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• CommentRowNumber1.
• CommentAuthorTobyBartels
• CommentTimeSep 7th 2010

I have written a disordered article lacking information. No, wait! I mean, I have written an article on a measure of disorder and lack of information: entropy.

This is from a very mathematical perspective, but I have also tried to indicate how this connects to the physical notion of entropy. (This is something that I would like to understand better!)

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeSep 8th 2010

• CommentRowNumber3.
• CommentAuthorEric
• CommentTimeSep 8th 2010

Oh yes. I love this subject. I could easily spend years thinking about this and little else… in another universe.

I’m looking forward to someone filling in information.

• CommentRowNumber4.
• CommentAuthorIan_Durham
• CommentTimeSep 8th 2010

Entropy! My favorite topic. I actually had a Math student who did a project with me a year or so ago on finding connections between entropy and symmetry and one of my colleagues in the Math department pretty much studies entropy and nothing else (mostly from a mathematical POV). I have strong opinions about the physical interpretation, however, so perhaps I’ll just be a fly on the wall and watch the rest of you duke it out.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeSep 8th 2010
• (edited Sep 8th 2010)

Hah! On the nLab.

I think the historical decision to introduce a unit for inverse mole is a bit questionable. But that’s just me. In any case I added to the entry a brief discussion to indicate that really Avogadro’s constant is nothing but a natural number.

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeSep 8th 2010

I’ve never understood whether we’re supposed to think of every appearance in maths of entropy as a manifestation of a common idea. Are, say, volume entropy, Kolmogorov–Sinai entropy, and Topological entropy, wiki all part of the family? There are a good range of sources here.

There’s even a Galois entropy, as discussed in All Entropies Agree for an SFT which relates entropy to the notion of “geometric asymmetry”. (SFT is ’shift of finite type’.)

• CommentRowNumber7.
• CommentAuthorDavidRoberts
• CommentTimeSep 8th 2010

Added paragraph on future SI definition of kilogram in terms of Avogadro’s number to Avogadro constant, and the analogy to how distance is defined in terms of the speed of light, rather than the other way around as it used to be.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeSep 8th 2010

Okay, David. I suppose that Freud missed one drive, namely the drive to behave like an encyclopedist. ;-) I followed it and created kg.

• CommentRowNumber9.
• CommentAuthorzskoda
• CommentTimeSep 8th 2010
• (edited Sep 8th 2010)

Encyclopedism is indeed one of the plagues of the western civilization; and its social diagnosis was one of the most frequent topics of conversations I had with my closest colleagues in grad school. As this project has some features akin to Bourbaki movement, then despite the differences, positive and negative ones, we should be aware of various aspects of it.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeSep 8th 2010
• (edited Sep 8th 2010)

Encyclopedism is indeed one of the plagues of the western civilization

Due to a recent new book by Grass, currently the feuilleton’s here had lots of reviews on the project of brothers Grimm, back then, to write a German dictionary. Or rather to write The German dictionary.

I was wondering about the encyclopedist drive in that case, too: lacking hyperlinks and anything, they just started straight with Aal and then continued with, dunno, aber and so forth. One of them died within F, the other within H. Just imagine.

I am wondering which nLab entry I will die over.

• CommentRowNumber11.
• CommentAuthorTodd_Trimble
• CommentTimeSep 8th 2010

Urs, it’s just as well that you started kg, as it calls for a page unit which should by all means be connected with Jim Dolan’s approach to algebraic geometry for category theorists.

• CommentRowNumber12.
• CommentAuthorzskoda
• CommentTimeSep 8th 2010
• (edited Sep 8th 2010)

The main problem with encyclopedism in wider sense is the spirit it brings: that having the systematic account of things is a satisfactory digest of life and culture, and from this satisfaction the drive to replace other aspects of life with tension to “finish” or continue with systematization and coding in the same vain. It forgets of the tension between eros and tanatos, the useful aspects of destruction, reconstruction, the existence of unknown and unknowable and importance of reinventing old and acknowledging unknown. It has corollaries like the notion of “general culture” which is of course logically unfounded and very relative notion, which usually impresses the monopoly of locally dominant collective notions, which are often inferior in depth and localistic in scope. It is also related to historical aspect of western culture (as opposed say to Indian culture which is not really aware of history and believes in huge cyclically repeated eras in which a human life has no special historical place) and pretends the dominance over history and globality of the view over the world. All of these are rather fertile to true and uncompromised intellectual drives. The Native Americans and other “primitive” people (see works of Eliade) mythical dances are actualization of eternal mythical stories, identification of human life with things people culturally, spiritually and emotionally are devoted to, as well as to their sources of food, their dead ancestors and other sources of spirituallity which one can not communicate with and exchange lively forces with in an ordinary life.

Fortunately, in the microencyclopedism, the Bourbaki movement was aware of its limits, in the words of Dieudonne, Bourbaki’s scope was to sistematize and simplify the dead mathematics, rather than to propose the way to reserach on alive one. Many of critics of Bourbaki movement were not aware of that.

Urs: As far as German encyclopaedia there was a math enc. started at about 1900, was it Felix Klein or somebody else involved ? It failed because it had that same unachievable goal of being thorough, all-encompassying, so when first few letters were done, they were already obsolete.

I am wondering which nLab entry I will die over.

Something so advanced we can not now even dream of…

• CommentRowNumber13.
• CommentAuthorzskoda
• CommentTimeSep 8th 2010

as to their sources of food

This is not a typo. The sources of life are from gods or other mythological sources. For example some of the tribes who had fish as a main food source in their winter dances become themselves fish to balance their tensions to have the eternal right to succeed in their survival by thanking to the fish for their effort by exchanging the role, namely becoming themselves fish within the ritual. Such ritualistic changes of role are important for establishing the balance with fears from chances given by gods, the one-sided role of food-taker and food-depender within the wider harmony explained in the myth.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeSep 8th 2010
• (edited Sep 8th 2010)

I forgot to add a punchline of the Grimm-story that feels like it has some meaning, though i don’t know if it does:

that immense project which they started, and of which they saw only the first 5th in their lifetime, was eventually finished: during cold war times linguists at universities in West Germany and in East Germany, disconnected by the Iron Curtain, continued working on this, word for word for word. it was finally finished 1971. Now it’s here.

Not sure what it all means. Maybe just that the encyclopedist drive is so powerful to unite even the class enemies. ;-)

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeSep 8th 2010
• (edited Sep 8th 2010)

Zoran,

concerning your criticism of encyclopedism: I entirely agree, though I could hardly have put it with the emphasis that you have.

But methink now with wikis, the situation is different. If an old entry we have is becoming outdate while we are working on the millionth entry, we just go back and update it. The wiki is alive (as long as there are people taking care of it) in a way that the classical encyclopedia never could be.

• CommentRowNumber16.
• CommentAuthorzskoda
• CommentTimeSep 8th 2010
• (edited Sep 8th 2010)

I entirely agree, though I could’t hardly have put it with the emphasis that you have.

Historically the encyclopedism was also related to the movement in French intellectual circles in 18th century, where the knowledge and rational point of view on the world has been a center axis. The understanding in which an intellect is reduced to its rational aspect is specific to western culture and the encyclopedistic movement has influenced it a lot, what is logical if you take the meaning of encyclopedism in its widest sense of kind of an approach to a relation between a human and its surrounding, society and the world (wolrd includes culture, inner self, nature and so on). It is really fruitful to see limits of western civilization by taking such a wide definition of encyclopedism; not just the literal narrow one.

• CommentRowNumber17.
• CommentAuthorzskoda
• CommentTimeSep 8th 2010
• (edited Sep 8th 2010)

The wiki is alive (as long as there are people taking care of it) in a way that the classical encyclopedia never could be.

This is true in a sense (it has some life and dynamics of its own), but the takeover of a spirit that the coding is everything and all its extensions are bad for the culture. Imagine of measuring importance of subjects by amount of wikipedia material devoted to it (you can imagine how many trashy subjects have lots of attention); also it depends very much on when the efforts are done. All the statements about unknown, forgotten etc. which make a difference between the real world and a hand-made collection of any kind are important to be aware of. Young generations will be educated as if those things which are not recorded never existed to mention just one of the effects of overly satisfaction with the attitude of a collector.

• CommentRowNumber18.
• CommentAuthorzskoda
• CommentTimeSep 8th 2010
• (edited Sep 8th 2010)

To add to this: there are communities related to wikis and they indeed live many aspects of what Moscow mathematical school called continuum of mathematical life; the danger of encyclopedic viewpoint is however to expect it as a measure of mathematical knowledge or mathematical life in general. That viewpoint, which is a targetted ideal of encyclopedism should never take over (even for semi-informed sick-ambitious readers).

• CommentRowNumber19.
• CommentAuthorTobyBartels
• CommentTimeSep 9th 2010

In principle, Avogadro’s number is not a natural number (merely a positive real number), because the ratio of the masses of a carbon-12 atom and the reference kilogramme in Paris is not is an exact multiple of 1000/12. At least, I assume not.

• CommentRowNumber20.
• CommentAuthorTobyBartels
• CommentTimeSep 9th 2010

Created unit.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeSep 9th 2010

In principle, Avogadro’s number is not a natural number (merely a positive real number), because the ratio of the masses of a carbon-12 atom and the reference kilogramme in Paris is not is an exact multiple of 1000/12.

That’s true. On the other hand, it seems that in the “Avogadro project” one approach is to define it to be some natural number, and then define the kilogram by this.

This should be better said in the entries. I can’t do it now. Maybe later.

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeSep 9th 2010
• (edited Sep 9th 2010)

Created unit.

I would have thought we need a genuine disambiguation page here, redirecting to unit of an adjunction , unit in measurement , unit of a ring . I see that you phrased a definition such that the last two are special cases of it, but I am not sure yet if I find this the best explanation:

I would have thought that the “right” definition for “unit of measurement” is: an isomorphism that identified an $\mathbb{R}$-torsor with $\mathbb{R}$.

• CommentRowNumber23.
• CommentAuthorMike Shulman
• CommentTimeSep 9th 2010

Crossing wires with Urs #22, I added a bit to Toby’s unit. I do like this point of view that unifies two or three of the concepts, but I’m also not sure that we want the page called unit to be only about it.

Is there a name for a rig in which every nonzero element is a unit? A “semifield”?

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeSep 9th 2010
• (edited Sep 9th 2010)

[edit: Isaid something here but I should better comment later when I have more leisure to be distracted]

• CommentRowNumber25.
• CommentAuthorTobyBartels
• CommentTimeSep 9th 2010
• (edited Sep 9th 2010)

@ Urs

On the other hand, it seems that in the “Avogadro project” one approach is to define it to be some natural number, and then define the kilogram by this.

Yes, but even there, there’s no reason in principle why this should be a natural number. (In practice, of course, it’s easiest if it’s not only a natural number but a multiple of a power of as large a power of $10$ as possible.)

I would have thought we need a genuine disambiguation page here

Yeah, I was a little surprised when I found myself writing it and not covering the unit of an adjunction. (Why is that called “unit”? What does it have to do with the other units?) I don’t know if it’s a coincidence that a unit of a ring and a unit of measurement are both special cases of a general concept, but it’s too good to pass up!

@ Mike

Is there a name for a rig in which every nonzero element is a unit? A “semifield”?

Apparently (2nd version). Note that the rig of tropical numbers is an important example.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeSep 9th 2010
• (edited Sep 9th 2010)

Yeah, I was a little surprised when I found myself writing it and not covering the unit of an adjunction. (Why is that called “unit”?

because it

$i : Id \to R L$

is the unit (= “identity element”) of a monoid $A = R L$ (here: monad)

$i : \mathbb{1} \to A$ $\mu : A \otimes A \to A$
• CommentRowNumber27.
• CommentAuthorTobyBartels
• CommentTimeSep 9th 2010

Ah, right. Well, I can fit that in!

• CommentRowNumber28.
• CommentAuthorTobyBartels
• CommentTimeSep 9th 2010
• (edited Sep 9th 2010)

I’ve written unit of an adjunction; next I’ll fix links to unit. (The only one seems to be at adjunction itself.)

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeSep 9th 2010

Well, I can fit that in!

Okay, thanks.

By the way: I was going to complain that units of measurements are not defined as torsor trivializations, but maybe you and Mike have convinced me that the way the entry now presents it is good.

• CommentRowNumber30.
• CommentAuthorTobyBartels
• CommentTimeSep 9th 2010

There is now a very general definition at unit.

• CommentRowNumber31.
• CommentAuthorUrs
• CommentTimeSep 9th 2010
• (edited Sep 9th 2010)

So should we remove the leading sentence “not to be confused with unit of an adjunction” now?

• CommentRowNumber32.
• CommentAuthorTobyBartels
• CommentTimeSep 9th 2010
• (edited Sep 9th 2010)

@ Urs #29

Well, I think that it’s perfectly reasonable to note that torsor trivialisations correspond precisely to elements of the torsor, that oriented $\mathbb{R}$-lines (or $\mathbb{R}_{\geq0}$-lines period) correspond precisely to $\mathbb{R}_{\gt0}$-torsors), and apply these insights to this situation. I just didn’t write that in.

• CommentRowNumber33.
• CommentAuthorTobyBartels
• CommentTimeSep 9th 2010

@ Urs #31

I think not; I’m afraid that somebody might well make a link to unit that should be moved to unit of an adjunction in the future. In fact, I’m surprised that there was only one such link to begin with! (I also briefly moved the page to units to check for links to that name; there were none.)

• CommentRowNumber34.
• CommentAuthorTobyBartels
• CommentTimeSep 9th 2010

I had to write rng too.

• CommentRowNumber35.
• CommentAuthorIan_Durham
• CommentTimeSep 10th 2010

What happened to the discussion of entropy? I was really curious to see what category theorists had to say on the topic. ;)

I’ve never understood whether we’re supposed to think of every appearance in maths of entropy as a manifestation of a common idea.

Me neither until I got working with my colleague Steve Shea. He did his PhD on entropy under Michael Keane. After discussing the mathematical, information theoretic, medical (yes, there are medical uses of the term!) and physics-related definitions of entropy, we concluded that they were all variations on the same thing - at least all the ones we were aware of. Certainly Kolmogorov entropy is a generalization of Gibbs-Boltzmann-Shannon entropy and there were several other “entropies” out there that we (and my student) found that all turned out to be variations on the same idea.

• CommentRowNumber36.
• CommentAuthorDavidRoberts
• CommentTimeSep 10th 2010

Comment at entropy.

• CommentRowNumber37.
• CommentAuthorUrs
• CommentTimeJun 5th 2011
• (edited Jun 5th 2011)

I have added two references to entropy (on entropy of states on operator algebras).

• CommentRowNumber38.
• CommentAuthorTobyBartels
• CommentTimeJun 5th 2011

I removed the discussion started by David in #36 from the page.

• CommentRowNumber39.
• CommentAuthorzskoda
• CommentTimeNov 10th 2012
• (edited Nov 10th 2012)

Somebody wrote within entropy

A discussion of entropy with an eye towards the presheaf topos over the site of finite measure spaces is in

There are some functors in the game, but as I did not read in detail, I do not see where really the properties of the whole category of such functors is used. Where is really in this paper the view toward topos point of view (more than just a definition) ? (this is not a complaint, but a question toward appreciation of the topos theoretic aspect there if any!)

I added the following reference to entropy

• William Lawvere, State categories, closed categories, and the existence (subtitle: Semi-continuous entropy functions), IMA reprint 86, pdf
• CommentRowNumber40.
• CommentAuthorUrs
• CommentTimeNov 10th 2012

I think I asked a similar question back then. I had glanced at the article, but not gotten any clear idea of what’s going on.

I had that phase for a while: wondering what all those “categorical” discussions of entropy actually do or aim for. I still don’t have a clear picture. Wish somebody would add a paragraph to the entry that explains it.

• CommentRowNumber41.
• CommentAuthorzskoda
• CommentTimeNov 25th 2014

Entropy considerations from thermodynamics generalize to many PDEs. I have added a note in the references section of entropy:

Entropy-like quantities appear in the study of many PDEs, with entropy estimates. For an intro see

• L. C. Evans, A survey of entropy methods for partial differential equations, pdf; (and longer course text:) Entropy and partial differential equations, pdf
• CommentRowNumber42.
• CommentAuthorUrs
• CommentTimeOct 20th 2018