WAS MATHETMATICS DISCOVERED OR CREATED?

Pythagoras (571-496 B.C.), a mathematician (perhaps best known for his "Pythagorean theorem") and philosopher born in Samos, Greece, believed that the answer to the big question "What is reality made of?" lay in mathematics. Pythagoras is famously quoted as having said "All is number." He even went as far as to declare that justice was the number 4 as it was a square number.

1. What do you think Pythagoras meant by "All is number."?
2. Do you agree with him? To what extent is it possible to explain the concept of reality through mathematics?


While most of us would find it difficult to entirely agree with Pythagoras, there is little doubt that mathematics has played a large role in our understanding of the world around us, and those we have yet to discover and explore.

When trying to define mathematics, several questions are raised. The big question concerning math, as far as ToK is concerned, is whether mathematics is a natural phenomenon or merely a game of logic invented by the human mind.

3. What is your "gut reaction" to this question? Was mathematics discovered or invented?

While Pythagoras believed that mathematics held all the answers,, PLATO,another Greek philosopher and mathematician whom we've discussed before (remember K=JTB?), held the belief that numbers held some kind of mystical existence, separate to the rest of the world.

Read the brief biography of Plato paying particular attention to the section on mathematics. In it you're reminded that Plato founded "The Academy" (an institution devoted to research and instruction in philosophy and the sciences) over which he presided from 387 BC until his death in 347 BC. Over the door of the Academy was written:

"Let no one unversed in geometry enter here."


4. What does this quotation reveal to us about Plato's view of mathematics? Summarize his belief about mathematics.
5. Why were so many of Plato's friends and students significant contributors to the field of mathematics despite the fact that Plato, himself, made no important mathematical discoveries?

A Mathematician's Apology

Read the following extract from G. H. Hardy book, A Mathematician's Apology

I began by saying that there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, and that the most important seems to me to be this, that the mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real'; but a very little reflection is enough to show that the physicist's reality, whatever it may be, has few or none of the attributes which common sense ascribes instinctively to reality. A chair may be a collection of whirling electrons, or an idea in the mind of God: each of these accounts of it may have its merits, but neither conforms at all closely to the suggestions of common sense.

I went on to say that neither physicists nor philosophers have ever given any convincing account of what 'physical reality' is, or of how the physicist passes, from the confused mass of fact or sensation with which he starts, to the construction of the objects which he calls 'real'. Thus we cannot be said to know what the subject-matter of physics is; but this need not prevent us from understanding roughly what a physicist is trying to do. It is plain that he is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics.

A mathematician, on the other hand, is working with his own mathematical reality. Of this reality, as I explained in §22, I take a 'realistic' and not an 'idealistic' view. At any rate (and this was my main point) this realistic view is much more plausible of mathematical than of physical reality, because mathematical objects are so much more than what they seem. A chair or a star is not in the least like what it seems to be; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it; but '2' or '317' has nothing to do with sensation, and its properties stand out the more clearly the more closely we scrutinize it. It may be that modern physics fits best into some framework of idealistic philosophy-I do not believe it, but there are eminent physicists who say so. Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is, because mathematical reality is built that way.
Who is G H Hardy?

Question 1 According to Hardy, who is in more direct contact with reality, a Mathematician or Physicist?
a) Physicist?
b) Mathematician?
Question 2 Is a physicist depended on
a) Maths
b) Religion
c) Chemistry
Question 3 Hardy regards himself as a . . . . .?
a) Realist
b) Idealist
Question 4 According to Hardy which is more real
a) A chair
b) The number 317

My Nose, My Brain, My Truth

Check out the article about the connection between senses, truth, and knowledge at the link below:
My Nose, My Brain, My Truth

Knowledge in different cultures

The first question is, in what ways do various languages classify the concepts associated with 'to know'? Research the two following languages - French and German and Hindi and Chinese.

In English, French, Spanish or Chinese, for example, what is the relationship between the different ways of expressing 'know': 'they know of it', 'they know about it', 'they really know it', 'they know that person', 'they know that this is so', 'they know how to do it'? Are there other ways of using the verb 'to know'?

Belief is an important part of knowing. Some would suggest that if you do not believe something that you cannot know it.

How do 'believing that' and 'believing in' differ?
How does belief differ from knowledge?"

Dangerous Knowledge

ToK Prescribed Titles (2010) Question 6

All knowledge claims should be open to rational criticism. On what grounds and to what extent would you agree with this assertion?

The essence of the Q: The Q reminds us of Socrates' saying: 'The unexamined life is not worth living.' What did he mean? Should we question everything, even at the expense of our own personal (and social) happiness and sanity? The openness of the Q allows you to look at numerous knowledge claims (make a list in advance) and what they attempt to establish. At first glance, wouldn't we tend to agree with the assertion? Surely, everything is open to critical questioning: we like to be certain about things and get to the truth of them. However, is reason the best method for reaching the truth of knowledge claims? Looking closer, you'll see that there are lots of things to ask yourselves. First, is the main assertion itself a knowledge claim and thereby open to rational criticism? Why? Second, what is the actual status of the assertion (think about the word 'should')? How does this affect our judgement? Finally, what does it mean to be 'rational' and what does 'criticism' involve? Presumably 'rational criticism' means to test all knowledge claims against the rigour of logic, giving grounds or reasons for the knowledge claims; that is, you will have to look at which claims are made through inductive reasoning and which through deductive (please don't simply re-gurgitate class notes!). The focus of the question is undoubtedly on the value of reason as a WoK, but you will need to look at how the other WoKs might be involved in any inductive or deductive process to establish the truth of knowledge claims. Look at the problem of induction and Popper's attempt to solve it.


Knowledge issues: Is reason alone the most reliable test for the truth of knowledge claims? Are the searches for truth and happiness mutually exclusive? Does rational criticism involve the sacrifice of emotion? Can subjective knowledge claims as in the Arts and Ethics ever be rationally criticised? What would a non-rational criticism of a knowledge claim look like? Must all knowledge claims have rational grounds for us to believe in them? In what way are inductive arguments driven by the human tendency to stereotype people? How and under what circumstances do we rationalise situations to our own advantage?

Approaches: Take different knowledge claims from each AoK and attempt to test them: which ones have good reasons to believe in them? Which ones do we believe without any rational grounds (and why)? Which ones are based on inductive arguments and which on deductive? Does the reasoning involve any logical fallacies? Try to choose knowledge claims from contemporary life, such as the Pope's recent statements the 'ecology of man' (why did the gay community get offended?); Bush's statements about sustaining the 'war on terror' (how did these serve to provide a smokescreen to carry out a personal agenda?) or the media's ongoing statements about the present financial crisis (how do these help to solve the crisis?)...You can take statements about historical events or even claims that purport to make knowledgeable statements about the future. Look at ethical statements - we should give a life ban to any sportsperson who takes drugs - is this open to rational criticism? Why? Mathematical knowledge claims are surely watertight - that is, once established, they are unquestionable: the internal angles of a triangle are equal to 180 degrees. Aesthetic statements are, however, beyond rational criticism, aren't they, since they are always based on personal opinion or taste (think about this!)? Is there any difference in the grounds given for knowledge claims in the Natural Sciences and those in the Human Sciences? Compare: 'Human creation and development can be explained by evolutionary genetic theory' and 'Eight out of ten men consider a sense of humour as the essential quality in an ideal partner'. Lastly, consider the status of knowledge claims about the supernatural: how far do these stand up to rational criticism?