|Posted by Holder of the Secrets on March 20, 2017 at 1:05 AM|
Side Note Regarding the Number Line and the Axiom of Choice
There is one other suggestion regarding the problems inherent within the number line that I did not get to as I am not sure it fits nicely here, but I have a potential solution to another quandary regarding the number line. When manipulating the number line, one is faced with a conundrum that begs the question “what is the line itself made of?” “How are all the numbers tied together?” No matter how deep you go, there is still a space between each number, no matter how miniscule. I suggest that the glue binding the numbers together on the number line are the number actions, such as +, -, x, /, =, <,>, % , $ etc. The symbols tie the numbers together by guiding or providing direction of how the numbers are to be used. A lot of the time we look at something without considering the actor. This is a problem inherent within Psychology as mentioned above, we seem to forget that we are an acting part of the equation; a variable forever confounded that can never be controlled for properly. This is why I am a firm believer in the axiom of choice.
Adding the axiom of choice creates a world of problems for mathematics because “[t]he method of proof that mathematicians have decided to use requires that results be obtainable by a finite number of steps” (Aczel, 2000, p. 176, italics in original). I am unconvinced that the best way to deal with an infinite number line is to place arbitrary limits onto it. This is why it was so hard to plot the Divided Line, the mathematics depended on how long the Line is because the axioms of mathematics are set up to necessitate limits and turn something infinite into something finite for analysis. (Much like using fixed Earth mathematics to launch rockets out of our atmosphere). A common critique of adding the axiom of choice is there is no guideline telling us where to go and what to do, making it randomized and arbitrary. However, I suggest that using hypotheses as one goes through mathematical procedures can help provide guidelines of where to go and what application to apply.
Another critique of the axiom of choice is that it is assumed if it were applied; the sequential order of the numbers would be ignored.
Consequently, there is no theoretical way to look at these numbers without
considering this property of order, i.e., it makes no sense to try to jumble up
the numbers and consider them as a unordered set. (Aczel, 2000, p. 177)
If, we apply the suggestion above to consider the number line as a spiral, rather than a linear line, this may help solve this conundrum. This does not involve jumbling the numbers out of order, it just involves placing them in an order where they can be seen as a set, or on an equal plane, rather than as individual numbers placed in a hierarchy.