Conclusion The version of social constructivism that I have been discussing is a naturalistic philosophy of mathematics that aims to provide an account of mathematics as it is practised, cognizant of both the social structures within the mathematical community and the historical development of the discipline. I have argued that in a particular sense of the term it is nominalist, because it regards mathematical objects as signs deployed within semiotic systems with sign rules and meanings. I have not denied that abstract objects exist, just located them in the realm of culture, alongside money, literature, and other human institutions and artifacts.
I have argued that mathematical knowledge is conventional in the sense that it is warranted by the rules of mathematics and the mathematicians’ understandings of logical necessity. However, I have claimed that these rules, and mathematicians decisions of acceptability based on them, is itself partly a result of historical contingency. By subscribing to these limitations and deviations from the traditional ideology of the purity, objectivity, and perfection of mathematics I aim to reclaim mathematics from the idealists. Bringing mathematics back down to earth, to the mundane reality of lived human life, is not to denigrate or besmirch it. Ironically, the aim is to offer an more accurate and a truer picture of mathematics as a part of lived human experience.