Non-European Mathematics

From George Gheverghese Joseph’s excellent book ‘The Crest of the Peacock: Non-European Roots of Mathematics’:

The concept of mathematics found outside the Graeco-European praxis was very different. The aim was not to build an imposing edifice on a few self-evident axioms but to validate a result by any suitable method. Some of the most impressive work in Indian and Chinese mathematics examined in later chapters, such as the summations of mathematical series, or the use of Pascal’s triangle in solving higher-order numerical equations, or the derivations of infinite series, or “proofs” of the so-called Pythagorean theorem, involve computations and visual demonstrations that were not formulated with reference to any formal deductive system. The view that mathematics is a system of axiomatic/deductive truths inherited from the Greeks, and enthroned by Descartes, has traditionally been accompanied by the following cluster of values that reflect the social context in which it originated:

  1. An idealist rejection of any practical, material(ist) basis for mathematics: hence the tendency to view mathematics as value-free and detached from social and political concerns
  2. An elitist perspective that sees mathematical work as the exclusive preserve of a high-minded and almost priestly caste, removed from mundane preoccupations and operating in a superior intellectual sphere

Mathematical traditions outside Europe did not generally conform to this cluster of values and have therefore been dismissed on the grounds that they were dictated by utilitarian concerns with little notion of rigor, especially relating to proof.

The Importance of Forgetting and Limited Memory

My memory, sir, is like a garbage heap.

— Funes the Memorious, Jorge Luis Borges

One popular conception of how systems such as Watson can aid human beings is by acting as a kind of extension of the database of the human brain and giving us better and speedier algorithms. So a doctor could instantaneously access all the information and data that he cannot possibly analyse on his own. Implicit in this conception is as assumption that we are better off if we can process and store more information and that our own limited, forgetful memory is not up to the task of dealing with complex domains such as medical diagnosis. And a robotic aid is surely so much better than memory-enhancing hormones or training to become a memory athlete. However, the assumption that more memory is better is unwarranted. As Gerd Gigerenzer notes “the philosophical world in which perfect memory would flourish is a completely predictable world, with no uncertainty” whereas human cognition is adapted to an unpredictable and uncertain environment.

The importance of limited memory in learning was highlighted in a study by cognitive scientist Jeffrey Elman. Elman demonstrated that under certain conditions, initial restrictions on the memory of an artificial neural network may improve its ability to comprehend the complex grammatical relationships that are key to learning a language. In Elman’s words:

one might have predicted that the more powerful the network, the greater its ability to learn a complex domain. However, this appears not always to be the case. If the domain is of sufficient complexity, and if there are abundant false solutions, then the opportunities for failure are great. What is required is some way to artificially constrain the solution space to just that region which contains the true solution. The initial memory limitations fill this role; they act as a filter on the input, and focus learning on just that subset of facts which lay the foundation for future success.

 It is in this context that the limited memory capacity of infants has a positive impact by acting “like a protective veil, shielding the infant from stimuli which may either be irrelevant or require prior learning to be interpreted.”

The most striking example of how perfect memory can malform human intelligence is the case of the Russian journalist and mnemonist Shereshevsky. While studying him, the neuropsychologist Alexander Luria found that Shereshevsky possessed a memory of almost unlimited capacity and durability. Luria tested Shereshevsky’s memory by asking him to repeat arbitrary series of numbers, words and syllables that Luria provided him with, a task that Shereshevsky completed without error no matter how long the series and how long back the series had been given to him. Indeed, he possessed a flawless recollection of series’ that Luria had given him as long as 15 years ago. In many respects, Shereshevsky’s mind resembles that of a computer. Luria notes that when asked to reproduce a particular word in the series, Shereshevsky “would pause for a minute, as though searching for the word, but immediately after would be able to answer my questions and generally made no mistakes” as if he were searching through a vast database with an incredibly accurate and efficient algorithm. Perfect memory however carried a high cost. Shereshevsky struggled to understand the meaning of simple passages of text (especially poetry or metaphors), “a struggle against images that kept rising to the surface of his mind.” He found it almost impossible to extract any true meaning from them or to be truly aware of anything at an abstract level. In this respect, Shereshevsky resembles Jorge Luis Borge’s famous character ‘Funes the Memorious’ whose prodigious memory meant that he was “incapable of ideas of a general, Platonic sort”.

The Common Origin of the Dreyfus Model of Skill Acquisition and Shu-Ha-Ri

The similarity between the concept of Shu-Ha-Ri and the Dreyfus Model of Skill Acquisition is well known. Both focus on a common theme – as one moves from being a novice at an activity to being an expert, the role of rules and algorithms in guiding our actions diminishes. They are instead replaced with an intuitive tacit understanding.

What is less commonly known is their common origin in the Chinese philosophical tradition of Taoism.  Hubert Dreyfus was deeply influenced by the work of Martin Heidegger who in turn was deeply influenced by Lao-Tzu. Shu-Ha-Ri is of course intimately connected with Zen Buddhism which is grounded in Taoist philosophy.

The Emotional Quarterback

The expert quarterback, like many other expert performers, utilises an intuitive approach rather than an algorithmic approach (ref. Hubert and Stuart Dreyfus).

Jonah Lehrer on the ‘Emotional Quarterback’

Laozi against the Computational Theory of Mind

The Tao that can be expressed is not the eternal Tao.”