Hacker News Reader

> I think this emphasis on precise definitions is actually not that helpful.

To the contrary I believe that it is very helpful. One cannot apply logic or any other highly formalized method when he lacks definitions. It is hard to find a good definition, but at the same time it is the most important step in creating a theory.

> I think that the guy the Socrates is examining is actually doing a better job of having someone learn what courage is by giving examples than Socrates is by trying to state a precise definition for it.

Yes, it is a very helpful technique. When scientist needs a definition for a term, she starts with observing different cases. (Or rather she'd start with reading what other wrote on topic, but let's ignore this in the name of clarity). But she'd do it not like Laches by proposing examples, and not like Socrates by proposing counter-examples, but like they both: seeking for examples and then seeking for counter-examples.

> Although not precisely like an ML neural net, our brains are enough like one that that this is likely to be similar as well.

Our brains also very different from an ML neural net. Because we could dig a problem, construct a formal description of it, and then devise something like E=mc^2. We can generalize experience, while ML can only interpolate.

Humans don’t learn from abstract definitions: I agree very very much with that.

The definition comes later. When people learn math concepts, the definition is in fact often of surprisingly little value. They need to see examples, make examples, play around with instances and applications of the “thing”, to gain intuition and finally come back to the definition.

The abstract definition is also often (usually? always?) something that the person who came up with it also only came up with after finding the abstraction of otherwise concrete things they were working with.

Try understanding Fourier transforms, or worse, Laplace, from their definition alone. Unless you have an idea of what’s happening already, it’s extremely hard. Work with the concept for a while, and the definition appears laughably simple!

Once you did that, look up the definition of a Hilbert transform. What is that?! But in practice, it’s just “multiply all complex frequencies by j” or, much more intuitively: “rotate all phasors by 90°… i.e. put them on their side”. Unlike with the previous example, you might happily work with this one and use it a lot, and still be mystified by its definition.

And, yes, I only finally understood and later fully embraced Monads by “just using them” for a while. Now the concept is clear and they are everywhere. Use them even more, and you run into situations where Monads are not enough in practice. And now you’ve made the first step towards Arrows!

On the other hand, I hate it when people take this mentality to the extreme and withhold the mathematical definition and insist on only using examples or "simplified" definitions. It's similar with language learning. Sometimes I just want to see the grammar rule (there often is one, at least to a very good approximation), but the teacher has this didactic philosophy that I wouldn't be able to "handle the truth" or something. If there is an adequate definition provide it. It's similar with computer science papers as well. On the one hand, a free-text description of an algorithm or a simplified pseudocode is helpful to get the gist, but very often I also need to see the actual unambiguous code to see exactly how it is and to check if I understood it right because no description covers all the corner cases.

Both are important, practical examples but also the abstract definition (if one exists to a good approximation).

That's also bad, yeah. We came to the abstract definition because it has value after all. (I think) OP and me just stated that it's not necessarily useful or sufficient to try to understand starting from the definition, but I do think there is still value in even just stating the definition already, for reference.

Luckily I can't remember encountering that didactic approach in math or language learning, but I'm not surprised at all if it exists and is somewhat widespread.

That was kinda Socrates' schtick: demand a definition, and then show that it wasn't a good one. Which was true, but not really all that useful.

The best I can say for it is that it shows the way people think they have definitions of things, but they don't. Push on pretty much any nontrivial definition and it will fail. That observation, at least, is kind of interesting, in that it shows the limits of the way we think about stuff.

But we do think about stuff, and both bio and silicon neural nets do a pretty good, messy job of it. If Socrates had pushed harder on that, he might have gotten some real work done about the nature of thought. Instead, he always just struck me as being kind of a jerk, rather than contributing anything.

The oracle at Delphi would beg to differ! ;) I take your point though, sort of… It’s maybe worth pointing out that we only know about Socrates from secondary sources: https://en.m.wikipedia.org/wiki/I_know_that_I_know_nothing You could maybe think of him as playing devil’s advocate, or employing ‘reductio ad absurdum’ methods (even moving towards a precursor to some sort of scientific method perhaps?) …or maybe as employing sometimes useful methods of liberating people from irrational superstitions/unexamined beliefs, which eventually helped in some way to lead us to more modern ways of thinking? Have a read of this if you fancy, see what you think: https://users.manchester.edu/Facstaff/SSNaragon/Online/texts...

I think he contributed something quite valuable - he pushed people into what we might call 'beginner mind', without which learning is impossible. His main thesis (as recounted by Plato) that any certainty is false certainty and that the only real wisdom comes from acknowledging one's own ignorance seems valuable indeed to me. I wonder what he would have made of today's statistical methods that try to bake that uncertainty in with a kind of recursive meta-certainty.

I partially agree with your initial statement. I think precise definitions are only helpful in certain contexts. In general they might not be necessary or even useful, but precision is useful when you're trying to examine your own beliefs and the beliefs of others.

Consider this: the soldier being willing to defend against an enemy when he doesn't fear death isn't necessarily displaying courage. This particular soldier could have a death wish for all we know.

If I posed this scenario, then maybe you'd dive a little deeper and say something like "Most people are afraid to die, so if you were a soldier that didn't want to die, and you choose to defend your post when confronted by an enemy that can kill you, that would make you courageous."

By probing for specifics, we learn a property of courage. We might say something like "Courage is the ability to act against the compulsions of fear." Whether that's satisfactory or not is another issue, but it allows us to have a better understanding of what we're seeing in the world. It gives us a better example of acts of courage versus acts of overconfidence, inebriation or apathy.

The downside to this definition is, if someone approached me in public and asked me to defined courage; if I used this definition, I would probably need to follow it with an example.

The reader presumably already knows what courage is, and does not need to be taught that. Rather, Socrates/Plato probably intended to guide us toward the eternal Form of courage. Laches' approach would indeed be better for teaching a neural net, but neural nets are not Socrates'/Plato's intended audience.

The reader already knows? Well, what is courage then? Can you explain it?

When I said "the reader", I meant other readers besides me. I am not a knowledgeable man, in fact just about all I know is how little I know. But as you are one of those other readers besides me, maybe you know what courage is, if so then please teach me :)

Off topic, but the specific content of Laches’ reply is interesting of itself for how much it says about Greek culture and war making.

It’s a very different reply than you might expect to get from, say, a Sioux warrior, or a European knight, or a samurai, etc. Or from the Greeks themselves, outside of this fairly narrow band of history!

We can argue that ML neural net does not know what a cat is.

We can also argue that a cat doesn't know what a cat is.

Humans believe they know what the human definition of cat is, but that doesn't mean there's a definitive ideal definition of cat that humans have privileged access to.

Some cats - like Schrodinger's Cat, and this cat - aren't cats at all. They're rhetorical devices about cats.

Which is convenient, because they don't need food or a litter box. (Although they can be messy in other ways.)

The point being that even a supposedly simple concept like "cat" is much more than the sum of some nominal properties and actions which are easy to enumerate. ("Has four legs, purrs.")

Both the ML and human "definition" of cat - which is more like an open list of weighted cat-related experiences - is much closer to defining what a cat is than the intersection of all the properties that supposedly define a cat completely.

In other words Plato was absolutely wrong. There are no Platonic ideal forms. (Even if there were - where would they live?)

There are only probability density fields in qualia space. And when there's enough qualia density in one location you can start to suspect that you're interacting with a cat/chair/philosopher/ML researcher/whatever, and then draw on your behavioural model of each object to predict what will happen as you continue to interact with it.