November 20, 2010
Posted by safeisrisky under Books
What is common between Michael Lewis, Malcolm Gladwell, Paul Graham , Clay Shirky ? If you have read any of the articles in the media written by the above authors, their articles are distinct. They are long, hyperlink free, engage our minds for a long period of time using arguments / interesting examples , make us a bit contemplative . In short we can DEEP READ their articles. They have the power to change our thought process too. However the other 99.9999999 % of the stuff that we come across on the net are basically short articles heavily laden with hyperlink , video links, multimedia etc. Exposure to such media is doing something to our brains and Nicholas Carr discusses the issue thread-bare in his new book , “ The Shallows”.
Medium is the message , goes the popular saying and Nicholas Carr argues that medium has greater impact than the content , on the way we think and act. The author begins his book by admitting that his capacity for concentration and contemplation has come down, after years of exposure to internet. He chose to ask around a few people , formally and informally , and found that this condition to be true for a large number of people. A wide range of people including, faculty from premier medical schools, students from various Ivy league colleges,corporate personnel increasing admit that they are in the same boat. They seemed to have lost the ability to read and absorb a longish article on the net. Some admit that they no longer read on the internet, they merely skim/ scroll. Few even go to the extent of justifying this change in the mindset by saying,”One no longer needs to read books, as the relevant content one is looking for is available as a blurb, short article, paragraphs in Google books, wiki pages , etc”. “Why waste reading a full length book when one can just skim a few articles / facts about whatever one is looking for ? “, goes the usual stance.
Basically for an increasing majority, net is rewiring their brains by making them think that the very idea of reading a book is old fashioned. There is absolutely no debate about the riches that internet can bring along. Its stupendous growth is a testimony for its utility. You can get to amazing amount of information with in a few clicks. However it looks like what we are trading away is “our old linear thought process” , Calm, focused, undistracted, the linear mind is being pushed aside by a new kind of mind that wants and needs to take in and dole out information in short, disjointed , often overlapping bursts- the faster the better! . Do we all realize it that net is rewiring our brains ? When we do a ton of things all at one go, check our email/gmail- allow the you tube video to load,-monitor the tweets-get notified of blog posts by RSS reader, what exactly is happening to our brains ? Most of us do seem to relate to the situation described above as we constantly multi-task on net. We might kid ourself and think that we are in control of the media. Are we truly in control ? Or is the Net that is controlling us ?
The book delves a bit in to the structure of brain. It traces the initial assumption of scientists and researchers who held a view that brain structure was akin to a single continuous fabric of nerve fibres which was kind of fully determined by DNA and the childhood experiences. The prevalent view was that structure develops early in a human’s life and there after it is pretty much fixed. Subsequent discovery of neurons, synapses, neuro-transmitters in the brain , and research shows that brain is massively plastic in nature. The space between neurons where the signals are transmitted changes constantly based on our life experiences, the way we think, the way we do things, etc. Umpteen number of examples are quoted in the book which drives home the point that mind is plastic but not elastic, meaning the mind does not snap back to the old structure after a while. It is constantly changing and massively elastic based on the way we think/do stuff. Ok, so if the brain is work in progress forever, can internet , which is basically a tool in our hands , change the entire structure of our brains ?
Yes says neuroplasticity research. The book traces out a bunch of tools starting from maps, clocks, ploughs, machines etc, and shows that in each case, the tool took us from the practical everyday usage to a situation where mind developed higher level of abstraction. Maps for example transformed experience in space to abstraction of space. Clocks which were just a tool to record the flow of time have had an unimaginable impact on our mind. We have managed to divide time in to greater precision , and have cut days in hours minutes and seconds, thus virtually changed the meaning of the word “time”. “I don’t have time for —- ” is a very common sentence you get to hear and if you pause and reflect on that statement, you are bound to realize that time controls us , a tool which we invented to control stuff. Same is the case with any tool which we start using. Internet is no different.
The book takes a stance that “The World of a screen is different from the World of a page”. Pages in the book can be deep read. Arguments , sometimes lengthy and winding can be read and pondered for a while. This is almost next to impossible while reading stuff on the screen . Do you remember any content that you have read for let’s say 1/2 – 1 hr hr on the net ? Unlikely. Why ? Because the medium itself is designed to give you short bursts of disjointed information. So, if you are constantly reading stuff ONLY online and not engaging yourself on deep reading, brain will get rewired soon and it is likely that your attention span is going to reduce in the activities not connected to net. That’s a little scary because true innovation comes from deep thought and something that robs deep thought is definitely not desirable.
The Very image of a book is changing. ebooks have hit the market which are being marketed at a deep discount price as compared to books/hard covers. The distribution of ebooks has also changed dramatically. Ipad, Kindle and other ebook readers, now enable us to get a book with in no time. But what is further dramatic is what we see on the screen. Inbuilt dictionaries, hyperlinked content, access to internet, blogs, magazines at now a click away. In the earlier times, if you had to disengage from a book, you had to atleast put the book away but now the distractions are built right in to the book!. So, will you be able to deep read a book ? Well, you can if you turn off the wireless and all the other distractions that come with ipad / kindle ? Economics will no doubt push the books to be published in ebook format, kindle format and the reading habits are bound to change. Infact I think we should welcome all these technologies but need to be AWARE that there are tons of distractions that the ebook readers come along with. Unless one turns off those distractions, you might not be able to read books in a silent and contemplative way.
Last week, my friend gifted me Kindle. It was a delightful surprise. The first thing I noticed that it had an awesome interface and easy access to tons of books and was extremely light. The moment I started reading I could not resist accessing kindle store, visiting hyperlinks in the book etc. However once I turned off the wireless and other distractions, Kindle was a charm. I could read for a long stretch of time with absolute ease. I think the new Kindle is going to be a game changer in the e-book reader business.
Another nice argument put forth in the book is that of memory. Research confirms that there are two types of memories, short term / working memory and long term memory. Long term memory usually comprises schemas, linkages , ideas and short term memory is like a scratch pad which gets rewritten quickly. However for something to go in to long term memory , it has to pass from the working memory. The problem with reading stuff on the net is that since it is disparate,hyperlink heavy , the information is tough to get in to long term memory. On the other hand , linear reading facilitates the passage between working and long term memory. Various experiments have shown that hyperlink , hypermedia text increases the cognitive load and most often times never gets to the long term memory. So, net by its very nature, distracts us. In other words, we get distracted from distraction by distraction.
An often quoted view is “ Why remember anything at all , if I can access stuff at the click of a button?”, The book gives a fitting reply to such a view by stating that
“We don’t constrain our mental powers when we store new long term memories. We strengthen them. With each expansion of our memory comes an enlargement of our intelligence. The web provides a convenient and compelling supplement to personal memory, but when we start using the web as a substitute for personal memory, by passing the inner process of consolidation, we risk emptying our minds of their riches”
The book draws upon various empirical and experimental research to prove that Net is subverting our capacity for concentration and contemplation , a by-product of linear reading. It does make one aware that conscious disengagement from the web is imperative in our daily lives.
In the choices we have made, consciously or not, about how to use computers and net, we have rejected the intellectual tradition of solitary, single-minded concentration, the ethic that the book bestowed on us. We have cast our lot with the juggler. Unless we consciously disengage from this medium for a few hours daily , we will become more distracted in our thoughts and actions.
November 10, 2010
Posted by safeisrisky under Books
This book is the sequel to the book A Radical Approach to Real Analysis, a book that I found to be the best historical narrative of Real Analysis. This book too in the same league, though applied to Lebesgue theory. Let me attempt to summarize this fascinating book. The book comprises 8 chapters structured in such a way that it takes the reader from Riemann integral to Lebesgue integral and finally answering a simple but revolutionary question in analysis, “ When does a function have a Fourier Series expansion that converges to that function? ”
Chapter 1 : Introduction
The first chapter starts off with emphasizing the five big questions that occupied the centre stage in the 19th century
When does a function have a Fourier series expansion that converges to that function?
What is integration?
What is the relationship between integration and differentiation?
What is the relationship between continuity and differentiability?
When can an infinite series be integrated by integrating each term?
One can say that the first chapter kind of summarizes quickly Volume I from the same author, titled “A Radical approach to Real Analysis”. Let me attempt to summarize the summary –:) . Basically the first enlightening counter example in real analysis was Dirchlet’s function which questioned the notion that function was some geometric concept, representable only as curves. Newton and Leibniz while grappling with physics problems came out with integration as a tool to help them in their quest. Most of the definitions were based on infinitesimals, which were looked upon by others with a skeptic eye. According to George Berkley, they were “ghosts of departed quantities”. In such an environment, Cauchy’s integral definition based on approximating finite sums was very elegant mathematically.
Riemann came along and then modified Cauchy’s approximating finite sums definition which proved a better way of integrating. Why is Riemann method superior to Cauchy’s? The author promises to answer this question at a later point in the book. Thanks to Fundamental theorem of Calculus, definite integral was viewed from the lens of anti-derivative function. Of course not all functions have anti derivatives that can be expressed in terms of standard functions. One can safely say that counterexamples paved the way for greater understanding in real analysis than any other field. Darboux example of continuous functions, not differentiable anywhere , challenged the prevalent view that “continuous functions would have to be differentiable at most points “ Weierstrass developed and worked on the concept of uniform convergence to bring sanity in to situations where term by term integration could be applied. Unfortunately , uniform convergence proved to be a sufficient condition than a necessary condition for term-by-term integration. Fourier series is a classic example where the series is not uniformly convergent but term-by-term integration makes sense. By the end of the introductory chapter, one realizes that clinging to Riemann definition would only cloud the linkages between continuity-differentiability-integrability-term by term integration issues.
Chapter 2 : The Riemann Integral
The second chapter talks about Riemann integrability. Riemann improved upon Cauchy and made it more generic by considering the value of the function at an arbitrary point in a partition. He then used Cauchy criterion for convergence to compute the integral value. Gaston Darboux digged in to the Riemann’s paper on definite integral and came up with conditions for Riemann integrability for any bounded closed function. Even though Riemann presented his workings, he was concerned about the linkage between continuity and differentiability. He presented a pathological function which was discontinuous at every rational number with an even denominator but was still integrable. This was followed up by a few more counter examples which showed that there existed functions which were everywhere continuous but nowhere differentiable functions. There were still many open questions that had to be worked on. The representation of Fourier series ranked high in the priority list of open problems. Given that a function can be represented as a sum of infinite trigonometrical series, Riemann turned the situation around and thought about the following questions:
Does a trigonometric series have to be integrable?
Is the Fourier series always identical to the series which we started?
Some of the interesting Fourier series converged to discontinuous functions and thus were not uniformly convergent. So, what can one say about functions which are not uniformly convergent ?
In this context, the chapter talks about three mathematicians whose work paved way to greater understanding. First was Eduard Heine. Heine was the first mathematician to describe a way out by coming up with “ Uniform convergence in general”, which he defined as a series with finitely many exceptional points that is uniformly convergence on any closed interval that does not contain one of these points. He thus went on to prove that a set of trigonometric functions which are uniformly convergent in general , no two distinct series converges to the same function.
Second mathematician mentioned is Hermann Hankel who grappled with the discontinuous functions and tried classifying them in to groups based on how Riemann integrable they are. He came up with a concept of set being dense and categorized discontinuous functions. His final conclusion was that a function that is point-wise discontinuous must be Riemann integrable. However this was a faulty conclusion, thanks to the counter example by H.J.S Smith. Georg Cantor is mentioned as the trio from the “Class of 1870” who realized that there is something far more important that had to be worked on, before talking about Riemann Integrability, continuity and differentiability – The Real Line.
Chapter 3 : Explorations of R
The third chapter is a brief summary of topology in relation to the metric space R. Basic definitions of neighbourhood, open sets, closed sets, compact sets, interior points, closure points, boundary points, dense set are given. The chapter then goes on to discuss means to accommodate algebra on Real line. A word based proof is given for Bolzano-Weierstrass theorem which is by far the best way to present the proof that I have come across. Concept of completeness is introduced subsequently where any ONE of the properties is equivalent to the other three, the properties/theorems being,
Every sequence of closed, nested intervals has a nonempty intersection that belongs to the set
Every bounded subset has a least upper bound in the set
Every Cauchy sequence converges to a point in the set
Every infinite bounded subset has a limit point in the set
The highlight of this chapter is to discuss the drama behind Heine-Borel theorem. The story starts off with Axel Harnack who carried a mistaken notion that the complement of a countable union of intervals is also a countable union of intervals. There was a major flaw in the statement which was finally resolved by Emily Borel. Borel proved that if the sum of the lengths of the open intervals is strictly less than 1, there must be points – infact , uncountably many points – that are not in any of these intervals. Lebesgue generalized Borel’s theorem. However the name that got stuck on this property( Any open cover on a closed and bounded set has a finite subcover ) is Heine-Borel Theorem.
The story then shifts to Dirichlet who was the first to discriminate between continuity and uniform continuity. However it was later published by Heine who never gave credit to Dirichlet. The proof used by Heine was very similar to Borel and Lebesgue’s proof. Another twist to the story is that, Arthur Schönflies, 1900, who claimed Borel’s result also holds for uncountable covers, pointed out connection to Heine’s proof of uniform continuity. He was the first to call it Heine–Borel theorem and the name was further popularized by Henry Young in his paper, “Overlapping intervals”. Heine-Borel theorem subsequently appeared in many formats such as Weierstrass theorem. One historian actually refers to it as “ Dirichlet-Heine-Weierstrass-Borel-Scoenflies-Lebesgue theorem”. Borel called this theorem as “the first fundamental theorem of measure theory“, even though the name never caught on. The chapter then gives a quick Cantorian Set theory where concepts such as Cardinality, Continuum Hypothesis , Power Sets are explained
Overall, a fair treatment to the topology of R is presented in this chapter.
Chapter 4 : Nowhere Dense Sets and the Problem with the Fundamental Theorem of Calculus
The chapter talks about SVC sets in the context of Smith-Volterra-Cantor contribution to calculus. It starts off with the construction of cantor set and gives a description of Devils’s staircase function, a function which has infinite steps between any two steps, derivative exists and is 0 at every point in [0,1] except at values in the cantor dust. Cantor set is uncountable and hence this function is such a strange function where F’(x) = f(x) and yet the evaluation part of FTC( Fundamental Theorem of Calculus) does not hold good.
Devil’s Staircase function
Volterra’s function is an example of a function which is pointwise discontinuous and not Riemann integrable. There is also a mention of the extent to which one can come up with pathological functions which Volterra proved. He conclusively proved that inverse of Ruler function cannot exist, a function which is continuous at rationals and discontinuous at irrationals cannot exist.
Developments from Baire are mentioned as they became the precursor to Lebesgue theory. Lebesgue had to say this
“ Baire showed us how to investigate matters; which problems to pose, which notions to introduce. He taught us to consider the world of functions and to discern there the true analogies, the genuine differences. In absorbing the observations that Baire made, one becomes a keen observer, learning to analyze common place ideas and to reduce them to notions more hidden, more subtle but also more effective. ”
So, what were Baire’s contributions ?
He introduced the concept of nowhere dense sets and categorized sets in to those that can be formed as a countable union of nowhere dense sets and those that can’t be. His genius lies in acknowledging that the first category sets are sparse in nature. He is famously known for his category theorem that an open interval cannot be expressed as the countable union of nowhere dense sets. Baire was trying to improvise on Hankel’s classification. Baire’s classification went like this: Continuous functions constitute class 0. Pointwise discontinuous functions that are not continuous constitute class 1. Inductively , if f is the limit of functions in class n , but it is not in any class k<=n, then we can say that f is in class n + 1. One outcome of such a classification is that , even though point wise discontinuous functions can have infinite discontinuities, there is enough continuity on a dense set of points. What was particularly not evident in Baire’s classification was : If you define an inductive based argument for classifying functions, is there a case when a function escapes all these classes?
Chapter 5: The Development of Measure Theory
The chapter starts off with listing the difficulties of Riemann Integral despite of the developments in 1880’s and 1890’s such as
It is defined for only bounded functions. While improper integrals had been introduced to deal with unbounded functions, this fix appears to be adhoc.
It is possible to have an integrable function with positive oscillation on a dense set of points and therefore the integral is not differentiable at any of the points in the dense set. This violates the anti-differentiation part of the fundamental theorem of calculus
It is possible to have a bounded derivative that cannot be integrated. This violates the evaluation part of the fundamental theorem of calculus
The limit of a bounded sequence of integrable functions is not necessarily Riemann integrable( Baire’s sequence)
The question of finding necessary and sufficient conditions under which term-by-term integration is valid was turning out to be extremely difficult
Despite these shortcomings, mathematicians did not part with Riemann integral but tried fixing the problem with improvements. Weierstrass came up with a modified version of Riemann integral but lost its steam very soon as it failed the additive property.
The chapter then introduces Giuseppe Peano, Camille Jordan and Emile Borel who formulated various concepts in set theory. Peano came up with the definition of content and showed that a set has content if its inner content was equal to the outer content. Jordan defined Jordan Measure of a set as its content and extended to finite set of disjoint measurable sets. Thus he was talking about finite additivity of content and increasing the concept to more sets. However using this concept of inner content and outercontent had one flaw. For far too many important sets , inner content was not equal to outer content. Emile Borel realized that the problem with Jordan measure was that it finitely additive. So , he introduced a new measure with countably additive property with three additional assumptions’
The measure of a bounded interval is the length
The measure of a countable union of pairwise disjoint measurable sets is the sum of their measures
If R and S are measurable sets with R being a subset of S , then m(S-R) =m(S) – m(R)
However there was a limitation to Borel Measure. It was applicable to a much smaller collection of sets than Jordan measure. The cardinality of Borel measurable sets was much smaller than that of Jordan measurable sets
The concept of countable union of pairwise disjoint sets was vastly improved by Lebesgue who revolutionized the field of integration with his Lebesgue Measure. Lebesgue added three more conditions to the measure
It is translation invariant
Countable additivity when applied to disjoint sets.
Measure of interval(0,1) is 1
With this definition of measure , he observed that outer measure and inner measure of a sets for all subsets of R were not equal. Also m(S-R) =m(S) – m(R) was not applicable to all the subsets of R. He then introduced a class of sets called measurable sets , which closely parallels Caratheodory’s condition. So , the slick presentation of Caratheodorys condition in most of current books is nothing but a statement of measure of a set and its complement , m(E) + m(E’) = 1. The chapter then goes on in to discussing the properties of measurable sets which maintain their status under Countable additivity, Finite unions and intersections, Approximations by finite number of open intervals. The chapter ends with a detailed construction of a non measurable set invoking “Axiom of choice”. Do non-measurable sets exist ? Like a lot of stuff in math, that is the wrong question to ask. We do not seek existence of mathematical objects, like lets say a 26 dimensional object. What we are interested in math is the rules that such objects follow. In the case of non-measurable set too, if we accept the axiom of choice, it exists!!
Chapter 6 : The Lebesgue Integral
This chapter introduced the concept of measurable function and lebesgue measure. One immediately notices that measurable functions are nice objects which preserve their properties under various operations, primary being the limit of measurable functions is measurable. The building block of any measurable function is termed as simple function and one can define any measurable function as a limit of simple functions. The chapter goes on then show that every Riemann integrable function is Lebesgue integrable and the value concurs by either of the approach. Lebesgue in his work subsequently realized the extent of pathology permitted for a function to be Riemann integrable . His lemma that says, “a bounded function defined on a closed and bounded interval is Riemann integrable if and only if it is continous almost everywhere “, put an end to the quest of finding the extent of discontinuity permitted for Riemann integrable function to an end.
Lebesgue integral by its very definition cannot be taught quickly to a beginner . A Riemann integral can be intuitively explained away to any undergrad with the concept of partitioning/area under the curve/ etc. In contrast, the very definition of Lebesgue integral assumes that the student is aware of concepts like simple functions, characteristic functions, supremum for a set of functions etc. I think it is only after carefully going over the real analysis concepts from scratch that one can begin to understand Lebesgue integral.
The chapter then goes on to talk about Monotone Convergence Theorem (MCT)and Dominated Convergence Theorem (DCT) both of which are essentially used to swap limit and integral signs in relation to a sequence of functions and its limit. MCT is relevant to monotonically increasing sequence of non negative measurable functions while DCT is relevant to a sequence of Lebesgue integrable functions. DCT is a sufficient condition not a necessary condition to swap limits and integral sign. The chapter then talks about various types of convergence such as convergence in measure, uniform convergence, point wise convergence, almost sure convergence etc and tries to give an insight in to the linkages between these types of convergence.
Chapter 7 : The Fundamental Theorem of Calculus
This chapter takes a closer look at differentiation. It starts off with defining Dini Derivatives and stating that a function is differentiable at a point if the four Dini derivatives at c are finite and equal. Dini also observed that if a function f has a Dini Derivative that is either bounded above or bounded below , then f can be written as a difference of two monotonically increasing functions. Camille Jordan found a simple characterization of Dini’s statement using a concept called bounded variation.
Every grad student taking a course on Brownian motion comes across bounded variation. However often times , there is no time for the faculty to go over the historical context of bounded variation. Books such as these are immensely helpful in giving a context. This book makes the reader aware of the connection between Dini Derivative and Bounded variation and thus pours flesh and blood in to seemingly dry definitions.
Coming back to this chapter, the next section covers a very important property about monotonic continuous functions, i.e they are differentiable almost everywhere. This was proved by Lebesgue. Weierstrass belief that some will find a monotonic continous function that was nowhere differentiable function was conclusively laid to rest. A few years later, a husband and wife team of William Young and Grace Young published an independent proof that continuity is not needed. Bounded Variation MEANT Differentiable Almost everywhere.
It then moves on to the problems with original FTC and explains the necessary conditions to move around the problems.
Antiderivative part of FTC
When is a function integrable? – Measurable functions for which the integrals f+ and f- are finite
If the integral exists, when can that integral be differentiated? – Always, almost everywhere
When does differentiating the integral take us back to the original function? – If the function in the integrand is lebesgue integrable
When is a function differentiable ? – If the function has bounded variation , then it is differentiable almost everywhere
If the derivative exists, when can that derivative be integrated ? – If the function is absolutely continous
When does integrating the derivative take us back to original function ? – If the function is absolutely continous
The biggest takeaway from this chapter is that
Lebesgue Integral => Absolutely Continous
Absolutely Continuity => Bounded Variation
Bounded Variation => Differentiable Almost everywhere
The chapter ends with FTC stated using the concepts of Lebesgue measurable functions, Absolute continuity where the former is used to state the antiderivative part of FTC while the latter is used to state the evaluation part of FTC.
Chapter 8: Fourier Series
This chapter tries to answer the question that was asked at the very beginning of this book and the prequel to this book “A Radical Approach to Real Analysis”
“When does a function have a Fourier Series expansion that converges to that function?”
Lebesgue proved a theorem according to which, if a function f is integrable on [-pi, pi] , the Fourier series of f converges to f almost everywhere atleast in the cesaro sense of convergence. It is not something we would have wished for as it is not convergence on the entire interval but almost everywhere and the convergence is not uniform but in the cesaro sense. However it was a breakthrough as finally there was some mathematical condition based on which one could talk about Fourier Series convergence to the function that generated it.
I found this chapter very dense (it goes in to Banach and Hilbert Spaces) . I found it difficult to get past a few sections in the first reading. I will revisit this chapter at a later date and try to understand the application of functional analysis.
A related link to this post : A Radical Approach to Real Analysis : Summary
Even though there are books out there which are a potpourri of measure theory , lebesgue integral and probability concepts, none of them will give you a historical context to the development of ideas focusing exclusively on lebesgue measure.
Before attempting any study of axiomatic probability, this book needs to be read to get an idea of the relevance of lebesgue measure to EVERYTHING in probability. Once you understand the historical context, the appreciation of as simple as a uniform distribution function on (0,1) takes a new dimension altogether.
November 9, 2010
Posted by safeisrisky under Books
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A nice workout typically jolts you out of bad day and makes you alive. For me,a good book does the same when I have no energy to step out of home. Last few days I have not been keeping well and this book has kept my mind active despite the dullness around.
In this delightful little book, Timothy Gowers delves in to the philosophical differences between people who are happy with the notions of infinity, curved spaces, square-root of –1, N dimensional spaces, etc, from those who find them disturbingly paradoxical. There are many amongst us who question the very purpose for thinking about quantities that do not exist in their experience. Why bother about to think about infinity? It’s just a mathematician’s symbol for something that is not finite. Why bother about anything beyond the 3 dimensional experience objects? Can you show me a 4 dimensional object? If not, why should I spend my time understanding a four dimensional geometry ?, goes the usual stance. They say, god created integers and rest of the math is basically human mind’s construct. So why bother developing a mathematical approach to our daily life situations, if it’s just a theoretical construct , a ghost of our imagination ? The book’s underlying theme is that math is a happy ghost which will only help you in understanding stuff better.
This book does not talk about history of math/ math disciplines, nor does it talk about various theorems or proofs of a specific subject. Its central purpose is to urge the reader to think mathematically, in other words, think abstractly. What is abstraction in mathematics?
One type of abstraction occurs in model building. When devising a model, one tried to ignore as much as possible about the phenomenon under consideration, focusing only those features that are essential for understanding the behaviour. In examples relating to Physics, let’s say a projectile motion of a stone, one abstracts away all the forces except the initial velocity, angle of projection and effect of gravity. While studying behaviour of gases, one abstracts away all the interactions of particles and considers individual particles moving in a chamber with no interactions. Despite removing most of the effects present in the real life scenario, the variables and the model used could just be the right kind of abstraction necessary to study it appropriately.
There is another type of abstraction which one finds in mathematics, which is the subject of this book. This abstraction is much deeper. One can easily relate this type of abstraction to let’s say a a chess piece, for example black king. If one were to ask whether black king exists, the question might make sense in the existential sense but goes no further. Yes, you see a chess piece which represents a black king.It begets an immediate question , “ What does a black king do in chess? ”. This is a far more interesting question as it talks about the role of black king than some platonic existential stuff. Mathematics is similar to the situation above where mathematical objects by themselves might not mean anything from a existential point of view. A mathematical object is what it does.
Take for example N, the set of natural numbers. For some numbers like 1, 2, 3, till some finite number you can probably visualize the number. Additions of these small numbers might make sense but once you get to slightly bigger numbers , a simple addition like 243 + 786 does not make any existential sense. Well, for any set of natural numbers, once you decide on the associate, commutative and distributive laws, then the result of 243 + 786 follows from these rules. There is nothing more than that. Numbers need not be very large before we stop thinking of them as isolated objects and start to understand them, through their properties, through how they relate to other numbers in a number system. So, after learning the basic number system at a school level, we are supposed to look at number “system” in the sense of what they do.
Similar is the case of negative numbers, fractions, irrationals, complex numbers. If you want to understand square-root of 2 , it is better to understand to what it does. Square root of 2 does not mean anything. It is not something we can see somewhere in reality. So, what should be the attitude towards such mathematical objects ? Well, it basically solves the equation X*X = 2. That’s it .Introduction of irrationals and defining properties is purely an abstract exercise but which gives us ways to solve equations. Same is the case with complex numbers. “i”, the symbol by itself does not carry any meaning, but in the context of solving an equation, X*X+ 1 = 0, it makes tremendous sense. i, a completely abstract thing , is used very heavily in the theory of quantum mechanics. It provides one of the best illustrations of a general principle: if an abstract mathematical construction is sufficiently natural, then it will almost certainly find a use as a model. There are umpteen such objects in math like infinity, logarithms, exponentials, fractional powers which need to be viewed from the perspective of, “What they can do? “.
Mathematics as a subject is built axiomatically. There are a set of axioms to begin with and mathematicians / scientists create statement called proofs that are built from these axioms. So in one sense, math is one subject where the disputes in principle, “can end”. If you and I debate about a theorem, we can dig deeper and deeper till all the axioms are laid on the table. As long as the logical structure is intact, we can say that theorem is proved. At the last stage where axioms are seen, mathematicians stop the argument. Now one can ask, “why not debate about axioms in mathematics?”. The most important thing that matters to mathematicians is less the truth aspect of axioms and more their consistence and usefulness. In that sense, it is very much different from let’s say economics. Two economists can debate about Monetarism and Neo-Keynesianism till their last breath and still not reach a conclusion.
Let me end my take on this book by giving a nice example which shows the usefulness of abstraction.
Visualize a 4 dimensional cube. How many edges does it contain?
To answer the above question, you got to understand the term “visualize”. In our everyday parlance, visualize is something we do with the help of our mind to “see” stuff which is not yet in our consciousness. For a mathematician, visualization has a different meaning. An object or concept that cannot be visualized by a mathematician, means it is something for which he needs to stop and calculate. I will try to make it clear in the context of this example. One can “just see” that for a three dimensional cube, there are 4 edges round the top, 4 edges round the bottom, 4 going from top to bottom,making it 12 all. Now a mathematician “can see” a 4 dimensional cube in this way. He would say :
“ I can think of a four-dimensional cube facing each other , with corresponding vertices joined by vertices (in the fourth dimension) just as a three dimensional cube consists of two squares facing each other with the corresponding vertices joined. Although I do not have a completely clear picture of 4 dimensional cube, I can still “see” that there are 12 edges for each of three dimensional cube, and eight edges linking the vertices together . This gives a total of 12+12+ 8 = 32. “
Thus an answer to the above question can be obtained even if such an object is beyond our practical experience.In the above case, visualization enabled an answer to the question. Not always. Mathematicians spend considerable time developing theorems/tools/lemmas to deal with higher and higher level of abstraction. These help in bridging the gap between “What mathematicians can see” Vs “What mathematicians cannot see ?”
So, next time you see a mathematical object described via properties and rules, this book will serve as a gentle reminder that that those rules and properties constitute the abstraction that is needed to work on them. Existential questions such as “what is i?”, “What is α?“, “What is logarithm?” are of no relevance.
If you want to understand from a mathematician’s point of view, “What is abstraction ? ” & “Why is it important to develop a certain sense of abstract thinking? ” , this book is spot on.