April 24, 2005

On Legal Mathematics(published in the Mathematics Journal, University of Warwick)

Christopher Columbus Langdell, Dean of the Harvard Law School and originator of the “Case Method” of teaching law, famously advocated that Law was a science, whose principles and doctrines could be ‘discovered’ in cases, much as biologists discover the principles of their science in their laboratories. To Langdell ‘science’ conjured up the ideas of order, system, simplicity, taxonomy and original sources. The science of law involved the search for a system of general, logically consistent principles, built up from the study of particular instances. Once the general principles have been found, it is then the task of scholars to work out, in an analytically rigorous manner, the subordinate principles entailed by them. When these subordinate principles have all been stated in propositional form and the relations of entailment among them clarified, they will, Langdell believed, together constitute a well-ordered system of rules that offers the best possible descriptions of that particular branch of law – the best answer to the question of what the law in that area is.

This ‘mechanical jurisprudence’, often criticised by American realists, resembles at first the methodology used in mathematics in deriving conclusions from basic axioms and logical deductions. When a lawyer writes a brief for a case in which he has to convince the judge that his argument should prevail, he structures it just like a geometric proof. He starts with all the given facts, then states the relevant laws and precedents that relate to the case. Then he makes his argument based on these facts using deductive logic, exactly as if he were doing a mathematical proof.

Mathematicians have the ability to analyse problems and principles just as lawyers have the ability to dissect dictums and rules from cases. Comprehending certain laws, for example taxation law, is as challenging as understanding some of the most complicated mathematical theories you will encounter. Most solicitors involved in civil cases, in which people are suing others, must be able to calculate percentages, interest rates and the like to determine what is or isn't a fair settlement for the parties involved. Likewise, lawyers involved in tax or corporate law have to perform a lot of computations involving money, interest rates, percentages and proportions. Patent lawyers who work on behalf of inventors generally must also have a degree in engineering because they must be able to understand the inventions and the mathematical formulas involved in the physics or chemistry applications of the product.

Although a comparative study of the relationship between the Law and Mathematics wouldn’t result in any offspring, there exists nonetheless methodological devices used in mathematics which corresponds to those used in the application of the Law. From its birth in ancient Greece, and for over two thousand years, mathematics has been viewed a body of collective truth, being the basis of innumerable scientific theories which describe the world around us. To achieve such powerful results, early mathematicians employed deductive reasoning in their examination of new hypotheses. This logical methodology created the assumption that mathematics is a certain science. But more recent realisations in the world of mathematics have revealed that it is not the body of truths once assumed to be, and further, that the very deductive reasoning used to create and develop these truths contain flaws.
In his book Mathematics: The Loss of Certainty, Morris Kline claims that ‘the current predicament of mathematics is that there is not one but ‘many’ mathematics and that for numerous reasons each fails to satisfy the members of different schools. Uncertainty and doubt concerning the future of mathematics have replaced the certainties and complacency of the past. The disagreements about the foundations of the 'most certain' science are both surprising and, to put it mildly, disconcerting. The present state of mathematics is a mockery of the hitherto deep-rooted and widely reputed truth and logical perfection of mathematics.’ However disorganised the world of mathematics may be today, contradictions have always existed in bodies of knowledge – especially in the Law – just after periods of major revision in which inevitable periods of uncertainty follow, new ideas are allowed to reach fruition.

Accordingly, although the promulgation of the law claims to guarantee its certainty and consistency, its application is a different matter. The courts, and indeed judges, play a significant role in applying the law in the real world in real situations, thus making the ambiguity of the Law vulnerable to ultra vires interpretation. ‘Statutes are not laws by virtue of their enactment. They only become law when applied by a decision of the courts’ argued J C Gray, 20th century American realist. Thus instead of being regarded as a body of abstract rules and principles, the law shall be understood from a broader angle. Legislation is therefore no more than a source of law: it is the courts that ‘put life into the dead words of the statutes’.

Likewise, mathematical laws and principles are no more than an instrument to an end: it requires real-world situations in order for its true efficacy to be understood. The efficiency of symbols and numbers in mathematics only becomes evident when assembled in theories and applied to real problems. Although both disciplines endeavour to be certain, they are both subject to the creative interpretation and ambiguity of human minds. In other words, they are both subject to the concept of relativity. Searching for the Truth in mathematics mirrors the search for Justice in the Law; objectives which appears to be unattainable for sceptics. Unfortunately, the ordinary citizen fails to contribute to this quest for Truth and Justice, as both disciplines remain perceptibly ‘inaccessible’ by the populace at large.

Perhaps I place too narrow a definition upon Truth and Justice – for despite the seeming contradictions of mathematics and the disagreements which characterise its past, one evident theme remains. Mathematics has always been and remains to be a remarkably effective method of describing the mechanics of the world around us. Accordingly, the Law has always been and remains to be the fundamental element that holds society together. Both disciplines are of respectable value and utility, even complementary at times, and both are consistently evolving towards the same uncertain future. Even if one is to disregard absolute certainty in mathematics, the Law, or in any body of knowledge, we must not give up the search for Truth and Justice, or allow our limitations to overcome us.


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  1. Lucas, this is a very interesting article to me. For the past couple of years I have been thinking about the philosophy of mathematics, and have actually used the very comparison with law that you use here. I myself am a mathematician, but I had a very interesting discussion with my brother (a barrister) about constitutional law and legal paradox (such as, can parliament legislate to limits its own powers? and the constitutional trickery used in instituting apartheid in South Africa). He pointed me towards the articles and books of Wade (I think that was the name) about constitutional law.

    Anyway, I've been attempting to develop an epistemological theory free from the use of the absolute sense of the notion of truth, inspired by pragmatism (James, Dewey, Peirce) and Wittgenstein. I think it might be of some interest to you given what you've said in your article. To give you an idea, here is something I wrote in an (unfinished) essay a year or two ago:

    "Rather than talking about truth at all, we should talk about statements occurring within systems of statements, and judgements of those systems of statements. Truth can be defined relative to a theoretical system, but the choice to apply, use and develop any particular system is a pragmatic one. A statement which is true relative to a theoretical system that has fallen into disfavour doesn’t become false, but the usefulness of the sense in which the statement is true fades. This gives us an alternative way of thinking about systems of thought which doesn’t involve using the word truth in a dubious absolute sense."

    Let me know if you'd be interested in exploring some of these ideas.

    24 Apr 2005, 15:35


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