Category: mathematical

Skepticism on present mathematical education

Sep 19, 2020
Modern children spend a lot of time learning arithmetic in school, chiefly the four operations on real numbers, solution of linear systems, operations on polynomials, and so on. Beginning perhaps at the high school level, quite many of them struggle, and others find themselves relatively good at it. Consequently, they either choose, or abandon, mathematics related university departments and occupations, depending on their performance.

The thing that troubles me is that, I suspect the traditional mathematics curriculum does not explore well, nor predict well, the innate talent of the students’. Indeed, viewed in a broader perspective, these operations are some straightforward algorithms in disguise. The students are really implementing a lambda-calculus or a Turing machine with their brain. If one is capable of basic programming, one would find the operations on a field, changing basis for a matrix, Euclid algorithm on a polynomial ring … they may be tedious to implement, but nothing more than a rigid following of rules. There is a lot more to explore in mathematics than these. Unfortunately, children surely do not know they are themselves Turing machines when they take an exam, and we cannot make them understand before they are exposed to concrete objects — which seems hard to debate too.

Still, the problem is, we practically require the children to invoke their vague intuition on computation. Those who return more correct answers in exams, win. What’s the big deal about that? The reason that some children are doing well is simply that they are quicker to realize, implicitly, the way to run a program with their brain; and the reason that some are not doing well, that they are slower. For the latter, they will never be revealed of the mystery, and will probably remain to think mathematics as an enigma.

What we can do about the situation, I don’t know for sure. There might, I hope, be a way that children may be introduced, somewhat, of the concept of computation, so that they are self-aware when they do it. If I am right, it might turn out in the future that the present way we classify students’ inclination is borderline silly, and far from unleashing their gifts.

The liminf and limsup for sets

Liminf and limsup for sets will be used in the proof of Borel-Cantelli Lemma. There are two equivalent definitions on Wikipedia, and it seems weird at first. Namely: liminf (resp. limsup) is defined as union of intersections (intersection of unions), and is also said as occurring infinitely often (occurring for all but finite indices). Let us verify their equivalence.

Let a sequence of sets, A_1, A_2, \dotsc be given. By definition,
\limsup\limits _{n \to \infty} A_n :=\bigcap\limits _{k =1} ^{\infty} \bigcup\limits _{n =k} ^{\infty} A_n … [1]
We claim
\limsup\limits _{n \to \infty} A_n := \bigcap\limits _{k =1} ^{\infty} \{\omega:\; \omega \in A_n \quad \mathrm {infinitely\; often}\} … [2]
By definition of union,
=\bigcap\limits _{k =1} ^{\infty} \{\omega:\; \exists n,\; n \geq k,\; \omega \in A_n\}
By definition of intersection,
=\{\omega:\; \forall k\; \exists n,\; n \geq k,\; \omega \in A_n\} … [3]
The meaning is completely the same as “infinitely often”, as claimed.

Alternatively, we may define
\limsup\limits _{n \to \infty} A_n =: \{\omega: \limsup \mathbf{1}_{\{A_n\}} [x] =1\}

Similarly, suppose
\liminf\limits _{n \to \infty} A_n :=\bigcup\limits _{k =1} ^{\infty} \bigcap\limits _{n =k} ^{\infty} A_n
We claim
= \bigcap\limits _{k =1} ^{\infty} \{\omega:\; \omega \in A_n \quad \mathrm {for\; all\; but\; finitely\; many}\; n\}
By definition of intersection,
=\bigcup\limits _{k =1} ^{\infty} \{\omega:\; \forall n,\; n \geq k,\; \omega \in A_n\}
By definition of union,
=\{\omega:\; \exists k\; \forall n,\; n \geq k,\; \omega \in A_n\}
The meaning is completely the same as “all but finitely many”, as claimed.

Alternatively, we may define
\liminf\limits _{n \to \infty} A_n =: \{\omega: \liminf \mathbf{1}_{\{A_n\}} [x] =1\}

Definite Description and Context

0. Background

It is, in a possible form, one of the central concerns of analysis philosophy to translate the ordinary utterances into predicate logic propositions. And I have increasingly felt that the goal, though important, might turn out to be inadequate and doom to fail. Indeed, I argue that the natural sentences may depend highly on context, and the context can be difficult or impossible to define by logic only.
Let us examine, for concreteness, Russell’s theory of definite description, namely the translation of the definite article “the” into predicate logic. The theory does not take account of linguistic context. Thus I am inclined to say that it is problematic.
According to Russell, a sentence that contains “the”, such as

(1) The F is G

Is identical to

(2) There is exactly one F, and F is G.

Thus,

(3) The capital of The UK is a beautiful city.

is identical, as of now, to

(4) There is exactly one entity so that: it is capital of The UK; and that entity is a beautiful city.

1. The example of incomplete description

The first objection, which is certainly not original, is of incomplete description. Say that: Alice has a dog, and the dog is ill. The sentence

(5) The dog of Alice’s is ill.

There are likely other people called Alice who keep dogs, and Alice may keep several dogs. And this is problematic even if we add other attributes that Alice’s dog possess:

(6) The dog of Alice’s, of black-dotted fur, of height 53.02cm, [etc….], is ill.

Possibly doesn’t point to a single dog either. In fact, the dog being referred is limited to the context (for example, what people in mind who is talking to Alice), and it is absurd to suggest the speaker of (5) intends (6).

2. The example of twin earth

Inspired by the Twin Earth Experiment, I think of a second objection. Consider that, in a possible world, there is another planet very far away, say in Andromeda, which is a Twin Earth, identical in every regard to our Earth, and every person on the Earth has an exact twin on the Twin Earth. Bob, from our Earth, arrives the Twin Earth in Andromeda, and says the sentence (3).

Now (3) does not translate to (4) in people in the Twin Earth. In the alien’s context, “The capital of The UK” is also called “London”, but it lies in the Twin Earth.

3. The example of physical reference

Moreover, I argue that the speaker who use the definite description usually does not really check the definition of the referred object, and may not even be capable in verifying the conditions that the description holds. Consider that Charlie is walking in the field and looks at a particular white duck. Later, Charlie saws and the duck again in a group of ducks. Charlie says to someone:

(7) I saw the white duck in the group this morning.

Russell would think that (7) can be translate to be

(8) There is only one thing so that: it is in the group, it is a duck, and was seen by Charlie this morning.

But it may be beyond Charlie’s capability to check (8), nor does he have good reasons to believe (8). The group of ducks may contain a white goose, which Charlie lacks the expertise to distinguish. Indeed, to justify (8), Charlie has to check, one by one, in the group, that “It is a duck” (which is difficult), and that he has saw it (which he did not really check). Most likely, Charlie does not know the precise definition of the duck, including that it is named under binomial nomenclature to be Anas platyrhynchos, that its bill width and wing length are within technical regulation, and so on. And in the group of duck there may be a white goose, which can be hard to tell from duck. In reality, the proposition “It is a duck” need not be evaluated as thus, for Charlie has physically followed the white duck, has pointed it out, and has good reasons to believe (7) is true.

4. Conclusion

In each case, I am inclined to think that it is unlikely that any countably-many conjunctions or disjunctions of predicate logic propositions can be faithfully translate the context of relevant dialogue. If the context is just “something more” than logical predication can deal with, will any theory of definite description ultimately fail?

5. References

  • Peter Ludlow, “Descriptions“, The Stanford Encyclopedia of Philosophy.
  • 王文方(2008)，形上學，台北:三民