Order of a study

Question

So i just recently had to drop two math courses, topology, math logic, because my math maturity wasn't up to the level needed to excel in them. I intend on taking them again, but not without first more preparation which leads to my question. Which order would i benefit more from in preparing for the courses? Topology first and then math logic? Or math logic first and then topology?

Thanks

Answer 1

Firstly I would like to clear up some things that I disagree with that have been mentioned in the comments.

To the contrary to some things mentioned above (in the comments), I think that taking topology before real analysis can be incredibly advantageous, since real analysis relies heavily on topological concepts (as do many areas of math).

As an undergraduate, I took my first analysis class before topology. I wouldn't recommend this, many of the tools involved were above my head, and many of the arguments used in analysis are topological in nature. I took topology before entering my second analysis class and the difference was amazing. Also a typical introductory course in topology does not really build off of the language of algebraic structures... it should be self contained (this is in response to a comment made). Although being familiar with abstract mathematical structures is advantageous, it shouldn't be necessary at all. Topology is, after all, one of the main building blocks of mathematics.

Now , in response to your inquiry:

Most introductory math courses are self contained in that they usually at least briefly discuss methods of proofs with some examples. However, having a firm grasp on logic will make any proof-based mathematics class a lot simpler. We have two cases:

Case 1: Topology before logic

I am trying to think of any benefits for this case. The only situation where I can see this as beneficial is if your math logic class were to draw examples from topology. For example, after discussing proof by contradiction maybe something like the following example could arise:

Prove by contradiction that there is no smooth injective map $f:\mathbb{R}^2\to \mathbb{R}.$

Such a question could be proven using topological concepts. This seems unlikely, and I would argue kind of silly, since one should understand at least some logic before moving onto such topics as topology. Developing topology requires understanding several methods of proof, whereas understanding basic methods of proof and logic does not require topology.

On the other hand I can think of countless drawbacks of not understanding mathematical logic before entering a topology course. It could be very difficult to produce proofs (which would be required on homework and exams), understand proofs (which will be in many ways essential to understanding topology), and even to read topological arguments without some understanding of mathematical logic.

Case 2: Logic before Topology

At this point it should be clear I favor this choice. I can't think of many drawbacks, except perhaps the unlikely scenario I have mentioned above where your math logic class used topological examples. I am sure this situation does on occasion arise, but you could always check the course syllabus to make sure this is not the case. Furthermore, if the course on logic just used a few topological examples, this shouldn't be a huge problem- the drawbacks of case $1$ seem much worse.

Understanding mathematical logic at least at some basic level is a formal prerequisite to understanding any higher level mathematics- since all rely heavily on the language of mathematical proof. Consider the following (I admit silly) example:

When I was an undergraduate a lot of mathematics instructors assumed we had been taught logic. I still remember my first honors mathematics class (at which point I had seen no logic), the professor kept writing "iff" everywhere. Having no experience with any formal logic, I assumed he meant "if" an was just a terrible speller. Of course I was wrong, and he meant "If and only if". This is of course an extreme example, and if you have taken any university mathematics you probably understand such statements. However many of the students had never seen this before, and for the first few days we were baffled.

Of course there is a lot more to mathematical logic than such statements as these, so the drawbacks of having no experience in logic can extend to far worse consequences.