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**Maths Invented or Discovered? **

The PBS Ideas Channel has just released a new video which discusses whether maths is invented by humans, or whether it is discovered (ie whether it can be said to really exist). It’s an excellent 10 minute introduction to a pretty complicated topic – and certainly accessible for students:

For those interested in more detail – here are some of the basics (some of this information is simplified from the Stanford Encyclopedia which goes into far more detail). You can also see more discussion of the topic on this site here

**1) Platonism**

The basic philosophical question in maths is ontological – ie concerned with *existence. *The Platonic school (named after Greek philosopher Plato) hold that mathematical objects can themselves be said to exist. Is there a “perfect circle” – in the realm of “ideas” upon which all circles on Earth are simply imitations? Is this circle independent of human thought? Does pi exist outside of human experience – and indeed space and time? The hard Platonists argue that mathematical structures themselves are *physically* real – and indeed that our universe may be a mathematical structure. Some other schools of mathematical philosphy include:

**2) Logicism**

Logicism seeks to reduce all of mathematics to logical thought – if all mathematics is reducible to logic does that mean that mathematics is purely an intellectual exercise? 20th Century efforts by Bertrand Russell and others to reduce mathematics to logical statements have not enjoyed much success.

**3) Intuitionism:**

“According to intuitionism, mathematics is essentially an activity of construction. The natural numbers are mental constructions, the real numbers are mental constructions, proofs and theorems are mental constructions, mathematical meaning is a mental construction… Mathematical constructions are produced by the *ideal* mathematician, i.e., abstraction is made from contingent, physical limitations of the real life mathematician. But even the ideal mathematician remains a finite being. She can never complete an infinite construction, even though she can complete arbitrarily large finite initial parts of it.” (Paragraph from Stanford).

Mathematics therefore does not really exist in any physical sense – it is merely a construction of the mind.

**4) Fictionalism:**

“Fictionalism holds that mathematical theories are like fiction stories such as fairy tales and novels. Mathematical theories describe fictional entities, in the same way that literary fiction describes fictional characters. This position was first articulated in the introductory chapter of (Field 1989), and has in recent years been gaining in popularity.” (Paragraph from Stanford).

This line of thought tries to explain the amazing effectiveness of mathematics in describing the real world in a novel way – by denying that it does! The reality that we think is being described by mathematics is nothing more than fiction – there is an underlying reality which we know nothing about. Think about Nick Bostrom’s Computer Simulation argument – if we were within a computer simulation, then our mathematical laws may very well explain the computer code – but the real reality would be that which existed outside the computer.

Like this topic? Then you might also enjoy:

Is God a Mathematician? – A Michio Kaku video which looks at how mathematics can be used to model the universe.

Simulations -Traffic Jams and Asteroid Impacts – An example of the power of mathematics in modelling the real world

Essential resources for IB students:

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams. I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers. These all have worked solutions and allow you to focus on specific topics or start general revision. This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

Plus Maths has a large number of great podcasts which look at maths ToK topics:

1) An interview with Max Tegmark (pictured above) about why he thinks that the universe is itself a mathematical structure.

2) An interview with physicists David Berman about how many dimensions exist.

3) A talk with cosmologist John Barrow about infinity.

4) A discussion with Roger Penrose about the puzzle of time.

And many more. Well worth a listen!

5) There’s also a good lecture by Professor Ray Monk on the University of Southampton page (see “useful downloads”) looking at the link between philosophy and mathematics – which takes a fascinating journey through the history of maths and the great ideas of great men.

6) Maths for Primates is a fantastic source of podcasts – 14 and counting, on fractals, Zeno, Hilbert’s hotel and more.

This is a huge topic – closely related to some of the Theory of knowledge concepts. It also gets very complicated. Here are some of the basics (some of this information is simplified from the Stanford Encyclopedia which goes into far more detail).

**1) Platonism**

The basic philosophical question in maths is ontological – ie concerned with *existence. *The Platonic school (named after Greek philosopher Plato) hold that mathematical objects can themselves be said to exist. Is there a “perfect circle” – in the realm of “ideas” upon which all circles on Earth are simply imitations? Is this circle independent of human thought? Does pi exist outside of human experience – and indeed space and time? The hard Platonists argue that mathematical structures themselves are *physically* real – and indeed that our universe may be a mathematical structure. (“Was mathematics Invented or Discovered”)

Some other schools of mathematical philosphy include:

**2) Logicism**

Logicism seeks to reduce all of mathematics to logical thought – if all mathematics is reducible to logic does that mean that mathematics is purely an intellectual exercise? 20th Century efforts by Bertrand Russell and others to reduce mathematics to logical statements have not enjoyed much success.

**3) Intuitionism:**

“According to intuitionism, mathematics is essentially an activity of construction. The natural numbers are mental constructions, the real numbers are mental constructions, proofs and theorems are mental constructions, mathematical meaning is a mental construction… Mathematical constructions are produced by the *ideal* mathematician, i.e., abstraction is made from contingent, physical limitations of the real life mathematician. But even the ideal mathematician remains a finite being. She can never complete an infinite construction, even though she can complete arbitrarily large finite initial parts of it.” (Paragraph from Stanford).

Mathematics therefore does not really exist in any physical sense – it is merely a construction of the mind.

**4) Fictionalism:**

“Fictionalism holds that mathematical theories are like fiction stories such as fairy tales and novels. Mathematical theories describe fictional entities, in the same way that literary fiction describes fictional characters. This position was first articulated in the introductory chapter of (Field 1989), and has in recent years been gaining in popularity.” (Paragraph from Stanford).

This line of thought tries to explain the amazing effectiveness of mathematics in describing the real world in a novel way – by denying that it does! The reality that we think is being described by mathematics is nothing more than fiction – there is an underlying reality which we know nothing about. Think about Nick Bostrom’s Computer Simulation argument – if we were within a computer simulation, then our mathematical laws may very well explain the computer code – but the real reality would be that which existed outside the computer.

Essential resources for IB students:

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams. I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers. These all have worked solutions and allow you to focus on specific topics or start general revision. This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

Godel, a 20th century, Austrian American mathematician attempted to use the rigour of formal mathematical logic to provide a proof for the existence of God. Whilst somewhat daunting, a more simplified version can be regarded as, “God, by definition, is that for which no greater can be conceived. God exists in the understanding. If God exists in the understanding, we could imagine Him to be greater by existing in reality. Therefore, God must exist.”

This logic was criticised by David Hume amongst others – as it jumps from the premise that because God could be greater in reality than imagined, therefore he must exist in reality – whereas no *must* is required.

Still it is an interesting example of the relationship between maths and philosophy, and how the underlying logical nature of mathematics can be applied to philosophical questions.