By Bobby Neal Winters
Anthony Esolen has written an excellent article for the Touchstone blog about numbers and the mysteries to be encountered therein. Dr. Esolen is a mathematical amateur in the original sense of the word: someone who pursues an activity out of love, not because it is his profession.
I am a professional, and I note that he perceives mysteries that have slipped past many of my colleagues. Numbers are not the straight-forward things they appear. What we teach to preschoolers, what we chide eight-graders for not understanding, contain puzzles as deep as any met by the doctors of the church.
Dr. Esolen wrote about rational numbers versus the irrational. I would like to mention simpler numbers that also, to me, to present profound mysteries. There are the counting numbers that we call the Natural Numbers or the Positive Integers, and then there are the Real Numbers.
We teach our children about the Natural Numbers before they even go to school. They sit counting pennies or pieces of candy as one, two, three, four, and so forth without questioning whether they should worry about the existence of something that applies to a collection of objects whether they are made of either candy or sugar.
These counting numbers are to be contrasted with the Real Numbers. The name “Real,” which probably arose to distinguish them from the so-called Imaginary Numbers, signifies no deeper reality than any other type of number. Students begin to make forays into the world of real numbers when they encounter decimals. They are a different sort of entity than the Natural Numbers.
Real numbers can represent portions of objects but Natural Numbers can only represent entire entities. I can eat 1.45 burritos. I can walk 0.456 miles. But, if I have 45 students in class, there cannot be 2.4 of them absent. There are not 3.7 birds on the wire outside of my window.
There was a time when the average family had 2.8 children in it, and a family of three could joke that little Billy was their 0.8th of a
child, but that was a joke. People laughed because you can’t have 0.8 of a child. We speak figuratively of a man having been render impotent as being left “half a man,” but that wording has its rhetorical power because of its impossibility.
Humans do not subdivide in that way.
We did in our country attempt to reduce slaves to three-fifths human status, but that is rightly abhorred now. A human with only one arm, one leg, one lung, and so forth is still one human and not half a human. That is human arithmetic.
One sperm from one man and one egg from one woman make one human being is also human arithmetic.
A baby human is not a fraction of a human, and neither is a handicapped human. We are whole entities than cannot be divided. That is our sacred arithmetic.
Anthony Esolen has written an excellent article for the Touchstone blog about numbers and the mysteries to be encountered therein. Dr. Esolen is a mathematical amateur in the original sense of the word: someone who pursues an activity out of love, not because it is his profession.
I am a professional, and I note that he perceives mysteries that have slipped past many of my colleagues. Numbers are not the straight-forward things they appear. What we teach to preschoolers, what we chide eight-graders for not understanding, contain puzzles as deep as any met by the doctors of the church.
Dr. Esolen wrote about rational numbers versus the irrational. I would like to mention simpler numbers that also, to me, to present profound mysteries. There are the counting numbers that we call the Natural Numbers or the Positive Integers, and then there are the Real Numbers.
We teach our children about the Natural Numbers before they even go to school. They sit counting pennies or pieces of candy as one, two, three, four, and so forth without questioning whether they should worry about the existence of something that applies to a collection of objects whether they are made of either candy or sugar.
These counting numbers are to be contrasted with the Real Numbers. The name “Real,” which probably arose to distinguish them from the so-called Imaginary Numbers, signifies no deeper reality than any other type of number. Students begin to make forays into the world of real numbers when they encounter decimals. They are a different sort of entity than the Natural Numbers.
Real numbers can represent portions of objects but Natural Numbers can only represent entire entities. I can eat 1.45 burritos. I can walk 0.456 miles. But, if I have 45 students in class, there cannot be 2.4 of them absent. There are not 3.7 birds on the wire outside of my window.
There was a time when the average family had 2.8 children in it, and a family of three could joke that little Billy was their 0.8th of a
child, but that was a joke. People laughed because you can’t have 0.8 of a child. We speak figuratively of a man having been render impotent as being left “half a man,” but that wording has its rhetorical power because of its impossibility.
Humans do not subdivide in that way.
We did in our country attempt to reduce slaves to three-fifths human status, but that is rightly abhorred now. A human with only one arm, one leg, one lung, and so forth is still one human and not half a human. That is human arithmetic.
One sperm from one man and one egg from one woman make one human being is also human arithmetic.
A baby human is not a fraction of a human, and neither is a handicapped human. We are whole entities than cannot be divided. That is our sacred arithmetic.
--Bobby Neal Winters
"I can eat 1.45 burritos. I can walk 0.456 miles. But, if I have 45 students in class, there cannot be 2.4 of them absent. There are not 3.7 birds on the wire outside of my window."
I believe this line of thinking can shed light on the proper of use of 'less' and 'fewer.'
Posted by: kkollwitz | September 10, 2010 at 08:02 PM
Thanks, kkollwitz and Bobby Winters, I have been wondering what to plan do in my ESL class (of more advanced folks) before I have even met them. The 'less' and 'fewer' error is prevalent and has theological and philosophical implications. Not to mention the Express Lane in the supermarket---12 or [fewer/less] items.
Posted by: Emil, Franklin, TN | September 11, 2010 at 05:41 PM
1 Husband plus 1 Wife = 1 Flesh
1 Father + 1 Son + 1 Holy Spirit = 1 God
Just semantics? Or Holy reality...
Posted by: Bull | September 12, 2010 at 01:30 PM
I posted this already, but I guess it got swallowed somehow:
The Constitution never asserted that a slave was three fifths of a person in a metaphysical sense. Three fifths of the total slave population (not three fifths of each individual, although I guess it comes to the same number) were counted for the purposes of representation in the House. It was actually those who favored slavery who wanted the full slave population to count for representation, since it would have increased Southern power. Sorry to seem nitpicky, but this is a pet peeve of mine.
Posted by: James Kabala | September 12, 2010 at 01:41 PM
...But, if I have 45 students in class, there cannot be 2.4 of them absent. There are not 3.7 birds on the wire outside of my window.
There was a time when the average family had 2.8 children in it, and a family of three could joke that little Billy was their 0.8th of a child, but that was a joke. People laughed because you can’t have 0.8 of a child...
Does this mean that statistical analysis, like calculating an average, is somehow frivolous or naughty?
Posted by: Benighted Savage | September 13, 2010 at 11:16 AM
What James Kabala said. . .
I understand the indignation inspired by the 'Three-Fifths Rule', but it was actually an attempt at an anti-slavery measure (perhaps not as well-thought-out as it might have been, but that was the aim).
It is one of the ironies of the outplay of slavery in 19th-century American politics that the pro-slavery side wanted to count slaves as full persons for purposes of the census, and hence to increase the size of their respective congressional delegations. On the other hand, and for the same reason, the anti-slavery side would have preferred not to count them at all. The 'Three-Fifths Rule' was a compromise. . .
Of course, as James said, it was not a 'metaphysical' statement. In fact, there is yet further irony to be found when one compares the 'pro-slavery' desire to count slaves at full population value, with the citizenship rights those slaves were actually accorded (or not). . .
Posted by: CKG | September 14, 2010 at 12:13 PM
Yes, the 3/5 thing was a bit of an over-reach on my part. Sorry to have brought it into the mix as a detractor, but I got caught in the rhetorical flow and couldn't help myself. It will be removed or accounted for in subsequent versions of the piece should there be any.
Posted by: Bobby Winters | September 19, 2010 at 03:13 PM