2. Did Einstein and Plato get it right? A TOK Presentation on the place of Mathematics in our world. A personal view

3. An essay title for May 2010 Discuss the claim that some areas of knowledge are discovered and others are invented. Discuss the claim that some areas of knowledge are discovered and others are invented. Note: this is on the 2010 list, so is not one of the titles this grade 12 will have.

4. Foundations This presentation looks into the very heart of mathematics. What is it? Where does it come from? This presentation looks into the very heart of mathematics. What is it? Where does it come from? This study might come under the heading “mathematical philosophy” or “meta- mathematics”, both of which sound a bit pretentious. I prefer the title of “foundations”. This study might come under the heading “mathematical philosophy” or “meta- mathematics”, both of which sound a bit pretentious. I prefer the title of “foundations”.

5. Two great thinkers Plato Plato Einstein Einstein

6. Plato’s ideal forms Plato thought ideal (or perfect) forms exist Plato thought ideal (or perfect) forms exist … such as perfect justice … such as perfect justice … or a perfect triangle … or a perfect triangle He understood a perfect triangle could not be drawn with ruler and pencil … He understood a perfect triangle could not be drawn with ruler and pencil … … but it still exists, as an ideal form, … but it still exists, as an ideal form, … as an abstract rather than physical object … as an abstract rather than physical object

7. From Plato to Platonism Modern Platonism enshrines three ideas: Modern Platonism enshrines three ideas: Existence Existence Abstraction Abstraction Independence Independence Applied to mathematical foundations this means Platonists believe mathematical objects exist as abstract realities independent of mankind Applied to mathematical foundations this means Platonists believe mathematical objects exist as abstract realities independent of mankind

8. Platonism’s great rival: Formalism Formalists essentially reject the idea of independent existence Formalists essentially reject the idea of independent existence They believe abstract ideas used by mathematicians are created by these mathematicians They believe abstract ideas used by mathematicians are created by these mathematicians Mathematics could not exist without the mathematicians Mathematics could not exist without the mathematicians Mathematics is a mental construction Mathematics is a mental construction

9. My thesis Mathematics is more than a game Mathematics is more than a game It is not simply a creation of mankind It is not simply a creation of mankind Rather, it is Nature’s language: it defines the ideal forms Plato described Rather, it is Nature’s language: it defines the ideal forms Plato described Mathematics actually describes how the universe works Mathematics actually describes how the universe works So, I am a Platonist So, I am a Platonist

10. Formalism v. Platonism Formalists believe mathematics is like a game of chess Formalists believe mathematics is like a game of chess Choose the rules carefully to avoid contradictions Choose the rules carefully to avoid contradictions Any match with reality is a fluke, or the result of a fortuitous choice of rules Any match with reality is a fluke, or the result of a fortuitous choice of rules

11. Formalism v. Platonism Platonists, (also called realists) however, believe Mathematics is describing the way the universe is. Platonists, (also called realists) however, believe Mathematics is describing the way the universe is. It’s very structure is independent of man- kind It’s very structure is independent of man- kind

12. Why is this division important? The nature of the foundations of Mathematics determines its uses and future development The nature of the foundations of Mathematics determines its uses and future development Therefore the issue is a contemporary one, with implications across the whole spectrum of knowledge Therefore the issue is a contemporary one, with implications across the whole spectrum of knowledge In particular, its relationship with the sciences is fundamental In particular, its relationship with the sciences is fundamental

13. My belief Mathematical objects (5, a triangle, sin30) are not man-made but are parts of nature Mathematical objects (5, a triangle, sin30) are not man-made but are parts of nature Mathematics is attempting to describe the way the universe is Mathematics is attempting to describe the way the universe is The rules of Mathematics are not created by mankind, but are rather discovered (or uncovered) by mankind The rules of Mathematics are not created by mankind, but are rather discovered (or uncovered) by mankind

14. Analogies to other Areas of Knowledge Science claims the same thing; it tries to explain the way the universe works Science claims the same thing; it tries to explain the way the universe works Artists attempt an identical feat Artists attempt an identical feat All three are giving us different perspectives on reality All three are giving us different perspectives on reality However, their means of approaching the subject vary considerably However, their means of approaching the subject vary considerably

15. Various approaches to reality The scientist relies on experimentation, observation, and the Scientific Method The scientist relies on experimentation, observation, and the Scientific Method They use reason and sense perception as their main Ways of Knowing They use reason and sense perception as their main Ways of Knowing

16. Various approaches to reality The artist works at a far more subjective level The artist works at a far more subjective level They appeal to emotion, sense perception and language to convey their vision of reality They appeal to emotion, sense perception and language to convey their vision of reality

17. Various approaches to reality The Mathematician uses only (mainly?) reason (perhaps guided by perception and emotion) The Mathematician uses only (mainly?) reason (perhaps guided by perception and emotion) Mathematical truths are the most objective truths from any area of knowledge Mathematical truths are the most objective truths from any area of knowledge

18. Einstein’s belief “The creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed. “ – Albert Einstein “The creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed. “ – Albert Einstein– Albert Einstein– Albert Einstein

19. Heisenberg’s thoughts “I think that modern physics has definitely decided in favor of Plato. In fact the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language.” “I think that modern physics has definitely decided in favor of Plato. In fact the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language.” – Werner Heisenberg – Werner Heisenberg– Werner Heisenberg– Werner Heisenberg

20. Knowledge Issues Platonists have a few significant problems to handle, namely: Platonists have a few significant problems to handle, namely: Gdel’s Incompleteness Theorem Gödel’s Incompleteness Theorem Paradoxes Paradoxes Axioms: an act of faith? Axioms: an act of faith?

21. Gödel’s Incompleteness Theorem In the 1930’s Gdel showed Mathematics could never be complete In the 1930’s Gödel showed Mathematics could never be complete This result was proved from within Mathematics itself This result was proved from within Mathematics itself There will always be true statements we can never prove true There will always be true statements we can never prove true Kurt Gödel

22. Paradoxes For a Platonist, a paradox, like the Russell Paradox, that leads to a contradiction can not be allowed to stand For a Platonist, a paradox, like the Russell Paradox, that leads to a contradiction can not be allowed to stand In severe cases they require mathematicians to re-write whole sections of the subject In severe cases they require mathematicians to re-write whole sections of the subject Bertrand Russell

23. Axioms At the most fundamental level, axioms, upon which all of mathematics is built, rely on no more than an act of faith as, by definition, they can never be proved At the most fundamental level, axioms, upon which all of mathematics is built, rely on no more than an act of faith as, by definition, they can never be proved Modern axiom sets are extremely complex and abstract in nature (in contrary to the simplistic and obvious laws of nature scientists and mathematicians hope to uncover) Modern axiom sets are extremely complex and abstract in nature (in contrary to the simplistic and obvious laws of nature scientists and mathematicians hope to uncover) The big question is, “are axioms invented, or are they the fundamental rules of nature waiting to be discovered?” The big question is, “are axioms invented, or are they the fundamental rules of nature waiting to be discovered?”

24. The Formalist Case We chose axioms, they are not God-given We chose axioms, they are not God-given If mathematics describes all of creation, how can it be incomplete, as Gödel proved If mathematics describes all of creation, how can it be incomplete, as Gödel proved Does Nature allow paradoxes? If not, how can mathematics? Does Nature allow paradoxes? If not, how can mathematics? David Hilbert – a formalist

25. The Platonist Defense Our axioms are not as well refined as they need to be. In time they will become clearer and more obvious Our axioms are not as well refined as they need to be. In time they will become clearer and more obvious Mathematics is ‘incomplete’ because Nature is ‘incomplete’: Heisenberg’s Uncertainty Principle shows this Mathematics is ‘incomplete’ because Nature is ‘incomplete’: Heisenberg’s Uncertainty Principle shows this We don’t allow paradoxes – we have not yet discovered the ideal axioms which avoids them We don’t allow paradoxes – we have not yet discovered the ideal axioms which avoids them David Hilbourne -- a Platonist

26. The Platonist’s Attack Besides, if mathematics is formalist, surely, being a creation of mankind, it would not apply universally. The argument goes, “Would a suitably advanced alien race discover (in their number system) the same value we have for  ?” Besides, if mathematics is formalist, surely, being a creation of mankind, it would not apply universally. The argument goes, “Would a suitably advanced alien race discover (in their number system) the same value we have for  ?” If it is no more than a type of game how comes it is so fundamentally useful in our technological world? Chess is not useful like this. If it is no more than a type of game how comes it is so fundamentally useful in our technological world? Chess is not useful like this. 

27. and then there is Neptune … Uranus was discovered in 1781, and by 1846 had almost completed one orbit of the Sun. But its position did not accurately match the prediction of Newton’s Law of Gravity. But the “error” could be explained by the existence of a more distant planet. Urbain Le Verrier, in Paris, predicted where this new planet would be found. Neptune was first observed on 23 September that year, where Le Verrier has predicted. Le Verrier had discovered Neptune “with the point of his pencil”. François Arago

28. Any other choice? There has been one more serious attempt to put mathematics onto a firm philosophical foundation: There has been one more serious attempt to put mathematics onto a firm philosophical foundation: Constructivism. Constructivism.

29. Constructivism Let by Brouwer in 1908. He maintained that the natural numbers were intuitive to all, and therefore could safely be used as the basis of all mathematics. Anything that could not be constructed in a finite number of steps from these natural numbers was not allowable. Let by Brouwer in 1908. He maintained that the natural numbers were intuitive to all, and therefore could safely be used as the basis of all mathematics. Anything that could not be constructed in a finite number of steps from these natural numbers was not allowable. L.E.J. Brouwer --- a constructivist

30. Constructivism – no thanks Nice idea, but a bit of a dramatic position to take; too draconian for most mathematicians. Nice idea, but a bit of a dramatic position to take; too draconian for most mathematicians. If you stick to constructivist rules many of the proofs of established mathematics would no longer work. If you stick to constructivist rules many of the proofs of established mathematics would no longer work. Result: Most of Maths would have to go! Result: Most of Maths would have to go! Constructivists are considered by many mathematicians as a bit of a fringe element – a weird sect, best ignored. Constructivists are considered by many mathematicians as a bit of a fringe element – a weird sect, best ignored.

31. Conclusion The arguments against the formalist school are so convincing that the formalist view must be rejected The arguments against the formalist school are so convincing that the formalist view must be rejected Constructivism is too restricting – nothing left Constructivism is too restricting – nothing left I believe in the Platonic school of thought that places Mathematics at the heart of our universe I believe in the Platonic school of thought that places Mathematics at the heart of our universe The problems Platonist’s face are temporary, and will be removed when suitable axioms are discovered The problems Platonist’s face are temporary, and will be removed when suitable axioms are discovered

32. The End