Sunday, 7 March 2010

Critique of Pure Mathematics

I remember when I was studying for the selective high schools tests, at about 11 or so. My sat down to help me grasp some mathematic principles. We had been focusing on how if you had three pencil sharpeners and took away two, how many would you have left.* All of a sudden, we crossed over into something else.

"If you have three, and take away two, what do you have left?" my mum read aloud.

I paused.

"...Three of what?"


"...Three... of what?"

I kept repeating that question, increasingly mystified. I later learnt that the number was referring to a 'pure abstraction', and I went with it because it was what I needed to do to score high on the exam, but some part of me never accepted this strange, context-free 'three'.  

Many years later, whilst writing essays for that school I gained admittance to, thanks to my results on the test, I discovered that a good example was the key to a good argument. A specific example which must be drawn on to illustrate a point. This discovery led me to believe that no example should go without a wider theory being developed out of it, and further intensified my belief that no abstract notion should be stripped of its practical muse(s).

Creating an abstract three, an abstract that applies to all sets of three and yet to none, never struck me as resonant. It struck me as an impossible arrangement.

I can see, from my vantage point, three copies of 'Lonely Planet Thailand' in my room. One was published in 2004. One was published two years later, and yet another was published in 2008. However, I could argue that it is really only one book that I possess - The 2008 edition is the condensed version of the last two, with some improvements and innovations. It is a continuation of the processes of information captured in 2005, and as such it cannot be said to be a book distinct and different from all the rest.

Or are there really many books within each book? How is a 'book' to be defined? Lonely Planet allows you to 'pick and mix' your Guidebook information if you purchase the chapters you desire over the internet. Is not then every section for every state in Thailand (Central Thailand, Isan, Chiang Mai province, etc) a book in its own right, possessing the information necessary for it to be an independent entity?

So how many books do I have again? 1 or 100?

The moment you recognise your examples, that's when pure abstraction fails, and things get more interesting. So no, I can never take two away from three. It's not pure and simple. It's not postmodern.

Would Baudrillard say that all mathematical formulas are simulations? Mathematics is just one more way in which experts on disappointment attempt to introduce universal systems which preclude the possibility of postmodern thought.

Don't be taken in. Deconstruct. Re-flower. Cross-pollinate. Love.


*Not exact question.


  1. Hi,

    This is Clive.
    good post. I think this shows that nothing, not even ideas, can exist in abstraction. I also think this is a good argument to believe that there is no reality outside of matter and specific material situations.

    Numbers are just tools that we use to explain concepts and ideas, much like a language. The good thing about mathematics is that it can allow us to be more precise, however it is not a be all end all "thing" (for lack of a better word) that exist outside the minds of human being.

    Have you looked much into critiques of abstract concepts of justice and human rights? They are well worth looking into, since many have argued that human rights themselves should not be looked at in abstraction and should be seen in their specific historical contexts.


  2. My view on abstraction is that generally it can serve as useful tool toward deeper understanding but that the journey up the levels of abstraction must always be complemented by the return journey. Not to do so allows one to remain in a place that bears little resemblance and has little or no relevance to the contingent, material world or to reality as we experience it everyday.

    I find that striving to understand the abstract essence of a concept enables me to see it within a broader context and I believe that everything has some context with which it exists (with the possible exception of God, which is a different discussion entirely). If we don't see the context that simply means that we don't see it, not that it doesn't exist. To diagnose the non-existence of something based on an untested assumption of knowledge\wisdom\understanding that we believe we possess is a best risky.

  3. Hi Clive,

    A great idea to empahsise the links between mathematics and language - mathematics doesn't denote reality any more than language does: they are both social constructions (which can have many valuable uses). I'm sure a lot of mathematicians would be appalled to hear that maths is another type of language, however their insistence on its reflection of truth is quite misguided.

    Thank you for bringing up the notion of human rights, I am indeed interested in cultural/moral relativity issues, and this is a good example of such.

    Let's take Thailand's political situation as an example; it refers to itself, has implemented many of the characteristics of, and is known abroad, as a democracy, however the right to the freedom of speech is such that if you say anything negative about the king (as a local or a foreigner), you could be put in jail. Now, I'm sure that nations like Australia, US and UK would like it to be otherwise, and may not see Thailand as a 'full' democracy, but compared to the rest of South-East Asia, where totalitarianism is more common than not, it is a radiant example of democracy in action. Any criticisms of Thai human rights by those nations would have to keep in mind the historical, economic, cultural, social and religious contexts. To attempt to impose Western legal norms universally, no matter how well-intentioned, is to disregard what makes Thailand a unique place.

    I wonder when/if the anti-freedom of speech law will change...

    There are those that would say that a lack of the universal human rights charter would be to allow societies to do whatever they like, but societies actually DO do whatever they like... attempts at outside control are resisted whenever possible. The way to influence Thailand for the better is not to ask it to conform to standards foreign to them, but work on the existing rights to make the society raise its own standards from within.

    Today I also read that Australia has become the first nation to officially recognise a person as being free of gender, or non-gender-specific. It should be noted that this innovation sets a new standard, even as Australia continues to ban same-sex marriage. I think it would be misguided to then call Australia either forward-thinking or backwards amongst the developed world, because it is a unique case study... just like any other.

  4. Hi Steve,

    Thank you for your comment, I think it's a good idea to agree to disagree in this case. :o)

    I take the view that everyone is subjective, without access to an objective reality. I find ways to communicate meaningfully with the people around me, however I think that I am responding to their realities in the best way I know how, and we are not, as some people believe, creating our own objective, shared reality. I don't even know myself, so how can I know another person's world? I think it's a positive thing to be aware of your personal context and to let other people be free to influence it without understanding it in the absolute sense. :o)