Mathematical Structure
in the Universe
(and in Judaism)
Jonathan Rosenberg
Mathematics in Jewish Sources
‫ הֵ ן הֵ ן גּוּפֵ י‬,‫ קִ נִּין וּפִ תְ חֵ י נִדָּ ה‬,‫ַרבִּי אֱ לִיעֶ זֶר בֶּן חִ סְ מָ א אוֹמֵ ר‬
:‫ פַּ ְרפְּ ָראוֹת לַחָ כְמָ ה‬,‫ תְּ קוּפוֹת וְגִימַ טְ ִריאוֹת‬.‫הֲ לָכוֹת‬
(‫)אבות פרק ג‬
“The Mathematical Writings of Maimonides”, Y.
Tzvi Langermann, The Jewish Quarterly Review, Vol.
75, No. 1 (Jul., 1984), pp. 57-65, points out that the
Rambam wrote Notes on the Conics (in Arabic),
commenting on and filling in some proofs left out of
the famous book by Apollonius (ca. 200 BCE), which
Rambam almost certainly read in Arabic translation.
Rambam’s letter to his student
R. Joseph Ibn Aknin
(Preface to the Guide to the Perplexed)
“Observing your great fondness for mathematics, I let you study
it more deeply, for I felt sure of your ultimate success.
Afterwards, when I took you through a course of logic, I found
that my great expectations of you were confirmed, and I
considered you fit to receive from me an exposition of the
esoteric ideas contained in the prophetic books, that you might
understand them as they are understood by men of culture.”
Mathematics:
The best known and the least
known of the sciences
A typical conversation:
So, what do you do?
I’m a mathematician.
I always hated mathematics.
Wasn’t mathematics worked out centuries ago?
So how do you spend your time, adding tables of
numbers? Endless calculations?
Do you just make up your own axioms and see
what follows from them?
What is Mathematics, Really?
• Looking for patterns.
• Abstraction, that is simplifying to try to
find the essence of a problem.
• Applying abstraction to relate what at
first seem to be very different subjects.
• Using these relations to play one
problem off against the other.
A Practical Example:
Geometry and Number Theory
• Here is an example of what I mean. Two
branches of mathematics which seem at first
to have nothing to do with each other are
geometry (here we will be looking at the
“shapes of graphs”) and number theory
(finding whole-number solutions to certain
equations).
• Modern (post-World War II) mathematics has
shown, however, that these are closely linked.
The Geometry Side
Geometry (cont’d)
Something even more interesting happens when we
plot the graph with x and y allowed to be complex
numbers. It turns out the graph has the shape of a
multi-holed torus, where the number g, called the
genus, is the number of holes and can be computed
directly from the polynomial f. In the example on the
last slide, g = 1 and the picture looks like:
The Number Theory Side
The associated number theory question is:
How many rational number solutions are there to
the equation?
For example, in the case of y2=x3+8, most rational
values of x will give irrational y, but we have the
integral solutions (x = -2, y = 0), (x = 1, y = 3), and (x =
2, y = 4). How many rational solutions are there?
Answer: infinitely many. For example, there are also
(x = 46, y = 312) and (x = -7/4, y = 13/8). Or try (x =
31073/2704, y = 5491823/140608). (Check it!)
Number Theory and Geometry
What do these two subjects have to do with each
other? Well, in fact, there is a very deep
connection. For example there is:
Theorem (Faltings, 1983): If g (the number of holes
in the graph) is at least 2, then y2=f(x) has only
finitely many rational solutions.
Another amazing connection comes if we count integer
solutions “mod p”, for p a prime. That means we require
only that y2 – f(x) be divisible by p.
Mod p Solutions
For example, suppose the equation is y2=x3+8 and
we take solutions “mod 11”. Then x = 5, y = 1 is a
mod 11 solution, since
53 + 8 – 12 = 125 + 8 – 1 = 132 = 11×12.
How many solutions does y2 = f(x) have mod p?
Answer: this again depends on the geometric
invariant g!
Theorem (Hasse 1936, Weil 1949) The number of
solutions mod p is p+1 to within an error of at most
2g√p. (The error bound cannot be improved, in
general.)
Why does anyone care?
Why do mathematics? There are really two
competing answers:
1. Because it’s interesting, beautiful, and
aesthetically pleasing.
2. Because it’s useful.
The subject of counting points mod p on curves was
originally thought to be beautiful but useless. Now
it turns out to be critical for certain applications to
computer security! These applications were only
developed about 50 years after the key
mathematical discoveries.
Is Mathematics Discovered
or Invented?
The Platonist approach: Mathematics is part of
Hashem’s way of creating the universe. We just
discover what is already there.
An example from the web page for one of my
grad courses:
MATH 745: Lie Groups II (Spring 2000)
In the beginning God created the simple Lie groups ...
The Other Point of View
On the other hand, mathematics as we know it
is definitely a product of human civilization.
Creatures or cultures that think differently from
us would probably develop somewhat different
mathematics.
Generatingrationalpointsonanellipticcurve
curve:=plot::Implicit2d(y^2=x^3+8,x=-3..5,y=-5..5):
points:=plot::Listplot([[1,3],[-7/4,13/8],[-7/4,-13/8]],LinesVisible=FALSE):
tanline:=plot::Line2d([1,3],[-7/4,13/8],LineColor=RGB::Red):
refl:=plot::Line2d([-7/4,-13/8],[-7/4,13/8],LineColor=RGB::Green,LineStyle=Dashed):
plot(curve)
y
5
4
3
2
1
-3
-2
-1
1
2
3
4
5
x
-1
-2
-3
-4
-5
plot(curve,points,tanline,refl)
y
5
4
3
2
1
0
-1
-2
-3
-4
-5
-3
-2
-1
0
1
2
3
4
5
x
2
Thisprocedureimplementsthisalgorithmto"double"onthecurvey =x3+ax+bstartingwiththepoint(x1,y1).
doubleEC:=proc(x1,y1,a,b)
localz,m,num,den,x3;
begin
m:=(3*x1^2+a)/(2*y1):
x3:=m^2-2*x1:
return([x3,(x1-x3)-y1])
end_proc:
pt:=[1,3];
forjfrom1to4do
pt:=doubleEC(pt[1],pt[2],0,8):
print(pt)
end_for
1, 3

- 7 , 4

21833
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64
1
4
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- 21929
64

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2034250577474418601
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7878754975744
7878754975744

155759067482812188395435194793776177668044818958351868620139460409141605673

,
1027505312661247764459969847266650040423716655560634470125010944
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- 155758536187945207787990075948421313970903719960541226076900896297421878569
1027505312661247764459969847266650040423716655560634470125010944
