## Thursday, October 20, 2011

### "You can't prove this"

The first subject I studied was financial mathematics. It was mostly mathematics with some economics attached which most of us found ridiculous. And rightfully so, as the year 2008 has shown. Anyway, mathematics is all about proofs.

(1)
A mathematical proof is perfect. You have assumptions and definitions which are all written out clearly. And then you have conclusions. The final conclusion proves what was to be proven. A simple example using the definition of an even number and the distribution law (an assumption) is this.
Consider two even integers x and y. Since they are even, they can be written as x=2a and y=2b respectively for integers a and b. Then the sum x + y = 2a + 2b = 2(a + b). From this it is clear x+y has 2 as a factor and therefore is even, so the sum of any two even integers is even. Quod erat demonstrandum.

After my first degree I decided to switch to physics. And physics was some kind of shock. Those physicists had a completely different idea about what a proof is! They would state a hypothesis and then make an experiment. And if the experiment was consistent with the hypothesis they would say that they had proven it. In my opinion that was stupid. Nothing was proven - if anything the hypothesis wasn't refuted.

And I have been right. It's just that those physicists, of course, knew that they hadn't proven anything. It was just an imperfect term they used. This is the reason why Mathematics isn't part of the natural sciences, but part of the humanities. Mathematics makes (arguably arbitrary) assumptions and then researches all possible logical conclusions which follow. While this is often extremely useful for the natural sciences, at the end of the day, Mathematics is not grounded in the natural world, but only in the human brain.

If you want to find something out about the natural world you usually don't have the luxury of knowing all circumstances. You can't just make assumptions, because, well, these assumptions can be wrong. Is the woman you consider your mother really your mother? You can't prove this! Yes, you can make a DNA test and the scientists may tell you that the probability of her being your mother is 99.99% percent, but that's not a proof. It is a typical result of the natural sciences.

It is the kind of logic humans use intuitively to some degree. It can also be defined in strict mathematical terms - using some assumptions which can't be proven.

(2)
Back to MMOs. Let's assume you are the developer of a successful game with exactly one million players. It is January and you change one feature. The player numbers drop to ten thousand within one week and stay there for the rest of the year. You are bankrupt. Was the change of the feature the reason for your failure?

You can't prove this! It could have been anything! Maybe aliens invaded the planet at just this time and stayed undetected. One of the things they did before they left was to make exactly 990.000 players of your game stop playing. Can you prove that this didn't happen?

When discussing things like this, one often encounters people arguing that correlation isn't proof. And, of course, they are right. Just because there is some chronological correlation between you changing the feature and your players stopping to play, doesn't mean that this is necessarily the reason. It could be anything!

But if humanity had only ever accepted mathematical proofs and not used Bayesian Analysis - in an intuitive way - we would still live in caves. Sure, maybe your computer doesn't work the way it does due to the scientific models applied. Maybe the models are completely wrong. Maybe it is just a coincidence and tomorrow the computer will stop working. Maybe you are just dreaming. Can you prove that any of this is wrong?
N. Taleb's famous example of the goose says that the goose looks at her past and concludes that she will continue to live a well fed life until a natural death. The next day she is butchered and eaten. Sometimes you miss critical information. Sometimes the people with doubts are correct.

But fact is that our common experience has shown that while correlation isn't proof, it is still a useful hint. Yes, sometimes it is wrong. Sometimes there is a common cause you didn't consider. Even though the number of people drowning increases at the same time that they like to buy ice cream, ice cream doesn't lead to people drowning. It's the Summer.

Maybe the aliens didn't only manipulate your players, but manipulated you, too! They made you change this feature which didn't cause anybody to stop playing. And then they also made your players stop playing! And now you think that the change of the feature is the reason, while, in fact, there is a hidden common cause that you didn't consider! Stupid you!

(3)
I think it's pretty obvious what I am saying here. First, I don't have all the information of the universe - and if I did, I arguably needed to have a brain as large as the universe. None of us have all the possibly relevant information. Certainly the developer does not! The trick is to make good decisions facing imperfect information.

Second, I don't want to prove anything on this blog unless I clearly state it. I am specifically looking at you, Azuriel :). All I do is making educated guesses - just like everybody else.

Third, my insights as a blogger can be interesting for developers not only in spite of me having less information, but because I have less information. Insights from the outside are useful for human beings, because we tend to become absorbed in our environment and ignore obvious things. We aren't computers. There's something called psychology. More information doesn't necessarily improve our decisions. Metrics aren't necessarily good: they can make you ignore more relevant but less quantifiable information. And those recent page hits from Anaheim tell me that Blizzard employees actually enjoy reading blogs written by bloggers who have less information than they do. I like that.

1. Since Plato's time, mathematics was thought of as the only true science; no need to go insult it by associating it with the humanities.

Post hoc ergo proctor hoc is common in the blogging community. In the same thread, I can read that the WoW subscription drop in Cata was caused by and proof that Cata was too hard while others say it proves Cata was too easy.

P.S.
A mathematician, physicist, and computer scientist are asked whether all positive odd integers > 2 are prime.

The mathematician says: "3 is prime, 5,7 are but 9 is not. It is false"

The physicist say "3, 5, 7 are prime, 9 is not, 11 is, 13 is. Nine must be an experimental error. It is true."

The computer scientist says "3 is prime, 5 is prime, 7 is prime, 9 is prime ..."

2. It's not an insult, Hagu. It is the truth. :)

Mathematics are grouned only in the human brain. Not in the outside world. That's why, in the past, people considered it 'pure'. But it is also the definition of the humanities. In fact, mathematics is the most pure of all the humanities, which often require outside (non-human) sources.

The astonishing fact probably is that mathematics turns out so useful for predicting the natural world, although it is only grounded in the human brain.

A mathematician doesn't need experiments. He only needs his brain to work - and some paper and pencil to enhance his memory.

3. Oh, one great quote by Einstein concerning this topic is this:

"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

4. @Hagu:

You say: "Since Plato's time, mathematics was thought of as the only true science; no need to go insult it by associating it with the humanities. "

So being placed the humanities is an insult?

Placing mathematics in humanities might be weird (never seen it happen before), but insulting?

@Nils: Humanities dont just sit around looking at their bellybutton.

A Physicist would say that Kristine must be a Humanities Major.

A Mathematician would still be unsure.

The computer scientist would be confused, because he still doesn't know what females are.

=)

6. Well, human brain evolved specifically to predict and model real world, so it's natural that it does sometimes succeeds in providing useful predictions or models by matching bits and pieces of memory (in form of math, or any other) to actual observed world phenomena.

The model is imperfect, but it's good enough to dominate other species :)

Real world pretty much never gives us perfect data. But a lot of data is "good enough" to make a decision.

7. Proof, in this forum, can best be described as data. And when people make statements, they should be prepared to come forth with the data they have (truth is not in question as yet and is irrelevant as yet). On the flip-side, the audience should with hold all assumptions and instead ask questions or make challenges based on the "proof" brought forward.

Now that's a perfect scenario. In reality, we all read into each others words--its not an attack. It's a matter of misinterpretation most of the time, which is why we should ask questions and challenge ideas. The act of asking for proof, therefore, is necessary to arriving at credible conclusions.

In the absence of proof, the person presenting the case might not persuade the audience. This is also normal. The audience might let the person know they weren't persuaded. The person whose case wasn't adequate shouldn't be offended at a request for proof. Again, proof in this forum means data and before the trial, data is neither truth nor lies. It's just data which the person believes has a strong enough correlation with this thesis to offer as proof that it's correct.

We just all have to strive to be clear and not object too much when others don't understand. It's a normal part of the process of having a discussion.

Oh ...and I'm on the side that math is in fact the strongest tool we have to explaining our world and universe ;)

8. "It's just that those physicians, of course, knew that they hadn't proven anything. It was just an imperfect term they used."
Science is in a hard spot in terms of marketing. It doesn't prove, it disproves, and perhaps if we repeatedly fail to disprove we accept it, but do not call it proven. But people like definite messages and if someone else offers it, they'll take that. You might have heard about the 'vaccine debate' in the US, where a really terrible study showed some minor links between one specific vaccine and autism. It was shown to be poorly-done and no further studies supported it. But ask a scientist directly "do we know that vaccines do cause autism?" and he will likely say "we have no evidence for it" rather than the slightly less accurate, but much more convincing "yes."

9. Well, for WoW the whole idea of "proof" is not really critical. Perception is everything, so if something feels "right" or "wrong" this is more important than proving that it's right or wrong. Your choice to play or not to play will be based on the perception and not the proof.
This is why on the WoW forums the questions coming from Blizzard are rarely about numbers or data (they have it, and better than yours), but how things "feel". For classes, the question is always "is it fun to play?", the theoretical DPS from simulationcraft cannot answer this.
If people from Blizzard go around to read blogs, it's not to see your conclusions from incomplete data, but to see how you feel the game is, because it's this "feeling" which will make you decide to resubscribe (which, BTW, pretty much means you like the game).

10. Helistar, you make it sound as if I am using incomplete data. I don't think I actually do this all that often. Moreover, I'm not writing this blog for developers; I am writing it for everybody, including my own pleasure.

I also doubt Blizzard is just interested in our 'feelings'. I think they are interested in understanding these feelings. And that requires reasoning.

Finally, companies might have one official opinion, but employees don't necessarily share it. They absolutely read blogs for their own pleasure and to find good arguments so that they can convince their fellow collegues to not repeat stupid mistakes, like replacing every casual group content of the game with a random LFD. ;)

11. Just as a way of explanation, there are two reasons I usually bring "proof" up:

1) Adding subjective qualifiers like "I think" or "I believe" completely eradicates all pretense of objective claims and allows you to say whatever you want. People usually do not do so, probably because it makes an argument sound weaker, but it is more accurate.

There is something to be said for the "implied 'I think'" anytime a blogger writes something, of course. Then again, if it becomes the norm, it will be difficult to ever know when someone is actually making a claim supported by fact/reasoning or relying on modern-day sophistry, ala most US politicians.

2) It doesn't make much sense to me why someone would attach importance to some niche factoid and not even consider other, equally valid (opposing) niche factoids. "The B Team took over a game growing at 2 million subs a year and ground it to a halt almost overnight." I do not remember who said that, but is that really a cogent argument when, market saturation aside, TBC actually represented a substantial decline in growth compared to vanilla by the same token? Did WoW really stop growing because of Wrath and no other reason? Is the endgame even particularly relevant to growth at all when far less than 20% of the US/EU playerbase kills 1 raid boss?

The above is merely an example for illustrative purposes.

WoW has a lot of moving parts, and I believe we can all agree with that. And it is fine to say that LFD (etc) was a bad move that made the game worse. No doubt it had some effect one way or the other. But what doesn't make sense is to say that WoW lost subs because of any particular thing or not. A piece of design can be bad and yet WoW gain subs in spite of that. And vice versa (see: vanilla WoW)!

It is a pet peeve of mine when people use overall subscription numbers as some sort of barometer for the effects of individual changes without even bothering to intellectually justify the isolation of the one item from the possible effects of the others. I quit my home insurance job in 2007 and the housing bubble popped soon after. I quit my job at a Honda plant in October of 2008, and then auto industry needed bailed out. Coincidence? I think not!

Anyway... carry on. ;)

12. @Azuriel: I think people hold on to proof, especially when talking about WoW, because blizzard divulges so very little of it. The spin machine is a monster and its all they use to communicate with players.

13. There is a huge difference between:

1) I made a change and my sub dropped by 990'000 player. It must be the change.

and

2) I would assume that making this change will drop my player base by 990'000. Let's make the change and see if that's correct.

14. Hey look, Azuriel missed the point again, even though you gave him a hint this time. here's a bigger hint:

Something one could derive from Nil's post is that the exclusive division of communication into subjective opinion and mathematical proof is a false dilemma.

I don't expect you to be able to understand that Azuriel, if you could, you would have stopped being so annoying years ago; but that's the issue here.