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Complex Numbers
Complex Numbers
A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.
The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.
Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.
Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.
April 20, 2024 at 11:45am April 20, 2024 at 11:45am
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I have almost no formal training in philosophy. This doesn't stop me from discussing philosophy. And today's article is, nominally, all about philosophy, so here I am, writing about it.
Philosophy is an art 
For Margaret Macdonald, philosophical theories are akin to stories, meant to enlarge certain aspects of human life
Well, that's one point of view. And as we're all about writing, here, the connection to "stories" is a fruit too low-hanging for me to ignore.
Note: I'm unclear on whether she styled her surname Macdonald or the more common MacDonald. Sources differ. I'm sticking to the linked article's style.
‘Philosophical theories are much more like good stories than scientific explanations.’ This provocative remark comes from the paper ‘Linguistic Philosophy and Perception’ (1953) by Margaret Macdonald. Macdonald was a figure at the institutional heart of British philosophy in the mid-20th century whose work, especially her views on the nature of philosophy itself, deserves to be better known.
Full disclosure: I'd never heard of Margaret Macdonald before reading this article, confirming the "deserves to be better known" bit above.
As for the actual assertion she made, I don't find it provocative. Maybe this is because, in the single human lifetime since that paper was published, some of that idea has permeated public consciousness; but if so, why would a 2024 essay call it thus?
Philosophy, stories, and science are all interests of mine, so that sort of statement is just begging me to comment on it. So I will:
I don't think I've ever held the view that philosophy is scientific. I have, I'm pretty sure, noted in previous entries that philosophy and science are different, and complementary: philosophy can guide science, and science can inform philosophy. I mean, sure, in the early days of science, what we call physics was known as "natural philosophy," but in those same early days, you had scientists studying alchemy or astrology or other subjects we consider mystical; things change.
For instance, science achieves results through, among other things, experimentation. But it's philosophy that tells us that certain kinds of experimentation, on humans or other animals for example, is an ethical violation. Even that philosophy, though, has changed over time as certain experiments have provided evidence that some nonhuman animals feel pain and suffer. (I have another article in my queue about that very thing.)
As for the asserted closer connection between philosophy and stories, I'm willing to listen to that argument.
I'm going to give the phrase "philosophical theories" a pass for now. The article gets to it, eventually.
Early proponents of the ‘analytic’ method in philosophy such as Bertrand Russell saw good philosophy as science-like and were dismissive of philosophy that was overly poetic or unscientific.
All due respect to Russell—some of his ideas contributed greatly to advances in computing, among other things—that's a narrow view.
But where would philosophy be if philosophers always agreed with each other?
Russell’s view of what counts as good philosophy was not one that Macdonald shared. In her 1953 paper, she embraces comparisons between philosophy and literature, poetry and art. For Macdonald, philosophical theories are very much like ‘pictures’ or ‘stories’ and, perhaps even more controversially, she suggests that philosophical debates often come down to ‘temperamental differences’.
I'm going to be using the word "art" in my commentary in the broadest sense, as in the article's headline. Literature (also in its broadest sense to mean "any work of fiction") and poetry are forms of art, as is music and dance.
For example, whether you are willing to believe (in accordance with thinkers like René Descartes) that we have an immaterial soul will come down to more than just the philosophical arguments you are presented with. Your view on this matter, Macdonald thinks, will more likely be determined by your own personal values, life experiences, religion and so on.
I think it's important to acknowledge that we all have different worldviews. Unsurprisingly, I have an article in the queue about that subject, as well. The truth of someone's idea—and I also acknowledge that "truth" can be a slippery concept—isn't necessarily dependent on other ideas the person has. What I mean by that is, for example, Descartes is best known for two things: "I think; therefore, I am" and the idea to plot coordinates on a grid. Those ideas seem unrelated. One can argue the truth of the first one without reference to the second, which, reductively, is simply a very convenient way to visualize data. And neither of them depends on mind/body dualism, though I can see how both could have come out of that worldview.
Anyway. The article also provides some background on Margaret Macdonald, who's the philosopher we're actually talking about here, and I won't quote it much. In brief, she was abandoned as a child and dealt with sexism in her professional career, both of which surely shaped her worldview. And it discusses how she got started in linguistic analysis.
Linguistic analysis involves paying attention to and drawing conclusions from the language used in particular contexts, including philosophical debates, scientific theories, and ordinary (common-sense) language.
That sound like the birth of postmodernism to you? It does to me. I object to the use of "common-sense," because I do not philosophically believe that it exists, but that's a quibble.
Putting the tools of linguistic analysis to work, Macdonald focuses her attention on the word ‘theory’. What do philosophers mean when they talk about philosophical ‘theories’? And is it the same thing that scientists mean when they use the word ‘theory’? Macdonald’s answer is a categorical ‘No’.
Gotta agree with her here. Also, both are different from what ordinary people think of as a "theory."
She claims that, when scientists put forward theories, they do so to explain empirical facts. Scientists put forward hypotheses (eg, ‘Earth is round’ or ‘physical objects are governed by laws of gravity’), which can then be verified (or falsified) by experiments and observations, leaving behind only plausible theories, and eliminating those that are refuted by factual evidence.
I'm not sure that states things as clearly as it could. A theory, in the scientific sense, is an explanation that can be, or has been, thoroughly tested and compared against evidence. It can be falsified, as with the old idea that heat is a substance in itself, or the luminiferous ether. It can be supported by overwhelming evidence, as with evolution. It can never be fully "proven," a common misconception about the purpose of science.
According to Macdonald, philosophical theories cannot be tested. Is that true? What might she mean by this? Once again, she uses the philosophy of perception as her example.
I mean, sure, it feels true. But we need to beware: if a philosophical theory can be tested, supported or falsified, it suddenly becomes a scientific theory.
Thus, the first step in Macdonald’s meta-philosophical argument is to show that philosophical theories are not ‘theories’ in a scientific sense since they lack the essential criterion of being confirmed or refuted by fact. For this reason, she argues, philosophical theories, unlike scientific theories, are not in the business of discovering new facts.
Fair enough.
Macdonald claims that philosophy’s value is much closer to that of art, literature or poetry than science. She explains that the arts inform us that ‘Language has many uses besides that of giving factual information or drawing deductive conclusions.’ A philosophical theory may not provide ‘information in a scientific sense’, she writes, ‘but, as poetry shows, it is far from worthless.’
Gonna have to side with Macdonald here.
At this point, one might think: enough is enough. It’s all very well to consider how philosophy overlaps with the arts, but surely Macdonald has gone too far when she suggests that philosophical theories are just ‘good stories’.
My problem there is the adverb "just." Not only is that a slippery word in itself, with several possible meanings, but in this context, it's dismissive. "He's just a kid." "She's just a woman." My worldview, it should come as no surprise, includes the idea that "good stories" are one of our most important and effective ways of communicating with each other. There's no "just" about it; calling something a good story is one of the highest forms of praise.
If philosophical debates come down to ‘temperamental differences’, then it looks like there’s no real right or wrong (or true or false) – any more than it’s right or wrong to prefer John Keats to Shelley, or Sally Rooney to James Joyce.
That's a leap, and the leap involves the idea that right and wrong are absolute and binary: something is either right, or wrong; it's either true, or false. Life doesn't usually work that way, much as some of us would like it to. We (justly) (see what I did there?) praise people for helping little old ladies across the street and refraining from kicking puppies, but what about someone who helps a little old lady across the street, kicking a puppy out of the way in the process?
And sure, an argument can be made that not helping someone across the street serves a greater public good, possibly eliminating someone who's a drain on society. That's a worldview commonly referred to as "evil." Delving into that philosophical morass is beyond my scope here.
Is this really true? Are judgments about art, literature and poetry purely a matter of subjective preferences? Some might be tempted to answer ‘Yes’. If I like my child’s hand painting more than a piece hanging in Tate Modern, I might be inclined to say that, for me, it is a better piece of art. Similarly, if I get more enjoyment reading Rooney’s novel Normal People than Joyce’s Ulysses, then who’s to say that Joyce is a better writer.
It is an unassailable fact that Joyce was shit, the literary equivalent of Jackson Pollock. Okay, okay, no, that's just my opinion. But I stand by it.
The worry might persist that surely there’s the matter of truth to contend with. Philosophical theories might be like good stories, but surely only one of those stories can be true, or at least closer to the truth than another?
It's been expressed by better minds than mine: Fiction always contains truth. Well, except maybe Finnegan's Wake. Fine: most fiction contains truth.
Well, I've banged on long enough, and there's a lot more at the link. You might not agree. That's fine. My main purpose here was to share my new knowledge of a philosopher I hadn't heard of, and look at what she was trying to express. Her ideas are at least worth contemplating... but so are Russell's. |
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