Thoughts on the paper
The paper appears to provide a high level overview of the role of mathematics in cognitive science. I'm not a sufficient expert in the overall field of cognitive science where I'd feel comfortable to truly judge the accuracy of the overall synthesis that Andler (2012) provides. That said, much of the paper is about providing examples of how mathematics integrates with cognitive science. And the examples seem reasonable. I could think of other examples that pertain to my work, but their absence does not really detract from the paper.
Andler also makes a number of distinctions about how mathematics can integrate with cognitive science. For example there is the kind of integration that statistics has with many experimental disciplines which is both fundamental and not very specific.
So in short, I think the paper provides a thought provoking big picture overview of the issues of interfacing maths with cognitive science.
Implications for you doing advanced work
This paper could potentially be motivating, but it might also be discouraging. It's so high level that it creates a vision, but the vision is so large that it might be overwhelming.
To do ground breaking research in cognitive science (or any area for that matter), you need to specialise. If you want to do research interfacing maths with cognitive science you would also specialise. The result is that you would only need a small subset of mathematics and cognitive science mentioned in the article. I use mathematics and statistics a lot in my research, and there's plenty of mathematics mentioned in that paper which I know little about.
If you are wanting to do research in this area, exposing yourself to a good undergraduate curriculum in mathematics, statistics, computing, and cognitive science would be a good start along with exposure to research. Then pursue a PhD with an appropriate advisor where you can hone your skills in a particular domain.
In terms of mathematics my own bias would favour learning calculus, linear algebra, probability, and statistics, but that's just my bias. It's also useful to learn how to code.