quarta-feira, 23 de novembro de 2011

Matemática e Cristianismo no blog de Steve Bishop

Matemática e Cristianismo no blog de Steve Bishop

Florence Nightingale on Statistics

"To understand God's thoughts, we must study statistics for these are the measure of His purpose" Florence Nightingale

Nightingale was the originator of the polar area diagram - the forerunner of the pie chart.

Saturday, 14 November 2009

Pi and mathematical reductionism

Last night I watched the film Pi (details here)

Here's the trailer.

It's a fascinating film. The 'hero' Max is a mathematical reductionist - and it screws him up. Idolatry does that.

Saturday, 7 June 2008

A bibliography for a Christian approach to mathematics


Gene B. Chase and Calvin Jongsma Bibliography of Christianity and mathematics 1910-1983 (Dordt, Iowa: Dort College, 1983)
An extensive annotated bibliography of papers and books dealing with Christian approaches and attitudes to maths.


Full length Books

Robert L. Brabenec A Christian Perspective on the Foundations of Mathematics (Wheaton College, 1977)
Brabenec has edited a number of 'Mathematics from a Christian Perspective' Conference Proceedings, of which this is the first. They contain a wealth of useful information of integrating math and a Christian worldview, however, they are concerned primarily with higher education. Many of the articles are now available on-line here.

Charis Mathematics Project Charis Mathematics (Units 1-9 & Units 10-19; A1 -A6) (John Shortt, director) (editors John Shortt and John Westwell) (Nottingham: The Stapleford Centre, 1996 and 1997)
Photocopiable materials for bringing out the 'spiritual and moral' aspects of mathematics at the secondary school level; written by Christian teachers of mathematics.
Chapter 7 looks at some theories in mathematics (number-world theory; J. S. Mill; Bertrand Russell;and John Dewey) and shows how they all depend on divinising certain aspects of creation.
Andrew M. Hartely Christian and Humanist Foundations for Statistical Inference (Resource Publications, Wipf and Stock, 2008)
Looks at four of the dominant paradigms in statistics and compares them with a Christian worldview
James Nikel Mathematics: Is God Silent? (Vallecito, CA: Ross House Books, 1990; 2nd edn 2000)
Written to show that God is not! Contains many useful examples.

Russell W. Howell and James Bradley (ed.) Mathematics in a Postmodern Age: A Christian Perspective (Eerdmans, Grand Rapids, 2001)
Examines the nature of mathematics, the influence of maths and faith perspectives in mathematics from a Christian perspective.

Calvin Jongsma and Trudy Baker The Shape and Number of Things: An Integrated Math Curriculum for the Elementary School (Toronto: Curriculum Development Center)
A complete primary school package.

T. Koetsier and L. Bergmans (eds) Mathematics and the Divine A Historical Study (Elsevier Science, 2004)

Chapters in books
Harro Van Brummelen Walking with God in the Classroom: Christian Approaches to Learning and Teaching (Ontario: Welch, 1988)
Though on Christian education in general, this book contains many insights into mathematics.

Gene B. Chase 'Has Christian theology furthered mathematics' In Facets of Faith and Science vol 2: The Role of Beliefs in Mathematics and the Natural Sciences: An Augustinian Perspective. Jitse M. van der Meer (ed.) University Press of America/ Pascal Centre for Advanced Studies: Lanham/ Ancaster, 1996.
Harold Heie and David L. Wolfe (editors) The Reality of Christian Learning: Strategies for Faith-Discipline Integration (Grand Rapids: Christian University Press/ Eerdmans, 1987)
Harold Heie and Gene Chase contribute two chapters on ways of integrating the Christian faith and mathematics.
Vern Poythress ‘A biblical view of mathematics’ in Foundations of Christian Scholarship: Essays in the Van Til Perspective (Vallecito, CA: Ross House, 1976)
Develops a [Cornelius] Van Tilian approach to maths
Geraldine J. Steensma and Harro W. Van Brummelen Shaping School Curriculum: A Biblical View (Terre Haute, Ind: Signal, 1977)
Contains a useful chapter on math education by Van Brummelen and some examples (Statistics and Deductive Geometry) of how this translates into classroom practice.
Danie F. M. Strauss 'A historical analysis of the role of beliefs in the three foundational crises in mathematics'. In Facets of Faith and Science vol 2: The Role of Beliefs in Mathematics and the Natural Sciences: An Augustinian Perspective. Jitse M. van der Meer (ed.) University Press of America/ Pascal Centre for Advanced Studies: Lanham/ Ancaster, 1996.

Danie F. M. Strauss 'Primitive meaning in mathematics: the interaction among commitment, theoretical worldview and axiomatic set theory'. In Facets of Faith and Science vol 2: The Role of Beliefs in Mathematics and the Natural Sciences: An Augustinian Perspective. Jitse M. van der Meer (ed.) (Lanham/ Ancaster: University Press of America/ Pascal Centre for Advanced Studies:, 1996).
Anthony Tol 'Counting , number concepts and numerosity' in Hearing and Doing John Kraay and Anthony Tol (ed.) (Toronto: Wedge, 1979)
Noel Weeks The Christian School: An Introduction (Edinburgh: Banner of Truth)
Chapter 12 deals with maths. He contrasts rationalist and romanticist views of math education and offers some pointers for a Christian approach.

Steve Bishop 'Beliefs shapes mathematics' Spectrum 28 (2) (Spring 1996)
Examines several philosophies of maths to show that maths is not neutral but shaped by beliefs. The contours of a Christian approach to maths are sketched.

Steve Bishop 'Mathematics and the myth of neutrality' Christian School Education 5 (4) (2001-2002): 19-21

James Bradley 'Two ways of knowing' Journal of the ACMS 2004 (Aug)

Edward Fackerell 'The Relationship Between Mathematics and the Christian Faith'. Christian Teachers Journal, Vol 11, No. 2, May 2003.
(The Christian Teachers Journal can be contacted at: PO Box 7000, Blacktown NSW 2148 Australia)

Geertsema, Jan C. A Christian View of the Foundations of Statistics PSCF 39 (September 1997):158-164

Jan Gormas 'A search for intellectual, relational and spiritual integrity: secondary mathematics from a Christian perspective' Education and Christian Belief 9(2) (2005) abstract
H. Harold Hartzler The Meaning of Mathematics JASA 1 (January 1949):13-19.

Willem Kuyk 'Some questions on the foundation of logic' Philosophia Reformata 34 (1969) 142-146

P. MacKenzie Entry Points for Christian Reflection within education (London: CARE, 1997) (Now available as an e-book)
Chapter 7 is on mathematics and looks at how it is approached from a UK Christian schools perspective.

Paul Marshall 'Mathematics and politics' Philosophia Reformata 44 (1979) 113-136
Vern Poythress Mathematics as Rhyme Journal of the American Scientific Affiliation 35/4:196-203. (See my comments here)
Vern Poythress 'Newton's Laws as Allegory' Journal of the American Scientific Affiliation 35/3:156-161.
Sharon K Robbert Christianity and mathematics
Devotionals with mathematics content!
Roger Slack Mathematics: An Historical Survey (Nottingham: ACT Mathematics Group 1981-83)
A useful 30-page booklet dealing with the history of math from the Pythagoreans to the present from a Christian perspective.

M D Stafleu 'The mathematical and technical opening up of a field of science' Philosophia Reformata 43 (1978).

Danie F. M. Strauss 'Number concept and number-idea' PhilosophiaReformata. 35 (1970): 156-177

Danie F. M. Strauss 'Number concept and number-idea (continued)' Philosophia Reformata. 36 (1971): 13-42.

Danie F. M. Strauss 'Dooyeweerd and Modern Mathematics' Reformational Forum, 1983, nr.2 (pp.5-17).

Danie F. M. Strauss 'Mathematical paradigms' Journal for Christian Scholarship, 3rd & 4th Quarter 1994 (pp.113-167).

Danie F. M. Strauss 'The three foundational crises in mathematics' Journal for Christian Scholarship , 1st & 2nd Quarter 1995 (pp.12-22).
Danie F. M. Strauss 'Mathematics and the Real World' Koers Year/Vol.65(1) April 2000, pp.95-121.

Danie F. M. Strauss Reductionism in Mathematics: Philosophical Reflections Journal for Christian Scholarship , Year 37, 1ste en 2nd Quarter, 2001, pp.1-12

Danie F. M. Strauss Is a Christian Mathematics possible? Tydskrif vir Christelike Wetenskap/Journal for Christian Scholarship, 2003(3&4):31-49.

Any other suggestions?

Friday, 17 August 2007

Turing on maths

This piece is taken from the play/ film 'Breaking the Code'. The screenplay was by Hugh Whitemore based on Alan hodges biography. Dilly Knox asks Alan Turing to explain in 'general terms' his research. This is Turing's response. It is a great summary of twentieth-century mathematics from Russell to Turing.

A few words of explanation ...in general terms?

Well, erm, it’s about right and wrong in ... general terms...It’s a technical paper in mathematical logic. but it’s also about the difficulty of telling wrong from wrong. You see people think, that well, most most people think that in mathematics we always know what is right and wrong, not so, not anymore. It’s a problem that has occupied mathematicians for forty or fifty years.
How do you tell wright from wrong?

Bertrand Russell has written a book on it his Principia Mathematica – his idea was to break down all mathematical concepts and arguments into little pieces and then show they can be derived from pure logic. Well it didn’t quite work out that way and after many years of intensive work all he was able to show was that it was terribly difficult to do anything of the kind; but it was an important book. Important and influential it influenced both David Hilbert and and Kurt Godel. It’s rather like what physicsts call splitting the atom. As analysing the physical atom has led to a new kind of physics so the attempt to analyse these mathematical atoms has led to a new kind of mathematics.

David Hilbert took the whole thing a stage further. I don’t suppose his name means much if anything to you? Wel there you are you see, it’s the way of the world people never really seem to hear about the really great mathematicians. But Hilbert looked at the problem from a completely different angle He said that if we are to have any fundamental system for mathematics, like the one Russell was trying to work out

It must satisfy three basic requiremts: consistency, compelteness and decidability.

Now consistency means you won’t ever get a contradiction in your own system, in other words you will never be able to follow the rules of your and end up with showing 2 + 2 = 5.

Completeness means that if any statement is true there must be some way of proving it by using the rules of your system

and decidability means, well decidability means that there must exist some method, some definite procedure or test which can be applied to any given mathematical assertion and which will decide whether or not that assertion is provable.
Now Hilbert thought this was a perfectly reasonable set of requiremtns to impose.

But within a few years Kurt Godel showed that no system for mathematics could be both consistent and compete and he did this by constructing a mathematical assertion, which said in effect: This assertion cannot be proved.
Classic paradox! This assertion cannot be proved. Well, either it can or can’t
If it can be proved we have a contradiction and the system is inconsistent
If it cannot be proved then the assertion is true. But it can’t be proved, which means the system is incomplete.
Thus mathematics is either inconstent or incomplete. It’s a beautiful theorm. It’s quite beautiful. I think Godel’s theorem is the most beautiful thing I know.

The question of decidabilty was still unsettled. Hilbert thought there should be one single clearly defined method of proving whether mathematical assertions were provable. The decision problem he called it the entscheidungsproblem. In my book on computable numbers I wanted to show that no one method can work for all questions, solving mathematical problems requires an infinite supply of new ideas. It is one thing to make this claim it is a monumental task to prove it. I needed to examine the provability of mathematical assertions past present and future. How on earth was it it be done?

Eventually one word gave me a clue – people had been talking a bout a mechanical process and process that could be applied mechanically to mathematical problems, a process without requiring any human invention or ingenuity – machine; that was the crucial word. I conceived the idea of a machine. A Turing machine which would be able to scan mathematical symbols, it read them if you like it would read a mathematical assertion and then arrive at a verdict whether that assertion was provable. I was able to show that Hilbert was wrong. My idea worked.

Thursday, 26 July 2007

Maths as rhyme?

I have recently read Vern Poythress's paper 'Mathematics as a rhyme'. Below are a few of my initial observations.

His idea as maths as a poem is an interesting idea, but I am not so sure as to how fruitful it is. It seems to be little more than Kepler/Galileo's idea that maths is the language of the universe.

I am not convinced of the metaphor of maths as a rhyme within the poem of the universe. It has too many resonances with the two books concept for my liking!

Not all poems rhyme!

If the poem is created by a poet, then the poet takes things he has not created (letters, words) to create the poem using rhyme etc. This suggest that God (poet) creates the universe (poem) using uncreated maths!

If maths is the rhyme what are the words?

He does makes some good points regarding mathematics; including:

Maths is a distinct science

It is not reducible to other sciences

Its regularities are a reflection of the faithfullness of God

There is at lest some degree of a posteri character within the knowledge of maths, even though it gives the impression that its truths are a priori

He subverts the view that maths is necessary rather than contingent. [ Though I would contend that it is necessary for creatures, but contingent for the Creator.]

Maths is personally structured [I'm not totally sure what he means here], for intelligibility there must be a personal interpreter.

There is a good discussion of reductionisms of logicism, formalism, intuition and empiricism - he shows the attractiveness of reductionism: "stimulating as a metaphor inadequate as ultimate explanation" [could this be said of maths as a rhyme within the poem of the universe?]

He shows the irreducibillity of the subparts of maths to one another.

Wednesday, 29 March 2006

Maths website

I've just started another website. This one is devoted to maths education resources and can be found here. At present it is a list of web links for pre-GCSE, GCSE (ages 14-16) and A-level maths (ages 16-18).

Sunday, 26 February 2006

Is there a maths gene?

The University of Cambridge Department of Experimental Psychology is looking into the question: Is there a maths gene? At present they are looking for volunteers with Grade A at A-level, or a score of 700 or more on the maths section of the American SAT 1.

The project comprises two phases. In Phase 1 you will be asked to complete an online maths test. This will take about 20 minutes. You will then be asked to invite your siblings to participate, if they also have a Grade A maths A Level. In Phase 2, you will be invited to donate a DNA sample via a simple, quick and non-invasive cheek swab.

Once you and your sibling have both returned a DNA sample, you will be entered into a prize draw to win £250 (or equivalent in other currency) worth of Amazon vouchers to share. The draw will take place once data collection for this project is complete.

All participants must be aged 16 years or over. This study has been approved by the University Ethics Committee, all information remains confidential, and in Phase 2, DNA will be stored anonymously. You may withdraw your participation at any time without having to give a reason.

This though does raise the question - if there is a maths gene, do we screen students before admitting them to A-level maths, or even before going to study a maths degree at university? That would certainly improve retention and achievement! Do genes really determine who we are?

There is no evidence to suggest that genes do determine who we are, although we are influenced by them. If they did determine who we are then someone with a kleptomania gene might be able to plead 'it wasn't me guv, wot did it, it was me genes.'

Thursday, 16 February 2006

Nestle get their maths wrong!

According to Nestle the makers of Smarties, Canadians eat enough Smarties to go round the Earth 350 times. A Grade 6 maths class in Ontario decided to put this claim to the test. It seems Nestle were wrong!

You can check their maths: according to Nestle Canadians eat 4 billion Smarties a year - the circumference of the Earth is approximately 40 000 km.

Saturday, 13 August 2005

Latin squares (aka Sudoku)

Sudoku (aka Su Doko, Sudoko, Soduko), is the latest craze to hit the UK. I first came across it about six months ago in a Times newspaper. The six out of the top 40 best sellers in The Guardian's bestseller's list are all Sudoko books! The concept is fiendishly simple: there is a 9 × 9 grid which is split up into nine smaller 3 × 3 grids, all the 3 × 3 grids, rows and columns must contain all nine numbers.

Sudoko apparently came to the
UK via Japan. However, it was first invented by the Swiss Leonard Euler around 1798, though he called it Latin squares. Euler was an extraordinary and prolific mathematican, see the Euler archive. He was also a committed Christian. Dan Graves in his Scientists of Faith has this to say of Euler:
Euler retained his firm Calvinist beliefs throughout life, holding daily prayer and worship in his home and sometimes preaching.

cited here.
It rather ironic that The Independent newspaper boasts that no mathematics is needed to solve it! I think they are getting confused with maths and arithmetic; there is a lot of maths going on! See here for example.
There are a number of Sudoko solvers on the web; books on how to solve them by, for example, Robin Wilson and Carol Voderman; as well as books of them; and there is no shortage Sudoko puzzles on-line to have a go at.

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