For people who are puzzled by the reference to “monads”, think about Schrödinger’s cat. Schrödinger’s cat has two classical states, “alive” and “dead”. The “monad” of quantum states is basically the vector space generated by this basis. From outside the vector space you can’t know whether the cat is alive or dead, and in fact the cat is in a superposition of these two states.
This vector space is a monad because:
you can “lift” elements of the basis into the space (put a live cat or a dead cat into a box)
you can “map” functions on the basis to functions on the entire vector space (put a device which kills the cat with certainty into the box)
and you can “bind” a function which takes classical states to quantum states to the entire vector space (i.e. make a device which randomly chooses whether to kill the cat based on a quantum event and put it into the box).
This is not a complete description; for instance, one needs to introduce a tensor product to account for the fact that independent (disentangled) monads can be effectively separated out. On the other hand this monadic description applies to all vector spaces; it’s not specific to quantum mechanics.
We are using monad in the functional programming sense? I thought Leibniz at first glance. I’ve never quite apprehended Leibniz’s monads, so I put off trying to grok the post fully. An introduction to them would be useful.
Leibniz’s monads were an attempt to resolve the mind-body problem. Supposedly, monads are something like atoms, and something like souls. They don’t interact with each other—all causality goes from God directly to the monads, not from one monad to the other—for example, perception is only accurate by God arranging for monads’ perceptions to be accurate.
It seems like really incredibly strange metaphysics / theology to me.
It seems like really incredibly strange metaphysics / theology to me.
I agree. Bertrand Russell explained it as due to Leibniz’s beliefs about causality—one substance could not affect another substance. By a monad I just mean an elementary “thing” which can have mental states.
It’s inscrutable to me too, but mainly because I never learned Haskell.
It does, however, contain a link to this interesting paper, which shows that straightforward application of Cox’s axioms allow you to have complex-valued probabilities in Bayesian inference, which makes the jump to quantum physics much easier.
I had a bit of a hard time following that too, but mainly because I’m not familiar with the ins and outs of mathematical fields and their notation.
Well, I suppose you have two questions to ponder. One, is it worth it to be more useful if this entails seeming less useful? Two, is this what will actually happen if you make that list?
At any time, the universe consists of a number of entities whose formal states inhabit Hilbert spaces of various dimension (thus |01>+|10> comes from a four-dimensional Hilbert space, while |1> comes from a two-dimensional Hilbert space), and the true dynamics consists of repeatedly jumping from one such set of entity-states to another set of entity-states.
But really, I was replying to whpearson’s question about bogus’s comment. (The FP monad at least makes more sense to me than Leibniz’s monads.)
FP monads are interesting, but I was taking my lead from Leibniz, not Haskell. That is, a monad here is an elementary entity (or at least, one without spatial parts, it may have some other form of internal structure) which can be the bearer of a state of consciousness. Though my monads differ from Leibniz’s in that they can interact with each other (Leibniz didn’t think relations are real and so disbelieved in causal interaction, which led him in peculiar directions).
Good post, thanks.
For people who are puzzled by the reference to “monads”, think about Schrödinger’s cat. Schrödinger’s cat has two classical states, “alive” and “dead”. The “monad” of quantum states is basically the vector space generated by this basis. From outside the vector space you can’t know whether the cat is alive or dead, and in fact the cat is in a superposition of these two states.
This vector space is a monad because:
you can “lift” elements of the basis into the space (put a live cat or a dead cat into a box)
you can “map” functions on the basis to functions on the entire vector space (put a device which kills the cat with certainty into the box)
and you can “bind” a function which takes classical states to quantum states to the entire vector space (i.e. make a device which randomly chooses whether to kill the cat based on a quantum event and put it into the box).
This is not a complete description; for instance, one needs to introduce a tensor product to account for the fact that independent (disentangled) monads can be effectively separated out. On the other hand this monadic description applies to all vector spaces; it’s not specific to quantum mechanics.
We are using monad in the functional programming sense? I thought Leibniz at first glance. I’ve never quite apprehended Leibniz’s monads, so I put off trying to grok the post fully. An introduction to them would be useful.
I don’t know much, but...
Leibniz’s monads were an attempt to resolve the mind-body problem. Supposedly, monads are something like atoms, and something like souls. They don’t interact with each other—all causality goes from God directly to the monads, not from one monad to the other—for example, perception is only accurate by God arranging for monads’ perceptions to be accurate.
It seems like really incredibly strange metaphysics / theology to me.
I agree. Bertrand Russell explained it as due to Leibniz’s beliefs about causality—one substance could not affect another substance. By a monad I just mean an elementary “thing” which can have mental states.
I think it must be the FP monad; here’s a sigfpe post which is, as usual, above me but which seems to treat vectors as Haskell monads: http://blog.sigfpe.com/2007/03/monads-vector-spaces-and-quantum.html
It’s inscrutable to me too, but mainly because I never learned Haskell.
It does, however, contain a link to this interesting paper, which shows that straightforward application of Cox’s axioms allow you to have complex-valued probabilities in Bayesian inference, which makes the jump to quantum physics much easier.
I had a bit of a hard time following that too, but mainly because I’m not familiar with the ins and outs of mathematical fields and their notation.
You should learn Haskell. Assuming you’ve already learned Erlang, of course.
I’ve got all kinds of advice on geeky things that are for most people complete wastes of time.
Do you have a list anywhere?
I’m thinking I should totally do that, but it might make me seem less useful. Like when Sherlock Holmes explains his deductions so they seem obvious.
Well, I suppose you have two questions to ponder. One, is it worth it to be more useful if this entails seeming less useful? Two, is this what will actually happen if you make that list?
But where does Mitchell Porter use anything from the functional programming side of things?
It might be this bit:
But really, I was replying to whpearson’s question about bogus’s comment. (The FP monad at least makes more sense to me than Leibniz’s monads.)
FP monads are interesting, but I was taking my lead from Leibniz, not Haskell. That is, a monad here is an elementary entity (or at least, one without spatial parts, it may have some other form of internal structure) which can be the bearer of a state of consciousness. Though my monads differ from Leibniz’s in that they can interact with each other (Leibniz didn’t think relations are real and so disbelieved in causal interaction, which led him in peculiar directions).