This is only mentioned in passing in the article, but the penny dropped for me when I read somewhere else that you should first think about negative numbers. They seem pretty real, but they're not. There's no negative quantity in nature. You can't have -5 rocks. Negative numbers are just the solution to equations of the form x + n = 0. Now after you let that sink in, imaginary numbers don't seem so weird after all. Just another kind of number defined as the solution to some equation.
Lots of replies to you are missing the point I think.
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....
You can owe someone five rocks (or dollars, to make it only slightly less abstract.)
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 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.
> A complex number is just a two-dimensional vector with real/imaginary components, whcih provides an amplitude and a phase (angle)
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).
You can for any positive n with geometric algebra (which encompasses the complex and quaternion algebras, the latter being much easier to understand in GA).
>> You can have 5 rocks destroyed in the future
You have 5 rocks now, and zero rocks in the future. At no point are there negative rock-shaped holes in the fabric of space.
The parent's comment still stands, he is talking about reality, not our mathematical interpretation of it.
You're not actually disagreeing. OP is saying the negative numbers are useful conceptually, which is what you're demonstrating here.
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't have -5 rocks. Negative numbers are just the solution to equations of the form x + n = 0.
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.
Exactly. Signed numbers for direction, complex numbers for 2D rotation, quaternions for 3D rotation (if you want to stretch the analogy to the breaking point.)
My big problem with imaginary numbers has always been their classification as "numbers" and not some other higher level structure with its own name. We don't call a matrix a number, even if it can be used in equations for instance. "Abelian group elements" ironically don't have a catchy ring to them.
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.
I get what you're saying, but complex numbers go right into literally any slot a real number goes into. That isn't true of a matrix. Plus, they implicitly include the reals as a subset, which lets them follow in the tradition of the integers, rationals, and reals. It's sort of hazy, but I'd only start to draw the line at the p-adiacs.
It is a bit hazy, yes. For me having a total order relationship is one of the intrinsic properties of intuitive numbers. Complexes don't have that, but reals do, so I'm sure if you meant you can put them in the slot, but you'd compare the modulus.
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.
That's a super fair point. I guess to me it matters more that it has all the usual field behaviors with the reals as a subset, but it is the first step "up" the hierarchy where you lose something instead of just gaining. I'm sort of used to substituting a metric in for order, but that obviously doesn't work everywhere.
Maybe. Consider for example Hossack (2020) Knowledge and Philosophy of Number. He indeed lays out a framework in which negative "numbers" aren't considered numbers, precisely because negative numbers combine a magnitude with a direction.
Maybe we should. Irrespective of any concerns on the philosophical underpinnings of mathematics, there is an active discourse on the shortcomings of pedagogical resources. See for example Hung-Hsi Wu and his negative view of "Textbook School Mathematics (TSM)". His critique is aimed at the US. I am unfamiliar with any cohesive argumentation across for example European curricula.
Negative numbers show up in nature all the time - they represent how much smaller a particular set of things is than what was defined as the 0 set. Sure, you can't have -5 rocks, but if you're counting how many more rocks you have than when you started counting gains and losses, -5 rocks is a reasonable quantity.
Negative numbers are more obvious though: if you have half a number line (positive numbers) and you want to talk about relative positions of things, negative numbers make a lot of sense.
“Imaginary” was an unfortunate choice of word. Positive integers and rationals feel “legit”* to most people, even (or especially) the non-mathematically trained.
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.
"That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question." - Gauss
I vote for “complete” numbers in place of “complex”, and we just don’t reference “imaginary” at all — it goes from “real” to “complete” when you add the missing root of unity.
My vote as an amateur mathematician would be “rotational” numbers. Everything having to do with complex numbers secretly (or not so secretly) has to do with angles and rotations, and it would be easy to describe to a class of middle schoolers that if you want to do math to describe how things rotate, it makes sense that you need a 2D space.
Well, IMO their rotational feature is kind of the accidental, happy and mysterious side-effect that starts from the "simple" computational trick of defining sqrt(-1) being i, and taking the calcuslations from there.
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.
I disagree but I don’t have an incredibly concrete argument, only an intuition that in the case of complex numbers, our mathematical formalism for solving equations proceeded our mathematical intuition about cyclical processes.
That's what I've finally settled on in my mind. Since we use imaginary numbers to model and measure real phenomenon, they're not imaginary in the sense that I would usually use that word. There's got to be a better name... Transcendental complement maybe.
> Positive integers and rationals feel “legit”* to most people, even (or especially) the non-mathematically trained.
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.
> But “imaginary” numbers are no more or less legit than, say, transcendental numbers.
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.
Well it's not that the square is a negative number but rather that you were only taught to do arithmetic with 1x1 matrices so you don't have a nice abstraction in your head to convince yourself. It's similar to people who insist that 0 is not a natural number because they haven't studied abstract algebra and don't care about the underlying algebraic structure (nevermind that peano arithmetic starts by postulating zero and succ...)
Why would abstract algebra suggest 0 is a natural number? The natural numbers aren't so much as a group, with or without 0. Peano starts with 0 because it's easier that way, but starting with 1 is easier for other things.
when I did my math degree in college the profs and grad students called them "complex numbers." I think they know there's a problem so they want to get the students who would go on to teach used to not calling them "imaginary"
FTA: it’s complex as in “military-industrial complex,” speaking of combination—of real and imaginary parts—rather than complication
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.
That doesn't quite get at it, though -- a complex number has an imaginary and real part. I guess we could call them "complex numbers with a zero real part," but
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
ai is a complex number and an imaginary, but not all complex numbers are imaginary numbers (like square/rectangle). a+bi is a complex number, but not an imaginary number. And a is a complex number and a real number. Which is the problem with trying to reduce it this way, you lose the ability to distinguish (by name) the two components of a complex number. What are the two parts of a complex number called if we drop real and imaginary?
> What are the two parts of a complex number called if we drop real and imaginary?
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.
I don't think that's a very fair comparison. You wind up with 1 when trying to make every number system from the naturals up. The imaginaries basically never come up except when using complex numbers in general, and certainly don't in the high-school and undergrad classes where people first learn about them.
Try Needham's "Visual Complex Analysis" - it's so aggressively intuition-y (and visual, duh) that you'll either join the cult or run for the hills (with good reason).
If I were in charge of math, I’d call them placeholder numbers, or something like that. They’re there to get you to a solution, but the solution to a legit problem will be a legit number.
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.
Only if you go in for a hidden-variable theory, which is absolutely cool but means you have to give up a lot of properties you'd really expect the universe to have
I like to call them "double-barreled numbers". I never liked the term "imaginary", because that gives mystical connotations and even hamper understanding, in my opinion.
You can make some progress by thinking of the imaginary numbers as being a two dimensional vector space over the reals, equipped with a very peculiar multiplication operator that is a combination of the usual stretch with a rotation.
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)
OK?
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),
which is two units out on the “imaginary” axis of our Argand plane.
I'm familiar with (and partial to) the geometric algebra literature, but I think it's a bit disingenuous/clickbaity to say that it implies imaginary numbers aren't real. I think of it as meaning that the algebra of the imaginary numbers arises as a slice through a more useful, bigger algebra.
What finally made imaginary numbers intuitively make sense to me is realizing that the number i just represents a 90 degree rotation on the complex plane. That's why i*i = -1 (it's rotating 180 degrees), why imaginary numbers are orthogonal to real numbers, why e^ix = cos(x) + i*sin(x) makes sense, everything.
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.
More relevant and less hacky, IMO, is that the real numbers aren't algebraically closed - there are polynomials with real coefficients that you cannot factor into getting real zeros.
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.
I majored in math and took a course on waves through the physics department. Everyone struggled with why imaginary numbers would show up and I was just like "we took the square root of this negative number, and the reals aren't closed when we take square roots, therefore imaginary."
I missed out on a lot of physical intuition and had basically no concept of what the imaginary numbers actually meant
Because with complex numbers you are implicitly defining a point on the complex plane. So how on the Cartesian plane you can have 2x + 3y, and that defines the point (2,3), you can have 2 + 3i on the complex plane defining the same point.
If you want to rotate the Cartesian coordinate -90 degrees,
A very helpful intuition is to consider a notion of "Compass numbers"; one might say, "the nearest coffee shop is East 2 blocks and North 3 blocks".
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.
To elaborate slightly; (1 + i) * (1 - i) = 1 - i^2. The left side is somewhere on the complex plane on a circle of radius 2 centered at the origin. The right side is somewhere on a circle of radius 1 centered at 1. The circles meet at one and only one place, namely 2. So 2 = 1 - i^2, or i^2 = -1.
I highly recommend video by Sabine Hossenfelder [1] on this topic. She's explaining the topic in layman terms, starting from what does it mean to exist (or be real) in physical sense.
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).
For those interested I also highly recommend Veritasium's video[1] as an excellent explainer as well, especially regarding the roots of imaginary/complex numbers in geometry.
Mathematicians have always sucked at naming their variables, and constants, and types...
This has given pundits of all ilk countless opportunity to debate...
> So let’s picture multiplication by –1 as half a rotation, anticlockwise, around a circle (in our case, the circle passes through 1 and –1). It’s actually a rotation by 180 degrees.
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...
I'm curious. I seem to be missing something, because I feel like what you're saying tracks very much with the "alternative interpretation". That is, I don't see the difference (in my mind's eye) at all. Is there maybe a different way to explain what you perceive to be critically different?
For me it is about expressing a concept as compactly as possible.
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.
Thank you for your response. I see where you are coming from. I think because I don't interpret their text as saying that i is "half of -1" but "half of the rotation that would lead to -1" I consider your description equivalent. In fact I am struggling to interpret their text to say "half of -1". Regardless, I see your point.
Can not recommend this series of videos from Welch Lab[1] enough on imaginary numbers. Deep dives into the history of how complex numbers were "invented" to enable mathematical representation which weren't possible by existing set of numbers. Also draws parallel to "invention" of negative numbers and zero to overcome the limitations of the then present number system (for eg. how to balance books without negative numbers).
Here's what I wonder: If somebody is hung up on complex numbers, will getting them over that hump get them any closer to understanding how or why complex numbers are used in physics or engineering?
The behavior of complex numbers under certain operations is seemingly the same mechanism as is seen in physical systems. The style of notation used in engineering usually denotes a periodic motion or a compound of many waves.
Of interest for physics is that without the imaginary part, real numbers alone are a leaky abstraction. This structure extends to quaterions and octonions.
Indeed, I'm a physicist with a background in electronics, so I'm fluent in that stuff. The question is whether explaining complex numbers to a non physicist is really going to bring them any closer to physics.
Considering complex numbers lead to group theory and group theory may model atomic structure, this metaphysical correlation is at least worthy of note. It is almost as if we are living in a distortion of truth.
I think they'll understand it if you get them to solve a few Fourier series/transforms with sinusoids, and then with complex exponentials, and observe the difference in difficulty.
> For some reason—whether a sense that there was some mistake, or someone copied something down wrong, or because it was so absurd—the manuscripts we have show that Heron ignored the minus sign and gave the answer as √63 instead.
Mathematicians have been constantly been giving bad names do their concepts...
Imaginary Numbers is one of the atrocious ones, Imaginary Numbers are part of the fabric of our reality... Meanwhile Real Number were axiomatized in a way that creates completely unrealistic numbers, "ghost-like" numbers, numbers that can never be given a name...
The first course in Analysis is ridden with examples of Continuous Functions that are not continuous in any sense humans would consider ever it...
Completely and utterly disagree, it may have been the case in the 1800s or whatever but the naming sense of modern mathematics is pretty on point and actually incredibly insightful. Consider the definition of a "connected" topological space and tell me it doesn't tell you something about society.
I'm not an expert here, but truly interested to hear responses to this question.
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.
I always imagine complex numbers by putting them on a Cartesian plane (2D linear space). Is there an extension of this for numbers that can be put on a 3D space?
Imaginary numbers are just a shorthand way of representing things in nature that relate to one another via the sine function, for example, charged particles in a magnetic field. You can bust out calculus to describe the motion, or you can use a convenient set of rules that represents the partially-solved equation.
Another way to think of it is that all numbers are imaginary.
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.
This view isnt "scientifically supported" because science is neutral on (indeed, even presupposes) the existence of abstract objects.
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.
I've thought about this problem quite a bit and, while my initial position was the same as above (math is not "real" per se) I had to concede that integers are real, because quantity is self-evidently real.
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.
Electromagnetic field changes are described by complex numbers. So not only you need fractions, you need irrational numbers and imaginary numbers to describe the universe. Why is counting oranges "self-evidently real" and describing electrons "kinda real"?
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.
> Electromagnetic field changes are described by complex numbers.
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.
Quantity has real concrete measurable effects that exist irrespective of the philosophical problem of classification. If I have two acorns I know I can potentially grow two very real trees. They are countable and that directly relates to the effect they can have on the world. I like to think that maybe every tree is one tree, or that all trees are part of a unity of "plants", but practically speaking seeds and trees are countable entities no matter how I classify them.
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.
> If I have two acorns I know I can potentially grow two very real trees.
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.
4 oranges are real because we have the neural architecture to classify the oranges as belonging to the same group according to whatever our classification criteria are.
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).
integers are real, because quantity is self-evidently real.
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.
This is one particular point of view, and it's extremely sloppy reasoning to say that it's "scientifically supported", given that mathematics is not a form of science.
There are plenty of very smart people, not just mathematicians but also physicists & scientists who are mathematical platonists.
There are plenty of smart people - scientists even - who believe in all kinds of deities.
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 are just assuming the point you're trying to prove
> 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.
You can boil down pretty much everything to "an accumulated edifice of metaphors designed by human brains, for human brains".
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.
But even if you deep dive enough, there are discrete values, like in Quantum Mechanics. And as long as you have discrete values, you have integers, no? So integers do not seem only like human models, they seem to me as something innate in the universe.
Quantum mechanics is a mathematical description of the behavior of the universe, so wouldn’t invoking this to prove mathematical objects exist be begging the question?
Not to say I agree with GP, but I don’t think it will be so easy to prove GP wrong either
"Discrete values" are also a human metaphor. You say there are two apples on your desk? I say there is a fuzzy quantum mess of probability distribution functions on your desk. "Two apples" is in your mind.
I’m baffled that you’re invoking QUANTum mechanics to ascertain that discrete values don’t exist. At any rate, nominalism has a rich history, so I doubt these hacker news comments will solve the issue…
are atoms discrete? we used to think so. we might never get a better model than quantum physics and it might still be wrong and fail to explain things. So there is a human idea of "discrete element" that we used to apply to everything - and as we look closer, it always breaks down. that doesn't mean it's a useless abstraction, but it is am abstraction - a tool for thought, a map, not the territory
I agree that this view it is not popular, but I also do not think that supporters often articulate their view/support well. I am a hard materialist and the amount of platonic-leaning discourse around the fundamentals of mathematics confuses me. I do not know how so many people (typical those outside philosophy and mathematical foundations) just assume a platonic style view.
I am currently reading "Where Mathematics Comes From" by George Lakoff and Rafael E. Núñez - the same Lakoff who authored the seminal "Metaphors We Live By", so I have a lot of time for him. At first it seems like they're just going to explore the pedagogical psychology of mathematics - how interesting! But then right at the end of the preface they hit you with "and by the way this is all there is to it, mathematical Platonism is a lie", which struck me immediately as straying out of their lane. But it seems their investigation into the titular question overwhelmingly led them to this conclusion. The argument is pretty simple - if there is a "platonic mathematics", we cannot have any direct experience of it. All mathematical thought, like all thought in general, is metaphorical. The predictive power of mathematics in the real world is unsurprising because we throw away the metaphors that don't work well.
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".
I don't think the issue has been as definitively settled as you have been persuaded to think. Let's take a look at the claim "if there is a 'platonic mathematics', we cannot have any direct experience of it" (I realize this is probably a paraphrase of a fuller argument, but it is what I have to work with here.)
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.
>The argument is pretty simple - if there is a "platonic mathematics", we cannot have any direct experience of it.
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.
I'll have to read the book, but in my mind, the (emprical) study of humans and their brains doesn't shed light on the metaphysical question of the nature of mathematics. What they find is how humans have developed to do mathematics. We could have evolved to be the way we are with or without mathematics being "out there". Survival in the physical world would lead us to "throw away the metaphors that don't work well". At any point in time a concrete human being would still be able to consider only a limited set of mathematical ideas i.e. for humans "mathematics bottoms out at "what goes on in human noggins"".
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.
Ah. I now see this comment too. I think I understand your other statement about "scientifically supported" better. I have also read the book, and I feel it makes a lot of sense. Like I said in my other post: most discourse only acknowledges nominalism or platonism. Neither sat well with me.
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.
Does their theory suggest anything about future AI mathematicians? I'm reminded of the bit in the Cyberiad where Princess Ineffabelle hums a simulated song, and you wonder, is that not a song?
1) I might be missing something, but as far as I can tell "scientific support" in this context seems ill defined. To the extent that it seems rather meaningless term (in this context)
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.
No it isn't scientifically supported because science does not include ontology and episemology as its domain. I'm not a platonist, but the reasons for not being platonists are philosophical.
>all numbers are imaginary.... mathematics is an accumulated edifice of metaphors designed by human brains
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.
The proofs are only correct in as far as you believe in the Axioms of mathematics that those proofs are built on. Stop "believing in" the axioms and the proofs are no longer true or even meaningful.
Oh, yeas, your proof is quite real. And it's completely meaningless, being all about imaginary things.
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.
