Category: mathematics, logic

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“When I imagine a triangle, even though such a figure may exist nowhere in the world except in my thought, indeed may never have existed, there is nonetheless a certain nature or form, or particular essence, of this figure that is immutable and eternal, which I did not invent, and which in no way depends on my mind.–Rene Descartes, Meditations on First Philosophy (1641) tr. John Cottingham, Descartes: Meditations on First Philosophy (1986)

This is the lovely response by the great mathematician J.J, Sylvester to Thomas Huxley’s muckety comment on the lack of imnagination in the mathematical sciences.  Huxley’s remarks were made at a meeting of the British Association for the Advancement of Science, stating that (Mathematics) “is that study which knows nothing of observation, nothing of induction, nothing of experiment, nothing of causation”¹ This quote is taken from Sylvester’s quick and elegant responses in two articles in Nature, December 30, 1869 (231-3) and January 6, 1870 (pp 261-3), as “A Plea for the Mathematician” and “A Plea for the Mathematician II”.

“For Sylvester, the ability to be able to imagine  what the experience of space would be like in dimensions other than three is sufficient to establish the empirical basis of geometry–the three-dimensional Euclidean is not the science of space in general, but the science of the space of our experience.”–Fact and Feeling: Baconian Science and the Nineteenth-Century Literary …by Jonathan Smith, pp 181-182

Ex nihilo nihil fit/Nothing comes out of nothing.–R Descartes, Principia philosophiae, Part I, Article 49

The original papers are available on this blog’s bookstore section.

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“How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.”--mnemonic device by James Jeans on remembering pi to 15 places, where each word length assoicates a number in pi.  Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag, pp 44-5.

There are simpler ways of remembering something than developing an odd mnemonic for remembering pi, especially when the memory device will get you only 30 places or so…and so it hardly feels worth the effort to remember some posie and then write it down to perform a pi-matic numerical translation.  This is especially so when you consider that the man recognized as being the world record holder but not so (Akira Haraguchi) committed the fist 100,000 places of pi and took nearly a day to recite it–megaefforts like this make the smaller accomplishments of remembering pi to 100 places seem fairly insignificant. 

Some of the people set to remembering pi (piphilologists) use methods similar to this as memory devices, including an entire book of 10,000 words constructed in just this way–many more, I think, use a memory palace/method of loci method, locating numbers and identifying sub-patterns and placing them in connecting “departments” in the brain.  (The Big Book on memory devices is by Frances Yates, The Art of Memory, University of Chicago Press, 1966; also Jonathan Spence, The Memory Palace of Matteo Ricci, Viking Penguin, 1984.)

I’m not sure what the need is for remembering the number to so many places when it seems as though the first seven digits will be sufficient for most (when “most” = “just about everything”) things. 

On remembering pi, from Nature, volume 72, p 558, 1905:

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This is a very special effort on behalf of logic and the law, and may have been the first effort to codify the idea of the law in symbolic logic.  It appeared in the journal Nature on 24 April 1913, and I think in spite of it being interesting and even pretty, it just doesn’t work for me.  But I’m reproducing the article (a long one for Nature) because it is really such an audacious thing, and a terrific idea, if not a good one.

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The question above is probably seldom asked and probably even less than so required, but it sure does look good for the title of a post.  The pillowed polyhedra came into light while I was breezing through some issues of the British Nature magazine for 1893/4–the title of the work of course was a lovely thought and the author of the paper was the mega-gifted and highly significant Lord Kelvin (William Thomson), but the bait that made me swallow the hook on the paper was the illustration.

There are many (i.e., 1,496,225,352)  different forms of the convex 14-faced polyhedron tetrakaidekahedron (see Wolfram Math
for a quick summary)–this is the first I can remember being displayed
on a 19th century pillow.  Kelvin was particularly interested in
displaying geometrical figures in three dimensional space, and would
return to the issue numerous times, particularly in the Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light (see page 606ff).

Notes:

Lord Kelvin, (William Thomson, 1824-1907), “On the Homogeneous Division of Space”, in Nature, two issues, as follows:  (1) 8 March 1894, pp 445-448; and (2) 15 March 1894, pp 469-471. A version of this paper appears slightly earlier (17 January) in the Proceedings of the Royal Society. 

Both papers are available for purchase via our blog bookstore, here.

These are remarkably interesting papers–especially considering their brevity.

It is interesting to note the list of what Thomson “is known for” in his Wiki entry:

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I am probably not understanding this wonderful effort¹ by the dead Lewis Carroll, establishing a new algorithm for doing long division.  It came into view while I was looking for a paper in Nature by A.A. Michelson on his analog computer, the harmonic analyzer, a fantastic device that he used to help measure the speed of light back in 1898.  Carroll (as Dodgson) appeared just a few pages away²; his obituary appears not much after the division article, thus making the paper the last of his career here on Earth. 

It is a very interesting effort, and it perhaps is even brilliant and of a wonderful construct, but the bottom line is that it makes the process of division harder to do–perhaps it is mire understandable as a process, but the process itself is decidedly not a preferred one.  Perhaps it is appropriate for his last work to have been on mathematics, since the majority of his 20 published books relate to mathematics or logic.

Notes

1.  DODGSON, Charles Lutwidge (‘Lewis Carroll’). “Abridged Long Division”, in Nature, page 269, in the weekly issue for 20 January 1898, with the original outer wrappers removed from larger bound volume. Very good copy. It is here that Carroll announces (in a long 3pp letter to the editor) his implementation of a new algorithm for division–it makes the process longer and more difficult, unfortunately.

The article is abstracted here, a few weeks after publication:

Article abstract:  Nature 57, 390-391 (24 February 1898), “Abridged Long Division” abstracted by ROBT. W. D. CHRISTI:

“HAVING been working on similar lines for some years, I was very much interested in the late Mr. Dodgson’s letter on abridged division in NATURE of January 20, and I should like to offer a few observations and to give a variation of the method which appears much simpler. It will be admitted that Mr. Dodgson’s plan is of limited application, and rather complicated for general use. There is nothing to hinder the method given below from being universally used, though it may not in all cases be the shortest. It also has the merit, I think, of directness and uniformity.”

2. This is one of three publications of Carroll’s mathematical works that appeared in Nature, being “Abridged long division”, Nature 57, 269-271; “Brief method of dividing a given number by 9 or 11”, Nature 56, 565-566; “Pillow problems. Curiosa mathematica. Part II”, Nature XLVIII. 564. 

The following list of Carroll’s mathematical works is taken from the table of contents of The mathematical pamphlets of Charles Lutwidge Dodgson and related piece, compiled, with introductory essays, notes, and annotations, by Francine F. Abeles. (New York : Lewis Carroll Society of North America ; Charlottesville : Distributed by the University Press of Virginia, 1994.)

Geometry

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It seems so unlikely that the Venn diagrams took so long to appear, that it hadn’t been Aristotle’s Diagrams, or Cartesian Diagrams; rather, these elegant creations–about a different way to represent propositions by diagrams– belong to Mr. Venn’s invention in 1880. Mostly. It was in that year that these famous more fully-developed diagrams first appeared in a paper entitled “On the Diagrammatic and Mechanical Representation of Propositions and Reasonings“ in the Philosophical Magazine and Journal of Science.  They seem so beautiful, and so simple, and so powerful, and so elegant. So necessary.  And so why-did-they-take-so-long-to-appear, so obvious.  In a way, they are in that category of The Great Obvious Invention, in a similar vein to Mr. Luke Howard first classifying the names of clouds  in 1803.  Sometimes things are not obvious until they are.  

[Source: Internet Archives, John Venn, Symbolic Logic, 1881, full text.]

Well, almost.  The diagrams did mostly make an appearance earlier–though not in the Venn form–in the work of Christian Weise (d. 1708) and then in that of Leonhard Euler (1707-1783), that according to the great William Rowan Hamilton, explained in his work Metaphysics and Logic (1858-1860). But the final form seems to be that of Venn.  

Hamilton’s work appears so:

They also appeared in the Proceedings of the Cambridge Philosophical Society, volume 4, 1880, in his article “On the employment of geometrical diagrams for the sensible representation of logical propositions” in which he discusses the history of the diagrams, including the Hamilton history.  Venn also includes this beautiful logic template of Bolzano:

I found a wonderful book review of Venn’s book Symbolic Logic that appeared the following year, appearing in the journal Nature on 14 July 1881, and reviewed by the inimitable W. Stanley Jevons (which is reproduced below).