Posts Tagged ‘cambridge’

(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.

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