are different ways of understanding numbers and the operations we perform on them, and shifting your mindset on how these things work is an important part of being able to grasp more complex concepts, like imaginary numbers.

Finally, I said my explanation does work in physical space, how is that? Well if your understanding of numbers is that they're like objects, and that addition and subtraction is like giving and taking, then it doesn't work in physical space, no. What you need to do is think about numbers as a distance, and think about addition and subtraction not as giving and taking, but instead about direction. So what that means is that numbers don't tell you how much of thing you have, they tell you how far away from zero you are. Now when people talk about the number 1, what they usually mean is positive 1, like 1 = 0+1. (We just don't always draw the addition symbol out front.) So now if 1 is the distance away from 0, then positive + is the direction you travelled away from zero in. And when you write -1, the negative symbol isn't saying you have an "opposite 1" or that you're "taking 1 away", it's that you're travelling a distance of 1 in the negative direction, which is exactly the opposite of the positive direction. On a number line, this idea is pretty easy to comprehend since you only have two directions you can travel.

However, i is defined as being the square root of -1, which doesn't make much sense if you stick with the traditional understanding of what a square roots is, since it seems impossible to take the square root of a negative number. You can take take the square roots of +1, which are just 1 and -1, since obviously 1*1 and -1*-1 both equal 1. I'd also like to point out that taking the square root of any positive number works like this, you always get two solutions which are both the same distance away from zero in opposite directions.

(Square roots and square numbers are usually taught using, well, squares, introducing exponent notation where, say, 5*5 = 5² = 25, which is explained visually with how a square of side lengths 5 naturally has an area of 25, then that the square root is the opposite of this process and that 5 = √25. As you may expect, there are other ways of defining the notion of a "square" and "square root", and I'm sure that some of those ways can help make the existence of imaginary numbers become much more obvious, it's just, to be completely honest I don't know them.)

Now, if you do try to take the square root of a -1 anyway, and you just don't accept that it looks impossible, you'll have to invent a new number that explicitly has the property of squaring into -1, and since this new number won't be able to fit on the number line what ends up happening is that the number line gets extended into the second dimension, as in, it used to be a number line but now its a number plane. And if this sounds ridiculous let me remind you, this isn't the first time we've invented new numbers and extended the number line, again: negative numbers didn't always exist.

Also, remember how I said +1 and -1 are both square roots of +1? well +1i and -1i (or just +i and -i) are both the square roots of -1, and as you can see, they also sit a distance of 1 away from zero in opposite directions. In fact, √16 = +4 and -4, and in the same way the √(-16) = +4i and -4i. I know this likely won't really be able to help you understand, but hopefully it's nice to see how it is self consistent in a way and that it does kinda help to think about numbers like that.

or anything useful at least- i'm no teacher i just like talking about this stuff:> fhsdkjhs

#viciously holding myself back from talking about the unit circle and more trig stuff and how e and pi are involved augahaugahuagh #and also from mentioning that like. antimatter exists. you literally can get -1 apples but that's so not the point fhsjdhks #oH ALSO (i) does have a use in physics too like it's used in a ton of electrical engineering it's not just a pure maths thing it's real #askmuck #maths moment #no joke i started writing this post hours ago. i literally got on my computer just to reply to this cause i knew it'd be a while lmaofdhkjs #ok but i. i think that's it. ouh.... i think i can finally post it ffdsgfks

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