The Beauty in Data Science
Who are you calling a sophist?
No, I don’t mean AI art, though that is beautiful. I mean the intrinsic, intellectual beauty of the field, before it’s ever taken into the realm of art. I suppose this is going to be my “why data science” post.
My last semester at college, I had the good luck to take a class that left me with so much to think about that I still think about it, years later. The serendipitous combination of my being interested in the material, the whole of the class seeming to get along well, and the whole of the class seeming to adore our young instructor — as well as it being a summer session, and all of us left feeling there was just a little extra time in the day to smell the roses — have left me with the memory of those last salad days being some of my favorite on campus. The class: a reading of Plato’s Republic.
It’s full of rich ideas and rich language. Armchair Socrates opens with, “I went down to the Piraeus yesterday…” and later delivers such gorgeous lines as these:
“The object of education is to teach us to love what is beautiful.”
“Nothing beautiful without struggle.”
More famous than any one quote, though, is his allegory of the cave. There’s a good chance it’s already familiar to you in some version. Plato tells the story of a man — all men, actually — chained to a wall in a cave, forced to watch the wall for shadows cast by the fire behind them. That’s their whole world until one man breaks free, ascends from the cave, and is blinded, literally, by truth, the light of day. He’s disoriented, in pain from too much sunlight, but gradually he adjusts and comes to a clearer understanding of reality. He returns to the cave to tell the others, but they’re afraid, angry even, at what he has to say: the shadows on the wall are their reality.
Plato, and through him Socrates, is the grandfather, the cornerstone of thought, on which so much of Western philosophy is built, and perhaps in no better way does the lineage show itself than in this idea: the allegory of the cave is about the effect of education on human nature; the idea that education is the way out of the cave for humankind; and the relationship, perhaps contrastive, between perception and reality.
Like all good metaphors, that’s not the only thing this is about. It’s also about Plato’s theory of forms, that our reality is populated by imitations of truth. For his theft, a burglar is made to pay a fine, an act of justice, but that is not the form of justice — merely an (imperfect) instance of it. One of the shadows on the wall. None of us really knows what perfect justice looks like. To make an analogy for this audience, think about object-oriented programming. The forms are like the classes themselves (perfect), and imitations are like the various instances of that class (imperfect). You never actually have the class (form) as an object — it’s untouchable, too abstract to be realized in a literal way — you have instances (imitations) of that class as objects. Not a perfect analogy, but a good enough one.
I felt myself stumble into some of these same ideas reading something that many here should be familiar with, An Introduction to Statistical Learning. Early on, the authors talk about how, given some quantitative response Y and z different predictors, X₁, X₂, …, X𝓏, there is an assumed relationship between Y and Χ = (X₁, X₂, …, X𝓏) defined as
Where f is some function on the input space and ε is a random error term with mean 0 (p.16 of An Introduction). In other words, the relationship between the input X and the output Y can be perfectly described, but for the most part, that knowledge is beyond us. What we can do instead is approximate the relationship between the input and the output as
since ε has mean 0, where f-hat represents an estimation for the function f and Y-hat represents the prediction that’s made from it (17, An Introduction). Even if we could make the predicted function f-hat a perfect match of the function f , there would still be some error in Υ-hat since we haven’t accounted at all for random error ε. In other words, some of the error in our predictions is irreducible — unfixable. In some ways, our approximation is very much an imitation in the Platonic sense; the form, Y, is unreachable. And yet! The imitation can be made very good.
That’s what the whole study of mathematical prediction is, in a nutshell: if nature has a blueprint, how close can we come to understanding it that we might anticipate it. The form is out there; how close can we get the imitation? When we train a model well, we’re approaching this perfect form that describes the relationship between our predictors and the outcome. We’re essentially picking the model that’s closest to the Platonic form of the data. Every time you train a model, every time you see your accuracy jump up or your RMSE plummet, you feel the thrill of knowing you approach something previously unknowable. Bit by bit, you can see your algorithm hacking away at the shadows, pulling you out of the cave, throwing you into the sunlight.
It’s always the right time to read a good book. Here’s a link to Republic (I’m partial to the Reeve translation, though there are myriad free versions available online).
