IB Theory of Knowledge

1. Mathematics

2. Mathematics
Mathematics, rightly viewed, possesses not only truth, but
supreme beauty: a beauty cold and austere, like that of
sculpture, without appeal to any part of our weaker nature,
without the gorgeous trappings of painting or music, yet
sublimely pure, and capable of a stern perfection such as
only the greatest art can show. The true spirit of delight,
the exaltation, the sense of being more than Man, which
is the touchstone of the highest excellence, is to be found
in mathematics as surely as in poetry.
BERTRAND RUSSELL, Study of Mathematics

3. • A truel is similar to a duel, except there are three participants
rather than two. One morning Mr Black, Mr Grey and Mr
White decide to resolve a conflict by truelling with pistols until
only one of them survives.
• Mr Black is the worst shot hitting his target on average only
one time in three. Mr Grey is a better shot, hitting his target
two times out of three. Mr White is the best shot, hitting his
target every time.
• To make the truel fairer Mr Black is allowed to shoot first,
followed by Mr Grey (if he is still alive), followed by Mr White
(if he is still alive) and round again until only one of them is
alive.
• The question is this: Where should Mr Black aim his first shot?
A Truel

4. What is Maths?
• ‘Mathematics is the science of quantity’ (Aristotle)
• ‘Mathematics is the language with which God wrote the
Universe’ - Galileo
• ‘Mathematics is the science of indirect measurement’
(Auguste Compte – French philiosopher, 1851)
• ‘All mathematics is symbolic logic’ – Bertrand Russell, 1903
• ‘Mathematics is the study of all possible patterns’ Walter
Warwick Sawyer – American mathematician, 1955

5. What is Maths?
• In pairs, you will be
assigned one of these topics
to research.
• You have 20 mins to make
some quick notes on the
mathematical knowledge,
and the origin (time/place)
• 10 mins for the class to
report back a very brief
summary of findings.
• Abacus and
Soroban
• Pythagoras
• Algebra
• Omar Khayyam
• Chaos Theory
• Ramanujan
• Calculus
• Zero

6. What is Maths?
• Maths appears to be the area of knowledge
which gives us greatest certainty
• It developed from humans’ innate ability to
reason
• You might therefore think of it as the ‘purest’
from of knowledge
• It allows us to see order in the Universe and
explain it in a kind of language
• In doing so it also has value in any area of natural
science
http://edrontheoryofknowledge.blogspot.mx/2014/05/eugene-wigner.html

7. Euclid
• Euclid was a Greek mathematician who died
around 300 BC
• He is often referred to as ‘The Father of
Mathematics’ and the inventor of geometry
• He developed a model of maths we now refer to
as the formal model:
• It consists of:
– Axioms
– Deductive reasoning
– Theorems (and proofs)

8. Axioms
• These are the starting points
• They are the things which we think we don’t need to prove
as they can be assumed to be self-evident
• If there was no starting point to maths we would be lost in
a chain of infinite regress (proving the proof of the proof of
the proof…)
• These are Euclid’s 5 original axioms of geometry:
– It shall be possible to draw a straight line to draw any two point
– A finite straight line may be extended without limit in either
direction
– It shall be possible to draw a circle with a given centre and
through a given point
– All right angles are equal to one another
– There is just one straight line through a given point which is
parallel to a given line

9. Deductive Reasoning
• This is reasoning in which we go from the
general to the specific:
• All ostriches are birds
• Carlos is an ostrich
• Carlos is a bird
• In maths, the premises are the axioms and the
conclusion is a theorem
• Therefore, maths is deductive rather than inductive in
nature
Premise 1
Premise 2
Conclusion

10. Theorems
• A mathematical theorem is a statement that
has been proven based on other accepted
statements (other pre-existing theorems or
axioms)
• However in a mathematical sense, the word
proof is difficult to define

11. Mathematical Proof
• A theorem has been reached by a series of logical steps that
all mathematicians can agree on
• A conjecture is unproven hypothesis that just seems to work
(the word was coined by philosopher Karl Popper)
• A conjecture does not constitute proof
• This is Goldbach’s Conjecture:
• Go through the first 20 even numbers and see if this is true?
• If it is true for the first 20 even numbers, have you proven
the conjecture?
• If it was true for the first 100 even numbers have you come
closer to proving it? What about for the first 100 000?
• Did you use deductive reasoning in this example?
Every even number is the sum of 2 prime numbers (1, 2, 3, 5, 7, 13, 17…)

12. Bertrand Russell (1872-1970)
• He tried to prove that all maths
is built on logic
• He failed, and nearly went mad
in the process
• It didn’t work because it is now
accepted that maths is built on
axioms that are intuitively true,
but impossible to prove through
logic

13. The Modern Axioms of Mathematics
• For any numbers m, n: m + n = n + m
• For any numbers m, n, k: (m + n) + k = m + (n + k)
• For any numbers m, n, k: m(n + k) = mn + mk
• There is a number 0 which has the property that,
for any number n: n + 0 = n
• There is a number 1 which has the property that,
for any number n: n x 1 = n
• For every number n, there is another number k
such that: n + k = 0
• For any numbers m, n, k, if k ≠ 0 and kn = km, then
m = n

14. Maths and Certainty
• As this is TOK, you might have been expecting
us to question the certainty of this area of
knowledge

15. Maths and Certainty
• The British philosopher John Stuart Mill (1808-
1873) claimed that our certainty in mathematics
arises from a vast number of empirical
observations (correspondence theory)
• In other words, we can feel certain that 2+2=4
because every other time a human has carried
out this calculation, it appears to be true
• Although maths itself is deductive in nature,
notice that this is an inductive conclusion
• However maths is also based on proving
theorems without prior experience (i.e. as well
as being a postori it also appears to be a priori to
some extent)
• It therefore seems to go beyond the empirical
The empirical argument

16. Maths and Certainty
• If you conclude that maths is not empirical, you
might think of it as being analytic (i.e. it is something
that is true by definition, independent of experience)
• This view is often attributed to the Scottish
philosopher David Hume (1711-1776)
• This suggests that maths is a truth that simply exists
‘out there’. Any mathematical operation is therefore
simply ‘unpacking the truth’
• However, there is an ongoing unresolved debate
between philosophers over whether maths is
discovered or invented
• Did even numbers exist before someone came up
with the definition of an even number?
• The analytic states that a conjecture must be true or
false by definition, however it seems that nature will
never allow us to determine the truth or falseness of
every conjecture
The analytic argument

17. Maths and Certainty
• If we decide that maths is neither
empirical or analytic, we might
conclude that it is therefore synthetic a
priori, i.e. knowledge that is found to be
true through observation, yet exists
independently of experience
• This view is attributed to the German
philosopher Immanuel Kant (1724-
1804), but is not really different to the
viewpoint of Euclid
• Of course, you might also decide that all
of this is a semantic argument that can
never really be resolved
The synthetic a priori
argument

18. Is Maths Discovered or Invented?
• There is a tribe in the Amazon that have never
developed the concept of numbers or counting. As far
as they are concerned, the concept of mathematics
does not exist
• Mathematical objects appear to us to exist in the real
world, but as soon as we try to capture them, they
seem to exist only in the mind
• No matter how hard you try to draw a perfect circle,
you will never achieve it because there will always be
imperfections in the pencil line you have made
• Similarly, you will never be able to perfectly measure
the length of a piece of string

19. Is Maths Discovered or Invented?
• The Platonist School gets around this problem by
inventing the world of forms
• In this, mathematics exists in a perfect form in a
realm which humans can never fully observe due
to the limitations of our sense perception (maths
is therefore transcendent)
• However the world of forms is also the basis of
our reality so we do perceive it, but in an
imperfect way
• I may be a nice model, but to many people this
just appears to be ‘magical thinking’
• After all, why should concepts like maths exist in
the world of forms and my chair doesn’t
• Since the existence of the world of forms cannot
by proven by definition it leads us to knowledge
through faith alone. Which you could argue is not
really knowledge