The Unapologetic Mathematician

Mathematics for the interested outsider

Additive Functions

I’ve got internet access again, but I’m busy with a few things today, like assembling my furniture. Luckily, Tim Gowers has a post on “How to use Zorn’s lemma”. His example is the construction of additive (not linear) functions from \mathbb{R} to itself.

In practice, as he points out, this is equivalent to defining linear functions (not just additive) from \mathbb{R} to itself, if we’re considering \mathbb{R} as a vector space over the field \mathbb{Q} of rational numbers! This is a nifty little application within the sort of stuff we’ve been doing lately, and it really explains how we used Zorn’s Lemma when we showed that every vector space has a basis.

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August 12, 2008 - Posted by | Algebra, Linear Algebra

9 Comments »

  1. Re: earlier pi nonsense

    I don’t mean offend your mathematical sensibilities

    I’m was asking doesn’t something have to be falsifiable to be knowledge? And don’t you need some way to measure things to falsify them? I just thought Pi would be that unit. I guess it can be any arbitrary constant. I was just trying to think of something independent of the number system itself which is how I ended up with Pi.

    I wasn’t trying to make an assertion, but ask a question. But I guess was expecting not just a yes or no, but also some alternative explanation for the basis of knowledge.

    I meant it as a thought experiment, sorry if caused too much exacerbation. Even worse than philosophy, I come from economics, where everything is wrong and concept is more important than accuracy.

    Again, sorry for the heckling from the peanut gallery I get carried away sometimes.

    Comment by Kurt Osis | August 13, 2008 | Reply

  2. \pi is far from independent of the number system. It is what it is exactly because of the way the real number system is structured.

    Comment by John Armstrong | August 13, 2008 | Reply

  3. I guess that is over my head. The relationship is dependent on the number system? I always thought, you know, if aliens imagined a circle regardless of their number system they would still have pi? if they don’t understand circles they’re going to have real trouble reading the plaque we put on Pioneer then.

    Ok well, i’ve given up, i’ve concluded knowledge doesn’t exist.

    Comment by Kurt Osis | August 13, 2008 | Reply

  4. Kurt’s error is based on the thought that Pi is only about circles. Historically, that is how it was found millennia ago, by empirical measurement of round objects in physical space. Then abstracted by Archimedes as the limit of a series of circle approximations as polygons. Then as an abstract mathematical object which emerges from hundreds or thousands of different mathematical processes. His question is not stupid, merely naive, and can be cured by more reading and thought and working out of examples. At the deeper level, it is a philosophical category error to confuse “truth” and “proof” in the empirical world (science, engineering, technology) with “truth” and “Proof” in the axiomatic world. Let alone with “truth” and “proof” in the aesthetic, politico-legal, or revealed/religious magesteria. I do not denigrate naive questions. Great thinkers such as Feynman and EInstein excelled at asking them, and finding novel answers.

    Comment by Jonathan Vos Post | August 14, 2008 | Reply

  5. JVP: Kurt is trying to drag a bunch of pseudo-mystical numerology here from where he started it over at Isabel’s place. There’s insightful-naïve questions like you’re talking about, and then there’s sheer nonsense. I’ll denigrate nonsense.

    Comment by John Armstrong | August 14, 2008 | Reply

  6. John:

    There’s one thing I’m not and that’s pseudo-mystical. (that is to say, I don’t accept other people’s nonsense, but accidentally stumble into creating my own nonsense to ignorance) I have one real question. What is knowledge, and when can it be said to exist and not exist.

    In math I can far to ignorant to distinguish between cause and effect and coincidence. You point out when I am wrong, but you don’t explain how you arrive at your conclusions, so I am forced to accept assertions as fact. But being a very skeptical I am loathe to do so. I would very enjoy to learn how you think and what is wrong with my nonsense. I just wish you would explain it.

    “pi is far from independent of the number system. It is what it is exactly because of the way the real number system is structured.”

    I have no idea what that statement means.

    I just started reading “What is Mathematics” http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192
    Maybe by the time I get to the end I will understand.

    Comment by Kurt Osis | August 15, 2008 | Reply

  7. sorry the typos above, should read:

    “…my own nonsense [due] to ignorance…”

    “In math I [am] far [too] ignorant…”

    “I would very [much] enjoy…”

    Comment by Kurt Osis | August 15, 2008 | Reply

  8. Big statements like “all knowledge comes from [foo]” are pseudo-mystical. Especially when [foo] is some silly little constant that people ascribe all sorts of significance too mostly because it’s the only recognizably “mathy” word they know. And people do that with \pi all the time.

    The full derivation of \pi from the ground up is much too long for a comment. I’ll get there when I get there. The basic idea is that it’s half the period of any solution to a certain natural differential equation. You don’t need any reference to the real world at all, especially not to idealized geometrical constructs which don’t actually exist in the real world in the first place.

    The questions that JVP referred to differ from yours in that they actually mean something. “Does all knowledge come from [foo]?” is a grammatically-correct sentence, but that’s about it. It has the veneer of epistemology, but it’s more at home with such classics as “Have you ever looked at your hands? I mean really looked at your hands?

    Comment by John Armstrong | August 15, 2008 | Reply

  9. [...] Then the discussion moved from an unrelated post on Michael’s weblog to an unrelated post on mine. [...]

    Pingback by Pi: A Wrap-Up « The Unapologetic Mathematician | October 16, 2008 | Reply


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