I think everyone who is interested in the interaction between mathematics and philosophy should be encouraged by the volume. The editors have brought together philosophers and mathematicians to try to increase interest in philosophy on the mathematics side. This is a difficult task, and I still have the impression that a philosophy-mathematics collaboration is more difficult than other kinds of interdisciplinary work, e.g. philosophy-physics or philosophy-cognitive science.
From the review:
Hopefully these brief summaries suggest how the editors have sought to link philosophy of mathematics more closely with the interests of mathematicians. There is certainly a need for more engagement between mathematics and the philosophy of mathematics and I believe that this volume marks a productive first step in this direction. It is worth briefly asking, though, what barriers there are to philosophy-mathematics interaction and whether this volume will do much to overcome them. As I have already emphasized, philosophers and mathematicians tend to approach a philosophical topic with different priorities. The mathematicians in this volume often emphasize examples and exciting developments within mathematics, while the philosophers spend most of their energy clarifying concepts and criticizing the arguments of other philosophers. When taken to extremes either approach can frustrate the members of another discipline. Philosophers rightly ask mathematicians to clarify and argue for their positions, while a mathematician may become impatient with endless reflection and debate. A related barrier is the different backgrounds that most philosophers and mathematicians have. Philosophers are typically trained through the careful study of their predecessors and are taught to seek out objections and counterexamples. While most philosophers of mathematics have an excellent understanding of foundational areas of mathematics like logic and set theory, for obvious reasons few have reached a level of specialization in any other area of mathematics. By contrast, most mathematicians will not have much of a background in philosophy and will be tempted to appeal to the most interesting examples from their own mathematics even if they are not accessible to philosophers, let alone many other mathematicians. I am happy to report that most of the philosophical and mathematical discussion in this volume should be fairly accessible to everyone, but this probably happened only because the editors were looking out for complexities that might put off the average reader. Finally, it would be a bit naive to ignore the substantial professional barriers that stand in the way of any substantial philosophy-mathematics collaboration. To put it bluntly, nobody should try to get tenure by publishing for a community outside their home discipline. That said, it is encouraging to see philosophers and mathematicians at least trying to engage each other's interests and I hope these efforts will be continued and expanded in the coming years.