Follow up to previous

August 23, 2011

> 1. I would like to solicit some clarification on Andrew’s meaning when he
> says, “mathematical innovations are [my emphasis] linguistic innova-
> tions.” I would rather say that mathematical innovation often entails
> attaching new technical meanings to words, e.g., ’gerb’, and sometimes
> introducing new words, e.g., ’surjection’, but that the mathematical
> innovations are not necessarily (and frequently are not) linguistic as
> perceived by the mathematician. In my experience, mathematical in-
> novation, which constantly accompanies mathematical work, is not
> generally linguistic; the mental imagery is geometric, diagrammatic,
> combinatorial, even kinesthetic (particularly for physicists), as well as
> aural (to use a Buddhist expression, mathematical “monkey chatter”)
> which is that part I would call linguistic. Perhaps Hadamard’s Psy-
> chology of Invention in the Mathematical Field would shed light on
> this issue.
>

Let me clarify.

Whatever the devices used by the individual mathematician, in order for the ideas to become part of a shared theory, some form of written communication is used. This is trivially “linguistic,” and, as I’m using it, includes diagrams and even geometric drawings. But are the choices that go into recording mathematical ideas merely style, more or less independent from content? I’m trying to argue that they are not, that a good deal of the content IS the organization of the (written account of the) collection of ideas.

I’m thinking of developments which dramatically progress
mathematics, not the personal phenomena that come with problem solving. The latter can be written about, but it’s mostly irrelevant to mathematics. Major innovations include the development of the decimal system, certainly this can be seen as both a linguistic and a mathematical innovation. Or Newton’s initial formulation of a differential equation, (just 9 years after calculus had been developed!), or for another example, Eilenberg-Maclane spaces. In both of these examples (and countless not mentioned) you have a heap of theory that goes toward defining objects which become *basic* objects of study. (Included in that theory are theorems, since frequently theorems are necessary to give the context for a definition). Now I can say “suppose I have this K(pi,n) space …” How do I translate that statement into the setting, a few hundred years prior, of Descartes or Newton? How is a differential equation translated into the language of the Greeks? It’s not just that the definitions would be long-winded, but much stronger: that for an adult of moderate intelligence to contemplate articulate questions about 4-dimensional manifolds before Gauss’s time, say, is absurd. In the 21st century it is not. To a great extent this is because us 21st century inhabitants have been given an organized language for asking (and *understanding*) such questions. In summary, language is a lot more than a list of labels, but instead is rich with structure. This structure is crucial to mathematics.

> 2. Isn’t the structural atomism Andrew mentions, “. . . statements and
> derivations in some atomic, axiomatic system of symbols” distinct
> from the structural aspects of the objects of mathematical theory? I
> guess we could take a nominalist position that there are no mathemat-
> ical objects, only the language that constitutes mathematical theories.
> Would a strict nominalist tell us that we are mistaken when we claim
> to be thinking in non-linguistic geometric imagery prior to enunciating
> mathematical statements about those images?

I can’t really address this, as I don’t really understand the
nominalist position. Am I confusing the structural aspects of the
“theory of mathematics” with “structural aspects of the objects of
mathematical theory.” Maybe I am. We can steer the ship back over to that topic.

> 3. Andrew kicks the hornets’ nest (or would have twenty years ago) in
> making points about “A Transformative Hermaneutics of Quantum
> Gravity”, i.e., the Sokol project. A question I would like to study is
> how the unintelligibility of Po Mo writing and the unintelligibility of
> mathematical writing compare. For example, three years ago or so,
> Gayatri Spivak wrote an article in “Art Forum” on Badiou. I couldn’t
> understand much of that article. Compare this with the fact that
> any of us wouldn’t understand much of Categories for the Working
> Mathematician without a great deal of work. Are these two cases
> essentially the same, or is there a significant difference lurking here?

I didn’t mean to make a point about that work, but the title serves as a great example of far-reaching abstraction (in natural language). If we know a priori that some work is mathematically sound, albeit requiring, say, a billion pages of difficult math to build up to, then perhaps, a priori, we can claim that the work is fundamentally different than a social philosophy text that requires a billion pages of background, although it might be more
interesting to note the similarities. The distinction between the
math and the philosophy is founded on the idea that math can be
formally correct, i.e., that it can be checked by automation.
Philosophy cannot, both for practical reasons and for the fact that social philosophy (any philosophy, really) requires a point of view (in our case human, of a certain culture, etc.). In math we can claim a conclusion is true even if we don’t understand the proof, provided, say, we understand the algorithm that computed the proof, and we understand the terms in the conclusion. The obvious example is the four colored theorem, or suppose the
Goldbach conjecture had been solved by a program. By contrast, what would it mean to say that a philosophical phrase (e.g., “God is Dead!”) is sound because some billion generated lines of code concluded it. Nonesense! Espousing such a slogan in philosophy is a way of referencing one’s own understanding of the justification, and is not the assertion of a FACT.

Of course, conclusions in mathematics can also be in terms which
require billions of pages of code to parse. So instead of something as simple as the Goldbach conjecture, the program
generates a line relating objects which it has defined, and which are incapable of being understood by a human, yet the human can check the code and conclude that, barring bugs, the conclusion
must be correct. This computer generated conclusion is much more akin to the philosophical assertion which requires billions of pages of justification, since in both cases the conclusion is essentially gibberish.

I won’t say anything more concrete about MacLane vs. Badiou, but would be interested in hearing another’s take on the comparison.

> 4. Meter, millimeter, micrometer, nanometer, . . . . This kind of termi-
> nology is similar to the would-be lengthette, isn’t it? The suffixes
> ’eme’ and ’etic’ seem to be used by philosophers of linguistics. Split
> infinitives are evolving to acceptable status, ’lite’ is understood as a
> variant of ’light’ albeit with a special meaning akin to that of ’je-
> june’. Dennis notes jargon such as texting abbreviations (in an email
> that you might not have received) that arises too quickly for me. I
> have heard that German grammar was reconstructed by grammari-
> ans (18th century?) after Latin grammar and that this constituted a
> major structural change. Ray, you can probably help us here.

The ‘ette’ suffix was an example of what we might call a structural innovation in our language. Indeed it does not exist as I have defined it (again, if I want accuracy but am unable to be precise, I should not use ‘a few nanometers’ to mean a relatively small length). In category theory we have the ‘co’ prefix which is extremely general. The introduction of such modifiers is clearly more substantial than the labeling of some very specific instance of some thing.

I was making a point about how language is only infrequently modified deliberately, to be more structurally robust. It’s mostly aside to the topic here.

> 5. Finally a point that is tangentially related to Andrew’s note. Struc-
> turalism was fashionable in several parts of the liberal arts in the 1950’s
> to 1980’s, but went out of fashion, declared by many to be irrelevant.
> Here is what Peter Caws says (Structuralism, 2000, p.105):
>
> In fact the stress on “formal models” in this statement of
> Chomsky’s points up a general problem for structuralism.
> It is perfectly true that outside some domains of linguistics
> very little progress has been made in developing interesting
> or fruitful formalisms: as we shall see in the next chapter,
> L´vy-Strauss’s attempts, which count among the most am-
> e
> bitious, seem often at once simplistic and contrived. But
> even within linguistics the results have not been much hap-
> pier, and my own view is that the whole attempt to conduct
> the structuralist enterprise in terms of rigorous mathemati-
> cal models is a case of what I call “spurious formalism.” A
> nonspurious formalism has to meet two conditions: what it
> deals with must be precisely specifiable in formal language
> (in the ideal case, moreover one frequently encountered in
> the physical sciences, it will be quantifiable), and this specifi-
> cation must make possible formulations and operations that
> would not be possible in ordinary language. This last con-
> dition is hardly ever met in structuralist studies; the main
> points can nearly always be conveyed discursively, and the
> introduction of formulas and technical expressions is often
>
> 2
>
> merely ornamental and plays no real role in the argument.
>
> In relation to Andrew’s (2), mathematics and mathematical physics
> could in principle be written in natural language, just as they could
> be written in some first-order formal language. It’s not that it is
> impossible, just that it is impractical in the extreme. Caws’s point is
> that the opposite is true of most structuralist efforts in the arts and
> human sciences, that is, natural languages serve better than various
> formalisms of a mathematical kind.

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Art is Shit

February 17, 2009

This evening I came across a facebook group promoting recognition and discussion of Wikipedia Art, a self referencing work/Wikipedia page (notable because of the cultural significance of inviting the controversy it baits on wikipedia for not being notable or culturally significant) whose creators insist is conceptual art. Throughout the discussion is the tireless debate of what constitutes art. I was intrigued and even a little bothered by the undeniable assertion and the implication. (Maybe it is enough to admit it is “art,” but if that means anything, then some implication should follow: I should care; I should support funding for it; I should support recognizing it on par with any other work of art, etc). So I did what any good unwilling participant observer would do: I went to vandalize the page, to highjack the work and reclaim my agency in spite of (and in homage to (ah, the levels of irony!)) my being enlisted as participant. Alas, the page had been deleted without so much as an archive’s history of the deletes and debates that the artist cited to justify the work. I didn’t get to vandalize the page, but I had too much fun posting a response on the facebook group, which I’ll boastfully repeat for you:

Wikipedia Art

Demanding a contrapuntal dialog of vigorous affirmation and inherent denial, the artist insists on the de facto status of Wikipedia Art as conceptual artifact, creating a dissonant ontological reassignment from extinct referent to extant rhetoric. While indisputably manifesting itself, recursively, as Art, both in referencing itself and in referencing that which does not exist, the audience’s apathy is commandeered as medium. Where previous artists have relied only on the milieu of controversy to maintain a similar status, here, the indifferent critic is slightly uncomfortably forced to ask himself the question “why should I care?”

Cf. Manzoni, Piero “Merda d’Artista” (1961) ; Tetazoo, James “No Knife. A study in mixed media earth tones, number three.” (1984)

See, I don’t mind. I’m having fun. Denying meaning is a meaningful way to engage with a piece. So I am guilty of justifying this work as I mock it, fine. As much as the view that criticism is part of art preempts serious criticism and absolves artist, it can also liberate me as critic and justify my objections. I just need to play by the rules, such as admitting it is art and I am a part of it. And in admitting this, I stretch the boundaries of what is art, because now art is a cheap laugh, a strawman soaked in fuel, a can of shit. Art is that which invites the novice to momentarily pontificate and jeer and ultimately something that he can forget. I’m okay with that. I understand that people will always highjack the symbols of virtue for cheap gain. But the symbols can’t keep themselves up. They sink down, to the low down things they are stuck to.

(See Christianity, peace, the swastika, art, Country music).


Theory and Emergence in a Deterministic System

January 18, 2009

A cellular automaton is, in a clear sense, the simplest of universes. As such, it is a useful setting to apply and test ideas from philosophy. The cellular automaton universe (CAU) seems the least allowing for emergence (without considering trivial universes, e.g., an empty universe), since every phenomenon follows from the evolution rule. I will argue, here, that in some sense language–and therefore theory and thought–organizes and communicates structure which is emergent, even in a cellular automaton universe.

Context-dependent definitions
Suppose our CAU contains a large number of collections of contiguous on cells, which are roughly the same size. Suppose these groups move in different directions and at different rates and that their design and the rule of evolution is such that when two collide either they are both destroyed (every cell is turned off) or they bounce off of each other. In this setting we could define a term particle for these groups of cells which appear to move. How do we define particle so that we can make unambiguous statements about the nature of particles? We might specify size: particles are between 1 and 10 billion cells. We could specify shape, say in specifying the size of the boundary, or in specifying the ratio of longest diameter to shortest diameter. Of course, there is necessarily some structural requirements for the property that particles bounce or mutually annihilate, but it may not be clear what exactly these requirements are. Even without these requirements known, we consider it meaningful to observe the bouncing and annihilating of particles.

If, on the other hand, our CAU had every possible bounded arrangement of cells, each an infinite number of times, (e.g., a random initial state on an infinite space) it would be extremely tedious to have a theory about large groups of cells and in fact it would not be of any use, since for any two differing bodies there would be many bodies which were intermediate (a geodesic might be a path of single flips which never flips a cell twice) and different macro behavior would necessarily come down to the difference of a single cell. In that setting a macro theory is pointless. In a more limited CAU, as the first described here, a macro theory is useful and meaningful.

We might notice that roughly half of the particles are hollow, that is, have a cavity of off cells near their center, and that particles are only annihilated when exactly one is hollow.

When a hollow particle and a non-hollow particle collide, the two are annihilated, otherwise a collision results in the particles bouncing off of each other. (1)

Are the terms of this statement shorthand for statements about individual cells? They are not. To reiterate, in translating this statement in terms of individual cells, one necessarily gets a ridiculously cumbersome statement which contains no more information than the rule for evolution, which we may imagine is a very small piece of information.

What happens if the cells are too small to be observed or detected? Certainly here the best we can do is make empirical statements, such as (1) above. We are unable to discern two arrangements which differ by a single cell, but frequently we don’t need to, because a vast set of arrangements is absent from our universe. In our universe we recognize some objects as existing on a continuum, such as photons, but do not concern ourselves with the continuum connecting any two objects, such as a continuum connecting photons to protons, because such continua do not exist in nature. With the absence of said continua, nature suggests particular objects for naming. We have a name for a species, say horse, and we do not have names for each intermediate object, say between horses and cows. In hypothetical contexts, the proposition “X is a horse” is fuzzy, not well defined. Yet in the context of animals on earth, this proposition is perfectly well defined.

Things are further complicated if you imagine the scientist and the audience to be immense arrangements of cells. Supposing our universe is a CAU, the human eye cannot detect a single cell, by a long shot, but recognizes a discrete alphabet on this page. The human ear is never hit with two identical sounds, yet discerns discrete phonemes of speech from extremely complex and subtle patterns of changing air pressure. The brain is in two distinct states every two utterances of the same word and every letter ever printed is a different configuration of ink on a different surface of paper. It’s a bit of a miracle we discern discrete structures at all.

What is the observation that some particles are hollow and some are not, if it is not shorthand for arrangements of individual cells? We’ve said it is somehow dependent on the observable arrangements of cells, it must also be dependent on the available variety of people and the fact that each one is affected similarly by viewing a picture of a hollow cell and by holding an aerobie frisbee, that some abstract quality is registered in both cases. In this way we can view the quality of being hollow not just as a simple property–albeit fuzzy–about a particular physical object, but in fact an extremely complex property of affecting the human brain in a particular way. Abstractly we can say hollow is a property of shape; contrasted with solid; it describes objects which are lacking a large part of their interior. In any of these definitions we need not make the mistake that those terms are well defined solely in terms of the physical form of the object. We can recognize that these terms, too, are meant to reference the experience of the reader, and that although there is a relationship between solid and hollow, it only makes sense to define hollow in terms of solid if the reader has had some experience with solid, and can conceive of its opposite.

A theory that has simple words with simple relationships (e.g., of hollow and solid particles), which is deconstructed into a much finer physical system (e.g., cells of an automaton) only in an incredibly complex and intractable way (e.g., by including the nature of human observation, consciousness and language itself, each having to be further described in terms of cells) is emergent in some sense, is it not?

The objection could be raised that “emergence” is just a point of view, a priori as valid as its contender: “all effect is the cause of the rule of evolution.” I don’t disagree that recognizing “emergence” is merely a point of view. Yet science is in the business of choosing an appropriate point of view. And the people interested in science are extremely complicated, physically. To these people, some layers of cause and effect are easy–easy to understand, with limitations. From these a foundation is lain, from which to understand the other layers. So “emergence,” as far as I understand, is not just the state of things being incomprehensibly complex, but also the fact that from incomprehensibly complex systems, simple structure can emerge.


For Lack of a Word

January 7, 2009

This from an encyclopedia entry:

*** is a notyetcoinedologism which refers to the practice of imposing structure on or between sets of data, in a more or less arbitrary fashion. Generally certain semantic cues are followed, resulting in a theory which is partially natural, while many connections will be found only after the choice of structure has been chosen. In its most natural degree ***ing is the process of observing structure that would be similarly observed by any from some large pool (e.g., humankind). As such, it is the bedrock for scientific theory of any kind. In its less natural degrees ***ing can be used to author mnemonic devices. It can also be used to mine new relationships between disparate sets of objects for the purpose of creating artwork, literature, music etc. An instance of *** is called a *** map. Pseudosciences may qualify as bodies of ***, although usage of the term *** presupposes a motive of playfulness and creativity. One does not *** to produce an irrefutable theory.

The origins of the word date to the year 2009 from a blog posted by Andrew Marshall, although the actual term was suggested by one of the blog’s readers. It may be a portmanteau of, or take inspiration from, the following words: algorithm, supervenience, append, arbitrary, map making, giving birth, theory weaving, superlogic, artificial, imposing, forcing, analogy, metaphor.

Examples where the modeling is mathematical include the imposition of a total or partial ordering on a set of real world items (or categories of items); the assignment of numbers to the members of a set, where qualities of the numbers are taken into consideration; the use of directed graphs to account for and suggest adjacency or local partial ordering. However, the product of *** is not usually entirely mathematical, as the natural language value of the objects involved is not forgotten. Frequently two sets with some similar interrelationships will be identified, whereby the modeling is almost entirely at the natural language level.

Concrete example: a piece of music with n movements will be identified with a region R containing n counties. The (standard) duration of each movement gives a natural order to the movements, and the land mass of the counties gives a natural order to the counties. The two are identified accordingly. It is then noted that duration and note count are roughly correlated, as land mass and population are. We may then ask to what accuracy is population correlated with note count. As we scour the data relevant to the music and the land we might find that the number of sections in each movement corresponds precisely to one greater than the number of large rivers passing through the interior of each county. Upon such an observation, we will find a way to identify the land masses between rivers with sections of movements, perhaps still according to our land mass:duration correspondence, perhaps according to some other distinction. Having made such an identification we will observe moods of the movements and find ways of seeing each county as embodying that mood. We might, finally, write a short story involving n people, each from a different county of R. These characters’ personalities will share perceived moods of their corresponding musical movements. We will give a clue or two in the story, possibly referencing the piece of music in some telling context, but the *** map is regarded, here, as scaffolding around a building or the wax prototype of a bronze sculpture. Some of it will not survive.