Sections :
When one learns mathematics, most of what one gains is knowledge how
to do something: how to solve a quadratic equation, how to prove a
trigonometric identity, how to find the area between the graphs of two
functions, and so on.
To justify a claim that one knows how to do one of
these things requires (i) a knowledge that certain truths
hold, i.e. a knowledge of mathematical laws or rules, together with (ii) the
ability to think logically.
Exercise 0.1
Someone claims that they know how to solve the quadratic equation x2  x  6 = 0 . To justify that claim, they could say that they know that ...
a.x2  x  6 = (x _____ )(x ______ ) , which can be checked using the distributive and the commutative law, (i.e. by 'multiplying out'.)
b.If a = b and c = b , then ___ = ___ , and therefore (x ______ )(x ______ ) = 0 .
c.If a b = 0 , then a = ___ or b = ___ , and therefore (x ______ ) = 0 or (x ______ ) = 0 .
d.If a + b = 0 , then a = ___ , and therefore x = 2 or x = ___ .
Can you think of other ways in which the person might justify that claim?
Note that in part a. of the Exercise, the factorization of the quadratic
expression does not follow from the distributive and the
commutative laws: it has to be 'invented', though its validity is then checked
by using those laws. (There is of course another method for solving quadratic
equations, 'completing the square', which does not require factorization  but
then that method too had to be thought of to start with, or invented.)
Thus, contrary to a common misconception, doing
mathematics requires not only (i) mathematical knowledge and (ii) the ability
to think logically, but also (iii) the ingenuity to put that knowledge to use,
to make up new arguments, to invent methods and answers: doing mathematics
often involves experimenting or proceeding by trialanderror, using one's
imagination or intuition, and so on  even if in the finished 'product', like
the solution of an equation or the proof of a theorem, these processes may not
be apparent. This need for ingenuity is the reason, of course, that people 'get
stuck' when they do mathematics.
In a certain sense, mathematics has been advanced most by those who are distinguished more by intuition than for rigorous methods of proof.
Felix Klein (1849 1925)
Our main concern in this section will of course with the basis on which we
can claim to know mathematical truths.
But to start with, we shall take a very brief look
at the two main areas into which the subject is sometimes divided.
The distinction is often made between pure and applied maths, though it would be an oversimplification to think of mathematics as divided in two.
Pure maths investigates abstract, though not necessarily 'useless', structures and the foundations of the subject. It includes areas such as set theory, theory of algebraic structures, number theory, geometry, analysis, graph theory.
Example 1:
A famous conjecture by Fermat (160165,) which has
only been proved in the last couple of years, states that if n is a natural
number such that n > 2, then there are no nonzero integers x, y, z such
that xn + yn = zn.
Example
2:
The FourColour Theorem, which has been proved only recently as well, states
that four colours will be sufficient to colour the countries on any plane map
such that no two adjacent countries have the same colour.
An important concept in pure maths is that of an isomorphism (from Gk. iso + morphé, equal shape,) when different structures agree in such a way that corresponding elements 'behave' the same way.
Example 3:
The set of equations of the form ax + by = c and the set
of straight lines in the Cartesian plane are isomorphic, so that we can find
the intersection of two lines by solving their equations simultaneously, and
two lines are parallel if the corresponding equations have the same ratio a/b.
Exercise 1.1
a.Give examples of nonzero integers x, y, z, such that x2 + y2 = z2.  What role do such number triplets play in geometry, and what are they called?
b.Draw a simple map to show that a 'ThreeColour Theorem' cannot hold.
c.Considering points, linesegments and squares as 'cubes' of 0, 1 and 2 dimensions, count, for n = 0, 1, 2 and 3,
i.the number of vertices of an ndimensional 'cube',
ii.the number of (n 1)dimensional sides, or 'faces', and
iii.the number of edges.
Try to deduce the numbers of vertices, of sides and of edges of a 4dimensional cube  which we can of course not imagine, but can work with mathematically.



In applied maths, theories are developed to serve as 'tools' for solving
problems which arise in other areas, such as physics, engineering or economics.
As a first step, this always requires the setting up a suitable model of the situation, i.e. a simplified representation or description, by making certain assumptions about the given situation and abstracting from some of its details.
Example 4:
In queuing theory, which models situations of waiting in
line for service, it can be shown, for instance, that when two 'phones are in
one place, the average waiting time is much less than when they are in
different locations.
Example
5:
Often the equations that describe a situation in physics, such as the movement
of three bodies under the gravitational forces they exert on each other, can be
set up easily enough, but can then not be solved in general.
Exercise 1.2
Based on your experience of different parts of the subject, what links do you think pure and applied maths may have?
In the natural sciences, as we saw, we say that we know a scientific law,
such as the law of gravitation, because even though it may initially have been
an invented hypothesis, the predictions we have deduced from it have been
thoroughly tested in experiments, which have corroborated it; and because it is
always open to further such testing.
Mathematics is not in this simple sense empirical,
i.e. based on sense experience. Instead we can say that we know a mathematical
truth if we know how to prove it, (which may require experience of proving
things mathematically: but this is not the same as being based on sense
experience.)
There are different methods of proof, classified according to the most conspicuous inference, such as
proof by logical deduction:
a sequence of propositions, each of which must either have been previously proved (or be evident without proof,) or follow by a valid logical argument from earlier propositions in the proof, such that the last proposition is what is to be proved.
proof by construction:
the objects asserted to exist are explicitly exhibited or constructed ( which is not actually necessary to just show that they exist.)
proof by (complete or mathematical) induction:
in contrast to the inductive reasoning in science, this does guarantee the truth of the conclusion; it goes as follows: if a proposition p(n) depending on a natural number n is true for some first number m, and we can show that whenever p(n) is true then so is p(n+1), then p(n) is true for all n ³ m.
proof by contradiction:
this works by assuming the negation of what one is
trying to prove and deriving a contradiction; so the logical form of the
argument is: ( ( ¬p => q ) & ¬q ) => p ;
(this is a variation of the modus tollens argument, as we saw in the
section on Logic and Reasoning.)
A proof by contradiction is also called indirect, while the others are
direct, and there is, or was, a school of mathematicians  called
Intuitionists  who 'don't approve' of such indirect proofs ...
Note that methods of proof cannot be classified and
catalogued easily, that there are other methods, and that many results can be
proved by different methods.
Exercise 2.1
Try to prove each of the following propositions by the method indicated.
a.2 x2  3 x = 0 has a solution between 1 and 2, (by construction.)
b.There is no smallest positive number, (by contradiction.)
c.n2 ³ 2n for all n ³ 2, (by induction.)
d.2 x2  7 x + 9 / x = 0 has a solution between 1 and 2, (by deduction: clearly state all premisses of your argument.)
Just as a valid logical argument must be such that whenever the premisses are true then so is the conclusion, a valid method of proof must be such that it always works.
Exercise 2.2
Contrary to the practice of many IB Mathematics students, the method of the following 'proofs' is not valid, (even when  unlike in the cases below  true conclusions are arrived at.) Describe the method clearly, and state which logical fallacy is committed. The conclusions of b. and c. are clearly false; give an example of an angle P in a. for which the conclusion does not hold.
a. 
to prove: 
b. 
to prove: 
c. 
to prove: 
square both sides: 
multiply by 1: 
square both sides: 

sin2 P + cos2 P = 1 true! 
adding: 0 = 0 true! 
so 1 = 1 true! 

hence: 
hence: 5 = 7 
hence: 1 = 1 
If we say that knowing a mathematical truth requires that we be able to
prove it, then what can we base our proofs on?
We can agree that in mathematical proofs valid
logical arguments are used, and perhaps also certain very basic notions, e.g.
that a part cannot be greater than the whole. But it is not possible to forever
keep returning to statements that have been previously proved, so mathematical
proofs must be representable as consisting of inferences from statements
assumed without proof. Such statements are called 'axioms' (from Gk. axioun, to consider worthy.)
The reduction of a given subject matter or theory
to a set of fewer basic propositions is called an axiomatization. An example
going back to antiquity is Euclidean geometry: in his famous Elements,
Euclid, in the 3rd century B.C., stated five axioms, which were considered
selfevident, and from these could be derived all the theorems of plane
geometry. Here are some examples of axioms, (though in a modern formulation,)
and a typical theorem:
A1 
Given any two points, P and Q, there is precisely one straight line l on which they both lie. 
A2 
Given any straight line l, there is some point P which does not lie on line l. 
A5 
Given any straight line l and any point P which does not lie on l, there is precisely one straight line l' such that P lies on l', and l and l' have no common point. (l' is called the parallel line to l through P.) 
T 
All straight lines perpendicular to a given line are parallel. 
Similarly there are axiomatizations of set theory (Georg Cantor, 1845
1918) and of the theory of natural numbers (Giuseppe Peano, 1858 1932.)
Of a set of axioms we can require more than that
each axiom be selfevident: the axioms should be
(Note that the axiomatization of a theory need not be unique, i.e. there can be different sets of axioms from which the theory can be derived: the axioms of one set are then theorems which can be proved from the axioms of the other. Thus we don't nowadays use Euclid's original five axioms as the basis of Euclidean geometry.)
We can now formulate the first of a series of views of what mathematical knowledge consists of:
First View: 
Knowing some mathematical truth requires that we be able to prove it from a set of axioms; these axioms are selfevident in the sense that we can have immediate knowledge of them and do not need to justify them further. 
However, in the middle of the 19th century the question was raised whether
axioms needed to be selfevident, for it was found that perfectly good theories
could be derived from sets of axioms which were not.
In particular Euclid's 5th axiom, A5 above, being the only one which concerned the 'global' rather than just the 'local' behaviour of the geometrical objects, had given rise to much unease amongst mathematicians; and it was found that if it was replaced by an alternative, such as A5' below, then a perfectly good nonEuclidean geometry could be derived, in which very different theorems can be derived:
A5' 
Given any straight line l and any point P which does not lie on l, there is no straight line l' such that P lies on l', and l and l' have no common point. 
T1' 
All straight lines perpendicular to a given straight line meet in one point. 
T2' 
Any two distinct straight lines enclose an area. 
It is clear that the meaning of some of the terms, such as ''straight
line'', cannot be the same here as the usual, intuitive one in Euclidean
geometry. But if we interpret ''straight line'' as referring to a great circle,
then this nonEuclidean geometry will be the geometry of the surface of a
sphere (Georg Riemann, 1826 1866.)
While someone might object that there is no reason
for studying such geometries, instead of one (Euclidean) geometry, it has in
fact turned out that physical space, on the large scale, has a nonEuclidean
geometry even though locally it is Euclidean.
Whereas previously the basic terms of geometry,
such as ''straight line'', had been considered intuitively obvious and the
axioms selfevident, the meanings of the basic terms are now defined,
implicitly, by how they are used in the axioms and the theorems derived from
them. This 'formalist' view goes back to David Hilbert (1862 1943,) who is
said to have commented, jokingly, that in his geometrical axioms
One must always be able to replace the words ''points'', ''straight lines'' and ''planes'' by ''tables'', ''chairs'' and ''beer mugs''.
The subject matter of mathematics is ... the concrete symbols themselves whose structure is immediately clear and recognizable. The formal conception of numbers ... does not ask what numbers are ..., but rather what is demanded of them in arithmetic. For the formalist, arithmetic is a game with signs, which are called empty. That means they have no other content (in the calculating game) than they are assigned by their behaviour with respect to certain rules of combination (rules of the game). The chess player makes similar use of his pieces; he assigns them certain properties determining their behaviour in the game, and the pieces are only external signs of this behaviour.
J. Thomae, 1898.
Exercise 3.1
To understand the formalist approach, and how terms can get meaning by the way they are used, consider the formal system K which is defined by the four axioms A1  A4.
A1 
Some DERs are KINDERs and some DERs are TENDERs, but no DER is both a KINDER and a TENDER. 
A2 
The result of GARring any number of DERs is a DER, and this does not depend on the order of the DERs. 
A3 
When two KINDERs or two TENDERs are GARred, the result is a KINDER. 
A4 
When a KINDER and a TENDER are GARred the result is a TENDER. 
a.Which of the following, if any, are theorems of the Ksystem? Try to derive any theorems from the four axioms above.
P1 
When three TENDERs are GARred, the result is a TENDER. 
P2 
When three DERs are GARred, the result is a TENDER. 
P3 
Every DER is either a KINDER or a TENDER. 
b.What would we say about the Ksystem if it included P1, P2 or P3 as a fifth axiom, A5?
c.Try to give an intuitive mathematical interpretation of the Ksystem.
With this new understanding of axioms, we can now formulate the second view of what mathematical knowledge consists of:
Second View: 
Knowing some mathematical truth requires that we be able to prove it in its theory from the set of axioms which determines the theory; the terms used to express our knowledge have their meaning by virtue of and in the context of that theory (Postulationism.) 
But even if mathematics can be thought of as a game, in which sets of axioms are invented and the theories that can be derived from them are investigated, some of these theories are of course useful:
Consider the way in which a great deal of mathematical thinking is actually done. The mathematician does not ask whether his constructions are applicable, whether they correspond to any constructions in the natural world. He simply goes ahead and invents mathematical forms, asking only that they be consistent with themselves, with their own postulates. But every now and then it subsequently turns out that these forms can be correlated, like clocks, with other natural processes.
Alan Watts.
Euclidean geometry, for instance, represents well our everyday experience of the space we live in, but a nonEuclidean geometry is needed in physics to describe the largescale structure of space.
The view expressed at the end of the last section is of mathematics as a
game, in which sets of axioms are developed into theories, just for their
interest, though some of the theories then turn out to be of practical use.
Many mathematicians in fact view their subject in
this way. However, for mathematical knowledge to be ultimately justifiable in
this way would require some guarantee that our mathematical theories are
'solid': in particular, we must require the sets of axioms to be (i)
consistent, i.e. free from internal contradiction, and (ii) complete, i.e. such
that all the theorems of a theory can be completely derived from the axioms.
Unfortunately for the formalist view  that mathematics is a game, albeit an at times useful one  Kurt Gödel (1906 1978) was able to show
Gödel's proof is not easy to follow, but in the end it hinges on something
like the liar's paradox known since antiquity  Epimenides was a Cretan who is
still famous for his statement: ''All Cretans are liars.''
That the consistency of mathematical theories, even
ones that have become well established, cannot be taken for granted was
demonstrated very clearly in 1902, when Bertrand Russell (1872 1970)
discovered a paradox (or 'antinomy') in Cantor's set theory, which had come to
play a fundamental role in mathematics.
In a library there are many books, some of which are catalogues of books or even catalogues of catalogues. And a catalogue may well list itself, too. Now consider the catalogue of all catalogues that do not list themselves: does it list itself or not?
The discovery of this paradox, hidden in a well established branch of their subject, came as a major shock to the mathematical community. Here are some other, related paradoxes.
Exercise 4.1
Investigate the following situations.
a.''The next sentence is false. The previous sentence is true.''
b.Adjectives in English can be divided into 'autological' ones, i.e. ones that are selfdescriptive, such as ''pentasyllabic'', ''awkwardnessful'' and ''recherché'', and 'heterological' ones, i.e. ones that are not, like ''edible'', ''incomplete'', ''bisyllabic''. Now, is ''heterological'' heterological? (Grelling's paradox)
c.The barber in a certain village is a man who shaves all men who do not shave themselves. Does the barber shave himself or not?
d.If it is true that ''the class of integers which can be expressed in less than 16 words is finite,'' then there must be a ''smallest integer not definable in less than 16 words''  the definition of which has only ten, and therefore less than 16, words! (Berry, Russell)
It seems therefore, that axioms do not provide a 'solid' enough foundation
on which to build our mathematics: if we cannot show them to be free from
contradiction, we cannot use them to guarantee truth, even within a theory. How
then can we justify any claims to mathematical knowledge?
One answer, put forward by W. V. O. Quine and
Hilary Putnam, amongst others, comes from basing mathematical knowledge
ultimately on facts of experience, like scientific knowledge.
The question of how
mathematics and science are related was of great concern to Albert Einstein
(1879 1955):
Here arises a puzzle that has disturbed
scientists of all periods. How is it possible that mathematics, a product of human
thought that is independent of experience, fits so excellently the objects of
physical reality? Can human reason without experience discover by pure thinking
properties of real things? ...
As far as the propositions of mathematics refer to
reality they are not certain; and as far as they are certain they do not refer
to reality. ... But it is, on the other hand, certain that mathematics in
general and geometry in particular owe their existence to our need to learn
something about the properties of real objects.
But Einstein would probably not have been happy with the position taken by
Quine, which is as follows:
When we
discussed science, we saw that we can never test an hypothesis in isolation: a
prediction is always derived from the hypothesis H to be tested in
conjunction with some 'auxiliary hypotheses'. So if such a prediction
turns out to be incorrect, it may actually be one of the auxiliary hypotheses
that has thereby been disproved rather than H. But conversely, if the prediction
was correct, not only has the hypothesis H been corroborated, but so have the
auxiliary hypotheses involved in making the prediction.
Certain of these auxiliary hypotheses are very general, and are tested in a wide range of experiments performed to test specific hypotheses. Thus any prediction in chemistry requires us to assume that matter does not simply disappear: the law of conversation of mass is one of these very general auxiliary hypotheses, and every time an experiment turns out as predicted, this law has been corroborated as well. Even our everyday 'predictions' of the outcomes of 'experiments', as when we are cooking, corroborate the law  so we are very sure of it.
Exercise 4.2
To see what such very general auxiliary hypotheses are, decide’what assumptions you would be prepared to reject if the following things happened, and which ones you wouldn't.
a.Having put together two amoebas and two amoebas in a petri dish, you then count five amoebas. (Is therefore 2 + 2 = 5 ?)
b.You put together two apples and two apples and later find that there are only three apples. (Is therefore 2 + 2 = 3 ?)
c.Mixing 1 pint of a liquid with 1 pint of another liquid gives you a volume of 1.9 pints.
d.A and B are twins. A goes on a space trip far away while B stays behind. When A returns she is considerably younger than B.
Now, according to Quine and others, some mathematical propositions are just
such very general auxiliary hypotheses: every time we use some mathematics in
physics, chemistry, and so on, or just in everyday life, that mathematics is
tested, and will be corroborated. And the 'useless' parts of mathematics, which
cannot be so 'tested', are the extensions  using the same rules of
derivation, and so on  of the 'useful' parts that have been 'tested'. Some of
these extensions, however, such as the theory of infinite sets, are far enough
from what we can know on an empirical basis to be dubious and open.
This has also been put as follows, and can be
formulated as our third and final view of what mathematical knowledge consists
of.
we regard our knowledge of basic principles [of mathematics] as resting on the widespread application of such principles in theories from the natural or social sciences which are in turn confirmed via sensory observations. In other words, ... basic mathematical principles are known via an inferential process which accords well with the hypotheticodeductive patterns [of the sciences.]
Hugh Lehman.
Why do we not confess that mathematics, like other sciences, is ultimately based upon and has to be tested in practice?
Lazlo Kalmar.
Final View: 
Knowing some mathematical truth requires that we be able to prove it in its theory; what constitutes a proof, and the foundations of some theories, have been tested, in conjunction with specific hypotheses, in science, and often in everyday life. Our knowledge of other parts of mathematics is less conclusive the further it is from what has been so tested. 
Exercise 4.3
There are various points at which a relation between mathematics and reality can give us a surprise:
Count in a
variety of tables and lists  in the Encyclopaedia Britannica,
say, or geography books, balance sheets in economics, astronomy books, etc. 
how many numbers start with each of the digits 1, 2, and so on, up to 9,
(excluding any that follow a definite pattern, such as dates and page numbers.)
Try to explain why the distribution is not even,
i.e. why the proportion for each digit is not 1/9, as one might have expected
it to be. (The pattern is called Benford's Law.)
Discussion:
It has been said, often unthinkingly, that ''mathematics is a language.'' By considering carefully how a language functions, what we use it for, how we learn it, and so on, find ways in which mathematics is, and is not, like a language.
EXERCISE 1.1.:
vertices 
faces 
edges 

point: 
1 
0 
0 
line segment: 
2 
2 
1 
square: 
4 
4 
4 
cube: 
8 
6 
12 
4D cube: 
16 
8 
32 
d.Even though we cannot imagine a 4D cube, we can draw a projection of a wire model of one in 2D, just as we can draw the projection of a wire model of a 3D cube:
(This exercise works even more easily with the following sequence: point, line segment, triangle, tetrahedron, and 4D tetrahedron.)
2. Mathematical Proofs :EXERCISE 2.2.:
This method assumes the proposition to be proved
and deduces from it a proposition which has previously been proved (or is
evident without proof.)
The form of the argument is: ( ( p => q ) & q ) => p ( which
is a case of 'affirming
the consequent', a logical fallacy.)
In a., if one starts with sin2 P + cos2 P = 1 , as one
should, what one can in fact deduce is that cos P = ± Ö(1  sin2 P) .
EXERCISE 3.1.:
EXERCISE 4.2.:
EXERCISE 4.3.:
While one might expect the distribution to be even, i.e. each digit to occur in 1/9 = 11.1 % of the cases, it will in fact be found to be very uneven and something close to the following:
first digit 
1 
2 
3 
4 
5 
6 
7 
8 
9 
proportion (%) 
30.1 
17.6 
12.5 
9.7 
7.9 
6.7 
5.8 
5.1 
4.6 
One explanation  the only one I have been able to
come up with ...  starts from the requirement that the distribution must be
such as to be independent of the units of measurement used: whether we measure
population by counting the individuals, say, or the number of their limbs, or
measure area in m2 or in m2/2.
It can then be shown that the proportion of numbers starting with the digit n
is p(n) = log (n+1)  log n .
Walter R. Fuchs, Modern Mathematics, 1966. Weidenfeld and Nicolson.
Behnke et al. (editors), Fischer Lexikon Mathematik, 2 vol.s, 1964. Fischer Bücherei.
Douglas R. Hofstadter, Gödel, Escher, Bach, 1979. Vintage Books.
Hugh Lehman, Introduction to the Philosophy of Mathematics, 1979. Basil Blackwell.
Andrew Maclehose, Th.o.K. Notes on Mathematics.