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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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Not sure what you would be the mathematical definition of "observe" but Tarski's undefinablity theorem and Godel's incompleteness theorem would be the analogue. –  Sniper Clown Apr 11 '13 at 6:59
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. –  V Welner Apr 11 '13 at 8:56
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) –  Rhett Butler Apr 11 '13 at 8:58
@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. –  Rhett Butler Apr 11 '13 at 10:26
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... –  Filippo Alberto Edoardo Apr 11 '13 at 23:07
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closed as not a real question by Andres Caicedo, Mark Sapir, Eric Wofsey, Todd Trimble, Andy Putman Apr 11 '13 at 14:53

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3 Answers

To answer your question on the observer-dependence of mathematics, we have at first to define what mathematics is. Since many brave and intelligent men have failed to accomplish this goal, I will consider only three different domains, basic subjects, basic tools, and advanced mathematics which appear to belong to mathematics by general consensus.

The basics of mathematics have been obtained from physics. 2 apples and 3 pears are 5 fruits. The pythagorean theorem holds, because it is always true in reality where not too big masses are around. Two coupled oscillators do what they do because each of them computes two harmonic functions which are added or subtracted to tell the trajectory. The three-body problem is always easily solved by three bodies.

With this understanding, I can answer your question as follows: The more we can observe and the more we can think and talk about it, the more mathematics we can do. With a simple abacus, only small numbers can be calculated. The last prime number calculated by hand is $2^{127} - 1$. With increasing computing capacity, more and more prime numbers will come into reach and more and more proofs will be done by automata, proofs that we probably cannot even understand, at least that we cannot completely read (think of the four-colour theorem or the sequence of digits of $\pi$). Fibonacci proudly published the prime numbers between 1 and 100, Newton did not yet know the first 100 digits of $\pi$, Brouwer asked whether there is a sequence of nine consecutive digits 9 in the decimal expansion of $\pi$. Today billions of digits of $\pi$ are available and Brouwer's question has been decided in the affirmative. So we can state that basic mathematics strongly depends on the facilities of the observer.

An observer observes and communicates his observation to other observers. Mathematics has been defined as discourse (about the universe of discourse) but of course it occurs inside of our universe with limited ressources. We should not forget that. It is undisputed that without signals, at least inside one brain, no mathematics is possible. Our thinking and talking is limited by the media available. That is also limiting the mathematics we can do. The maximum number of steps of a proof is limited by the memory space of the proving system like the maximum number of digits of a number.

In addition to these material restrictions, we have the relativity in logic of set theory that has been discovered and advocated by Skolem, who asserted that there is no possibility of introducing something absolutely uncountable, but by a pure dogma. In 1929 he wrote already (in German): It seems that Hilbert wants to maintain Cantor's ideas in their old absolutistic sense. I find that remarkable. It is strange that he never has considered the relativism that I have proved for every finite formulation of set axiomatics.

It is rarely mentioned, but we have a similar relativity in proof theory discovered by Gödel (who did not prove that some propositions are undecidable but that their decidability is relative to the system applied). Here it may be even most obvious that mathematics depends on the observer, in that possible proofs depend on the level of theory attained by the observer.

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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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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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