Archive for the ‘Mathematics’ Category

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Mathematical hands

Published by peter on 2013-01-16 in Computer technology, Creativity, Design, Hands, Human intelligence and Mathematics. 0 Comments

With MOOCs fast becoming teaching trend-du-jour in western universities, it is easy to imagine that all disciplines and all ways of thinking are equally amenable to information technology.   This is simply not true, and mathematical thinking  in particular requires hand-written drawing and symbolic manipulation.   Nobody ever acquired skill in a mathematical discipline without doing exercises and problems him or herself, writing on paper or a board with his or her own hands.   The physical manipulation by the hand holding the pen or pencil is necessary to gain facility in the mental manipulation of the mathematical concepts and their relationships.
Keith Devlin recounts his recent experience teaching a MOOC course on mathematics, and the deleterious use by students of the word-processing package latex for doing assignments:

We have, it seems, become so accustomed to working on a keyboard, and generating nicely laid out pages, we are rapidly losing, if indeed we have not already lost, the habit—and love—of scribbling with paper and pencil. Our presentation technologies encourage form over substance. But if (free-form) scribbling goes away, then I think mathematics goes with it. You simply cannot do original mathematics at a keyboard. The cognitive load is too great.

Why is this?  A key reason is that current mathematics-producing software is clunky, cumbersome, finicky, and not WYSIWYG (What You See Is What You Get).   The most widely used such software is Latex (and its relatives), which is a mark-up and command language; when compiled, these commands generate mathematical symbols.   Using Latex does not involve direct manipulation of the symbols, but only their indirect manipulation.   One has first to imagine (or indeed, draw by hand!) the desired symbols or mathematical notation for which one then creates using the appropriate generative Latex commands.   Only when these commands are compiled can the user see the effects they intended to produce.   Facility with pen-and-paper, by contrast, enables direct manipulation of symbols, with (eventually), the pen-in-hand being experienced as an extension of the user’s physical body and mind, and not as something other.   Expert musicians, archers, surgeons, jewellers, and craftsmen often have the same experience with their particular instruments, feeling them to be extensions of their own body and not external tools.
Experienced writers too can feel this way about their use of a keyboard, but language processing software is generally WYSIWYG (or close enough not to matter).  Mathematics-making software  is a long way from allowing the user to feel that they are directly manipulating the symbols in their head, as a pen-in-hand mathematician feels.  Without direct manipulation, hand and mind are not doing the same thing at the same time, and thus – a fortiori – keyboard-in-hand is certainly not simultaneously manipulating concept-in-mind, and nor is keyboard-in-hand simultaneously expressing or evoking concept-in-mind.
I am sure that a major source of the problem here is that too many people – and especially most of the chattering classes – mistakenly believe the only form of thinking is verbal manipulation.  Even worse, some philosophers believe that one can only think by means of words.     Related posts on drawing-as-a-form-of-thinking here, and on music-as-a-form-of-thinking here.
[HT:  Normblog]

The pure mathematical universe

Published by peter on 2012-11-08 in Mathematics, Matherati and Music. 0 Comments

Somewhere on his blog, the indefatigable Cosma Shalizi has written about the disciplinary universe of mathematics – that in addition to pure mathematics itself, pure mathematics is used in (and is essential to) the disciplines of Statistics and Computer Science.  This idea struck a chord, and I began to wonder exactly what particular aspect of pure mathematics was being used in each of these other disciplines and where else such methods or approaches were being used.  Of course, having trained as a pure mathematician who turned to mathematical statistics and then eventually to computer science, I know precisely what parts or theories of  pure math were being used in these two disciplines, so this is not my question.    For example, the theory and practice of mathematical statistics draw on probability theory (which itself draws on measure theory and the theory of integration, which in turn require Cantor’s theory of infinite collections), and, in statistical decision theory,  on the differential geometry of information.    (Indeed, I recall being strongly annoyed in my introductory statistics courses that so often proofs of theorems were postponed until “after you know measure theory.”)  Rather, what interests me is what abstract processes – what we might call, mathematical styles of thinking (mathmind, as distinct from, say, the styles of thinking of anthropology or history or chemistry)  – were being used, and where.
Not for the first time, I considered an input-process-output model.  From this viewpoint, we can view pure mathematics itself as a process of (mostly) deductive reasoning that transforms facts about abstract formal objects into other facts about abstract formal objects.   The abstract formal objects may have a basis in some (apprehension of some manifestation of) some real domain or objects, but such a basis is neither necessary nor important to the mathematics.   Until the mid 20th century, people used to say that mathematics was the theory of number, although why they thought this when the key examplar of this theory was Euclidean geometry, a theory which is mostly number-free and scale-invariant, I don’t know.    Since the mid-20th century, people have tended to say that mathematics is the theory of structure and relationship, which better describes most parts of pure mathematics, including Euclidean geometry, and better describes the potential applications and utility of the subject.
Many of the disciplines in the mathematical universe use the same processes – essentially deductive reasoning and, sometimes, calculation – to transform different inputs to certain outputs.   Here is my list (to be added to, when I think of others).
Pure Mathematics

  • Input = Abstract formal structures and objects
  • Process = Manipulation based on deductive reasoning (and, occasionally, calculation)
  • Output = Knowledge about abstract formal structures and objects

Theoretical Physics

  • Input = Mathematical models of physical reality
  • Process = Manipulation based on deductive reasoning and calculation
  • Output = Knowledge about (mathematical models of) physical reality

Mainstream Economics

  • Input = Mathematical models of economic reality
  • Process = Manipulation based on deductive reasoning and calculation
  • Output = Knowledge about (mathematical models of) economic reality

Computational Economics

  • Input = Computational models of economic reality
  • Process = Manipulation based on deductive and inductive reasoning, calculation and simulation
  • Output = Knowledge about (computational models of) economic reality

Exploratory Statistics:

  • Input = Raw data
  • Process = Processing and manipulation
  • Output = Information

Computer Processing:

  • Input = Information
  • Process = Processing and manipulation, including operations derived from both deductive and inductive reasoning, and simulation
  • Output = Information

Statistical Decision Theory (quantitative decision theory)

  • Input = Information
  • Process = Processing and manipulation, both inductive and deductive reasoning
  • Output = Knowledge, Actions

Computer Science:

  • Input = Abstract formal structures and objects, intended as models of computational processes
  • Process = Manipulation based on both deductive and inductive reasoning, and simulation
  • Output = Knowledge about (abstract formal structures and objects, as models of) computational processes

Engineering:

  • Input = Physical objects and materials
  • Process = Manipulation based on deductive reasoning and calculation
  • Output = Physical objects and materials

(Formal) Logic

  • Input = Formal representations of statements and arguments
  • Process = Manipulation based on deductive reasoning
  • Output = Formal representations of statements and arguments

AI Planning

  • Input = Information, actions
  • Process = Manipulation based on deductive reasoning and simulation
  • Output = Knowledge, actions, plans

Qualitative Decision Theory

  • Input = Information, actions
  • Process = Processing and manipulation, both inductive and deductive reasoning
  • Output = Knowledge, actions, plans.

Musical composition

  • Input = Abstract formal structures and objects with a sonic semantics
  • Process = Manipulation, based on deductive-like reasoning or simulation-like generation
  • Output = (Plans for the production of) sounds

Some of the statements implied by these input-process-output schemas are contested.  I would argue that, for instance, any knowledge gained by mathematical economics is only ever knowledge about the mathematical model being studied, and not about the real world which the model is intended to represent.   But this is not the view of most economists, who seem to think they are talking about reality rather than their model of it.  Perhaps this view explains why economics seems peculiarly immune to the major revision or rejection of models on the basis of their failure to predict or describe actual empirical data.
A word on the last schema above:   The composition, performance and even the auditing of music may involve thinking, as I argue here.  Some of the specific modes of musical thinking involved have much in common with deductive mathematical reasoning, in the sense that they can involve the working out of the logical consequences of musical ideas, where the logic being used is not Modus Ponens or Reductio ad Absurdum (as in pure mathematics), but a logic of sounds, pitches, rhythms, timbre and parts.

Polygon construction

Published by peter on 2012-10-09 in Mathematics. 0 Comments

Mathematician Sean Eberhard has a nice post about constructible regular polygons, giving a proof of a characterization of the n-sided polygons (aka n-gons) which are constructible only with a ruler and a compass. Those which are so constructible correspond to n being decomposable into a power of 2 and a product of primes of a certain form:

Theorem The regular n-gon is constructible by ruler and compass if and only if n has the form p_1 * . . . . * p_l * 2^k, where p_1, . . . , p_l are distinct primes of the form 2^{2^m} + 1.

That physical geometric actions should map to – and from – certain prime numbers is a good example of some of the deep interactions that exist between different parts of mathematics, interactions that often take us by surprise and usually compel our wonder.
One question that immediately occurs to me is whether there are other instruments besides ruler and compass which, jointly with those two instruments, would enable n-gon construction for other values of n.   Indeed, is there a collection of instruments (presumably some of them “non-constructible” or infinite in themselves) which would eventually garner all n, or at least other interesting subsets of the natural numbers?

De mortuis nil nisi bonum

Published by peter on 2012-10-06 in Economics, Mathematics and Obituaries. 0 Comments

In a posthumous tribute to one of my late university lecturers, I read:

His [name of university] years were characterised by his love and enthusiasm for teaching.  His dedication to his students was reciprocated in their affection for him. The large Economics I classes that he taught (numbering in some cases up to 400 students) were legendary.”

Although I would prefer not to speak ill of the dead, these words are a distortion of the historical truth, or at the least, very incomplete.   The lecturer concerned was certainly legendary, but mostly for his vituperative disdain for anyone who did not share his extreme monetarist and so-called “economic rationalist” views. It is true that I did not know ALL of my fellow economics students, but of the score or so I did know, no one I knew felt they received any affection from him, nor did they reciprocate any. Indeed, those of us also studying pure mathematics thought him innumerate. He once told us, in a thorough misunderstanding of mathematical induction, that any claim involving an unspecified natural number n which was true for n=1, n=2, and n=3 was usually true, more generally, for all n. What about the claim that “n is a natural number less than 4“, I wondered.

As I recall, his lectures mostly consisted of declamations of monetarist mumbo-jumbo, straight from some University of Chicago seminar, given along with scorn for any alternative views, particularly Keynesianism. But he was also rudely disdainful of any viewpoint, such as many religious views, that saw value in social equity and fairness. Anyone who questioned his repeated assertions that all human actions were always and everywhere motivated by self-interest was rebuked as naive or ignorant.

In addition to the declamatory utterance of such tendentious statements, his lectures and lecture slides included very general statements marked, “Theorem“, followed by words and diagrams marked, “Proof“. A classic example of a “Theorem” was “Any government intervention in an economy leads to a fall in national income.” His proof of this very large claim began with the words, “Consider a two-person economy into which a government enters . . . ” The mathematicians in the class objected strongly that, at best, this was an example, not a proof, of his general claim. But he shouted us down. Either he was ignorant of the simplest forms of mathematical reasoning, or an ideologue seeking to impose his ideology on the class (or perhaps both).

I remained sufficiently angry about this perversion of my ideal of an academic discipline that I later wrote an article for the student newspaper about the intellectual and political compromises that intelligent, numerate, rational, or politically-engaged students would need to make in order to pass his course. That such a lecturer should be remembered as an admirable teacher is a great shame.

Halmos on learning mathematics

Published by peter on 2012-10-06 in Mathematics. 0 Comments

Paul R. Halmos:

Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

 

Embedded network data

Published by peter on 2012-09-24 in Forecasting and Mathematics. 0 Comments

In June, I saw a neat presentation by mathematician Dr Tiziana Di Matteo on her work summarizing high-dimensional network data.  Essentially, she and her colleagues embed their data as a graph on a 2-dimensional surface.   This process, of course, loses information from the original data, but what remains is (argued to be) the most important features of the original data.
Seeing this, I immediately thought of the statistical moments of a probability distribution – the mean, the variance, the skewness, the kurtosis, etc.   Each of these summarizes an aspect of the distribution – respectively, its location, its variability, its symmetry, its peakedness, etc.  The moments may be derived from the coefficients of the Taylor series expansion (the sum of derivatives of increasing order) of the distribution, assuming that such an expansion exists.
So, as I said to Dr Di Matteo, the obvious thing to do next (at least obvious to me) would be to embed their original network data in a sequence of surfaces of increasing dimension:  a 3-dimensional surface, a 4-dimensional surface, and so on, akin to the Taylor series expansion of a distribution.     Each such embedding would retain some features of the data and not others.  Each embedding would thus summarize the data in a certain way.   The trick will be in the choice of surfaces, and the appropriate surfaces may well depend on features of the original network data.
One may think of these various sequences of embeddings or Taylor series expansions as akin to the chain complexes in algebraic topology, which are means of summarizing the increasing-dimensional connectedness properties of a topological space.  So there would also be a more abstract treatment in which the topological embeddings would be a special case.
References:
M. Tumminello, T. Aste, T. Di Matteo, and R. N. Mantegna [2005]:  A tool for filtering information in complex systems.  Proceedings of the National Academy of Sciences of the United States of America (PNAS), 102 (30) 10421-10426.
W. M. Song, T. Di Matteo and T. Aste [2012]:  Hierarchical information clustering by means of topologically embedded graphs. PLoS ONE, 7:  e31929.

Imaginary beliefs

Published by peter on 2012-08-21 in Creativity, Forecasting, Mathematics, Planning, Religion and Uncertainty. 0 Comments

In a discussion of the utility of religious beliefs, Norm makes this claim:

A person can’t intelligibly say, ‘I know that p is false, but it’s useful for me to think it’s true, so I will.’ “

(Here, p is some proposition – that is, some statement about the world which may be either true or false, but not both and not neither.)
In fact, a person can indeed intelligibly say this, and pure mathematicians do it all the time.   Perhaps the example in mathematics which is easiest to grasp is the use of the square root of minus one, the number usually denoted by the symbol i.   Negative numbers cannot have square roots, since there are no numbers which when squared (multiplied by themselves) lead to a negative number.  However, it turns out that believing that these imaginary numbers do exist leads to a beautiful and subtle mathematical theory, called the theory of complex numbers. This theory has multiple practical applications, from mathematics to physics to engineering.  One area of application we have known for about a  century is the theory of alternating current in electricity;  blogging – among much else of modern life – would perhaps be impossible, or at least very different, without this belief in imaginary entities underpinning the theory of electricity.
And, as I have argued before (eg, here and here), effective business strategy development and planning under uncertainty requires holding multiple incoherent beliefs about the world simultaneously.   The scenarios created by scenario planners are examples of such mutually inconsistent beliefs about the world.   Most people – and most companies – find it difficult to maintain and act upon mutually-inconsistent beliefs.   For that reason the company that pioneered the use of scenario planning, Shell, has always tried to ensure that probabilities are never assigned to scenarios, because managers tend to give greater credence and hence attention to scenarios having higher-probabilities.  The utilitarian value of scenario planning is greatest when planners consider seriously the consequences of low-likelihood, high-impact scenarios (as Shell found after the OPEC oil price in 1973), not the scenarios they think are most probable.  To do this well, planners need to believe statements that they judge to be false, or at least act as if they believe these statements.
Here and here I discuss another example, taken from espionage history.

Carnival of Mathematics #85

Published by peter on 2012-04-10 in Mathematics. 0 Comments

The latest Carnival of Mathematics, number 85, is now published, here.
As usual, the selection emphasizes posts about puzzle-solving rather than ones about structure and form, but that (unfortunately) is how most mathematics is.   Not the best part, but.  The best is about structure and form.
 

Black Fields Medallists

Published by peter on 2012-04-08 in Anthropology and Mathematics. 0 Comments

US journalist John Derbyshire has published a screed comprising racist advice to his son.  Among the tendentious statements contained in it is this one:

(5) As with any population of such a size, there is great variation among blacks in every human trait (except, obviously, the trait of identifying oneself as black). They come fat, thin, tall, short, dumb, smart, introverted, extroverted, honest, crooked, athletic, sedentary, fastidious, sloppy, amiable, and obnoxious. There are black geniuses and black morons. There are black saints and black psychopaths. In a population of forty million, you will find almost any human type. Only at the far, far extremes of certain traits are there absences. There are, for example, no black Fields Medal winners. While this is civilizationally consequential, it will not likely ever be important to you personally. Most people live and die without ever meeting (or wishing to meet) a Fields Medal winner.

It is true that there are no black Fields Medallists.  There are also no women, of whom there are rather more in the world than the 40 million black Americans.   There are also no Canadians, no Spaniards, and no Poles among the winners.    This is particularly surprising given the major and disproportionate contribution that Polish mathematicians, for example, have made to mathematics and related disciplines.   And there are many more New Zealanders and Belgians than their populations would lead one to expect.  Perhaps the list of medal winners more reflects the knowledge and biases of the people awarding the prizes than the ability of the potential candidates.    Such a social construction provides a more logical explanation than what Derbyshire implies.  But of course logic is never a strong point of racists.
 

Michael Dummett RIP

Published by peter on 2011-12-29 in Mathematics, Matherati, Obituaries and Philosophy of Language. 0 Comments

The death has just occurred of the philosopher Michael Dummett (1925-2011), formerly Wykeham Professor of Logic at Oxford.    His writings on the philosophy of language and the philosophy of mathematics have influenced me, particularly his thorough book on intuitionism.   Having been educated by pure mathematicians who actively disparaged intuitionist and constructivist ideas, I found it liberating to see these ideas taken seriously and considered carefully.  The precision of Dummett’s writing and thought clearly marked him out as a member of the Matherati, as also his other formal work, such as that on voting procedures.
POSTSCRIPT (2012-01-21):  The logician Graham Priest remembers Dummett as follows:

It is clear that Dummett was one of the most important — perhaps the most important — British philosopher of the last half century. His work on the philosophy of language and metaphysics, inspired by themes in intuitionist logic, was truly groundbreaking. He took intuitionism from a somewhat esoteric doctrine in the philosophy of mathematics to a mainstream philosophical position.
Perhaps his greatest achievement, as far as I am concerned, was to demonstrate beyond doubt the intellectual respectability of a fully-fledged philosophical position based on a contemporary heterodox logic. Philosophers in the United Kingdom, even if they do not subscribe to Dummett’s views, no longer doubt the possibility of this. Dummett had an influence in Australia, too. It was quieter there than in the U.K., but the relevant philosophical lesson was amplified by logicians who endorsed heterodox logics of a different stripe (for which, I think, Dummett had little sympathy). The result has been much the same.
In the United States, though, Dummett had virtually no significant impact. Indeed, I am continually surprised how conservative philosophy in the United States is with regard to heterodox logics. It is still awaiting a Dummett to awaken it from its dogmatic logical slumbers.
Graham Priest, City University of New York Graduate Center, and the University of Melbourne (Australia)

References:
His Guardian obituary is here.  An index to posts about members of the Matherati can be found here.
M. Dummett [1977/2000]: Elements of Intuitionism. (Oxford: Clarendon Press, 1st edition 1977; 2nd edition 2000).