It's often said that mathematics is "the universal language," and I think it's true that the underlying concepts that mathematics describes are universal and not culture-dependent^{1}. However, the language and notation that we use to describe these concepts, and the particular structures and classifications that we choose to study, can vary between cultures.
As an example, let's look at some math from the Etymologiae (Etymologies) of Isidore of Seville. This text was written in the 7th century AD, and was very popular in the Middle Ages.
Even and Odd (and Evenly Even, Oddly Even, Evenly Odd, and Oddly Odd)
Isidore writes^{2}:
An even number is one which can be divided into two equal parts, like 2, 4, and 6. An odd number is one which cannot be divided into equal parts, with a one in the middle either lacking or unnecessary, like 3, 5, 7, 9 and the rest.
This is fine, I guess.
An evenly even number is one which is evenly divided into even numbers until it reaches an indivisible unity; as, for example, 64 has as a half 32, and this however has 16, then 8, 4, 2, and finally 1, which is singular and indivisible.
So evenly even numbers are just powers of 2.
An evenly odd number is one which can be divided into equal parts, but its parts immediately remain indivisible, like 6, 10, 38, and 50. For as soon as you divide such a number, you will come to a number which you cannot cut in two.
So evenly odd numbers are just 2 times an odd number. Note that contrary to the name, evenly odd numbers are not odd.
An oddly even number is one whose parts can be divided but which do not arrive at unity, like 24. For this, divided in half, gives 12, then 6, then 3; and that part does not accept further division, but before unity it comes to an end, which you cannot cut in two.
But is there also an "oddly odd"? Don't worry, of course there is!
An oddly odd number is one which can be divided into odd numbers an odd number of times, like 25 or 49, which, while they are odd, are divided from odd parts, as 49 is seven sevens and 25 is five fives.
At this point we have a bit of a problem on our hands, because I don't know what this is supposed to mean. The text kinda suggests that any odd number which can be divided into an odd number of odd numbers is oddly odd, which should encompass all odd composite numbers. However, both of the examples given are squares, so maybe "oddly odd" specifically refers to odd squares.
Unfortunately, neither of these explanations works. The "odd composites" explanation fails because in Isidore's system the term "composite" only applies to odds in the first place, so composite and oddly odd should mean the same thing^{3}. The "odd squares" explanation fails because squares are mentioned later^{4} with no reference to oddly odd numbers.
I would also like to mention at this point that the Latin words for even and odd are par and inpar, so to fully convey what reading this is like I really ought to be writing "odd" as "uneven."
So, to summarize what we have so far, consider a number n = d2^k, where d is odd.
k = 0 | k = 1 | k > 1 | |
---|---|---|---|
d = 1 | not a number^{5} | evenly even (n = 2) | evenly even |
d > 1 | (unevenly?) uneven | evenly uneven (but not uneven) | unevenly even |
Prime and Composite (and Mediocre?)
Odd numbers are either simple, or composite, or mediocre^{6}. Simple numbers are those which have no other part besides unity, as 3 has only 3, 5 only 5, and 7 only 7. For these have only one part.
Composite numbers are those which are divided not only by unity, but are also produced from another number, as 9, 21, 15, and 25. For we say these are three threes, three sevens, five threes, or five fives.
So we have primes and composites. Isn't that everything?
Apparently not...
Mediocre numbers are those which in some way seem to be simple and not composite, but in other ways composite; for example, 9 is prime and not composite when compared to 25, which does not have any numbers in common except for one; but if you compare it to 15, it is secondary and composite, since there are numbers in common besides one, namely 3, since 9 is three threes and 15 is five threes.
So 9 and 25 are relatively prime, but 9 and 15 are not, and this means that 9 is "in some ways prime, but in some ways not."
There are two problems with this.
The first is that literally every composite number has this property of being relatively prime to some numbers but not others.
The second is that Isidore gave 9 as an example of a composite number, but now he's also giving it as the only example of a mediocre number, and these two categories seem like they're supposed to be mutually exclusive.
In conclusion, I have no idea what a mediocre number is.
Some More Nonsense
So far, we've only seen the "first division" of numbers.
Isidore goes on to describe the relationships between numbers.
All numbers are compared either with respect to themself, or to another number. The latter are divided thus: some are equal and some are unequal. The latter are also divided thus: Some are greater, and some are lesser.
So far, so good.
Greater numbers are divided thus: multiples, superparticulars, superpartients, superparticular multiples and superpartient multiples.
Lesser numbers are divided thus: submultiples, subsuperparticulars, subsuperpartients, subsuperparticular submultiples, and subsuperpartient submultiples.
Oh no.^{7}
Isidore also gives the following explanation of equality:
Numbers are said to be equal when, with respect to quantity, they are equal...
Unequal numbers are those which, when compared to each other, demonstrate inequality.
In other words, equality is shaped like itself.
I realize that this post has been an impenetrable wall of text, so I should probably include this helpful diagram^{8}:
Hopefully this answers all remaining questions.
- In case anyone was wondering, when it comes to philosophy of mathematics I think I lean towards some form of structuralism, particularly one of the "non-eliminative" structuralisms, but I don't feel qualified to go into more detail about that. ↩
- My translation ↩
- Although we haven't gotten to mediocre numbers yet... ↩
- As "circular numbers," with cubes being "spherical numbers," although I should mention that Gelius mentions "cubic" numbers (κύβοι or quadrantalia) in the Noctes Atticae 1.20 ↩
- Isidore says "One is the seed of a number, but is not a number" ↩
- The Latin is mediocris, and since I have no idea what mathematical concept it's supposed to express, I'll just stick with "mediocre" ↩
- These terms seem to badly approximate the idea of equivalence classes in modular arithmetic, but I really don't want to bother translating all of this nonsense. ↩
- This diagram accompanies the version of the text found here ↩