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July 24, 2008

Real versus Complex Numbers

Posted by David Corfield

Over here, we’re looking at the differences between two famous infinitely large fields:

\mathbb{C} is uncountably categorical, that is, it is uniquely described in a language of first order logic among the fields of the same cardinality.

In case of \mathbb{R}, its elementary theory, that is, the set of all closed first order formulae that are true in \mathbb{R}, has infinitely many models of cardinality continuum 2 02^{\aleph_0}.

In naive terms, \mathbb{C} is rigid, while \mathbb{R} is soft and spongy and shape-shifting. However, \mathbb{R} has only trivial automorphisms (an easy exercise), while \mathbb{C} has huge automorphism group, of cardinality 2 2 02^{2^{\aleph_0}} (this also follows with relative ease from basic properties of algebraically closed fields). In naive terms, this means that there is only one way to look at \mathbb{R}, while \mathbb{C} can be viewed from an incomprehensible variety of different point of view, most of them absolutely transcendental. Actually, there are just two comprehensible automorphisms of \mathbb{C}: the identity automorphism and complex conjugation. It looks like construction of all other automorphisms involves the Axiom of Choice. When one looks at what happens at model-theoretic level, it appears that “uniqueness” and “canonicity” of a uncountable structure is directly linked to its multifacetedness.

Apparently, there is something Galoisianly model theoretic going on.

Now, I see Jamie Vicary has a new paper – Categorical properties of the complex numbers, and I am wondering whether bridges can be built.

Abstract:

Given the success of categorical approaches to quantum theory, and given that quantum theory is underpinned by the complex numbers, it is interesting to study the categorical properties of the complex numbers directly. We describe natural categorical conditions under which the scalars of a monoidal \dagger-category gain many of the features of the complex numbers. Central to our approach is the requirement that the \dagger-functor be compatible with the construction of particular limits in the category; we show that this implies nondegeneracy of the \dagger-functor, as well as cancellable hom-set addition. Our main theorem is that in a nontrivial monoidal \dagger-category with finite \dagger-biproducts and finite \dagger-equalisers, for which the monoidal unit has no proper \dagger-subobjects, the scalars have an involution-preserving embedding into an involutive field with characteristic 0 and orderable fixed field, and therefore embed into the complex numbers if they are at most of continuum cardinality.

In the discussion he writes,

We set out to find a set of properties of a category that imply the category contains an algebraic structure similar to the complex numbers. We have achieved this, to some extent. The most significant missing property is that, although we have shown that the self-adjoint scalars admit a total order, this will not necessarily resemble the order on the real numbers. As a consequence, although the scalars will embed into the complex numbers (as long as they are at most of continuum cardinality), we cannot guarantee that an embedding can be found that is involution-preserving, sending the self-adjoint scalars into the real numbers.

The problem seems to be that while the reals can be characterised as

the ‘biggest’ Archimedean order: for every field with a specified Archimedean order, there is an order-preserving embedding into the real numbers. In particular, any such field is at most of continuum cardinality. We know that our self-adjoint scalars will admit an order, but it is possible that they will fail to admit an Archimedean order, which is a necessary property for them to admit an embedding into the real numbers.

Posted at July 24, 2008 2:14 PM UTC

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33 Comments & 1 Trackback

Re: Real versus Complex Numbers

This confused me. (It confused me over at your other blog too.)

Algebraically closed fields of characteristic zero are uncountably categorical, right? So it’s nothing particular about C. And from that point of view, I can explicitly construct other automorphisms. For example, I can take the algebraic closure of the p-adics; I can probably find an explicit element of the Galois group there. (I don’t actually know that’s the case, but I’m guessing it’s true.)

What makes C special is that its constructed from R. So in this argument, you are explicitly choosing your first-order model of R, i.e. R itself. You could take one of the other models, construct the equivalent of C over that model, and viola you have another algebraically closed field of characteristic zero. This field is isomorphic to C, but I would guess that you would need the axiom of choice to prove it.

Posted by: Walt on July 25, 2008 4:08 AM | Permalink | Reply to this

Re: Real versus Complex Numbers

I’m confused too. I’m hoping that my understanding of ‘sameness’ could be helped by bringing model theory and category theory together. I see you have thoughts on this.

A new fact for me. The algebraic closure of the pp-adics, I read, is not metrically complete. Its metric completion C pC_p is isomorphic to the complex numbers, so may be regarded as “the complex numbers endowed with an exotic metric”.

Posted by: David Corfield on July 25, 2008 9:32 AM | Permalink | Reply to this

Re: Real versus Complex Numbers

Just had a look at Jamie’s article. Some random comments:

p. 2:

the \dagger-functor, first ma[d]e explicit in the context of categorical quantum mechanics by Abramsky and Coecke

Ah, I hadn’t know that they have been the first to talk about \dagger-functors in QM.

p. 4:

If a Frobenius algebra in Hilb\mathbf{Hilb} has its multiplication related to its comultipplication by the \dagger-functor, then it is a C *C^*-algebra [6, 11]

Is this implciitly assuming that Hilb\mathbf{Hilb} denotes the category of finite dimensional Hilbert spaces?

General:

It is often said that what makes QM behave so “strangely” as compared to classical mechanics is the superposition principle. But really it is the fact that it is the superposition over the complex numbers. Superposition over the real numbers is not strange but addition of probabilities.

So I was waiting if the article ended with a punchline such as:

By theorem 3.1 every time evolution monoidal \dagger-functor QM:1Cob RiemC QM : 1Cob_{Riem} \to C for CC a nontrivial monoidal \dagger-category with finite \dagger-biproducts and finite \dagger-equalizers and in which the monoidal unit has no proper \dagger-subobjects exhibits “quantum weirdness” in the sense that xyz.

I didn’t see a comparable statement. Could one be given?

Posted by: Urs Schreiber on July 26, 2008 10:31 AM | Permalink | Reply to this

Re: Real versus Complex Numbers

So I was waiting if the article ended with a punchline…

We were chatting about what’s strange about \mathbb{C} back here. Bob talks about ‘incompatible observables’. Can that be expressed in Jamie’s language?

Posted by: David Corfield on July 26, 2008 1:55 PM | Permalink | Reply to this

Re: Real versus Complex Numbers

In fact Bob was utterly wrong here , … there ARE two incompatible observables on the two element set in Rel, according to the definition Ross Duncan and I wrote down in our ICALP’08 paper. Roughly, compatible obsevables are a pair of dagger Frobenius algebras which mutually form `scaled’ dagger bialgebras. One checks that on the two element set in Rel, highly remarcably, such a pair exists, namely

delta_g = 0 ~ (0,0); 1~(1,1)
epsilon_g = 0 ~ *; 1 ~ *

and a misterious one

delta_r = 0 ~ {(0,0),(1,1)}; 1~{(0,1),(1,0)}
epsilon_r = 0 ~ *

My student Bill Edwards and I have constructed a categorical version of Rob Spekkens’ toy model — the quest for such a categorical reformulation was launched by John in
this week’s finds 251
. Without realising, we were making use of that pair of complementary observables in Rel (Jamie Vicary pointed this out to us).

Recent discussions which Bill, Rob Spekkens and I have been having seem to indicate that the key issue which separates categories of relations from quantum mechanics is the fact that categories of relations, by definition, admit local hidden variable representations. And there seems to be a close connection of this with having something to do with complex numbers. We are currently trying to make this claim solid.


Posted by: bob on July 28, 2008 1:28 PM | Permalink | Reply to this

Re: Real versus Complex Numbers

Thanks for those comments, Urs, and thanks to David for the post — it’s great that you guys are interested in this!

Ah, I hadn’t know that they have been the first to talk about \dagger-functors in QM.

Well, I’m sure lots of people realised that taking adjoints of morphisms in Hilb gives an interesting endofunctor. But I think Samson and Bob were the first to suggest this might be something worthwhile to axiomatise and study. Of course, John used the the **-functor in FdHilb to great effect in HDA2 over a decade ago, for reasons deeply connected to quantum mechanics, so I should certainly put in a mention to that as well.

Jamie said:

If a Frobenius algebra in Hilb has its multiplication related to its comultiplication by the \dagger-functor, then it is a C*-algebra

which led Urs to ask:

Is this implicitly assuming that Hilb denotes the category of finite dimensional Hilbert spaces?

Everyone who reads the paper seems to ask this, so I suppose I should make it a bit clearer!

My statement is true even when Hilb contains infinite-dimensional Hilbert spaces, because every Frobenius algebra is necessarily finite-dimensional — the infinite-dimensional spaces don’t contribute. The proof is quite neat: if you have a Frobenius algebra on an object, you can show that the object is self-dual, but in Hilb, the only self-dual objects are finite-dimensional.

There are certainly infinite-dimensional C*-algebras, of course — but unfortunately, these never arise as Frobenius algebras, even in a category with infinite-dimensional objects! This is a real pain, since infinite-dimensional algebras are so useful… it’s just the way the cookie crumbles.

Your comment about 1Cob Riemm{}_{Riemm} is interesting. Unfortunately, I can’t immediately see a natural way to talk about functors from this category into our target category C — we would really want these functors to be ‘smooth’, and I’m not sure how you would define that in this context.

But you say that, for you, ‘quantum weirdness’ has to do with a sort of ‘destructive interference’. Although my scalars won’t necessarily have this, I show that the scalars embed in a nice way into a field, which will give you all the destructive interference you can handle. Hopefully that’s ‘weird’ enough for you!

Posted by: Jamie Vicary on July 26, 2008 10:17 PM | Permalink | Reply to this

Re: Real versus Complex Numbers

Hi Jamie,

thanks for the reply!

Concerning the \dagger-functor axiomatics: it’s probably one of those cases where a certain concept is clear to various people working in the area but somebody must be the first to officially baptize it.

(Happens a lot in internal category theory for instance: it has been clear since Ehresmann that there should be symplectic groupoids, supergroupoids, holomorphic groupoids and the like, but sombody will be the first to actually mention these terms as a definition in a paper.)

Your comment about 1Cob Riemm\mathbf{1Cob}_{Riemm} is interesting. Unfortunately, I can’t immediately see a natural way to talk about functors from this category into our target category CC — we would really want these functors to be ‘smooth’, and I’m not sure how you would define that in this context.

Right, one would have to add in the list of properties of CC that there is a smooth structure of sorts on it.

But that’s not the point I was trying to get at. For the purpose of my question we could consider QM with discrete time evolution and a functor from B\mathbf{B}\mathbb{Z} to CC, for instance, which is nothing but a choice of one-step time evolution propagator.

I am just asking: given a category CC of the kind you are considering in theorem 3.1 as the context for quantum theory, which of the usual quantum effects are you bound to see?

I am asking because in many other discussions we had here on the blog, not the least in relation to Jefrrey Morton’s article we had seen that many aspects of quantum theory lend themselves to a very fundamental categorical description, while the appearance of “phases”, the fact that the path integral sums up unimodular complex numbers instead of real numbers, is at least more subtle from the point of view of natural toposophy.

Posted by: Urs Schreiber on July 27, 2008 9:53 AM | Permalink | Reply to this

Re: Real versus Complex Numbers

Urs asked:

Given a category C of the kind you are considering in theorem 3.1 as the context for quantum theory, which of the usual quantum effects are you bound to see?

Well, first, I should say that the main point of the theorem is to show that lots of important, basic properties follow from the existence of these ‘\dagger-limits’ — for example, the self-adjoint scalars admit a total order, just like the complex numbers; and the norm of a nonzero state is never zero, just like for vectors in a Hilbert space. I’m not sure if these count as ‘quantum effects’, but they’re pretty important!

It’s important to point out that there’s nothing here that actually requires the complex numbers; the nonnegative real numbers, for example, will do. If by ‘quantum effects’ you mean something more than what you get in a category of free modules over the nonnegative reals, then since these are a model for the axioms, I’m afraid you’ll be disappointed. The way that the complex numbers come into the game is that, up to cardinality, they are the ‘largest’ semiring that can emerge.

David mentioned a concrete ‘weird’ property of Hilb: objects can have two different bases which are, in a sense, maximally imcompatible. I haven’t replied to his comment yet since I haven’t worked out a good answer, but this is a good, ‘quantum’ property to tinker around with and try to prove.

But, anything that requires the complex numbers, or phases — that’s still out of reach. I have some quite concrete ideas for how to get at this sort of thing categorically, but they’re still very much at an embryonic stage.

Posted by: Jamie Vicary on July 27, 2008 11:08 PM | Permalink | Reply to this

Re: Real versus Complex Numbers

If we were to say what’s going on in what Borovik writes in terms of sketches, would the following be correct?

There is a finite discrete sketch for fields (see Chapter 8 of Category Theory for Computing Science by Michael Barr and Charles Wells). Models are suitable graph maps to Set. For any two models of continuum cardinality, satisfying the properties of the complex numbers, there is a natural equivalence between them. In fact, there are 2 2 02^{2^{\aleph_0}} such equivalences.

On the other hand, there are infinitely many models satisfying the properties of the reals which are pairwise nonequivalent.

Then what morphisms are there from any model satisfying the properties of the reals (can an FD sketch capture these?) to a version of the complex numbers? 2 2 02^{2^{\aleph_0}}?

Posted by: David Corfield on July 26, 2008 2:37 PM | Permalink | Reply to this

Re: Real versus Complex Numbers

Am I alone in really preferring links to arxiv papers go to the abstract rather than the PDF? This lets you more easily poke around at their other work, or other versions (including later ones) of this paper.

Posted by: Aaron on July 27, 2008 6:33 AM | Permalink | Reply to this

Re: Real versus Complex Numbers

preferring links to arxiv papers go to the abstract rather than the PDF

I have been trying to follow the practice of linking to the arXiv abstract when referring to the paper as a whole and linking to the pdf when pointing out specific parts of it (sections, equations, etc.)

Often I would want to give a link to a particular page of a pdf. Is there any command one can append to the PDF’s URL such as to control at which point the pdf reader will open it?

Posted by: Urs Schreiber on July 27, 2008 9:24 AM | Permalink | Reply to this

Re: Real versus Complex Numbers

Aaron wrote:

Am I alone in really preferring links to arxiv papers go to the abstract rather than the PDF? This lets you more easily poke around at their other work, or other versions (including later ones) of this paper.

I always link to the abstract rather than the PDF. In addition to the reasons you mention, there are also people who prefer Postscript to PDF.

Indeed, when people here post links to the PDF, I feel free to change their links to point to the abstract.

If people figure out how to point to a particular page of the PDF, I would be glad to let them do that.

Posted by: John Baez on July 27, 2008 11:20 AM | Permalink | Reply to this

Re: Real versus Complex Numbers

I’m pretty sure that it’s impossible to link to a specific page on a PDF. Opening PDFs in a browser at all requires a plug-in of some sort. Technically all http knows from PDFs is “it’s some other sort of file to download”, and the default behavior of any browser (without some sort of plugin set to catch PDF files) is to just download it to disk.

Posted by: John Armstrong on July 27, 2008 1:52 PM | Permalink | Reply to this

Re: Real versus Complex Numbers

It requires a plug-in or an external application. The external applications generally only get the downloaded file names, but most plugins get passed the http string, which they can then parse.

Actually, for PDF plugins, you can append the html-specified “#” section anchors to get particular pages.

See:
http://kb.adobe.com/selfservice/viewContent.do?externalId=317300
or
http://www.planetpdf.com/enterprise/article.asp?ContentID=6426

Posted by: Aaron on July 28, 2008 12:57 AM | Permalink | Reply to this

Re: Real versus Complex Numbers

Cool! So this link will take you (in your browser’s PDF plug-in) to page 17 of Reconstructing the Universe by Ambjorn, Jurkiewicz and Loll. (IMHO, links to the arXiv abstract page would generally still be preferable, but if someone wants to focus on a particular figure or section of a paper, it’s great to be able to do that too.)

Posted by: Greg Egan on July 28, 2008 5:14 AM | Permalink | Reply to this

Re: Real versus Complex Numbers

Hmm, on Macs it’s a bit of a headache to get PDF files to appear within the Firefox browser, as noted here:

Adobe Reader is available for the Mac platform, but the included browser plugin does not work with Firefox. Adobe Reader’s browser plugin must be disabled, and the third-party PDF Browser Plugin must be installed.

Maybe the Linux people will have more luck with that.

Posted by: Greg Egan on July 28, 2008 5:50 AM | Permalink | Reply to this

Re: Real versus Complex Numbers

Not only that, but the PDF reader that comes built-in for Safari doesn’t recognize the # argument. Just opens page 1 like every other time.

Oh wait, is this another time that the Café is going to specify which software we must use to read it properly?

Posted by: John Armstrong on July 28, 2008 6:32 AM | Permalink | Reply to this

Re: Real versus Complex Numbers

Not only that, but the PDF reader that comes built-in for Safari doesn’t recognize the # argument. Just opens page 1 like every other time.

I’m not sure what you’re referring to there; is “the PDF reader that comes built-in for Safari” some kind of PDF-savvy plug-in that’s present if you don’t install Adobe Reader? I’ve installed Adobe Reader on my Mac (though I generally use the Preview app to read PDF files), and that’s put a plug-in in the system that Safari can use – and which does obey the #page=nn argument.

Oh wait, is this another time that the Café is going to specify which software we must use to read it properly?

So far it’s just a few people exploring the possibilities and seeing what the kinks are. I’m sure our hosts will be interested to read about the various glitches people report; the reason I put that link up was so people could test it out in different browsers and operating systems. Maybe I should have said that explicitly.

Anyway, for the record: my experience on a Mac is that the #page=nn trick works for Safari with the Adobe plug-in … and that I’m too lazy to hunt for the third-party PDF plug-in that works with Firefox, to see if that honours the #page argument.

Posted by: Greg Egan on July 28, 2008 6:58 AM | Permalink | Reply to this

Re: Real versus Complex Numbers

I haven’t installed anything else to read PDFs since I got my new MacBook Pro. Out of the box, Safari will open PDF files in the browser, but it doesn’t recognize #page=nn

If it’s going to become common practice here to link to a PDF with that extra argument, I think it would be polite to mention which page you’re going to in the link text, since this behavior is clearly not as universal as Aaron implied it is. But I don’t really expect that to happen. The technical attitude here is “we’re going to set up the server end for our convenience and demand that you choose your client to comply”.

Mostly I’m thinking of the MathML here, that if I’m really interested in (and can’t piece together from context) I have to switch browsers to view. Or, since I haven’t yet, download an entirely new browser and keep it kicking around my hard drive just so I can read this one weblog. Sure, it would be better if Safari read MathML, and many people have requested that it support MathML, but at the moment it doesn’t, so the choice to go with MathML instead of a server-side LaTeX renderer basically says “we don’t want your Safari-browsing kind around here”.

Posted by: John Armstrong on July 28, 2008 2:35 PM | Permalink | Reply to this

Re: Real versus Complex Numbers

I did not mean to imply it was universal, or even nearly so. It’s a semi-standard, that’ll work for the plugins that support it and won’t work for those that don’t, or for “external applications”. As it happens, it doesn’t work on my setup, and I wish people would only use it when there are other clues in the text as to what part I should be looking at.

Posted by: Aaron on July 29, 2008 8:41 AM | Permalink | Reply to this

Re: Real versus Complex Numbers

John A. wrote:

The technical attitude here is “we’re going to set up the server end for our convenience and demand that you choose your client to comply”.

Actually, I think the attitude of David Corfield, Urs Schreiber and myself is something like:

“We were too lazy to set up blogging software that displays TeX, but Jacques Distler did it and offered to let us use his, so we took advantage of that kind offer. Now there are lots of other options that are easier to use — but so far the advantages of switching are outweighed by the advantages of being lazy and sticking with what we’ve got.”

As for referring to individual pages or equations inside PDF files, I’m unlikely to do that much until lots of other people agree that it works. Eventually something like this will become standard practice in the scientific literature. It’s sort of sad that it hasn’t yet, not even on the arXiv. But I’ll gladly sit back and let other people fight about the technical implementation.

Posted by: John Baez on July 29, 2008 11:52 AM | Permalink | Reply to this

Re: Real versus Complex Numbers

I am not sure what the worries are about.

We have been – and surely will continue to – commonly point each other to parts of pdf documents in our discussions: “see page nn here, see equation mm there, have a look at figure pp over there”.

This is always a slight nuisance since it requires to “leave the room” as John used to say.

Now, thanks to Aaron, we have figured out to make following such pointers at least a tiny bit more convenient – for some readers on some equipment, yes, but still – by making the electronic house-elve open the pdf at least at the right page number.

In an even better world, I imagine it would be possible to include a pane much like an embedded picture in a comment inside which the relevant part of the pdf is visible. Or something like that.

Posted by: Urs Schreiber on July 29, 2008 1:02 PM | Permalink | Reply to this

Re: Real versus Complex Numbers

Urs wrote:

I am not sure what the worries are about.

As usual, the worry is that some people will start relying on technology that doesn’t work for other people, cutting those other people out of the discussion.

I had forgotten, for example, that John Armstrong has never managed to read the math symbols on this blog.

My usual solution to this problem is to lag behind the cutting edge, so lots of people can read what I write.

Stuck with a text-only web-browser? No problem! You can still read This Week’s Finds in ASCII.

And for a fee I can have it sent to you by mail, telegraph, or pigeon.

My experiment with videos of math lectures was an attempt to be cutting-edge — but I concluded that it’s not worth bothering with videos (yet).

Posted by: John Baez on July 29, 2008 1:33 PM | Permalink | Reply to this

Re: Real versus Complex Numbers

As usual, the worry is that some people will start relying on technology that doesn’t work for other people, cutting those other people out of the discussion.

Exactly. This is why every professional web developer has five computers, each with eight browsers on it. Now I’m not advocating that sort of extremism, but before adopting new habits it’s useful to slow down and consider people who don’t interact with the technology exactly as you do.

I’m not saying not to use this tag in links to PDFs, but have in the back of your mind the fact that many people don’t use a combination of browser and plugins that will support it.

It’s the difference between saying “theorem 5” and saying “theorem 5 on page 23”. The latter has the information right out there in the text and doesn’t make me search my address bar to find out what you expect my browser to know.

Posted by: John Armstrong on July 29, 2008 3:18 PM | Permalink | Reply to this

#page=

Hmmm… funny. I have exactly the opposite reaction to this news.

Y’see, up till now, I have always hated the plugin(s) that make PDF files open in the browser. I have much preferred to read PDFs in Acrobat Reader.

Now I learn that there’s something that the plugin can do, that my usual procedure (of disabling the plugin, and causing the PDFs to open in AcroRead) cannot.

This new feature would cause me to re-evaluate my antipathy to PDF plugins, and perhaps to seek out a plugin solution for Mozilla/Firefox. But only if people actually use the new feature, and use it enough to make switching worthwhile.

This is why every professional web developer has five computers, each with eight browsers on it.

Maybe I’m funny that way, but I have five different web browsers installed on my laptop, and I use three of them regularly. As I type this, two of the three are running, and I will probably launch the third, when I sit down to work on a talk that I am preparing.

There are certainly users (Eric Forgy, for one), who are stuck with a certain technology, and unable to change it. But the vast majority of users do have the capability to upgrade, and generally would be much happier overall if they did (see, for instance, the discussion about resizing <textarea>s or the discussion about “print selection”)

Urs wrote:

In an even better world, I imagine it would be possible to include a pane much like an embedded picture in a comment inside which the relevant part of the pdf is visible. Or something like that.

I don’t think we’re going to allow <iframe>s in comments. But popping up the PDF, at the relevant page, in a separate window, is definitely doable.

Posted by: Jacques Distler on July 29, 2008 4:44 PM | Permalink | PGP Sig | Reply to this

Re: #page=

There are certainly users (Eric Forgy, for one), who are stuck with a certain technology, and unable to change it.

It’s not a question of being stuck. Yes, I can install FireFox. But there is only one site that I read for which this would make any difference. You’re actually asking me to either burn that much more hard disk space, or to convert my existing patterns, all because you think you know what’s best for everyone.

See, this is the attitude I’m talking about: “You users should alter your behavior to match our development choices. We know better than you what will make you happy.”

Posted by: John Armstrong on July 29, 2008 5:08 PM | Permalink | Reply to this

Re: #page=

So, does the same hold true for other categories of software?

  • Would you refuse to download a bittorent client, if the only thing you intended to do with it was download a copy of the arXivs to your computer?
  • Do you eschew the use of BibDesk, because the only thing it does is handle bibtex files (a task that you can perfectly well do, albeit less conveniently, in your existing text editor)?

Even sticking to the web, eschewing the use of any web technology (in recent years, <canvas>, XMLHttpRequest, SVG and, yes, MathML come to mind), until every browser implements them, means that those technologies will never get used, and hence will never get implemented.

It’s just a fact of life that somebody has to move first. Some browser has to be the first one to implement a new technology, and some websites have to precede others in using the new technology.

Otherwise, we’d be permanently stuck with the Web, circa 1994, without so much as <table>s.

The objective isn’t to shut people out, or to create deliberate hardships. The objective is, ultimately, to make the web experience for everyone better.

Granted, commercial sites, which stand to lose money from users who can’t use the new-fangled technologies, are likely to be more conservative in deploying them than non-commercial sites, like this one. But even big commercial sites sometimes require certain browsers, or certain plugins be installed.

Posted by: Jacques Distler on July 29, 2008 5:54 PM | Permalink | PGP Sig | Reply to this

Re: #page=

Would you refuse to download a bittorent client, if the only thing you intended to do with it was download a copy of the arXivs to your computer?

Yes. I hardly see the need to have (a snapshot of) the entire arXiv on my computer. That’s why they store it: so we don’t all have to.

Do you eschew the use of BibDesk, because the only thing it does is handle bibtex files (a task that you can perfectly well do, albeit less conveniently, in your existing text editor)?

I haven’t yet seen the need to use such a utility, and when I do I’m sure it will be much smaller than a browser.

Otherwise, we’d be permanently stuck with the Web, circa 1994, without so much as <table>s.

And when tables were new, good web developers wrote their pages to provide alternates for those browsers that couldn’t render them. You say “use a browser that supports MathML or you get nothing.”

It’s not that you intend to shut people out. It’s that you clearly don’t even see a reason to consider the experience of people who interact with the technology differently than you do.

John provides an ASCII version of TWF. It’s not as good as the HTML version rendered in a modern browser, and that provides a motivation to use such a browser. But he considers that some people have their own reasons for using a text-only browser like Lynx, and he makes the effort to provide for their experience.

Posted by: John Armstrong on July 29, 2008 7:03 PM | Permalink | Reply to this

Re: #page=

Would you refuse to download a bittorent client, if the only thing you intended to do with it was download a copy of the arXivs to your computer?

Yes. I hardly see the need to have (a snapshot of) the entire arXiv on my computer. That’s why they store it: so we don’t all have to.

Perhaps a poorly chosen example, then (though I do have one word for you, in that regard: SpotLight). But the principle remains.

I haven’t yet seen the need to use such a utility, and when I do I’m sure it will be much smaller than a browser.

Camino is 45 MB. Firefox is 60 MB. How many GB of free space do you currently have on your hard drive?

I’d be surprised, indeed, if that were the consideration. If it were, removing the non-English localizations from your MacOSX Installation, using Monolingual, would free up at least 5 GB of space, enough for over a hundred copies of Camino (or 200 copies of BibDesk — all of the aforementioned applications are roughly in the same ballpark of size).

You say “use a browser that supports MathML or you get nothing.”

Actually, no. You get all the characters which comprise the equation. They’re just not arranged properly. For many (perhaps even most) inline equations, that’s good enough to be able to decipher them.

There’s even a CSS stylesheet which one could install, and which would give a much better approximation to “real” MathML layout.

I, personally, have my doubts about this CSS-approach to laying out Math. And, of course, it runs afoul of your position that you shouldn’t have to do anything. But I’d venture that, even not doing anything, the result is as good as John’s ASCII Math.

It’s that you clearly don’t even see a reason to consider the experience of people who interact with the technology differently than you do.

Exactly the opposite is true.

MathML is accessible to blind users (either being read aloud in a screen reader, or converted to braille).

For the (vastly larger) number of people who are not blind, but merely need a larger font-size, MathML rescales seamlessly, along with the text, so it remains perfectly readable at large font sizes (unlike pictures of the equations, which either don’t rescale at all, or turn into illegible crap if they do).

Those are the sort of people who “interact with the technology differently” and it is in no small part for their benefit that we are actually using MathML in the first place.

Posted by: Jacques Distler on July 29, 2008 8:16 PM | Permalink | PGP Sig | Reply to this

Re: Real versus Complex Numbers

Hey, nobody says that we’ll start making page numbers a secret that is to be well hidden in a URL suffix. All we said is that there is trick to make getting to a certain page number faster.

Posted by: Urs Schreiber on July 29, 2008 5:33 PM | Permalink | Reply to this

Re: Real versus Complex Numbers

I wrote:

I’m too lazy to hunt for the third-party PDF plug-in that works with Firefox, to see if that honours the #page argument.

I got over my laziness, but the third-party PDF plug-in that works with Firefox (on Macs) is only for PowerPC Macs, and I’ve got an Intel Mac.

The documentation on Linux plugins seems to suggest that the Adobe plug-in should work with Firefox.

Posted by: Greg Egan on July 28, 2008 12:33 PM | Permalink | Reply to this

Re: Real versus Complex Numbers

I use Linux+Firefox+(Adobe plug-in) in my office, and the #page argument sends me the correct page.

At home I use (PowerPC Mac)+Firefox+(3rd party PDF plug-in), but I haven’t had the chance to test that combination yet; maybe I’ll report on that later.

Regardless, I’ll second the suggestion above that anyone making such a link to a specific page of a PDF should also identify the page number in the text of the message. Oh, and I also generally prefer linking to an arXiv abstract when there isn’t a good reason to do otherwise.

Posted by: Mark Meckes on July 28, 2008 6:31 PM | Permalink | Reply to this

Re: Real versus Complex Numbers

At home I use (PowerPC Mac)+Firefox+(3rd party PDF plug-in), but I haven’t had the chance to test that combination yet; maybe I’ll report on that later.

That combination also honors the #page argument.

Posted by: Mark Meckes on July 29, 2008 4:00 AM | Permalink | Reply to this
Read the post Category Theory and Model Theory
Weblog: The n-Category Café
Excerpt: Category theory and model theory
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