Game (theory) on trust

[Nazzzgul666]

Just found this little game, it's really more about showing game theory in a playful way than an entertaining game, but i found it educational and if somebody has some thoughts about it, maybe we can get a discussion.

[Content Consumer]

Interesting...

Generally I don't particularly agree with people who attempt to apply mathematics to human interaction, primarily because such theories tend to make sweeping generalizations and preconceived assumptions about how and why people behave the way they do. I was all set to get irritated when I saw:

"Of course, real-world trust is affected by much more than this. There's reputation, shared values, contracts, cultural markers, blah blah blah."

and

"...as game theory reminds us, we are each others' environment. In the short run, the game defines the players. But in the long run, it's us players who define the game."

The maker seems to be at least aware of this, even if he only put part of it in as a footnote. Ah well.

 

One thing that struck me was this comment though:

"I think our modern media technology, as much as it's helped us increase communication... has increased our miscommunication much more."

Speaking wholly from personal experience (and attempting to use personal anecdotes as evidence - please don't hate me too much ), this tends to be true. Partly because "modern media technology" generally means text-only communication, which is very open to misinterpretation and invites pedantry which often seems insulting.

[Nazzzgul666]

I agree on you with pretty much anything.

From my experience, studying math and reading a lot about sociology, economics and some other stuff... mathmaticians usually know very well that their models have limits, where those limits and (potentially) weak points are, etc. But when those models get outside the faculty... i have the feeling it's more often than not that somebody who doesn't understand the model takes it, makes assumptions about it's use, then uses it with this assumptions and gets some kind of results he expected. But in those cases, the difference between a mathematical model and reading the future from bowels is often smaller than the "scientific" results and reality. Therefore, your anecdotic evidence might not be worth much... but it's still as good as a bunch of big studies. And at least you know that this is vulnerable.

 

And yes, i'm talking about you, economists, psychologists and social scientists. Physicists at least have usually a good idea how those models work and what they can be used for or not, even if they don't understand all implications.

*edit: I have to admit that mathmaticians aren't entirely innocent that our work is not properly used. One of my profs told me that it started during ~1925-50 that the language used in math became more and more complicated until the 80ies-90ies when even physicists listening to a mathmatician about a talk of the very own special field of those physicists, they didn't understand it anymore. Since then they try to revert that somehow, but with limited success. It's not only all the other guys who have to work on that problem, but they should at least accept that their work might have quite limited value.

 

When reading that modern media part my first thought was "Twitter" because of the lengths restriction and i'm not sure if i should consider forums - or even text in general - modern.

I think smilies can do a pretty good job to improve communication, compared to a old school letter without them for example. But that's the mathmatician in me... there are good reasons that we insist on careful definitions of everything before we talk about it. Even 1+1=2 depends on the definition/the system we're talking about. Your CPU would only ask "2? What's that? I only know 1 and 0, and 1+1 obviously equals 10."

I also could imagine that our miscommunication just increased because our communication increased, because we have - or take - less time for a single conversation. Nothing beats a face-to-face talk, but even there miscommunication happens often enough. And i often blame haste. 

 

 

[zzz72w3r]

Technically this game is more behavioral economics than game theory.  Game theory deals with situational outcome while behavioral economics try to solve problems that traditional economics has failed because human behaviors are not always rational. 

 

Take this game for example.  If the game is framed such a way that: "Someone will gain 3 if you are willing to give 1 and you could gain 3 as well if someone reciprocate the kindness but there is no guarantee" then the outcome with large enough samples will surely yield far more win-win outcomes than not, thus creating a virtuous cycle mainly because 3:1 gain is ridiculously high net positive outcome within a rational framework.  However, by changing the question to "You could gain 3 by losing your own 1 only if someone else is willing to lose one without guarantee of return." then the outcome will yield far less "trust" even if the society net-gain is still the same ridiculously high return.  The math is the same but how it is processed changes the result.

 

This actually has been tested in the real world.  Organ donation is definitely very net positive for the society but hardly rewarding for the donors.  After isolating donation statistics of culturally and economically similar countries where the variation can be as high as 90%+ and as low as single digits, it turns out that the difference is opt-in vs. opt-out in the drivers license. 

[Nazzzgul666]

I don't know what you mean. It's just a slightly variation of the default showcase prisoners dilemma. Game theory deals with situations where the decisions of several different people influence (the results) of each other. And yes, of course you change the result if you change the process. The game shows that for several conditions, including gain. You can change quite much at the last panel to try and test if you want.

[zzz72w3r]

Game theory predates and begets the much broader behavior economics but I would put this exercise more in the scope of behavior economics as its math is tilted for a specific outcome (3:1).  

 

EDIT: So in classic prisoner dilemma, the math is 0,0 or -1,+1 hence the dilemma between the two players.  On the other hand, as in the case above, the outcome is clear -1,3=2, yet in aggregate it is much lower than 2, figuring out this irrational behavior of what should be a simple rational decision is behavioral economics.

[Nazzzgul666]

Game theory predates and begets the much broader behavior economics but I would put this exercise more in the scope of behavior economics as its math is tilted for a specific outcome (3:1).  

I don't know how or why you come to the conclusion that behavior economics is somewhat broader than game theory. Maybe i should be more explicit and say mathematical game theory, which is a bit misleading because it Isn't that much about games. Behavior economics may use game theory, and so do games, but by no means game theory is limited to those two.

Behavior economics makes assumptions about humans, game theory doesn't. 

And the game isn't limited to 3:1, you can change it later. 3:1 is just one example to explain stuff which is not related to the gain, and changing more than one variable while explaining only one of them would just be confusing. 

[zzz72w3r]

 

Game theory predates and begets the much broader behavior economics but I would put this exercise more in the scope of behavior economics as its math is tilted for a specific outcome (3:1).  

I don't know how or why you come to the conclusion that behavior economics is somewhat broader than game theory. Maybe i should be more explicit and say mathematical game theory, which is a bit misleading because it Isn't that much about games. Behavior economics may use game theory, and so do games, but by no means game theory is limited to those two.

Behavior economics makes assumptions about humans, game theory doesn't. 

And the game isn't limited to 3:1, you can change it later. 3:1 is just one example to explain stuff which is not related to the gain, and changing more than one variable while explaining only one of them would just be confusing. 

 

 

Not sure if you read my earlier EDIT before your reply.  A large number of behavioral economics problems can be based in game theory, which this particular game is, but the application of it is a behavioral economics problem in that we want to promote trust and why we are not getting more.

 

EDIT: Sorry, again with anther edit.  Behavioral economics does NOT and cannot make assumptions about human behaviors.  If it does then it's psychology and not economics.  The reason that it is so prominent now is precisely in its ability to mathematically shown irrational behaviors in what is supposed to be a rational framework, which is what this game was designed for.

[Nazzzgul666]

That human behavior is supposed to be rational is an assumption, afaik pretty common in economics. Math doesn't do that. And in general it doesn't deal with specific numbers. The math for the classical prisoners dilemma is a:b, with a,b element of |N. You confuse math with calculating, but that's not the same.

[zzz72w3r]

No, I am not sure you dug at what the game you posted was for.  So here is my shitty attempt in explaining why this game is behavioral economics and not game theory: 

 

In game theory, rational behavior MUST be assumed.  Thus by changing the value, the outcome also needs to change.  In prisoners' dilemma we have two players choosing 0,0 or -1,+1.  Now if we change the math to choose between 0,0 or -1,+2 then the outcome should be much more predictable.  That the outcome of prisons' dilemma may not be 50/50 between 0,0 and -1,+1 is because humans are not rational.  This is where game theory stopped.  It says irrational behaviors cannot be quantified. 

 

In this "trust" game, the math is 3:-1 which should according to classic game theory yield an average with large numbers of players willing to play thus yielding an average closer to 2 and the theory does not care nor know how to explain how the reality could differ.  Again, this is where game theory stopped in that irrational behaviors is just that and the difference is no longer a math problem.  How can anyone rational choose not to play a game with payoff of 3:-1?  Not in behavioral economics, what if by changing how the math is processed, the outcome also changes.  In a classic behavioral economics technique, displaying the 3 before informing the -1 could yield an aggregate average closer to 2 than showing -1 before informing about the 3.  So you think this is just psychology but it's not.  This is quantifiable and mathematically modifiable behavior. 

 

EDIT: I don't think the people who put up that page actually know what that test was designed for.

[Nazzzgul666]

No, I am not sure you dug at what the game you posted was for.  So here is my shitty attempt in explaining why this game is behavioral economics and not game theory: 

 

In game theory, rational behavior MUST be assumed. Nope  Thus by changing the value, the outcome also needs to change.  In prisoners' dilemma we have two players choosing 0,0 or -1,+1.  Now if we change the math to choose between 0,0 or -1,+2 It is impossible to change the math by changing numbers. The math is a,b, not 0,0 or any other number. then the outcome should be much more predictable.  That the outcome of prisons' dilemma may not be 50/50 between 0,0 and -1,+1 is because humans are not rational.  This is where game theory stopped. That's where it begins.  It says irrational behaviors cannot be quantified. How do you came to this?

 

In this "trust" game, the math is 3:-1 It's not. The math is still a:b which should according to classic game theory yield an average with large numbers of players willing to play thus yielding an average closer to 2 and the theory does not care nor know how to explain how the reality could differ. It doesn't have to, because it doesn't care about any numbers.  Again, this is where game theory stopped in that irrational behaviors is just that and the difference is no longer a math problem.  How can anyone rational choose not to play a game with payoff of 3:-1?  Not in behavioral economics, what if by changing how the math is processed, the outcome also changes.  In a classic behavioral economics technique, displaying the 3 before informing the -1 could yield an aggregate average closer to 2 than showing -1 before informing about the 3.  So you think this is just psychology but it's not.  This is quantifiable and mathematically modifiable behavior. 

I've never said this game is about psychology. I said economics is about psychology, and the game doesn't have anything to do with economics. You may draw some conclusions out of this which may be useful for economics, but that is not what the game is about. I don't think i want to continue this, you don't answer my questions, you don't understand that math is not calculating, and you don't seem to care.

[zzz72w3r]

Whatever you want, it's your thread.  Economics is not psychology and there is zero conflict between seeing this exercise as game theory or behavioral economics. 

 

Have fun.

[Bazinga]

No discussion from me. Nice little interactive presentation there, and quite informative too. Hope the math is solid. It obviously has its limits when it's applied to messy stuff like economics and sociology but I have no problems with that as long as they do proper science. But many sociology papers at least seem to suggest the opposite, constantly confusing correlation with causality etc.

 

I didn't understand the "simpleton" player though, they didn't describe that one in a way that I can grasp.

[Nazzzgul666]

No discussion from me. Nice little interactive presentation there, and quite informative too. Hope the math is solid. It obviously has its limits when it's applied to messy stuff like economics and sociology but I have no problems with that as long as they do proper science. But many sociology papers at least seem to suggest the opposite, constantly confusing correlation with causality etc.

 

I didn't understand the "simpleton" player though, they didn't describe that one in a way that I can grasp.

Yeah. Simpleton isn't that easy, his description at start confused me too, but later i got it. 

Ok, he starts with coop, but he may slip and drop the coin, accidently cheating (later he can plan to cheat but accidently give the coin).

If you cooperated, he repeats his last move even if it was an accident. 

If you cheated, he does the opposite of his last move, even if it was an accident.

Remember that he reacts to the round before.

 

As an example: he plays with copycat, both start coop, nobody slips.

round 2: copycat copies simpleton, gives coin. Simpleton plans to repeat his own move (coop), but slips and cheats.

round 3: copycat copies the (accidential) cheat from round 2. Simpleton copies his own (accidential) move from round 2, he cheats.

round 4: copycat copies cheat. Simpleton does the opposite of his own move, he ccoperates.

round 5: copcat cooperates. Simpleton does the opposite of his own last move, he cheats.

round 6: copycat cheats. Simpleton repeats his own move, he cheats.

round7-99: repeat round 4, 5 and 6 until another mistake happens and the order changes.

 

This game is still pretty basic game theory, the math is proven and solid. The doubt that remains is: what happens if there are other personalities (or other conditions) not covered?

For example me, the very (un)forgiving copycat? Cheat me once and i assume a mistake. Cheat me twice (in a row) and i still assume a mistake. Cheat me three times and i you'll have to coop something like 10+ times before i give you a little trust again. Or more likely i'll stop playing with you.

 

That you have covered all personalities can never be proven. Probably one can prove that you can ignore most personalities because they don't make a difference (in the long run), but to prove that there is no personality that is superior to the forgiving copycitten is like god... you can't prove that such things don't exist. At least i think so, i'm no expert in game theory, but denying existence is difficult in general even if humans aren't involved. For example you can mathematical prove that a piece of code does what it is supposed to do, but you can't prove that it doesn't do anything else.

[Bazinga]

Well at least some types that are important to most people are covered. For example a lot of people I know probably think that the cheating type will always be successful. Especially in Eastern Germany that sort of bitter prejudice is very common, "you have to be a crook to get rich", you know.

It's reassuring that math doesn't support that claim, even though cheaters are successful in the short run. As seen in the diesel scandal that's stirring up Germany atm. Short term profits increased but now that it came out they basically burned all the money they earned in the last years, and not just the plus they got through cheating.

 

... Well, I hope people won't hold it against them for too long, the cars are still pretty solid, even though the people designing them are (or were) cheaters.

Not that my job depends on that, but Germany and the cars built here, they often seem like companions in fate. Probably millions of jobs in that sector.

[Nazzzgul666]

Well at least some types that are important to most people are covered. For example a lot of people I know probably think that the cheating type will always be successful. Especially in Eastern Germany that sort of bitter prejudice is very common, "you have to be a crook to get rich", you know.

It's reassuring that math doesn't support that claim, even though cheaters are successful in the short run. As seen in the diesel scandal that's stirring up Germany atm. Short term profits increased but now that it came out they basically burned all the money they earned in the last years, and not just the plus they got through cheating.

 

... Well, I hope people won't hold it against them for too long, the cars are still pretty solid, even though the people designing them are (or were) cheaters.

Not that my job depends on that, but Germany and the cars built here, they often seem like companions in fate. Probably millions of jobs in that sector.

Yes, to pretty much everything^^

The problem with those "you don't know if they exist" parts in complex systems like society is: let's say, there is a type that beats the copycitten and most others, not only slightly but he pushs it out of the game immediatly. Then it's also possible that the only one who can counter him is the grudge who always cheats. That's not very likely, and probably there are ways to prove that at least every type that would beat copycitten also beats the grudge. Such "dancing around topics/possible exceptions we don't really know much about" and increasing knowledge with small steps is what mathmaticians do when they fail with a prove for a big step. 

But this only works - at least the way it's done - because proves in math are based on logic, not on empirical data like all other science. A proff in math will never be (completly) reverted because once proven correctly, it's true for all time. You might find an example where it doesn't work, for example it's not totally impossible that you find a triangle where Pythagoras theorem does not apply. It's just quite unlikely because that would mean you've found a triangle no mathmatician since Pythagoras could imagine. But even then it's not completly wrong, you just have to add that this theorem does not apply to this kind of triangles you've found.

 

This happened for example for something you've probably learned in school: the shortest link between two points is a straight. Can't imagine where it's different? But astrophysics use that all the time: Stars bend the space, and the shortest way now follows this bent space which isn't straight anymore. This may not have that much influence on our daily lives on earth, but it's an impressive example where, after thousands of years, an important exception is found for something that sounded very simple and the exception created a whole new field in math.

 

About the car companies... imho it's the responsibility of our government to make sure the managers are punished, for several reasons. It would allow the public to go on and buy cars without grudge, it would prove the proverb that crime doesn't pay off, and afteral those guys tell us they get that much money because they are accountable. If that's not true, they don't deserve more than their secretary.

[Bazinga]

 

About the car companies... imho it's the responsibility of our government to make sure the managers are punished, for several reasons. It would allow the public to go on and buy cars without grudge, it would prove the proverb that crime doesn't pay off, and afteral those guys tell us they get that much money because they are accountable. If that's not true, they don't deserve more than their secretary.

 

*insert some quip about bankers*

 

I think you're spot on about the difference between math and science. Doesn't make the application of game theory on people interactions invalid of course, science just has a habit to cherry-pick the tools it needs and disregard the exact formalism which never stops to outrage mathematicians. 

 

Doesn't matter as long as they can use these tools to make predictions that can be falsified though.

Maybe the same experiments will have different outcomes in different frameworks (like other cultures, or 100 years from now, or any other hidden factor that might influence the results). Meaning you probably need to repeat the same experiments/studies over and over until the end of time no matter if you think you proved the related theories right or not. But if the results look convincing I'm inclined to accept them... for now.

[Nazzzgul666]

 

 

I think you're spot on about the difference between math and science. Doesn't make the application of game theory on people interactions invalid of course, science just has a habit to cherry-pick the tools it needs and disregard the exact formalism which never stops to outrage mathematicians. 

 

Doesn't matter as long as they can use these tools to make predictions that can be falsified though.

 

What makes me more concerned than outraged is how strong they often believe in their empirical data (although they dismiss the formalism). For mathmaticians those are nothing more than a clue, something that says "yes, that might be worth a look." There is a famous unsolved problem, the "Riemannsche Vermutung", something about prime numbers. After they found that it's correct for the first 300.000.000 prime numbers somebody was finally willing to bet two bottles of wine that it's right. But that's what those 300 million fitting tries are worth... not much more.

 

If you tell any other scientist that you want 300mio experiments and if every single one has the same result you'll give his thoughts a try... they might be more willing to accept our formalism.

Game theory is btw a subfield of math, that's why all the guys who think "Oh great, i'll learn how to earn money in Las Vegas." leave the courses screaming.

It depends on the university/the prof if you need at least to finish basic studies in (pure) math before you can visit one, but it's unlikely you understand it if you didn't. And at least at TU Berlin, math courses for economics/engineers don't really prepare you for that, might be different in other countries or even other universities, not really sure. You'll probably know most words, but without really understanding all the implications of the basics i think you'll miss half the content of game theory as well. And therefore the limitations and possibilities.

[Bazinga]

I don't know, math never seemed to be a field of science to me, more like the language used for physics, and mathematicians mainly flesh out the grammar and vocabulary. As described here for example.

Some get a bit carried away and create their own worlds instead of trying to apply it to ours, somewhat similar to linguists like Tolkien. ^^

And physics is the only real science there is.

At least in my mind everything else are just different levels of dilution, with chemistry being close to physics level of exactness, biology not too far but already full of questionable mumbo jumbo and other fields of study getting increasingly worse the more complex the systems they describe. With the human mind and human social dynamics being the least scientific in the bunch, it barely resembles proper science anymore except for the lingo.

Some of it can still be valuable though, at least if these experts know the limits to their theories.

 

or in the words of one of my biggest idols:

 

and about cargo cult science (maybe jump to 5 min in):

[Nazzzgul666]

I don't know, math never seemed to be a field of science to me, more like the language used for physics, and mathematicians mainly flesh out the grammar and vocabulary. As described here for example.

Some get a bit carried away and create their own worlds instead of trying to apply it to ours, somewhat similar to linguists like Tolkien. ^^

And physics is the only real science there is.

At least in my mind everything else are just different levels of dilution, with chemistry being close to physics level of exactness, biology not too far but already full of questionable mumbo jumbo and other fields of study getting increasingly worse the more complex the systems they describe. With the human mind and human social dynamics being the least scientific in the bunch, it barely resembles proper science anymore except for the lingo.

Some of it can still be valuable though, at least if these experts know the limits to their theories.

 

 

So i'm (actually taking a break after burn out) studying math, my sister is a junior prof for linguistics... you are good at making friends?

 

No, you may have a point, depending on how you define science. If it depends on experiments to you, math isn't one. I'm pretty sure Feynman would disagree, though.

Didn't read the whole essay, i think he already starts with a wrong assumption: math doesn't count as natural science, never did and i never met anybody who claimed that. Imho the german word "Geisteswissenschaften" fits much better than the english translation "Humanities" (didn't find another one?) because math doesn't care much about humans. But it's all in your mind, math does not need any relation to the physical world (but doesn't exclude it, of course).

If natural science is the only science you accept then math isn't science, it's a subfield of philospohy, but... so is everything else. The difference is how you prove things, and imho a proof that is true for all time* because it isn't based on empiric data is better than one that completly relies on such data. I think physics has the advantage that it not only uses both math and data, but relies on math a lot. You can know for sure that your data has to be wrong if it violates the rules of math. Your assumptions about which formula fits may be wrong, but not the laws of math itself.

 

It's true that to do math you have to learn the language, but that is... only a fraction of the basics. You can't do physics either if you don't know the meaning of the words, you need to know what exactly a frequency is before you can talk learn about it. In theory you could make up your own words, do your own experiments and come to the same conclusions as all physicists before you, but you would need to do so. If you want to read and understand what they are talking/writing about there is no other way then learning what their words mean first.

 

What doing math (in opposite to using it) is about is proving things right or wrong (and if it's actually possible to find a proof). And in doing so, the first thing you do is thinking about possible exceptions, or how to prove that there can't be an exception. And because we often deal with infinity, empiric data doesn't matter much. As an example the question "Are all prime numbers >2 odd?" There is an infinite amount of prime numbers, it is impossible to try them all. If you find one that is >2 and even you'd be finished, but only because you can't find an example doesn't mean there is none. You have to try another way, so you look at the definition of prime numbers and say

"well, what would it mean if there is an even prime number? Let's say there is a prime number a, and it's even. Then what happens if i calculate a/2? That would still be a natural number [add definition of even & natural numbers], but that conflicts with [definition of prime numbers], so that is impossible. Therefore, all prime numbers >2 must be odd."

 

The proof expected in the essay is similar, but this time you have an example. The word "all" is a trigger for mathmaticians, one example is enough to proof it wrong no matter if it's the only example. So you take, for example, 9 and divide it by 3. 3 is a natural number that is neither one nor 9 itself, therefore 9 doesn't fit the definition of a prime number and the assumption that all odd numbers are prime numbers is wrong. Q.e.d.

 

The joke in the essay ignores the existence of the proof and says mathmaticians "leave that to the reader." And yes, books or scripts for students of math often do that. But that's not how math works, a "good book" has a proof somwhere for anything it claims.  The only things which don't need a proof are definitions and axioms. The problem with that is that sometimes simple things are not simple to proof. I can't remember the exact number, but it took Bertrand Russel a few hundred pages to prove that 1+1=2 in our common system. And that's the short version, not easy to follow. You don't want to read that proof every time you add two numbers, that's why i put "good book" in quotes. At least i wouldn't want to read that book.

You have to make compromises at some point, but if you start claiming stuff that hasn't been proven at least somewhere you're not a mathmatician anymore and everybody with a minimum of dignity would call you out for that.

 

There is a "workaround" or however you may call it. The mentioned "Riemannsche Vermutung" is not only famous but important for some other conclusions, and some people don't want to stop there just because nobody could proof it during the last ~150 years right or wrong. So, they start their proof with "Assuming the Riemann Hypothesis is true/false..." and draw some conclusions from there. That's legit because it's totally clear that their conclusions stand and fall with that assumption, and it has the potential to actually proof it wrong. If one of your conclusions is that reality would have collapsed one year ago, you can tell that your assumption is wrong.

 

And that many mathmaticians don't care about "reality" or practical uses... well. The british mathmatician G.H. Hardy was very proud that his field, the number theory, had no connection to the real world. He called it the purest math because he could be sure he isn't under any pressure to do a certain research or even find a certain result because it might be needed in the industry. And it didn't matter until he died in 1947, but the crypto algorithms for RSA are based on his work and wouldn't have been possible without it. That nobody knows if or how parts of math could be useful yet doesn't mean it isn't.

 

*my example with Pythagoras theoreme might be proven wrong was misleading, that won't happen. Ever. There already is the limitation to triangles with a right angle, and to euclidian geometry , but within those limits you can not find a triangle where this rule does not apply.

 

And yes, Feynman is also one of my idols. Not that much as a mathmatician although he did some good work here, but definitly as a scientist.

[Bazinga]

Well, informatics are a field where math can often be applied 1:1, mostly because at the end of the day computers are deterministic machines that somewhat work like mathematical operators, they get binary input (so the input doesn't need to be filtered, abstracted or otherwise reduced), do something clearly defined with it and then give out some binary result.

They are completely deterministic and completely logical, or in other words math simulators.

So yeah, maybe every field of math can have useful applications, I kinda ignored the power of computers, forgive me.

 

But good luck trying to apply anything math related to something else (like even basic particle interactions) outside of such a closely defined man-made machine without using experiments or controlled studies.

Of course we can argue about definitions all day but for me the scientific process always includes guessing a theory, trying to make predictions with it and comparing these predictions to the real world through a controlled experiment. Always has been and always will be.

 

Feynman agrees with me btw:

God I love to hear him talk, he's such a stand up comedian and you still learn a lot when you listen to him. Incredibly charismatic speaker.

 

 

And you're right, there are 2 languages involved of course, math and the one we use in our everyday lives. Unlike many folks from the humanities like philosophers etc I don't believe that our own language makes science questionable (because culturally determined), the definitions have to make sense, you need to be able to formulate cause and effect relations and the grammar has to be logical but I'm pretty sure that those requirements are easy to fulfil and are universally given for German, English and most other languages.

 

@ about making friends:

Not that easy for me tbh, you might not guess it but IRL I don't talk a lot or even need a lot of human company. I have friends but I'm still a bit of a loner compared to most people.

Seems like I'm compensating something when I'm online. Should ask a shrink about it I suppose. 

[Nazzzgul666]

Well, informatics are a field where math can often be applied 1:1, mostly because at the end of the day computers are deterministic machines that somewhat work like mathematical operators, they get binary input (so the input doesn't need to be filtered, abstracted or otherwise reduced), do something clearly defined with it and then give out some binary result.

They are completely deterministic and completely logical, or in other words math simulators.

So yeah, maybe every field of math can have useful applications, I kinda ignored the power of computers, forgive me.

 

But good luck trying to apply anything math related to even basic particle interactions outside of such a closely defined man-made machine without using experiments or controlled studies.

Of course we can argue about definitions all day but for me the scientific process always includes guessing a theory, trying to make predictions with it and comparing these predictions to the real world through a controlled experiment. Always has been and always will be.

And you're right, there are 2 languages involved of course, math and the one we use in our everyday lives. Unlike many folks from the humanities like philosophers etc I don't believe that our own language makes science questionable (because culturally determined), the definitions have to make sense and the grammar has to be logical but I'm pretty sure that both requirements are easy to fulfil and are universally given for German, English and most other languages.

 

 

I still don't think he agrees with you, he talks about using math to do physics, not doing math. It is not easy to explain the difference, you actually have to do it yourself and that's why we math students have the option to visit courses or not, but ~70-90* hours per week of homework are mandatory. 

 

But i will try. Doing math means discovering, proving and formalizing the rules, how and when (not) to use them. Every child learns that a*b is the same like b*a. But it's not always true, that fits only if you're in a system called group. Matrices in general are not a group, order matters when multiplying them. Mathmaticians don't create any formulas to calculate stuff, they define if/when a formula is actually legit, and how you can calculate with(in) that formula. Imagine you want to calculate some stuff related to speed. So the physicist says: speed=distance/time. 

The mathmatician tells you, that if you want to calculate the time you need to reach a place, you do not need to invent a new formula. You just can multiply both sides with distance and it will be right. He knows that because he has a proof that in this system multiplikation is a legit operation, and it will always create a legit result. Without that knowledge, physicists would either have to do the work of mathmaticians (what Feynman actually did in some cases because he considered it necessary) or they would be lost. 

We do not care if you use that knowledge to calculate with speed and time, volume and mass, or gods and sausages, as long as you can prove that gods are within a group where multiplication is allowed.

 

From another point of view, we work the exactly same way, we just don't call our experiments experiments. We make a guess, and then we think how we could falsify it, then we try. The difference is that we have the possibilities to actually proof we are right, something a physical experiment can never do, those can only prove for sure that you're wrong. If your experiment fails to prove that you're wrong it's more likely you're right, but you don't know for sure. 

 

If you're lacking some case of doubt in math... don't worry. We have that, we just couldn't find many cases we can't prove wrong or right yet. My math idol Kurz Gödel proved (technically he called himsel a logician) that there must be some cases. Afaik the only known example is about the question if there are more than two different sizes of infinity, something that - with our current systems - is both proven true and wrong at the same time and therefore not deterministic.

 

Informatiks is one example where "useless" math was actually needed, the algebraic geometry (where a straight isn't the shortest way) is another. I'm honestly not sure what was first: a mathmatician who said "intresting thought, let's think about it" or a physician who said "the geometry we know doesn't work in space", both is possible. But then the mathmaticians take that thought and research how such a system has to look like (mathematical), which old rules still apply or not, if they can find replacements for old rules that don't work in that system or completly new rules.

While i think there is little difference between a theoretical physicist like Feynman and somebody in applied maths, an example: for approximations, usually done by computers because it's so much calculation but not exclusive for IT: according to Moores law PCs in the last ten years became faster about the factor one thousand. In the same time mathmaticians improved the algorithms used for (some difficult) approximations about the factor 1 million. Well, pretty much anybody could have done that, most part of it was just to say "let's leave this step away". What mathmaticians did was to find steps which weren't that improtant for accuracy and proof that less than 2% (end result) accuracy are lost. You can skip 90% of the calculations if you follow the algorithm found in applied math. That's something most IT guys can't do, some physicists might but don't care enough because it's too far from their field, and it's pretty useful for both i think.

 

What most people consider math when it comes to language is only a short part. Usually they mean the parts where mathmaticians have been lazy, or how we call it, efficient.

That i prefer f(x) over "(Placeholder for the) function with the definition ... that is applied to the (placeholder for the) variable with the definition ..." is because i'd never get anything finished if i'd use common language for this stuff. Common language can be deterministic, but often it has too many synonyms and assiciations you want to avoid in math, personal or cultural. If you want to explain those things without even the possibility of misunderstandings, you need a lot of time.

On the other hand my examples for proofs were totally legit and quite representative, in fact ~75% of my proofs for homework didn't include more calculations than an "a/2 equals prefered(or conflicting) result", and only 25% were what you usually consider as "math language". 

That i - like most mathmaticians - neither like calculating nor i'm i really good at it might be a reason, if a prosa proof isn't exaggerated much work i'll often use prosa. On the other hand... when you know the rules you have to follow, pushing some formulas until they fit is just routine. Calculating is mindless boring, and if you're drowning in difficult homework sometimes you're glad about some boring tasks where you just have to watch not falling asleep instead of a real challenge.

 

What indeed is something that makes me upset: seeing words well defined in math mindless used in sociology. If a sociologist (actually happened) talks about homomorphisms and simply points to the mathmatical definition, that creates a raw imagination of what he's talking about. But since the words used to define a homomophism don't make much sense outside of math, at least not by default, i see it as feeding a bullshit generator with terms from math, not as an indication he actually knows what he's talking about. In the end he wasted my time by abusing something i care about.

 

*That is, if you try to stay in the "Regelstudienzeit", not even sure if that word exists in english or has the same meaning. In my case it was important because of Bafög.

 

*edit: one of my favorite math jokes might give you an insight if i failed so far: 

The cash register at the supermarket is broken, a math student, an assistent and a prof are in the queue. While the cashier usees pen &paper, the student checks his stuff, calculates in mind and tells the result. The cashier a minute later is impressed and says "That's correct!"

The assistent sees what the cashier does and says "That's an addition." The cashier grumbles something about not helpful.

The prof sees the paper and says "It is possible to solve that problem." and doesn't even recognize the cashiers response.

 

We are no alternative to physicists or engineers or economists. We do what is necessary for those to do their work. But we don't do it because they need it, we do it because we can and enjoy it.

 

*edit2: same for me with friends and talking. I don't hink it's difficult to make friends, but talking to people IRL is exhausting. Online it doesn't matter if you need a minute, an hour or a day to answer, i can follow my sometimes weird thoughts and come back on topic later. Which is more relaxing for me, even compared to the company of people who actually enjoy even my weird thoughts.

[dnoah]

I think the easiest way to explain mathematics is to refer to abstractions. Plato/Socrates had a "Theory of ideas", stating that those abstractions, or "forms", exist a priori, and the physical reality is merely derived from them (oversimplified, of course). The essence here is that the brain deals primarily in abstractions. Every math term, from the simplest ones like equality, as in "1=1", to the infinite complexity things like Julia fractal sets, is an example of such an abstraction. We don't have equality in physical reality, since we cannot perform measurements with infinite accuracy, and thus "1 nearly equals 1" [abstraction] emerges. From the point that abstractions is the only thing human brain deals in and communicates in (again oversimplified), it can be concluded that math is the only real science there is. Until we come to exactly defining "physical reality", that is. Penrose does a pretty good job at explaining all that - "The Emperor's New Mind" the book was called, if memory serves.

(Math) I wouldn't go so far as to assume uncomparability of cardinal numbers to illustrate the Godel's theorem in action. That uncomparability essentially boils down to falsehood of the AC (Axiom of Choice), which nullifies quite a bit of the fascinating aspects of set theory, functional analysis, math logic etc. (Please correct me if I'm wrong, it's been too long.) We have a very beautiful method called Cantor's diagonal argument (Russell liked it a lot, btw) that allows to state that the set of all subsets of an infinite set (math notation 2 to the power of set, i.e. 2^(|R)) has a greater cardinal number (is bigger in terms of comparability of infinite sets) than that original set. The simplest case is uncomputability of all sets of natural numbers, which in turn leads to the "halting problem", the absence of proof for which (called undecidability, and is an instance of the Godel's theorem) can be proven rather simply in original Turing's terms.

Going back to OP, that "game" is a good simple representation of some concepts of game theory. It is also a good example of what is wrong with pseudo-science, if that game were to be treated seriously (which it was not meant to be, obviously). As for social sciences (IMO), they could do well better without indulging themselves in make-believe. Society as a system is bound to be unstable, and hence any model describing it has better be unstable, and hence uncomputable (in any practical sense). Meaning, an infinitesimal change in initial conditions can lead to enormous change in solution for any meaningful time period. Even if we had such a model, making a prediction would require unrealistic amount of computations, like double the computation to advance the projection for each second while retaining accuracy (way beyond NP-fullness, I mean). But that is past the point, because what we see in social sciences, is that they are looking for models with nice continuous stable solutions because they *want* them nice, because of the inherently trigger-happy pattern recognition ability of their brain.

_{[Edited typos]}

[Nazzzgul666]

I think the easiest way to explain mathematics is to refer to abstractions. Plato/Socrates had a "Theory of ideas", stating that those abstractions, or "forms", exist a priori, and the physical reality is merely derived from them (oversimplified, of course). The essence here is that the brain deals primarily in abstractions. Every math term, from the simplest ones like equality, as in "1=1", to the infinite complexity things like Julia fractal sets, is an example of such an abstraction. We don't have equality in physical reality, since we cannot perform measurements with infinite accuracy, and thus "1 nearly equals 1" [abstraction] emerges. From the point that abstractions is the only thing human brain deals in and communicates in (again oversimplified), it can be concluded that math is the only real science there is. Until we come to exactly defining "physical reality", that is. Penrose does a pretty good job at explaining all that - "The Emperor's New Mind" the book was called, if memory serves.

I have to think about that if i consider that easy.

 

(Math) I wouldn't go so far as to assume uncomparability of cardinal numbers to illustrate the Godel's theorem in action. That uncomparability essentially boils down to falsehood of the AC (Axiom of Choice), which nullifies quite a bit of the fascinating aspects of set theory, functional analysis, math logic etc. (Please correct me if I'm wrong, it's been too long.) We have a very beautiful method called Cantor's diagonal argument (Russell liked it a lot, btw) that allows to state that the set of all subsets of an infinite set (math notation 2 to the power of set, i.e. 2^(|R)) has a greater cardinal number (is bigger in terms of comparability of infinite sets) than that original set. The simplest case is uncomputability of all sets of natural numbers, which in turn leads to the "halting problem", the absence of proof for which (called undecidability, and is an instance of the Godel's theorem) can be proven rather simply in original Turing's terms.

 

This is a nice example of misunderstandings that may happen if you don't use the formalized language. It's not about Cardinal numbers but the difference between (i can't find a formalized translation? ) "countable infinite" and "over-countable infinite", the different amount of natural numbers and real numbers. Cantor always wanted to know if there is an infinity between those two and, went tp psychatry while he tried to figure it out. This question is beyond uncomputability, it's actually undecidable. I can't remember or  find right know who proved that, it was a couple of years ago but not that long ago, 20 years max. 

 

Going back to OP, that "game" is a good simple representation of some concepts of game theory. It is also a good example of what is wrong with pseudo-science, if that game were to be treated seriously (which it was not meant to be, obviously). As for social sciences (IMO), they could do well better without indulging themselves in make-believe. Society as a system is bound to be unstable, and hence any model describing it has better be unstable, and hence uncomputable (in any practical sense). Meaning, an infinitesimal change in initial conditions can lead to enormous change in solution for any meaningful time period. Even if we had such a model, making a prediction would require unrealistic amount of computations, like double the computation to advance the projection for each second while retaining accuracy (way beyond NP-fullness, I mean). But that is past the point, because what we see in social sciences, is that they are looking for models with nice continuous stable solutions because they *want* them nice, because of the inherently trigger-happy pattern recognition ability of their brain.

_{[Edited typos]}

I think it can actually be treated seriously, it shouldn't be treated as an... final answer? Not sure how to say it. It's a serious example, but not a study might fit. For social science... i like the idea of Assimov (i didn't read that much, only 2-3 books) that you can predict the behavior of a society pretty accurate. But even in those books with his scifi math, it only applied to a society with the population of one planet. Anything smaller and chances decrease from 99,x% for a certain event to to a 50% chance (with 2 options, it happens or doesn't happen) pretty fast. If you don't know what i mean, blame me for a poor explanation, not him for writing BS.

 

To be more specific for real world examples, sweden is a good example i think. It has one of the most happy populations on earth. I think it's sure to say, there won't be a revolution soon. But nevertheless, their government is currently in a crisis and it's possible they all will resign (currently 3 secretaries) because they outsourced some data (including names members of the special forces, police suspects,...) to IBM and those published it accidently in the cloud. What i mean is: you CAN make predictions which are correct. But usually they are not accurate enough to be helpful. Or you can be accurate, but then it's probably wrong.

Another "save" example for predictions: There will be more cars on the streets during the rush our. Here you may even get some pretty accurate predictions if you have enough data from past. Probably still not accurate enough to prevent all traffic jam, but at least to reduce the chances and sizes.

[dnoah]

 

 

I think the easiest way to explain mathematics is to refer to abstractions. Plato/Socrates had a "Theory of ideas", stating that those abstractions, or "forms", exist a priori, and the physical reality is merely derived from them (oversimplified, of course). The essence here is that the brain deals primarily in abstractions. Every math term, from the simplest ones like equality, as in "1=1", to the infinite complexity things like Julia fractal sets, is an example of such an abstraction. We don't have equality in physical reality, since we cannot perform measurements with infinite accuracy, and thus "1 nearly equals 1" [abstraction] emerges. From the point that abstractions is the only thing human brain deals in and communicates in (again oversimplified), it can be concluded that math is the only real science there is. Until we come to exactly defining "physical reality", that is. Penrose does a pretty good job at explaining all that - "The Emperor's New Mind" the book was called, if memory serves.

I have to think about that if i consider that easy.

 

(Math) I wouldn't go so far as to assume uncomparability of cardinal numbers to illustrate the Godel's theorem in action. That uncomparability essentially boils down to falsehood of the AC (Axiom of Choice), which nullifies quite a bit of the fascinating aspects of set theory, functional analysis, math logic etc. (Please correct me if I'm wrong, it's been too long.) We have a very beautiful method called Cantor's diagonal argument (Russell liked it a lot, btw) that allows to state that the set of all subsets of an infinite set (math notation 2 to the power of set, i.e. 2^(|R)) has a greater cardinal number (is bigger in terms of comparability of infinite sets) than that original set. The simplest case is uncomputability of all sets of natural numbers, which in turn leads to the "halting problem", the absence of proof for which (called undecidability, and is an instance of the Godel's theorem) can be proven rather simply in original Turing's terms.

 

 

This is a nice example of misunderstandings that may happen if you don't use the formalized language. It's not about Cardinal numbers but the difference between (i can't find a formalized translation? ) "countable infinite" and "over-countable infinite", the different amount of natural numbers and real numbers. Cantor always wanted to know if there is an infinity between those two and, went tp psychatry while he tried to figure it out. This question is beyond uncomputability, it's actually undecidable. I can't remember or  find right know who proved that, it was a couple of years ago but not that long ago, 20 years max.

You're refer... (*Clears throat*) Sorry. You're referring to the continuum hypothesis, which states that there is no cardinal number between <edited typo> |N and |R. Its independence of other axioms of set theory is notoriously hard to prove (like you said, it incapacitated Cantor himself), that's why I mentioned the halting problem. Halting problem for Turing machines is easily unprovable, by the same process I've shown below.

 

The fact that |R is more "powerful" that |N is provable, assuming some form of AC is correct, by a process called Cantor's diagonal argument. Here it is, somewhat simplified for shortness.

 

Let's assume that we can "count" real numbers, say, those ranging from 0 to 1. Let's "count" them and write them down, starting from first, in their decimal form. What we get is an infinite table:

 

(1st real number) .1491775.....

(2nd real number) .3523057.....

(3rd real number) .14097154018....

(...)

 

Now, let's take the diagonal of that table: 1st column in 1st row, n-th column in n-th row, etc. Let's change each number we've got to anything else (i.e. +1 mod 10, so that 9 becomes 0). What do we have? A decimal representation of some real number between 0 and 1. But it's not the first number that we've written down, since the its first decimal sign is different, because we've changed it. It's not the second, since it's 2nd decimal sign is different. Et cetera. So what we've got is a decimal number not in our table. Which cannot be, since we've written down them all!

 

Ergo, our initial assumption is incorrect and real numbers cannot be counted, whichever method we use.

 

With this true, the Continuum hypothesis states, that if a set cannot be counted (with natural numbers, i.e. computable) it must be at least as large as |R (the corresponding cardinal number is called continuum).

 

 

Going back to OP, that "game" is a good simple representation of some concepts of game theory. It is also a good example of what is wrong with pseudo-science, if that game were to be treated seriously (which it was not meant to be, obviously). As for social sciences (IMO), they could do well better without indulging themselves in make-believe. Society as a system is bound to be unstable, and hence any model describing it has better be unstable, and hence uncomputable (in any practical sense). Meaning, an infinitesimal change in initial conditions can lead to enormous change in solution for any meaningful time period. Even if we had such a model, making a prediction would require unrealistic amount of computations, like double the computation to advance the projection for each second while retaining accuracy (way beyond NP-fullness, I mean). But that is past the point, because what we see in social sciences, is that they are looking for models with nice continuous stable solutions because they *want* them nice, because of the inherently trigger-happy pattern recognition ability of their brain.

_{[Edited typos]}

I think it can actually be treated seriously, it shouldn't be treated as an... final answer? Not sure how to say it. It's a serious example, but not a study might fit. For social science... i like the idea of Assimov (i didn't read that much, only 2-3 books) that you can predict the behavior of a society pretty accurate. But even in those books with his scifi math, it only applied to a society with the population of one planet. Anything smaller and chances decrease from 99,x% for a certain event to to a 50% chance (with 2 options, it happens or doesn't happen) pretty fast. If you don't know what i mean, blame me for a poor explanation, not him for writing BS.

 

To be more specific for real world examples, sweden is a good example i think. It has one of the most happy populations on earth. I think it's sure to say, there won't be a revolution soon. But nevertheless, their government is currently in a crisis and it's possible they all will resign (currently 3 secretaries) because they outsourced some data (including names members of the special forces, police suspects,...) to IBM and those published it accidently in the cloud. What i mean is: you CAN make predictions which are correct. But usually they are not accurate enough to be helpful. Or you can be accurate, but then it's probably wrong.

Another "save" example for predictions: There will be more cars on the streets during the rush our. Here you may even get some pretty accurate predictions if you have enough data from past. Probably still not accurate enough to prevent all traffic jam, but at least to reduce the chances and sizes.

 

I generally agree with the rest. Like I said, it's just my opinion, about how things work with society, that whatever the model is, however good it seems to be, there's always one (IRL more like trillion) small, unaccounted for, factor, that will cause this model to collapse, getting completely out of touch with experimental data.

 

Yeah, and Asimov gave some excellent examples with his Robot series, of how a simple model like the Three Laws of Robotics can cause a shitstorm when colliding with real world situation. He did not seem to aim to answer those questions (rather to hypothesize), and neither do I.

_{[edited typos]}

([Edit2] @Nazzzgul: Just re-read this exchange, and... Talk about pedantry, huh. In my defense, I've only just now sufficiently understood what you had replied in green.)