Interpreting informal mathematical language
by Arnold Neumaier
October 31, 2009
Most people think that mathematical language is very precise. Indeed, we usually understand each other easily when talking about familiar mathematics, and can easily recognize and resolve misunderstandings.
Within our project FMathL -- Formal Mathematical Language, we are trying to teach the computer abstract mathematics written in the usual informal language (such as in a Latex document). This requires to become very conscious of the precise meaning of mathematical statements, since they must bee unambiguously communicated to the computer.
It turns out that the interpretation of an informal text depends very much on which formalization of the informal mathematical language one has in mind when thinking about a mathematical subject -- even about very elementary things such as the natural numbers.
Therefore it is important to find out what is the dominant interpretation mode among the many possibilities.
I had naively supposed that my mode were the dominant mode, since I had seen lots of different kinds of mathematics during my life and never had any problems understanding other mathematicians, once the terminology was introduced.
However, recently I came across some category theorists who have a completely different interpretation framework. (See the links in my essay on Foundations of Mathematics.)
Therefore, on October 5, 2009, I posed to our faculty and to all research assistants of our mathematics department the following questionnaire.
Eine Frage zur mathematischen Vorstellung
A question on mathematical intuition
Liebe Kolleginnen und Kollegen,
bitte nehmen Sie sich eine Minute Zeit, um die folgenden beiden Fragen
(ohne grosses Nachdenken) zu beantworten:
please take a minute of your time to answer (from the top of your head) the following two questions:
1. Sind Primzahlen Bestandteil der mathematischen Struktur der
Are prime numbers part of the mathematical structure of natural numbers?
2. Bilden die natürlichen Zahlen eine additive Halbgruppe?
Do the natural numbers form an additive semigroup?
Falls Ihre Zeit knapp ist, können Sie hier zu lesen aufhören und
If you are short of time, you may stop reading here and reply directly.
Thanks a lot!
Für die Neugierigen:
For the curious:
Die Fragen zielen darauf ab, ob unsere Intuition der informellen Praxis
oder der rigoros formalisierten Präzision folgt.
The questions aim at whether our intuition follows more cloesly informal practice or strictly formalized precision.
Bis heute hatte ich nämlich gedacht, dass beide Antworten intuitiv
selbstverständlich Ja seien.
For until today, I had thought that both answers are trivially yes.
Heute stellte sich heraus, dass es Mathematiker gibt, deren Intuition
hier meiner genau engegengesetzt ist.
Today it turned out that there are mathematicians whose intuition is here exactly opposite to mine.
Ich möchte mir daher ein statistisches Meinungsbild über die
intuitiven Vorstellungsinhalte machen, die Mathematiker(innnen) zur
Beziehung zwischen diesen Begriffen formen.
Therefore I want to form a statistical spectrum of opinion about the intuitive contents of the imagination that mathematicians form about the relation between these concepts.
Antworten mit einem Informationsgehalt von mehr als zwei Bits sind
natürlich auch willkommen.
Replies with an information content of more than two bits are of course welcome, too.
I got 34 answers within 10 days (from an estimated number of 170 mathematicians asked), and no further answers later. The tally came out to be
To define prime numbers, one needs the multiplication. Thus the first question can be answered with yes only if multiplication (and further definitions based on it) is considered to be part of the ``mathematical structure of natural numbers''.
Some people also think that a set N on which operations other than + are defined, is, strictly speaking, not a semigroup, though the pair (N,+) is one. These would have to answer no to the second question if they answer yes to the first -- unless they automatically strip N from its multiplicative structure and translate the second question to formally mean ``Is (N,+) a semigroup?''
Some people think of the natural numbers as being defined by the Peano axioms only. Then there is only a distinguished zero and a successor function, not enough structure to have either primes or a semigroup. (This would require what is commonly called a conservative extension of a Peano system.) Then both questions are answered with no.
Thus the answer depends very much on which formalization of the informal mathematical language one has in mind when thinking about natural numbers.
The overwhelming majority seems to think like me that the natural numbers have all the structure one usually learns about it, and that saying that an object is an additive semigroup is the same as saying that one has an associative addition (no matter what other structure is present).
Comments received from others:
FMathL -- Formal Mathematical language
my home page
Arnold Neumaier (Arnold.Neumaier@univie.ac.at)