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.

A Wonderful Idea: Metagrammatical Numbers

October 17, 2008

Apply Cantor’s diagonal argument to the list of describable numbers, and one arrives at a description of a number which is not describable.  A paradox? Not at all.  Merely proof that higher orders of description exist. That is, some numbers are metagrammatical.  They can be described and computed to arbitrary precision, but necessarily the description is meta to whatever grammar one begins with.  This is delightful!

Details:  We call a number describable if it can be described by a finite expression, with a specified alphabet and a specified grammar, which decides how to compute the number from the description if it is grammatical, and discards expressions which are not grammatical (i.e., we are using ”grammar” to mean both what is admissible as a description, and given an admissible description, what is the protocol to compute the number which is described).  For example, consider the set

$\{1,-,(,),/\}$

With this set we can describe every rational number, since we can write a rational number as an expression using just these 5 characters.  (If we are careful we can get rid of the parentheses). The grammar throws out nonsensical expressions like $1--/-$, and computes the others. Allowing iterative operators (e.g., $\Pi, \Sigma$), quantifiers ($\forall,~\exists$), a countable number of variable ($v_1,\ldots v_n,\ldots$), et cetera, we get descriptions of algebraic numbers and a lot of common transcendental numbers. Still, even with a countable alphabet we can enumerate the number of finite strings of characters, so that the set of describable numbers is countable. This is not too surprising, but if you have never thought about it, it is worth pondering. This set of numbers is a field, for example, and not a trivial example either! Indeed (almost) all numbers that are useful at all are describable. Of course, picking one at random (with equal distribution) is impossible, so probabilists, among others, demand the existence of indescribable numbers.

Let us make a list of all descriptions on the left hand side of our paper, and on the right we will make the corresponding list of numbers, written in binary (which frequently do not terminate, but that’s allowed). Now we’ll construct a number, $\chi$, that can be computed but is not on our list. Let $\chi$ have as its $n$th diadic (“diadic” is like “digit” but for base two and not base ten) a 0 if the $n$th description has a 1 in its $n$th diadic, and a 1 if the $n$th description has a 0 in its $n$th diadic. That is, go down the diagonal of the list of numbers and make the number which has switched each diadic in the diagonal. Have we not described $\chi$? But it cannot be on our list, since it differs from every number on our list in at least one diadic.

The point is we can never nest a description of a grammar within the grammar itself, although we can nest one grammar inside another. If we could nest a grammar inside itself, we could, in grammar $G$, write the description

the $n$th diadic of this number is the opposite (base 2) of the $n$th diadic of the $n$th description in $G$,

and we would have produced the paradoxical situation of having a number which both is and is not on a countable list.

More specifically, let some huge $N$ be the number in an ordering of $G$, so that the $N$th description is the one in the above blockquote, and let it correspond to the real number $\chi$, which it describes. If such a description can be written, it must be written in a finite number of characters, and so there must be some such number $N$. Now we ask what is the first diadic in $\chi$? No problem, our grammar just asks itself what the $1$st diadic in the first description is, and adds 1 (mod 2) to it, next it asks what the $2$nd diadic in the second description is, and adds 1 (mod 2) to it. Eventually it gets to $N$ and it asks itself about the $N$th description, but the $N$th description requires it make a list of the first $N$ descriptions to get to the $N$th diadic in that number, so the grammar tries to reference itself, and in doing so must reference itself again, in a nested infinite recursive loop. So forget about the fact that the grammar is paradoxically trying to switch one of its own diadics. Forget the paradox that the number is both on and not on the countable list. The first problem is that the grammar goes into an infinite recursion.

In fact, lets go back and do the same thing except instead of switching the $n$th diadic of the $n$th describable number, we use the $n$th diadic of the $n$th describable number as the $n$th diadic in our number, say $\chi_0$. Can you think of a more agreeable number to have on our list? $\chi_0$ could be any of the countable numbers on our list. Suppose it is the $N$th number. What does it have for its $N$th diadic? Well, either 0 or 1, since this number must merely be equal to itself. But then $\chi_0$ is not describable, since its description does not produce a unique value!

This is, essentially, how one approaches the Berry Paradox. In short,

definable in this system

is just never a logical definition (or clause thereof) in that system.

However, once we agree on what a definition or a description is, then these examples tell us that there are deterministically producible, unambiguously definable, finitely expressible numbers and strings which can only be expressed with a grammar meta to the one first chosen.

What a delight!