Do imaginary numbers exist?
Pysiycists are not sure what to make of imaginary numbers. There is no problem with real numbers. We use positive integers for counting. Fractions are needed when we use numbers for measurement. Friction causes things to decelerate, creating a need for netative numbers.
But what in the real world corresponds to i = square root of -1. When working with waves, complex numbers simplify calculations, but are they really necessary? Recent work in quantum mechanics suggest that complex numbers may not be dispensible. They may be a fundamental part of reality.
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38 Answers
This is a metaphysical question. Does “blue” exist? or do some things that exist simply have the property of “blueness” to some degree? or does it physically exist in the neurons of those who are thinking of blueness at any give moment? The academic discipline that addresses these kinds of questions of existence is known as ontology.
I’m certainly no expert, but I suspect you’ll find that most theories of ontology will not see any meaningful difference between the existence of an imaginary number and a real, whole number. For example, “does five exist?” is no less complex in many of the approaches to ontology than “do imaginary numbers exist?”
But 5 is meaningful. We could create a universal standard of 5 objects. Anything in one to one correspondence with the standard would have the 5 property. There is nothing comparable for imaginary numbers.
I remember from high school ” i X i = -1 ”
It’s not true that physicists are “not sure” what to make of imaginary numbers. The term “imaginary” in this case is a mathematical term and not a metaphysical term. They are numbers that are used all the time in complex mathetmatics.
@LostInParadise “But 5 is meaningful”
As @Caravanfan points out, imaginary numbers are also meaningful. They accomplish real things when used in calculations.
@LostInParadise “We could create a universal standard of 5 objects…”
But does it exist? where? what does it mean for a thing like 5 to exist without having mass or position? All of those kinds of questions don’t get any harder/easier if the number we’re discussing is imaginary or a real/whole number. That’s my point.
And there are answers that philosophers have come up with. Some (you get one guess who) have proposed a “Platonic heaven” of ideals that actually exist somewhere, others claim things exist in our minds in the form of electochemical data in our neurons, others say they don’t exist. And like most areas of philosophy, there’s no singular theory that doesn’t have problems and unintuitive consequences.
Check This Article Out.
I am going to consider these questions while I go take a number 2.
@gorillapaws , Talklng about objects is too limiting. We need to talk about processes. The natural world is full of complex interactions. My way of defining 5 is as a process. Create a form with 5 parts and any collection that maps one for one with our five standard can be said to have 5 parts.
@LostInParadise “My way of defining 5 is as a process.”
For arguments sake, let’s assume that’s true. Does 5 exist or not?
I always thought that imaginary numbers were just a higher dimensional location on a XYZ plus grid. On a two dimensional graph.
It’s a slippery question but if you look at something real such as a chair you will find it is really made up of molecules and atoms and if you look closely at those atoms you will see that they are composed of quarks and gluons and the only way you can understand how these ultimate particles behave is through mathematics and specifically imaginary numbers.
So, in a way, imaginary numbers are all that exists.
@gorillapaws , Yes 5 exists. Any collection that can be mapped one to one to the 5 standard has the property of 5.
@LostInParadise Where does it exist? Does it contain atoms? If not, how can something exist with a mass of 0?
@Jeruba I don’t have an answer. I have studied this puzzle in school and never came to a satisfying answer—as have many philosophers much smarter than myself. The thing I know with certainty is that there are no easy answers to this (and similar) question and anyone who thinks there are, almost certainly hasn’t studied the problem enough. That article I linked is a great into to some of the issues involved, but it’s really just scratching the surface. There are entire courses dedicated to the subject.
@gorillapaws, I’ve had some exposure myself: philosophy minor, father a philosophy prof, son a phil major, lots of conversations and reading over the years.
I would tend to approach that question as a parallel to “When you blow out a candle, where does the flame go?”—i.e., it’s a process, not an entity, and when you extinguish it, the process stops. Aren’t your thoughts a process too?
And so I’m wondering if we can construe the concept of “this many” (in the present case, a countable value equal to the number of fingers on one hand) as a process of either actual or implicit enumeration.
@Jeruba Ok, so if five exists insofar as it’s the process of counting to five, does five only exist then when a person is thinking of enumerating five things? Are we saying its existence is contained in the neurons of the person thinking about that process? In other words, if nobody happens to be thinking of the number 9,765,966,686,096 then it doesn’t exist until someone happens to be thronging of it? That would mean numbers flash in and out of existence constantly.
@gorillapaws , Does a tree exist only when somebody is looking at it?
@LostInParadise Of course. A tree has mass and position. These other abstract things we’re talking about do not.
Physicists and Electrical Engineers often need imaginary numbers in their calculations to get correct answers for real phenomena. So imaginary numbers have to be as real as the phenomena the calculations predict.
But most of the equations can be done wih real numbers, even though the calculations are more difficult. The complex numbers have no intrinsic meaning.
That was the point I was trying to make. There are problems in quantum mechanics which seem to be unsolvable without complex numbers, so complex numbers appear to be baked into the universe.
@gorillapaws, well, I was just speculating, saying “what if,” and not taking a position. (Around here, not too many make that distinction.) What if the process is not one of counting but one of being five? I don’t think being one of five, or part of a composition or group enumerable as five, is a defining property of anything—not even of fingers, because the fiveness in which they participate is not an attribute of anything. Right? It’s a temporary construct imposed upon them and does not inhere in them.
Even the fairly solid faces on Mount Rushmore, presumed to be enduring for some considerable age together in that spot (and longer than my fingers will remain a group of five), do not possess a defining quality of fourness.
So the state of being one of five is transitory and relative, right? You can’t look at this oak tree per se and say that it is one of five on this property. All you’re looking at is the one. And if you redefine the boundaries (this street, this neighborhood, this planet), the number changes. But the act of composing the countable quantity is a process—one that halts when something changes the configuration and not when you stop counting.
How’m I doing?
@Jeruba ”...and not taking a position”
That’s kind of how these things have to be approached, with an open mind and from all directions. It sounds similar to what was discussed at the end of the article I linked above.
“A note on Maddy’s naturalized platonism:”
“Maddy actually thinks that we perceive sets. Number theory, as many logicians are proud to point out, can be reduced to set theory — i.e., numbers can be reduced to sets, which are, of course, generally seen as just another sort of abstract object. Maddy’s move is to bring those sets into the natural world. So that when we see an egg, we are perceiving that egg, but are also perceiving the set containing that egg. (A set containing an object is different from the object itself, you may recall from your math studies.) And that set containing the egg is a natural object, different from the egg itself. But now we run into trouble. Certainly there must be something different between an egg and a set containing that egg; otherwise ‘set containing that egg’ is just a proper name denoting the egg in question, and nothing metaphysical hangs on the distinction. (If you call me “Alec” or “author of this post”, you are not positing the existence of two people — these are just two different names for the same person.) Well, the usual distinguishing feature of abstracta is that they are not spatiotemporally located; but on Maddy’s scheme sets are spatial objects. The problem: Our egg and the set containing it necessarily co-exist in the same exact region of space-time, and yet they are supposed to be different things. In what does this difference consist? Well, certainly nothing physical, otherwise they wouldn’t co-exist in the exact same region of space-time. But then the difference must be something non-physical — i.e., something about the set must be abstract. And if this is the case, then we’re right back to all of the problems inherent in platonism, particularly the problem of how we can have any knowledge of such abstracta.”
@Jeruba “How’m I doing?”
Probably better than me. It’s been so long since I’ve studied this one. I bet @SavoirFaire could have this whole topic clearly explained in a couple of paragraphs, including all of our failures in reasoning. :P
@gorillapaws, I doubt that I’m surpassing you in this question. I never studied it as a matter of particular focus, and when I took my last philosophy class, Nixon was president. But if @SavoireFaire is going to come and set us straight, I want to get one more lick in first. It’s about the imaginary numbers, which I know nothing about.
Got to be tomorrow, though. My left brain signs off around this time of night.
I do not think that imaginary numbers are any less or more real than other mathematical concepts, like spheres, circles, pi or infinity.
@ragingloli – or irrational vs rational numbers. All real.
Here is a question. Did math exist before humans discovered it?
Well, cells have been dividing by 2 for billions of years before humans came around.
We most certainly discover math. We invent notation to describe it though.
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