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Posts tagged with complex numbers

Suppose you are an intellectual impostor with nothing to say, but with strong ambitions to succeed in academic life, collect a coterie of reverent disciples and have students around the world anoint your pages with respectful yellow highlighter. What kind of literary style would you cultivate?

Not a lucid one, surely, for clarity would expose your lack of content. The chances are that you would produce something like the following:

We can clearly see that there is no bi-univocal correspondence between linear signifying links or archi-writing, depending on the author, and this multireferential, multi-dimensional machinic catalysis. The symmetry of scale, the transversality, the pathic non-discursive character of their expansion: all these dimensions remove us from the logic of the excluded middle and reinforce us in our dismissal of the ontological binarism we criticised previously.

This is a quotation from the psychoanalyst Félix Guattari, one of many fashionable French ‘intellectuals’ outed….

scientist and polemicist Richard Dawkins, Postmodernism Disrobed. A review of Intellectual Impostures published in Nature 9 July 1998, vol. 394, pp. 141-143.

 

Above we read an assertion without evidence. Dawkins posits that an intellectual impostor with nothing to say would write in a certain way. But where’s the proof? I guess whoever’s reading this book review is assumed to already know what Dawkins (Sokal/Bricmont) are talking about and agree with his implications: namely, that postmodernists have nothing to say, and that they cultivate an obtuse literary style to obscure the fact (and that this somehow also attracts followers).

Who says “chances are”? Dawkins’ attack amounts to a flame.

 

Here is a not-unusual passage written in that other famously obtuse jargon, mathematics:

The prototypical example of a C*-algebra is the algebra B(H) of bounded (equivalently continuous) linear operators defined on a complex Hilbert space H; here x* denotes the adjoint operator of the operator x: H → H. In fact every C* algebra, A, is *-isomorphic to a norm-closed adjoint closed subalgebra of B(H)….

That’s from Wikipedia’s article on C* algebras. I think the language is similarly impenetrable to Guattari’s. But mathematics = science = good and humanities = not science = bad, at least in the minds of some.

Here is an excerpt (via @wtnelson) written for teachers of 4–12-year-olds, 40 years ago, by Zoltán Pál Dienes:

psychologically speaking, relating an object to another object is a very different matter from relating a set of objects to another set of objects. In the first case, perceptual judgment can be made on whether the relation holds or not in most cases, whereas in the case of sets, a certain amount of conceptual activity is necessary before such a judgment can take place. For example, we might need to count how many of a certain number of things there are in the set and how many of a certain number of these or of other things there are in another set before we can decide whether the first and the second sets are or are not related by a certain particular relation to each other.

Clear as mud! Clearly Z. P. Dienes was an intellectual impostor with ambitions to collect a coterie of reverent disciples.

 

I don’t know enough about postmodernism to opine on it. I just get annoyed when putatively sceptical people casually wave it off without proving their point.

(And if you’re going to point me to the Sokal Affair or Postmodernism Generator CGI, I’ll point you to At Whom Are We Laughing?.)

 

In Lacan: A Beginner’s Guide, Lionel Bailly describes his subject as “a thinker whose productions are sometimes irritatingly obscure”. He goes on:

Most Lacanian theory [comes from his]  spoken teachings…developed in discourse with…pupils…. [Various modes of presentation which are appropriate in speech] make frustrating reading. …leading the reader toward an idea, but never becoming absolutely explicit…difficult to discover what he actually said…thought on his feet—the ideas…in his seminars were never intended to be cast in stone…freely ascribes to common words new meanings within his theoretical model…Lacan, despite the fuzziness of his communication style, strove desperately hard for intellectual rigour….at the end of the day, it is … clinical relevance that validates Lacan’s model. [Lacan being a psychoanalyst and his ideas coming out of that work.]

So there’s an alternative hypothesis from an authority. Bailly admits the communication style was poor and gives reasons why it was. But rather than judging the work on rhetorical grounds, we should judge it on clinical merit—the ultimate empirical test!

Compare this to Dawkins. Besides the suppositions I already mentioned, he chooses words like: “intellectuals” within scare quotes; ‘anoint’, ‘revere’, ‘coterie’—to undermine the intellectual seriousness of his targets. Who are the empiricists here and who relies on rhetoric?

(Source: members.multimania.nl)




Consider ℂ, the field of complex numbers, as a 1-dimensional vector space. The balanced sets are ℂ itself, the empty set and the open and closed discs centered at 0 (visualizing complex numbers as points in the plane). Contrariwise, in the two dimensional Euclidean space there are many more balanced sets: any line segment with midpoint at (0,0) will do.

As a result, ℂ and ℝ² are entirely different as far as their vector space structure is concerned.

(Source: Wikipedia)




A road map of mathematical objects by Max Tegmark, via intothecontinuum: The arrows generally indicate addition of new symbols and/or axioms. Arrows that meet indicate the combination of structures. For instance, an algebra is a vector space that is also a ring, and a Lie group is a group that is also a manifold.

A road map of mathematical objects by Max Tegmark, via intothecontinuum:

The arrows generally indicate addition of new symbols and/or axioms. Arrows that meet indicate the combination of structures.

For instance, an algebra is a vector space that is also a ring, and a Lie group is a group that is also a manifold.


hi-res




Just playing with z² / z² + 2z + 2

g(z)=\frac{z^2}{z^2+2z+2}

on WolframAlpha. That’s Wikipedia’s example of a function with two poles (= two singularities = two infinities). Notice how “boring” line-only pictures are compared to the the 3-D ℂ→>ℝ picture of the mapping (the one with the poles=holes). That’s why mathematicians say ℂ uncovers more of “what’s really going on”.

As opposed to normal differentiability, ℂ-differentiability of a function implies:

  • infinite descent into derivatives is possible (no chain of C¹ ⊂ C² ⊂ C³ ... Cω like usual)

  • nice Green’s-theorem type shortcuts make many, many ways of doing something equivalent. (So you can take a complicated real-world situation and validly do easy computations to understand it, because a squibbledy path computes the same as a straight path.)
 

Pretty interesting to just change things around and see how the parts work.

  • The roots of the denominator are 1+i and 1−i (of course the conjugate of a root is always a root since i and −i are indistinguishable)
  • you can see how the denominator twists
  • a fraction in ℂ space maps lines to circles, because lines and circles are turned inside out (they are just flips of each other: see also projective geometry)
  • if you change the z^2/ to a z/ or a 1/ you can see that.
  • then the Wikipedia picture shows the poles (infinities) 

Complex ℂ→ℂ maps can be split into four parts: the input “real”⊎”imaginary”, and the output “real“⊎”imaginary”. Of course splitting them up like that hides the holistic truth of what’s going on, which comes from the perspective of a “twisted” plane where the elements z are mod z • exp(i • arg z).

a conformal map (angle-preserving map)

ℂ→ℂ mappings mess with my head…and I like it.










Real numbers are imaginary, and imaginary numbers are real.


[I]maginary numbers describe a physical state of something, so as much as a number can exist, these do. But … real numbers, [being ideal], are imaginary.

David Manheim

(I changed some parts that I don’t agree with but the phrasing and initiative are his.)

The “rational” numbers are ratios and the “counting” numbers are, um, what you get when you count. But “real” and “imaginary” numbers have nothing to do with reality or imagination (each is both real and ideal in the same sense).

 

How about we start referring to them this way?

  • ℝ = the complete numbers. ℝ is the Cauchy-completion of the integers, meaning that ℝ has completely fills in enough options so that any sequential pattern will be able to dance wherever it wants and never need to step its shoe on another element outside the system in order to fulfill its pattern.
  • Any field adjoined to the √−1 becomes “twisting numbers”. This derives from the “twisting” feeling one gets when multiplying numbers from ℂ. For example 3exp{i 10°} • 5exp{i 20°} = 15exp{i 30°}, they spiral as they multiply outwards. Keep multiplying numbers off the zero line and they keep twisting. Tristan Needham coined the word “amplitwist” for use in ℂ.
  • ℂ = the complete, twisting numbers. Since ℂ=ℝ adjoin √−1.
  • “Complete spiral numbers” sounds nice as well.

Just to give a few examples of other acceptable numbers systems:

  • ℚ adjoin √2
  • the algebraics
  • ℚ adjoin √[a+√[b+c]]
  • sets
  • DAGs
  • square matrices … with many kinds of stuff inside
    Magma to group2.svg
  • special matrix families
  • certain polynomials (sequences) … taking many kinds of things (not just “regular numbers”) as the inputs
  • clock numbers (modulo numbers)
  • Archimedean fields and non-Archimedean fields
  • functions themselves … and the number of things that functions can represent boggles the mind. Especially when the range can be different than the domain. (Declarative sentences can have a codomain of truth value. Time series have a domain of an interval. Rotations of an object map the object to itself in a space. And more….)
  • And many, many more! Imagination is the limiting reagent here.




The definition of toposes has surprisingly powerful consequences. (For example, toposes have all finite colimits.)

Probably the best analogy elsewhere in which a couple of mild-sounding hypotheses pick out a very narrow and interesting class of examples is the way in which the Cauchy-Riemann equations select the analytic functions from all smooth functions of a complex variable.

Michael Barr & Charles Wells, Toposes, Triples, and Theories (p 64)




Tachyons

As every sci-fi geek knows, matter may travel faster than the speed of light as long as its mass is imaginary (a multiple of √−1). A so-called tachyon would not overturn special relativity—and it would provide a handy way of resolving any conflicts in a given Star Trek plot.

  • 14th Law of How to Write Star Trek: Whenever you’ve written yourself into a hole, instead of re-writing the show so that it’s better, simply make characters issue the word “tachyon” several times toward the end. Everything is magically resolved, returning all aspects of life to the way the show started with no long-term consequences for the characters—which by the way is a great lesson to teach to young adults—and then Spock or Data has an “a-ha!” moment wherein he throws around jargon to further justify the deus ex machina.

The only problem with tachyons, as any sci-fi geek can attest, is that “imaginary” mass is pure fiction! How could anything weigh an imaginary amount?

 

Well, I’m not sure that tachyons do exist—although if someone wants to post some arXiv links to relevant papers that would be awesome—but, I will say that “imaginary mass” isn’t that ridiculous of a concept.

As Tristan Needham said in the best book about complex numbers ever, the “imaginary” descriptor only reflects the historical prejudice against √−1.

Do imaginary numbers exist? No. But neither do counting numbers. Numbers are linguistic entities that humans communicate with. Sort of like how trees, flowers, bushes, shrubs, brambles, and vines all exist in nature, but those classifications, concepts, words, groupings are human-language mental constructs. “Five” doesn’t “exist” per se, but mathematical models built with the-thing-that-satisfies-the-properties making five five, do wonderfully at prediction of physics experiments.

Anyway, imaginary numbers exist just as much as other numbers. Just like rational numbers, they’re generated by an operation that comes up as a matter of course in algebra. And algebra seems to have something to do with nature. God knows why. (ohh! which way did I mean it?!)

So I’m not saying imaginary mass exists, but here are some good ways to think about imaginary numbers.

  • imaginary numbers are twisted numbers
  • imaginary numbers are phase-shifted like a sine wave versus a cosine wave
  • an imaginary current heats up a wire but does no useful work

If the mass of a particle is an imaginary number, then … that might help you make sense of tachyons.

 

Nerdy side note: E=MC² is not the real equation to describe the conversion of energy into matter or vice-versa.

  • E=MC² tells you how to convert stationary matter into energy.
  • The real equation is E² = [mc²]² + [pc]².
  • (p is momentum.)
  • (Notice that the real equation is of the form A²+B²=C². i.e., Energy is the hypotenuse (C) to the triangle sides B=mc² and A=p•c)

You can casually start/interrupt conversations with this knowledge the next time you attend a kegger / black-tie affair. Doing so will win handsome glances from potential sex partners. Also, there is a 0% chance that anyone will think you’re an insufferable know-it-all.