How the modern world arose from imaginary numbers
nautil.us
nautil.us
> You can't have -5 rocks
You can have 5 rocks destroyed in the future. That to me, is what -5 physically means: a guarantee that the object associated with the number will be eliminated out of existence in the future and decrement the negative number by 1.
Some people call it debt, but to me, emotionally that word feels too financial. So I prefer “a guarantee to be eliminated out of existence in the future “, or something like that.
Conversely, 5 cows means: 5 cows currently in existence.
5 cows - 5 cows means: I see 5 cows and now they don’t exist any more and there is nothing.
I wish my math skills were better, I am optimistic that I’d find a similar thing for imaginary numbers and maybe even complex numbers.
With that said, I do get where you’re coming from and I find it a compelling perspective as well. It’s simply that I feel the perspective I described as well.
Exactly, you use it to store some information that has no real quantity but may be converted to a real quantity in the future through some other process.
I'm a polytechnic university student, we use imaginary numbers extensively in all sorts of places, especially whenever there is any oscillatory behaviour, such as an electrical signal or a light wave. A complex number is just a two-dimensional vector with real/imaginary components, whcih provides an amplitude and a phase (angle). An oscillating sinusoidal signal/wave may appear to be zero and completely static if you freeze time at the right moment, but as time progresses, it will continue oscillating, like a swing in a park.
In a way, the magnitude represents the built up "momentum" of the system, whilst the real quantity is the immediate physical value at any given point in time (given by the phase). The amplitude is always the same at any given moment, even when the swing is vertical, it has momentum which will help it reach its maximum height.
Personally, I still think they are just "invented", but I think the vast majority of engineers much prefer them to the alternative, manipulating trigonometric functions (every engineer's nightmare). They're a neat way to represent the exchange of potential and mechanical/electrical energy with a single value and some simplified mathematics (this is an engineer's, not a mathematician's, point of view). Like negative numbers, we could have chosen to have two positive quantities, balance and debt, instead we find use in merging these definitions, whether negative values make sense or not. We have become used to to negative numbers representing the "inverse" action, which makes sense when representing a phyiscal quantity such as velocity.
Note that, while the complex numbers are isomorphic to two-dimensional vectors, they are not used the same way. In particular, there is no equivalent to complex multiplication or division that is normally used with two dimensional vectors (though you could define them of course, they are not normally used).
The difference is also reinforced by the fact that there is no equivalent of the complex numbers for 3-dimensional vectors, or really any other n-dimensional vectors. That is, you can't define a multiplication and addition operation for n-tuples of real numbers for n>2, with the usual semantics of multiplication and addition (associativity, distributivity, inverse, neutral element).
Also thinking about a number line is useful when talking about both negative numbers AND complex ones: negative numbers are to the left of 0, but complex numbers are up and down from the number line.
You can go five miles east instead of west (more of a vector, but still makes sense if your movements are limited to a number line instead of a number plane).
Imaginary numbers are more of the same with a twist. i is just a 𝛑/2 rotation on a plane instead of a negative number, which can be thought of as rotation by 𝛑 on a line.
The big mistake with imaginary numbers is calling them imaginary. There's nothing imaginary about them. They're a very specific kind of operation which can be expanded with very little thought or effort to complex numbers, which have incredibly useful properties in engineering.
Calling them "imaginary" is cripplingly confusing for almost everyone, and many never get over it.
You can also go five miles south instead of west
You are right, negative numbers are a useful concept which can be mapped to a number of real world operations, but that in some fundamental way not the same thing as positive integers. The same can be said of zero, of course.
This is probably why it took humans ages to come up with such concepts, the jump is bigger than it seems after you think of this as "normal".
On a related note, real numbers are far weirder than most people think....
The best mental model of negative numbers that works for me is to treat it as direction.
In your example, if I have "-5" rocks it means I owe +5 rocks to someone else, let's say John. If, after a few days Rachel were to give me +5 rocks and then you square it off with John you are left with no rocks. Directionally Rachel → (5 rocks) You → (5 rocks) John;
So +/- stand for things flowing into and away from you respectively.
Numbers in laymans terms imply something that can be quantified and compared, which in the end I think leads to a lot of confusion when introducing the term.
That's why I'd prefer to relabel them like "2-d numbers" for instance, to make it clear that some properties are affected like going from points on a line to points on the plane.
But “imaginary” numbers are no more or less legit than, say, transcendental numbers. But in middle school you learn this confusing thing and maths starts to fall off the rails.
* Had to find a synonym for “real” here.
1) that's quite a mouthful
i) it implies that the other part is something other than real
-1) it seems weird to describe something sort of... subtractively like this
Real and imaginary! I'm not saying we should never use the work imaginary, I'm just saying that what makes i interesting is not that it's an imaginary number, but that it's a complex number. The imaginaries on their own aren't a number system at all, and rarely come up in isolation. Objecting in introducing sqrt(-1) as a complex number seems a little silly.
From first introduction in middle school math all the way through undergraduate university math to reading this tonight at age 55, no teacher or professor or book I recall has ever stated that "complex" meant "multi-part." I always thought of it as "complicated." Yes we learned "real part" and "imaginary part" but that was never connected to the name "complex." An exercise left to the reader, I suppose.
Math education has a long way to go.
A complex number should be thought of as the quotient of plane vectors: a quantity z = v/u which when multiplied by vector u yields v: that is zu = v.
(Important note: multiplication of a complex number by a vector is not commutative, zu ≠ uz.)
For more: http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
I would philosophically say "Completeness" is a more important property (since now negative roots have a meaning attached to them). But their rotational aspect is perhaps more mind blowing.
edit: to match with existing terminology you might call the reals the "completes" and the complex numbers the "algebraically-closeds" but... meh.
It allows you to use the same algorithm to solve polynomials, and stuff like that, but you end up with these placeholders that will vanish along the way to the final solution.
I disagree? The ratio of a circumference to a diameter is clearly a positive number between 3 and 4, and it sure feels more legit than something whose square is a negative number.
Probably because they are understandable quantities in their daily lives whereas complex numbers never are. I really doubt the name has anything to do with it.
Addition is the usual addition of components, but the multiplication (using polar coordinates) looks like
C = ABexp(T1+T2),
where the T are the phases of A and B.
To multiply two positive reals, since their phases are zero, it’s only a stretch: AB
To multiply two negative reals, since their phases are pi, it’s ABexp(pi+pi). But exp(2pi) is unity, so we have AB again, a positive number.
To find the square root of a positive number D, find a positive real (zero phase) that, when multiplied by itself, gives D. AA, for example.
So, real number arithmetic is a subset of complex number arithmetic, with zero phase. And therefore zero rotation.
Now, find the square root of a negative number, say -4. To do this, we have to step off the real axis for the first time: out into the Argand plane!
-4 is represented as
4exp(pi)
So we are looking for a number out here on the Argand plane that when multiplied by itself following our rule gives 4exp(pi).
And that number is
2exp(pi/2),
2exp(pi/2)2exp(pi/2)
equals 4exp(pi)
QED
When you are dealing with 2 dimensions, complex numbers are a kind of hack for representing both dimensions without any kind of vectors or pairs or anything, just numbers.
I missed out on a lot of physical intuition and had basically no concept of what the imaginary numbers actually meant
This is a big problem when you have something like a spring-mass-damper system. The springiness, damping, and mass of the system generate coefficients of a degree-two polynomial, and the zeros of that polynomial correspond to how the system moves over time.
If you want to rotate the Cartesian coordinate -90 degrees,
x' = x*cos(π) - y*sin(π) = -y
y' = y*cos(π) + x*sin(π) = x
2x + 3y => -3x + 2y
(2 + 3i)*i = 2i - 3
One might say the the real numbers are restricted to just two directions. Extend them to allow the "sign" of a number to be any direction.
It is intuitive to imagine what addition of "Compass numbers" might look like, and what multiplication by a positive scalar real number might look like.
The question is, what should multiplication of two compass numbers neither of which is real be?
One of the earliest developments of complex numbers was done by a Danish cartographer, Caspar Wessel. He wrote a lovely paper that was published in an obscure forum in 1797. He is now credited as the first person to understand the correspondence of complex numbers and vectors on a plane.
It seems quite natural that a cartographer would be interested in "numbers" that can point in any direction.
If you posit the existence of a multiplicative identity and (arbitrarily) label it "1", and then take a compass number 90 degrees away from it and (arbitrarily) label it "i", (and furthermore assume field axioms), the formula "(1 + i) * (1 - i)" forces the conclusion that i * i is the compass number pointing in the opposite direction of the multiplicative identity.
Complex numbers, as useful as they are, just an abstraction, a tool which can be replaced with different forms of calculations. That makes the question of "reality" rather complex (pardon the pun).
It's from Veritasium: https://www.youtube.com/watch?v=cUzklzVXJwo
Sorry but this just didn't work for me. My mental model is to use +/- as directional indicators. For me multiplication by -1 is same as multiplication by +1 but in the opposite direction.
I would rather think of "i" as rotation by 90°. In this realm +/- stand for anti/clockwise. So +/- continue to stand for direction and "i" tells me if I need to rotate or not.
It neatly lines up with different operations. As an example (+i) × (-i) == -i² == 1. In terms of movements you are rotating 90° anti-clockwise followed by 90° clockwise bringing you back where you started.
On the other hand, Nautilus's article is bit too much of mental gymnastics for me to follow their reasoning.
> So let’s picture multiplication by –1 as half a rotation..It’s actually a rotation by 180 degrees..What happens if we only do half of this rotation? It’s halfway to multiplying by –1, which you can think of as the same as multiplying by √–1...
In case of Nautilus they first ask us to imagine -1 as 180° and then proceed say half of it as 90° rotation and finally give it a name i. For some this may work fine as a way to understand. However it gets awkward to further explain. If 𝑖 stands for half of -1 (as they say) then does that mean 𝑖 == -(1/2)? If not then why not? and so on.
Compare that with:
𝑖 is a unit of rotation which is 90°. That's it. Using this as a base it's easier to explain why 𝑖² = -1 (two units of anti-clockwise rotation) or why 𝑖³ = -i (three units of anti-clockwise rotation is same as one unit of clockwise rotation) and so on.
More fundamentally, they ask us to imagine an operation on 1-d (-1) as happening through 2-d (180° rotation). It seems convoluted way to introduce 𝑖. I'd rather explain 𝑖 from the first principles and proceed to show its consequences such as 𝑖² = -1.
Apparently there's not much to say about how that mistake came up, though.
[1] https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...
Of interest for physics is that without the imaginary part, real numbers alone are a leaky abstraction. This structure extends to quaterions and octonions.
I wonder if the minus sign was in use in the time of Heron (First century AD). I couldn't tell from this Wikipedia page https://en.wikipedia.org/wiki/Plus_and_minus_signs
To say that 1+1=2 is "true", does that not require a corollary in "reality" to something fundamental that can be called a "one" object? I believe this is called mathematical constructivism.
Imagine, hypothetically, that we cannot identify something that is physically fundamental and individual. My question is whether any mathematics in that scenario could be considered "true" without such constructivism, in other words, without a physical correspondence to an unquestionably, physically fundamental "one" object.
Numbers aren't real. Platonism is wrong. Imaginary numbers aren't "out there" somewhere. The whole system of mathematics is an accumulated edifice of metaphors designed by human brains, for human brains, and there's no god "behind the curtain". It's just a tool of thought. It reflects the "reality" of the universe only insofar as we've looked at the universe, noticed patterns, and constructed metaphors around them.
This is not a popular viewpoint! But it is the only scientifically supported one.
Not to say I agree with GP, but I don’t think it will be so easy to prove GP wrong either
I do not like this conclusion. Mathematics has always been something of a religion for me. But I can find no flaw with the argument. From a scientific perspective, mathematics bottoms out at "what goes on in human noggins".
Aside, but this is also Aristotle's exact argument against Platonism in general, though when he makes it in the Nichomachean Ethics he is specifically talking about ethical Good (if the definition/actual taking place of the Good lies in some other plane, we can't participate in it so no one is or can be good), but the idea is the same even when he's talking about what a soul is in De Anima. Aristotle doesn't believe in 'souls' in the way we think of them as religio-spiritual entities that exceed the capacity of the body; a 'soul' for Aristotle is the body but in a way that radically challenges the idea of a body as mere shell or vessel - soul is what any form of life repeats doing, as a body, in order to continue being itself. It should be noted that a lot of time at Aristotle's Academy was spent in Zoology, studying animals and their anatomy.
Firstly, note the word "direct" here. If it has any relevance, then the authors have assumed the burden of explaining either that there are only direct experiences, or why indirect experiences don't count.
Secondly, what are the premises here? If this is supposed to be axiomatic, then there is literally no reason to either accept or reject it, and claims that the issue has been settled are just statements of belief; otherwise, the argument needs to have premises that are not begging the question in some way. As it stands, this claim is not an argument; it is more of an intuition pump.
Metaphysical discussions tend to (always?) end up as being about the meaning of words like 'real' and 'true'. Whether such discussions can really tell us anything about what must be true is arguably the most meta question in metaphysics.
I'd say the patterns you mentioned in an earlier comment are a way for math (or parts of it e.g. some integers) to be "out there". If humans embody mathematics, then analogously so do those patterns.
Now following Hume and Locke "induction" is often treated as something "invalid", a problem to be solved. If induction is however is not a problem (see for example, Groarke, 2009, An Aristotelian Account of Induction). Aristotelian approaches are reasonable. Hence, numbers and other mathematical concepts can be very real.
No one believes abstracta have a physical location -- they lack physical properties. The claim "2 + 2 = 4" is true -- and clearly not true invirute of anything anyone thinks... if we kill that person (/people), it is no less true.
Indeed, if numbers don't exist (for example), do we suppose that we can't communicate issues of quantity with other species (, & possible alien life, etc. etc.) ? (If we can, what shared things are we talking about when we quantify?)
It seems deeply implausible to say that our use of number is circumstantially psychological -- any description of reality is going to be indispensably quantitative --- quantity is what we are talkng about. We are not talking about ourselves.
If you have four oranges, the quantity "four" is right there. If you take away one of those oranges you know that the result cannot be split evenly without a remaining orange because of the properties of odd numbers.
If you cut the remaining orange in half then you get a rational number, but is that self-evidently real? The halves of the orange are only "halves" because we consider them in relation to their origin, which we consider to be "one" orange. So rational numbers necessarily involve the human action of relating some quantity to a reference quantity, therefore they are a higher-level abstraction built on top of the fundamental physical property of quantity.
In the end I decided that math is based on a foundation of quantity (and maybe "space" as well?) and everything else was a derived abstraction. I am very curious if anyone else has a good argument for other parts of math being fundamental.
But the there are more integers than there are quantifiable 'things'[1]. Are integers that are a lot larger than, say the size of the power set of all fundamental particles in the universe still "self-evidently real".
[1] Assuming a finite universe (or a finite number of finite universes) and a few other things.
I'd argue the opposite - oranges never appear in the laws of physics. They are just our description of a collection of atoms sharing some pretty loosely-defined characteristic. Oranges aren't perfectly equivalent to each another, so whether you count 1 small and 1 big orange as 2 or 1.5 oranges depends on your arbitrary decision. How about 1 orange and 1 hybrid species between orange and grapefruit? How close you need to be to be considered orange? Classes of equivalence are determined by us not by the universe, and numbers are derived from that.
Electrons on the other hand are as undeniably real as anything in this universe can be.
You can do this, but there's no need to. You can describe electromagnetism using only real numbers.
A better argument for imaginary numbers being necessary to describe the universe is quantum mechanics, since quantum interference (in particular destructive interference) means that two possible events that each have a positive probability taken in isolation can cancel each other out, implying that probabilities can combine with a minus sign. And that means that probability amplitudes, which are square roots of probabilities, can have nonzero imaginary parts.
If there are two planets, we can discuss philosophically that one might be a "moon" and not a "planet", or in some sense that the planet is "continuous" with the space dust or whatever. But the existence of two distinct bodies in space will still create very specific gravitational fields from their interactions. Tides are different if you have one vs two moon, Lagrange points etc.
As for electromagnetic fields, I am not smart enough to make a judgement on that. They are described by complex numbers, but does that mean they reflect a physical embodiment of complex numbers? Or is it just that we require complex numbers in order to resolve their behavior into something measurable? I love to learn about electricity but sadly the math is beyond my ability.
There are 2-seeded acorns. And you can get more than 1 tree from 1 seed in some species by asexual reproduction. I guess it depends on how you count trees. All the possibilities I see (number of trunks, distinctive DNA, unconnected cliques of cells) are fuzzy and have unintuitive counterexamples.
What if you can’t classify but only be conscious of input? Kinda like being in a super dreamy state (or psychedelic one). From that state of consciousness, numbers aren’t real but reality can be (in the psychedelic case).
Just brainstorming
Groups. You can stay in your kitchen (the neutral element) or go into the bedroom, then come back (inverses). In my mind, this is as real as quantity.
Really? What is the empirical evidence for it?
have you looked there already? :)
In the game Hearts, if you take most of the spades you lose. However, if you manage to take all the spades you win, and they call it "shooting the moon".
In a similar fashion, when you reject everything as an unreal system of metaphors, Platonism "shoots the moon" by having us reexamine what we thought we meant by "real" in the first place.
There are plenty of very smart people, not just mathematicians but also physicists & scientists who are mathematical platonists.
Mathematics is indeed not a form of science. But the existence and shape of mathematics is an observable phenomenon, and so metamathematics - the study of what it is and where it comes from - can be studied scientifically. How do you know mathematics exists? Well, there's a textbook right there. Who wrote the textbook and why? A human, expressing metaphors inside their heads. How did those metaphors get inside that human's head? Ah, well, that's the interesting bit - the answer of course transpires to be "a combination of innate ideas imprinted by genetic evolution by natural selection, and sociology". And you don't have to stop there, you can explore in glorious detail exactly where each idea comes from, what innate monkey-ish tendency is being deployed, how exactly ideas like "infinity" fit in a mind designed for finding fruit and chasing things.
We can similarly bring all manner of religious beliefs under the anthropological knife. It's not a pretty process though, to the people who believe in them.
You're begging the question presupposing the non-Platonic viewpoint here. How do we know that metaphors are "inside the head"?
> There are plenty of smart people - scientists even - who believe in all kinds of deities.
Okay? This is supposed to make me feel - how exactly? I'm not inherently disdainful towards theism or theists, but if I were, I guess your remark would make me like science less, or something?
> We can similarly bring all manner of religious beliefs under the anthropological knife
I'm not really sure we can, actually. At least not in some kind of non-contentious, "objective" sense. I don't really trust individual humans to give an accurate account of why they believe their beliefs, but I trust "anthropology" and "sociology" even less. My distrust for this on an individual scale comes from the fact that many beliefs & memes exist for purposes of social signalling, group identification, etc, and it might not actually be in your interest to know exactly why you believe what you do.
But these auxiliary functions of beliefs, such as signalling etc, seem to me to scale up as you introduce groups and larger-scale activities such as "anthropology" and "sociology". Without some feedback loop keeping them honest, why would I expect anthropologists or sociologists to tell me a true story about why someone believes what they do, any more than that person or anyone else? In aerospace engineering, the feedback loop is that if your design is bad, your jet engine won't work. As a result, I generally trust aerospace engineers about jet engines. But what is there to stop sociologists, anthropologists, etc from just settling on some bullshit that agrees with their preconceived beliefs or flatters their group status and promoting it forever?
But back to math. The history of mathematical ideas is complicated and interesting, but it isn't really that relevant to the question of whether the things those ideas are about are "real", which is equivalent to asking whether mathematical platonism is true or not. The question of platonism comes down to the definition of words like "real" and "exist". It is very easy to equivocate using these words, which is why most discussions about mathematical platonism are so low quality. I think the overall question isn't that meaningful so I'm not really a platonist or an anti-platonist. In most parts of human life, when I say "x exists", I mean that I can reach out and touch x, that it has a mass, temperature, surface texture, etc. In math, when I say "x exists" I just mean that I can talk about x without creating any logical contradictions. The square root of -1 may not exist in the same sense as my laptop here, but it exists in the sense that I can do things with it, such as add, multiply, raise to powers, etc, without reaching a contradiction in my formal system. So the whole "out there" thing doesn't really matter. There doesn't need to be an "out there" in order for me to meaningfully say that the square root of -1 exists.
I think that a lot of philosophy is like this too, when you mentally zoom in really closely on a problem, it often reduces to some kind of equivocation or inconsistent language usage.
Btw I don't really consider anthropology or sociology to be real intellectual disciplines, and I'm pretty on the fence about psychology and economics. I realize that is an unpopular opinion but I've thought about it a lot and I'm pretty certain that it's correct. Aerospace engineering is real because it attaches to some fundamental reality, namely that of the spinning fan blades, the combusting fuel, etc. If you get your engineering wrong, the fan blades won't spin. Likewise, math is attached to systems of axioms. When your do your math wrong, you get a contradiction. Sociology and anthropology don't attach to anything, they're like a closed loop, like theology. If you get your anthropology wrong, nothing really happens.
Even if you stop “believing in” math, your proofs are still either correct or incorrect.
It can only have any meaning if you adhere to some scientific model.
At this point you've deviated so much from the OP discussion that you could as well talk about angels dancing in pinheads. Any quantification of them is as real as your proof.
Or you can say they exist but in a different way to physical reality.
I mean pi probably still was 3.14159... before humans evolved so it's not our fault really.
Personally I think maths not only exists but physical reality is a subset. I mean why else is there something rather than nothing? Scientifically it's the only hypothesis that works for that really.
2) While the vast majority of the discourse on the interpretation of mathematics oscillates (fruitlessly) between Platonism and Nominalism, I tend towards a more Aristotelian view. See for example
- Franklin: An Aristotelian Realist Philosophy of Mathematics
- Keith Hossack: Knowledge and the Philosophy of Number.
Note that these two authors do not converge on exactly the same interpretation.
Apropos: North (2021): Physics, Structure, and Reality explores the relationship between "mathematical structure" reality and theoretical physics.