In fact there is, take A to be the space of differential forms on some manifold, and M the wedge product. Given two differential forms eta, mu, there is no way to recover eta and mu from M(eta,mu). Even simpler take two arbitrary numbers multiply them, unless they were prime you have no way of telling from the product what the two original numbers were. In other words multiplication destroys information.
In the case of numbers this is fine because there is in fact a way of copying them beforehand by an operation Delta : A -> A x A, which sends a number a to (a,a), so if given (a,b) you want both their product and the numbers themselves, you need a map A x A -> A x A x A given by (for example)
(id x M x id) . (id x id x Delta) . (Delta x id)
In the case of differential forms, there is no such (natural) map Delta.
No, applying a map does not mutate or consume the parameters, and therefore no information is destroyed. Function application in mathematics is referentially transparent.
To make this point perfectly clear: Whenever you encounter an expression such as "f(x)", you may freely re-use the expression "x" in a different place. This is a matter independent from the category you choose to work with -- for example, the expression "psi \otimes psi" makes sense in the monoidal category of Hilbert spaces.
This largely depends on your viewpoint, the map M : A * A \to A was meant to be a morphism in an arbitrary monoidal category. An element like x, would be interpreted as a morphism x : I -> A (in general the objects A in a monoidal category, or any category for that matter have no notion of elements). The expression f(x) is interpreted as the composition f . x, so no you are not free to use x a second time, once you formed that composition. This is swept under the rug in most cases because monoidal categories like Set are cartesian and allow you to freely copy morphisms like x : I -> A, via the Delta map Delta : A -> A * A. Explicitly you have Delta(x) = Delta . x = x * x. In general it might not be possible to even determine if two morphisms x : I -> A and y : I -> A are the same.
In some cases you might be able to assume from the start that you have a certain number of equal morphisms x : I -> A "in reserve", for example in linear logic you have the operator ! ("of course") which gives you an arbitrary number of identical morphisms to play with.
In any case this distinction is quite subtle and I understand, why you might assume that I'm simply misunderstanding things. In particular I should emphasize that almost no programming languages work this way, although with some effort you would be able to recast typical cryptographic / numerical code in this language.
It is also really easy that to see for example in the case of addition that indeed information is destroyed, clearly the map
(a,b) -> (a+b,a-b)
has an inverse if the characteristic isn't 2, the subsequent projection to the first factor destroys information. Categories in which that is possible have maps d_A : A -> I.
What tel is pointing out above, is that Delta and d together form a comonoid.
While psi \otimes psi certainly makes sense, the map psi \to psi \otimes psi is not linear and therefore not a morphism (Physicists call this the no cloning theorem for some reason).
I understand where you are come from, but you are conflating different meta-levels (external vs. internal language/logic): You are absolutely free to use any formal expression "x" a second time, and - crucially - it will have the same mathematical meaning if you choose to do so, as can be seen by pondering over the tautology "P(x) <-> P(x)". This is not a property of the category that you chose to work with, but rather of the formal language that we use when engaging in mathematics, as you have demonstrated yourself when you had permitted that "psi \otimes \psi" made sense.
The non-existence of a morphism "psi -> psi \otimes psi" and the notion of "destroyed information" that you are discussing in the rest of your post is independent from all of this. If you wish I can elaborate on the "no cloning theorem".
Well ok, then this was a simple misunderstanding, or miscommunication. My knowledge is largely based on reading of http://ncatlab.org/nlab/show/internal+logic and several other pages on that wiki, plus graduate training in math and physics. Based on your last reply I'm confident you have a similar background. Logic and category theory are not my specialty and I probably haven't expressed myself as clearly as I would in professional communication.
To check if I understand the argument: I'm guessing you're talking about the internal logic or language of some category, in contrast to the external one used to talk about that category.
Right I think that was the point of misunderstanding between me and asuidyasiud. If you think of x as an assumption in linear logic for example, then there is a difference if you assume x or x^2 and so on. Since in programming languages such x : I -> A are values of type A, this then ties in with duplication and destruction I was going on about above. I tried to express the sentiment that in many ways it is actually natural to think of those kinds of logic as more fundamental, because it makes the duplication of values (unnatural for things like resources, spaces of state, etc.) explicit or even impossible.
Yes, not injective essentially, which is not the same as not having an inverse. The notion can be extended though, for example functors that aren't faithful are certainly destructive .
Mathematicians arguably do not think of the first example as a loop with a mutating variable, but rather as a short-hand notation for f(m) + ... + f(n) - as a macro, if you wish.
> It is completely irrelevant, ecologically speaking.
How so? It makes the box unrecyclable; worse still, as soon as such an item is detected in the recycling plant it will often cause the whole batch of material to be rejected and dumped[1].
No-one has time to pick-through cardboard removing batteries and circuitry.
[1] Source: anecdote from a colleague who worked in a 'recyclarium'. Contaminated material causes the line to be stopped, batch dumped and machines cleaned.
The actual pre-processing sorting is quite fascinating, using spectrometers to detect the type of materials.
From one of the comments above: "It's not the box, but a small plastic enclosure"
And while I sympathise, consider that this is part of packaging for a computer, where the manufacture of the computer itself will consume many times as many resources, and the power to use the computer over its lifetime will consume many times as many resources. With respect to the overall environmental impact of the purchase of that device, the inclusion of a device like that will be a rounding error.
It's not really an excuse to waste more resources though. Saying "this only wastes a tiny amount of resources compared to this other thing" isn't justification to waste it anyway.
Let's say you are an airline and I am an oil company. You want to have a predictable cost for your fuel so you can plan, I want to charge whatever is the market price at the time. So a bank comes along and agrees, for a modest fee, that in a years time, you can buy fuel at a fixed price. In a year, they buy fuel from me and give it to you. If my price is lower, they pocket the difference as profit, but if it's higher, then they have to eat the cost and therefore make a loss (which is unlikely to be entirely covered by the fee you already paid them, but that's fine, because what you were really paying for was peace of mind).
People have been doing this with agricultural products for at least 3000 years.
Agriculture and airlines tend to use futures to hedge, not options.
A (long) futures contract is the right AND obligation to purchase a commodity at a set price at a set time (and place).
A (call) options contract is the right BUT NOT the obligation to purchase a commodity at a set price within a set time (for the most typical American-style option).
The point is that invoking an impossibility theorem oftentimes - and also in this case - demonstrates that the formalization one has chosen to work with is not a desirable one.
For example, if a group of people by some social process comes to a consensus then arguably this represents the "will of the people". Thus it makes sense to reason about this concept without requiring the existence of ranked preferences.
So for you, "will of the people" represents a consensus preference? And following this idea, if there is no consensus (i.e., at least one person in Portland wants to ride an Uber), there is no "will of the people"?
The whole point of Arrow is that you need some very strong assumptions (e.g., cardinal preferences) to define a "will of the people". The only real world expression of cardinal preferences is a set of supply&demand curves, however - based on this the "will of the people" says Uber should exist.
"If there is no 'will of the people', then there is no consensus"? That doesn't make a lot of sense to me. Groups make decisions all the time, but that doesn't mean those decisions were all endorsed by everyone in the group. Was splitting off from the Catholic Church to form the Anglican Church really the will of the people of England? Because there was a consensus.
If a group of people comes together, discusses, and comes by some process to a unanimous decision ("consensus") then it does usually make a lot of sense to regard the outcome as the "will of these people".
The point I am trying to make through the last n posts is that Arrow's theorem does concerns the impossibility of a certain, narrow-minded formalization. It is therefore incorrect to conclude that 'the "will of the people" is a nonsensical concept by Arrow's Impossibility Theorem', which is what you had claimed.
I have nothing to say about the people and churches of England.
Your claim is that if a unanimous decision has been arrived at, it constitutes the will of the people. No one disputes this - the case where everyone agrees is trivial and uninteresting. It also does not describe the situation with Uber in Portland, Delhi, or anywhere else. I want Uber in Delhi, some politicians don't. Hence there is no consensus.
You either have a coherent definition of "will of the people" that goes beyond consensus, or you don't. If you do, give the definition.
> EDF was granted permission by the regulator in the summer to relax its graphite weight-loss limit at the Dungeness reactor in Kent from 6.2% to 8% after it came close to breaching the original safety margin.
> Hinkley-B and Hunterston-B are also getting close to their higher 15% limits, too.
Then what? There's nothing magical about DOIs. You need someone to store the citation metadata. And generate / deposit citation metadata. And maintain the persistence of the DOI. What precisely does the DOI represent? A codebase? A fork of it? A file? A file at a particular revision? A changeset?
