An imaginary Number is a seemingly paradoxical name. Aren't all numbers really the product of our imagination? Numbers are the conceptual representations of things in the world, but are numbers themselves real? Almost everybody asks this question at some point in learning mathematics, usually around the time they encounter negative numbers or complex numbers. At first glance, the number 3 seems much more “real” than numbers like -1, 3i, or even polynomials such as “3x + 6y”. The reason is, obviously enough, we see the manifestation of the number three in our day-to-day lives: three cars on the street, three minutes until I'm late, three dead cats on my front lawn, et cetera. Although perhaps less obvious, we can also see some examples of negative numbers fairly often, in situations like five minutes late to class, ten feet below see level, and two months before my birthday. But unbeknownst to many, practically every “number”, including polynomials, equations, and whatnot, are all physically represented in one way or another. Complex and imaginary numbers are integral to the mathematical formulations of probabilities in quantum mechanics, and the probabilities that we encounter at the atomic scales directly display the properties of complex numbers; in knot theory and coordinate geometry, polynomials are often used to express quantities that are mathematical but not able to be expressed by single numbers.
So all numbers are equally real; if the number 1 is real, then the number 3i is just as real, and so is the number -3. But the question really goes much deeper than that. Numbers are properties that we assign to physical objects, yet not physical objects themselves. We can observe three people walking down the street, yet we never see a number three or the concept of three walking into Starbucks, just as we can observe red sweaters and scarlet dresses but not the color red or the shade of scarlet by itself. In other words, numbers cannot exist “in the real world” in isolation to the objects it is referring to at the time, just as colors cannot exist in the world unless as a part of a description of a physical object. So does being a descriptive factor of a physical object make the property real? To this particular question, I would argue that the answer is that the property is just as real as the physical object it is describing; the number three is just as real as the people whose numerousness it represents, and the color scarlet is just as real as the dress whose visual hue it attempts to describe.
Alright, so numbers are real when related to concrete objects. But in the large majority of the time, numbers are discussed only as concepts. So is the idea of a number, in isolation from concrete representations of the idea, real? To address this deeper question, I believe it is beneficial to look at the distinction between specific and generic concepts. For example, the concept of an “animal” is significantly more general than the concept of the particular bald eagle that lives in the Indianapolis Zoo. There's almost no doubt* that the bald eagle is a concrete object. But is an animal a thing? Can an animal be without it being simultaneously a specific species of an animal? My point is that an animal cannot be (exist) in isolation of a more specific physical thing it represents; i.e. a general concept cannot be (exist) in isolation from the specific, real-world objects it describes. All animals are, within themselves, physical, existent objects; and an animal cannot exist without it also being a specific individual of a specific species. So analogously, it would seem that a number, such as three, cannot be without the physical representations of it. In other words, by the logic of this analogy, the number three only has any meaning at all in light of the set of things it represents, and hence the very idea of three cannot exist in isolation from the physical representations of three. So the question of whether or not a concept in isolation from the specific representations of it is real becomes irrelevant, as all concepts are mandatorily related to physical objects.
Thus far, our conclusion is that no idea has any meaning or definition whatsoever without the set of things that the idea describes. Is est, a concept is necessarily defined by a set of things that the concept describes; concepts and ideas are not defined a priori and completely rationally, but only empirically, resting on examples and reasoning by analogy**. So, the concept of three does not exist and cannot exist independently from its physical manifestations – its real-world counterparts – and therefore the concept of three is real. Or as real as the concepts of those physical objects, and as real as the concept of reality itself.
Ultimately, each idea upon whose real-ness we can argue is nothing more than a human creation. We invent these words and ideas around examples to make communication and storage of information and knowledge much easier. Therefore the question of the real-ness, or the existence, of these ideas that we made up for our ease of use are, in my view, rather purposeless. There are certainly things in this world that are real, but to give each and every one of them names and to classify all things by naming them perfectly is a difficult and rather impossible task. But having made up these ideas, we have given ourselves the tools to think more critically about the world, and that's the purpose of ideas and concepts.
Concepts are merely human creations that are used as tools to think more carefully about everything that each of us observe, and numbers are no exception. I don't think there's much benefit to debating the real-ness of something we as humans created for our sake; I do find benefit, however, in using these ideas carefully to observe the world around us. Ideas are merely tools that we have at our disposal to find each of our goals in life and to reach them to the best of our abilities. So although I do think imaginary numbers are just as real as any other concept we encounter every day, I find more value in knowing not that imaginary numbers are real, but that we can use the idea of imaginary numbers to achieve goals that are otherwise impossible. The existence of the world, the universe, our consciousness, or the permanence thereof are to me the wrong questions to be asking; because whether or not we find these arbitrary relations between the products of our thoughts, we can benefit from them. And I think that's the power of ideas. Our ideas are never the most accurate representations of the world, and they aren't meant to be. Rather, each of our set of ideas are a filter through which we observe the world, and they help us understand the complex and multifaceted truth to the best of our abilities. So rather than think, “I think, therefore I am”, I would argue that a better direction of philosophy is to ask, “How can I think to improve the status quo?” In that, I think, we can find a much more worthwhile journey.
* Sparing, for the sake of the brevity of the argument, the possibility of the brain-in-a-vat scenario
** Incidentally, this is why, without a supreme God (whose existence or properties are, by definition, defined a priori) the Cartesian method of philosophy resting entirely on rationalism and attempts to do away with empiricism, fails. But that's a topic for its own post.
If you enjoyed this piece, you might also enjoy my next post, The binary enlightenment.
Have a comment or response? You can email me.