Importantly, however, the complex numbers have no total ordering that respects addition and multiplication. In other words, there's no large set of "positive complex numbers" closed under both operations.
This is also the reason why the math in this XKCD strip doesn't actually work.
You can still find divisors for Gaussian integers. If x, y, and xy are all Gaussian integers, which will be trivially fulfilled for any x when y=1, then x, y both divide xy.
You can then generalize the \sigma function by summing over all the divisors of z and dividing by |z|.
The resulting number \sigma(z) lies in C (or maybe Q + iQ), not just Q, but it's perfectly well defined.
I think you are confusing the word "color" that identifies a certain type of visual experience, with the word "color" that identifies a certain set of light-frequencies. This is much like confusing the word "sound" which means "auditory experience", with the word "sound" which means "acoustic vibrations".
You see certain frequencies in a different way than people with red-green colour blindness; in short these frequencies lead to different qualia, different visual experiences. That's rather obvious and rather useless in discussing the deeper philosophical point.
But to say that you experience certain visual experiences differently than others experience them, may even be a contradiction in terms -- unless it's meant that the atomic qualia trigger in turn different qualia (e.g. different memories or feelings) in each person. Which is probably also trivially true...
On the common sense view that qualia are the kolors generated by our minds, which do so based on sensory input about the colors in the world, it makes sense that color-to-kolor conversion (if you will) should be imperfect even among people with properly functioning sight.
Its possible my writing wasn't clear enough to convey this point (or that you were objecting to CCC, not me), but I was getting at the idea that we probably do experience slightly different kolors. It was never my intention to be philosophically "rigorous" about that, just to raise the point.
If you believe this, then you must similarly think that Mary will learn nothing about the qualia associated with colors if she already understands everything about the physics underlying them.
That doesn't follow. Figuring out the behaviour of a programme is just an exercise in logical deduction. It can be done by non-superscientists in easy cases, so it is just an extension of the same idea that a supersceintist can handle difficult cases. However, there is no "easy case" of deducing a perceived quality from objective inormation.
Beyond that, if all you are saying is that the problem of colours is part of a larger problem of qualia, which itself is part of a larger issue of experience, I can answer with a wholehearted "maybe". That might make colour seem less exceptional and therefore less annihilaion-worthy, but I otherwise don't see where you are going.
I'm not just talking about behavior. The kinds of things involved in experiencing a program involve subjective qualities, like whether Counter-Strike is more fun than Day of Defeat, which maybe can't be learned just from reading the code.
It's possible the analogy is actually flawed, and one is contained in its underlying components while the other is not, but I don't understand how they differ if they do, or why they should.
How about telling me whether you actually had?
I maintain that the difference between code & a running program (or at least our experience of a running program) is almost exactly analogous to the difference between physical matter & our perception of it. The underlying structure is digital, not physical, and has physical means of delivery to our senses, but the major differences end there.
I don't see where you are going with that. If you are a superscientist, there is nothing you can learn from running a programme that you cannot get from examining the code. But M's R proposes that there is something you can get from seeing a colour yourself. The analogy doesnt seem to be there. Unless you disagree with the intended conclusion of M's R.
If you are a superscientist, there is nothing you can learn from running a programme that you cannot get from >examining the code.
If you believe this, then you must similarly think that Mary will learn nothing about the qualia associated with colors if she already understands everything about the physics underlying them.
In case I haven't driven the point home with enough clarity (for example, I did read the link the first time you posted it), I am claiming that there is something to experiencing the program/novel/world inasmuch as there is something to experiencing colors in the world. Whether that something is a subset of the code/words/physics or something additional is the whole point of the problem of qualia.
And no, I don't have a clear idea what a satisfying answer might look like.
Coincidentally I am in the process of writing my final advanced logic assignment as we speak (I wouldn't call myself a working logician as a) I'm undergrad, and b) rarely working). My module focuses on the lead up to Godels incompleteness theorem, so overlaps with set theory related stuff a lot. I might be able to answer some general questions but no guarantees.
Know how you feel about doing very different things simultaneously, done both political philosophy and logic recently, odd shift of gears.
Random question You wouldn't know how to show the rand of an increasing total recursive function is a recursive set would you? Or why if a theory has a arbitrarily large finite model it has an infinite model?
Odd thing about doing high level stuff is realising that the infrastructure you get used to doing lower level stuff (wikipedia articles, decent textbooks, etc.) ceases to exist. I feel increased sympathy for people pre information age.
You'd have to explain what the rand function is, since that is apparently an un-Google-able term unless you want Ayn Rand (I don't), the C++ random return function, or something called the RAND corporation.
The second question is due to compactness.
I'm the kind of person who reads things like Fixing Frege for fun after prelims are over.
Edit: Oh, & I don't mean to be rude, but I probably wouldn't call anyone a working mathematician/logician unless they were actively doing research either in a post-doc/tenure position or in industry (eg at Microsoft).
As far as Mary's Room goes, you might similarly argue that you could have all of the data belonging to Pixar's next movie, which you haven't seen yet, without having any knowledge of what it looks like or what it's about
I dont see how you could fail to be able to deduce what it is about, given Mary's supercientific powers.
Or that you can't understand a program without compiling it & running it.
Ordinary mortals can, in simple cases, and Mary presumably can in any case.
Or that you can't understand a program without compiling it & running it.
You''re not a superscientist. Can I recommend reading the linked material?
It's possible I already had & that you're misunderstanding what my examples are about: the difference between the physical/digital/abstract structure underlying something & the actual experience it produces (eg qualia for perceptions of physical things, or pictures for geometric definitions, etc).
I maintain that the difference between code & a running program (or at least our experience of a running program) is almost exactly analogous to the difference between physical matter & our perception of it. The underlying structure is digital, not physical, and has physical means of delivery to our senses, but the major differences end there.
I'm not sure what the prolem of distingusihing colours analytically is supposed to relate to. The classic modern argument, Mary's Room attempts to demonstrate that the subjective sensation of colour is a problem of materialism, because on can conceviably know everything about the neuroscience of colour perception without knowing anything about how colours look. That could sort-of be re-expressed by saying Mary can't analytically deduce colour sensations from the information she has. And it is sort-of true that once you have a certain amount of experiential knowledge of colour space, you could gues the nature of colours you haven't personally seen. But that isn't very relevant to M's R because she is stipulated as not having seen any colours. So, overall, I don't see what you are getting at.
It's just another cool problem about colors.
As far as Mary's Room goes, you might similarly argue that you could have all of the data belonging to Pixar's next movie, which you haven't seen yet, without having any knowledge of what it looks like or what it's about. Or that you can't understand a program without compiling it & running it.
I'm not entirely sure how much credibility I lend to that. There are some very abstract things (fairly simple, yes) which I can intuit without prior experience, and there are many complicated things which I can predict due to a great deal of prior experience (eg landscapes described in novels).
But I mostly raised it as another interesting problem with a proposed [partial] solution.
I don't know about excruciating detail, but I think the general idea is this:
One would not predict the existence of evil in a universe created by a benevolent God.
One would not predict the existence of intrinisically subjective qualities in an entirely physcial, and therefor entirely objective, universe.
I wouldn't predict the existence of self-replicating molecules either. In fact, I'm not sure I'm in a position to predict anything at all about physical phenomena without appealing to empirical knowledge I've gathered from this particular physical world.
It's a pickle, all right.
Granting this, why do we all see the same colors, if we do?
I can quickly and easily prove that some people see colours in a different way to the way that I do.
To my eyes, red and green are visibly and obviously distinct. I cannot look at one and consider it to be the other. Yet, red-green colour blindness is the most common version of colourblindness; these people must see either red, or green, or both in some way differently to the way that I see these colours.
You'll notice that the next few sentences of my post address this same idea for fully functional members of different species. But it doesn't technically refute the claim for qualia, only that we're not all equally responsive to the same stimuli.
It is, for example, technically possible (in the broadest sense) that color-blind people experience the same qualia we do, but they are unable to act on them, much in the same way that a friend with ADD might experience the same auditory stimuli I do, but then is too distracted to actually notice or make sense of it.
I note, however, that the physical differences in color-blindness (or different species' eyes) are enough reason to lend little credibility to this idea.
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If you sum over all the divisors of z, the result is perfectly well defined; however, it's 0. Whenever x divides z, so does -x.
Over the integers, this is solved by summing over all positive divisors. However, there's no canonical choice of what divisors to consider positive in the case of Gaussian integers, and making various arbitrary choices (like summing over all divisors in the upper half-plane) leads to unsatisfying results.
That's like saying the standard choice of branch cut for the complex logarithm is arbitrary.
And?
When you complexify, things get messier. My point is that making a generalization is possible (though it's probably best to sum over integers with 0 \leq arg(z) < \pi, as you pointed out), which is the only claim I'm interested in disputing. Whether it's nice to look at is irrelevant to whether it's functional enough to be punnable.