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April 11, 2007

Category Theory as Esperanto

Posted by David Corfield

From Ross Street’s obituary of Max Kelly in The Sydney Morning Herald:

Professor Emeritus Max Kelly was solely responsible for introducing into Australia a branch of mathematics known as category theory, which pervades almost all research in the fundamental structures of mathematics, allowing people in one branch of maths to understand others in a common form, not unlike Esperanto in languages. It is used in theoretical physics, computer architecture, software design, and banking and finance to connect ideas and streamline the management of information.

The Wikipedia entry on Esperanto reckons it has “enjoyed continuous usage by a community estimated at between 100,000 and 2 million speakers”, and that there are about a thousand native speakers. It looks, then, that category theory is a more successful language, so long as we restrict ourselves to the mathematical community. Even at the upper limit, only 1 in 3000 of the world uses Esperanto, and only 1 in 6 million speaks it as ‘a native’.

How many mathematicians speak category theory as a native?

Posted at April 11, 2007 10:10 AM UTC

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Re: Category Theory as Esperanto

Pure category theory? I’m not sure. “Vulgate” category theory, probably more than one might think.

I like the parallel the article draws with the “common, unifying structure”, but I don’t think Esperanto is a good comparison in the long run. Esperanto was cobbled together to unify the people, not the languages. Category theory is more like the classical Latin nobody’s spoken in ages, but from which many modern languages descend. Uncovering classical Latin sheds light on modern Catalan.

Apropos or not, the comparison puts me in mind of something Tolkien said about Esperanto: that it failed to take off because it’s missing a mythopoesis. Language, in his view, is inextricably bound with poems and songs and stories and culture. Without the stories to bind the culture together, the language is just a curiosity. These days it’s easy to look back and see this as the reason why there are more speakers of Quenya or of Sindarin today than of Esperanto.

So, what is a mathematical mythopoesis? What are the analogues of stories and songs that bind together a mathematical discipline? I think it ends up having a lot to do with the “stories” David talks about, especially in light of the recent comments about Moufang loops not being part of any good story (yet).

Posted by: John Armstrong on April 11, 2007 1:43 PM | Permalink | Reply to this

Re: Category Theory as Esperanto

These days it’s easy to look back and see this as the reason why there are more of Quenya or of Sindarin today than of Esperanto.

Not to drag the conversation off of category theory, but do you have a source for that? I wouldn’t have thought there would be any speakers, at least in any meaningful sense, as there just wasn’t enough vocabulary.

Posted by: Aaron Denney on April 11, 2007 3:49 PM | Permalink | Reply to this

Re: Category Theory as Esperanto

I don’t have a good source offhand, but I can try looking when I get a chance.

As for there being minimal vocabulary, I’ll admit there isn’t that much in the well-known works. However, if you go to The Lost Road and Other Writings (volume 5 of The History of Middle-Earth) there’s a document called the “Etymologies”, which lists quite a lot. Remember that Tolkein didn’t invent the languages to use in his stories – he wrote the stories as a place to use his languages.

Beyond that source, enthusiasts have greatly expanded both languages’ lexicons beyond Tolkein’s original work. There’s a lot more vocabulary than you might think just from reading the standard quadrilogy.

Posted by: John Armstrong on April 11, 2007 8:40 PM | Permalink | Reply to this

Re: Category Theory as Esperanto

Remember that Tolkien didn’t invent the languages to use in his stories – he wrote the stories as a place to use his languages.

Very true. And that’s what gives these stories the quality that is missing in essentially all other (of the far too many) attempts to mimic them.

But that means that it’s the other way around as compared to math, doesn’t it? In math you are looking for the right language to tell your stories, not the other way around.

(Of course it should not come as a surprise that literature, or mythology even, follows different principles than math does.)

Posted by: urs on April 12, 2007 9:34 AM | Permalink | Reply to this

Re: Category Theory as Esperanto

But that means that it’s the other way around as compared to math, doesn’t it? In math you are looking for the right language to tell your stories, not the other way around.

Actually, I wouldn’t be so sure of that. Voevodsky worked out motivic cohomology for its own sake, and then went looking for where he could “make a splash”, as he put it.

Not to compare myself to his caliber, but I’m working out a nice little language of spans and cospans in knot theory, trying to find an old problem to solve to show that the language is worthwhile.

Posted by: John Armstrong on April 12, 2007 4:08 PM | Permalink | Reply to this

Re: Category Theory as Esperanto

There’s some interesting work by Harvard historian of science, Peter Galison, in Image and Logic, which likens the interaction between experimenters, instrument designers and theorists in physics to trading activity between different peoples. Without fully understanding the other’s culture, one can still manage to trade successfully, using Pidgins.

Sometimes a mediating community takes on a life of its own, and its children, taught the Pidgin, develop it into a full-blown Creole.

I suppose we might say that mathematicians don’t start out from such mutual incomprehension, but that, all the same, dialects do form which are hard to translate. And even when the dialect is comprehensible, the culture, its myths and rituals, may not be.

Posted by: David Corfield on April 11, 2007 4:04 PM | Permalink | Reply to this

Re: Category Theory as Esperanto

I see that John A. is trying to taunt me to respond to the comments about Moufang Loops and racks. Are they part of a good story? I think so. Moufang Loops were invented to study finite geometry and finite geometry is groovy even if I and my friends don’t study them.

Quandles, as part of the story, are a big part. The story line is facinating. They are discovered, forgotten and rediscovered. They are cleverly named by David Joyce, and the naming allows us to confuse David with James who coined the name “quark.” Even though Colin Rourke thinks this a silly name, the name sticks. They encode conjugation in a reasonable algebraic system. The representation of (homogeneous) quandles as cosets of the stabilizer of the automorphism group captures the knot quandle and its automorphism group as the fundamental group. Quandle cohomology and cocycle invariants is a successful categorification. John B, Alissa C, and Derek W’s work indicates there is physical significance.

More generally, we (n-cat cafe regulars) cannot describe the mathematical mythopoesis because this cafe is now part of that mythopoesis. It is part of the story being told. The sad thing is that we are talking about Esperanto when we should be talking about Max Kelly. Those who knew him must be sad. I didn’t know him. My loss.

I am also saddened by the passing of Vonnegut. He taught me to read fiction, and that humor was important even in the face of tragedy. So it goes.

Can someone tell us a story about Max Kelly?

Posted by: Scott Carter on April 12, 2007 5:52 PM | Permalink | Reply to this

Re: Category Theory as Esperanto

We had a post and comments about Max Kelly back here.

Posted by: David Corfield on April 12, 2007 8:35 PM | Permalink | Reply to this

Re: Category Theory as Esperanto




Posted by: Scott Carter on April 12, 2007 11:11 PM | Permalink | Reply to this

Re: Category Theory as Esperanto

These days it’s easy to look back and see this as the reason why there are more of Quenya or of Sindarin today than of Esperanto.

This is preposterous, Quenya has no speakers, while Esperanto has an estimated number of speakers between 100.000 and 2 million speakers, including native speakers as well. I just wanted to set this straight for the record and for people coming here from Google.

Posted by: Andreas van Cranenburgh on April 27, 2010 12:48 AM | Permalink | Reply to this

Re: Category Theory as Esperanto

John where did you find that information about Tolkien’s opinion on Esperanto. I’m doing a research paper and am trying to collect sources. Thanks!

Posted by: Jackie on November 20, 2011 10:44 PM | Permalink | Reply to this

PNAS, Sci-Fi, History; Re: Category Theory as Esperanto

3 quick comments on Artificial Languages (none, sadly, using Catgory Theory):

(1) Kirby, S., Dowman, M., and Griffiths, T. L. (2007) Innateness and culture in the evolution of language. PNAS, 104(12):5241–5245.

Related links
Authoritative: (Publisher’s PDF… likely be available here.)
Web search: Google Web Search :: Google Scholar


Human language arises from biological evolution, individual learning, and cultural transmission, but the interaction of these three processes has not been widely studied. We set out a formal framework for analyzing cultural transmission, which allows us to investigate how innate learning biases are related to universal properties of language. We show that cultural transmission can magnify weak biases into strong linguistic universals, undermining one of the arguments for strong innate constraints on language learning. As a consequence, the strength of innate biases can be shielded from natural selection, allowing these genes to drift. Furthermore, even when there is no natural selection, cultural transmission can produce apparent adaptations. Cultural transmission thus provides an alternative to traditional nativist and adaptationist explanations for the properties of human languages.

Keywords: cultural transmission, iterated learning, Bayesian learning, nativism

(2) Watson’s “Embedding”

[Ian Watson’s degree and teaching were in Linguistics]

The Embedding, the remarkable first novel by Ian Watson, is one of the
most sophisticated novels to focus on issues of human and alien
linguistics. Watson argues that there may be patterns common to every
human language except one (of the roughly 5,445) and that that one might
be the very one needed to communicate with ETs. Ironically, in his novel,
the essential language is spoken by a tiny Amazon rainforest tribe (the
Xemahoa) about to be destroyed by a huge dam project. Indeed, at least
1,000 of the human languages are “endangered species,” and when they die
within another century, 1,000 views of the universe die with them.
In The Embedding, three different linguistic plots are intertwined. (1)
Children in an experiment live in an artificial environment are speak a
wholly artificial language. (2) The Xemahoa, under the influence of a
certain drug, “maka-i,” can speak and understand a language (Xemahoa B)
normally incomprehensible to them. (3) The Sp’thra, extraterrestrials,
offer space travel secrets in return for the “widest possible knowledge of
language” to ensure their imperiled communications system. Ian Watson
graduated from Oxford with a First in English, did research in comparative
literature, and has taught in African and Japanese universities. His
insights about alien linguistics in this novel include:
¥ The involvement of Rand, the Hudson Institute, NASA, and
the National Security Agency [pp.21-22]
¥ “Ever since [MIT Professor Noam] Chomsky’s pioneer work, we
all assume that the plan for language is programmed into the
mind at birth. The basic plan of language reflects our
biological awareness of the world that has evolved us….
so we’re teaching three artificial languages as probes at the
frontiers of a mind.” [p.45]
¥ “Speech processing depends on the volume of information the
brain can store short-term …. but a permanent form isn’t
practical for every single word – we only need remember the
basic meaning. So you’ve got one level of information –
that’s the actual words we use, on the surface of the mind.
The other permanent level, deep down, contains highly
abstract concepts – idea associations – linked together
network-style. In between these levels comes the mind’s plan
for making sentences out of ideas. The plan contains the
rules of what we call Universal language – we say it’s
universal, as this plan is part of the basic structure of
mind and the same rules can translate ideas into any human
language whatever …. All [human] languages being cousins
beneath the skin…” [p.49]
¥ “We shall have found out something about the mind’s idea of
all possible langauges….. All languages spoken by beings
evolved on the same basis as ourselves. I can’t vouch for
languages that silicon salamanders elsewhere in the universe
have dreamed up…” [p.53]
¥ “Honeybees evolved their communication system away from the
direction of sound to that of dance. Only primitive bee
still use noises. Evolved bees developed the
aerial dance to express themselves more logically.” [p.56]
¥ When the alien first lands on Earth, in Nevada, it says:
“Nice planet you have here. How many languages are spoken?” [p.129]
because it has learned English in three
days, from audiotapes. “You can imprint a language directly
into the brain then?”
“Good guess – provided it conforms to … the rules of
Universal Grammar!” [p.130]
¥ “We call ourselves collectively the Sp’thra. You do not
hear the ultra and infrasonic components of the word so I
drop them. It means the Signal Traders. Which is what
we are – a people of linguists, sound mimics and
communicators…. Besides being expert
communicators in many modes, we use language machines…”
¥ “There are so many ways of seeing This-Reality, from so
many viewpoints. It is these viewpoints that we trade for.
You might say that we trade in realities…” [p.137]
¥ “The Sp’thra make the following offer for what we want to
buy…. We will tell you the location of the closest
unused world known to us, habitable by you. The location
of the nearest intelligent species known to us ready to
engage in interstellar communications, together with an
effective means of communication using modulated tachyon
beams. Finally, we offer you an improvement on your current
technology for spaceflight within your solar system…
[in return for] working brains [sliced from their bodies]
competent in six linguistically diverse languages….”
¥ The aliens are in an obsessive search for the Change
Speakers: “They are variable entities.
They manipulate what we know as reality by means of their
shifting- value signals. Using signals that lack constants
– which have variable referents…. They are free.
They shift across realities. Yet when we have successfully
superimposed the reality-programmes of all languages …
we too shall be free.”
¥ The aliens, on hearing about the Amazonian native Xemahoa
now offer the much more valuable “interstellar travel
The novel does not have a conventional happy ending, and yet it does
provide an impressive range of notions about the limits of language, and
the possible value of human languages to extraterrestrials.

(3) The first experimental work on whether children raised by deaf and dumb nurses would spontaneously speak in Hebrew or some other “Edenic” language was circa 1200 A.D. by the Holy Roman Emperor Frederick II, patron of Fibonacci, and experimental biologist.

Posted by: Jonathan Vos Post on April 11, 2007 5:18 PM | Permalink | Reply to this

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