# Causality and operationalism: from sets and functions to monads

When working in a laboratory, the most basic behaviour is to turn a knob or dial and then see a transformation of some data output. An example is increasing a magnetic field and seeing Zeeman splitting. We normally use this behaviour to create a function, thinking of the system as being composed of a set of states. I am interested in a program which borrows some of the assumptions of quantum gravity. Namely, I am working towards a picture where states are not fundamental, but instead processes are. This leads us to the following picture. We take the turning of the knob as a morphism and the change in the data output as another morphism. The experiment, then, is a map from an arrow to an arrow and this is just an endofunctor on the category of the apparatus. Can we then use this endofunctor to create a monad and subsequently an algebraic theory for the system under investigation?

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"Of course, states don't exist, only processes do." - That's one hell of a statement. Perhaps we would be better sticking to physics than philosophy. –  Joe Fitzsimons Nov 27 '11 at 12:25
Sounds me as a sohisticated rehearsal of the old debate of operator against wave function, or Heisenberg vs Schroedinger pictures. –  user135 Nov 27 '11 at 21:34
You haven't defined the category of the apparatus. A definition seems to be necessary before looking for a monad structure. –  Scott Carnahan Nov 28 '11 at 5:49

(Disclaimer: I can't even attempt answering the stated question as such because I have too little command of the terminology of "categorizing". But I still like to think that even the stated premises themselves deserve more comment than fit in 600 characters, and more chance to perhaps form an answer of this in ... the process.)

When working in a laboratory, the most basic behaviour is to turn a knob or dial and then see a transformation of some data output.

"Turning the knob" seems to describe a certain "transfomation", too; perhaps of some "input setting". Accordingly, (taking part in) such "transformations" seems to be the most basic behavior ("when working in a laboratory").

But of course there may be plenty such basic "transformations" acquired "on a long day/night in the lab"; and (subsequently) they ought to be put into some relation to each other to eventually ... "have something to show for it".

An example is increasing a magnetic field and seeing Zeeman splitting.

There is a (proverbial) "world of difference" between "turning the knob" and asserting that a magnetic field had increased (within the particular lab region);
and no less between "seeing some data output" and asserting an instance of ... this (including references in there).
And no less either, at least in principle, between "touching the knob" and asserting that the knob had been turned, for instance.

The difference is (closely related to): Evaluation. You know (even better):

Given the (rather) basic data, $\psi$, and having designed and selected some particular way, $\hat{A}$, of carrying out the evaluation, then going to work and (ideally) obtaining one value (that's "worth showing off"): $\lambda_a$.

This applies if the data "was (any) good" enough to carry out this evaluation; and it might be accordingly referenced as $\psi_a$. Or, if the data "wasn't (any) good (for that)", then not. In any case this data is retained for subsequent evaluations; be it in the same way, or in other particular ways, perhaps also involving supersets of this data, or subsets.

We normally [...] create a function

Sure: once we have obtained values we may map them to each other. But "working in the lab" apparently starts well before that.

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( See disclaimer on top.)

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