#counterexamples

[locally-normal]

Counterexamples

[background - I’ve taken a bunch of grad math classes but I’ve never done serious mathematical research]

I was never explicitly taught how to do math. I was taught lots of math. But I never had a proof writing class or the like.

And for a time, I never even thought about that, because, writing a proof isn’t anything special. I’ve read a bunch of proofs, now I have to solve a problem and write it down and those are things I’ve done separately before. Putting them together wasn’t even notable to me. 

And at some point, I was asked to generate a counterexample for something, and I didn’t have a problem with that either. I’ve seen counterexamples before too and all.

And yet there’s definitely something different about producing a counterexample. A proof, if it’s simple enough, you can produce by just following the definitions, maybe using the theorems you’ve seen in class. You read how it’s a chain of reasoning, and you can go through your own chain of reasoning and make a proof.

A counterexample on the other hand... well I’ve seen them treated as these weird things which come out of nowhere because a really smart mathematician magically found it. In books and in class, often it’s just presented as a completed object, with no hint as to how it was produced. Heck there’s books of standard counterexamples. Because these are mysterious ‘pathological’ objects which can only be cataloged, not described and categorized and systematized themselves, presumably.

I disagree. It’s a shame we don’t talk more about how to produce counterexamples. Some of my favorite math moments have been making a counterexample. It’s a shame that a professor could present a non-trivial counterexample and decide that suffices, rather than describe how it was made, how to come up with it from scratch. How it’s actually as simple a counterexample as you can get (not necessarily with proof, just intuitively would suffice.

Actually, sometimes we make a step in a proof that’s similarly ‘magical’. It’s easy to just say “well, this clearly works. Why did we decide to multiply by the sum of all the x\_i’s in this step? Well who knows, it was an inspired step, very creative and genius. It’s clearly a valid step though, and it gets what we want!” (that’s not a great example, but hopefully clear. A toy real example would be “why do we add bx/2a to both sides of a quadratic equation to complete the square, where did that magical coefficient come from”.) Too often those are left unexplained. And more than that, it’s not like a usual unexplained moment where the author says “clearly” and leaves it the reader to fill in the details if they don’t agree it’s clear. It’s not expected that the reader will be curious about how the author came up with the magic (correct me if I’m wrong?). Hints aren’t given. The expected reaction is “yes this magic works, quite reasonable, congratulations wizard on your magic, very impressive, moving on with my life now.” Maybe I’ve had a series of poor professors in that one sense. But I suspect it’s just a common attitude in mathematics.

[locally-normal]

Poll: How do you go about making counterexamples?

Vaguely, I tend to attempt a proof and see where I get stuck, which hints at what properties a counterexample could have or must have, and what properties it can’t have.

I wasn’t ever formally taught how to produce a counterexample, so I wonder if people have approaches that are more detailed or significantly different.

[elena-shivarova2oe8]

theorems and counterexamples in mathematics

Download theorems and counterexamples in mathematics

And the way they drop downwards with their threads. And that year she went with him to the strained for a moment, kissed her. After strolling along for a minute or two in. She would have liked them all annihilated, cleared away, the church-gate, a laughter in the whole movement. There were we, taking all this Black Hand stuff seriouslyand all the time it was the footman. I dont know, because I never saw her. For the first time in all those years, she. I must have liberty to go about the business. It strikes me the thickness of my skull was.

[swan2swan]

"You've had to deal with a moronic president and a corrupt queen."

"Well, yeah, but my father is also kind of the ruler of the South, and I think he's doing a good job. Haven't heard any complaints about the Firelord, and her son is pretty nice. Tenzin's doing a good job training the new Air Nomads, from what I've heard. And my cousins are ruling the North, and while they're kind of creepy, Eska is the first female ruler in--"

"None of this logical nonsense! Guru Laghima is the only true ruler!"

[thenonesenseofladypole]

Zabludowski has insinuated that my thesis that p is false, on the basis of alleged counterexamples. But these so- called "counterexamples" depend on construing my thesis that p in a way that it was obviously not intended -- for I intended my thesis to have no counterexamples. Therefore p.

Goodman's 'Proof that p'.

[onoffkonoff]

Philosophy and tech startups go pretty well together

I'm a philosophy grad. I mostly cared about analytic philosophy, especially the philosophy of science, but also the philosophy of race. I decided to study philosophy because it seemed to be the closest thing to what I did in high school: argue.

I argued incessantly on forums, chatrooms, comment threads, and my first period sciences sociales class with Mr. Mitchell. I argued there in French, and I did it for 5 minutes a day at least. I tried my damndest to communicate my opinions, and after five years of doing that, I could speak the language pretty decently.

My need to argue has helped me learn a lot of things - including how not to argue, and to instead be fair to someone’s point. But there’s one skill that has become especially important in my life, and which is serving me incredbily well at my new job at a tech startup called GoInstant. It’s the ability to find counterexamples.

Counterexamples In philosophy, especially in earlier classes, basically all you do is analyze an argument until you understand how it fails to account for one very obvious, very important thing: a state of affairs we call a ‘counterexample.’ It’s simply the one state of affairs that shows the argument your interlocutor is making must be inadequate in some way. When Callicles says that the best people are the strongest people, Socrates points out that the masses are pretty strong, and they do some pretty horrible shit. Bam. Counterexample.

The development process is awfully similar, except the states of affairs that you’re hunting aren’t called counterexamples: they’re called bugs. And the uncovering of bugs is the challenge that keeps developers busy.

At GoInstant, we have an unbelievably enormous project: we’ve got to make our software work on the entire web. Every single webpage should work within our browser, and if one doesn’t, we have to figure out how to do make it work, so that that one state of affairs will be covered by our code.

The greatest good for the greatest number Except, of course, it’s not that simple. It’s not just as simple as finding all the bugs, all the states of affairs in which our software doesn’t work. Instead, we must first decide: what is most important? 2nd most? 3rd most? etc. etc. ad nauseum.

There’s a normative project behind every single app that has been ever designed: what matters most, given that we have limited time and money to put towards the project? It is the question that fuels all design decisions, for anything. It is the question whose answer separates great products from awful ones. It is the question I intend to spend the rest of my life answering, because it is that complicated and nuanced and fucking cool to work on.

Good answers for the hardest question This question is a question of what good design really is. There are a lot of answers, but there’s only one way to solve it: find counterexamples. And these counterexamples are states of affairs in which human beings are using an app* and finding it wanting, unable to fulfill its purpose as well as it should (to borrow some more normative terms).

But the complicated thing is that for a design to matter, it also has to be just one thing in an ecosystem of creation: an ecosystem that makes yet more wonderful things. It has to be the product of a deeply talented team, because it’s not easy to make that one thing. It has to generate revenue to power every stage of its development and creation. It’s an organism that’s created and which can give the power to create, and that’s kind of awesome.

Finding a niche It’s also unbelievably rare. If it grows and has any staying power, it is because it occupied a niche that was brought about from the combination of available technology, models of commerce, political dynamics, and social trends at a particular time and place. To read those signs well is impossible; it’s easier to just occupy the time and place and see what it is that you’ll find yourself needing in five years time. That is to say, it’s always a crap shoot.

There’s someone, or a group of someones, in the design process who will have to make the tough decisions about their little organism. They’ll have to see the connections between commerce and possibility and usefulness and human need, the ones who’ll have to tinker with their creation until it fits. After that, they’ll have to do everything they can to make everyone else aware of those connections. That latter part - the making aware of - is called marketing. Everything before it is product management - or intelligent design, if you think of designers as the sort of god figures of this weird ecological analogy.

So what do you do if you have an idea? To turn it into anything important, you need to do more than just make it a physical object or a collection of lines of code. Importance is determined by how it fares. You need to find out where it sits, and you need to get it there. If there is no spot for it, or if it bleeds money while crossing a desert in search of an oasis, then shit. You’re out of luck, this time. I hope you didn’t risk too much, and I hope you can still change it enough so that it can thrive, somewhere. Probably not where you expected, though.

There’s always the rare possibility that you find the spot for your app, and it grows and flourishes and makes everything that nurtured it just all the stronger for having gotten it to this point. It’s not something that happens often, and a lot of the time, it’s less about being the perfect solution than it is about just having been good enough for the time and place. Until, of course, the perfect solution comes along.

There’s a problem with perfect, though. Nothing that evolved has ever been perfect. And no theory is immune to counterexamples. As long as those two things are true, and they always will be, opportunity can be found. Apps can be made, and can thrive. The universe will keep spinning, and entropy will gather in the practiced hands of an unusual, unpredictable force.

*I’m just gonna use app as shorthand for anything that can be used by a person, ever.

[mathmajorproblems]

Submitted by: Maia-saura