A sharp reader has uncovered an omission in the section in abstractmath.org on sets (letter quoted below). In the section on the rigorous view (http://www.abstractmath.org/MM//MMImagesMetaphors.htm#rigorousview) I said that math objects (http://www.abstractmath.org/MM//MMMathObj.htm) don't change over time. He points out that the set of Presidents of the USA changes over time, and that the relation of fatherhood does too.
I expect to repair this soon. Clearly I have to say that a set is certainly an abstract object(http://www.abstractmath.org/MM//MMMathObj.htm#abstractobject), but it is a mathematical object only if its elements are math objects. Similarly a relation is a math object only if it is a set of ordered pairs of math objects.
Note that I am NOT proposing that we should change the definition of set to exclude non-math objects. Common usage allows sets of other than math objects. If any reader knows of a text that allows sets to contain only math objects I would be interested to learn about it.
In making these distinctions, I am hung up on letters of the English alphabet. Computing scientists talk about these all the time and clearly treat them as math objects. But letter of the English alphabet as a phrase can be interpreted in several ways. Historically the set of letters has certainly changed over time (i and j were distinguished only starting in the sixteenth century). The c.s. treatment of letters is implicitly taking a fixed set as given once and for all. (The c.s. literature distinguishes upper case and lower case, too. If you asked the person on the street how many letters are in the English alphabet they will say 26, clearly not distinguishing case. This is a separate issue from the one I have raised here.)
Comments on all this will be welcome.
Charles Wells
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The letter:
I have been perusing your website. I had a few questions concerning the differences between a mathematical object and an abstract object.
You said a math object does not change over time.
But what about the following scenario:
The set of Presidents of the United States will change over time as new Presidents get elected. Is this a math object or an abstract object?
But the set of the first 30 presidents of the United States is static. It does not change. Is this a math object, I assume?
I thought all sets were automatically math objects.
the same for relations: Is friendship a mathematical relation? If so, how? the truth or falsity of friendship between two people can change over time. but what about being the father of someone. Is this a mathematical relation between two individuals. fatherhood should never change. Finally, can you even have an abstract or mathematical relationship between physical objects or do the objects themselves need to be abstract when applying the relation/function?
What good books can you recommend on the above topics?
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