Wednesday, January 25, 2012

Russell’s Paradox in the Social Sciences


This is not a rigorously researched academic essay. It is a work in progress. I attempt here to draw a parallel between Russell’s Paradox which shook the foundations of mathematical logic in the 1900s and the values that arise out of the necessity of negotiating our daily lives in today’s postmodern world. I have not defined terms such as ‘postmodern’, ‘tolerance’, ‘secularism’, ‘worldview’ and ‘value’. I am aware that these terms can be variously understood, but for my purposes definition can only serve to complicate my argument. Therefore, I rely on an intuitive understanding of these concepts.

This parallel that I attempt to draw is merely an analytical tool which can be used – I am not sure if it has already, but I could not find any references to it in my reading- to look at dilemmas we face in our daily lives thanks to globalization, increased social mobility coupled with a postmodern outlook on life. I do not in any way, mean this as a dismissal of postmodernism. I have not tried to propose a solution to this paradox, and am not sure if one exists. Mathematics found a way to avoid this paradox, I am not sure if this is possible in the social sciences, though I do hope so.

In 1901, Bertrand Russell, a staunch believer in mathematics and logic as being infallibly rational and subject to proof, discovered to his dismay, a hole in the very foundations of both. Set Theory (now called naïve Set Theory) had been lauded until then as the tool to understand the basis of mathematics. Russell himself had been thrilled by the advent of Set Theory in the mathematical world and set about to explore them with much zeal, until he discovered what would later come to be known as ‘Russell’s Paradox’.  (Ernest Zermelo is said to have discovered the paradox the previous year, but failed to publish his findings resulting in Russell’s getting the credit for it).

The paradox itself is easy to understand if one is familiar with Set Theory. Simply put it asks, if X is a set of all sets that do not contain themselves (also called ‘normal sets’), will X contain itself?* If the answer is yes, then by containing itself, it fails to be eligible to contain itself (it becomes an ‘abnormal set’). If not, it is no longer the set of all normal sets, since it is, itself, a normal set. Either way, it is a contradiction or a paradox.

This paradox becomes particularly interesting when we look at the essence of postmodernism, which I consider to be, the denial of objectivity and acceptance of multiplicity and pluralism. I shall attempt to demonstrate a parallel between values which arise out of a postmodern outlook - tolerance and secularism- and Russell’s paradox.

If pluralism is accepted as a fact of life and tolerance is required in today’s society to deal with pluralism in daily life, where does tolerance end? How much intolerance can tolerance tolerate?


This becomes clearer when we look at religion and secularism. If we consider an individual, A who subscribes devoutly to a certain religion, say Zoroastrianism and another individual, B, who subscribes devoutly to a different religion, say Christianity, and if both A and B live in the same society, our postmodern worldview requires that both be allowed to practice their religion unhindered. Secularism, the postmodern ‘value’ in question requires that we ‘tolerate’ both worldviews. (To digress: In Indian usage -and here I am relying purely on my own experience- secularism implies not merely tolerance of the other religion, but a degree of respect for it as well. - I am aware of the problems of declaring this, but am writing from my understanding of both tolerance and respect. 'Respect' I believe, carries a connotation of critical engagement which 'tolerance' does not. If however the reader disagrees feel free to replace 'respect with any other word which carries this sense.) However, by accepting that both religious worldviews are equally valid –accepting the multiplicity of worldviews-, it follows too that secularism can be seen as a worldview distinct from others, say a Hindu religious worldview or an Islamic religious worldview, instead of an overarching category, as it was intended. And indeed, each of these religious worldviews were intended as overarching categories. Seeing secularism this way, as a subset of itself, reduces it to the level of the religious worldviews it was paternalistically trying to encompass. But the logic of secularism itself dictates that it not be considered absolute. This paradox is strangely reminiscent to Russell’s Paradox.

Interestingly, this analogy can be extended to the practice of Social Sciences today, where despite our newfound zeal for reflexivity, we still attempt a binary separation between thought and action, theory and practice, subject and object. For instance, a religious academic appears today to be a contradiction in terms, or else s/he is expected to separate these two spheres of his/her life, all this despite our focus on plurality/multiplicity of worldviews and arguments favouring interdisciplinarity. If we were to put postmodern ‘values’ into practice this should not be the case. And although we draw our research from the world around us, research tending the world over (or so our researchers would have us believe) to be more participatory and reflexive, universities and academics still style themselves as neutral zones for the exalted pursuit of knowledge (strangely reminiscent of the Enlightenment whose logic we constantly critique).

The logic of postmodernism would require us to see research and the conclusions drawn from it as equally valid to the conclusions drawn by farmers from their lived experience and conclusions drawn by religious leaders and so on. (And I am not saying that this is not so) If however, we accept that, research ceases to be an overarching framework of enquiry, and becomes a subset of itself. If we do not accept it, we do not hold true to our avowed acceptance of plurality.

Based on these analogies, I argue that any binary categories we create (in a context that accepts locationality and plurality of paradigms) is subject to the same paradox. Thus under postmodernism we come up against this stumbling block repeatedly, and not only when putting our theories/’values’ into practice. This quirk of logic is present even in the realm of pure theory (eg: How much intolerance can tolerance tolerate?)- the logic of distinguishing theory and practice, itself comes up against the same paradox!

I conclude in the hope that this analogy adds in some way to our analytical tools. I am not sure if a solution to these paradoxes is possible- in fact I am quite pessimistic about it- at the level of the grand narrative. However, all of us continue to negotiate these apparent contradictions in our daily lives. Perhaps that is the level at which solutions exist, and perhaps a reflexive awareness of this paradox might help us negotiate these contradictions better.


I do not have a bibliography, because as I have mentioned earlier, this is not a rigorously academic essay. It is based on my thoughts, general readings, lectures, discussions with my friends and the general atmosphere of the University. Due credit however goes to Sanjit Basu, who introduced me to Russell's Paradox.

*A more detailed explanation of Russell’s Paradox:

·         Any definable collection is known as a set. The individual items within that set are known as members of the set. Thus X is a set, of which r,k,l and 7 are members.
X = {r,k,l,7}
·         A set can contain other sets.
Y = {1,3,(a,6),y,(2,4)}
Here B is a set, of which 1,3,(a,6), etc; are members. This can also be represented as: B = {1,3,C,y,D}; where C and D represent the sets (a,6) and (2,4) respectively.
·         A set can contain itself. Sets which contain themselves can be called abnormal sets. Thus Z is an abnormal set containing elements a, f and G as well as itself.
Z = {a,f,Z,G}
This means: Z = {a,f,(a,f,[a,f,…………..,G]G)G}
·         A set which does not contain itself is called a normal set.
·         If we consider the set of all normal sets, will it contain itself?

2 comments:

ukvvvw said...

Wow G, this is a really interesting post. I quite enjoyed the way you brought in both mathematics and post-modernism. I do wonder though, if tolerance by its very nature can exist devoid of respect. Isn't that the rather vague difference between tolerating and tolerance? But perhaps that's just how I think due to the Indian point of view.

-V

Daughter of the Night said...

Thanks!

I think tolerance can exist devoid of respect. It can be partonising - you allow for deviance from what you consider the norm. It can be completely neutral as well. The brand of tolerance I prefer is respectful, but tolerance does not necessarily imply that (I think).