Surviving Upper Division Math

It’s that time of the year. Classes are starting up. You’re nervous and excited to be taking some of your first “real” math classes called things like “Abstract Algebra” or “Real Anaylsis” or “Topology.”

It goes well for the first few weeks as the professor reviews some stuff and gets everyone on the same page. You do the homework and seem to be understanding.

Then, all of a sudden, you find yourself sitting there, watching an hour-long proof of a theorem you can’t even remember the statement of, using techniques you’ve never heard of.

You panic. Is this going to be on the test?

We’ve all been there.

I’ve been that teacher, I’m sad to say, where it’s perfectly clear in my head that the students are not supposed to regurgitate any of this. The proof is merely there for rigor and exposure to some ideas. It’s clear in my head which ideas are the key ones, though I maybe forgot to point it out carefully.

It’s a daunting situation for the best students in the class and a downright nightmare for the weaker ones.

Then it gets worse. Once your eyes glaze over that first time, it seems the class gets more and more abstract as the weeks go by, filled with more and more of these insanely long proofs and no examples to illuminate the ideas.

Here’s some advice for surviving these upper division math classes. I’m sure people told me this dozens of times, but I tended to ignore it. I only learned how effective it was when I got to grad school.

Disclaimer: Everyone is different. Do what works for you. This worked for me and may only end up frustrating someone with a different learning style.

Tip Summary: Examples, examples, examples!

I used to think examples were something given in a textbook to help me work the problems. They gave me a model of how to do things.

What I didn’t realize was that examples are how you’re going to remember everything: proofs, theorems, concepts, problems, and so on.

Every time you come to a major theorem, write out the converse, inverse, switch some quantifiers, remove hypotheses, weaken hyphotheses, strengthen conclusions, and whatever you can think of to mess it up.

When you do this you’ll produce a bunch of propositions that are false! Now come up with examples to show they’re false (and get away from that textbook when you do this!). Maybe some rearrangement of the theorem turns out to be true, and so you can’t figure out a counterexample.

This is good, too! I cannot overstate how much you will drill into your memory by merely trying unsuccessfully to find a counterexample to a true statement. You’ll start to understand and see why it’s probably true, which will help you follow along to the proof.

As someone who has taught these classes, I assure you that a huge amount of problems students have on a test would be solved by doing this. Students try to memorize too much, and then when they get to a test, they start to question: was that a “for every” or “there exists?” Does the theorem go this way or that?

You must make up your own examples, so when you have a question like that, the answer comes immediately. It’s so easy to forget the tiniest little hypothesis under pressure.

It’s astounding the number of times I’ve seen someone get to a point in a proof where it looks like everything is in place, but it’s not. Say you’re at a step where f: X\to Y is a continuous map of topological spaces, and X is connected. You realize you can finish the proof if Y is connected.

You “remember” this is a theorem from the book! You’re done!

Woops. It turns out that f has to be surjective to make that true.

But now imagine, before the test, you read that theorem and you thought: what’s a counterexample if I remove the surjective hypothesis?

The example you came up with was so easy and took no time at all. It’s f: [0,1] \to \{0\} \cup \{1\} given by f(x) = 1. This example being in your head saves you from bombing that question.

If you just try to memorize the examples in the book or that the professor gives you, that’s just more memorization, and you could run into trouble. By going through the effort of making your own examples, you’ll have the confidence and understanding to do it again in a difficult situation.

A lesser talked about benefit is that having a bunch of examples that you understand gives you something concrete to think about when watching these proofs. So when the epsilons and deltas and neighborhoods of functions and uniform convergence and on and on start to make your eyes glaze over, you can picture the examples you’ve already constructed.

Instead of thinking in abstract generality, you can think: why does that step of the proof work or not work if f_n(x) = x^n?

Lastly, half the problems on undergraduate exams are going to be examples. So, if you already know them, you can spend all your time on the “harder” problems.

Other Tip: Partial credit is riskier than in lower division classes.

There’s this thing that a professor will never tell you, but it’s true: saying wrong things on a test is worse than saying nothing at all.

Let me disclaimer again. Being wrong and confused is soooo important to the process of learning math. You have to be unafraid to try things out on homework and quizzes and tests and office hours and on your own.

Then you have to learn why you were wrong. When you’re wrong, make more examples!

Knowing a bunch of examples will make it almost impossible for you to say something wrong.

Here’s the thing. There comes a point every semester where the professor has to make a judgment call on how much you understand. If they know what they’re doing, they’ll wait until the final exam.

The student that spews out a bunch of stuff in the hopes of partial credit is likely to say something wrong. When we’re grading and see something wrong (like misremembering that theorem above), a red flag goes off: this student doesn’t understand that concept.

A student that writes nothing on a problem or only a very small amount that is totally correct will be seen as superior. This is because it’s okay to not be able to do a problem if you understand you didn’t know how to do it. That’s a way to demonstrate you’re understanding. In other words: know what you don’t know.

Now, you shouldn’t be afraid to try, and this is why the first tip is so much more important than this other tip (and will often depend on the instructor/class).

And the best way to avoid using a “theorem” that’s “obviously wrong” is to test any theorem you quote against your arsenal of examples. As you practice this, it will become second-nature and make all of these classes far, far easier.

Advertisements

The Three-Body Problem is Awesome

If you’ve been around this blog this year, then you know I fell into a bit of a slump. I was reading things, but nothing seemed to connect. In fact, it all seemed derivative, flat, and downright bad.

I’ve gotten out of that somehow, and I seem to have hit a period where most things I read (or movies I see) draw me in immediately and seem imaginative and fresh. I’m not planning on making a bunch of “…is Awesome” posts, but that’s where I’m at right now.

The Three-Body Problem by Liu Cixin is unlike anything I’ve read before. It’s pretty difficult to explain why, because I don’t want to spoil anything. Part of the fun of this trilogy is that there are M. Night Shyamalan type twists (things that make you rethink everything that happened before and make it all make sense).

When these types of plot twists happen once at the end of a book or movie, it feels like a cheap gimmick and can be off-putting. When they happened dozens of times across this book trilogy, they left me in awe of the structure of the narrative.

You’ll think you’ve finally got a grasp on things near the end of Book 2, and then you learn that you had no idea what was really going on. Like I said, there are dozens of these, and each time you think it can’t happen again, it somehow does.

The books are also filled with lots of neat ideas (even if not scientific). I can describe one that happens in the first book that won’t ruin any plot points.

The first idea is to notice what happens if you “unfold” a two-dimensional object into one dimension. Here’s an example of a solid square being pulled into a string:


Now, convince yourself this is the case whenever you take a higher dimensional object and “unfold” it into lower dimensions. You’ll always get an arbitrarily large new thing.

Next, he takes the concept of string theory seriously and says: what if a proton is actually a six-dimensional string curled up into compactified dimensions? Well, with super good technology and a full understanding of the physics, maybe the proton could be unfolded into an arbitrarily large three dimensional object.

In that case, we could store infinite amounts of information in it. We could even make it the best supercomputer AI ever made. Then we could fold it back up, and it would be roughly the size of a proton again. Just imagine what that could do!

The trilogy is truly an “ideas” book. It’s kind of fascinating how strong the ideas alone were to keep me wanting to read. The plot definitely waned at points and character motivations were weak, but I didn’t really care.

To me, this book was essentially the opposite of Seveneves. Seveneves was a bunch of cool ideas that got tedious to read, because none of them served the plot. They were just Neal Stephenson spewing every idea he ever had into a plotless mess.

In contrast, every single cool idea in The Three-Body Problem series advances the plot in a meaningful way, and wow, there’s a ton of them.

I can’t recommend this trilogy enough if you’re into hard sci-fi (and my warning about character/dragging plot doesn’t turn completely alienate you).

Maybe Infinite Jest is About Addiction

And so but I’ve been re-reading Infinite Jest in this strange, almost purely subconscious way, where I take on just a few pages (seriously, like 2-3 pages) every night right before sleeping. I’ve done the calculations, and so you don’t have to tell me it will literally take years to finish it this way.

I’m in no rush. I’ve read it before.

If you’ve never read it, you really must. It’s terrifying how prescient it is. How could someone in the mid 90’s have seen the coming technology that would be so entertaining it would totally consume our lives? I’m thinking Twitter and Facebook and our phones and the games on them. But DFW actually has a Netflix-like system where people can watch any TV they want at any time. That was unthinkable back then.

It also predicts that we’d come to live in an opioid epidemic.

And all of the below, etc.

Anyway, I digress.

This weird thought occurred to me around page 300 (yes, I’ve been doing this for 100+ days already):

Maybe Infinite Jest is about addiction.

Hear me out. This is one of those things that’s so obvious it requires justification.

I know, Don Gately is in a halfway house for Demerol addiction, and the opening scene is of Hal’s (supposed) reaction to taking DMZ destroying his life, and the kids at the Tennis Academy do pot and alcohol and amphetamines and have tricks to pass urine tests.

I know, the title refers to entertainment so infinitely addicting you pee and poop yourself and then die rather than pull yourself away, and that one character, whose name I can’t remember, holed up in the bathroom stall of a library and drank cough syrup every day to avoid withdrawal but had to go out at some point and ends up having a massive DT withdrawal on a train and probably dies.

I know DFW, himself, had addiction problems and was in AA.

Etc, etc.

But hear me out. It’s not as obvious as it seems. Addiction is everywhere in the novel, but what is the novel about?

What if someone said to you: Breaking Bad is about addiction.

You’d say: Whoa! Hang on. Addiction is everywhere in that series, sure, but that’s not at all what the show is about. Not. Even. Close.

DFW is famous for complaining about the reviews (even (especially) the positive ones!!) when it came out, because no one could possibly have read it in the two-week window (or whatever it was) and actually understood what it was about.

I owed it to him to understand what the book was about if he would rather have crappy reviews than positive reviews by people too intimidated by it to admit they were clueless as to what the book was even about.

I took his comments to heart.

Infinite Jest wasn’t about addiction. That was too obvious. Everyone would immediately understand the book if that’s what it was about.

DFW was also obsessed with literary theorists and philosophy and Wittgenstein and psychiatry and math and semiotics and postmodernism and irony, etc. I looked to these for answers, and I found a treasure trove of ideas.

I won’t try to go into depth on what I came up with. You can see early thoughts in some other posts I’ve done: Westward and Preparation for Infinite Jest among others.

Basically, one can read Infinite Jest as a critique of the psychological theories of Jacques Lacan. The “Entertainment,” at least as much as we can see in the novel, is an on-the-nose manifestation of his ideas.

Language is central to our subconscious, and Saussure’s signifier/signified distinction live on different layers. Wallace thought these poststructuralists were brilliant but flawed. Infinite Jest wants to use postmodernism to show why they were flawed.

Now we’re on to what the novel might actually be about!

Many scenes support this reading, mostly having to do with the various recovery methods. Wallace wants to say: how do we break free of our addictions? Well, it’s obviously not what these theorists were saying! Look what that would look like.

DFW presents a parody as refutation.

This view is also supported by all the circumstances under which characters literally lose their ability to speak. Sure, drugs are the proximate cause, but think through the other circumstances of their lives at that time. Think about Hal’s encounter with the therapist after finding his father having committed suicide.

Why was Wallace upset at people calling the novel funny?

Maybe it’s that things that were supposed to be deep references to Lacan were seen as surface-level jokes.

Corporations subsidize years in the future. Most of Infinite Jest takes place in the Year of the Depend Adult Undergarment (YDAU). We laugh, thinking about what it would be like to have to sign checks with the year being the name of an incontinence product.

No!

It’s more than that. The year wasn’t chosen purely for humor. It’s saying that when our society progresses to this point, adults will have regressed back to babies. All we think of is: want, want, want. We rage at the TV like a baby when Netflix goes out for, heaven forbid, 30 seconds.

In Wallace’s version of the future, terrorists use this entertainment as a tool of both terror and placation. In our reality, we entertain ourselves to death with Facebook while our adversaries use it to elect our presidents for us.

And so but then we don’t care. We want reality stars to be our leaders. It keeps us entertained.

What in the world was this post even about anymore? How did I start talking about real life when this is supposed to be about a book published over 20 years ago?

Focus.

I thought Infinite Jest was about this brilliant refutation of heady philosophers. It cleverly uses addiction to get these points across in multiple ways. It invents its own language to poke at the signifier/signified hypothesis.

Then I woke up in the middle of the night with cold sweats, heart pounding, disoriented (probably withdrawal), and I thought to myself:

Maybe Infinite Jest is about addiction.

Then I realized it doesn’t matter.

Your phone notified you of 10 more interesting things since you started reading this. You haven’t made it this far, and we can’t progress. Our eyes are stuck to the screen. We won’t be able to pull ourselves away. We will poop and pee ourselves and wish we had put on our Depends until it doesn’t matter, because we’ll all be dead.

The thought brought me comfort, and I went back to sleep.

Critical Postmodern Readings, Part 3: Baudrillard

Jean Baudrillard is one of those postmodernist philosophers that people can name but probably don’t know much about. He’s most famous for his work Simulacra and Simulation, in which he argues we’ve replaced everything real in our society by symbols (more on this later). If you’re thinking of the movie The Matrix, then you’ve understood. That movie gets misattributed to a lot of different philosophers, but the underlying concept is basically a fictionalization of Baudrillard’s ideas.

I thought we’d tackle his paper “The Masses: The Implosion of the Social in the Media” published in New Literary History, Vol 16, No 3. The paper appeared in 1985, and it tackles issues relevant to our current media environment. I thought it’d be interesting to see how it holds up now.

He begins with an observation about the media:

…they are what finally forbids response, what renders impossible any process of exchange (except in the shape of a simulation of a response, which is itself integrated into the process of emission, and that changes nothing in the unilaterality of communication).

In the 80’s, as well as traditional media today, this is certainly true. There’s no way to comment on or engage in a dialogue with the people presenting information on TV or radio or even podcasts or newspapers and blogs with closed comments. Traditionally, media gets to define the conversation, and when there is “response” to what they say, it’s still controlled by them, and they still distribute that response to you.

Baudrillard wants to frame this as a power imbalance. The media have a monopoly on information. When a response is allowed, the exchange of ideas becomes more balanced.

Baudrillard brings up the case of an opinion poll as an example to motivate the next part of the paper. He points out that this distribution of information is merely symbolic of the state of opinion. There is a complicated interaction where the information itself changes opinion, rendering itself obsolete. This type of distribution of information introduces uncertainty on many fronts:

We will never know if an advertisement or opinion poll has had a real influence on individual or collective wills—but we will never know either what would have happened if there had been no opinion poll or advertisement.

Here, I have to say this analysis is a bit dated. This statement was probably accurate in the 80’s, but with Google, and other analytic big data companies, tracking so much of our lives, we can be quite certain if certain advertisements or polls have caused some sort of influence on both individual and collective wills.

This point is mostly not important to the overall thesis of Baudrillard in the article, though. He goes on to make an astute observation that can cause a bit of anxiety if you dwell on it too much. We don’t have a good way to separate reality from the “simulative projection in the media.”

It’s a complicated way to say that we just can’t check a lot of things. If we see on the news that there was a minor earthquake in Japan, we believe it. But we weren’t there. All we get is the simulation of reality as provided by the news. Of course, there are other ways to check that fact by going into public seismic activity records, etc.

But there are other narratives and simulations that are much harder to check, and in any case, we are bombarded by so much information that we don’t have time to check it. We believe the narrative as presented. If we come across a competing narrative, we only become uncertain. It doesn’t actually clarify the situation (here we get back into Lyotard territory).

Baudrillard would later write a book-length analysis of this about the Gulf War (entitled The Gulf War Did Not Take Place) in which he claims that the American public only received propaganda about the war through the media. The war took place, but the simulated reality the public received did not accurately reflect the events that occurred. Moreover, there were pretty much no sources outside this propaganda to learn about the actual events.

We live in an age of hyperinformation, and the more we track how everything is changing, the worse our understanding gets. This isn’t Baudrillard’s wording, but I can see how this makes sense: we confuse noise for signal when we pay too close attention. We also get trapped in our own little information bubbles when we pay too close attention. “Hyperinformation” (his term) can lead to more uncertainty, not less.

I think we’ve come to a point where hyperinformation is at least somewhat good. Yes, for the reasons listed, it can be paralyzing if you want the truth. But at the same time, it means the truth might be out there to discover. We don’t only get the corporate media narrative now. There are independent reporters and journalists working hard to present viable alternatives. It isn’t hopeless to see through the noise now (as it was back in the 80’s).

Baudrillard says we can get out of the despair of all this by treating it like a “game of irresponsiblity, of ironic challenge, of sovereign lack of will, of secret ruse.” The media manipulates and the masses resist, or better yet, respond.

I’ll just reiterate that what Baudrillard identifies as the central problem here has been partially solved in modern day. The masses have twitter and facebook and comments sections and their own blogs and youtube channels. The masses have a way to speak back now. Unfortunately, this has opened up a whole new set of problems, and I wish Baudrillard were still around. He’d probably have some interesting things to say about it.

Now that I’ve been doing this Critical Postmodern Reading series, I’m coming to believe these postmodernists were maligned unjustly. I’m coming to believe we should keep two terms distinct. The “postmodernist philosopher” analyzes the issues of the postmodern condition. The “postmodern academic” utilizes the confusion brought on by the postmodern condition to push their own narrative.

It’s easy to look at the surface of Baudrillard and claim he’s some crackpot history denier that thinks there’s no such thing as objective reality so we all make our own truth.

But if you read him carefully, he seems to be saying some very important true things. He thinks there is an objective, true reality, and it’s dangerous that we all simulate different versions of it (i.e. we filter the news through an algorithm that tells us the world is how we think it is). The truth gets hijacked by narratives. He sees the monopoly the media has on these narratives as damaging and even simulating a false reality.

His writing doesn’t even slip into incomprehensible, postmodernist jargon to obscure the argument. I thought this article was illuminating despite and comprehensible. The only parts that don’t still feel applicable are where he didn’t predict how technology would go.

Year of Short Fiction Part 6: Cosmicomics

I’ve sort of been dreading this one, but it’s the only thing remaining on my short fiction list that I own. Three years ago I wrote up my interpretation of Italo Calvino’s If on a winter’s night a traveler. Calvino can be strange and highly symbolic, but that book’s meaning jumped out at me with little effort. He had constructed a condensed history of critical theory through the story.

I had a vague familiarity with Cosmicomics, so I knew it would be harder. The stories all feature or are told by a character named Qfwfq. Each story starts with a tidbit of science such as:

Situated in the external zone of the Milky Way, the Sun takes about two hundred million years to make a complete revolution of the galaxy.

The story that follows is usually related to this somehow. The collection as a whole can be read as a symbolic retelling of the history of the universe. Calvino has taken real science and created mythologies that actually fit the data.

But it’s more than that. The stories often have a moral to them or a symbolic quality. They aren’t just fictionalizations of the events of the early universe. They’re almost parables like classic mythology. He’s achieved something odd with these.

The collection came out in 1965, fairly early in Calvino’s career, and well before the highly experimental If on a winter’s night a traveler. Calvino believed realism to be dead, and these stories mark his foray into a new type of fiction. He held on to pieces of realism but incorporated highly fantastical elements.

That’s enough of an overview, let’s dig into my favorite story to see these elements at work. “All at One Point” is a story about the Big Bang. More specifically, it’s about the time when the universe existed in a single point.

The beginning of the story comically plays with the idea that “we were all there.” On a scientific level, this is obviously true. Every atom in the universe existed in the singular point “before” the Big Bang. This includes every atom in our bodies, so we were physically there.

Calvino cleverly takes this statement to its extreme form and personifies us as actually existing at one point. The narrator, Qfwfq, says, “…having somebody unpleasant like Mr Pber^t Pber^t underfoot all the time is the most irritating thing.”

The story spends quite a bit of time in a Flatland-type thought experiment. Through humorous interactions, Calvino teases apart a lot of odd ideas about what it actually would mean to collapse the universe to a single point. For example, one couldn’t count how many people were there, because that would require pulling apart, no matter how slightly.

One family, the Z’zu, got labelled “immigrants.” This, of course, makes no sense, because there is no such thing as outside or inside the point. There is no such thing as before or after the point. Time only started at the Big Bang. So the family couldn’t have come from somewhere else.

The humor in this surface-level reading of the story is already worth it, and I won’t spoil any of the other awkward moments shared by these people from all occupying the same point.

Then the story turns its attention to Mrs Ph(i)Nk_o. She is one of the Z’zu, the family everyone hated. But she’s different. She is pure happiness and joy, and no one can say anything bad about her.

In an act of epic generosity, despite what people say about her family, she says:

Oh, if I only had some room, how I’d like to make some tagliatelle for you boys!

That’s what causes the Big Bang. The universe is made and expands and the Sun and planets and everything. It all happened because of a true act of selflessness and love. The phrasing of the final paragraph is very moving. I won’t quote it here, because I think it must be read in context to be appreciated.

The theme, when condensed to a pithy phrase, is something like “love can make universes.” It sounds really cliche and cheesy, and I think this is one of the things that makes these stories so brilliant. In the moment of reading, they feel profound and fresh.

Calvino’s use of vivid space imagery takes you on a grand journey. These cliche themes are the same that one can find in all the great ancient stories. They only feel tired when done in modern stories. By creating his own mythology, Calvino is able to revisit these sorts of themes without embarrassment.

For the Year of Short Fiction, I do want to return to the question of: why short? In other words, does great short fiction have a genuine uniqueness to it, or is it essentially the same as a novel, just shorter?

I think here we can definitively say that this type of writing can only work in short stories. Even expanding one of these to a novella length would be too much. These stories each revolve around a conceit and a theme. The conceit would grow tiresome if done for too long. I cannot imagine a novella of jokes about everyone existing on top of each other. They would lose their impact.

What excites me about Cosmicomics is that this is the first thing I’ve read this year that I feel this way about. I could imagine the novellas I’ve read and even Cthulhu working as full novels. They wouldn’t be as tightly written, but they’d still work. The very nature of Cosmicomics is that they are short stories. I’m glad to have finally found this.

I should stipulate, though, that one can read the entire collection of stories as a novel: an autobiography of Qfwfq’s life and fictionalization of the history of the universe. This is also an interesting and unique aspect, because almost every short story collection I can think of has separate, unrelated stories. This full collection should be read together to get the best experience.

Critical Postmodern Readings, Part 2: Finishing Lyotard

Last time we looked at the introduction to Lyotard’s The Postmodern Condition: A Report on Knowledge. That introduction already contained much of what gets fleshed out in the rest of the short book, so I’m going to mostly summarize stuff until we hit anything that requires serious critical thought.

The first chapter goes into how computers have changed the way we view knowledge. It was probably an excellent insight that required argument at the time. Now it’s obvious to everyone. Humans used to gain knowledge by reading books and talking to each other. It was a somewhat qualitative experience. The nature of knowledge has shifted with (big) data and machine learning. It’s very quantitative. It’s also a commodity to be bought and sold (think Facebook/Google).

It is a little creepy to understand Lyotard’s prescience. He basically predicts that multinational corporations will have the money to buy this data, and owning the data gives them real-world power. He predicts knowledge “circulation” in a similar way to money circulation.  Here’s a part of the prediction:

The reopening of the world market, a return to vigorous economic competition, the breakdown of the hegemony of American capitalism, the decline of the socialist alternative, a probable opening of the Chinese markets …

Other than the decline of the socialist alternative (which seems to have had a recent surge), Lyotard has a perfect prediction of how computerization of knowledge actually affected the world in the 40 years since he wrote this.

Chapter two reiterates the idea that scientific knowledge (i.e. the type discussed above) is different than, and in conflict with, “narrative” knowledge. There is also a legitimation “problem” in science. The community as a whole must choose gatekeepers seen as legitimate who decide what counts as scientific knowledge.

I’ve written about why I don’t see this as a problem like Lyotard does, but I’ll concede the point that there is a legitimation that happens, and it could be a problem if those gatekeepers change the narrative to influence what is thought of as true. There are even known instances of political biases making their way into schools of scientific thought (see my review of Galileo’s Middle Finger by Alice Dreger).

Next Lyotard sets up the framework for thinking about this. He uses Wittgenstein’s “language game” concept. The rules of the game can never legitmate themselves. Even small modifications of the rules can greatly alter meaning. And lastly (I think this is where he differs from Wittgenstein), each speech act is an attempt to alter the rules. Since agreeing upon the current set of rules is a social contract, it is necessary to understand the “nature of social bonds.”

This part gets a little weird to me. He claims that classically society has been seen either as a unified whole or divided in two. The rules of the language games in a unified whole follow standard entropy (they get more complicated and chaotic and degenerate). The divided in two conception is classic Marxism (bourgeoisie/proletariat).

Even if it gets a bit on the mumbo-jumbo side through this part, I think his main point is summarized by this quote:

For it is impossible to know what the state of knowledge is—in other words, the problems its development and distribution are facing today—without knowing something of the society within which it is situated.

This doesn’t seem that controversial to me considering I’ve already admitted that certain powers can control the language and flow of knowledge. Being as generous as possible here, I think he’s just saying we have to know how many of these powers there are and who has the power and who legitimated that power before we can truly understand who’s forming these narratives and why.

In the postmodern world, we have a ton of different institutions all competing for their metanarrative to be heard. Society is more fractured than just the two divisions of the modern world. But each of these institutions also has a set of rules for their language games that constrains them.  For example, the language of prayer has a different set of rules from an academic discussion at a university.

Chapters 7-9 seem to me to be where the most confusion on both the part of Lyotard and the reader can occur. He dives into the concept of narrative truth and scientific truth. You can already feel Lyotard try to position scientific truth to be less valuable than it is and narrative truth more valuable.

Lyotard brings up the classic objections to verification and falsification (namely a variant on Hume’s Problem of Induction). How does one prove ones proof and evidence of a theory is true? How does one know the laws of nature are consistent across time and space? How can one say that a (scientific) theory is true merely because it cannot be falsified?

These were much more powerful objections in Lyotard’s time, but much of science now takes a Bayesian epistemology (even if they don’t admit to this terminology). We believe what is most probable, and we’re open to changing our minds if the evidence leads in that direction. I addressed this more fully a few years ago in my post: Does Bayesian Epistemology Suffer Foundational Problems?

… drawing a parallel between science and nonscientific (narrative) knowledge helps us understand, or at least sense, that the former’s existence is no more—and no less—necessary than the latter’s.

These sorts of statements are where things get tricky for me. I buy the argument that narrative knowledge is important. One can read James Baldwin and gain knowledge and empathy of a gay black man’s perspective that changes your life and the way you see the world. Or maybe you read Butler’s performative theory of gender and suddenly understand your own gender expression in a new way. Both of these types of narrative knowledge could even be argued to be a “necessary” and vital part of humanity.

I also agree science is a separate type of knowledge, but I also see science as clearly more necessary than narrative knowledge. If we lost all of James Baldwin’s writings tomorrow, it would be a tragedy. If we lost the polio vaccine tomorrow, it would be potentially catastrophic.

It’s too easy to philosophize science into this abstract pursuit and forget just how many aspects of your life it touches (your computer, the electricity in your house, the way you cook, the way you get your food, the way you clean yourself). Probably 80% of the developed world would literally die off in a few months if scientific knowledge disappeared.

I’ll reiterate that Lyotard thinks science is vastly important. He is in no way saying the problems of science are crippling. The above quote is more in raising narrative knowledge to the same importance of science than the devaluing of science (Lyotard might point to the disastrous consequences that happened as a result of convincing a nation of the narrative that the Aryan race is superior). For example, he says:

Today the problem of legitimation is no longer considered a failing of the language game of science. It would be more accurate to say that it has itself been legitimated as a problem, that is, as a heuristic driving force.

Anyway, getting back to the main point. Lyotard points out that problems of legitimating knowledge is essentially modern, and though we should be aware of the difficulties, we shouldn’t be too concerned with it. The postmodern problem is the grand delegitimation of various narratives (and one can’t help but hear Trump yell “Fake News” while reading this section of Lyotard).

Lyotard spends several sections developing a theory of how humans do science, and he develops the language of “performativity.” It all seems pretty accurate to me, and not really worth commenting on (i.e. it’s just a description). He goes into the issues Godel’s Incompleteness Theorem caused for positivists. He talks about the Bourbaki group. He talks about the seeming paradox of having to look for counterexamples while simultaneously trying to prove the statement to be true.

I’d say the most surprising thing is that he gets this stuff right. You often hear about postmodernists hijacking math/science to make their mumbo-jumbo sound more rigorous. He brings up Brownian motion and modeling discontinuous phenomena with differentiable functions to ease analysis and how the Koch curve has a non-whole number dimension. These were all explained without error and without claiming they imply things they don’t imply.

Lyotard wants to call these unintuitive and bizarre narratives about the world that come from weird scientific and mathematical facts “postmodern science.” Maybe it’s because we’ve had over forty more years to digest this, but I say: why bother? To me, this is the power of science. The best summary I can come up with is this:

Narrative knowledge must be convincing as a narrative; science is convincing despite the unconvincing narrative it suggests (think of the EPR paradox in quantum mechanics or even the germ theory of disease when it was first suggested).

I know I riffed a bit harder on the science stuff than a graduate seminar on the book would. Overall, I thought this was an excellent read. It seems more relevant now than when it was written, because it cautions about the dangers of powerful organizations buying a bunch of data and using that to craft narratives we want to hear while deligitimating narratives that hurt them (but which might be true).

We know now that this shouldn’t be a futuristic, dystopian fear (as it was in Lyotard’s time). It’s really happening with targeted advertising and the rise of government propaganda and illegitimate news sources propagating our social media feeds. We believe what the people with money want us to believe, and it’s impossible to free ourselves from it until we understand the situation with the same level of clarity that Lyotard did.

The Ethics of True Knowledge

This post will probably be a mess. I listen to lots of podcasts while running and exercising. There was a strange confluence of topics that seemed to hit all at once from several unrelated places. Sam Harris interviewed Neil deGrasse Tyson, and they talked a little about recognizing alien intelligence and the rabbit hole of postmodernist interpretations of knowledge (more on this later). Daniel Kaufman talked with Massimo Pigliucci about philosophy of math.

We’ll start with a fundamental fact that must be acknowledged: we’ve actually figured some things out. In other words, knowledge is possible. Maybe there are some really, really, really minor details that aren’t quite right, but the fact that you are reading this blog post on a fancy computer is proof that we aren’t just wandering aimlessly in the dark when it comes to the circuitry of a computer. Science has succeeded in many places, and it remains the only reliable way to generate knowledge at this point in human history.

Skepticism is the backbone of science, but there is a postmodernist rabbit hole one can get sucked into by taking it too far. I won’t make the standard rebuttals to radical skepticism, but instead I’ll make an appeal to ethics. I’ve written about this many times, two of which are here and here. It is basically a variation on Clifford’s paper The Ethics of Belief.

The short form is that good people will do good things if they have good information, but good people will often do bad things unintentionally if they have bad information. Thus it is an ethical imperative to always strive for truth and knowledge.

I’ll illuminate what I mean with an example. The anti-vaccine people have their hearts in the right place. They don’t intend to cause harm. They actually think that vaccines are harmful, so it is the bad information causing them act unethically. I picked this example, because it exemplifies the main problem I wanted to get to.

It is actually very difficult to criticize their arguments in general terms. They are skeptical of the science for reasons that are usually good. They claim big corporations stand to lose a lot of money, so they are covering up the truth. Typically, this is one of the times it is good to question the science, because there are actual examples where money has led to bad science in the past. Since I already mentioned Neil deGrasse Tyson, I’ll quote him for how to think about this.

“A skeptic will question claims, then embrace the evidence. A denier will question claims, then deny the evidence.”

This type of thing can be scary when we, as non-experts, still have to figure out what is true or risk unintentional harm in less clear-cut examples. No one has time to examine all of the evidence for every issue to figure out what to embrace. So we have to rely on experts to tell us what the evidence says. But then the skeptic chimes in and says, but an appeal to authority is a logical fallacy and those experts are paid by people that cause a conflict of interest.

Ah! What is one to do? My answer is to go back to our starting point. Science actually works for discovering knowledge. Deferring to scientific consensus on issues is the ethically responsible thing to do. If they are wrong, it is almost certainly going to be an expert within the field that finds the errors and corrects them. It is highly unlikely that some Hollywood actor has discovered a giant conspiracy and also has the time and training to debunk the scientific papers that go against them.

Science has been wrong; anything is possible, but one must go with what is probable.

I said this post would be a mess and brought up philosophy of math at the start, so how does that have anything to do with what I just wrote? Maybe nothing, but it’s connected in my mind in a vague way.

Some people think mathematical objects are inherent in nature. They “actually exist” in some sense. This is called Platonism. Other people think math is just an arbitrary game where we manipulate symbols according to rules we’ve made up. I tend to take the embodied mind philosophy of math as developed by Lakoff and Nunez.

They claim that mathematics itself is purely a construct of our embodied minds, but it isn’t an “arbitrary” set of rules like chess. We’ve struck upon axioms (Peano or otherwise) and logic that correspond to how we perceive the world. This is why it is useful in the real world.

To put it more bluntly: Aliens, whose embodied experience of the world might be entirely different, might strike upon an entirely different mathematics that we might not even recognize as such but be equally effective at describing the world as they perceive it. Therefore, math is not mind independent or even universal among all intelligent minds, but is still useful.

To tie this back to the original point, I was wondering if we would even recognize aliens as intelligent if their way of expressing it was so different from our own that their math couldn’t even be recognized as such to us. Would they be able to express true knowledge that was inaccessible to us? What does this mean in relation to the ethics of belief?

Anyway, I’m thinking about making this a series on the blog. Maybe I’ll call it RRR: Random Running Ramblings, where I post random questions that occur to me while listening to something while running.