A possible meaning of “meta” – Mathematical writing

I posted one of my exponent fractals on Dan Meyer’s 101qs.com, with the question, “how many dots is that?”  I was sort of disappointed that I had to pose the first question.  That seems to defeat the point, if I’m interested in what questions the photo prompts on its own.  I did get a few interesting questions though…

The question I’m a little more interested in, the one I posed in my first post on this stuff, is this one; If 9 is a meta-triangle (a triangle of triangles), is 27 a meta-meta-triangle?  I’ve often called it that, but I can see a different interpretation, so I asked the meaning.

I got a bite and this comment from Max Goldstein, but I wanted to share what I thought was in interesting answer, and once I started writing, the math started to push back, and I noticed new things, and on and on.   This is me doing math, so I thought I’d publish some of my work here.

* * *

Question: What’s a meta-meta-triangle?

Like most math, this depends on the meaning of the terms. Let’s take “triangle” as understood and consider “meta.”  A meta-movie is a movie about a movie.  When faced with the challenge of writing, young poets often write about writing.  That’s meta-poetry.  9, then, is a meta-triangle, because it’s a triangle of triangles.

Then what’s a meta-meta-triangle? It’s a meta-triangle of meta-triangles!  So in the powers of 3, that would be a 9 of 9’s.  That’s 81, not 27!

[Do you buy that?  Is it clear?]

* * *

Here’s an argument by notation:

Definition: meta(x)= x(x).
Examples: meta(play)=play(play)     —–>  Hamlet
meta(triangle)=triangle(triangle)     —–>  9

Let’s use m instead of meta.  It starts to sound weird if you say it too much.  Now m(string)=string(string), and m(asdf!3!)=asdf!3!(asdf!3!).

Note: The definition of meta(x) relies on x(x) making sense. If x is in meta’s domain, then x must be in its own domain! This is just spooky to me…

Then if “meta” is in its own domain, we know m(m)=m(m), by definition.  So meta(meta) is itself!  (I’m not making this up.)  As above, m(triangle)=triangle(triangle)=9.  In short, m(t)=9.  Get ready…

meta-meta-triangle is m((m(t))=m(9)=9(9)…  81?

[Does the notation make this clearer or obfuscate the ideas?]

* * *

Well that’s it.  It’s a little piece of mathematics that I spent time carefully wording for clarity and communication’s sake.  Choosing a single notation, while proper, can be a little austere and hard for the reader, so I picked and chose which representations to use on each line, in the same way I choose punctuation.  I hope that comes across.

There’s so much unreadable mathematics in this world.  It breaks my heart.  I’m convinced it’s helping to kill mathematics.  This means I have a responsibility to try and work on quality writing – concise, elegant, clear, and convincing mathematical arguments that can be read widely.  We’ve been putting extra effort towards this in my department.  In short, this is the essence of proof – “convincing” argument.  (note: unreadable symbology and jargon is often NOT convincing at all.)

Maybe you’ll share your own mathematics with the world.  The goal is the clear and simple communication of ideas.  Two-column proof is NOT the only way.

Anyhow, with MArTH Madness and everything else going on, life is really rushing straight at me right now.  Somehow it feels amazing!


7 responses to “A possible meaning of “meta” – Mathematical writing

  1. I think that when your photo comes up in the random-next-photo page, your question isn’t there. We only see your question when you link specifically to that image from your profile-thing or something.

    • Oh good! That’s a bit better.

      • A bit? What part of your original criticism still stands?

      • Hey, Dan. It would be nice to have the uploader’s question as an optional part of the upload process. I guess I don’t ever want to spoil it for them.

        I might have had all of the questions open only after a “show me what other people wonder” click, regardless of how the page is viewed.

        Gripes aside, cool site! Many of my students really liked it, and they did enjoy seeing questions of people they know!

  2. Prof. L.Radhakrishna

    “Meta” is a prepositional prefix. You have used only one meaning ‘about’.
    How about using other meanings? Try “behind”,and “beyond”!
    Now do the recursion!!
    Meta-meta engineering; metaphysics! Metamathematics does not exist, for ‘beyond’ as the meaning of ‘meta’.

    • All interesting, but the post is titled, “A possible meaning…” We can always appeal to convention or precedence to answer questions like this, but the thing that feels most mathematical is this; Simply choose a meaning and make a mathematical argument.

      I chose a meaning that fit most snugly with my concept of a meta-triangle as a triangle of triangles, but naturally other choices could be made.

      We run into the same problem every time a question about the nature of infinity arises. We have to choose what we mean and simply be clear about it.

      Honestly I’m not sure how the other definitions could apply to the triangle case. Can you explain?

    • I just retread your examples; Very provocative! The engineering behind the engineering…m the math beyond the math. Cool ideas.

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