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Logic and its sub-fields are closely related to philosophy. There is an undeniable mutual interaction between one's philosophical point of view and his/her approach in teaching mathematical logic. In fact you can teach a particular course material in essentially different ways based on your philosophical interests. Some logic teachers teach logic with implicit/explicit indoctrination of their personal philosophical point of views to their students.

As an example once I watched a course video of one of my intuitionist colleagues in his undergraduate "Foundations of Math" class. In that lecture he described the Axiom of Choice as follows:

The Axiom of Choice is not intuitively reasonable because it suggests assuming the existence of an invisible entity. This axiom doesn't say anything about the way which one can produce a choice function. It just says that such a function exists somewhere in the ideal mathematical universe.

Some years later some of the students of that class participated in my "Advanced Set Theory" graduate course as a part of their voluntary graduate courses. It was clear that they don't like the uncountable cardinals at all. Also everywhere which I proved the essential use of Axiom of Choice in a usual mathematical theorem using forcing construction, they said something like this:

...Thus this theorem is not intuitively reasonable and we should remove it from our mathematics.

None of these students were working on logic. They just passed a few elementary courses in their undergraduate curriculum. I don't know if this is just a coincidence or it is an impact of participating in a logic course of an intuitionist teacher. Based on such facts there are arguments among logicians of our department about possible impacts of this ideological teaching on students.

Those who are agree with ideological teaching in logic courses say:

  • It is an undeniable right of each teacher to introduce the teaching material by their own philosophical point of view.
  • The impacts of a teacher's personal philosophical point of view on the students is minor. They can see different arguments in different references which are accessible everywhere.

Those who are not agree with ideological teaching in logic courses say:

  • It is nice to introduce philosophical aspects of mathematical theorems in logic courses but the teachers should be neutral in explaining these ideas because a logic teacher's duty is teaching logic to students not indoctrinating his/her personal point of views.
  • The impacts of a teacher's personal philosophical point of view on the students could be essential because the teacher is students' main reference and so he/she has a great influence on forming their research interests and scientific future. Thus by an ideological teaching, a teacher restricts the freedom of his/her students to choose their own philosophical point of views.

Question. What are possible (dis)advantages of the ideological teaching in logic courses? Could it be harmful/useful for students? Does it have a notable impact on students' research interests in future? Is there any math education long term research to determine the similarity of students and teachers' point of views in logic and philosophy of mathematics?

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    $\begingroup$ I really know nothing about this, but I liked this comment on Mathoverflow by Charles Stewart: > But the best starting point is probably Proofs and Types [by J.-Y. Girard] [...] reading at least up to the proof of cut elimination. A warning: Girard's style is a little slippery, and it is common for students to say they have read it, who turn out to have absorbed the opinions but little of the results. $\endgroup$ – Marius Kempe Mar 31 '14 at 18:30
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    $\begingroup$ Personally I think it's fine as a teacher to tell students one's opinions on whether certain axioms apply to the real world, as long as one also encourages them to question their teacher's opinions. Of course this means that one must be able to handle discussions in a fair manner, and in such a way so that students do not assume that teachers are ideologically superior in any way. $\endgroup$ – user21820 May 31 '14 at 10:34
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What are possible (dis)advantages of the ideological teaching in logic courses? First I'd say that ideological teaching is not restricted to logic. Granted, it is more likely to encounter ideology in a logic class, but not only there. I've had a lecturer in differential equations who told us that all of mathematics is basically concerned with solving differential equations (this exaggeration did not serve to make the course more interesting than it was (and it was, perhaps not to all students, but it was)). I've heart algebra enthusiasts bad-mouth the concreteness of analysis and analysts bad-mouth the abstractness of algebra. I guess it is only natural, but should be avoided. I think it is obvious that some of the (dis)advantages of ideological teaching is that it may alienate some students and draw others to the subject matter. The thing is you don't know what will happen to which student.

Could it be harmful/useful for students? Yes, it can be both harmful and useful. If your ideology closely fits that of the mainstream and you thus push a good student to pursue working on a problem many people appreciate, then you (probably) increase their potential for landing a job. On the other hand, if you discourage a good student from following a seemingly tangential area of mathematics that ends up being really important then it may do harm. As said above, it is very difficult to know in advance. But, in my opinion, ideology should be presented as such. I may like or dislike the axiom of choice, but what I teach should reflect (largely) the mainstream. I may provide my own opinion of course, but present it as such.

Does it have a notable impact on students' research interests in future? It certainly might (as the cases you describe would indicate). In my experience, some students are highly influenced by peers and experts while others are less so.

Is there any math education long term research to determine the similarity of students and teachers' point of views in logic and philosophy of mathematics? I have no idea.

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    $\begingroup$ Thank you very much. Your answer is really good. $\endgroup$ – user230 Apr 2 '14 at 0:47

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