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I want to ask whether something not from math can be modeled in any way using math, and particularly in a way that serves to prove/disprove some hypotheses or affirmations about that particular something.

Specifically I want to ask about meritocracy, and whether it can be disproved through math. My thinking was that if it can be proven through math that it's a system that depends on initial conditions, and that initial conditions are several variables that each assume values in a continuous spectrum, then with a limited number of combinations (say 8 billion) you won't get the same initial conditions for any two runs (any two people), thus for any two people it's false that initial conditions are irrelevant and only their actions affect the outcome.

It's odd and sketchy and iffy, but I thought I might learn something from asking the question. But it seems way too different from other questions I've found on this site, so I thought it would be polite to first ask in meta whether it's a valid question here.

If it's not, and you can point me to any other site you think I can try asking, I would be grateful.

Also, terribly sorry for using the word "math" so liberally, and with such a tone of "something magical that proves stuff"

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    $\begingroup$ While discussing concrete mathematical models sounds like a good idea, one should be careful about making statements such as "I have a mathematical proof that X is wrong", where X is a complex social or political concept. $\endgroup$
    – supinf
    Sep 25 '20 at 16:07
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    $\begingroup$ I don't think math stackexchange is the right forum for this. The math stackexchange is meant to answer specific questions about mathematics, but not to explore new unformed theories, even in a purely mathematical context. In your case, this is not even purely mathematical, but more about modeling sociological or political concepts. This belongs more in a sociology/politics forum than here. $\endgroup$
    – PatrickR
    Oct 3 '20 at 1:35
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    $\begingroup$ @supinf: Indeed, there is an inescapable gap between mathematics and the real world that can only be crossed by (non-mathematical) interpretation; I tried to give some explanation of that in my answer. In other words, one cannot have a purely mathematical proof that some real-world concept is good/bad/whatever. $\endgroup$
    – user21820
    Oct 7 '20 at 13:32
  • $\begingroup$ From your description, it sounds like something that is unanswerable from a mathematical perspective, and really rather a stretch to even consider it as a mathematical question. $\endgroup$ Oct 8 '20 at 14:21
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It sounds broad, but if you narrow it down, I think it could be very interesting. My recommendation would be to split it up into multiple questions.

What you want is to formulate "meritocracy" explicitly so that questions about it can be studied mathematically. If you already have a way of doing this, that's awesome. If not, your first question should be something along the lines of "How can I formulate 'meritocracy' mathematically?" which would include some properties you think must be satisfied by your definition of meritocracy, as well as a general description of the questions you're interested in answering about it. Questions to develop a definition may be "soft", but they are still acceptable (desirable, even) when asked with focus and proper context.

After you have a good mathematical definition for meritocracy, you want to come up with some explicit mathematical questions to ask about it, and ask these separately. You already have one:

  • Can two participants in a meritocracy have identical initial conditions?

Note that once you have your definition, this question may wind up being somewhat meaningless, and develop into something more like

  • What is the difference in expected outcome between those of equal natural ability but different initial conditions?

To address these, you'll need to decide how to incorporate the notions of initial conditions, actions, natural ability, and outcomes into your definition. Are actions probabilistic? Are outcomes real numbers (e.g. total wealth over a lifetime)? Are initial conditions (or natural ability) inherited from parents? How are outcomes determined? These would be good things to consider when constructing your meritocracy definition, but try to keep it simple at first, to prevent overwhelm.

Some other examples of questions you might want to consider using your definition:

  • What is the long term behavior of meritocracy? Is it stable?

  • Does meritocracy necessarily lead to a generational accumulation of power?

  • How much can a participant in a meritocracy be affected by random chance events in life?

Again, if you already have a few of these questions in mind, it would be relevant to include them as context in your initial formulation question, since these may affect what goes into the formulation, e.g. to study generational affects of meritocracy, your mathematical formulation of meritocracy ought to be compatible with generations. (But make sure you state explicitly that they are there for context, not to be answered, so you don't get closed for being too broad.)

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Do be aware that even if you have a precise mathematical definition that you think captures (the non-mathematical) meritocracy, and you have rigorous mathematical proofs of certain theorems of this mathematical concept, there still is a disconnect between the mathematical theorems and the real-world concept. One will always need some non-mathematical interpretation of any given mathematical theorem when it is 'applied' to the real-world. So one can (justifiably or not) object to your application of whatever theorems about "meritocracy" to actual meritocracy.

I say this because your entire question seems to be based on the implicit premise that mathematics can be used to prove things about non-mathematical things and that would settle the matter. Well, it does only to people who accept your real-world interpretation of the mathematics. In particular, suppose that I accept that classical first-order logic holds in the real world, and you state precisely some axioms over some first-order language, and write down a proof of some theorem from those axioms, then you would still have to explain to me how you wish to interpret the symbols in your chosen language, and convince me that your axioms are actually true in the real world under your chosen interpretations. If you fail to do so, all you have is a theorem of a mathematical theory that I may not think is relevant to the real world.

By the way, I think that the underlying non-mathematical idea in your 2nd paragraph is trivially true, and needs no mathematics to prove it. Consider that even in an ideal meritocratic society, a child who happens to contract a disease that cannot be treated can never have exactly the same merit-based opportunities that many other children have, whether or not it leads to leadership roles. One has to think very carefully about what true fairness means, before even talking about popular notions such as "meritocracy".

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  • $\begingroup$ Meritocracy means that among people in the same initial conditions and subject to the same external perturbations, the winner is who reaches the best score. You have to think about a 100m run. That is a fair meritocracy test, and that's why no one can criticize Bolt results. But anyone can criticize a football team that wins the league. $\endgroup$
    – Kosh
    Oct 7 '20 at 14:39
  • $\begingroup$ Just to avoid nonsense answers. One can ask if other ones were given the same possibility given to Bolt to become a champion. But this is another question. $\endgroup$
    – Kosh
    Oct 7 '20 at 14:42
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    $\begingroup$ @Kosh: You need to think more carefully. Nobody in a 100m race has exactly the same initial conditions. Some have better shoes than others. Some wear more comfortable running attire than others. Some have longer legs than others... $\endgroup$
    – user21820
    Oct 7 '20 at 15:33
  • $\begingroup$ @user21280 "just to avoid nonsense answers" did not work. Cheers. $\endgroup$
    – Kosh
    Oct 7 '20 at 21:01

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