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Is there any field of mathematics that deals with the role of the observer? E.g., some formulation in which a set is changed, in some unspecified way, when it is observed? Or maybe some philosophy of mathematics that addresses this? (I am more interested in formal mathematical systems than philosophical expositions, though.)

I was thinking a bit about the Axiom of Choice and the Well Ordering Principle. The Axiom of Choice feels true, and the Well Ordering Principle feels false. However, one might envision mathematics differently as an observer dependent practice, if we posit (as some foundational axiom) that a set exists only if it has been observed, or its definition has been observed, or something (?). In that case, the order of observation might impose an ordering on any set of sets, and this ordering might differ among observers. Though, that is just one way I can imagine a set being affected by observation.

Let me zoom out a bit. In the 20th century, several fields of human endeavor moved in a direction broadly known as postmodernism where subject matter was no longer considered independently of the observer. E.g., postmodern literature, and observer dependent physics of the 20th century. I am wondering if there has been anyone who's considered what an observer dependent mathematics might look like.

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    $\begingroup$ Wow, really Mark? Do you think that is what I was asking here? I am asking whether there's a mathematical formalism in which an observer, of some sort, is made explicit. I suppose that the Turing machine in which a mathematical framework is implemented is one type of observer. But I'm more interested in whether there is a set theory that explicitly includes an observer, rather than questions about computability per se. I was not really asking about how math papers are refereed. $\endgroup$
    – V Welner
    Apr 11, 2013 at 8:56
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    $\begingroup$ Here is another aspect: Skolem has proved that the notion of uncountability is observer-dependent. Thoralf Skolem: "Über die Grundlagendiskussionen in der Mathematik", Den Syvende Skandinav. Matematikerkongr. Oslo (1929) 3-21. (In German) $\endgroup$
    – user112109
    Apr 11, 2013 at 8:58
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    $\begingroup$ @V: You may also be interested in this great book by Ebbinghaus on "Ernst Zermelo, An Approach to His Life and Work", Springer (2007): Skolem's conclusion was that the notions of set theory are relative to the universe of sets under consideration: The axiomatic founding of set theory leads to a relativity of set theoretic notions ... Later Skolem even strengthened his opinion by what can be regarded the essential motto of so-called "Skolem relativism": There is no possibility of introducing something absolutely uncountable, but by a pure dogma. $\endgroup$
    – user112109
    Apr 11, 2013 at 10:26
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    $\begingroup$ Voted to close because this is not a mathematical question, it's a question of philosophical interpretation. You could put an "observer-dependent" gloss on what we think we are doing when (say) working internally in a countable model of set theory, but it's totally unnecessary, and irrelevant to formal mathematical systems. If a computer generates a proof in a formal mathematical system, did it involve any "observations"? $\endgroup$
    – Todd Trimble
    Apr 11, 2013 at 16:43
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    $\begingroup$ I am very disappointed by this question being closed, especially because it received two answers, one of which is pretty long and refers to a seemingly serious paper (although in Philosphy and Logic). @Mark Sapir: You thought it should be Community Wiki, other thinks it might not: you could complain with the author, vote it down, but if it were legitimate as Community Wiki, it stays so... @Todd Trimble: you seem to say it is not a question, and then use your comment to give an answer... $\endgroup$ Apr 11, 2013 at 23:07

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Since you mentioned physics I think you may be thinking of the so-called "measurement problem":

In math, maybe the idea can be worked into an explanation of the difference between Bayesian and frequentist interpretations of probability. Something like: the Bayesian interpretation says random variables are random til actually observed.

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Perhaps this idea can be used to motivate the phenomenon of "local triviality, global nontriviality". Here the role of "observer" is played by a (local) coordinate chart, where, say, the bundle over a manifold looks trivial, whereas global nontriviality transcends each individual observer. That's why you can't comb the sphere.

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