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Great mathematical ideas

Published
by
peter
on 2009-09-23
in Logic, Mathematics, Matherati, Probability theory and Uncertainty
. Closed

Normblog has a regular feature, Writer’s Choice, where writers give their opinions of books which have influenced them.   Seeing this led me recently to think of the mathematical ideas which have influenced my own thinking.   In an earlier post, I wrote about the writers whose  books (and teachers whose lectures) directly influenced me.  I left many pure mathematicians and statisticians off that list because most mathematics and statistics I did not receive directly from their books, but indirectly, mediated through the textbooks and lectures of others.  It is time to make amends. 

Here then is a list of mathematical ideas which have had great influence on my thinking, along with their progenitors.  Not all of these ideas have yet proved useful in any practical sense, either to me or to the world – but there is still lots of time.   Some of these theories are very beautiful, and it is their elegance and beauty and profundity to which I respond.  Others are counter-intuitive and thus thought-provoking, and I recall them for this reason.

  • Euclid’s axiomatic treatment of (Euclidean) geometry
  • The various laws of large numbers, first proven by Jacob Bernoulli (which give a rational justification for reasoning from samples to populations)
  • The differential calculus of Isaac Newton and Gottfried Leibniz (the first formal treatment of change)
  • The Identity of Leonhard Euler:  exp ( i * \pi) + 1 = 0, which mysteriously links two transcendental numbers (\pi and e), an imaginary number i (the square root of minus one) with the identity of the addition operation (zero) and the identity of the multiplication operation (1).
  • The epsilon-delta arguments for the calculus of Augustin Louis Cauchy and Karl Weierstrauss
  • The non-Euclidean geometries of Janos Bolyai, Nikolai Lobachevsky and Bernhard Riemann (which showed that 2-dimensional (or plane) geometry would be different if the surface it was done on was curved rather than flat – the arrival of post-modernism in mathematics)
  • The diagonalization proof of Gregor Cantor that the Real numbers are not countable (showing that there is more than one type of infinity) (a proof-method later adopted by Godel, mentioned below)
  • The axioms for the natural numbers of Guiseppe Peano
  • The space-filling curves of Guiseppe Peano and others (mapping the unit interval continuously to the unit square)
  • The axiomatic treatments of geometry of Mario Pieri and David Hilbert (releasing pure mathematics from any necessary connection to the real-world)
  • The algebraic topology of Henri Poincare and many others (associating algebraic structures to topological spaces)
  • The paradox of set theory of Bertrand Russell (asking whether the set of all sets contains itself)
  • The Fixed Point Theorem of Jan Brouwer (which, inter alia, has been used to prove that certain purely-artificial mathematical constructs called economies under some conditions contain equilibria)
  • The theory of measure and integration of Henri Lebesgue
  • The constructivism of Jan Brouwer (which taught us to think differently about mathematical knowledge)
  • The statistical decision theory of Jerzy Neyman and Egon Pearson (which enabled us to bound the potential errors of statistical inference)
  • The axioms for probability theory of Andrey Kolmogorov (which formalized one common method for representing uncertainty)
  • The BHK axioms for intuitionistic logic, associated to the names of Jan Brouwer, Arend Heyting and Andrey Kolmogorov (which enabled the formal treatment of intuitionism)
  • The incompleteness theorems of Kurt Godel (which identified some limits to mathematical knowledge)
  • The theory of categories of Sam Eilenberg and Saunders Mac Lane (using pure mathematics to model what pure mathematicians do, and enabling concise, abstract and elegant presentations of mathematical knowledge)
  • Possible-worlds semantics for modal logics (due to many people, but often named for Saul Kripke)
  • The topos theory of Alexander Grothendieck (generalizing the category of sets)
  • The proof by Paul Cohen of the logical independence of the Axiom of Choice from the Zermelo-Fraenkel axioms of Set Theory (which establishes Choice as one truly weird axiom!)
  • The non-standard analysis of Abraham Robinson and the synthetic geometry of Anders Kock (which formalize infinitesimal arithmetic)
  • The non-probabilistic representations of uncertainty of Arthur Dempster, Glenn Shafer and others (which provide formal representations of uncertainty without the weaknesses of probability theory)
  • The information geometry of Shunichi Amari, Ole Barndorff-Nielsen, Nikolai Chentsov, Bradley Efron, and others (showing that the methods of statistical inference are not just ad hoc procedures)
  • The robust statistical methods of Peter Huber and others 
  • The proof by Andrew Wiles of The Theorem Formerly Known as Fermat’s Last (which proof I don’t yet follow).

Some of these ideas are among the most sublime and beautiful thoughts of humankind.  Not having an education which has equipped one to appreciate these ideas would be like being tone-deaf.

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Myopic utilitarianism

Published
by
peter
on 2009-09-23
in Computer Science, Computer technology, Forecasting, History, Logic, Mathematics and Rhetoric
. Closed

What are the odds, eh?  On the same day that the Guardian publishes an obituary of theoretical computer scientist, Peter Landin (1930-2009), pioneer of the use of Alonzo Church’s lambda calculus as a formal semantics for computer programs, they also report that the Government is planning only to fund research which has relevance  to the real-world.  This is GREAT NEWS for philosophers and pure mathematicians! 

What might have seemed, for example,  mere pointless musings on the correct way to undertake reasoning – by Aristotle, by Islamic and Roman Catholic medieval theologians, by numerous English, Irish and American abstract mathematicians in the 19th century, by an entire generation of Polish logicians before World War II, and by those real-world men-of-action Gottlob Frege, Bertrand Russell, Ludwig Wittgenstein and Alonzo Church – turned out to be EXTREMELY USEFUL for the design and engineering of electronic computers.   Despite Russell’s Zen-influenced personal motto – “Just do!  Don’t think!” (later adopted by IBM) – his work turned out to be useful after all.   I can see the British research funding agencies right now, using their sophisticated and proven prognostication procedures to calculate the society-wide economic and social benefits we should expect to see from our current research efforts over the next 2300 years  – ie, the length of time that Aristotle’s research on logic took to be implemented in technology.   Thank goodness our politicians have shown no myopic utilitarianism this last couple of centuries, eh what?!

All while this man apparently received no direct state or commercial research funding for his efforts as a computer pioneer, playing with “pointless” abstractions like the lambda calculus.

And Normblog also comments.

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Science and poetry

Published
by
peter
on 2009-09-16
in Logic, Poetry and Rhetoric
. Closed

The Asian scholar Arthur Waley once wrote:

All argument consists in proceeding from the known to the unknown, in persuad­ing people that the new thing you want them to think is not essentially different from or at any rate is not inconsistent with the old things they think already. This is the method of science, just as much as it is the method of rhetoric and poetry. But, as between science and forms of appeal such as poetry, there is a great difference in the nature of the link that joins the new to the old. Science shows that the new follows from the old according to the same principles that built up the old. “If you don’t accept what I now ask you to believe,” the scientist says, “you have no right to go on believing what you believe already.”   The link used by science is a logical one. Poetry and rhetoric are also concerned with bridging the gap between the new and the old; but they do not need to build a formal bridge. What they fling across the intervening space is a mere filament such as no sober foot would dare to tread. But it is not with the sober that poetry and eloquence have to deal. Their te, their essential power, consists in so intoxicating us that, endowed with the recklessness of drunken men, we dance across the chasm, hardly aware how we reached the other side.”    (Waley 1934, Introduction, pp. 96-97)

Reference:

Arthur Waley [1934]: The Way and its Power: A Study of the Tao Te Ching and its Place in Chinese Thought. London, UK: George Allen and Unwin.




Guerrilla logic: a salute to Mervyn Pragnell

Published
by
peter
on 2009-09-13
in Computer Science, Heroes, History, Logic, Mathematics and Matherati
. 5 Comments

When a detailed history of computer science in Britain comes to be written, one name that should not be forgotten is Mervyn O. Pragnell.  As far as I am aware, Mervyn Pragnell never held any academic post and he published no research papers.   However, he introduced several of the key players in British computer science to one another, and as importantly, to the lambda calculus of Alonzo Church (Hodges 2001).  At a time (the 1950s and 1960s) when logic was not held in much favour in either philosophy or pure mathematics, and before it became to be regarded highly in computer science, he studied the discipline not as a salaried academic in a university, but in a private reading-circle of his own creation, almost as a guerrilla activity.

Continue reading ‘Guerrilla logic: a salute to Mervyn Pragnell’

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Computer science, love-child: Part 2

Published
by
peter
on 2009-09-12
in Computer Science, Computer technology, Fiction, Logic and Mathematics
. Closed

This post is a continuation of the story which began here.

Life for the teenager Computer Science was not entirely lonely, since he had several half-brothers, half-nephews, and lots of cousins, although he was the only one still living at home.   In fact, his family would have required a William Faulkner or a Patrick White to do it justice.

The oldest of Mathematics’ children was Geometry, who CS did not know well because he did not visit very often.  When he did visit, G would always bring a sketchpad and make drawings, while the others talked around him.   What the boy had heard was that G had been very successful early in his life, with a high-powered job to do with astronomy at someplace like NASA and with lots of people working for him, and with business trips to Egypt and Greece and China and places.  But then he’d had an illness or a nervous breakdown, and thought he was traveling through the fourth dimension.  CS had once overheard Maths telling someone that G had an “identity crisis“, and could not see the point of life anymore, and he  had become an alcoholic.  He didn’t speak much to the rest of the family, except for Algebra, although all of them still seemed very fond of him, perhaps because he was the oldest brother.

Continue reading ‘Computer science, love-child: Part 2′




Thinkers of renown

Published
by
peter
on 2009-09-06
in Books, Culture, Logic, Marketing strategy, Mathematics, Matherati, Music, Obituaries, Politics, Post-Industrial Nomads, Religion and Science
. Closed

The recent death of mathematician Jim Wiegold (1934-2009), whom I once knew, has led me to ponder the nature of intellectual influence.  Written matter – initially, hand-copied books, then printed books, and now the Web – has been the main conduit of influence.   For those of us with a formal education, lectures and tutorials are another means of influence, more direct than written materials.   Yet despite these broadcast methods, we still seek out individual contact with others.  Speaking for myself, it is almost never the knowledge or facts of others, per se, that I have sought or seek in making personal contact, but rather their various different ways of looking at the world.   In mathematical terminology, the ideas that have influenced me have not been the solutions that certain people have for particular problems, but rather the methods and perspectives they use for approaching and tackling problems, even when these methods are not always successful.

To express my gratitude, I thought I would list some of the people whose ideas have influenced me, either directly through their lectures, or indirectly through their books and other writings.   In the second category, I have not included those whose ideas have come to me mediated through the books or lectures of others, which therefore excludes many mathematicians whose work has influenced me (in particular:  Newton, Leibniz, Cauchy, Weierstrauss, Cantor, Frege, Poincare, Hilbert, Lebesque, Godel, and Kolmogorov).  I have also not included the many writers of poetry, fiction, history and biography whose work has had great impact on me.  These two categories also exclude people whose intellectual influence has been manifest in non-verbal forms, such as through visual arts or music, or via working together, since those categories need posts of their own.

Teachers & lecturers I have had who have influenced my thinking include:  Leo Birsen (1902-1992), Sr. Claver Butler RSM (d. 2009), Burgess Cameron, Sr. Clare Castle RSM,  John Coates, Dot Crowe, James Cutt, Bro. Clive Davis FMS, Tom Donaldson (1945-2006), Sol Encel (1925-2010), Claudio Forcada, Richard Gill,  Myrtle Hanley (1909-1984),   Chip Heathcote,  Hope Hewitt (1915-2011), Alec Hope (1907-2000),  John Hutchinson, Marg Keetles, Joe Lynch, Robert Marks, John McBurney (1932-1998), David Midgley, Terry O’Neill, Jim Penberthy* (1917-1999), Malcolm Rennie (1940-1980), John Roberts, Gisela Soares, Brian Stacey (1946-1996), Frank Torpie,  Neil Trudinger, David Urquhart-Jones, Frederick Wedd (1890-1972), Gary Whale, Ted Wheelwright (1921-2007), John Woods and Alkiviadis Zalavras.

People whose writings have influenced my thinking include: John Baez, Ole Barndorff-Nielsen, Charlotte Joko Beck (1917-2011), Johan van Bentham, Mark Evan Bonds, John Cage (1912-1992), Albert Camus (1913-1960), John Miller Chernoff, Sam Eilenberg (1913-1998), Paul Feyerabend (1924-1994), George Fowler, Kyle Gann, Alfred Gell (1945-1997), Herb Gintis, Jurgen Habermas, Charles Hamblin (1922-1985), Vaclav Havel (1936-2011),  Jaakko Hintikka, Eric von Hippel, Wilfrid Hodges, Christmas Humphreys (1901-1983), Jon Kabat-Zinn, Herman Kahn (1922-1983), Maynard Keynes (1883-1946), Paul Krugman, Imre Lakatos (1922-1974), Trevor Leggett (1914-2000), George Leonard (1923-2010), Brad de Long, Donald MacKenzie,  Saunders Mac Lane, Karl Marx (1818-1883), Grant McCracken, Henry Mintzberg, Philip Mirowski, Michel de Montaigne (1533-1592), Michael Porter, Charles Reich, Jean-Francois Revel (1924-2006), Bertrand Russell (1872-1970), Pierre Ryckmans (Simon Leys), Oliver Sacks, George Shackle (1903-1992), Cosma Shalizi, Rupert Sheldrake, Raymond Smullyan, Rory Stewart, Anne Sweeney (d. 2007), Nassim Taleb, Stephen Toulmin (1922-2009), Scott Turner, Roy Weintraub, Geoffrey Vickers (1894-1982), and Richard Wilson.

FOOTNOTES:

* Which makes me a grand-pupil of Nadia Boulanger (1887-1979).

** Of course, this being the World-Wide-Web, I need to explicitly say that nothing in what I have written here should be taken to mean that I agree with anything in particular which any of the people mentioned here have said or written.

Myrtle Hanley (1909-1984),




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