Earlier I said I wanted to speak of mathematics and never got to it. I wish to speak about the whole art and science of mathematics. Mathematics is, in fine, a science, but it is also one of the arts of the quadrivium; this is why I say I wish to speak of the whole art and science.
I do not wish to speak of the whole in one sitting, so I will speak of it in parts by way of examples. I have questions, and I want to ask these questions of all, not only of mathematicians and of math teachers (for these are not necessarily the same; another example of this is that not all priests are theologians though all priest are teachers of theology). I hope many, lovers of math and its haters, come with other questions and answers. I hope that if my questions are not clear or not well asked, that someone will help me better ask them. I think these are questions that should be asked, for we are trying to educate our children. I only ask these to ask, are we doing it well.
I have taught mathematics. I truly enjoy the study of mathematics, the learning of it, and the teaching of it. Through my experience quite a few questions of those writers of textbooks and makers of curricula mathematica have come to mind.
Why is it, for example, in geometry, qua geometry, in the postulate for right-angles there is this number ninety, as in ninety degrees? Is there need for it?
Why ninety? Why not one? Why not one to four? Why not three-hundred sixty? Is it because Ptolemy (or someone else) divided the circle into three-hundred sixty parts, and the quarters of these are ninety? What does a circle have to do with a right-angle or any angle? Can I not know what an angle is without thinking about a circle? Etc.?
I ask these questions for a few reasons: my first is it is an example of other problems that come to mind when I have read and taught from these textbooks that our students use.
My second is that I can know what a right-angle is without knowing anything about number. Ninety, or any number for that matter, is not essential to right-angle. Number is not in the nature of right-angle. Angle is not number in its nature. It is true that every angle is a quantity, but not all quantity is number, for example, a line is a quantity, but a line can be two; three, one-hundred three, or some other number. No one number belongs to it. I do not have to cut up a line or apply some other cut-up-line (a ruler) to it to know that it is a quantity and to learn about its other properties as quantity.
My third is that the division of an angle (whether it be divided into two, three, ninety, etc.) is not a postulate but a proof*. There are postulates, and they do have a quality of a proof, but they are not proofs, and proofs are not postulates.
This brings up another two questions: why is it that modern textbooks of geometry call these things postulates that can and have been proved? And the last is, why do these books need so many? I know works of geometry that have fewer postulates but have done more in teaching this art and science than these new textbooks ever have. In fact these new textbooks would not exist were it not for these other works, and all these new textbooks do is make more difficult the learning of these works what was better said by these works.
I do not wish to speak of the whole in one sitting, so I will speak of it in parts by way of examples. I have questions, and I want to ask these questions of all, not only of mathematicians and of math teachers (for these are not necessarily the same; another example of this is that not all priests are theologians though all priest are teachers of theology). I hope many, lovers of math and its haters, come with other questions and answers. I hope that if my questions are not clear or not well asked, that someone will help me better ask them. I think these are questions that should be asked, for we are trying to educate our children. I only ask these to ask, are we doing it well.
I have taught mathematics. I truly enjoy the study of mathematics, the learning of it, and the teaching of it. Through my experience quite a few questions of those writers of textbooks and makers of curricula mathematica have come to mind.
Why is it, for example, in geometry, qua geometry, in the postulate for right-angles there is this number ninety, as in ninety degrees? Is there need for it?
Why ninety? Why not one? Why not one to four? Why not three-hundred sixty? Is it because Ptolemy (or someone else) divided the circle into three-hundred sixty parts, and the quarters of these are ninety? What does a circle have to do with a right-angle or any angle? Can I not know what an angle is without thinking about a circle? Etc.?
I ask these questions for a few reasons: my first is it is an example of other problems that come to mind when I have read and taught from these textbooks that our students use.
My second is that I can know what a right-angle is without knowing anything about number. Ninety, or any number for that matter, is not essential to right-angle. Number is not in the nature of right-angle. Angle is not number in its nature. It is true that every angle is a quantity, but not all quantity is number, for example, a line is a quantity, but a line can be two; three, one-hundred three, or some other number. No one number belongs to it. I do not have to cut up a line or apply some other cut-up-line (a ruler) to it to know that it is a quantity and to learn about its other properties as quantity.
My third is that the division of an angle (whether it be divided into two, three, ninety, etc.) is not a postulate but a proof*. There are postulates, and they do have a quality of a proof, but they are not proofs, and proofs are not postulates.
This brings up another two questions: why is it that modern textbooks of geometry call these things postulates that can and have been proved? And the last is, why do these books need so many? I know works of geometry that have fewer postulates but have done more in teaching this art and science than these new textbooks ever have. In fact these new textbooks would not exist were it not for these other works, and all these new textbooks do is make more difficult the learning of these works what was better said by these works.
I will leave this here. I think I have asked enough questions.
This question about the different kinds of quantity (number and magnitude) brings about another distinction that is lost in our speech today. I do not know how many times I hear people of all ilk, from the most uneducated to the most educated, say something such as, "there are less troops...", "there are less customers buying..." etc. "Less" is an adjective that strictly speaking belongs to magnitude, not number. What should be said is, "there are few troops...", "there are few customers buying..." etc. "Few" is the adjective that strictly speaking belongs to number. There are exceptions to this, e.g., the number ten is less than the number thirty-three. This exception is based on the fact that both of these numbers are being considered as one (like a whole, in the manner of a magnitude) and not as the multitude of units.
The point is that this loss of distinction in speech is a sign of the loss in mathematics of the distinction between number and magnitude. Dedekind aside, they are still different. There are still no odd and even lines. There are no prime lines and ones that are not prime. Every number is either even or odd. Every number is either prime or it is not. They are still not magnitudes and magnitudes are not numbers. Etc.
So I give another reason for my above question.
*Proof here is being used generally. There are two basic (some might say three with Q.E.I.'s): ones that show a things property and ones that show that a thing exists. The first usually ends in Q.E.D., the other with Q.E.F. Q.E.D. stands for, "quod erat demonstrandum", which was to be shown, and Q.E.F. stands for, "quod erat faciendum", which was to be done. Q.E.I., by the way, signifies a proof that is to find something; this to me is the same as showing something exists, so I do not make the distinction, though Sir Isaac Newton does. I am still thinking about it. Oh, Q.E.I., as one might have guessed, stands for, "quod erat inveniendum" which was to be found.
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