Monday, September 15, 2014

Is math "falsifiable"?



Ooooh, another chance to babble on about philosophy of science!

Kevin Bryan writes:
Arrow’s (Im)possibility Theorem is, and I think this is universally acknowledged, one of the great social science theorems of all time. I particularly love it because of its value when arguing with Popperians and other anti-theory types: the theorem is “untestable” in that it quite literally does not make any predictions, yet surely all would consider it a valuable scientific insight.
But Arrow's Theorem is a math result. It takes some definitions of mathematical objects as given, and it shows a relationship between those mathematical objects. Of course you can't "falsify" it with empirical observation, any more than you can "falsify" Jensen's Inequality.

I really hope there aren't "Popperians and other anti-theory types" running around out there complaining that math results are useless because they're non-falsifiable. There are at least two reasons that would be silly.

Reason 1: Pure math results tell us about what we can and can't do with math itself. For example, suppose we knew that it was impossible to factor an integer in polynomial time. That would have important implications for cryptography. Math itself is a technology, so math results can give us useful technological advancements.

Reason 2: Pure math results are necessary for math to be useful for engineering. One big reason - the main reason, I'd argue - that we make mathematical theories is because the math seems to correspond to real observable phenomena. As long as that correspondence holds, then we can predict things about the world just by doing math. To "use" a theory means to assume that the correspondence holds - to simply do the math and assume that it's going to give you useful results, without having to go re-test the theory each and every time. If you don't let yourself make that assumption, then all mathematical theories are useless for engineering purposes.

Modern engineers can do a hell of a lot of cool stuff just by doing math using theories from physics and chemistry, without re-testing those theories every time they want to build an airplane or synthesize a polymer. And computer scientists can do a hell of a lot of cool stuff just by telling their computers to do pure math. So if there are "Popperians" going around saying pure math isn't useful, they should think again.

Anyway, the rest of Kevin's post is quite interesting, and the philosophy-of-science literature it links to is even more interesting - here are a couple more papers in that literature: (paper 1, paper 2, paper 3). And here's the paper that started the discussion. Neat stuff. As blog readers must have already guessed, I've actually considered just quitting finance and working on this stuff instead, and maybe someday I will. When I'm old, perhaps...

60 comments:

  1. Please, go. No one dies when math professors play Sudoku. Greece and Spain will chip in for the movers if you take a few other economists who don't understand the way science uses math with you.

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    1. Are you one of the Austrian trolls, then?

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    2. No, I don't make your bestiary. You might be better with the nick names.

      Anyway, here's my creation story:
      http://thorntonhalldesign.com/philosophy/2014/5/8/change-your-mind-and-see-what-was-always-there

      Here's someone who clearly had the exact same experience I did:
      http://fixingtheeconomists.wordpress.com

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    3. I believe Noah had "Unclassifiable" in his bestiary.

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    4. But I'm definitely classifiable!

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    5. Anonymous11:21 AM

      Classified under "Who Gives A Shit"

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    6. Lars Syll gives a shit:
      http://larspsyll.wordpress.com/2014/09/16/krugman-and-mankiw-on-loanable-funds-so-wrong-so-wrong/

      Nancy Cartwright gives a shit:
      http://youtu.be/KZtT9J2vfps

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    7. Nick Rowe *should* give a shit:
      http://econospeak.blogspot.com/2014/09/explaining-other-things-equal-clause-to.html

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    8. I forgot two of the best ones!
      Steve Randy Waldman gives a shit:
      http://www.interfluidity.com/v2/5537.html

      The head of the Duke Philosophy Department gives a shit:
      http://www.3ammagazine.com/3am/whats-wrong-with-paul-krugmans-philosophy-of-economics/

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    9. So I created an EconoTroll entry for myself...
      http://thorntonhalldesign.com/philosophy/2014/9/16/econotroll-entry-dilettante-philosopher

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    10. Anonymous4:36 PM

      I give a shit, too, Thornton.

      I just took a crap and dedicated it to you. With admiration, of course.

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  2. Julian6:47 PM

    What you are supposed to do is to test first if the axioms hold in reality before trying to apply the other results. That's why you have to flatten the surface before building anything, otherwise your blueprints will be worthless.

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    1. What does it mean for axioms to "hold in reality"? An axiom is just an assumption about mathematical objects.

      Do you mean that one should test how well the mathematical objects correspond to real observable things?

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    2. I think he's trying for some formulation of modus ponens, with axioms standing in for P.

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    3. axioms holding in reality... hmmm: how about parallel lines cross at exactly three places. How about "a one to one and onto map of the real numbers to {5} exists." Or the old standby: "This statement is a lie"

      More internal consistency than "hold in reality" I guess.

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    4. Anonymous10:09 AM

      A agree, and I would answer Noah's question above this way:

      I can see that the mathematics themselves have relevance. The question we should ask is not whether the axioms hold in reality, but whether they hold for the systems we're interested in.

      I could easily see applications of this theorem to small biological systems, perhaps at the level of DNA or cellular grouping, whereby the actors fit the simplistic assumptions about choice and the meaning of choice.

      However, it seems obvious to me that the axioms are not applicable to our national election system, and most likely they wouldn't even be relevant to smaller groups of people. People don't fit the assumptions about choice that these axioms make.

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    5. Julian3:01 PM

      Let me rephrase it (and actually change the argument slightly). A mathematical theory is not falsifiable by its very nature but the statement "the mathematical theory X is applicable to a real life situation Y" _is_. Basically, you define an equivalence between real life objects and mathematical objects and then all the statements of the theory become predictions about real life that can be true or false. Trying to test all these predictions is impractical and maybe even impossible, but you can (and in certain circumstances, you should) test at least some of them. Ideally, you should check that the axioms are applicable since they're the foundation of the theory (but I realized after posting my comment that this may be impractical).

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    6. Anonymous6:44 PM

      I like where you're thoughts are going here, Julian.

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  3. If you really want to learn about the difference between math and science, here's a great read:
    The Courtier and the Heretic: Leibniz, Spinoza, and the Fate of God in the Modern World
    http://www.amazon.com/gp/product/B001RO6P7U?btkr=1

    Every economist thinks he's Newton, when he's really Leibniz, who was such an epic fool that he was immortalized as Dr. Pangloss in Candide by Voltaire.

    First Tycho Brahe, then Kelpler, then Leibnitz, then and only then, Newton. The quote about the shoulders of giants wasn't just whistling Dixie.

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    1. no no doctor no math have imaginary numbers6:59 PM

      like economy ...

      they burn Pangloss is Lisbon n'est pas....
      math have plenty of meta in physics and blue efedrin too....

      if you really want to learn don't quote others please

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  4. Noah, you are a philosopher and haven't realized it yet, although it is beginning to dawn on you.

    Philosophy has gained a bad name as a result of being confused with "metaphysics." But what philosophers do is study foundations. Their chief tool is philosophical logic.

    This involves the investigation of criteria among other things logical in the broadest sense of understanding how language works to do what it does, both as language and in relation to the world.

    Math is a formal language, or collection of formal languages. The criterion of logical (syntactical, analytic) truth is logical consistency and the criterion of falsifiability is contradiction. Math is about proofs, and these proof hang on establishing logical pedigree to axioms and postulates, which are definitions, through the application of logical operators, that is, through formation and transformation rules.

    Theories in physics are expressed mathematically, and their necessity arise from logical truth, specifically deduction of theorems. These theorems then are "interpreted semantically" to connect them with the world being described in such as way that they are testable, that is, there is one and only way that what is described as semantically true fits observation (data). Importantly, semantic interpretation describes a possible state of affairs which is true if and only if a fact corresponds to it. Thus, semantic truth is said to be "theory-laden."

    The falsifiability in this case is the correspondence of representation (a putative state of affairs to what is represented (an actual fact), usually through some sort of observed data, directly by inspection or indirectly through inference from other observations. This is the correspondence theory of semantic truth, and the criterion is an empirical warrant. Wittgenstein formalized this view in the Tractatus (1921) based on Hertz's Principles of Mechanics.

    However, language is much more complex than math and physical science, and its logic is difficult to penetrate. Moreover, without being clear logically, it is easy to be "bewitched" (Wittgenstein) by language. Wittgenstein explored this in Philosophical Investigations (1953) and his other later work published posthumously.

    However, even physical science can't be forced into a fixed logical/epistemic paradigm. That is to say, all specific cases don't conform precisely to the general case. Social science much less so. Social science, including economics, is a mixture of math, theory that generates testable hypothesis and narrative. There is a lot of narrative involved, and it is subject to the vagaries of ordinary language. Moreover, the narrative is based on ideology that is often presumed so that many assumptions, including foundational matters like criteria, remain hidden. Since economics is also a "policy science," normative assumptions are often involved and remain hidden unless specified or exposed by analysis.

    This is what your post is about, and it is also relevant to other posts you have written. You are already asking the questions that define foundational issues.

    There has been relatively little done specifically in the philosophy of economics, that is, the foundations of economics. It's a wide open field for pioneering. Go for it.

    Here is an article you might enjoy reading.

    J. B. Davis, Sraffa, Wittgenstein and neoclassical economics, Cambridge Journal of Economics 1988,12, 29-36

    Mark Blaug's The Methodology of Economics, or How Economists Explain (Cambridge, 1980) lays out some of the history, if you haven't seen it.

    Btw, a lot of the kerfuffle between orthodox and heterodox economics is over foundational issues, which is what disagreement over methodology is based on. Lars Syll's blog regularly examines the application of math in economics. Lars is a critical realist.

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    1. A lot could be done to force social sciences into the physical science realm if we just tossed ethics out the window. Step 1 to any experiment: get a bunch of twins. Just grab em off the street if you have to: but best of all clone them in a special laboratory twin "nursery." Next we'll be doing things like locking one twin in a darkened sound proof box (that's the control twin), while we do experiments with the other one. Of course we need to perform a vivisection on both at the conclusion of the experiment for completeness.

      Why not extend the idea to whole "nations" of twins: separate them into two identical bio-domes, into one of which we inject a new variable: like fill it with chlorine gas or some such. Then we can measure objective realities about their circumstances, like how much C02 each "society" produces, etc.

      Stupid ethics is always holding back progress!... think if Dr. Frankenstein had listened to the hand wringing ethicists and their incessant whining about "creating a monster." Incurious, Luddite kill-joys!

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  5. Theorems are literally tautologies. Of course they can't be falsified!

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  6. Anonymous8:44 PM

    There's a conflation of the word theorem/theory in the mathematical sense with the scientific sense going on as well. Pure math doesn't make assertions about physical reality, it's the application or appropriateness of a particular modeling convention that can be falsified.

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  7. Todd Kreider11:45 PM

    I hope you eventually get out of finance and work on "this stuff”

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    1. Oh, I won't really. When I get old though...

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    2. Todd Kreider12:59 AM

      OK, but I bet you will be out of finance and doing "other stuff" within ten years, a.k.a. "old" (45 in 2025?) The "other stuff" field is waiting for you, man.

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    3. Heh. Well, we'll see!

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  8. Hey Noah - this is Kevin, whose post you linked to. I assure you the Popperians are out there! Write a theory paper and with probability one you will be asked about testable implications every time you present or submit. That theory has roles other than providing testable implications (e.g., elucidating a mechanism, clarifying a concept, clarifying the author's thinking, considering counterfactuals, allowing empirical analysis about unmeasurable concepts like welfare or impossible-to-measure concepts like income before direct income surveys were taken, etc.) and that these other roles tend to be the important ones - well, let's say this isn't completely accepted. To wit, your commenter Julian above.

    The philosophers of econ are actually pretty good on this score. Both Mary Morgan and Nancy Cartwright have a view of theory which is compatible with modern phil. of science (which is very much not Popperian, as you know, at least in the "naive Popper" sense).

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    1. I mean, I love testable implications as much as the next guy, but what's wrong with math? A lot of what econ theorists do is just math, and I don't see a problem with that. It's true for computer scientists, and no one complains about that, do they?

      And ironically, a lot of the most pure-mathy things in econ are the things that end up having the most useful engineering applications. Stable matching algorithms, or discrete-choice models, or auction theory. Don't you think that's the case?

      Yeah, I don't think Popper got it all right, though I think his ideas were a lot more nuanced than the naive version that people think of. Hard to tell, with those wordy old European dudes...

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    2. Computer scientists don't write WSJ op-eds demanding society acknowledge the true implications of their math.

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    3. Noah: it's perfectly obvious that Kevin knows far more about Popper and related issues than you do.

      Kevin: I've been reading Morgan, Cartwright, et. al. lately. I'm making my way through The World in the Model now; great stuff.

      "There has been relatively little done specifically in the philosophy of economics"

      There are, however, at least three journals devoted to the subject.

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    4. StudentAlephNull: Yes, it is. I read Popper when I was 15 and haven't met any of these "Popperians", except when they come and troll my blog comments.

      I read The World in the Model, and didn't get anything out of it that I didn't already know - it was basically just a description of how economists play around with models. I saw Cartwright give a speech once, and I remember it being good, but haven't read anything by her...I'll check her out.

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  9. Falsificationism was Popper's response to Hume's problem of induction. Maths uses deductive reasoning from axioms not inductive reasoning. So surely there is no reason it needs to be falsifiable?

    Unless you want to argue that the axioms are based on inductive reasoning. Or if you are claiming that the maths describes the real world somehow.

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    1. Anonymous7:28 AM

      Cut and dry response that summarizes the whole debate in a few words that also idiots should be able to understand, including=ng economists.

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    2. Anonymous10:09 AM

      I'd simply like to second that Mr. Duigan's response should address the issue. It is a more concise version of what I was intending to note as I read the initial post.

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  10. This sounds a bit like someone saying that a hammer is useless for a housing builder, because you can't build a house out of hammers. He's confusing the tools with the building materials.

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  11. Math is almost magical. Isn't it wonderful how a formula can express, for example, the waning power of gravity over a distance? I'm not a physicist, so please forgive me if I'm wrong, but isn't it great that gravity is caused by mass, i.e. billions of atoms, quants or waves or energy, and yet the tiny effects of each physical unit aggregate to something that exhibits such a regularity, such stability, that it can be express by a formula?

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  12. Anonymous7:25 AM

    Failed in Physics, failed in Math, failed in MBA, failed in Enginerring is the road that leads to Marketing or Philosophy.

    A testable fact.

    BTW, math is falsifiable in the sense that arbitrary math statements like 3+2 = 10 are falsifiable. Axiomatic systems are also falsifiable if one of the acioms is proven false.

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    1. Attempts were made to disprove Euclid's Parallel Postulate. What we got instead was (two kinds of) non-Euclidean geometry.

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    2. Actually I recently saw some data indicating that philosophy majors had higher IQs than anyone except physics and math majors...not sure if it was reliable though.

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    3. philosophy majors had higher IQs than anyone except physics and math majors.

      Probably an artifact of how the tests were constructed. If you construct a test that focuses on certain types of abstract reasoning then physics and math majors will score high because those are skills they need. Throw in a little reading comprehension and all of a sudden the philosophy majors surge forward. If, instead, you tested for the ability to solve three dimensional motion problems in your head in real time, the football quarterback would be the smartest guy on campus.

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    4. 3+2 = 10 is in itself not falsifiable. If I stated that above there are any number of reasons the equation/statement is perfectly cromulent.

      Pretty sure that 3+2=10 is true in a base 4 number system, also assuming that the symbols represent predefined concepts as well.

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    5. Correction, 3+2=10 in base 5. See anything can be true by adjusting some basic assumptions. All statements presuppose existence in some form... Blah blah blah philosophy jazz hands!

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  13. There are alwasy going to be theorists who spend most of their time deducing conclusions from existing theories, formulating new theories, and comparing one theory with another. Every science needs such people.

    There are also supposed to be a bunch people attempting to bring this theoretical activity to bear on the study of the scientific study of actual world by identifying empirical consequences of the theories, and determining whether those consequences do in fact hold through various forms of empirical observation and analysis.

    Much of the recent criticism of economics is that the field appears to have become become top-heavy with with people of the former type, and not enough of the latter type.

    Some other criticisms:

    Economists sometimes incorporate simplifying assumption into their theories, engage in no serious effort to empirically test their theories, and then over time they seem to "forget" that these assumptions are simplifying and that the theory hasn't been vigorously tested. No problem, really, except these folks sometimes then go out and start telling people and policy makers what they should be doing.

    There is also the kneekerk "positivism" response. (It's actually more of a kneejerk holism response, that for some reason economists call "positivism"). When assumptions built into the theory are challenged as being either false outright or crudely oversimplifying, the economist sometimes comes back with the observation that the theory as a whole is what is tested against reality, and that an empirally adequate or useful theory can incorporate assumptions that are not true in isolation. That's a fine reponse in practice. But the fact is that most of the theories they are defending in this way really haven't been empirically tested in a very serious way, and the only justification for the assumptions was supposed to be their independent plausibility.

    It still seems to me that economics, although supposedly a social science studying the actual social world, and useful for the insight it goves us into that actual world, is a field that has evolved to contain far too many people who have wandered in from other sciences, who do not even like the social world much, are utterly impatient with the careful study of actual institutions and their mechanisms, have little sense of history and the way in which the nature of institutions and norms of social behavior are highly contingent and dependent on the ways in which they have evolved; and also have little sense for the metamorphic flux (as opposed to mere cycles) displayed by living societies over time. In other words, there are a bunch of people in economics who seem relatively ignorant of the human sciences and the methodological wisdom that has been built up in them over time, but who have somehow wandered into a human science.

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    1. Anonymous3:22 PM

      A very good explanation of the problem. There are other good ones above that deal with the nuts and bolts of philosophy, math and logic, but the key to relevancy is the philosophy binding it to human experience as Hickey above explains so well.

      It is very easy to explain why Arrow's theorem is irrelevant to economics. Its axioms postulate that the system under study is most relevantly defined by the extremes of outcomes and the failure to provide for a guaranteed avoidance of such. Human systems almost never operate at these extremes for obvious reasons.

      Just one example regarding this axiom: a single player should not dictate the outcome. Why not? I'm actually quite certain that in many small voting systems, a single person can dominate and determine the outcome. In human systems, we call this a leader, and it isn't to be washed away as harmful because it is confused with dictatorship.

      Having a single person determine the outcome of a vote is not what people would generally consider to be dictatorship. Yet for people with very little experience in complex social situations, they might think this is the definition of dictatorship. It isn't.

      There's nothing wrong with studying this kind of thing, but as you point out, there needs to be a much stronger gatekeeper that prevents people who study such things from affecting policy debates directly. I see this as a bad habit of almost all economists, that they tend to fail in the long run at modulating their own input to policy debates.

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  14. Anonymous9:23 AM

    This line of 'scientific reasoning' worries me. Why? Because it purports to provide some sort of insight into complex social choices by using words that have deeper meaning to people that simple scientific investigations:

    Arrow's impossibility theorem talks about non-dictatorship as if it has a mathematical meaning. To most people, dictatorship doesn't have a mathematical meaning, it only has a social meaning. I suspect the biggest weakness in creating such a theorem or evaluating it is in creating the right axioms, but I believe the interpretation used in selecting the axioms for this theorem are based in social fear rather than mathematics, therefore the mathematical definition of dictatorship probably doesn't meet the social interpretation of that concept.

    Therefore, in my brief look at this theorem, I completely reject its results as non relevant to large voting systems. The magnitude of the numbers involved in most real systems make it inapplicable. If people don't like dictatorship, they are always free to form voting strategies that can avoid them. I don't believe there is any social relevance of such a mathematical concoction.

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    1. Anonymous9:32 AM

      Let me explain my worries a bit better:

      I could see Arrow's theorem being used by some people as an argument that an instant runoff system of voting cannot meet our goals of a fair election system any better than our current system. However, the theorem doesn't say anything about how the choices themselves change the psychology of the voters, nor does it talk about the fact that many people's preferences can be changed based upon their own perception of the popularity of a candidate, the results of which are only available through the voting system itself.

      While I find this a very interesting line of thought, I see it all too likely that pursuing it for too long could lead a person to very false conclusions just based upon relevancy bias -- and I just created this term -- which is to say that people want things they spend time upon to be relevant.

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  15. With regards to Math I like simple explanation by Eliezer Yudkowski: logic (and math) is not about creating truth but about preserving truth.

    Math is a very useful tool that says that if you start with some premises that are true/false and manipulate them in some manner what will happen in the end. But the "trueness" of the premise - in relation to reality - is a task for other sciences to determine.

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    1. Anonymous10:43 AM

      This is right. Too often I see social scientists using mathematics without analyzing the applicability of the premises or axioms to human social systems. I suspect this happens in economics more than other social sciences because there are financial benefits that go along with seeming relevant.

      If an economists finds something new they think is relevant, but it is only relevant to low-level biology, they are likely to ignore the things that make it irrelevant to economics and politics. Economics and politics are where the money is. There's not that much money in biology.

      I understand why economists find this stuff interesting. I find it very interesting, but given our current political dynamic, we can't afford to allow the interest of social scientists to trump our common sense. This is part of the problem, and part of the reason some politicians seem to ineffective -- if they rely too much upon political science, they're likely to seem artificial, untrustworthy, arbitrary and bizarre. I can think of some prominent examples.

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  16. Anonymous1:25 PM

    "I really hope there aren't "Popperians ..." running around out there complaining that math results are useless because they're non-falsifiable."
    if there are, they haven't read much Popper. (World 3 anyone?)

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  17. Anonymous3:57 PM

    Just one more thing to add:

    I don't understand the references to Popper in these discussions. Popper talks about falsifiability, but the things we're talking about here are not even testable at the level they claim to be relevant.

    This isn't a question of falsifiability to me, it is a question of testability. These are 2 different things. The language of these theorems suggest a relevancy to human systems. Yet not only are they not falsifiable, they aren't testable for positive confirmation through natural experiments.

    Really, I think this is the crux of the confusion and disagreement here. Many things economists talk about are testable for positive results through natural experiments, and that gives them a leg up to ideas such as this which aren't.

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  18. You really ought to test the triangle inequality empirically - and don't you dare impose an inner product on my beloved metric space.

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  19. Economic theory: A tautology, derived from a set of assumptions, that is resistant to falsifiability.

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  20. Anonymous11:48 AM

    Is it intuitive, given Arrow's theorem, that in practice the number of coalitions formed by candidates will be less than or equal to the Nakamura number of the contest in question?

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  21. Math is not science and science is not math. Math is a useful tool to scientists that find that creating theories using mathematical methods leads, in some cases to, to models that make very accurate predictions. If the application of math didn't help then it wouldn't be used.

    Also, I think the closure argument about mathematical reasoning has been dealt with by Kurt Goedel in his most famous work.

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    1. I was waiting for someone to bring up Godel (Goedel?)!... (I knew it shouldn't be me though because I'm just barely familiar with a few of his ideas)

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  22. Math or at least some of it is falsifiable if there are blunders or if exceptions can be found. Even when falsified math can be useful. Two non contentious (except amongst mathematicians) examples are the Dirac delta function and Feynman graphs, although ways were belatedly found around the nay sayers or at least some of them

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  23. The author of the blog probably intended this other blog of his as context:

    https://afinetheorem.wordpress.com/2013/05/02/the-axiomatic-structure-of-empirical-content-c-chambers-f-echenique-e-shmaya-2013/

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