Wednesday, October 17, 2012

What is math, and why should we use it in economics?


In my last post, I pointed out that the Nobel Prize-winning work of Lloyd Shapley and Al Roth, makes heavy use of mathematics, and indeed would be completely impossible without math. This, I said, is evidence against the idea that economics doesn't need (or shouldn't use) math.

But then some commenters asked me: What do you mean by "math"? And I thought that was an interesting question.

There is no "correct" definition of the word "math", any more than there is a correct definition of the word "art", or the word "love". There are many different definitions, all of which are drawn from similar connotations; in other words, people look at a bunch of things, say "This is math, and that is math", and then try to distill and formalize the similarities between the things that seem like math. For example, the definition I tended to like in college was called the "formalist" definition: 
"Mathematics is the manipulation of the symbols of a language according to explicit, syntactical rules."
Basically, this just means "math" = "logic". Philosophically, I'm fine with that. It's an expansive definition. But it's not very helpful when talking about economic methods, since it includes lots of stuff that people wouldn't normally call "math".

So what do I think is a useful definition? When it comes to scientific methodology, I think of "math" as basically being the same thing as "precision of meaning." This working definition is not a yes-or-no sort of thing; it's a sliding scale. Methods can be more math-y or less. 

So what do I mean by "precision of meaning"? Basically, something with a precise meaning has fewer alternative things that it could mean. For example, compare the two scientific propositions:

1. If you push something, it will push you back.

2. Momentum is conserved.

The second statement has a more precise meaning than the first. For example, the first statement could mean "If I push on something with a force of 5 Newtons, it will push on me with a force of 5 Newtons in the exact opposite direction that I pushed." Or, it could just as easily mean "If I push on something with a force of 5 Newtons, it will push on me with a force of anywhere between 1 to 1,000,000 Newtons, in a direction 15 degrees east of the direction I pushed." But the second statement can only mean the first of those two things, not the second. 

Therefore, I would say that the second statement is more mathematical than the first. Note that both of these statements are logical statements; for example, you can apply the rules of first-order logic to either statement to rule out the situation where I push something and it doesn't push me back at all. By the formalist definition, we can do "math" with either statement. But my "precision" definition makes a distinction between the two.

So by this definition, are probabilistic statements less mathy than deterministic ones? No, as long as they are explicit about the fact that they are probabilistic statements.

Are qualitative statements less mathy than quantitative statements? Not necessarily ("The sign of the first derivative is positive" is qualitative but is precise in its meaning), but in practice, this often tends to be the case. Quantitative statements must be precise, while qualitative statements may or may not be. This is just due to differences in the languages we use for expressing qualitative and quantitative statements. And this tendency is why people usually think math is about numbers and/or symbols that stand for numbers.

What, then, to raise the old question once more, is mathematics? The answer, it appears, is that any argument which is carried out with sufficient precision is mathematical, and the reason that your friends and ours cannot understand mathematics is not because they have no head for figures, but because they are unable [or unwilling, DRH] to achieve the degree of concentration required to follow a moderately involved sequence of inferences. This observation will hardly be news to those engaged in the teaching of mathematics, but it may not be so readily accepted by people outside of the profession. For them the foregoing may serve as a useful illustration.
So there you go. Great minds think alike...and mine occasionally happens to stumble to the same conclusions.

So why should we use math in economics? Well, I can think of a number of reasons:

1. We may want to make precise predictions about what will happen in a market.

2. We may want to make precise predictions about the conditions under which things will happen in a market.

3. Precise statements often help resolve debates, avoiding the phenomenon of "talking past each other".

4. Precise statements often lead to unintuitive but logically inescapable results.

5. It is usually easier to check sets of precise statements for logical inconsistencies.

I think all of these reasons are good reasons sometimes and bad reasons sometimes (note how imprecise of a statement that is!). I have no hard-and-fast rule about how much precision to use, and when. But I do know that if you tried to implement a Shapley-Roth matching algorithm without mathematically precise statements about what happens when, it would be hopeless. 

And I also know that in the blogosphere, many debates go on and on without being resolved, when both sides are really just talking past each other. Egos get bruised, grudges develop, and understanding is not advanced, even when the different sides' positions are not mutually incompatible or even that far off. That's why, when debates get really long and confusing, I think it's time to whip out the math, define terms, and get really precise. (By the way: In my experience, defining terms is really the critical piece of this. It's very very hard to make imprecise statements when all your words are precisely defined!)

So are there times when we should use less math in economics? Sure. Sometimes we understand a phenomenon so little that imprecise statements are more valuable than precise ones; precise formulations, if we believe them, give us the illusion of understanding, while imprecise statements, by pointing us in many directions at once, give us a menu of options for seeking the truth. And I also suspect (without proof) that some authors use excessive precision as a form of obscurantism, cloaking simple ideas in daunting reams of equations, or performing byzantine manipulations of simplistic assumptions, in order to deter outsiders from entering their hyper-specialized sub-field and criticizing their work. 

But these are cases in which the purpose of imprecision is to lead us to greater future truth. And that truth, if it is found, will certainly be expressed with great precision - i.e., if there is an economic theory that really works, it's going to use some math. The only time not to use math in econ is when we haven't found the right math yet.

And in practice, I find that a few of the people calling for less math in economics (You know who you are!) don't seem to have any such goal in mind. There are a few people out there who would rather econ stay imprecise forever - so that nobody will ever be proved wrong or right, and we can let a million flowers bloom, and everyone's scholarly opinion about the economy will be equally valid. Paul Krugman discusses these folks when he says:
[Some people] claim to reject neoclassical economics, but their alternative is not an alternative model but a lot of verbiage; they talk at the economy, and imagine that by so doing they achieve a higher level of sophistication and realism than economists who try to express their ideas in terms of little models. 
And they’re kidding themselves; all they’ve done is hide their implicit models and prejudices behind a dust cloud.
Agreed. Math is not always the most appropriate tool in economics. But the more real successes economics achieves, the more good math it will use.

Update: And here is a useful reminder that the things people call "math" don't always meet my definition...computer-generated gibberish was accepted for publication in a math journal. Gibberish, of course, has no precision of meaning at all.

Update 2: Alex Marsh has a good post that discusses the pitfalls of using math in economics. The main pitfall he identifies is that people start to believe in their own math because it's simple. Marsh is absolutely right. Making simplifications is a necessary evil, and when people do it, sometimes they forget - or decide not to believe - that the things they left out of the model still exist. Believing that your own oversimplifactions are the Laws of the Universe is easy, seductive, and deadly. Only empiricism - the relentless insistence on matching models to real-world data - can provide an effective check on this tendency.

117 comments:

  1. The formalist definition of math as language is one I understand. My native language is music: I can hear music simply by looking at it on a page. But I don't necessarily expect others to be able to do that - it is a talent. Why is it that people whose native language is math think that the reason people struggle to understand math is either lack of concentration or laziness? Shapley's remark is pig-headed.

    Not that this has anything to do with use of math in economics, of course. But Coppola's Law states that the wrongness of an economic theory is positively correlated with the complexity of the math used to explain it.

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    1. Why is it that people whose native language is math think that the reason people struggle to understand math is either lack of concentration or laziness? Shapley's remark is pig-headed.

      Well, I've often wondered if math is a universal talent (like spoken language) or a rare one (like music). I'm not sure which it is.

      Coppola's Law states that the wrongness of an economic theory is positively correlated with the complexity of the math used to explain it.

      Coppola's Law is wrong (though not well defined, since "complexity" and "wrongness" are both hard to quantify). Shapley and Roth's theory is right; it works quite nicely. And it used considerable math.

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    2. Frances,

      Are you saying that you were born with your ability to read music? You never did any practice or otherwise exercised patience to acquire skill?' That *would* be quite remarkable!

      I know I acquired my music reading ability in much the same -- rather old fashioned -- way that I acquired my grasp of mathematics. It took work. Talent may have played a role: certainly both came more easily to me than to some other people who made a similar effort. But I did put in quite a lot of time.

      I don't think Shapley meant to do anything more than 'demystify' the math -- to claim that anyone could learn it who was suitably motivated. 'Talent' is often just a magic word introduced to provide a fictitious 'explanation' for whatever motivates some people to do something others can't be bothered with.

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    3. No, Seth, I'm not saying that. But as far back as I can remember I've been able to look at a page of music and "hear" it, in much the same way that some mathematicians can look at an equation and "see" the shape. And it isn't, it just isn't about practice or hard work. I've done at least as much math as music in my life, but for some reason I can't make the same transition into "fluency".

      I'm afraid I fundamentally disagree with Shapley. He's someone who "gets" math and therefore doesn't understand why others don't. I've had to tackle the same problem myself: I can sight read anything, because I can hear the music on the page, but the majority of people I teach can't do this, no matter how hard they try. I don't "understand" why they can't, but I don't have the right to accuse them of laziness or lack of concentration. And I don't think mathematicians have that right either.

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    4. Noah, I'm not being entirely serious, of course....but you really can't tell me I'm "wrong" and then criticise me for not defining "wrongness"!

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    5. Heh.

      It's me I'm criticizing for not defining "wrongness".

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    6. Anonymous4:58 PM

      Mathematics is not a language by any commonsense definition of the term.

      For example, all languages throughout the world allow us to ask for a cup of tea. Mathematics does not.

      Language can be used allusively - "yeah, right!" can mean "no, wrong!". But in mathematics 2+2=4 can never mean 2+2=47.

      The distinction lies in the words "according to explicit syntactical rules". Maths has explicit syntactical rules; language has syntactical rules but they're not explicit.

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    7. Language can be used allusively - "yeah, right!" can mean "no, wrong!". But in mathematics 2+2=4 can never mean 2+2=47

      I see someone hasn't taken a Modern Algebra class... ;-)

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    8. Anonymous12:26 PM

      Touche. But even in modern algebra, 2+2=4 can never "mean" 2+2=47, even though they may both be contemporaneously accurate within differing syntactical requirements.

      I'm conscious that I'm laying myself open to attack by linguists... But that's for another day.

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  2. For example, the definition I tended to like in college was called the "formalist" definition:

    "Mathematics is the manipulation of the symbols of a language according to explicit, syntactical rules."

    Basically, this just means "math" = "logic". Philosophically, I'm fine with that. It's an expansive definition. But it's not very helpful when talking about economic methods, since it includes lots of stuff that people wouldn't normally call "math".

    So what do I think is a useful definition? When it comes to scientific methodology, I think of "math" as basically being the same thing as "precision of meaning." This working definition is not a yes-or-no sort of thing; it's a sliding scale. Methods can be more math-y or less.

    So what do I mean by "precision of meaning"? Basically, something with a precise meaning has fewer alternative things that it could mean.


    If this is your definition of mathematics, then not only do I agree with you, but I would actually go further insofar as I see no reason why any academic work should be "nonmathematical" (all work should be as definitionally precise and operationally transparent as possible).

    But in reality, this is not what economists use a lot of math in their work seem to mean by mathematical. Because Paul Krugman seems to think, as you note, that "a lot of verbiage" is not "mathematical".

    [Some people] claim to reject neoclassical economics, but their alternative is not an alternative model but a lot of verbiage; they talk at the economy, and imagine that by so doing they achieve a higher level of sophistication and realism than economists who try to express their ideas in terms of little models.

    And they’re kidding themselves; all they’ve done is hide their implicit models and prejudices behind a dust cloud.


    If I am expositing my "model" as precisely as I can using English instead of substituting the English out for mathematical symbols, what difference does it really make? Certainly, translating ideas back and forth between mathematical symbols and plain English aids in clarifying my own thinking, but other than brevity I think that clear-thinking is equally good in both languages.

    I propose that the symbolic language known as "mathematics" is not the same thing as "rigour" (what you are defining as "mathematics"). I am all for rigour in every dimension of thought except regarding art (what Nietzsche defined as Dionysian, etc). But as for the symbolical language of mathematics? I think that that is useful too, but I don't think that it is by any means characteristically superior to plain English.

    It is possible to express an idea very badly or clumsily using mathematical symbols, and it is possible to express an idea very precisely and rigorously using plain English.

    Indeed, one of the Fathers (alongside Walras) of the mathematicization of economics, Alfred Marshall wrote:

    Use mathematics as shorthand language, rather than as an engine of inquiry.

    Keep to them till you have done.

    Translate into English.

    Then illustrate by examples that are important in real life.

    Burn the mathematics.

    If you can’t succeed in 4, burn 3. This I do often.

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    1. The two methods of expression seem to have specific problems related with their improper usage. Generally, I find the bad usage of mathematical symbols to be clear, but unrealistic. Generally, I find the bad usage of verbiage to be unclear and imprecise. Returning to Shapley's piece that you referenced:

      We assert that this set of marriages is stable. Namely, suppose John and Mary are not married to each other but John prefers Mary to his own wife. Then John must have proposed to Mary at some stage and subsequently been rejected in favor of someone that Mary liked better. It is now clear that Mary must prefer her husband to John and there is no instability.


      This example is verbiage translated out of mathematical symbols, and translating it into English reveals how problematic the clarity of mathematics can be when applied to the real world.

      What if John never proposed to Mary because he was too shy, for example? What if he did propose to her and she wanted to marry him, but her parents pressured her into marrying someone else?

      Human preferences are far, far more complex than their characterisation in matching theory. Matching theory is interesting and practically useful to varying degrees, of course, but there are many, many applications where matching theory will describe as stable something that is deeply instable. Indeed, I would propose that the only theory that can truly model reality in perfect detail is reality itself. That doesn't mean to say that I am against simplified predictive mathematical models (i.e. "neoclassical models) but their use must always be qualified.

      The most worrying dimension of Shapley's position is that it acts as an excuse for expressing ideas in a language that most of society find impenetrable. Is this simply a matter of concentration and pure thinking, or is this symbolical? I think it's the latter. I can almost always grasp ideas — no matter how complex — when they are explained in English, but I find understanding most symbolical mathematical formulations to be very, very hard work. (Not to say that some hard work is not good for me — it is — but I feel like a fish thrashing around out of water with mathematical symbols, and not so with English or even with musical notation, which is the same view that Frances Coppola has expressed).

      So I am all for rigour in every realm of intellectual enquiry, without exception. On the other hand, I favour the minimisation of mathematical notation in economics. An accountable science expresses ideas in nonspecialist language so that work is open to nonspecialist scrutiny/understanding.

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    2. If I am expositing my "model" as precisely as I can using English instead of substituting the English out for mathematical symbols, what difference does it really make?

      None at all, provided your English is as precise as the symbols would have been. Of course, English can sometimes be clunky for certain things. For example, if you need to differentiate an expression or evaluate an infinite sum. Each language is optimized for certain tasks. Symbols are, of course, generally useless for assigning real-world meanings to other symbols.

      Indeed, one of the Fathers (alongside Walras) of the mathematicization of economics, Alfred Marshall wrote:

      Well, he had an opinion, but I don't put much credit in argumentum ad verecundiam...

      In physics, unexpected results often emerge from the symbolic manipulations. To assert that this can never happen in economics is, in my opinion, the sheerest arrogant folly.

      What if John never proposed to Mary because he was too shy, for example? What if he did propose to her and she wanted to marry him, but her parents pressured her into marrying someone else?

      Human preferences are far, far more complex than their characterisation in matching theory. Matching theory is interesting and practically useful to varying degrees, of course, but there are many, many applications where matching theory will describe as stable something that is deeply instable. Indeed, I would propose that the only theory that can truly model reality in perfect detail is reality itself. That doesn't mean to say that I am against simplified predictive mathematical models (i.e. "neoclassical models) but their use must always be qualified.


      Of course.

      I can almost always grasp ideas — no matter how complex — when they are explained in English, but I find understanding most symbolical mathematical formulations to be very, very hard work.

      Well, there are times when that work is necessary to work out the implications of one's results. For example, if I told you the assumptions of a New Keynesian model in English, I'm sure you could easily understand them. But could you then tell me what the optimal monetary policy would be in that model? I'll bet you couldn't...almost no one on the planet could!

      On the other hand, I favour the minimisation of mathematical notation in economics. An accountable science expresses ideas in nonspecialist language so that work is open to nonspecialist scrutiny/understanding.

      I do really like fields to be open to scrutiny by nonspecialists. Ironically, though, the people who often come up with the most trenchant and penetrating criticisms of mathematical econ models are mathematicians and physicists, who have the mathematical faculties to understand the limitations of the approaches economists are using. It's not just economists who know math!

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    3. I don't put much credit in argumentum ad verecundiam...

      In physics, unexpected results often emerge from the symbolic manipulations. To assert that this can never happen in economics is, in my opinion, the sheerest arrogant folly.


      I'm not engaging in argumentum ad verecundiam, but I think Marshall's opinion is interesting and relevant to the discussion, which is why I included it.

      And I'm certainly not asserting that unexpected and interesting and useful results can't result from manipulating equations. If that's how a school of thought chooses to do economics, then great. The only real implication of my view is that economics papers should translate the mathematical manipulations into common English like Marshall suggests so that an idiot like me can understand the conclusions and thought process quickly instead of having to reach for a mathematical dictionary, read ten Wikipedia articles and sometimes even watch Khan Academy videos on differential equations...

      I do really like fields to be open to scrutiny by nonspecialists. Ironically, though, the people who often come up with the most trenchant and penetrating criticisms of mathematical econ models are mathematicians and physicists, who have the mathematical faculties to understand the limitations of the approaches economists are using. It's not just economists who know math!

      True enough. But given that we both agree that ideas can (given enough rigour and clear thinking) be expressed equally well in English and mathematics, it is better to be even more inclusive. Academic debate and expression were once limited to those who spoke Latin, and that was a bad thing, and I feel to some degree like mathematical symbols are the new Latin. Certainly mathematical symbols, like Latin, are more adept at expressing certain ideas and manipulating ideas in a certain way than plain English, but so is Japanese....

      One reason why I think the economics blogosphere is so praiseworthy, of course, is that it is forcing academic economists into explaining their ideas in plain English. Heck, I wouldn't know what Wallace Neutrality was without the economics blogosphere...

      Well, there are times when that work is necessary to work out the implications of one's results. For example, if I told you the assumptions of a New Keynesian model in English, I'm sure you could easily understand them. But could you then tell me what the optimal monetary policy would be in that model? I'll bet you couldn't...almost no one on the planet could!

      I couldn't tell you precisely what the mathematical manipulation would yield without actually doing the math, that's true. But that's the nature of a mathematical model. If you gave me the same model's assumptions in English, I am confident I could reason out the gist of the implications, and I am sure you could too.

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    4. The only real implication of my view is that economics papers should translate the mathematical manipulations into common English like Marshall suggests so that an idiot like me can understand the conclusions and thought process quickly instead of having to reach for a mathematical dictionary, read ten Wikipedia articles and sometimes even watch Khan Academy videos on differential equations...

      I agree with that (for most cases)...

      we both agree that ideas can (given enough rigour and clear thinking) be expressed equally well in English and mathematics

      Actually I'm not quite sure that's correct for all ideas...

      I couldn't tell you precisely what the mathematical manipulation would yield without actually doing the math, that's true...If you gave me the same model's assumptions in English, I am confident I could reason out the gist of the implications, and I am sure you could too.

      Well, that actually is true for most of the intuitive, simplistic, rigged models that you see in macro. But actually, that's one of the very reasons I think those models should use less math - they don't need much of it to reach their conclusions from their assumptions, they just stick it in there to look smart.

      BUT, on the other hand, there are some models where you or I couldn't even get the gist of the result without the symbolic manipulations. Really! In fact, sometimes the symbolic manipulations end up completely reversing one's intuition about what the result should be! That's the kind of model where I think the math is absolutely essential. They tend to be in micro, btw.

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    5. BUT, on the other hand, there are some models where you or I couldn't even get the gist of the result without the symbolic manipulations. Really! In fact, sometimes the symbolic manipulations end up completely reversing one's intuition about what the result should be! That's the kind of model where I think the math is absolutely essential. They tend to be in micro, btw.

      I think you need to illustrate your answer with some mathematics!

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    6. OK, that's a subject for another post! ;)

      I'll explain some models of herd behavior...those give really neat results.

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    7. I'll explain the (very similar) models in these papers:

      http://snap.stanford.edu/class/cs224w-readings/banerjee92herd.pdf

      and

      http://www.unc.edu/~fbaum/teaching/articles/Bikhchandani_etal_1992_JPE.pdf

      And then I'll explain the really crazy extension in this paper:

      http://www.sciencedirect.com/science/article/pii/S0022053183710744

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    8. I read the Banerjee paper, and in my opinion the level of herding in the model seems to flow very neatly from the models' explicit assumptions. It was an interesting model, though, even though I think that the penultimate and final sections (criticisms of assumptions) made many valid points. I look forward to your post on this.

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    9. First I have to take down John Taylor, discuss tax fairness with Miles, and write two new Atlantic articles! It'll come though...these models are one of my very favorite kinds (despite the difficulty of finding evidence for them in the real world!)...

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    10. Oh but as for one's ability to forecast the conclusions just from the assumptions without using any math, take a look at the Bannerjee paper's assumptions and setup and ask yourself these questions:

      1. What happens if people can delay their actions and watch what other people do?

      2. What happens if, given an even split between one's own signal and someone else's, one would choose to follow the other person's signal half the time and one's own signal half the time (instead of one's own signal all the time, as the author assumes)?

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    11. 1. All things being equal that sounds like it would tend to encourage herding (although as Bannerjee states, that makes the mathematics greatly more complicated).
      2. All things being equal that would tend to encourage herding.

      From personal experience, though, if I had a signal (say the choice is which tech stock you will spend $1,000 on) I am not going to give a damn whether 100 people in front of me have chosen to buy AAPL. So I tend to think that Bannerjee conceptualises the kth decision maker's dilemma in a herding-inducing ("herdogenic"? hahaha) manner, whether or not there are various other assumptions in the model that might could be changed to increase the degree of herding....

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    12. 1. All things being equal that sounds like it would tend to encourage herding (although as Bannerjee states, that makes the mathematics greatly more complicated).

      Actually, it turns out not to make a difference.

      2. All things being equal that would tend to encourage herding.

      Seems like it would, but once you work out the math, you find it actually discourages herding!

      Neat, huh?

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    13. Well 1. probably also depends a lot on the order of those with a signal vs those without, so it not making a difference is not that surprising and makes some sense.

      2. is very surprising and jarring, and I guess you win a cookie for pointing that out to me. Intuition fail. The math for this was not in the Bannerjee paper, was it? Can I see the math?

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    14. The math for this was not in the Bannerjee paper, was it? Can I see the math?

      Nope, I just made it up. OK, so imagine that the right restaurant is B and the first guy in the line gets a (wrong) signal for A. First guy goes to A. Second guy gets a (wrong) signal for A and goes to A. In the old version, third guy would know for certain that the first two guys both got A signals (because if second guy had gotten a B signal he would've gone to B). But now, when 3rd guy sees 2nd guy go to A, he knows there's a chance that A actually got a B signal. So if third guy gets a B signal, he's now more likely to go to B, thus breaking the chain...

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    15. That's one specific example of where it could break the chain, but aren't there other eventualities where the 50/50 split leads to building a chain when it would have instead been broken (that's almost definitionally true... e.g. 1 gets an A signal, goes to A, 2 gets a B signal, goes to A on a 50/50 split)?

      Or perhaps I am misinterpretating how the model applies to restaurants given that the actual model is for investment amounts....?

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    16. That's one specific example of where it could break the chain, but aren't there other eventualities where the 50/50 split leads to building a chain when it would have instead been broken (that's almost definitionally true... e.g. 1 gets an A signal, goes to A, 2 gets a B signal, goes to A on a 50/50 split)?

      Yes, actually! Herding chains (i.e. chains of incorrectness) become easier to start, but easier to break. In the long run, though, the fact that B is the right restaurant (and thus more people get B signals) will make it impossible for any new bad chain to start. Whereas with an unbreakable chain, the utility loss from one chain starting will be infinite. So the utility gain from making people a little more conformist can be very large...

      For investment, as opposed to restaurants, it gets a little more complicated, but you get a similar result; adding noise, or randomness, breaks chains.

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  3. I found your article a little difficult because you are talking about at least three types of precision:

    * precision of term definition
    * precision of mathematical manipulation of those terms
    * precision of quantitative information.

    Ironically, you were not always precise about which precision you meant.

    And I'm not sure what to do with your statement "Quantitative statements must be precise", because in the sciences there is no such thing as precise in a measurement: it is made with a particular precision, with a range of error (or uncertainty.) I suppose the question is whether it is sufficiently precise so that the results of the operations upon it will be of useful precision.

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    1. Ironically, you were not always precise about which precision you meant.

      Yup! I just meant precision of definition.

      I'm actually not sure what your second type means...

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    2. Mathematical manipulation consists of use of operations/properties/logic/etc. in a non-fallaceous manner. 2+2, set operations, conditional/boolean logic, yada yada. You obviously know it, and know that it is necessary for moving beyond verbiage.

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  4. "That's why, when debates get really long and confusing, I think it's time to whip out the math, define terms, and get really precise. (By the way: In my experience, defining terms is really the critical piece of this. It's very very hard to make imprecise statements when all your words are precisely defined!)"

    have you noticed the recent debate on the blogosphere about whether this recovery is longer than other recoveries from financial crises?? it turns out that it all depends on what you mean by "financial crisis", "recession" and "recovery". i had never thought those could be imprecise terms, but apparently they are!!

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    1. have you noticed the recent debate on the blogosphere about whether this recovery is longer than other recoveries from financial crises?? it turns out that it all depends on what you mean by "financial crisis", "recession" and "recovery". i had never thought those could be imprecise terms, but apparently they are!!

      This is going to be the subject of my next post... ;-)

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  5. Noah,

    Please provide concrete examples other than matching theory of economic models that have suggested precise, quantifiable predictions.

    McCloskey's criticism of mathematics in economics is that the vast majority of theoretical models make only qualitative statements, proving the existence and uniqueness and existence of equilibria, without suggesting any magnitude to test. Would you comment on that?

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    1. Please provide concrete examples other than matching theory of economic models that have suggested precise, quantifiable predictions.

      1. Vickrey-Groves-Clarke auctions (which are used to sell Google ads)

      2. McFadden's Random Utility Model (which predicted BART patronage before the train was built)

      3. Dave Agrawal's model of local sales taxes (predicted the amount by which local sales taxes reduce differences in sales taxes across state borders)

      4. Black-Scholes-Merton option pricing model (predicted the relationship between the price of options and various other asset prices)

      Note that those are all out-of-sample predictions, too.

      There are many, many, many more. They're mostly in tax econ, labor econ, public econ, financial econ, environmental econ, agricultural econ, and other areas of microeconomics.

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    2. McCloskey's criticism of mathematics in economics is that the vast majority of theoretical models make only qualitative statements, proving the existence and uniqueness and existence of equilibria, without suggesting any magnitude to test. Would you comment on that?

      Sure. First of all, purely qualitative predictions can be conceptually and definitionally precise, as I mentioned before - "f'(x)>0" is a purely qualitative prediction. This kind of prediction can be useful, although not as useful as an equally reliable quantitative prediction (which necessarily conveys more information). For example, "aspirin significantly reduces headaches" is a purely qualitative piece of information, but very useful to know, even if you don't have a mathematically precise measurement of how much it reduces headaches.

      Second of all, proving the existence and uniqueness (and stability) of equilibria is not a value-less activity, though in practice I find that the assumptions required to prove existence and uniqueness tend to be either patently false or untestable.

      Third of all, I agree with McCloskey that quantitative predictions are the ultimate goal.

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    3. Thanks, that's great.

      I think what really gets missed in the qual v. quant debate, which is by no means exclusive to economics (soc and poly sci having big fights about it, and I understand the phil of science oriented guys in physics discussing it), is that what we're conventionally calling qualitative isn't qualitative -- it's a *degree* of quantification. The argument here is one of degree, not kind.

      So in social science, we have always talked about what is more and what is less, and which way variables lean and tend.

      What we're interested then is the relative margin of benefit to applying varying degrees of quantitative precision to different applications. And that's a matter of appropriate tolerances (also McCloskey's point, which she's always invoking Herb Simon and his being influenced by comp scientists and engineers in his loosening to bounded rationality).

      What I think we get with the qual v quant dichotomy is the idea that mathematics provides better tolerances on almost every problem, almost all of the time. That it's maybe ok to have some qualitative theory, if some laggards are going to *insist* on a methodological hissy fit, but that mathematics would be preferable in most cases.

      That's sadly a product of disciplinary fashion and signaling, much like writing books in Latin in the 16th century in order to establish intellectual ethos.

      Oh, and I don't buy the argument that people use math in deliberate obfuscation any more than I buy the argument that Foucault and Derrida used their vocabularies for deliberate obfuscation. I just don't think people are that small and knavish. I think people get genuinely convinced that hyper-technical vocabularies are necessary to establish the ethos of their argument. And that of course creates inefficiencies and monopolies in disciplinary discourse which people I think get justifiably upset about (holding constant the Occupy complaining; talking academics here).

      Until academics take a hard look at the degree to which prestige and aristocratic ethos dominate their institutional equilibria, I don't think we're going to get a lot more cross-talk, competition, and gains from trade that we might otherwise realize.

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    4. Oh and on Black-Scholes-Merton: the residuals on that model when first implemented were huge, something like 40% I think. Then over a few months you saw that gap close miraculously. To me that suggests firms and traders merely started playing to the model's predictions. I think I got that tidbit from Gigerenzer in Rationality for Mortals -- incredible book.

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    5. Oh, and I don't buy the argument that people use math in deliberate obfuscation any more than I buy the argument that Foucault and Derrida used their vocabularies for deliberate obfuscation.

      Hmm, well, maybe you're right...I always kind of suspected Derrida of that especially...

      Oh and on Black-Scholes-Merton: the residuals on that model when first implemented were huge, something like 40% I think. Then over a few months you saw that gap close miraculously. To me that suggests firms and traders merely started playing to the model's predictions.

      Well, 40% ain't that bad! But the convergence effect is extremely interesting, and is the subject of a paper I am trying to write.

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    6. Foucault and Derrida definitely, definitely did this. I can do it too. If I was more of a dick, I am sure I could have had a nice career in academia as an English Professor writing obscurantist screeds on gender identity, sexual politics, and projecting sexual/Freudian/gender connotations onto just about everything.

      The only literary-theory-ish idea I use regularly is the problem of the difference between the sign and the signified, i.e. arguments about semantics. I think that that is just about the only English idea that the sciences would benefit from engaging with more.

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    7. Anonymous3:37 PM

      Can anyone provide references which document the Black-Scholes-Merton convergence? Sounds very interesting.

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  6. Anonymous12:11 AM

    What is notable is the focus on "precision", and the entire failure to include ANY mention of the distinction between "precision" and "accuracy".

    Precision without accuracy is useless.

    And any system that promotes precision but ignores or dismisses accuracy is worse than useless, it is misleading (and often dangerous).

    Indeed, the focus on precision (while ignoring accuracy) is the hallmark not of science, but of pseudoscience. To wit, astrology is very "precise" (even "logical" and "mathematically elegant/complex"), yet it is utter nonsense (well, except of course to those who BELIEVE in Astrology) -- likewise with a host of other similarly inane things: Biorythms, Phrenology, etc. Precise measurements and calculations, mapping, mathematics, etc -- yet ultimately little more than deceptive nonsense.

    And the same of course can be true of "science" and theories concerning things that are scientifically observed -- entire elegant, logical, and mathematically complex and "precise" systems can be built up (Cf Ptolemaic Astronomy) -- and yet be entirely wrong in their basic fundamental assumptions; rendering all of the "math" as a worthless exercise in nonsense.

    So, again, the failure to mention (or apparently even consider) the importance of accuracy, and it's distinction from "precision" -- combined with the over-emphasis on the latter -- is really rather informative concerning the dogmatic blindness of the author.

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    1. I was talking about conceptual precision, not mathematical precision, in this post. But of course you're right about mathematical precision vs. accuracy. Assessments of the accuracy of scientific theories require out-of-sample predictions, which is why math alone is not science, even though it is a key tool of science.

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    2. Anonymous1:13 AM

      Precision without accuracy can be very useful. This comes up in my work often. I have precise but inaccurate measurements of A and B, but since I only need to know how A relates to B that is enough.

      Precise and accurate information would be more useful than just the precise information, but that doesn't make just the precise information useless.

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    3. I think you may just not be getting what I'm talking about... ;-)

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    4. I think Noah's referring to what we sometimes call "clarity" when he says "precision".

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  7. i forgot to mention another debate that's been going on in the blogosphere, the one about the burden of debt. it turns out that it hinges on the definition of "future generations"!!

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    1. Exactly, which is why I linked to that debate in this post... ;-)

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  8. I saw what you did there Noah!
    There is no "correct" definition of the word "math", any more than there is a correct definition of the word "art", or the word "love".
    You are equating (pun intended!) math with art and love. And non-math people were complaining about Shapley's pig-headed remarks!

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    1. And you unfortunately have completely missed the point I was making with regard to Shapley. I can definitely agree that math is in many ways more like art than science. And that wholly explains why hard work alone is not sufficient to gain fluency in it, and why Shapley's dismissal of people who struggle with math as "lacking concentration" is pig-headed. No-one would deny that music, painting etc. require innate talent as well as the ability to work hard. Why should math be different?

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    2. You are equating (pun intended!) math with art and love.

      No, I'm just pointing out that words do not have "correct" definitions.

      And that wholly explains why hard work alone is not sufficient to gain fluency in it, and why Shapley's dismissal of people who struggle with math as "lacking concentration" is pig-headed. No-one would deny that music, painting etc. require innate talent as well as the ability to work hard. Why should math be different?

      Frances, go read Shapley's comments again. He says that people don't do math because they lack the ability. Then he adds a parenthetical to allow for the possibility that some people have the ability and lack the desire. But clearly he thinks that lack of ability is the main thing holding people back from doing math.

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    3. "the reason that your friends and ours cannot understand mathematics is not because they have no head for figures, but because they are unable [or unwilling, DRH] to achieve the degree of concentration required to follow a moderately involved sequence of inferences."

      Noah, being "unable to achieve the degree of CONCENTRATION required" is not the same as "lacking ability". Concentration is required in many fields, not just math.

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    4. I wouldn't focus on the word "concentration". He just means the mental ability in general.

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  9. So math is really just logic? Or would it be better to jettison the "logic" condition and just emphasize precision on its own? What do you think of Feynman's comment that he thought his contemporaries (philosophers of science) were wrong in thinking that meanings in science always had to be precise?

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    1. I agree with Feynman on this.

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    2. Precision is not necessary in mathematical models of reality. Precision is an inherent characteristic of mathematical methods, but there are entire fields of mathematics devised to get around this problem (such as statistics), and students are warned that precision does not mean accuracy.

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  10. Doesn't it depend on what the field is trying to do? Physics, for instance, is trying to describe the properties of the material world. What we do with the understanding is up to us, and the description is true or false by reference to the material world, so it doesn't matter to us non-physicists if the description is in mathematical terms or in Sanskrit or whatever.

    But economics as a field is concerned with human behaviour. Its descriptions of behaviour matter, and are true or false to the extent we understand and follow them. So its inescapably value-laden: it's a form of moral philosophy. Assumptions couched in mathematical terms are invisible to the people which they affect, which is surely a moral issue in itself?

    Also, economics is isolating some particular aspects of behaviour, but people act from lots of motives, some individual, some social, some genetically disposed. If economics is to tie in with those other fields of understanding, it needs to have a common language with them. the discourse of people about their motives is not mathematical, although formalising some statements may help elucidate the discussion. How, for instance, does one capture "power" mathematically? It's an imprecise but pervasive driver in human affairs.

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    1. I don't understand what you mean, sorry...can you try giving some examples?

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  11. Anonymous5:46 AM

    I am not as enthusiastic as you seem to be about the use of mathematics in economics. It is not that for economics as a science it is good to make imprecise statements, quite the contrary. It is that the commonly accepted standards of mathematics in economics very often make it impossible to ask the right questions. Why are the microfoundations in macro so bad? Why didn't DSGE modelling focus more on financial frictions in the past? I would argue that it's mostly because it is difficult to incorporate these things into general equilibrium models from a mathematical viewpoint.

    Another thing is that even if you can find a couple examples where good predictions were made by economists using maths (Shapley-Roth, auctions, etc), there are also many examples of laymen making great predictions without any maths at all. Rubinstein once said that he had never seen a single example where a game theorist could give advice, based on the theory, which was more useful than that of the layman (http://thebrowser.com/interviews/ariel-rubinstein-on-game-theory). I tend to agree. Non-mathematicians can come up with reasonable matching algorithms even without proving that they are stable or incentive compatible or whatever, which is what the maths used in economics is all about.

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    1. Brett7:13 AM

      This isn't quite true, even the most difficult of math used in DSGE is nothing compared with, say, solving diracs equation (and I've dobe both). The problem isn't too much relaince of maths in ecomics it's that we don't teach economics undergrads enough math and therefore economics grad students/professors often don't have to tools to solve more difficult problems.

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    2. Anonymous8:40 AM

      Maybe DSGEs were a bad example as they are clearly not the most mathematically complicated animals one can find in economics. In general, I think you would be hard-pressed to find many fields of mathematics that have been successfully used in physics but never used in economics or econometrics or mathematical finance in one way or another. If economists really were so terrible at mathematics and physicists so great at it, econophysics would have conquered top economics journals a long time ago! My general point is that some things are very hard to model given the current state of the art, and our inability to model them mathematically often precludes us from even asking about them in the first place.

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    3. It is that the commonly accepted standards of mathematics in economics very often make it impossible to ask the right questions. Why are the microfoundations in macro so bad? Why didn't DSGE modelling focus more on financial frictions in the past? I would argue that it's mostly because it is difficult to incorporate these things into general equilibrium models from a mathematical viewpoint.

      I agree, and I've made this point in earlier posts about DSGE...

      This isn't quite true, even the most difficult of math used in DSGE is nothing compared with, say, solving diracs equation (and I've done both).

      Oh, totally true!! The barrier presented by DSGE is not that it is mentally hard to add interesting features (it's not!), it's that it's incredibly clunky...that makes statistical estimation of the models impossible, makes "calibration" of the models less credible, and causes a negative publication bias.

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    4. In other words, the problem with DSGE is not that it's hard to do. It's that it's hard to do in a way that successfully captures any features of the real economy.

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  12. Great post, Noah. I particularly agree with the point on maths being able to illuminate counterintuitive results. On that note, you may be interested in this older post which covers very similar ground. (As you'll see, I actually tried to go through two examples that illustrate this principle at work.)

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  13. Brett7:16 AM

    I think you alluded to but missed a very cruical role math plays in economics: Accounting.

    If you have a few simple relationships it's relatively easy to keep track of whats going on in your head or with 'wordy agruments'. But as soon as you introduce more than a few simple relationships the purpose of maths becomes to keep track of them.

    As you point out this is espeically important when perfectly logical (wordy) arguments provide different answers to the full mathemetcial exposition.

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  14. Excellent post--I will have to refer all my econ-skeptic friends here.

    One point I would add--and this is something that Krugman said--is that a big part of why we use math in econ is actually because it makes things a lot easier. I suppose you could do econ without math in the same sense that we could have accomplished the moon landing by trial-and-error, but it is way easier to derive a mathematically precise model that will tell us exactly where our logic fails.

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  15. Just a little warning:

    Yes, there is a language of mathematics (predicate logic).
    It needs operational semantics to make it "understood".

    In axiomatic mathematics two functions are seen as identical if their input-output relations are identical.
    In algorithmic mathematics one looks into performance, resources, accuracy and robustness..

    Mathematicians like closed form solutions, they allow for exact results. But closed form solutions are usually only valid in small subdomains ..

    In computer mathematics you have symbolic and numerical commutation ... asymptotic mathematics is the combination of both (decompose into subdomains, where exact solutions are possible and recombine ...)

    However, in any domain, it is highly recommended to organize problems, models, solvers and their implementations orthogonally.

    Take the Black Scholes PDE .. It has a closed form solution with much to simple assumptions. In its extensions you need numerical schemes. If you naively use, say, tree base methods your system will blow up when calculating the greeks of a barrier option, even worse with calibration ... Not to speak of stochastic volatility models with jumps ...

    So, math is great, but is bears traps that can become horrible in interplay ....

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  16. Anonymous8:49 AM

    What was Keynes' attitude to math?

    Better to be vaguely right than precisely wrong?

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    1. Yep. I agree with that, and I tried to re-state that idea near the end of my post...but Keynes was far pithier.

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    2. Blue Aurora2:42 PM

      Just for everyone's reference, if you want more information on John Maynard Keynes's attitude to mathematics, I highly recommend reading this article in Synthese.

      http://www.springerlink.com/content/l5457w0721nhrrx5/

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  17. If you keep math in your theory and never try to apply it to reality you'll be just fine leaning on mathematics. But economic reality can't be modeled.

    It's been proven time and time again from long term Capital to the London whale. The same way the nautilus shell diverges from mathematical precision. That's a simple example but This is why economist must always prefix things with "all else being equal" but all else is never equal. the economy has real live thinking participants. These participants can't base their decisions on knowledge because when it comes to anything in the economy you can never get an interdependent variable. Contrary to popular belief supply & demand or anything else in the real economy is independently given.

    If everything is dependent on the participants views and participants views are dependent on say, supply and demand, which is dependent on other participants views, where do you get an independent variable from? And if you can't get an independently given variable how can any of your functions be trusted? Without that independent variable none of your functions can truly be uniquely determined.

    In other words you have all dependent variables so sooner or later your model will break down... .

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    1. We are well aware of the problem of simultaneity, which is what you are describing. In empirical research, the way to deal with it is through instrumental variables, which is just a fancy way of extracting exogenous variation (ie, independence) from an otherwise endogenous (dependent) variable. In theory, we simply reduce models down to "structural" parameters, which are exogenous, and this is where the whole microfoundations debate comes from. But for practical purposes it is often quite acceptable to ignore a lot of the endogeneity that exists in the real world. For example, in reality government policy is endogenous to economic conditions, since voters and politicians base their policies on what's happening to the economy. But the political process is sufficiently slow that this is turns out in most cases to be almost identical to the case where we assume that policy is exogenous.

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    2. soolebop2:00 PM

      thing is, by doing so you can't account for Rflexivity. This means you may be able to successfully model government policy but extracting exogenous variations may be the reason that in practice government policy seldom if ever get's it's intended results...

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    3. In science math is a modelling tool. Economists routinely use the wrong sorts of mathematical models, FWIW.

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  18. correction *supply & demand is NOT independently given**

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  19. An interesting question- what is math?

    In my view, math is the language of quantity. The validity of quantitative analysis, barring the discovery of ultimate quantum, is built on (not unjustified, to an extent) faith in common behavior of qualifications (apples, by and large, behave life apples).

    In other words, math is only as precise as the qualifications one attempts to quantify.

    This view might shed some light on Einstein's quote: As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality.

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  20. Math does not give precisely accurate predictions about n-body systems or n-compartment models. You can prove that general equilibrium "exists", but you cannot derive what the prices will be. If you conform your economics discussion to the math, that is one thing. If you cause people to believe that you will always derive real answers to real questions, then that is quite another.

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    1. "Math does not give precisely accurate predictions about n-body systems"

      Math does not give a closed form solution but it can give highly accurate computational approximations. The practical limit on the accuracy comes from the limit on one's ability to parametrize the properties of the bodies. If you are dealing with idealized point masses, you can get as much accuracy out of the calculation as you are willing to pay for in terms of computer resources. Years ago I was playing with idealized three body problems and it was relatively easy to get to 12 digits of accuracy.

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    2. 12 digits of accuracy applying to a real world problem, using which real-world measurement techniques for verification?

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    3. Lee

      Inferred accuracy from the fact that energy, momentum and Jacobi integral were all conserved to very high accuracy and the method had internal checks on accuracy.

      Implicit in my post is the fact that when you are dealing with the real world an answer accurate to 12 digits is completely accurate because it is beyond anything you are ever reasonably going to be able to measure. Even if you had a closed form solution you would not both evaluating it to twelve decimal places.

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    4. The question is, how did your model perform against the real 3-body system it modeled? You have explained that it is accurate because theory says it is so.

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    5. Sorry, Mr. Arnold, but Absalon is probably too busy helping land rovers on Mars to respond further.

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  21. This debate just illustrates so much of what is wrong about economics. People talking past each other arguing over definitions. Mathematics is a huge field (from counting how many lions in a field to the work of Godel, with number theory, algebra, geometry, calculus, graph theory, etc in between). It may make no sense to try to define mathematics.

    Noah - your statement number 4. "Precise statements often lead to unintuitive but logically inescapable results." scares the shit out of me. One of the things I see is that people come up with a simplified model, ignoring what we can loosely call second order effects, and then say that their model leads to some inescapable logical result. They then seek to impose their model on the world, forgetting that the model was always only an approximation to the real world and a model that would inevitably break down as you move away from the status quo (as the perturbations get bigger, the second order effects become important).

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  22. Anonymous1:28 PM

    "We may want to make precise predictions about what will happen in a market."

    Hehehehe.

    Mathematical systems (neo-classical) fail to say anything useful if they do not relate to reality. And they dont.

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    1. Ah, but some "neo-classical" systems describe reality very very well!

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    2. Not some; most of the results of price theory in the 20th century are extremely robust. Supply and demand curves are everywhere -- the rest is just details, details one usually doesn't even need to talk in competent terms about supply and demand, opportunity cost, etc.

      Here's a neoclassical idea that escapes most people still: a transfer moves resources from one set of hands to another without any value added. An arbitrage creates value. The vast majority of people in the world continue to intuit that economic exchanges are zero-sum transfers, and to these people the idea that the theory of arbitrage leans entirely on fancy and irrelevant mathematics is appealing.

      Ooof.

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    3. Well, "neo-classical" (and I use the term loosely because other people seem to use it loosely) ideas generally fail when applied to decision-making under risk and/or uncertainty - basically, anything that is both dynamic and stochastic. We just do not understand much about this type of behavior.

      But when the situation is static and/or deterministic, neo-classical descriptions of human behavior are generally very accurate, in lab experiments and in empirical studies.

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    4. It sounds like the theory of arbitrage is the reality of specialization and trade. However, if it is "financial arbitrage", it looks like there can be an increase in money value without a corresponding increase in real goods and services. Any price-bubble demonstrates this. On the other hand, if you are speaking about "transfer" such as in a welfare transfer, it may be considered an increase in "social" value, IF the alternative is disadvantage or destitution which imposes a lessening of human capitall, or a social cleanup-cost.

      In other words in social transfers you may be arbitraging wealth-positions in society, thus avoiding greater social transaction-costs later.

      Taking it very abstractly, the only two ways we can economize are 1. specialization and trade, and 2. reducing transaction costs.

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    5. Anonymous9:04 AM

      @Graham - but what about the negative externalities generated by arbitrage.

      (As an aside - a big problem with economics is that it notices effects like "arbitrage can be positive" but is really bad at quantifying them. And worse, economists make huge claims on the back of the noticing.)

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    6. Actually, Graham, I've seen some very solid arguments, supported by evidence, stating that supply curves don't exist in most markets.

      Most markets really have sellers as "price makers" defining quantity and price *well in advance* with buyers as "price takers". If sellers get into a position of competition, the first reaction is product differentiation and the second is advertising.

      So, there is no supply curve. Supply curves appear to exist only in commodities markets (coal, bulk grain, etc.)

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  23. Wonks Anonymous2:09 PM

    Math is the "universal language". It is not analogous to latin.

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    1. Mathematical symbols — like Latin, which was once the universal language of the academy — are only universal to the extent that other people understand them. I am not saying that mathematical symbols are not a convenient means of expression in certain circumstances. But I do wish that certain wonkish papers spent more time clarifying so that interested amateurs such as myself can grasp what a paper is trying to say.

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    2. Anonymous6:38 PM

      Learn math?????????

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    3. Learn latin???????

      Then you can read the holy scriptures...

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  24. Anonymous8:01 PM

    Math does not equal logic.

    Never read Russell or Godel it seems clear.

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    1. Argumentum ad verecundiam? Weak sauce.

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    2. Depends on what you mean by equal. There's logicism, which (roughly) is the idea that mathematical propositions are propositions about logic, that math as we know it can be reduced to basic tautologies about the necessary nature of reality, and there's formalism, (which Noah explicitly name dropped) which supposes that math consists of the logical manipulations of what are ultimately arbitrary formal systems. (Although less arbitrary when you decide what systems to put to work to apply to the real world.) Logicism runs into some serious problems because of the work of Russell and Godel and others, but formalism seems less effected by such things.

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  25. Anonymous12:44 AM

    Who could argue that math is not useful in economics?

    Rather than ask if math should be used in economics, wouldn't the more relevant question be, should it be the only tool of analysis used in economics? Unless you argue it should be the only tool, I'm not sure what is being added here (but, I think the graphic with the brain head is pretty cool and the book looks interesting as well)

    Warming to the topic, I wonder, In economics, is one example usually sufficient to prove a universal statement? Since Roth's theory required mathematics, and it produced a useful practical application in economics, is this proof that math is always useful in economics? Have all economic theories that used mathematics produced useful practical applications? If some of that math did not produce some useful practical application or actually produced something approximating a load of haddock and butter beans, was it immediately recognized as such and eaten with potatoes?

    Given the volumes of haddock and butter beans that are passionately baked and fried in economics, Doesn't Roth's trifecta seem more of an outlier here, anyway?

    I've enjoyed your blog, but it strikes me as funny (looking) to use words to argue that math is superior way to formulate analysis. I guess it really depends upon what you are trying to study, convey, and the issue at hand. Which means, again, that we need more of both words and math (and pictures of brains) and not all of one and none of the other. Well, maybe not the pictures of brains, but its a nice touch and worth the triple A rating I gave this blog post.

    I wonder, just because I'm not really sleepy yet, has mathematical analysis ever been misused in economics? Ever? At least in my experience, mathematical explanations can produce a rigorous logically consistent analysis, and end up producing useless results because of flawed assumptions.

    Are economists really good and clear about always evaluating the assumptions they use to build their models? Are there any examples where they have not done this? If these are plausible scenarios, could it be that individuals who criticize the use of mathematics in economics are critical of the idea that 'math fixes everything' in economics?

    If math is the perfect language that allows us to accurately analyze the economic system, why can't the physicists fix our economy? Or, at the very least, could they explain how it works well enough to stop the thing from blowing up every 5-7 (Dimon) years?

    Would you argue that modern finance, and the rise of mathematical finance over the last 40 years, has fixed finance and made markets that were more efficient at pricing and allocating risk? Why or why not?

    Most importantly, could we use mathematics to rid the world of Tucker Carlson? I find him endlessly tiresome. Is Tucker so annoying because he is bad at math, or is there some other reason? Is Tucker Carlson as annoying as an environment that allows random people to review a really reasonable short essay (as in your blog post) and then squat all over it with a series of random non sequiturs and challenges that you really aren't even being paid to consider? For instance.

    In using math, Do economists ever go back and just fundamentally review their assumptions about seemingly innocuous things that could potentially be conventions with unintended or unrecognized implications, things like the law of demand, or equilibrium, or the natural rate of output? Would it be possible to rebuild economics on an entirely scientific and non-value laden analysis if we eschewed all language other than mathematics? Would it be relevant to humans? The gleisner robots?

    Is it necessary to defend the current practices of economics from the math critics? Why? To rub the non-economist's noses in the fact that they aren't being paid to do math? To present examples of the use of mathematics that actually tasted better than a hillock of baked horsepaddies and pinto beans?

    Anyway, my work is done

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  26. math is fine providing that a particular representation remains true to the subject matter. When Keynes talked about effective demand, expectations and uncertainty there was an implict causal ordering. Hicks replaced Keynes description with a system of simultaneous equations and lost the causal ordering; See Pasinetti's essay "The economics of effective demand" for the steps Hicks took and how the proper interpretation of Keynes description got lost.
    On how to represent Keynes on the topic, do it in algorithmic form as in a computer program.

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  27. Anonymous9:07 AM

    The problem with mathematics and economists (note, not necessarily economics, but economists) is usually and repeatedly about false precision. The mathematics induces two beliefs:

    1) The math is so beautiful, it must be correct, who cares about the evidence?

    2) If we express this in mathematics we can calculate X with precision. However, in reality, there is often no precision in the variables. This is where words are often a more useful way of expressing something, because they have more degrees of nuance about imprecision, especially for unwary economists.

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    1. Anonymous11:29 AM

      ^ Have you actually read a real empirical paper? A ton of work shows how the real world deviates from simpler models, and an enormous part of econometrics is about calculating proper errors on your estimates.

      Every time I hear someone say "economists must take mathematical results as the be-all end-all," it's usually someone who doesn't actually know what economists do.

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  28. It is better to be generally accurate then precisely wrong.
    3.1416 is more generally accurate than 3.14155 because the next digit after the first 5 is 9 not another 5, and 3.1416 is closer to the true value, less 'precise' but more accurate.
    Precise statements are understood to be qualitatively more accurate than imprecise statements, the meaning of 'accurate' and 'precise' changes between the context of statement versus numbers, it's a use of language distinction.
    In the original post you could substitute 'accurate' for 'precise' in (1,2) and be making a more precise statement which is also more accurate, we don't want a theory that predicts reduced unemployment down to the last person if the actual result is increased unemployment in tens of thousands.
    The usefulness of math in economics is whether it allows proper representation of the underlying reality. Hicks (accidentally) mathematically misrepresented Keynes argument; not the fault of maths per se; but if there is causal orering among variables you better watch out on how to structure the math representation.
    (3,4,5) of the post require accurate representation of the underlying proposition before you start making predictions. Hicks missed out there. We all (or most) thought Hicks was explaining Keynes, and he wasn't.
    Theory and its mathematial representation can go so far: Capitalism may be inherently unstable so there may be barrier triggers, "and where you cross this value, someone better do something or we are in one deep mess".
    The "do something" might involve judgements about moral hazard, future generations, ...beyond the everyday business of economics managing within the target boundaries.
    One problem i see with current economic theory is that it is apparently unacceptable to say, "and if you do what this theory says you may run off the rails", so the theories we get are structured so the predictions never run off the rails, and are wrong, so we do run off the rails.

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  29. Noah

    You commented that you did not understand my points above at 2:10 am.

    Point 1 is that economics is not just descriptive, but also - inescapably - prescriptive. Any argument about that's the most efficient way to produce or allocate something has to reckon with "why produce that? "what makes efficiency your preferred criterion?" "who gets what?" and similar issues. In other words, the assumptions are social and moral. Extended casting of economic arguments in mathematical terms both obscures this, and removes the ability to participate from the people affected. When economists argue that, for instance, the maths "proves" that the financial system is safe, or that markets will allocate more equitably than the alternatives, it's much harder to take the argument apart (and it carries more authority) than if it were put in English. It seems to me that quite often the economist themselves miss the assumptions behind the math.

    Point 2 is that economics explains only a small part of human behaviour. For large parts we have to turn to sociology, history, psychology and other related disciplines. If economics walls itself off behind a mathematical facade, It narrows our ability to explain any particular instance.

    I am not arguing against mathematical testing of economic arguments. I am arguing for Marshall's point - that the words better enable you to see the foundations and linkages.

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    1. Point 1 is that economics is not just descriptive, but also - inescapably - prescriptive.

      Actually, I don't think that's true at all. Suppose I use a computer to create a random combination of economic variables, and call that combination "blubglub". Now you ask me what sort of market institutions, utility functions, etc. would maximize blubglub. I could give you that answer. But "blubglub" is not prescriptive - it's just some random crap a computer spat out.

      In other words, the assumptions are social and moral.

      But I don't think that follows, logically. Even when economics is prescriptive, the assumptions don't have to be social and moral. In fact, I think they should not be - they should simply be as realistic as possible. Why? Because accurate, realistic assumptions are the best tool to achieve our social and moral goals.

      Extended casting of economic arguments in mathematical terms both obscures this, and removes the ability to participate from the people affected.

      If what you mean is that formalism can obscure the prescriptive content of an econ paper, then you're right.

      Point 2 is that economics explains only a small part of human behaviour. For large parts we have to turn to sociology, history, psychology and other related disciplines. If economics walls itself off behind a mathematical facade, It narrows our ability to explain any particular instance.

      But I think narrowing is exactly the point. Narrow phenomena can be understood more completely than broad phenomena.

      Note that disciplines like history produce very few usable conclusions.

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  30. I`m using some of the ideas found here (http://www-bisc.cs.berkeley.edu/Zadeh-1973.pdf) in some of my research on technological development in renewable energy tech. It`s quite applicable to this discussion. Basically, the more complex and dynamic a system is the less value precise statements will have.

    Stable systems probably can be represented quite accurately with mathematics, but it might not be so useful in understanding their evolution. However this assumes an ontological problem with mathematical representation.

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  31. "the assumptions don't have to be social and moral. In fact, I think they should not be - they should simply be as realistic as possible."

    And people are not, of course, moral or social animals. So assuming away morality and sociality is, therefore, entirely realistic. or perhaps economics is not meant to about about humans, but just a disembodied theory of constrained choice (without actual, you know, choosers). In which case the conclusions are usable exactly how?

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    1. And people are not, of course, moral or social animals. So assuming away morality and sociality is, therefore, entirely realistic.

      But wait...including descriptions of real people's actual morality in economic theory (which lots of people do) is different than imposing the economist's own personal morality on the conclusions of the theory (which, unfortunately, some people do as well). Those are just very very very different things.

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  32. I miss it back in the day when I was the only one who commented. Then when you ignored me I felt special. Now I'll just feel like everybody else.

    On the plus size...I'm really hoping that you think about the opportunity cost concept when trying to read all the comments. I'm too lazy to look up your definition...something about adding up the reverse reciprocal value of all the other possible activities in the world? I think it required more math than I could handle.

    The other day you gave me a gift when you mentioned that your adviser believes in the Great Pumpkin. Of course you know that there's no such thing in economics as a gift. You were just burdening me with the obligation to reciprocate. Here's what I'm giving you in return. Oh man, it's really good. It's an epic milestone. It's the world's first liberal blog entry dedicated entirely to the opportunity cost concept...Forced to Choose: Capitalism as Existentialism. Because economics is all about adding value...the value that I'll add is my response...Helping Liberals Understand the Opportunity Cost Concept

    Well, I've often wondered if math is a universal talent (like spoken language) or a rare one (like music). I'm not sure which it is.

    I guess it's just that I have trouble understanding what you write...

    Maybe not quite universal...cause...honestly I talk worser than I write.

    You know what really bugs me? Eskimos have 100 words to describe "snow" and we don't have one word that describes the opposite of "to waste". For example, "don't waste your time". What's the opposite word? There IS no opposite word. There's produce...but we're not going to say, "be sure to produce your time"...we're going to say, "be sure to use your time productively". Another example, "a mind is a terrible thing to waste." What's the opposite, "a mind is a wonderful thing to produce."? That only produces an image of an assembly factory that produces brains.

    Yeah yeah...it's driving me nuts. I want to write my congressperson or something. But believing in congresspeople is like believing in the tooth fairy...but way worse. So what can I do? Do you know any wordsmiths? I'll pay top dollar* for somebody to coin a word that is the opposite of "to waste".

    *$10 (after it's added to the dictionary)

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  33. Anonymous2:02 AM

    "Note that disciplines like history produce very few usable conclusions"

    So, Noah Smith, no conclusion can be drawn from the Nazi holocaust of WW2?

    Are economic forecasts not often conclusions based on historic (repeat, historic) time series data?

    Math only useful for understanding models and not how real-world economies work. Which math model was able to predict the global downturn?

    Math is not used but rather abused in economics to obfuscate and enhance an author's chance of publishing in neoclassical oriented journals. And, of course, to show how 'smart' the author is. How many poliy makers actually read those journals?

    Mathematics is not science; it is only a tool of science and just like any other tool is often abused (usually by neoclassical economists)by those economists imbued with huge doses of physics envy.

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  34. Noah

    To put this as simply as possible, to be realistic and usable, economics has to incorporate politics and morality in any model - just because no description of how people produce or distribute wealth is or can be politically or morally neutral. Because politics and morality are, inter alia, ABOUT the creation and distribution of wealth. Pareto optimality, for instance, is a deeply politically conservative notion. The math is just one way of working out the logical correlates of some particular set of assumptions about behaviour (hence some set of political and moral assumptions). If these correlates are to be applied in the real world to real people, those real people who will bear the consequences need to be able to test the reasoning. Which they cannot do if it is availble only in mathematical terms.

    An appealing alternative is that all economic models are tested on a remote island populated only by economists.

    History is not there to produce conclusions other than on what actually happened. It's there to test conclusions. Preferably before application.

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    1. Incorporating politics and morality into descriptive models of the world does not mean that economists must make prescriptive statements.

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  35. Business people are skeptical about math because a simple rule of thumb gets you very close to the correct answer. Moreover, theory is sometimes overkill in the context of making empirical predictions.

    In Corporate Finance, there is the Tirole model which uses math to model managerial moral hazard. The result is that positive NPV projects sometimes aren't funded if the manager doesn't have sufficient capital. Then you make empirical predictions and test the model.

    A business person would say "you shouldn't invest if the manager doesn't have enough skin in the game". The empirical predictions would the same as above. So does the Tirole model really add any value?

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  36. If only I could get so many comments on a post!

    Simon Stevin, a financial mathematician, pretty much created modern western mathematics when he established the Dutch Mathematical School around 1600. As Stevin's maths was practical, rather than academic, today the Dutch use terms rooted in their language, rather than in Greek or Latin. The term for mathematics is is wiskunde, which literally means `the art of true knowledge'. Descartes was inspired to leave France for Holland after becoming disgusted by the cynicism of contemporary sophists, angered by the blindness of dogmatism and frustrated by the barrenness of scepticism, the result was the Discourse on the Method and the rest is history, as they say.

    Mathematics can be moral, certainly pre-nineteenth century mathematicians (and economists) would have done so. See my paper "Ethics and Finance: The Role of Mathematics" (http://papers.ssrn.com/abstract=2159196)

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  37. Mathematics is distinguished from science precisely because of the formalism of mathematics: in mathematics, you define all your terms precisely -- without reference to empirical matters or the real world -- and follow the symbol manipulations through precisely.

    Computer science is a branch of mathematics.

    Mathematics is *useful* for science entirely because it enables precisely defined models with precise, clear predictions. So, in response to David Rashty, the purpose of the mathematical model is to define the term "enough skin in the game".

    Otherwise the statement "you shouldn't invest if the manager doesn't have enough skin in the game" is unfalsifiable, because however much skin the manager has, it could be "enough" or "not enough".

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  38. I think you may be confusing precise with unambiguous; an approximation with bounded error is unambiguous, but (unless the error is negligible) is not precise. Mathematical expressions are necessarily unambiguous (up to what the symbols stand for), but I'm not sure that all unambiguous expressions are equivalent to a mathematical expression.

    Many large/complex systems can be approximated well most of the time, with no surprising differences between the English description and the mathematical description, but a more detailed mathematical model that correctly predicts some unintuitive aspect of the real systems behavior may disagree with the English description while agreeing with the less detailed mathematical description. Much of the progress of Physics follows this pattern.

    Your definition of math reminds me of Paul Graham's definition: "it would not be a bad definition of math to call it the study of terms that have precise meanings" (http://www.paulgraham.com/philosophy.html ¶10, I constantly misremember it as "specific meanings"). I've come to think of math as a set of notations for describing relationships between quantities, and methods for analyzing those relationships. Math is useful because it can be used to describe relationships between quantities that are measurable in the real world, and when those descriptions are accurate enough they can be used to make useful predictions.

    The advantage to expressing a model mathematically is that it forces you to be unambiguous, even to be unambiguously imprecise. If you can't express your model mathematically, you probably haven't finished figuring out what your model is, and you certainly can't test it against any real-world data. If you haven't found the right math yet, you haven't figured out what's going on, and I don't think avoiding math is going to help.

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