A symbol is not a word. Doing algebra problems is not like reading a sentence, it’s like playing checkers or chess- although it is immeasurably more powerful. This symbol can do this. What is a number to algebra? There is no answer; for a number is really anything that can be used in an equation. Pi, i, 3, 3.5, 1, -2; all are numbers as far as algebra is concerned- but try definining number in such a way as to include all of these symbols. A symbol does not say what something is; it tells us how something can be used.

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4 Comments

  1. Holopupenko said,

    September 2, 2006 at 3:49 pm

    Shulamite:
         Just catching up on some reading, and I noticed your post.
         You note “This symbol can do this.” Could you please clarify what you meant?
         It seems to me symbols can’t actualize anything by themselves or in and of themselves. In some analogical sense, it may be understood that symbol “x” in vector calculus “tells me” I should take the cross product of two vectors… assuming I know how to carry out the process. But, in fact, this symbol really doesn’t “tell me” anything I didn’t already know: I know (understand) the meaning and the process, and it is I who conceptually (with the probable assistance of scratching out further symbols on paper to help organize and guide the process) “do this.”
         Moreover, if you think about it, even the conceptualization preceded the actual process. Why? First, I had to understand the concept of cross product before being able to perform it. Second, once I understood the concept, I could turn the work over to computer I’ve programmed to perform the task for me (which I’ve actually done).
         Algebraic symbols are really conventional (as opposed to natural or formal) signs that owe their signification to human reason and (deliberate) will, and are therefore human artifacts that are wholly arbitrary. Words are not so arbitrary because there is something remotely natural about them: their meaning is imposed by human institution. This is different from the mathematical symbol for infinity (which is a designator) or the mathematical symbol for vector cross product (which is an “operator”) because any symbol whatsoever could be used. And so, there is an appropriateness in the signification of words, which is absent in a pure symbol such as a “sideways eight” signifying infinity.
         So, just like traffic lights or stop sign don’t “do this” or cause that, neither do algebraic signs.
         I’m sorry for nipping so closely at your heels, but I’m a bit sensitive that in the scientific world similar issues crop up that end up importing to much of reality into the equations describing physical phenomena. For example, I’m sure you’ve heard something like the assertion, “Newton’s equations govern the behavior of freely-falling bodies.” That’s not correct. First, the equations themselves were correlated from observations that obtained certain data… which means they were formulated by abstracting away from real objects. Second, equations don’t actually anything because they merely describe (in an already abstract way) the trajectory of the body. So, to say, equations “govern” the behavior of physical objects is not only sloppy, it turns reality on its head.
         Your thoughts would be greatly appreciated.

  2. shulamite8810 said,

    September 3, 2006 at 9:26 pm

    Both symbols and words have in common that they signify by convention: they are imposed. The difference is both obvious and yet easy to underappreciate: a symbol doesn’t say what something is, but a word does. A symbol stands for something, it’s like a little representation of the thing itself.

    My old master compared symbols to word by taking the first kind of symbol: a coin. Coins are symbols that signify by convention, and the manipulation of coins gives us an idea of what is going on in algebra: although Algebra is immeasurably stronger

  3. November 10, 2006 at 6:38 am

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