Imaginary Numbers Rant

August 23, 2011

I’m preaching to the choir here, if to anyone at all, I suppose. Still, it should be said. Descartes coined the term “imaginary number” to deride the invention of a number such that squared becomes -1. The name stuck and has done countless harm to the notion, fooling pupil and school teacher alike into implicitly accepting the formalist position: “that which can be defined, precisely, exists. End of story.” It is no wonder that many have been suspicious of the invention of imaginary constructs that can do the impossible. I have a rhetorical question to justify imaginary numbers to my students. The question is:

Can you think of contexts where a particular class of number is inapplicable?

Think about this for a moment. By “class of number” I mean, for example, negative numbers. Negative numbers, we learn, are great for expressing debt. They apply perfectly well, also, to linear orders where there is no smallest element (temperature isn’t the best choice, since Kelvin has a smallest possible value… year is better, since although we estimate a beginning of the universe, we can’t count days since then, so BC years are appropriately negative). What negative numbers don’t seem to apply to are numbers of family members, times you can iterate a process, dimensions of a space, distances. It is no wonder they too were suspect when first conceived.

How about the class of numbers which are not 1? Possibly it makes no sense to talk of multiple universes (I am of this camp, nevermind the fashionable physics which suggests the idea). For if there is a vantage point from which to observe 2 universes, then causally the two are part of your universe, even if they are entirely independent of each other. We might (should) define universe to be all those things which cause change in what we can perceive (to my mind this is possibly an asymmetric relation: a realm can exist which is broadcasted to us, upon which we have no impact, so that it is part of our universe but we are not part of its). Then to any observer there is, by definition, but one universe.

For number of family members and iterations of a process, it doesn’t generally make sense even to apply positive rational numbers. Particularly if the process is instantaneous. Can you wake up 2 and a third times? Well, not without defining what it means to wake up a non-integer number of times, it certainly is not given. Is it okay that certain numbers have limited scopes to the context to which they apply?

The idea of iterating a process, when put in mathematical terms, gives rise to the old subject of solving functional equations. If I have a function f:R->R from the real line to itself, I can define the square of the function, that is f^2, to be the function composed with itself. Then I can ask if there exists a function g so that g^2=f, i.e., does f have a square root? If f is the function of sending x to -x then normally there exists no function which halves this process (for example, sending x to the average of x and -x certainly doesn’t work). For 1×1 matrices, multiplication and composition are the same thing. So for linear maps from R to R, finding half of the inverting map, [-1], is exactly finding a square root of -1. If you extend the real line to be sitting in a plane, then it becomes clear that you can rotate the plane 90 degrees in either direction to give a process which when iterated twice sends x to -x. This mysterious fake number, i, is nothing more than rotation of a plane by 90 degrees. Do rotations by 90 degrees exist? If anything does, they damn well do. Teachers have been heard saying “i is important because it solves x^2=-1, which you want to be able to do for engineering problems” It is a good teacher that says “i is important because it is a compass point for any object with a quality that takes values in the circle, which certainly includes many physics and engineering problems” So, any phenomenon involving waves involves complex numbers. If waves exist, complex numbers do too.

From above: let us know what it is we are doing with language, so that we may deliberately do much more of it.

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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.