Mathematics

Mathematics

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

• 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

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

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

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

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)

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

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

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

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…)

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

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

Maths and Certainty

• As this is TOK, you might have been expecting us to question the certainty of this area of knowledge

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

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

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

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

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