About metamathematical statements in a recent post by Timothy Gowers

(This is a comment about this post by Professor Timothy Gowers, in a series for mathematical undergraduates he has started there.).

I was delighted to rediscover several distinctions about mathematical discourse in ordinary english I usually do not keep consciously in mind when writing.

But you write in the first part:

“Here are a few metamathematical statements.

-1- “There are infinitely many prime numbers” is true.

-2- The continuum hypothesis cannot be proved using the standard axioms of set theory.

-3- “There are infinitely many prime numbers” implies “There are infinitely many odd numbers”.

-4- The least upper bound axiom implies that every Cauchy sequence converges.

In each of these four sentences I didn’t make mathematical statements. Rather, I referred to mathematical statements.”

I beg to differ slightly.

First, many metamathematical statements are mathematical statements in a larger theory and can often be treated as mathematical objects (for example Model theory).

Second, I would have drawn important distinctions between those four (but it was not exactly the subject of your post which is already very detailed).

Further, each of your four sentences implies a specific mathematical universe with minimal logical and set-theoretic axioms for it to be meaningful and unambiguous. I think this is important to point out to young mathematicians.

In these sentences we have most of the time silent implications together with an explicit “implies” (see below).

There is a topological analogy: most properties of, say a knot, depend on the space it is embedded in.

Not that these sentences do not have the same implied strength or the same immediate relevance for the mathematician, undergraduate or not. So I prefer to rephrase them with parameters and implicit hypothesis.

-1- Theorem A is true (in implied theory T)

-2- Axiom C is independent from Axiom-System S

-3- Theorem A has Corollary B (in implied theory T common to A and B)

-4- Axiom L (added to implied Theory R) gives it the strength to prove Theorem V.

The first sentence is of the most common kind for a mathematician.

The third sentence is very common as well and is a very small step from -1-.

Both -1- and -3- are used so frequently that the distinctions between mathematics and metamathematics is blurred as in common metalinguistic sentences people use every day : “Please, can you finish your sentence?” or “Do not answer this question!”

The fourth one is of strong metamathematical character and of interest to most mathematicians, because Theorem V is useful and a common way to express continuity. It could be paraphrased/expanded : one of the solutions to create a mathematical universe where you can have a notion of continuity for your analysis theorems is to have a Theory R consistent with Axiom L and add this axiom L to R, creating Theory R2 and go on with finding limits.

But the second one is the strongest of all, the most “meta” and the only one to be explicit about its metamathematical context. It is part of a family of statements of about “relationships between logical contexts in which you can do mathematics”. You can call that meta-trans-peri-mathematics or meta-meta-metamathematics.

It would be very difficult to find an equivalent to -2- in a non mathematical situation. It would be considered at best very subjective or dogmatic such as “You cannot speak about the “Gestalt” philosophical concept in english without using the german word “Gestalt” or another philosophical german word of equivalent depth and power. You will always fail if you try.”

The remarquable thing about mathematics is that we can reach a so strong level of implication in our discourse about it.

Read Full Post »

Mathematical acids

I just learned there is an abscisic acid as well as a phaseic acid . A pity these plant hormones are not simple substances you could give to students of analytical geometry to make them understand theorems and concepts better.

Read Full Post »

Correlating Composers

If you use one of the newest Google tools, Google Correlate, you can look at typical associations between keywords based on the geographical search pattern.

Here is an example for classical music , looking for debussy and getting liszt :

The list of correlated terms is not really surprising:

0.8860 liszt
0.8473 tchaikovsky
0.8251 g minor
0.8100 c minor
0.8090 tetrachord
0.7887 maurice ravel
0.7856 beethoven
0.7789 well-tempered clavier
0.7774 brahms
0.7760 symphony 5
0.7742 bach
0.7735 prelude in c
0.7689 bach toccata
0.7665 stravinsky
0.7631 bach fugue

This is just a list of very common keywords for students of classical music. A few composers, a few tonalities. The really strange point is why does Google receive a concentration of these keywords from a single state? Why are Utah users so much more interested (or clueless but wanting to know) about Debussy, Liszt or Bach?

Is there something like a geographically pertinent “classical America”, where a significant part of the public is interested in classical music?

this list of correlates to Franz Schubert demonstrates the relevance of this kind of correlation measures in simple (and culturally segregated) cases. All the members of this short list are relevant and could be used for feeding a simple expert system. The first non musical search term is for “diabetes symptoms” later in the list.

Read Full Post »

Science and the seasons

I made a little Google Trend request, here is a snapshot:

What is striking is the seasonality. People look up science-related terms when they feel compelled to do so, when classes are going to start or restart (September, the very end of December, before going back to school, trying to find a private tutor, a book, a quick answer on wikipedia).

In fact you can even guess the location of class holidays and exams in, say, France, that way

The large down bowl fits the summer vacation and the accentuated peak before is the time of the Baccalauréat (high-school final/college entry exam). The small down trends fit the starts of the various school vacations quite well, the largest and easiest to find being the year-end holidays (usually 15 days, starting a few days before Christmas).

The news volume increase for physique in the lower part is a consequence of the word being used in english for body shape (unfortunately a topic that news media find more interesting or easier to sell than physics).

Read Full Post »

Harry Potter and the Methods of Rationality

That’s the first I time about a fanfiction, but this one is certainly worth it. Its size and quality has been already recognized by many readers and has suscited more than 12000 reviews.  The first chapters were published in February this year (2010). EDIT 2011/06/01: there are now 72 chapters.

Harry Potter and the Methods of Rationality

It is an alternative history of Harry Potter, with a few initial different hypothesis who have far reaching consequences compared to the choices made by Mrs Rowling.

It is a very good mix of (among other things):

- Harry Potter (mainly the first three books) by you-know-who for the basic context and overall plot and characters

- The non-A world by van Vogt (explaining rationality and sanity)

- Ender’s Game (battle school and growing for an incoming battle) by Orson Scott Card and its sequels

- Dune (a few allusions to the first book, but especially The Emperor of Dune and its reflection on power)

 

As the author himself says, it is a good idea to have read at least the first volume of the original Harry Potter, but I would even says that you enjoy the alternative history even more if you have read at least to the third one (or seen the films).

The author’s site is there as well as one of his co-projects.

Read Full Post »

algebra, abstraction and Marcel Proust

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.

Read Full Post »

Blog and opinion republishing

I used to have personal webpages, opinion pages, archived scientific and philosophic correspondence, even blogs on the web on various periods but most of them are now unreachable, deleted or non editable. This forms what I use to call volume XVI of my collected works (volume XIV being notes taken during seminars and lectures). All the posts in this blog with a date before April 1st 2010 are republications of previous contributions of this kind. New information or new ideas added after the original publication will be outlined with the date.

Read Full Post »

Splendid Cello wood box wins “Fine Woodworking Magazine” contest

The other entries are really worth looking at

http://www.finewoodworking.com/item/10986/cello-box-wins-tablesaw-prize

 

Read Full Post »