medical condition might be given a clean bill of health. A slight overestimate
and a completely healthy patient might be subjected to invasive and risky
medical intervention unnecessarily.

##  10 Exercises: Physics

  1. 1.

Topic: Physics
Year level: 1
Ethical pillars: 1,5,9

Write a MATLAB function that plots

$y=y_{0}+\frac{g}{k}t-\frac{g}{k^{2}}\left(1-e^{-kt}\right)$
$x=\frac{v_{0}}{k}\left(1-e^{-kt}\right)$

in 2-dimensional space, the variables $k$, $v_{0}$ and $y_{0}$ should
adjustable by the user. Plot this for $v_{0}=800$ and $y_{0}=4000$ and varying
values of $k$. Can you guess where these equations are coming from? Based on
the variable names and your plots, try to reverse-engineer the source of the
problem. What are the variables $v_{0}$, $y_{0}$ and $k$?

Solution and comments: This question is designed to show students that they
might be working in an information-constrained environment where their
employer asks them to solve mathematical problems without giving them proper
contextual knowledge. It teaches them to closely look at all variable names
and produce plots of equations to reverse-engineer the source of the problem.
The reverse-engineered problem is that of aerial bombing. Most students will
only see it after having completed the task (i.e. coded the function). The
variable $y_{0}$ is the height of the bomber, $v_{0}$ its velocity when
dropping the bomb and $k=\frac{D\rho A}{2m}$ is coefficient of the drag force.

We created this question by using the work of Pepper and Wilson363636Pepper,
C. and Wilson, C. (2009). Aerial Bombing Techniques. Available online at
https://mse.redwoods.edu/darnold/math55/DEproj/sp09/WilsonPepper/CPandCWfinal.pdf.

Given the complexity of including source code and plots, we omit these in this
solution.

  2. 2.

Topic: Dynamics
Year level: 1
Ethical pillars: 4,5

A projectile of mass $m$ is fired from a point $\mathbf{r}_{0}$ with velocity
$\mathbf{v}$, while a wind blows with constant velocity $\mathbf{u}$. The
gravitational acceleration is $\mathbf{g}$ and the air exerts a drag force on
the projectile equal to $mk$ times the velocity of the wind _relative to the
projectile_.

    1. (a)

Write down and solve the equation model to find the trajectory of the
projectile.

    2. (b)

Suppose that $\mathbf{v}$ is vertical and $\mathbf{u}$ is horizontal. Show
that the travel time $T$ of the projectile satisfies the implicit equation

$1-\mathrm{e}^{-kT}=\frac{kT}{1+\lambda},$

where $\lambda=kv/g$. Verify that $\lambda$ is nondimensional, and give a
physical interpretation of $\lambda$.

    3. (c)

(*) Find asymptotic expressions for $T$ in the limits $\lambda\gg 1$ and
$\lambda\ll 1$.

    4. (d)

Although no analytic form is available for $T$, find an expression for the
range $R$ in terms of $T$ and the other parameters.

    5. (e)

Real measurements can never be perfectly accurate or precise. Given that $u$,
$v$ and $k$ are uncertain, estimate the uncertainty in $T$ and $R$.

Hints:

       * •

Treat each parameter separately.

       * •

Linearise the equations about the true values by writing, say,
$u=u_{0}+u^{\prime}$, where $u_{0}$ is the true value and $u^{\prime}$ is the
uncertainty. Find an equation relating $T^{\prime}$ to $u^{\prime}$.

       * •

Give your answers in terms of relative uncertainties, expressing
$T^{\prime}/T_{0}$ in terms of $u^{\prime}/u_{0}$.

    6. (f)

(*) What does the drag model assume about the projectile?

    7. (g)

(*) Long-distance projectiles must also account for the rotation of the earth.
Estimate the lengthscale over which rotation becomes important.

Solution and comments: Experimental scientists are well aware of the
uncertainties that their experiments can have. On the other hand,
mathematicians are somewhat suspicious of uncertainty, but they should not
fall into the trap of thinking that ‘exact’ answers are ever exact! This is
especially important when the application, in this case the aiming of a
projectile, could have serious effects on safety.

Due to time constraints, we were not able to provide a full solution to this
question.

##  11 Exercises: Optimisation

  1. 1.

Topic: Optimisation
Year level: 1
Ethical pillars: 5,6

Let $S$ be an n-dimensional random vector of asset returns. Let
$U\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}\rightarrow\mathbb{R}$ be the
concave utility function of the investor and let $r\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}$ be the convex risk
function associated with an investment $x\in\mathbb{R}^{n}$.

    1. (a)

Set up an optimisation problem that maximises the investor’s expected utility
given an upper bound $b\geq 0$ for the risk. You may assume that the investor
has no other monetary constraints.

    2. (b)

Set up an optimisation problem that minimises the risk given a lower bound
$u\geq 0$ on the expected utility. You may assume that the investor has no
other monetary constraints.

    3. (c)

You can assume that the optimal solutions of (a) and (b) all lie on the same
frontier in space. By choosing the appropriate values for $b$ and $u$ the
optimisation problems lead to the same values. Are the optimisation problems
the same? Put into words how we use and understand risk in each case. Are we
using risk in the same way?

Solution and comments: This question is designed for students to explore
different optimisation models. It shows them that they can solve one problem
using two different approaches, they can mathematically get the same result.
But the approaches differ at a psychological and communicative level and thus
elicit different responses from managers, customers, etc.

    1. (a)

$\displaystyle\underset{x}{\text{maximize}}$
$\displaystyle\mathbb{E}\left[U\left(S^{T}x\right)\right]$ subject to
$\displaystyle r\left(x\right)\leq b$

    2. (b)

$\displaystyle\underset{x}{\text{minimize}}$ $\displaystyle r\left(x\right)$
subject to $\displaystyle\mathbb{E}\left[U\left(S^{T}x\right)\right]\geq u$

    3. (c)

In the first case we see risk as a constraint and in the second case
minimizing risk is the objective. The student should understand that
equivalent mathematical models can elicit different responses in everyday
situations when the underlying variables, terms and ideas are used in
different ways. Model (a) is commonly understood as more “optimistic”, while
model (b) is typically seen as more “pessimistic”. Many people intuitively
prefer to maximise their utility function, instead of minimising risk. To
many, it feels as if they get more out of it.

  2. 2.

Topic: Optimisation
Year level: 1
Ethical pillars: 1,5,9

An oil company wants to build a pipeline connecting an offshore oil platform
to a refinery (on land). The coastline is straight. The oil platform is a
distance $D_{1}$ from the coast. The refinery is on the coastline, distance
$D_{2}$ from the point on the coast closest to the platform. Building the
pipeline has a cost per unit length $c_{1}$ at sea, and $c_{2}$ on land. How
should the pipeline be built? What are the factors that need to be considered
when providing a response to this question?

Solution and comments: The cost-minimising path is given by Snell’s law, and
is an exercise in trigonometry. But who said we were optimising over cost?
This is an assumption that is trained in to mathematicians while they are
students, but it need not always be the right way to optimise. Economic
actions almost always have externalities, such as possible damage to the
environment (the pipe may go through a coral reef or protected habitat) or to
existing infrastructure (it may go through a school, or a site of
archaeological significance). Policymakers have to take a holistic view of
these effects, and the mathematicians that advise them need to be honest about
the simplifications and the assumptions that go into their models.

The idea for this question comes from Shulman373737B. Shulman (2002). Is There
Enough Poison Gas to Kill the City?: The Teaching of Ethics in Mathematics
Classes. The College Mathematics Journal 33(2), pp. 118-125.
https://www.tandfonline.com/doi/abs/10.1080/07468342.2002.11921929 .

The cost-minimising path (ASSUMING there are no other associated costs) will
be a diagonal straight line under water from the platform to some point on the
shoreline, then a straight line down the shoreline to the refinery. So let us
take ANY such pipe layout (diagonal under water, then straight along the
shoreline). Let $x$ be the length of the pipe along the shoreline. Then the
length of the (diagonal) pipe under water, $L$ is given by Pythagoras:

$L=\sqrt{D_{1}^{2}+(D_{2}-x)^{2}}$

Thus the total cost of the pipe in terms of $x$, $C(x)$, is given by

$C(x)=c_{2}x+c_{1}\sqrt{D_{1}^{2}+(D_{2}-x)^{2}}$

Differentiating with respect to $x$, and looking for a minima, we get

$0=\frac{dC}{dx}=c_{2}-\frac{c_{1}(D_{2}-x)}{\sqrt{D_{1}^{2}+(D_{2}-x)^{2}}}$

And thus

$\frac{c_{2}}{c_{1}}=\frac{(D_{2}-x)}{\sqrt{D_{1}^{2}+(D_{2}-x)^{2}}}$

Let $\theta$ be the angle the underwater pipe makes with the shoreline.
Observe that we have

$\cos(\theta)=\frac{(D_{2}-x)}{\sqrt{D_{1}^{2}+(D_{2}-x)^{2}}}=\frac{c_{2}}{c_{1}}$

Thus

$\frac{D_{1}}{D_{2}-x}=\tan(\theta)=\tan(\arccos(\frac{c_{2}}{c_{1}}))$

Rearranging gives

$x=D_{2}-\frac{D_{1}}{\tan(\arccos(\frac{c_{2}}{c_{1}}))}$

so the minimum cost pipe has 2 segments:
Segment 1: a straight pipe on the shore, going up distance $x$ from the
refinery.
Segment 2: a straight pipe under water, from the end of the segment 1 pipe to
the platform.

But what we have computed is “the minimum cost pipe, assuming there are no
other costs involved, and not factoring in any other impacts or concerns”,
which is not what the question asks. The question asks “How should the
pipeline be built?”; this is not a mathematics problem, it is a problem about
building a pipe. Mathematics is a useful tool in providing an answer, but
there are many other things to consider. Figuring out where and how to build
the pipe involves many considerations, including: What is under the sea (coral
reef, etc)? What is on the shoreline (protected habitat, important buildings,
areas of cultural or religious significance, etc)? What are the risks of
having a pipeline in that area (oil spills, etc)? What are the ongoing
maintenance costs of underwater pipes, compared to overland (rust, difficulty
in fixing leaks, etc)? How susceptible to damage or sabotage is an underwater
pipe, compared to overland (damage from protest groups, accidental impact from
vehicles, etc)? What are the political consequences of building an oil
refinery (protests, reputational damage, etc)? Is there a better thing to be
building instead (an offshore wind farm)?

There are numerous factors to consider here, more than can be listed. The
point is to get students to realise that there are at least SOME factors, and
that optimising over (naive) cost, with no other considerations, is not at all
a good way to answer the question.

  3. 3.

Topic: Optimisation
Year level: 1
Ethical pillars: 5,6

    1. (a)

You have just been hired as an intern by an investment firm and are given $n$
possible investment opportunities, each having a deterministic profit $p_{i}$
but random investment costs $a_{i}$. You are told that you have a fixed budget
constraint $b$. Someone else has already modelled this for you as a Knapsack
problem using the mean $\mu_{i}$ of $a_{i}$.

$\displaystyle\underset{x}{\text{maximize}}$
$\displaystyle\sum\limits_{i=1}^{n}p_{i}x_{i}$ subject to
$\displaystyle\sum\limits_{i=1}^{n}\mu_{i}x_{i}\leq b$ $\displaystyle
x_{i}\in\\{0,1\\},\;i=1,\ldots,n.$

Briefly discuss why this might not be a good idea.

    2. (b)

A simple way to tackle the randomness of the problem is by considering chance
constraints. In this case we accept that side conditions are violated with a
probability $\epsilon.$

$\displaystyle\underset{x}{\text{maximize}}$
$\displaystyle\sum\limits_{i=1}^{n}p_{i}x_{i}$ subject to
$\displaystyle\mathbb{P}\left(\sum\limits_{i=1}^{n}a_{i}x_{i}\leq b\right)\geq
1-\epsilon$ $\displaystyle x_{i}\in\\{0,1\\},\;i=1,\ldots,n.$

Show that it can be rewritten as

$\displaystyle\underset{x}{\text{maximize}}$
$\displaystyle\sum\limits_{i=1}^{n}p_{i}x_{i}$ subject to
$\displaystyle\Phi^{-1}\left(1-\epsilon\right)\sqrt{\sum\limits_{i=1}^{n}\sigma_{i}^{2}x_{i}^{2}}+\sum\limits_{i=1}^{n}\mu_{i}x_{i}\leq
b$ $\displaystyle x_{i}\in\\{0,1\\},\;i=1,\ldots,n.$

You can assume that the $a_{i}$ are independent and follow a normal
distribution $a_{i}\sim N\left(\mu_{i},\sigma_{i}^{2}\right)$. How would you
describe the special form of the first constraint in the second version in
simple language?

Solution and comments: The question is designed to show students that ignoring
randomness and assuming idealised deterministic conditions can lead to extreme
costs in optimisation conditions. Students should learn that this can often be
avoided through very simple concepts. The student should understand that it
may be necessary to question the validity of an optimisation model they are
given by a 3rd party, and not blindly rush to solve it.

    1. (a)

The classical formulation of the knapsack problem assumes that the $a_{i}$,
$p_{i}$ and $b_{i}$ are known and fixed an known in advance. The student
should intuitively understand that if randomness is involved, naively solving
the classical problem might no longer be a good idea. Random variations of
parameters can lead to large violations of the constraints.

Note to Instructor: Ben-Tal et al.383838Ben-Tal, A., and Nemirovski, A. (2000)
Robust solutions of linear programming problems contaminated with uncertain
data. Mathematical Programming 88(3),pp. 411–424. found in their study of the
NETLIB library of optimization problems, that a deviation in the coefficients
of as little as $0.1\%$ can lead to the violation of the inequality
constraints by $450\%$. In $18\%$ of the cases the violations were larger than
$150\%$. The classical problem formulation is often unusable in practice when
the parameters are uncertain.

    2. (b)

We can rewrite the constraint by considering

$\displaystyle\mathbb{P}\left(\sum\limits_{i=1}^{n}a_{i}x_{i}\leq b\right)$
$\displaystyle=\mathbb{P}\left(\frac{\sum\limits_{i=1}^{n}a_{i}x_{i}-\sum\limits_{i=1}^{n}\mu_{i}x_{i}}{\sqrt{\sum\limits_{i=1}^{n}\sigma_{i}^{2}x_{i}^{2}}}\leq\frac{b-\sum\limits_{i=1}^{n}\mu_{i}x_{i}}{\sqrt{\sum\limits_{i=1}^{n}\sigma_{i}^{2}x_{i}^{2}}}\right)$
$\displaystyle=\mathbb{P}\left(Z\leq\frac{b-\sum\limits_{i=1}^{n}\mu_{i}x_{i}}{\sqrt{\sum\limits_{i=1}^{n}\sigma_{i}^{2}x_{i}^{2}}}\right)\geq
1-\epsilon$

where $Z\sim N\left(0,1\right)$. After using the cumulative distribution
function $\Phi$ of the standard normal distribution and some rearranging, we
obtain the solution:

$\Phi\left(\frac{b-\sum\limits_{i=1}^{n}\mu_{i}x_{i}}{\sqrt{\sum\limits_{i=1}^{n}\sigma_{i}^{2}x_{i}^{2}}}\right)\geq
1-\epsilon$
$\Rightarrow\frac{b-\sum\limits_{i=1}^{n}\mu_{i}x_{i}}{\sqrt{\sum\limits_{i=1}^{n}\sigma_{i}^{2}x_{i}^{2}}}\geq\Phi^{-1}\left(1-\epsilon\right).$
$\Rightarrow
b-\sum\limits_{i=1}^{n}\mu_{i}x_{i}\geq\left(1-\epsilon\right)\sqrt{\sum\limits_{i=1}^{n}\sigma_{i}^{2}x_{i}^{2}}\Phi^{-1}$
$\Rightarrow\Phi^{-1}\left(1-\epsilon\right)\sqrt{\sum\limits_{i=1}^{n}\sigma_{i}^{2}x_{i}^{2}}+\sum\limits_{i=1}^{n}\mu_{i}x_{i}\leq
b$

Taking a closer look at the constraint reveals its special form (in simple
language):

$\text{penalty from randomness }+\text{ classical cost when using the mean
$\mu_{i}$ of $a_{i}$}\leq b.$

##  12 Exercises: Probability

  1. 1.

Topic: Probability
Year level: 1
Ethical pillars: 2,5,6

You are on holiday with your friend Kelly, and the two of you go hiking.
Unfortunately, Kelly falls on a rock and breaks her right foot. She is
transported to a small remote hospital and needs to stay at the hospital for
another night. The power of the hospital fails each hour independently with
probability $p$.

    1. (a)

Derive a formula for the expected number of hours until the hospital loses
power.

    2. (b)

How small does $p$ have to be, such that the expected number of power failures
in a given year is less than 1?

    3. (c)

Kelly has to stay in the hospital for another $x$ hours. If the power does not
fail, she will be discharged in time and the two of you still catch your
return flight. What is the probability that the hospital loses power in the
first $x-1$ hours?

    4. (d)

Kelly asks you what value of $p$ would make you feel comfortable. How do you
decide?

Solution and comments: This question is designed to show students that
mathematics is not always black and white, but that there are many subjective
components in setting up models and choosing values.

    1. (a)

Let $N$ be the number of hours until the hospital first loses power. Then we
need to calculate
$E\left[N\right]=\sum\limits_{i=0}^{\infty}P\left(N>i\right).$ The probability
that the power does not fail in the first $i$ hours is given by
$\left(1-p\right)^{i}$. Hence,

$E\left[N\right]=\sum\limits_{i=0}^{\infty}\left(1-p\right)^{i}=\frac{1}{1-(1-p)}=\frac{1}{p}$

    2. (b)

The expected number of power failures in a year is given by

$365\cdot 24\cdot p.$

Solving $365\cdot 24\cdot p<1$ gives $p<\frac{1}{365\cdot 24}$.

    3. (c)

The probability that the power fails in the $i$-th hour for the first time is
given by $\left(1-p\right)^{i-1}p$. Hence,

$P\left(N<x\right)=\sum\limits_{i=1}^{x-1}P\left(N=i\right)=\sum\limits_{i=1}^{x-1}\left(1-p\right)^{i-1}p.$

    4. (d)

The final part of the question is designed to test if the student gives a
solution only considering their friend, the chance of catching the return
flight or if they also think about the other patients. As mathematicians, we
often need to choose values for variables that make us feel comfortable or
seem right. The question is designed to show students that outside of pure
mathematics, “finding a value that feels right” can be tricky. Some will
decide to optimise $p$ according to either (a), (b) or (c), others might
compare different solutions or tell you that this “question is not well
defined”. When mathematics is applied in a social context, questions are
rarely well-defined and leave plenty of room for different interpretations.

  2. 2.

Topic: Probability
Year level: 1
Ethical pillars: 5

Two players A and B flip a biased coin alternately with A going first, and the
first player to flip a heads wins the game. Each flip is independent of all
previous flips, and has a probability $p$ of coming up heads. Find an
expression in terms of $p$ for $\Pr(A)$, the probability that A wins, and
sketch $\Pr(A)$ against $p$. What happens as $p\rightarrow 1$? What about at
$p=1$? What happens as $p\rightarrow 0$? What about at $p=0$?

Solution and comments: The purpose of this question is to help mathematicians
see that their mathematics might still yield a solution outside their domain
of definition of the problem. In this instance, there is no winner when $p=0$,
even though in the limit as $p\rightarrow 0$ the chance of player A winning
approached $1/2$. It is easy to forget what the original problem was, and its
constraints, after it has been ‘mathematised’, and this can lead to a
mathematical model being misused and giving nonsensical outputs that are
harmful if relied upon in the physical world.

This can be solved by seeing that the chance of B winning is the chance of A
flipping tails x the chance of A winning (as, after flipping a tail, A passes
the coin to B, and B now has the same chance of winning as A originally did).
So this can be solved using two simultaneous equations:

$\Pr(A)+\Pr(B)=1$ $\Pr(B)=(1-p)\Pr(A)$

Which gives

$\Pr(A)=\frac{1}{2-p}.$

(This can also be solved by summing an infinite series; it gives the same
answer). But this is only valid if there is an actual chance of heads; that
is, $0<p\leq 1$. If $p=0$, then the equation $\Pr(A)+\Pr(B)=1$ is not valid,
as no-one can win, so in that case $P(A)=0$.

In the limiting case $p\to 1$ we have $\lim_{p\to 1}\frac{1}{2-p}=1$. That is,
as $p$ gets closer to $1$, A stands an overwhelming chance of winning. And
moreover it is clear that when $p=1$ we have $P(A)=1$ (as soon as A flips the
coin, they will get heads and win).

In the limiting case $p\to 0$ we have $\lim_{p\to
0}\frac{1}{2-p}=\frac{1}{2}$. That is, as $p$ gets closer to $0$, the game
becomes closer and closer to being perfectly fair (the advantage of going
first is greatly reduced). But, when $p=0$ we have $P(A)=0$, as it is not
possible for A to ever flip heads. The limiting behaviour of the model is is
no way indicative of the outcome of the system at that limit point.

This is a very important lesson: after mathematising, the initial assumptions,
domain of definition, etc can be easily lost or forgotten, and the model
misused as a result. Just because the expression $\frac{1}{2-p}$ has a well-
defined limit as $p\to 0$, and is even defined at $p=0$, does not mean it is
indicative of the physical behaviour of the system at $p=0$.

  3. 3.

Topic: Probability
Year level: 1
Ethical pillars: 4,5,6

    1. (a)

Consider the following methods of randomly choosing a chord in a circle:

      1. i.

Fix a point on the circumference as one end of the chord, and choose an angle
$\Theta\sim\text{Unif}(0,\pi)$ to be the angle subtended by the chord at the
centre.

      2. ii.

Fix a radius to intersect the chord at right-angles, and choose a distance
from the centre $H\sim\text{Unif}(0,1)$ to be the midpoint of the chord.

      3. iii.

Choose a uniformly distributed point in the circle $(X,Y)$ to be the midpoint
of the chord (to do this, choose uniform variables $D\sim\text{Unif}(0,1)$,
$\Theta\sim\text{Unif}(0,2\pi)$, then set $X=\sqrt{D}\cos\Theta$,
$Y=\sqrt{D}\sin\Theta$).

For each of these, find the distribution function $F_{L}(l)$, and the
corresponding probability density function $f_{L}(l)$, for the length $L$ of
the chord. In each case evaluate the probability that $L$ is greater than the
side length of an inscribed equilateral triangle.

    2. (b)

How would you answer the following question: “Given a circle of unit radius,
what is the probability that a random chord of the circle is longer than the
sides of an inscribed equilateral triangle?”

Solution and comments: This is a crucial observation for students studying
probability; how we define “random” determines/changes what we can deduce
about random processes. To say “A randomly chosen item from a particular set
has a $50\%$ probability of possessing property X” is a meaningless statement
unless it is fully qualified with how that item was “chosen randomly”. This is
one of the ways mathematics can, and is, used to dupe the public, especially
in when probability theory and statistics are involved.

The final question is (deliberately) not well-formulated, because there are
many ways in which a chord could be chosen ‘randomly’. When we informally say
‘pick a number from 1 to 10 randomly’, we usually implicitly mean ‘under a
uniform distribution’; that is, $Pr(n)=0.1$ for all $n\in\\{1,\ldots,10\\}$.
This interpretation does not work in this problem, where a chord has multiple
degrees of freedom.

    1. (a)

Computing the probability functions and densities:

      1. i.

We have $L=2\sin(\Theta/2)$, thus

$\displaystyle F_{L}(l)$ $\displaystyle=\mathbb{P}\\{L\leqslant
l\\}=\mathbb{P}\\{2\sin(\Theta/2)\leqslant l\\}$
$\displaystyle=\mathbb{P}\\{\Theta\leqslant
2\arcsin(l/2)\\}=F_{\Theta}(2\arcsin(l/2))$
$\displaystyle=\frac{2}{\pi}\arcsin(l/2)~{}~{}~{}(0<l<2)~{},$

giving

$f_{L}(l)=\frac{d}{dl}F_{L}(l)=\frac{1}{\pi}\frac{1}{\sqrt{1-l^{2}/4}}~{}~{}~{}(0<l<2)~{}.$

      2. ii.

We have $L=2\sqrt{1-H^{2}}$, giving

$\displaystyle F_{L}(l)$ $\displaystyle=\mathbb{P}\\{L\leqslant
l\\}=\mathbb{P}\\{2\sqrt{1-H^{2}}\leqslant l\\}$
$\displaystyle=\mathbb{P}\\{H\geqslant\sqrt{1-l^{2}/4}\\}=1-F_{H}(\sqrt{1-l^{2}/4})$
$\displaystyle=1-\sqrt{1-l^{2}/4}~{}~{}~{}(0<l<2)~{},$

and

$f_{L}(l)=\frac{d}{dl}F_{L}(l)=\frac{l}{2}\frac{1}{\sqrt{1-l^{2}/4}}~{}~{}~{}(0<l<2)~{}.$

      3. iii.

We have $L=2\sqrt{1-(X^{2}+Y^{2}}=2\sqrt{1-D}$, giving

$\displaystyle F_{L}(l)$ $\displaystyle=\mathbb{P}\\{L\leqslant
l\\}=\mathbb{P}\\{2\sqrt{1-D}\leqslant l\\}$
$\displaystyle=\mathbb{P}\\{D\geqslant 1-l^{2}/4\\}=1-F_{D}(1-l^{2}/4)$
$\displaystyle=l^{2}/4~{}~{}~{}(0<l<2)~{},$

and

$f_{L}(l)=\frac{d}{dl}F_{L}(l)=\frac{l}{2}~{}~{}~{}(0<l<2)~{}.$

The probability that $L$ is greater than the side length of an inscribed
equilateral triangle is $1-F_{L}(\sqrt{3})$, giving the results:

      1. i.

$1-F_{L}(\sqrt{3})=1-\frac{2}{\pi}\arcsin\left(\frac{\sqrt{3}}{2}\right)=1-\frac{2}{3}=\frac{1}{3}~{}.$

      2. ii.

$1-F_{L}(\sqrt{3})=\sqrt{1-\frac{3}{4}}=\frac{1}{2}~{}.$

      3. iii.

$1-F_{L}(\sqrt{3})=1-\frac{3}{4}=\frac{1}{4}~{}.$

    2. (b)

The difficulty with this question is that even when we interpret the term
‘random’ – here as some form of ‘uniformly distributed’ – it is still not
enough to determine how exactly to sample a chord. The mathematical choice of
which definition of random to use will heavily influence the outcome of the
computation, as we saw above.

One might see this more clearly when sampling between sets. For ease of
measurement, a set S of people might be split into two subsets A,B. Then, to
sample height from the whole set S, we might uniformly choose between A and B
(probability 0.5 in each case), and then uniformly choose one member from that
set and measure their height. If set B had fewer people than set A, but had a
lot of tall people, then this would give a different (taller) average than
just sampling uniformly from the whole set S.

  4. 4.

Topic: Probability
Year level: 1
Ethical pillars: 2,5

Colourblindness is usually caused by a defective gene in the X chromosome. Let
the defective allele, which is recessive, be written as ‘x’, and the healthy
allele, which is dominant, be written as ‘X’. Most women have two X
chromosomes, while most men have one X and one Y chromosome. Epidemiologists
have determined that, globally, $p\approx 8\%$ of men are colourblind.

    1. (a)

If a man is not colourblind, what is his genotype? If a woman is colourblind,
what is her genotype?

    2. (b)

Approximately what proportion of women are colourblind? Approximately what
proportion women are carriers of the colourblindness gene, but not colourblind
themselves?

    3. (c)

Neither of Sam’s parents are colourblind. What is the probability that Sam’s
child is colourblind?

    4. (d)

What other issues do you need to consider to provide meaningful and helpful
probabilities here, and why?

Solution and comments: This question is all about implicit assumptions, how
they can be erroneous, and the complications that can arise from them.
Implicit assumptions (a particular name implies a particular gender) can seem
trivial to solve or address, from a mathematical perspective, but only IF they
are actually identified. If they remain undetected, then it doesn’t matter how
easy they would have been to address; they remain unaddressed, and the problem
the create thus persists.

    1. (a)

A non-colourblind man must have a non-defective X chromosome, and his genotype
must be XY. A colourblind woman must have two defective chromosomes, so her
genotype must be xx.

    2. (b)

Assuming that birth and death rates are independent of colourblindness (so
that the value of $p$ does not change between generations), about $p$ of all X
chromosomes carry the defective allele. Therefore, a woman has probability
$p^{2}\approx 0.6\%$ of having genotype xx, and probability $2p(1-p)\approx
15\%$ of being a carrier, with genotype Xx.

    3. (c)

Did the student assume that Sam is male? (It is important to be aware of one’s
implicit biases, such as ‘male by default’, especially in a community that is
majority-male.) Therefore both cases must be considered to account for the
fact that Sam may either be male or female.

Sam’s father must have genotype XY, and their mother’s genotype is distributed
with probabilities

XX (healthy) | Xx (carrier) | xx (colourblind)
---|---|---
$\displaystyle\frac{1-p}{1+p}$ | $\displaystyle\frac{2p}{1+p}$ | $0$

when we exclude the possibility that she is colourblind.

(1) If Sam is male, then it is the X chromosome that he receives from his
mother that determines whether Sam is colourblind. Sam can only get the
defective chromosome if his mother is a carrier, so his genotype distribution
is

XY (healthy) | xY (colourblind)
---|---
$\displaystyle\frac{1}{1+p}$ | $\displaystyle\frac{p}{1+p}$

(So, he has a better chance than the population of not being colourblind.) We
don’t know anything about Sam’s partner, so we assume her genotype follows the
population distribution for females

XX (healthy) | Xx (carrier) | xx (colourblind)
---|---|---
$(1-p)^{2}$ | $2p(1-p)$ | $p^{2}$

Now consider Sam’s children. For a male child, the genotype depends purely on
the mother, and does not depend on Sam (or Sam’s parents), so Sam’s son has
genotype distribution

XY (healthy) | xY (colourblind)
---|---
$1-p$ | $p$

i.e. no different from a population male. For a female child, we begin by
calculating the probability that she is colourblind: this can occur only if
Sam is colourblind, and his partner has at least one copy of the defective
allele:

$\Pr(\text{Sam's daughter is
xx})=\frac{p}{1+p}\left[p^{2}+p(1-p)\right]=\frac{p^{2}}{1+p}.$

Similarly,

$\Pr(\text{Sam's daughter is
XX})=\frac{1}{1+p}\left[(1-p)^{2}+p(1-p)\right]=\frac{1-p}{1+p}.$

Hence the genotype distribution for Sam’s daughter is

XX (healthy) | Xx (carrier) | xx (colourblind)
---|---|---
$\displaystyle\frac{1-p}{1+p}$ | $\displaystyle\frac{2p-p^{2}}{1+p}$ | $\displaystyle\frac{p^{2}}{1+p}$

(2) Now suppose that Sam is female. She inherits at least one healthy X
chromosome from her father, and so Sam’s genotype distribution is

XX (healthy) | Xx (carrier) | xx (colourblind)
---|---|---
$\displaystyle\frac{1}{1+p}$ | $\displaystyle\frac{p}{1+p}$ | $0$

(which is even better than her mother’s). Sam’s partner can be assumed to have
the population distribution for males,

XY (healthy) | xY (colourblind)
---|---
$1-p$ | $p$

Now consider Sam’s children. For a male child, the genotype depends purely on
Sam, and does not depend on Sam’s partner. So

$\Pr(\text{Sam's son is xY})=0.5\frac{p}{1+p}=\frac{p}{2(1+p)}$

So Sam’s son has genotype distribution

XY (healthy) | xY (colourblind)
---|---
$\displaystyle\frac{2+p}{2(1+p)}$ | $\displaystyle\frac{p}{2(1+p)}$

For a female child, we begin by calculating the probability that she is
colourblind: this can occur only if Sam’s partner is colourblind, and Sam
provides one copy of the defective allele:

$\Pr(\text{Sam's daughter is xx})=0.5\frac{p}{1+p}p=\frac{p^{2}}{2(1+p)}$

Similarly,

$\Pr(\text{Sam's daughter is
XX})=(1-p)(\frac{1}{1+p})+0.5\frac{p}{1+p}=\frac{(1-p)(2+p)}{2(1+p)}$

Hence the genotype distribution for Sam’s daughter is

XX (healthy) | Xx (carrier) | xx (colourblind)
---|---|---
$\displaystyle\frac{(1-p)(2+p)}{2(1+p)}$ | $\displaystyle\frac{3p}{2(1+p)}$ | $\displaystyle\frac{p^{2}}{2(1+p)}$

  5. 5.

There are several issues that need to be considered. For example:

     * •

There are other (rarer) causes of colourblindness (we have only said that
colourblindness is usually caused by a defective X chromosome, but there may
well be other genetic, or environmental, causes, which will affect the
probabilities calculated. These need to be factored in at the start of the
problem.

     * •

There may be variability by ethnicity or region. This is important, because
people do not choose random partners from the entire population to have
children with; they will normally choose someone who is geographically quite
near to them, and they may also have a preference for choosing someone of the
same ethnicity. In some regions, and/or within some ethnicities, $p$ may
differ substantially, so a more targeted value of $p$ (rather than the global
average) would be needed to compute meaningful probabilities.

     * •

Are ‘healthy’ and ‘defective’ good terms? Could they stigmatise people? This
is common usage in this case, but could be more controversial in some other
cases. There are various genetic conditions (e.g., autism, dwarfism) which
many people deem as their own unique character traits, rather than a “defect”.

     * •

Gender and chromosomal sex are not equivalent. For example, perhaps as many as
$1.7\%$ of people are intersex.393939Intersex Human Rights Australia (2013).
Intersex population figures. https://ihra.org.au/16601/intersex-numbers/ This
is not necessarily a small number compared to $p$.

  6. 6.

Topic: Probability - Littlewood’s Law
Year level: 1
Ethical pillars: 5

Suppose we define a ‘miracle’ as an exceptional event that occurs with
frequency $10^{-6}$ (‘one in a million’). Suppose also that a person observes
$1$ ‘event’ every second that they are alert, and that they are alert for $9$
hours of each day. Approximately how many miracles would they expect to
observe in one month? How sensible or meaningful is such a computation?

Solution and comments: This problems demonstrates why many quantitative
statements are not in a sense ‘mathematical’ – here for example, an ‘event’ is
not very well defined. When we teach students to model or mathematise
something, we often use physical examples or examples where we know that it
makes sense in a positivist or materialist sense.
However, you can mathematise many things where the underlying “object” or
“event” is not very well defined or ambiguous, and the mathematisation or
model then often hides that fact. In this question, we ask the students do
calculate the rate of “miracles”, but you can very well use other social,
psychological or theological phenomena instead. Even ambiguous physical or
biological events work very well if you highlight to students the ambiguity
that can be hidden behind or through mathematical formulations.
Hence, problems can arise when mathematical rigor is used to give (undeserved)
authority that is built on imprecise terms or unjustified assumptions.

The (approximate) expected rate of ‘miracles’ will be

rate
$\displaystyle=\frac{1~{}\text{miracle}}{10^{6}~{}\text{seconds}}\cdot\frac{60~{}\text{seconds}}{1~{}\text{minute}}\cdot\frac{60~{}\text{minutes}}{1~{}\text{(alert)
hour}}\cdot\frac{9~{}\text{alert
hours}}{1~{}\text{day}}\cdot\frac{30~{}\text{days}}{1~{}\text{month}}$
$\displaystyle=\frac{60\cdot 60\cdot 9\cdot
30}{10^{6}}\frac{\text{miracles}}{\text{month}}$ $\displaystyle=\frac{3\cdot
3\cdot 9\cdot 3}{250}~{}\text{miracles per month}$
$\displaystyle=\frac{243}{250}~{}\text{miracles per month}$
$\displaystyle\approx 1~{}\text{miracle per month}~{}.$

For the reasons discussed above, this is not a terribly meaningful statement.
One could instead describe or quantify “events” in a different way (eg: A
person sees one event per minute, or per hour. Or an event could be something
entirely different, such as “An item presented on the 6 O’Clock news”.) Under
these different definitions, the rate of miracles would differ dramatically.

  7. 7.

Topic: Probability - Simpson’s Paradox
Year level: 1
Ethical pillars: 2,4,6,9

In a particular admissions cycle, a mathematics department observes a higher
success rate for male applicants than for female applicants. To investigate
whether this is the same across the two sub-departments of Pure Mathematics
and Applied Mathematics, the following year the department asks each applicant
to give their preference for pure or applied (they are not allowed to be
ambivalent), and records the resulting statistics as follows:

Total:
---
| Applications | Successful
Female | $300$ |  $30$
Male | $1000$ |  $210$

Prefer applied:
---
| Applications | Successful
Female | $270$ |  $18$
Male | $350$ |  $15$
Prefer pure:
---
| Applications | Successful
Female | $30$ |  $12$
Male | $650$ |  $195$

    1. (a)

Compare the success rates for male and female applicants that prefer applied;
the success rates for male and female applicants that prefer pure; and the
success rates for male and female applicants over all.

    2. (b)

What do you notice? Why is this possible? This is known as Simpson’s Paradox.

    3. (c)

If possible, find the admission statistics by gender and mathematics
preference (pure/applied) in your mathematics department, and see if the same
phenomenon occurs.

Solution and comments: The purpose of this problem is to demonstrate Simpson’s
paradox, in which a trend appears in several different groups of data but
disappears or reverses when these groups are combined. It also attempts to
highlight the immense gender disparity in many mathematics departments around
the world.

    1. (a)

We calculate the success rates:

| Prefer applied | Prefer pure | Total
---|---|---|---
Female | $\frac{18}{270}=\frac{14}{210}$ | $\frac{12}{30}=\frac{4}{10}$ |  $\frac{30}{300}=\frac{10}{100}$
Male | $\frac{15}{350}=\frac{9}{210}$ | $\frac{195}{650}=\frac{3}{10}$ |  $\frac{210}{1000}=\frac{21}{100}$

    2. (b)

We note that females with a given preference have a higher success rate than
males with the same preference, but lower overall. This is Simpson’s Paradox.
The heuristic reason for why this is possible is that the largest male cohort
(prefer pure) has a much higher acceptance rate (0.3) than the largest female
cohort (prefer applied) which is about 0.067. So what dominates the overall
acceptance of men is for those who prefer pure (0.3), while what dominates
overall acceptance for women is those who prefer applied (0.067).
This is a great lesson in why it is usually a terrible idea to “take averages
of averages”.

    3. (c)

Students may have access to this data. The main point of the question is not
so much to redo the calculation (it is not a given that Simpson’s paradox will
always arise here), but rather to illustrate the immense gender disparity in
many mathematics departments around the world, which is likely (but not
certain) to include the department the students are currently in. Observations
like this resonate more when they are closer to home.

  8. 8.

Topic: Probability
Year level: 1
Ethical pillars: 6

Discuss the following statements, and how they use different notions of the
word “chance”:

    1. (a)

‘There is a $1/2$ chance that a flip of a fair coin gives heads.’

    2. (b)

‘Given that I am playing as White, there is a $1/3$ chance that I will beat
you at chess.’

    3. (c)

‘There is a $1/2$ chance that the foetus is male.’

    4. (d)

‘If you sample a random voter from a referendum where $52\%$ of voters voted
‘yes’, then there is a $52\%$ chance that they voted ‘yes’. ’

    5. (e)

‘If you sample a random voter before an upcoming election, there is a $30\%$
chance that they will vote for the incumbent.’

    6. (f)

‘There is currently a $40\%$ chance that AI will wipe out humanity in the next
5 years.’

Which of these events would you bet on (either for or against), if any?

Solution and comments: These statements are meant to highlight the different
ways in which a probability (ie: chance) can be interpreted by mathematicians,
or by the population at large.

    1. (a)

This is an example of _aleatoric uncertainty_. The probability can be
determined by conducting many trials and observing that $1/2$ of them produce
successes.

    2. (b)

Chess has no chance elements by itself (apart from the process to determine
who plays White, hence the conditioning). The probability expresses
uncertainty about the relative skills of the two players, as well as
uncertainty about the factors (e.g. psychological) that might affect the
quality of their play.

    3. (c)

This is an example of _epistemic uncertainty_ : the foetus’ sex has already
been determined. It is presently unknown, but could be measured.

    4. (d)

Each voter’s choice in the referendum has already been fixed, but the
randomisation comes from the sampling process.

    5. (e)

In contrast, voters have not yet fixed their individual choices for the next
election yet.

    6. (f)

This is a statement about an uncertain event that cannot be repeated and has
no precedents. The $40\%$ probability is a subjective assessment.

As these examples demonstrate, probabilities should sometimes be interpreted
as subjective quantities (‘degrees of rational belief’), which are to be
updated as further information becomes available (using Bayes’ formula).

The interpretation of randomness sometimes has legal consequences. In some
jurisdictions, gambling laws apply to wagers on games of chance but not on
games of skill, and the division can be somewhat arbitrary: for example, poker
is considered to be a game of skill in New York, but a game of chance in
Germany.

##  13 Exercises: Statistics

  1. 1.

Topic: Statistics
Year level: 1
Ethical pillars: 3,4,5,6

Your city council aims to redesign the city centre to make it more bike-
friendly if at least 75% of the citizens agree. The city administration claims
that this is the case. A competing political initiative believes that the
actual percentage of supporters is less than 75%. To test the claims, a
hypothesis test is conducted at the 5% significance level with a sample size
of $n=100$.

    1. (a)

How does the city council test? For which results is it confirmed?

    2. (b)

How does the competing political initiative test? For which results is it
confirmed?

    3. (c)

For which results can neither party reject the null hypothesis?

    4. (d)

Assume that the null hypothesis could not be rejected. Without further
comment, one of the parties approaches you with another 100 data points and
asks you to perform the test again by combining the old and new samples to
reach a sample size of $n=200$. What do you do?

Solution and comments: The purpose of this question is to teach students some
basics about good data sampling. It also teaches students to explore competing
ethical interests in data collection and analysis, and the pressures that a
mathematician can face in scenarios like these.

     * •

The student should understand the difference between a right- and left-sided
hypothesis test and be able to explain why the parties chose different tests
in questions (a) and (b).

     * •

The student should see the problem of p-hacking and lay out some of its
properties in this context. They should be able to discuss the dangers of
using an existing sample multiple for multiple decisions: to perform the
hypothesis test, to use the result to decide that a larger sample is needed,
and then to use the data points again in a later test on combined data. The
student should also be able to question the source of the new sample and if it
was properly sampled. They should understand that competing ethical interests
could mean that the standards of good sampling/interviewing may have been
violated.

Due to time constraints, we were unable to include a full solution to this
exercise.

  2. 2.

Topic: Statistics
Year level: 1
Ethical pillars: 6,9,10

In 2020 the UK government tried to explain their new COVID alert levels by the
following equation:

$\text{COVID Alert Level = R (rate of infection) + number of infections}.$

where the alert level is a discrete scale from $1-5$, and $R$ is the average
number of people each infected person infects.

Why does this not make much sense? Why could it still be good to communicate
this way? Why could it be dangerous to communicate this way? How would you
communicate it?

Solution and comments: This question teaches students that it is often
necessary to break down difficult mathematics into everyday language. It
teaches them about what information is lost along the way and to be wary of
the fact that others will have a lot less mathematical training (and hence may
need a simpler form of communication). You can find a full discussion in the
article by Woolley 404040T. Woolley (2020). Coronavirus: Why the maths behind
‘COVID alert levels’ makes no sense. The Conversation.
https://theconversation.com/coronavirus-why-the-maths-behind-covid-alert-
levels-makes-no-sense-138634.

    1. (a)

The units don’t match: The alert level is unitless and ranges from 1-5, R is
usually also unitless while the rate of infection is not, the number of
infections is unitless. All three parameters are also defined on different
ranges (R is usually quite small, while the number of infections can be in the
thousands).
Students might observe that actually there might have been some predetermined
function $f\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{R}\to\mathbb{N}$ giving

COVID Alert Level = f(R,number of infections)

ant that this is actually a sensible statement and probably close to what was
being done at the time. They should then consider how much the general
population would understand this mathematically formalised statement.

    2. (b)

The benefit of this style of communication is that people quickly learned to
worry about both the “rate of infection” and “the number of infections”. It
breaks down difficult concepts into everyday language.

    3. (c)

People can be confused as it doesn’t properly weigh the parameters. They want
to tell you that both the rate and number of infections are important, but are
they equally important? Is one or the other more important? Is the importance
constant over time or does it vary?

    4. (d)

This is hard, and that is the first thing students should acknowledge. Not
everyone has a degree in mathematics, so one is constrained in the language
and terminology that can be used.
They might lean towards variants of “The COVID alert level depends on the
infection rate as well as the number of infections”. But how does it depend on
these? How are they weighed? And how can one communicate this to public in a
quick and effective manner, that fits on one static PowerPoint slide? What is
the actual point of this communication; what is it trying to achieve? (it is
not there just as an education segment for the population; it is there to
effect some change in society - what change?
Ultimately, who is harmed if this communication is not effective, and how
badly are they harmed? And who will take the blame, or be blamed, if the
communication is ineffective and there is resulting harm?

##  14 Exercises: Numerical Methods

  1. 1.

Topic: Numerical Methods
Year level: 1
Ethical pillars: 6

Let $F\mathrel{\mathop{\mathchar 58\relax}}[a,b]\rightarrow\mathbb{R}$ be a
smooth function with $F(a)<0<F(b)$, and $F^{\prime}(x)>0$. Then the equation
$F(x)=0$ has a unique solution $x=a$. Numerical approximations to $a$ can be
found using Newton–Raphson iteration, which converges quadratically.

Convey the above information to someone who has no post high school
mathematical training. State the formula for Newton–Raphson iteration and
explain it graphically.

Solution and comments: The question is designed for students to break down the
mathematics into non-technical language and to communicate in a way that
requires minimal mathematical knowledge. It teaches them that pictures and
graphic representations are often a useful tool to explain aspects to non-
technical people.

Due to time constraints, we were unable to include a full solution to this
exercise.

##  15 Exercises: Pre-Calculus and Functions (catch-up / introductory)

  1. 1.

Topic: Pre-Calculus
Year level: 0-1
Ethical pillars: 5,6
As you have seen during school, text problems are often highly dependent on
their historical and social context. But this context naturally changes
throughout time or when you go somewhere else. This is one of the reasons why
textbooks are regularly updated to not just reflect the newest research in
pedagogy, but also to ensure that their ethics goes with the time.

Read carefully, but don’t solve, the following question from a German textbook
during World War II. You can find this and other examples in Shulman414141B.
Shulman (2002). Is There Enough Poison Gas to Kill the City?: The Teaching of
Ethics in Mathematics Classes. The College Mathematics Journal 33(2), pp.
118-125. https://www.tandfonline.com/doi/abs/10.1080/07468342.2002.11921929 :

“According to statements of the Draeger Works in Luebeck, when using
pesticides on a field, only 50% of the chemicals are effective. The atmosphere
must be covered up to a height of 2 metres in a concentration of 45 mg per
cubic metre to cover all the crops. How much chemical is needed to cover crops
of four square kilometres?”

Now solve the following version of this question:

“According to statements of the Draeger Works in Luebeck, when gassing a city,
only 50% of the evaporated poison gas is effective. The atmosphere must be
poisoned up to a height of 20 metres in a concentration of 45 mg per cubic
metre. How much phosgene is needed to poison a city of 50,000 inhabitants who
live in an area of four square kilometres?”

The mathematics you’ve used is almost identical. We call this the dual-use
problem of mathematics: many mathematical equations and results can be
simultaneously used for beneficial and harmful applications. Can you find yet
another example for the same mathematics?

Solution and comments: A word of warning is necessary for this question. The
dual use of mathematics is usually not explored at the high school level. When
using this question, be aware that students have very little background and
have almost never seen a mathematics question that actively asks them to do
something bad. This is why we urge you to not have students solve the original
problem directly, but rather to explore the mathematics and ethics of this
question indirectly by solving a similar problem and finding other
applications for the same mathematics.
Before setting this question, you should also be aware of who your students
are, their background and how they’d react to seeing mathematics in this
context. Care and empathy are needed when going into the very dark areas of
mathematical applications and education. To not leave students alone with
their emotions, we suggest that this question is best suited for classroom
discussions rather than as a homework problem.

Due to time constraints, we were unable to include a full solution to this
exercise.

##  16 Projects: Communicating your Mathematics

  1. 1.

Communicating and Documenting mathematics in a team - The Ethics Memo

This question can be added on to an existing question or exercise from
previous sections. It teaches students a way to communicate and document their
ethical concerns.

Additional question: Communicate the problem at hand to a relevant party (e.g.
your boss) in an Ethics Memo, as illustrated below:

The Ethics Memo
To:
From:
Re:
Date:
Issue: Summarise the ethical challenge that you have encountered during your
mathematical work. This includes translating professional jargon into readable
prose. The level of detail should be adopted to how familiar the recipient of
this memo is with the problem and the mathematics. Generally speaking: The
shorter the better, often a few lines is enough.

Interests: In this section you should address why the recipient ought to care
about the issue. It should address the most important ethical and other
interests. You should rank all interests according to a pre-defined standard
(e.g. necessary legal obligations, vital ethics/norms/interests, and other
important interests that are not vital).
_Legal:_
_Vital:_
_Important:_

Analysis: This section analyses the general shape of the ethical challenge.
You should briefly explain its dynamics and point out key causal factors. Try
to explain some of the necessary background in a way that the recipient of
this memo can understand and wants to read.

Objectives: Consider what resources and capabilities can be mobilized, and
formulate achievable goals: what can you feasibly do to address the ethical
issue? Do not aim for too much. If the problem is too involved, make this
clear. Be humble and aware of how much you can actually do.

Options: Identify and briefly describe distinct (non-)mathematical solutions
that address the issue. This section must include an assessment of the pros
and cons of each solution. Only consider realistic and feasible solutions.
Avoid writing sandwich memos where you present two unreasonable options and
the only reasonable recommendation is the one you prefer. Whenever possible,
you should present strong alternatives. Understand that  not doing
mathematics, or doing less mathematics, is often an option. Not every problem
requires a mathematical solution.

Recommendation: This section is a concise summary of the option you recommend
and why you recommend it. Try to get your point across as succinctly as
possible. Be aware of any preconceptions that your recipient might have about
your recommendation.

Implementation: Summarise the initial steps that need to be implemented.
Whenever possible, avoid professional jargon while accurately describing each
step. These steps can (and often do) include the consultation of other
specialists and domain experts. Briefly describe how success could be
measured.

Notes to Instructor: Ethics memos are an experimental idea which we’ve
internally tried and discussed. It is best suited for problems that have
multiple feasible solutions (e.g. optimisation problems that allow for
multiple ways to model it or when a non-mathematical solution is a good
option) or face competing interests (e.g. as in the case of the oil pipeline:
environmental constraints vs business interests). The style of this memo is
adapted from policy memos which are used to analyse political problems and
make recommendations.

To assist students in seeing what aspect of ethical issues might be worth
signposting in such a memo, it might be useful to suggest that they consult
the Manifesto for the Responsible Development of Mathematical Works424242M.
Chiodo & D. Müller (2023). Manifesto for the Responsible Development of
Mathematical Works – A Tool for Practitioners and Management.
https://arxiv.org/abs/2306.09131.

  2. 2.

Group project: How your science can end up in the newspapers.
When you read about scientific research in the media, it has gone through
several steps before the information has reached you. Consider David
Spiegelhalter’s 434343D. Spiegelhalter (2017). Trust in numbers. Journal of
the Royal Statistical Society Series A: Statistics in Society, 180(4), pp.
948-965. https://academic.oup.com/jrsssa/article/180/4/948/7068387 pipeline
for how statistical results end up in the news:

Figure 2: Pipeline of Science Communication (Spiegelhalter, 2017).

Step 1: Understanding the diagram

    1. (a)

Discuss the diagram in your group.

    2. (b)

Try to understand the work, interests and standards of each entity in the
diagram. Do you see any conflicts of interest arising between the different
entities?

    3. (c)

Do you know what all the filters are and how they work?

Step 2: Analysing a case study

    1. (a)

Read the scientific paper [INSERT TITLE] and summarise its main results.

    2. (b)

Find and read newspaper articles about this research. Summarise how they
presented the results to the public.

    3. (c)

Understand if the presentation by the news is accurate. If possible, try to
identify where the problem started.

This question is supposed to teach students that they might not always have
control over how other people perceive their published work.

Notes to Instructor:  It is best to select a research paper such that its
(mathematical) results are not just suitable for your course, but for which
students can find articles for the entire pipeline, including communication
from the university’s press office and a range of newspaper articles. This
enables them to properly understand how research is publicised and broken down
to fit different audiences.

##  17 Projects: Open-ended discussion questions

The following questions, based on particular case studies, require a little
more thought, discussion and some research. They are well suited for seminar-
style discussions.

  1. 1.

This question is about the Diffie–Hellman key exchange protocol.

    1. (a)

Explain to a non-mathematical audience why the Diffie–Hellman algorithm is
important.

In 2015, a major flaw was discovered in the implementation of the
Diffie–Hellman algorithm by Adrian et al.444444D. Adrian et al. (2015).
Imperfect Forward Secrecy: How Diffie–Hellman Fails in Practice. CCS ’15:
Proceedings of the 22nd ACM SIGSAC Conference on Computer and Communications
Security. https://dl.acm.org/doi/10.1145/2810103.2813707 , which affected tens
of thousands of popular websites.

Read Adrian et al. (2015) and pay particular attention to §§5–7.

    1. (b)

What were the reasons for the flaw?

    2. (c)

Do you think the authors were right to publish this work?

Solution and comments: This question deals with two main ideas:
1\. What are the ethical issues associated with creating large-scale tools to
break encryption en masse, especially when the tool itself is prohibitively
expensive to produce (large lookup table), but comparatively easy to use and
to lose/leak/steal (how “safe” can one keep such a tool)?
2\. Regarding responsible disclosure, what are the ethical issues with
exposing state surveillance capabilities, and does the fact that a state
enabled the vulnerability in the first place justify full disclosure (which
might incentivise other states to develop, or just outright steal, this
technology)?

  2. 2.

The statistician Ronald Fisher (1890–1962) made many important contributions
to statistics, including the concepts of hypothesis testing and $p$-values.
However, Fisher’s work on statistics was partially motivated by a desire to
justify his beliefs in parapsychology and racialism. This motivation does not
necessarily invalidate Fisher’s mathematical work. But how can the scientific
community safeguard against misuse of statistical techniques? Start by reading
about Fisher, his mathematics and views on eugenics and race.454545W. Bodmer
et al. (2021). The outstanding scientist, R.A. Fisher: his views on eugenics
and race. Heredity 126, pp. 565–576.
https://www.nature.com/articles/s41437-020-00394-6

Solution and comments: The consequences of Dual use technology are well-
explored in science, and this is something that mathematicians should also
start to be aware of.

The example of Fisher’s work here is a unique one, as it is the reverse of
dual-use. Usually, a new piece of technology is discovered/developed with
benevolent intentions, and then later on is reappropriated/misappropriated for
more nefarious use. In Fisher’s case, the statistical tools he developed were
for very harmful purposes, with almost no regard for ethics or morality.
However, these same tools turned out to be very useful in doing good, helpful,
and socially beneficial statistical analysis.

This is an example where students might identify and highlight the good,
positive uses of these mathematical tools, while at the same time
acknowledging and scrutinising their problematic origins. Distinguishing
between “the scientist” and “the science” is often a delicate topic, and one
that this question allows students to explore in some depth.

  3. 3.

Read about the Sally Clark case.464646The corresponding Wikipedia article and
press release by the Royal Statistical Society are a good start.
Wikipedia about Sally Clark: https://en.wikipedia.org/wiki/Sally_Clark
Statement by the Royal Statistical Society:
http://www.inference.org.uk/sallyclark/RSS.html Explain the chain of faulty
reasoning that led to her wrongful conviction. Can you explain it to a
layperson?

The Royal Statistical Society has released several guidelines and
reports474747These are all available at: https://rss.org.uk/membership/rss-
groups-and-committees/sections/statistics-law/ regarding “Statistics and the
Law”, and in particular for the use of statistical in medical misconduct
cases. These might be useful a useful starting point for your investigations.

Solution and comments: Sally Clark’s conviction, based on several pieces of
very bad mathematics, is an example of the power that mathematics has over
people. Mathematical results are often seen as ‘neutral’ or ‘unquestionable’ —
and mathematicians are sometimes to blame for advertising their subject as
that way. While that description might apply to high school geometry (most
people’s encounter with ‘real’ mathematics), it certainly does not apply to
the mathematics that is used in human contexts: like any other science,
mathematics suffers from ‘garbage in, garbage out’.

Clark was not the only person to be wrongfully imprisoned by Roy Meadow’s bad
testimony: two other mothers, Trupti Patel and Angela Cannings, were convicted
under similar circumstances.

##  18 Projects: Longer Essay Topics

###  18.1 Introduction to essay tasks

####  18.1.1 Aims

These essays serve many purposes, and there are many new skills that you can
learn, and knowledge that you can gain. Ultimately, what you get out of this
is entirely up to you and what you choose to put in; there are no set
expectations. As a guide, here are some things that you might learn or gain
from doing the essay:

  1. 1.

Learning to use LaTeX.
Yes, this is important! You’re mathematicians, or are in the process of
becoming mathematicians. Our community communicates in documents written
almost exclusively in LaTeX. This is excellent practice of LaTeXformatting.
Better to do it now in a non-examinable relaxed setting, than later in a
masters or PhD thesis! For example, when you first try and do quotations marks
like “this”, you’ll probably end up doing ”this”.

  2. 2.

Looking up research papers.
It is actually a non-trivial task to find research papers and preprints
online. Various publications are placed behind paywalls, and although you all
have access to most of these (in theory) via a university account login,
getting to them is never very easy. You’ll also learn about arXiv, which is
where mathematicians deposit preprints of papers before they go through the
(long and arduous) refereeing and publication process. It is not uncommon for
an arXiv preprint to go up online in year $N$, yet for the published paper to
appear in year $N+3$.

  3. 3.

Writing prose.
When was the last time you wrote an entire paragraph of text that didn’t
include any numbers, variables or subscripts? And what proportion of the
sentences that you’ve written in the last 3 years have begun with words other
than “If, then, else”? If we want to communicate mathematically-based ideas to
non-mathematicians, then we need to learn how to write prose! This is
important: we can’t send politicians or upper management lines of C++ code or
complicated formulae and expect them to extract the same meaning that we do.
We need to write for them, and they simply won’t read code or formulae.

  4. 4.

Looking into something with technical and non-technical aspects.
Some of this will involve looking at the technical details of the work of
various mathematicians; you’re probably already quite competent at that. But
you’ll also need to do things like make judgement calls, interpret the
actions, and analyse and predict things in a qualitative way (“The effects
were harmful”, without having a precise definition of “harmful”.) You’ll need
to discuss what the mathematicians did, and why it may have been a bad idea!
You’ll need to make an argument, not a proof. This is a different skill to
what is usually taught in a mathematics degree, and now is as good a time as
any to practise it. Remember, mathematics lecturers say “Here are true
things”, while lecturers in all other faculties simply say “Here are things,
and here is my argument as to why they are worth considering”.

  5. 5.

Doing research on Ethics in Mathematics.
This is hard. There are very few well-defined resources on this. So you may
find it somewhat unproductive to simply Google “Ethics in Mathematics”. You’ll
need to find and read the existing works, and then analyse them in your own
mind (think of this as rotating them to get a different/better/more useful
point of view), and then discuss them from that particular point of view.
There are no right or wrong answers here. The challenge is that you need to
find something sensible to study, and then figure out something sensible to
say about it.

####  18.1.2 Format and length

Obviously, this should be formatted in LaTeX. Which style file you use is up
to you (if you don’t know what that means, then look it up). As for length:
try not to write a phone book here. Perhaps aim for 3000–4000 words, and have
a reasonable hard upper bound (e.g. 6000 words) that prevents you from going
overboard with your ideas.

Don’t feel pressured to complete everything that is listed in an essay
description; you can chop and choose as you like. So long as you present
something resembling a complete argument (so if an essay has 3–4 parts,
perhaps you cover 1–2 of them). If you’re feeling creative, you can add/modify
part of an essay, provided it makes sense to do so.

Try and lay out your essay as you might lay out an undergraduate/masters
essay, with a title, abstract, table of contents, introduction, conclusions,
and proper (full) referencing. There are places where it may be necessary to
insert some formatted mathematics; that’s fine, and you can be the judge how
much detail to put in. A couple of lines is ok, but a page of algebra should
be relegated to an appendix (or removed entirely), as this is not a
mathematics essay!

###  18.2 Essay: Algorithmic fairness

With the rapid development and deployment of algorithmic processes in society,
many people have raised concerns about potential bias in these algorithms, and
the ways in which that bias may affect people’s lives. Some mathematicians
seem to have “come to the rescue”, creating systems that identify and measure
bias in algorithms, but this has spawned even more (technical and non-
technical) debate about how reasonable the bias-checking processes are! The
issue is complicated.

We start by looking at a particular example of algorithmic decision making:
the prison sentencing algorithm known as COMPAS. You can find a reconstruction
of the algorithm in the paper by Rudin et al. (2019) The age of secrecy and
unfairness in recidivism prediction: https://arxiv.org/pdf/1811.00731.pdf.

1\. Summarise what the COMPAS algorithm does, how it works, how it’s derived,
and how it’s applied.

The work of ProPublica was one of the first pieces to give a thorough analysis
and testing of the COMPAS algorithm. It can be downloaded from:
https://www.propublica.org/datastore/dataset/compas-recidivism-risk-score-
data-and-analysis

2\. Download the COMPAS data set and explore it. Which variables seem to be
ethically relevant and which variables were not collected but could have given
insight into ethical issues?

You could then read the original ProPublica articles:
https://www.propublica.org/article/machine-bias-risk-assessments-in-criminal-
sentencing
https://www.propublica.org/article/how-we-analyzed-the-compas-recidivism-
algorithm

3\. Compare the insights of ProPublica with your own perspective on the data
set and your own ideas on algorithmic fairness.

There are several technical, and non-technical, papers which deal with
fairness in algorithms. Here are a few that you can read over:

  * •

Cynthia Dwork et al., “Fairness Through Awareness”. Technical paper giving a
mathematical way of measuring if an algorithm is fair.

  * •

Cynthia Dwork et al., “Fairness Under Composition”. Technical paper showing
how composition of fair algorithms need not be fair.

  * •

Sorelle Friedler et al., “On the (im)possibility of Fairness”. Technical paper
showing that various mathematical definitions of “fair” are inconsistent.

  * •

Michael Skirpan, “The Authority of ‘Fair’ in Machine Learning”. Non-technical
paper discussing the fairness of trying to use an algorithm for a particular
decision-making task.

  * •

Alexandra Chouldechova et al., “The Frontiers of Fairness in Machine
Learning”. A semi-technical paper surveying existing works which discuss
algorithmic bias and fairness.

  * •

Reuben Binns, “Fairness in Machine Learning: Lessons from Political
Philosophy”. A non-technical paper discussing how mathematicians are trying to
re-invent the wheel when it comes to describing fairness, and that
philosophers have already thought about this.

4\. Try and make a summary of the algorithmic fairness debate, or at least
certain parts of it.

Remember, you are free to pick and choose whichever aspects of the essay
interest you the most.

###  18.3 Essay: Mathematicians and cryptography

In recent years the work of mathematicians in state surveillance and
cryptography has come under some degree of public scrutiny. This essay aims to
explore some of that scrutiny, and look at the arguments that mathematicians,
along with computer scientists, are making that might be changing the way
mathematicians view their own involvement in cryptographic work.

We take as our starting point the Snowden revelations of 2013. There it was
revealed that the NSA, in collaboration with many other surveillance agencies
collectively known as the “Five eyes” network, had engaged in systematic mass
digital surveillance of practically the entire internet-connected world. This
is a very very large topic, and not one that can be covered in depth over a
matter of weeks. However, the technical details are still of interest. You can
find a list of news articles here: https://github.com/iamcryptoki/snowden-
archive , and the complete document archive here:
https://search.edwardsnowden.com . Thus, the first part of the essay is:

1\. Discuss some of the technical capabilities uncovered in the Snowden
revelations, give technical details of how these were carried out, and discuss
the sorts of technically-trained people who may have done this work.

From this treasure trove of information spawned many discussions, and many
pieces of serious academic work. An early discussion piece was written by Tom
Leinster in 2014 titled “Maths spying: The quandary of working for the
spooks”, where he begins to discuss the merits and drawbacks of the
involvement of mathematicians in such work. So the second part of the essay
is:

2\. Analyse the points that Leinster raises in his piece, look at who else is
making these points, and see if you can document a “community” of vocal
objectors and their specific objections or arguments.

A much more detailed academic paper written on this was the (now very well-
known) work by Phillip Rogaway titled “The Moral Character of Cryptographic
Work” in late 2015. In it, he gives a history and overview of the role of
cryptography and cryptographers in society; past, present, and future. This is
a long piece, and covers many sub-points and case studies. So the third part
of the essay is:

3\. Read through the piece by Rogaway and summarise his main arguments, both
technical and non-technical. Perhaps choose one or two particular case studies
that he mentions, and investigate them further.

If choosing a case study from the piece by Phillip seems daunting, here is one
I have pre-selected that involves a very technical academic paper by David
Adrian et al. from 2015 titled “Imperfect forward secrecy: how Diffie-Hellman
fails in practice”.

4\. Discuss the technical aspects covered in the piece by Adrian et al., as
well as their practical consequences.

And finally, see if you can find out who the following quote is from: “The
laws of mathematics are very commendable, but the only law that applies in
Australia is the law of Australia.”

Remember, you are free to pick and choose whichever aspects of the essay
interest you the most.

###  18.4 Essay: The Ofqual grading algorithm

In mid-2020 the UK Office of Qualifications and Examinations Regulation
(Ofqual) produced an algorithm to standardise exam grades across England.
Though Ofqual had been involved with standardisation algorithms in the past,
this was the first time that all student grades had been completely predicted
by the agency, done so in response to the cancellation of all A-level exams as
a result of COVID-19. This algorithm was somewhat complicated, relied on many
assumptions, took in various data sets, and was designed to achieve certain
objectives. Ofqual have produced a handy 319 page report on the algorithm,
“Awarding GCSE, AS & A levels in summer 2020: interim report”, which can be
found here:
https://www.gov.uk/government/publications/awarding-gcse-as-a-levels-in-
summer-2020-interim-report

1\. Analyse the Ofqual algorithm and discuss what it did, how it worked, and
what it was designed to achieve. Comment on how various assumptions and
parameters were made and set, how well it reflected what it was designed to
achieve, and how reasonable those aims were.

The public reaction to the grades produced by Ofqual was one of extreme
criticism, and even the UK prime minister eventually referred to it as a
“mutant algorithm” when he announced that all grades would revert back to
those predicted for students by their teachers.

2\. Give a commentary on why the public reacted so badly to the output of
Ofqual’s algorithm. How did Ofqual defend it as “fair”, and what public
criticisms were made arguing that it was “unfair”? How accessible to the
public was their $319$ page report? What ultimately led to the UK government
abandoning the algorithm?

This algorithm was built at the request of the UK government. But ultimately,
it was designed and implemented by mathematicians, tasked with “solving” a
hard and important problem. The mathematical modelling process can sometimes
be flawed, with rather bad consequences, as outlined in Chiodo and Müller’s
Manifesto for the Responsible Development of Mathematical Works, found here:
https://arxiv.org/abs/2306.09131. In particular, this article identifies 10
steps common to most mathematical development process (and in particular,
modelling) which might lead to a problematic output.

3\. Explain what parts of the algorithm and its development process led to
such an unacceptable output. What where the problems with the assumptions and
data sets used by the mathematicians? Identify and comment on the mathematical
and non-mathematical aspects of development you see as being problematic. You
might use the 10-point list in the above manifesto as a starting point.

Ultimately, the Ofqual algorithm had no impact on the eventual grades received
by students. Thus, it achieved nothing, apart from creating a great deal of
angst for the students for a week or so.

4\. What might the mathematicians at Ofqual, and the agency as a whole, done
differently? What sorts of oversights or mistakes did they make, what
pressures and constraints were they under, and was it even possible to produce
a “reasonable” algorithm or was this outcome a forgone conclusion?

Remember, you are free to pick and choose whichever aspects of the essay
interest you the most.

###  18.5 Essay: Modelling COVID-19

In April 2020 the Virtual Forum for Knowledge Exchange in the Mathematical
Sciences (V-KEMS) hosted a 2 day study group which undertook some mathematical
modelling on how to re-open industry in UK after the national lockdown (to
prevent COVID-19 transmission) ended. At the time it was still unknown how to
safely open up shops, factories, etc, so that they could operate in a somewhat
effective manner and yet still keep COVID-19 transmission rates low. A summary
document from that study group, titled “Guiding Principles for Unlocking the
Workforce - What Can Mathematics Tell Us?” can be found here:
https://gateway.newton.ac.uk/news/2020-05-12/10198

1\. Summarise the aims and purpose of the study group, and who it comprised
of. What problems they did they approach, why did they choose those problems,
and what tools did they use to try and produce meaningful output.

Understanding and modelling a new disease is a very difficult task, especially
when one does not have precise or complete information about the spaces and
places where disease transmission is of concern, such as in workplaces.

2\. Give an outline of some of the work done in this study group, and the
preliminary results obtained. Comment on how the problems were set up and
“mathematised”, what assumptions were made, and what data was used to support
the modelling.

Often, mathematicians look to “help” in situations and scenarios where they
believe their work might be of assistance. However, their perspectives and
insights might not be sufficient to fully appreciate the problem at hand, yet
their output is given great authority as it is presented as “The mathematics
says X”. Some mathematicians will claim that “Mathematical modelling always
improves a situation or problem.” But The mathematical modelling process can
sometimes be flawed, with rather bad consequences, as outlined in Chiodo and
Müller’s Manifesto for the Responsible Development of Mathematical Works,
found here: https://arxiv.org/abs/2306.09131. In particular, this article
identifies 10 steps common to most mathematical development process (and in
particular, modelling) which might lead to a problematic output.

3\. Comment on as many potential flaws or shortcomings as you can find in this
modelling output, and/or in the process that led to it. How might the use of
this work “go wrong” and potentially lead to harm? You might use the 10-point
list in the above manifesto as a starting point.

The same organisation (V-KEMS) ran another study group in June 2020, this time
focusing on reopening UK universities in a COVID-secure manner. They produced
a working paper titled “Unlocking Higher Education Spaces - What Might
Mathematics Tell Us?”, which can be found here:
https://gateway.newton.ac.uk/event/tgmw82

4\. Summarise the objectives and output of this second report, and comment on
any potential flaws or shortcomings you see in it. How does it differ from the
first report, in terms of the issues raised in the 8-point list in the above
SIAM article? If there are any noticeable differences, then what might have
happened in the few months between the reports to effect such change?

Remember, you are free to pick and choose whichever aspects of the essay
interest you the most.

###  18.6 Essay: Medical Statistics and Probability

Retinal detachment (RD) is a serious medical emergency in which the retina
(the light-sensitive tissue at the back of the eye) separates from the eye. RD
can lead to permanent blindness if not treated quickly. Myopia (short-
sightedness) is a risk factor for RD.

An optician says: ‘Retinal detachment affects about $1$ in $10,000$ people,
but $40\%$ of RD cases happen in people with severe myopia.’

You have severe myopia. How worried should you be about RD?

Hints:

  * •

You will need supplementary information, such as Haimann et al. (1982)
(https://www.ncbi.nlm.nih.gov/pubmed/7065947)
and Kempen et al. (2004) (https://doi.org/10.1001/archopht.122.4.495 ).

  * •

The optician has actually misrepresented the findings of Haimann et al.
(1982). Can you see why?

Solution and comments: This problem gets students to further investigate what
statistics actually mean, and to see that they can be very easily
misrepresented.

The optician’s statements by themselves do not tell you anything about your
own probability of RD: you will need to know the prevalence of severe myopia
in order to calculate your own probability, using Bayes’ formula.

Haimann et al. (1982) report that $1$ in $10,000$ people is the _annual
incidence_ of RD. The lifetime risk of RD is rather higher; a crude estimate
is to multiply the annual incidence by the number of years one is alive.

The statistics reported in those papers apply to very particular populations:
for example, the subjects in Haimann et al. (1982) were all from Iowa.

Myopia depends on both genetics and environment, so more information about
incidence broken down by sex, ethnicity, age, etc. would be needed to give a
more detailed probability for an individual.

###  18.7 Essay: A topic of your choosing!

Yes, that’s right: you get to choose your own topic! This is a horribly
double-edged sword, and not for the faint-hearted. You can choose any topic or
point relating to Ethics in Mathematics that you have come across in the news,
in the literature, or that you have even been part of yourself. It really is
up to you.

There is a course given at MIT on “The Ethics and Governance of Artificial
Intelligence”, and they have a very comprehensive reading list which might
give you some starting points. It can be found here:
https://www.media.mit.edu/courses/the-ethics-and-governance-of-artificial-
intelligence/

There is also the now-popular (though somewhat devoid of technical detail)
book by Cathy O’Neil titled “Weapons of math destruction”, which you can read
through and look for inspiration.

And there is the following “Manifesto for the Responsible Development of
Mathematical Works”, where you might take one of the sub-questions and
reverse-engineer it to find examples of mathematical work where that
particular aspect has gone terribly wrong:
https://arxiv.org/abs/2306.09131

Whatever topic you choose, be sure to keep it as close to ethics in
mathematics as possible; it can be very easy and tempting to wander in to
other neighbouring disciplines such as computer science, engineering and
physics.

##  19 Handout: Why do we need Ethics in Mathematics?

Notes to Instructor: This handout is a version of a handout that we
distributed at a mathematics departmental open day for prospective students.
It is written with the intention to be fun to read and to make students aware
of where modern mathematics is used. It serves well as a leisurely
supplementary reading for students if you decide to use some of the example
sheet questions in this document.

Have you ever thought about the ethical issues that mathematicians and people
using mathematics might encounter? Have you ever thought about all those
issues that are specific and peculiar to mathematics?

Since primitive humans first began to use language and organise themselves
socially, they have also counted and used simple arithmetic. It is a way of
talking about and knowing about the world that is as much a part of what it is
to be human as anything else that we might characterise as being an essential
part of being human. Elementary mathematics – arithmetic, simple geometry, and
the logic of problem solving have always been part of human activities and
forms of knowledge of almost all human societies. But in the 21st century
things are a bit different. High-end mathematics done by highly trained
professional mathematicians is ubiquitous, and is unavoidable in a digital
world. Most of the technology around us does not just use elementary
arithmetic, algebra, and geometry – it often uses the most cutting-edge
mathematics, the products of recent research and of the most technically
sophisticated and abstract mathematics that humanity has produced.
Mathematicians seem to be taking over the world, except that nobody has
noticed. For example:

  * •

The global financial markets are run by computer programmes that use
mathematical models and mathematical analysis of financial products and
markets. Modern financial mathematics is cutting-edge, highly developed,
sometimes Nobel Prize winning and seriously hard. When the models fail, the
economic consequences can be disastrous. A lot of mathematicians go and work
for financial institutions, developing tools that come straight from research
mathematics.

  * •

The mathematics that keeps an aeroplane in the sky and brings you to your
holiday destination is breathtakingly complex. Airlines optimise over
passenger fares, fuel consumption, supply and demand of seats, and the
logistics of moving aircraft, staff, hangar space, repairs and spares, food,
fuel costs, as well as a thousand other things using computer-run optimisation
software. Doing so requires data mining, multi-variable analysis and custom-
made, sophisticated mathematical algorithms that only professional
mathematicians can construct. And when they go wrong, aeroplanes don’t fly.

  * •

Tomorrow’s weather predictions and models of global climate change both
require mathematical methods and massive multiple-equation-solving techniques
that were unimaginable in the 1950s. And, as you know, they still will get the
weather wrong when you organise a barbeque for Saturday afternoon. By the way,
supermarkets will automatically adjust their assortment of barbeque meats like
beef burgers, sausages, and chicken (and therefore the whole supply chain) in
real-time based on mathematical algorithms, data mining, and those weather
predictions. Even your Saturday afternoon barbeque relies on unbelievably
complex modern mathematics.

  * •

From the mathematics behind the physics of the new and super-bright high-
resolution flat screen on your mobile, to the way your phone signal jumps
seamlessly from mast to mast as you move around, to the CPU chip in your phone
and the encryption (or not) on your phone and who can break that encryption,
your mobile phone represents hundreds of thousands, perhaps millions of
mathematician-hours. So when you arrange your barbeque, you are doing it on
the back of a vast amount of PhD (and beyond) level mathematics. How many of
today’s undergraduates will be developing the mathematics used in the next big
high-tech thing?

A hundred years ago, even 70 years ago, mathematicians could get away with
thinking that mathematics was value free and pure and removed from any ethical
question or issues of social responsibility. But today mathematicians are
amongst the most powerful and socially significant professional groups in
society today. They play a part in everything we do (including the photocopier
that printed this page). Don’t we need to make all mathematics conscious of
their particular power and social responsibilities?

This doesn’t mean don’t do mathematics (any more than ethical issues mean you
shouldn’t do medicine, or law, or physics, or computer science, or AI …), but
it does mean we need to make everyone in the community of (aspiring)
mathematicians (that’s you, by the way) aware that mathematics, too, exists in
a socially responsible and ethical world and that there are very special
issues and problems that only mathematicians can see and think about.

Mathematics can be a great tool for good – and it is, of course; admit it: you
couldn’t live without your smartphone or laptop, or holidays in the sun, or
your pension fund. But like any powerful tool, mathematics can also be used
for ill, and there are enough examples of it around us to be very concerned.
Every professional and practicing mathematician needs to be sensitive to the
specific ethical problems raised by mathematics. Every mathematics student
needs to think about it, too.

That’s why we have designed mathematical exercises that incorporate some form
of ethical teaching, or lessons, into the mathematics. These questions might
feel a bit different at first, but we urge you to come at them with an open
mind. They are intellectually stimulating and quite challenging, and a lot of
fun!

##  20 Handout: On the interaction of science, technology and society

Notes to Instructor: This handout sets the scene and briefly discusses some of
the big shifts in societies that came from scientific revolutions. It is more
philosophical in nature and is well suited to get students thinking about how
their future profession can (and will) impact society.

Have you ever considered the big societal and technological shifts that have
happened throughout history? Have you ever considered how scientific research
and technological advancements change societies in the long-run?

Since the onset of civilisation, mankind has weathered several periods of
upheaval driven by an improved understanding of nature and the concordant
developments in technology that follow. These have had major consequences
regarding lifestyle, occupation and social structure; from the advent of
sedentary, densely populated settlements after agriculture to the explosion of
trade following the invention of the steam engine, from the need to adopt a
system of standardized currency in order to replace simple barter to the
discovery of IVF to help struggling couples with conception.

We hope the reader will not find it controversial for us to postulate that, in
our pursuit for knowledge and the search for patterns, underlying principles
and ultimate causes, science and technology have drifted towards the abstract,
and thus so have our occupations and the ways in which we add value. As
perfunctory evidence, note that development in an economy is strongly related
to the split of its output amongst the primary (agrarian), secondary
(manufacturing) and tertiary (services) sectors of industry. Technologically
underdeveloped societies are primarily agrarian, developing societies
generally focus on manufacturing and highly developed societies add value by
virtue of abstract services like finance, consulting, and insurance. Consider
the transition from Babylonian merchants offering lenders cattle as collateral
for merchandise to contract underwriters who met in coffeehouses in Amsterdam
to insure transatlantic trade in the 1500s to corporate juggernauts like AIG
who churn out insurance requests and claims by the hundreds of thousands. It
is not difficult to observe similar evolutions for other industries or point
out enormous industries that rely heavily on the development of new
technology, like energy or IT.

And we make our first observation: that of the increasing clout, influence and
power that mathematically and technically trained people are steadily gaining
in society as a result of this gradual abstractification of industry and
value. Whereas mathematicians and physicists and computer scientists used to
remain cloistered away at the research frontier in their disciplines, we are
now finding increasingly that their skills, talents and expertise are being
progressively leveraged not only in order to propel modern industry, but also
to sustain and garner fresh insight in other fields of research. While
marketing students in the ‘60s pored over portfolios and design theory, those
of today are also schooled in data science and statistics. Students of ecology
are now learning how to manipulate complex and abstract dynamical systems.
These are but a few examples; we can and will provide more.

Our second observation is easier to defend; we note that the pace at which
scientific research is taking place is accelerating, that its scope is now
breathtakingly vast, and that the specificities are only getting increasingly
technical and esoteric. It is getting steadily more difficult for researchers
and specialists to keep track of developments in even slightly orthogonal
fields to theirs. A consequence of this is the widening gap between the
cutting edge of technical research and general public understanding. As basic
evidence, consider your understanding of the inner functioning and mechanics
of the various technologies that you use over the course of your daily life
and correlate this with the age of their development. Cars and mortgages are
hard enough to understand, but their complexity couldn’t hold a candle to the
intricacy internal to an iPhone or to the Google Search Engine.

To summarise: we see an inordinate amount of power being steadily shifted to
highly technically trained people particularly in mathematics and computer
science, both by virtue of their increasing relevance and impact to industry,
but also as a result of the exacerbated isolation of their activities from
public understanding.

But, why do we care?

Each technological advance raises fresh ethical concerns about its impact on
mankind, from the debate surrounding nuclear power, the concerns of gene
splicing or the debates surrounding governmental surveillance. Historically
speaking, these concerns have been voiced both internally (within the research
community) and externally, both sides of which contribute to the erection of
systems of governance, a code of good practice or the implementation of
checks-and-balances to prevent things from going awry. This can be in the form
of governmental legislation, the existence of professional oversight bodies,
public education, and so on. We have privacy laws that forbid web-giants from
exploiting personal data; the Council for Engineers; the need for large
companies to undergo external audits; the Profession Standards Authority for
Heath and Social Care. Sociologically speaking, research and development
inculcates a shift of power, often shockingly quickly, within society, and
society often repositions itself naturally and adapts slowly, like a gigantic
organism, in order to preclude a large-scale abuse of this new-found power.

Our contention and fear is this: we are now reaching a point where this
natural societal adaptation to ensure accountability in developments in
technology is struggling to keep up. That the pace of advance is now so quick
that it can outstrip the politics that would seek to ensure that it does not
inadvertently cause harm. That the technicalities are now so obscure and
difficult that it is becoming nigh-impossible for external forces to keep
track of the ideas and mechanisms, much less the social impact that these will
cause. That these advancements and the consequent power to impact society is
now being shifted to the mathematically-trained and that the checks-and-
balances already in place to ensure ethical practice, accountability and to
promote thought and discourse vis-a-vis the moral ramifications of their work
are not functioning as effectively as they should. It is therefore of critical
importance that the technical research community is made aware of the enormous
ethical aspects that surround their work. As the mathematically and
technically trained are best placed to keep abreast of the tide, it is
critical that they play a fundamental role in any process to maintain
accountability within the community of the mathematically and technically
trained.

##  21 Handout: Further Reading

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