Imaginary Numbers
I always thought of the concept of "imaginary numbers" as hilarious.
So, in math one day, we learned about the imaginary number i, which stands for the square root of negative 1 (√-1). According to the teacher, this number is imaginary because it's technically impossible. √-1 is only used to represent numbers that cannot exist, because, obviously, negative square roots can't exist.
The reason imaginary numbers are so funny is because math itself doesn't exist. Math is only taught to little schoolkiddies because it is a commonly accepted system that most humans need to use. All the other subjects taught in school - science, history, literature, music, languages, etc. - are concrete topics that can be proven to exist via the senses. Mathematics, on the other hand, is simply a complex theory that humans invented to help simplify and explain the universe in a way that we can understand. One might argue that all of science also fits this description, but for the purposes of this post, I'm not going to get into that.
Of course, the reason imaginary numbers are called what they are is because within the realm of math, imaginary numbers should not be able to exist. i is just a figurehead for something impossible to express using "ordinary" mathematics. Yet the irony is that math itself is "impossible" to express in some ways, because math itself does not exist either.
There is no such thing as a number; likewise, there is no such thing as multiplication, geometry, or subtraction. The reason math doesn't exist is because we say that things only in our heads do not exist. Therefore, the concept of imaginary numbers is redundant, because math itself is entirely imaginary and conceptual and epic.
Does mathematics exist? To answer this question, the parameters of existence must be defined. Obviously this is no easy task, because there are several different opinions on the subject. Obviously, because people are just like that. Here are some views:
1. Some would argue that mathematics does in fact exist, because enough people need mathematics to exist. This argument is essentially the majority rule. For example, leprechauns are commonly accepted as fictional because the majority of the population believes they are fictional. Similarly, math is commonly accepted as real because the majority of humans believes it is!
2. Existence could also encompass that which only dwells in abstraction: metaphor, color, number, imagination, other fun stuff. So via this definition, yes, mathematics does exist.
The only problem I have with this argument is that if everything conceptual doesn't exist, then doesn't everything not exist? Can't everything encountered and pondered over count as some sort of concept? Everything from time to sidewalks to cucumbers can be considered a "concept". I do not understand this argument.
I suppose what people are actually trying to say with this argument is that anything that exists only in one's head is nonexistent. That seems very...harsh and untrue. If I accuse someone of lying, and the supposed liar says they told the truth, which point of view is correct? Both perceptions of the situation exist in each person's head. Does that make them both nonexistent?
Well, most people would probably say yes, both points of view exist. But in my opinion, only the people who consider both points of view to exist are those who know about this lying-accusing example. Everyone else doesn't consider either point of view to exist, because they don't even know about this example yet. Here, existence is defined as a subjective state: anything that enters an individual's consciousness. This definition is the most exciting one. To me, anyway.
3. Related to the purely-conceptualists are the people who claim that math does not exist, because it is subjective and created by humans. This argument frames "existence" in relation to some sort of "true", objective base. According to this opinion, mathematics exists to humans, but not to any alien races out there. The problem I see with this argument is that any kind of truly objective base is impossible.
Taking this argument a bit further, one could argue that nothing can actually be proven to exist, because objectivity is impossible to reach. Humans are inherently subjective: we see everything from the point of view of a human. Thus anything humans create only exists to humans, and anyone/anything else that stumbles upon it. But that's a post for another day.
4. As an offshoot of the subjectivity argument, some claim that math can exist because it can explain natural phenomena. Think the Fibonacci sequence and nautilus shells: the sequence explains the growth pattern of the shell:
This logic goes something like:
So, in math one day, we learned about the imaginary number i, which stands for the square root of negative 1 (√-1). According to the teacher, this number is imaginary because it's technically impossible. √-1 is only used to represent numbers that cannot exist, because, obviously, negative square roots can't exist.
The reason imaginary numbers are so funny is because math itself doesn't exist. Math is only taught to little schoolkiddies because it is a commonly accepted system that most humans need to use. All the other subjects taught in school - science, history, literature, music, languages, etc. - are concrete topics that can be proven to exist via the senses. Mathematics, on the other hand, is simply a complex theory that humans invented to help simplify and explain the universe in a way that we can understand. One might argue that all of science also fits this description, but for the purposes of this post, I'm not going to get into that.
Of course, the reason imaginary numbers are called what they are is because within the realm of math, imaginary numbers should not be able to exist. i is just a figurehead for something impossible to express using "ordinary" mathematics. Yet the irony is that math itself is "impossible" to express in some ways, because math itself does not exist either.
There is no such thing as a number; likewise, there is no such thing as multiplication, geometry, or subtraction. The reason math doesn't exist is because we say that things only in our heads do not exist. Therefore, the concept of imaginary numbers is redundant, because math itself is entirely imaginary and conceptual and epic.
Does mathematics exist? To answer this question, the parameters of existence must be defined. Obviously this is no easy task, because there are several different opinions on the subject. Obviously, because people are just like that. Here are some views:
1. Some would argue that mathematics does in fact exist, because enough people need mathematics to exist. This argument is essentially the majority rule. For example, leprechauns are commonly accepted as fictional because the majority of the population believes they are fictional. Similarly, math is commonly accepted as real because the majority of humans believes it is!
2. Existence could also encompass that which only dwells in abstraction: metaphor, color, number, imagination, other fun stuff. So via this definition, yes, mathematics does exist.
The only problem I have with this argument is that if everything conceptual doesn't exist, then doesn't everything not exist? Can't everything encountered and pondered over count as some sort of concept? Everything from time to sidewalks to cucumbers can be considered a "concept". I do not understand this argument.
I suppose what people are actually trying to say with this argument is that anything that exists only in one's head is nonexistent. That seems very...harsh and untrue. If I accuse someone of lying, and the supposed liar says they told the truth, which point of view is correct? Both perceptions of the situation exist in each person's head. Does that make them both nonexistent?
Well, most people would probably say yes, both points of view exist. But in my opinion, only the people who consider both points of view to exist are those who know about this lying-accusing example. Everyone else doesn't consider either point of view to exist, because they don't even know about this example yet. Here, existence is defined as a subjective state: anything that enters an individual's consciousness. This definition is the most exciting one. To me, anyway.
3. Related to the purely-conceptualists are the people who claim that math does not exist, because it is subjective and created by humans. This argument frames "existence" in relation to some sort of "true", objective base. According to this opinion, mathematics exists to humans, but not to any alien races out there. The problem I see with this argument is that any kind of truly objective base is impossible.
Taking this argument a bit further, one could argue that nothing can actually be proven to exist, because objectivity is impossible to reach. Humans are inherently subjective: we see everything from the point of view of a human. Thus anything humans create only exists to humans, and anyone/anything else that stumbles upon it. But that's a post for another day.
4. As an offshoot of the subjectivity argument, some claim that math can exist because it can explain natural phenomena. Think the Fibonacci sequence and nautilus shells: the sequence explains the growth pattern of the shell:
 |
| Nautilus shells: explained via mathematics. |
Nature exists --> humans exist --> humans invented mathematics --> mathematics can explain nature --> mathematics exists
So this argument claims that anything that can sufficiently explain something considered reality exists as well. Thus science is widely accepted to exist. Of course, this argument varies depending on one's definition of "sufficient". One could say that to Copernicus-era Europe, the geocentric universe was considered reality, because it explained all that Europeans knew about astronomy and so forth. And, yes, to those Europeans, the geocentric model was reality. We really have no right to laugh at them (although we can berate the Church for locking up Copernicus).
So in conclusion, it's really up to you whether or not mathematics exists, and whether or not you laugh at imaginary numbers.
P.S. I invented all the views of these "some people". I didn't actually go around asking people if math exists or not. Well, I did, but I didn't get any workable answers. Feel free to leave me some workable answers. I'd love that.
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