| CARVIEW |
Am I the only one to find this earth being created under the compass out of chaos looking a lot like a Mandelbrot Set M ?
The image of God using a compass is famous (see for instance the much later William Blake) but this one is curious by the image of the world. It dates from the 13th century in a splendidly illustrated and fastuous kind of Bible digest and commentary called “a bible moralisée” made for royals of France.
(excerpted from A. Borovik's blog Mathematics under the microscope)
This work by William Blake could suggest that the mathematician and its mathematics emerge from geological sediments, old life fossilized, crystals and stones, through a geometrical body, allegorically as perfect as his theorems, and reminiscent of the Renaissance artists such as Leonardo da Vinci associations between perfect body proportions, art, architecture and the Universe.
It also reminds of the much later and severely misleading “path of evolution” images going from fish to men.
In this passage, after having received a letter from Albertine, the narrator makes a comparison between the abstraction that mail, faces, even bodies represent of the thoughts of people we know and the ideal thought we have of them, with the successive abstractions that in mathematics arise from objects to number sums and from numerical expressions to algebraic ones.
Cependant, je relisais sa lettre et j’étais tout de même déçu du peu qu’il y a d’une personne dans une lettre. Sans doute les caractères tracés expriment notre pensée, ce que font aussi nos traits : c’est toujours en présence d’une pensée que nous nous trouvons. Mais tout de même, dans la personne, la pensée ne nous apparaît qu’après s’être diffusée dans cette corolle du visage épanouie comme un nymphéa. Cela la modifie tout de même beaucoup. Et c’est peut-être une des causes de nos perpétuelles déceptions en amour que ces perpétuelles déviations qui font qu’à l’attente de l’être idéal que nous aimons, chaque rendez-vous nous apporte, en réponse, une personne de chair qui tient déjà si peu de notre rêve. Et puis quand nous réclamons quelque chose de cette personne nous recevons d’elle une lettre où même de la personne il reste très peu, comme, dans les lettres de l’algèbre, il ne reste plus la détermination des chiffres de l’arithmétique, lesquels déjà ne contiennent plus les qualités des fruits ou des fleurs additionnés. Et pourtant, l’amour, l’être aimé, ses lettres, sont peut-être tout de même des traductions (si insatisfaisant qu’il soit de passer de l’un à l’autre) de la même réalité, puisque la lettre ne nous semble insuffisante qu’en la lisant, mais que nous suons mort et passion tant qu’elle n’arrive pas, et qu’elle suffit à calmer notre angoisse, sinon à remplir, avec ses petits signes noirs, notre désir qui sait qu’il n’y a là tout de même que l’équivalence d’une parole, d’un sourire, d’un baiser, non ces choses mêmes.
Proust, Albertine Disparue, A la Recherche du Temps Perdu.
My own english translation of the outlined passage:
As one does not find anymore in the letters of algebra the determinate values of the figures of arithmetic, themselves already removed from the qualities of the fruits and flowers being summed.
The purpose of models is not to fit the data but to
sharpen the questions.
Samuel Karlin
Even for the industrial engineer trying to control a production process, the design of a model is just a step in the continuous process of understanding what happens in the plant. One can compare this quotation with another famous one:
The purpose of computing is insight, not numbers.
Richard Hamming, opening quote of Numerical Methods for Scientists and Engineers, McGraw-Hill, 1962
A familiar concept in adaptative models is overlearning. If your model is (temporarily) fitting the data really very well, it will probably not be robust : it depends too much on transient components and properties of the data. It has stopped learning and started recording, memorizing, compressing the data, not abstracting it. That can easily happen if you don’t have enough data with sufficient dimension and sufficient volatility in regard to the complexity of the underlying mechanism and the precision you require in one learning step or one hierarchical level. Feeding more data is not the solution, especially if you are not prepared to redesign your model or improve and filter your inputs.
That is the same in educating humans: in many cases, confronted with a series of exercices, it will be easier for the pupil to learn a false and chaotic series of local rules and exceptions about what to do than to reshape and extend his understanding. The next series of exercices will not be a motivation to simplify the previous system of beliefs, it will just mainly be erased to leave room to the new urgent version about tomorrow’s homework, and so on. What will remain is certainly not sorted according to certainty : all rules remembered are more or less equal in status, all knowledge will a series of unconnected and fading islands with a few if any lighthouses.
When one asks students struggling with some material, to explain aloud their reasoning when solving a problem, a large part of their troubles are enshrined coincidences taken as infaillible shortcuts. They project all feelings of confusion or complexity on the subject matter and on the teacher, not on the current state of their learning process, exactly as they have been prepared to do by student folklore.
I have created a page with a small bibliography I will try to update and comment. Please post comments here for addition and suggestions. I welcome external reviews.
Another commented list of popular or scholarly treatment of contemporary mathematics.
I started this for MathOverflow recently.
Please post comments for suggestions.
Two books by Olivier Keller fit quite nicely with my interrogation about prehistorical mathematics.
AUX ORIGINES DE LA GÉOMÉTRIE : LE PALÉOLITHIQUE
Monde des chasseurs cueilleurs
Vuibert editions, Paris, 2004
LA FIGURE ET LE MONDE : UNE ARCHÉOLOGIE DE LA GÉOMÉTRIE
Peuple paysans sans écriture et premières civilisations
Vuibert editions, Paris, 2006
I intend to write a review of each of them that I will post here.
Harry Dym ‘s definition of Hell:
Hell is a boring incomprehensible math talk that goes overtime .
As quoted by Doron Zeilberger for instance here.
Other personal definitions ?
Reading one recent version of the book by Alexander Borovik, Shadow of the Truth made me remember a pet project of mine I used to call “Prehistorical Mathematics”, almost twenty years ago.
It has a lot of intersection with several area of studies:
But with a different emphasis. While I am interested in the way our ancestors have been developing the mathematical culture of humanity, my focus is on what kind of mathematical intuition and development we have been forbidding ourselves when we have developed the precursors of our current mathematical views.
Biface de Saint Acheul - face and front view of a stone tool
What are the kind of questions that should be investigated ?
Just a reminder of different forms of the most common names for quasigroup theory in various languages.
| en | quasigroup | loop | latin square |
| fr | quasigroupe | boucle / loop | carré latin |
| de | Quasigruppe (n) | Loop (!) | lateinisches Quadrat |
| it | quasigruppo | loop (!) | quadrato latino |
| ru | Квазигруппа | лупой (?) / петля | Латинский квадрат |
| ch | 拟群 | 圈 | 拉丁方陣 |
| fi | kvasiryhmä | ||
| cz | kvazigrupa | latinským čtvercům |
One can see that in some cases the english original “Loop” has not been translated. I have doubt for the current practice in russian. What I have seen looks like a wordplay on the fact that the russian word for spyglass (coming from the french loupe) is used for loop. Do any mathematical article use петля or a name of the same family ? In chinese, the character that has been adopted is the one for circle.
All suggestions, corrections and comments welcome.
Tasteful algebra and quasi objects of study