Objectives

In reading this chapter you will:

• ask whether animals are sentient and/or rational;

• consider whether deontologists would impose moral constraints on our use of animals;

• reflect on whether animals must be counted in the utilitarian calculus;

• consider whether animals have rights and if so what sort of rights they have;

• ask whether speciesism is comparable to racism and sexism;

• reflect on whether animals are good experimental models for human beings;

• reflect on how the moral status of animals impacts on our use of them for research purposes;

• consider how animal rights activism affects the public perception of the use of animals in scientific research.

Factual information: The use of mice for research in the UK

From the 1970s until the 2000s there was a steady decline in the use of animals inresearch, possibly due to the activities of animal rights activists. The boom in geneticresearch reversed this decline.

Mice are particularly valuable for research because 99% of their 30,000 genes havedirect counterparts in the human genome.

By 2003 there were already 3,000 GM strains of mice. It has been predicted therewill be 300,000 by 2020. Hundreds of mice are needed to produce each, though mostof the mice are killed because they do not develop the needed mutation.

In the UK it is estimated that several million mice are used annually in geneticresearch. Some have suggested that mice are now little more than ‘catalogue entries’like paperclips or biros to be routinely ordered for the proper functioning of theinstitution.

Objectives

In reading this chapter you will:

• consider whether or not something that is unnatural is also immoral;

• reflect on the arguments for and against this claim;

• consider whether morality is a matter of emotion or feeling rather than reason;

• reflect on the reasons for and against thinking that our intuitions are a good guide to morality;

• consider why many people think that it is immoral to take risks;

• reflect on the extent to which it is immoral to take risks;

• reflect on the common belief that morality is all a matter of opinion;

• consider the extent to which morality is a matter of opinion.

In this section we shall consider those arguments that we will come across again andagain as we work through this book. There are four such arguments: (i) it’s notnatural; (ii) it’s disgusting; (iii) it’s too risky; and (iv) it’s a matter of opinion. Thesegeneral arguments are intuitively attractive, they often underpin discussions aboutethics in the media and they almost certainly feature in your own ethical thinking. Inreading this part of the introduction you may find yourself questioning some of thethings you believe to be obviously true. Questioning such assumptions is an import-ant part of thinking critically.

Objectives

In reading this chapter you will:

• learn to distinguish ethical issues from social issues;

• reflect on the requirements for the smooth running of society;

• consider the nature of social decision-making;

• learn to distinguish the moral and the legal;

• consider the principles that govern just societies;

• briefly consider the nature of political authority.

In thinking ethically we are trying to decide which actions are right and wrong, whichactions we should or shouldn’t perform. But no man is an island, and the decisions wemake about howtoactmust bemade inthe context of the lawsof the land in which we live.Some of themost important ethical decisions, therefore, are not primarily decisions abouthowindividuals should or shouldn’t act, but rather decisions about whether a given action:

• should or shouldn’t be illegal

Nearly every country in the world has made it illegal to clone a human being forreproductive purposes. Even if an individual believes that human cloning is morallyacceptable, therefore, he cannot rationally clone a human being without taking intoaccount the fact it is illegal and that the state will punish him if it discovers what he isdoing. (We shall be considering reproductive cloning in Chapter 8.)

• should or shouldn’t be regulated by law

In Britain and in some US states (e.g. Rhode Island, California and New Jersey) it islegal to clone a human being as far as the blastocyst stage of embryo development forthe purposes of research (so-called ‘therapeutic’ cloning). Anyone wanting to clone ahuman being for such purposes, however, must jump through the myriad hoops bywhich such activities are regulated by the law. They will, for example, in the UK, needa licence from the Human Fertilisation and Embryology Authority (HFEA), whosejob it is to subject requests for licences to close examination, then they will need to obey the various regulations governing the activity itself, then finally they will have todestroy the clone by the 14th day. (We shall be considering therapeutic cloning inChapter 7.)

Objectives

In reading this chapter you will:

• reflect on the claims of those who promise a significant increase in longevity;

• consider whether you would like to live for 1000 (or even 150) years;

• examine the social consequences of everyone's having the choice of living much longer;

• examine the moral consequences of a significant increase in lifespan;

• consider whether a normal lifespan is part of what makes us human;

• ask yourself whether normal humans and ‘immortal’ humans could belong to the same species.

Having considered the ethical and social issues generated by biotechnology at thebeginning of life, we shall now turn to the issues that arise at the end of life. In thischapter we shall consider the issues generated by the possibility of significantlyextending our lives.

The possibility of ‘eternal’ life

If someone were to offer you the gift of eternal life, would you cry ‘let me at it!’ orwould you be suspicious?

It wouldn’t be unreasonable to be suspicious. Since the beginning of time peoplehave sought the fountain of youth, and so far all have been disappointed. Quite afew scientists, however, believe the gift of all-but-eternal life is within our grasp notthrough magical potions but through genetic know-how.

The bio-gerontologist Aubrey de Grey of Cambridge University believes we are onthe very verge of conquering death.He believes the first ‘immortal’ might already be 60.5Once we conquer death, according to de Grey, we will live on until, inevitably, we have anaccident. He calculates (by extrapolating from teenage lifespans and accident ratesgenerally) that the average lifespan will then be 1,000 plus. Not quite immortality,but close. Perhaps if you were very risk-adverse you could achieve the real thing.

Objectives

In reading this chapter you will:

• learn about a major cause of infertility;

• reflect on the reasons for the shortage of donor gametes;

• consider what it would be like to have no idea who your parents are;

• reflect on ways to alleviate the gamete shortage;

• ask whether we should use eggs from aborted foetuses to alleviate the egg shortage;

• ask whether it is right to use artificial sperm to alleviate the sperm shortage;

• consider whether paying for reproductive resources is a morally unacceptable ‘commodification’ of the human body;

• reflect on artificial wombs and the extent to which they will relieve women of the burden of carrying babies to term.

A major cause of infertility is the inability to produce fertile gametes. Sincewomen have been having babies later this type of fertility has increased. Manymen have low sperm counts, or even no sperm at all and sperm counts quitegenerally seem to be decreasing. Sometimes the problems are not with gametesbut with difficulty in providing a womb hospitable to a developing foetus. Formany sub-fertile people, therefore, their chance of a child still depends on theirability to secure fertile gametes, and/or a hospitable womb. Securing these‘resources’ is not easy.

In this chapter we shall consider the ethical and social issues emerging from thedemand for and supply of gametes and of hospitable wombs. We shall start byconsidering gamete donation then turn to surrogacy.

We now turn our attention to special types of functions between vector spaces known as linear transformations. We will look at the matrix representations of linear transfor mations between Euclidean vector spaces, and discuss the concept of similarity of matrices. These ideas will then be employed to investigate change of basis and change of coordinates. This material provides the fundamental theoretical underpinning for the technique of diagonalisation, which has many applications, as we shall see later.

7.1 Linear transformations

A function from one vector space V to a vector space W is a rule which assigns to every vector v ∈ V a unique vector w ∈ W. If this function between vector spaces is linear, then it is known as a linear transformation, (or linear mapping or linear function).

Definition 7 .1 (Linear transformation) Suppose that V and W are vector spaces. A function T : V → W is linear if for all u, v ∈ V and all α ∈ ℝ:

1. T(u + v) = T(u) + T(v), and

2. T(αu) = αT(u).

A linear transformation is a linear function between vector spaces.

A linear transformation of a vector space V to itself, T : V → V is often known as a linear operator.

In this chapter, we look at orthogonal diagonalisation, a special form of diagonalisation for real symmetric matrices. This has some useful applications: to quadratic forms, in particular.

11.1 Orthogonal diagonalisation of symmetric matrices

Recall that a square matrix A = (ai j) is symmetric if AT = A. Equivalently, A is symmetric if ai j = aji for all i, j ; that is, if the entries in opposite positions relative to the main diagonal are equal. It turns out that symmetric matrices are always diagonalisable. They are, furthermore, diagonalisable in a special way.

11.1.1 Orthogonal diagonalisation

We knowwhat itmeans to diagonalise a square matrix A. Itmeans to find an invertible matrix P and a diagonal matrix D such that P-1 A P = D. If, in addition,we can find an orthogonal matrix P which diagonalises A, so that P-1 AP = PT A P = D, then this is orthogonal diagonalisation.

Definition 11.1 A matrix A is said to be orthogonally diagonalisable if there is an orthogonal matrix P such that PT AP = D where D is a diagonal matrix.

As P is orthogonal, PT = P-1, so PT A P = P-1 A P = D. The fact that A is diagonalisable means that the columns of P are a basis of ℝn consisting of eigenvectors of A (Theorem 8.22). The fact that A is orthogonally diagonalisable means that the columns of P are an orthonormal basis of ℝn consisting of an orthonormal set of eigenvectors of A (Theorem 10.21).

In this short chapter, we aim to extend and consolidate what we have learned so far about systems of equations and matrices, and tie together many of the results of the previous chapters. We will intersperse an overview of the previous two chapters with two new concepts, the rank of a matrix and the range of a matrix.

This chapter will serve as a synthesis of what we have learned so far, in anticipation of a return to these topics later.

4.1 The rank of a matrix

4.1.1 The definition of rank

Any matrix A can be reduced to a matrix in reduced row echelon form by elementary row operations. You just have to follow the algorithm and you will obtain first a row-equivalent matrix which is in row echelon form, and then, continuing with the algorithm, a row-equivalent matrix in reduced row echelon form (see Section 3 .1.2). Another way to say this is:

• Any matrix A is row-equivalent to a matrix in reduced row echelon form.

There are several ways of defining the rank of a matrix, and we shall meet some other (more sophisticated) ways later. All are equivalent. We begin with the following definition:

Definition 4.1 (Rank of a matrix) The rank, rank(A), of a matrix A is the number of non-zero rows in a row echelon matrix obtained from A by elementary row operations.

Linear algebra is one of the core topics studied at university level by students on many different types of degree programme. Alongside calculus, it provides the framework for mathematical modelling in many diverse areas. This text sets out to introduce and explain linear algebra to students from any discipline. It covers all the material that would be expected to be in most first-year university courses in the subject, together with some more advanced material that would normally be taught later.

The book has drawn on our extensive experience over a number of years in teaching first- and second-year linear algebra to LSE undergraduates and in providing self-study material for students studying at a distance. This text represents our best effort at distilling from our experience what it is that we think works best in helping students not only to do linear algebra, but to understand it. We regard understanding as essential. ‘Understanding’ is not some fanciful intangible, to be dismissed because it does not constitute a ‘demonstrable learning outcome’: it is at the heart of what higher education (rather than merely more education) is about. Linear algebra is a coherent, and beautiful, part of mathematics: manipulation of matrices and vectors leads, with a dash of abstraction, to the underlying concepts of vector spaces and linear transformations, in which contexts the more mechanical, manipulative, aspects of the subject make sense.