The Philosophical Implications of Mathematics

Book Signing @ Kepler's in Menlo Park

2007-09-23T09:36:00.000-07:00

I'll be speaking and signing books at Kepler's in Menlo Park on Thursday Sept 26th, at 7.30pm.

http://www.artsopolis.com/event/detail/24097

Meanwhile the book is selling well and getting a lot of good press. Here's the amazon link:

http://www.amazon.com/Certain-Ambiguity-Mathematical-Novel/dp/0691127093/ref=pd_bbs_sr_1/104-9722234-0627160?ie=UTF8&s=books&qid=1190471905&sr=8-1

The book is available for pre-order

2007-05-24T20:47:00.000-07:00

A Certain Ambiguity is available for pre-order.

You may go to Princeton University Press or to Amazon to pre-order it.

Thankfully, the reviews have been very good:

Martin Gardner : A Certain Ambiguity is an amazing narrative that glows with a vivid sense of the beauty and wonder of mathematics. The narrator is deeply troubled by the ancient question of whether the objects and theorems of mathematics have a reality independent of human minds. Mixing fiction with nonfiction, A Certain Ambiguity is a veritable history of mathematics disguised as a novel. Starting with the Pythagorean theorem, it moves through number theory and geometry to Cantor's alephs, non-Euclidean geometry, Gödel, and even relativity.

Eli Maor, author of "e: the Story of a Number" and "The Pythagorean Theorem: A 4,000-Year History" : This is a truly captivating thriller that will take you on a whirlwind tour to infinity--and beyond. But be warned: once you start reading, you won't be able to put it aside until finished! A masterly-told story that weaves together criminal law, ancient and modern history, a young man's quest to know his deceased grandfather-and some highly intriguing mathematics.

Keith Devlin, Stanford University, author of "The Math Gene" : This rich and engaging novel follows the path that leads one young person to become a professional mathematician. By deftly blending the young man's story with mathematical ideas and historical developments in the subject, the authors succeed brilliantly in taking the reader on a tour of some of the major highlights in the philosophy of mathematics. If that were not enough, the book also examines, through the minds of its characters, the natures of faith (religious and other) and truth. I am strongly thinking of building a university non-majors math course around this novel.

Joan Richards, Brown University : A Certain Ambiguity is a remarkably good effort to work through some fundamental issues in the philosophy of mathematics in the context of a novel. Crucial to the success of such a venture is creating characters and a plot that are strong enough to hold a reader's interest. Suri and Bal succeed particularly well in the story of Vijay Sahni and Judge Taylor. This well-written book will, I believe, find readers not only among mathematicians, but in a wider audience that is intrigued by mathematical meaning.

Alexander Paseau, University of Oxford : Suri and Bal convey the beauty and elegance--as well as the fascination--of basic mathematical concepts.

Shameless Plug

2005-08-28T14:39:00.000-07:00

Here's a brief description of my (mathematical) novel that will be out in the next 12-18 months (2 of you sent mail asking for it and I don't need much more incentive than that!):

The human heart yearns for absolute truth and certainty. But can we be truly certain about anything—or is everything we believe accidental and meaningless, shaped by the happenstance of genetic and social inheritance? Perhaps mathematics alone, with its uncompromising rigor, can lead us to certainty. In our 90,000 word novel, we examine where mathematics can and cannot take us in the quest for certainty.

Our book will show the reader the following: First, that mathematics can be deeply beautiful—in this regard it is not unlike music or painting; second, that mathematics has profound things to say about whether absolute truth is obtainable; and lastly, that a novel is the best medium through which to convey the excitement and meaning of doing mathematics

Our protagonist, Vijay Sahni, an Indian mathematician, has glimpsed the certainty that mathematics can provide and does not see why its methods cannot be extended to all branches of human knowledge, including religion. Arriving to pursue his academic career in a small New Jersey town in 1919, his outspoken views land him in jail, charged under a little-known Blasphemy law (on the state statute books to this day). His beliefs are challenged by Judge John Taylor, who does not believe that mathematical deduction can be applied to matters of faith. In their discussions the two men discover the power—and the fallibility—of Euclid's axiomatic treatment of geometry, long considered the gold standard in human certainty. In the end both Vijay and Judge Taylor come to understand that doubt must always accompany knowledge.

Big News!

2005-08-19T15:26:00.000-07:00

As some of you know, I've authored a novel that examines whether absolute certainty is achievable through Mathematics. I'm thrilled to report that it has been accepted for publication. More details coming...

Numbers and Biology

2005-08-02T14:43:00.000-07:00

Watching Vir, my 2 year old, attempt to count I realize that numbers may appear more natural to us (human adults) than they really are. Vir makes 3 kinds of mistakes in his counting:
1) He will count an object more than once before moving on to the next one
2) He ignores some items in front of him completely, or
3) He’ll continue counting even though he has accounted for every item on the list. So unless I stop him he often ends up at nineteen (the largest number he knows) even when I have asked him to count the 3 apples in the fruit basket.

To be sure, he’s got the sequence down. He understands that 1 is followed by 2 is followed by 3 etc. He even understands that counting somehow refers to the number of objects. And he understands the idea of ‘many’. “So many cars,” he’ll observe on the freeway.

But that’s it. He hasn’t grasped yet that counting a set means ticking off each element exactly once. Which if you think about it, is quite an advanced idea. We’re so familiar with the idea, however, that we tend to forget that numbers are merely shorthand notations for the cardinality of sets, and are ultimately the creations of our intelligence. In this they are exactly like groups or rings or transfinite cardinals.

And in a few months as Vir understands the rules governing numbers, he too will think that theyare woven in the fabric of the universe…and are not products of human biology.

Wait I hear you say—won’t any intelligence, at the very least, have to be able to count? I’m not so sure any more. Not sure at all.

Numbers and Experience

2005-07-10T23:27:00.000-07:00

It is often argued that while geometry is unable to adequately describe the world around us, numbers are more reliable, more certain. 2 cows plus 2 cows, always equal 4 cows. Unlike non euclidean geometries, there are no non standard arithmetics. Gauss, for a time at least, believed that ‘truth resides in number.’ In a similar vein Jacobi said “God ever arithmetizes” (as opposed to eternally geometrizing).
However, as Kline observes in Mathematics, The Loss of Certainty, the sharpest attack on the truth of arithmetic came from Hermann von Helmoholtz, a superb physicist and mathematician. In his Counting and Measuring he observed that the problem in arithmetic lay in the automatic application of arithmetic to physical phenomena. Some kinds of experiences suggest whole numbers and fractions, while others don’t: one raindrop added to another does not make two raindrops. Two pools of water, one at 40◦ another pool of water at 50◦ when mixed together do not make a pool of water at 90◦. Lebesgue facetiously pointed out that if one puts a lion and a rabbit in a cage, one will not find two animals an hour later! Helmoholtz gives many (more serious) examples but his overarching point is that only experience can tell us where to apply, and not apply, standard arithmetic.

Like Euclidean Geometry, arithmetic is not absolutely applicable to the physical world.

Evolutionary Mathematics

2005-06-13T17:21:00.000-07:00

Chaitin, as he often does, has got me thinking. He writes:

Von Neumann also said that we ought to have a general mathematical theory of the evolution of life... But we want it to be a very general theory, we don't want to get involved in low-level questions like biochemistry or geology... He insisted that we should do things in a more general way, because von Neumann believed, and I guess I do too, that if Darwin is right, then it's probably a very general thing.
For example, there is the idea of genetic programming, that's a computer version of this. Instead of writing a program to do something, you sort of evolve it by trial and error. And it seems to work remarkably well, but can you prove that this has got to be the case? Or take a look at Tom Ray's Tierra... Some of these computer models of biology almost seem to work too well---the problem is that there's no theoretical understanding why they work so well. If you run Ray's model on the computer you get these parasites and hyperparasites, you get a whole ecology. That's just terrific, but as a pure mathematician I'm looking for theoretical understanding, I'm looking for a general theory that starts by defining what an organism is and how you measure its complexity, and that proves that organisms have to evolve and increase in complexity. That's what I want, wouldn't that be nice?

And if you could do that, it might shed some light on how general the phenomenon of evolution is, and whether there's likely to be life elsewhere in the universe. Of course, even if mathematicians never come up with such a theory, we'll probably find out by visiting other places and seeing if there's life there... But anyway, von Neumann had proposed this as an interesting question, and at one point in my deluded youth I thought that maybe program-size complexity had something to do with evolution... But I don't think so anymore, because I was never able to get anywhere with this idea...

Tons of interesting stuff to chew on, but I'll limit myself to this: Imagine a simulation where you have two entities: organisms and resources. The organisms are just data structures which reproduce when they have been getting enough resources. The resources are re-generable and are of various types.

Now let's add on a few complexities: Assume that an organism 'eats' only certain types of resources. So Organism 42 can only live on Resource 118 for example. Further assume that the quantity of Resources stays relatively stable...with exceptions of rare time units of plenty and others (also rare) of drought. Also assume that there can be more than one type of Organism that consumes a certain type of Resource, and also that there are Resources that are not consumed by any organism when the simulation starts.

An Organism will then have the following data elements: Its type [corresponds to the species it belongs to]; its number [i.e. its name]; the Resource number(s) it consumes; its wellness number - a measure of how well fed the organism is - if the wellness number goes over a limit the organism will reproduce; an organism competitive index which will measure how well the individual competes within his species; and a species competitive number that measures how well the species competes with other species vying for the same resource. Reproduction passes on the competitive indices to the progeny. When the wellness index falls below a certain level, the organism dies.

Now also imagine that you have random mutations. A random mutation could change the type of resources an individual consumes and/or its competitive indices (either up or down).

These are only the barest details...but I hope you believe that it is possible to capture the main points of Darwin theory in a reasonable simulation.

Hit start and run the simulation: You will probably see organisms dying and being born; species will be created by the right mutations - they will also thrive or struggle - but eventually all will die out. The world itself may reach some kind of stable equilibrium, but more likely than not...at some point we'd hit zero organisms or zero resources.

All this is worth doing in its own right (in fact I'd be shocked if someone hasn't already done it), but now, just for fun, imagine one last externality: Say that organisms of a certain complexity level can perceive a proportional complexity of mathematical truths. So for example an organism of complexity index 1088 could really 'get' that there can be no largest prime (but other, more difficult theorems are beyond it), and an organism of complexity index 4063 could 'get' the prime number theorem ('get' = a deep understandig that does not allow for the result not be true. Similar, but not equal to proof).

It seems to me then that there will always be mathematical statements that we humans couldn't get, no matter what.

This is far from air tight, but there may be something to chew on here.

--Gaurav Suri

The Voynich Manuscript

2005-06-07T23:03:00.000-07:00

The Voynich manuscript is a very old 230+ page manuscript written in a code that no one has been able to crack. Here's the Wikipedia entry:

The Voynich manuscript is a mysterious illustrated book of unknown contents, written some 600 years ago by an anonymous author in an unidentified alphabet and unintelligible language.
Over its recorded existence, the Voynich manuscript has been the object of intense study by many professional and amateur cryptographers — including some top American and British codebreakers of World War II fame — who all failed to decipher a single word. This string of egregious failures has turned the Voynich manuscript into the Holy Grail of historical cryptology; but it has also given weight to the theory that the book is nothing but an elaborate hoax — a meaningless sequence of random symbols.
The book is named after the Russian-American book dealer Wilfrid M. Voynich, who acquired it in 1912. It is presently item MS 408 in the Beinecke Rare Book Library of Yale University.

The book has strange drawings of flowers, alien looking plants and naked women. Its history is utterly fascinating. Find out more here. Almost as fascinating as the book itself is the history of the men who have attempted to decipher the symbols. In many cases they have bolted on semi-plausible theories even though they were few supporting facts to be had.

All this forces us me to ask - where does the meaning of anything lie? Surely it is not in the symbols we use to communicate the ideas. No, meaning must lie in the mind of the humans deciphering the language/symbol. The human mind has the power to give anything meaning; it also has the power to force meaning where there is none to be had. Be it the notion of the color red, the undecidability of the Continuum Hypothesis or the rhythms of the Voynich manuscript.

--Gaurav

First thoughts on Rebecca Goldstein’s, Incompleteness: The proof and paradox of Kurt Gödel

2005-05-20T11:33:00.000-07:00

I bought this book despite myself. I’ve carefully studied Gödel’s Incompleteness Theorems and expected Goldstein to give a soft, non rigorous, largely biographical treatment which wouldn’t teach me anything new. I bought the book almost out of duty – it is after all in the subject I care most deeply about – I should read it just in case. I am glad I got took the chance. This is a great book, and I am not one to use the term loosely.

The power of the book doesn’t come from its treatment of the theorem itself (she does an adequate job, but others have done better. See for example Nagel and Newman’s classic, Gödel’s Proof for a fine non-technical treatment); rather the books achievement is that it puts Gödel’s work in context. Goldstein successfully (and finally) gives Gödel’s theorems the philosophical interpretation that he himself would have intended.

Before reading Incompleteness I often wondered why Gödel, an avowed Platonist, did most of his work in Mathematical Logic, the most formalist of all mathematical fields. Also, why did he join the Logical Positivists of Vienna who in their way were the most extreme kind of Formalists; and lastly why did Gödel associate himself with a group who revered the teachings of Wittgenstein – the very same Wittgenstein who essentially claimed that all of a mathematics was a mere tautology (a claim that was almost surely quite repulsive to Gödel, and to almost every other mathematician).

Goldstein answered all of this (and more). She gets her answers not from the mathematics, but from the story of Gödel’s life and the philosophical battles that drove him.

In brief, the story is that the Logical Positivists essentially believed that truth lived in the precise, meaning-aware use of language. According tho them, it is only possible to identify a statement as being true or false by proving or disproving it by experience. Logic and mathematics was excluded from this rule; they claimed that mathematics was a branch of logic and was for all intents and and purposes a mere tautology.

Gödel on the other hand was a Platonist; he believed that mathematicians uncovered truths about the universe, and mathematical concepts were merely communicated by—but not contained within—its equations and symbols. Yet, confusingly, Gödel belonged to a Positivist group. He largely stayed silent through their meetings, neither objecting nor agreeing, for that was not his way.

But the internal storm of disagreement that welled within him did lead him to prove that Positivists were wrong. He proved that the structural manipulation of mathematical symbols could not yield all statements that we know to be true. He demonstrated a ‘true’ statement that was not provable—which should have banished Logical Positivism for ever.

Yet it didn’t; for Godel, before Goldstein’s book, was never well understood.

I’ll have a lot more to say about all this in the coming weeks.

5 good uns

2005-05-15T23:23:00.000-07:00

Ah, Why, ye Gods, should two and two make four --Alexander Pope

In mathematics there are no true controversies. --Gauss

Logic is the art of going wrong with confidence --Anonymous

Let us suppose there are things like the truth --Xenphanes

We know truth, not only by reason, but also by the heart --Pascal

--Gaurav Suri

Infinity in the physical world

2005-05-12T21:03:00.000-07:00

Here’s a section from an essay on Infinity by Hector Parr (which has several inaccuracies within it). I bring it up because it brings many of our beliefs about infinity into high relief:

“The world of the Pure Mathematician is far removed from the real world. In the real world there is no difficulty finding the length of the diagonal of a square and expressing this as a decimal, but in the perfect world of Pure Mathematics this cannot be done. In the real world we know the process of counting the natural numbers can never be completed, so that the number of numbers is without meaning, while the mathematician finds it necessary to say that if the process were completed, the number would be found to be Aleph-0. These are harmless follies; what the mathematician gets up to inside his ivory tower need not concern those outside.
But the mathematician's ideal world did impinge on reality at the beginning of the twentieth century when Russell (1872-1970) attempted to reduce all mathematical reasoning to simple logic. Even the natural numbers themselves could be defined in terms of a simpler concept, but to make this possible Russell found it necessary to assume that the number of real objects in the universe is itself infinite. Here again I find it astonishing that he made this assumption so glibly. He called the principle his "Axiom of Infinity". Now an "axiom" is something which is self-evident, unlike a "postulate", which is assumed for convenience even though it is not self-evident. Why did Russell not refer to the principle as his "Postulate of Infinity"? To me it is far from self-evident that there are an infinite number of things in the universe; in fact I cannot see that the statement has any meaning. Infinity is an indispensible concept in Pure Mathematics, but is it not meaningless when applied to the number of real things?”

I disagree with a lot of this selection and I’ve reproduced it here because Parr’s view is a fairly commonly held one. First, he says that in the real world there is no ‘difficulty’ expressing the diagonal of a square as a decimal, but in pure mathematics ‘this cannot be done.’ This statement is either vague or meaningless. Mathematicians have no difficulty assigning a decimal to a diagonal; but they do so as an approximation, not as a precise value. In the ‘real world’ the exact same condition holds – it’s just that for most purposes an approximation suffices. Second, Parr writes that in the real world the process of counting cannot be completed, while the mathematician says that if the process were completed, the number would be Aleph-Zero. This is incorrect. Mathematicians know that counting the Naturals is an unending process. The need for Aleph-Zero arises because Cantor proved that there are several levels of infinity and that he named the beginning level to be Aleph-Zero.

The idea of the ideal world impinging on reality with Russell’s reduction of mathematics to logic is incorrect. None of Russell’s theories required there to be an infinite number of objects in the physical universe. But here finally we are at the real question Parr is attempting to raise: Is there a true infinity in the physical universe? I tend to think not. The universe is likely finite (although unbounded); it likely has a finite life, so time is finite; Some people may argue that God is infinite, but we’re getting into mysticism there. So let’s say that despite being wrong in all the steps of his argument, Parr’s conclusion is correct – that infinity does not exist in the physical world.

Does that somehow make the mathematics of infinity a mere game?

I think not. If one acknowledge 1, 2, 3…, one must acknowledge (countable) infinity and from there it’s a few elegant steps to many levels of infinity. Formalists argue that infinity is meaningless; but it easy for me to be on Godel’s side who argued that if we perceive something about the universe with our mind (e.g. infinity), then it is at least as real the stuff we perceive with our senses (e.g. my 2 year old throwing my phone on the floor)

--Gaurav Suri

The biological roots of mathematical proof

2005-05-07T22:12:00.000-07:00

A great article about the biological roots of music proposes that: “music is merely "auditory cheesecake," or "an evolutionary accident piggy-backing on language," as Daniel J. Levitin at McGill University explained in a recent issue of the journal Cerebrum. But many scientists—Levitin among them—don't agree. "Some researchers are finding that listening to familiar music activates neural structures deep in the ancient primitive regions of the brain, the cerebellar vermis," Levitin writes. "For music so profoundly to affect this gateway to emotion, it must have some ancient and important function.””

I wonder the same thing about the human ability to do proofs. Proof is a recent invention which—if you really examine it—seems to be based on some pretty shady ground. Some professional mathematicians think the emphasis on strict proof is a mystification. G.H. Hardy, one of the most eminent pure mathematicians, commented: "There is strictly no such thing as mathematical proof; proofs are what Littlewood and I call gas, rhetorical flourishes, devices to stimulate the imaginations of pupils". Lets assume for a second that Hardy is right (I believe he is). This would make proof remarkably similar to music: a profound activity, beautiful in its way, but without any obvious function.

What then are the biological roots of proof? Are they some accidental piggy backs on our traditional reasoning abilities? Or are they a direct consequence of language?

--Gaurav Suri

Beautiful Equation

2005-05-03T08:02:00.000-07:00

Consider the series of the reciprocal of squares. If you have not seen it before, you’ll never believe what it converges to:

1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + 1/49 + … = π2/6 (this is pi squared divided by 6)

Yes, π2/6!! That is the same π that you are familiar with: the ratio of a circle’s circumference and diameter. How did it get into this equation? What could a ratio pertaining to circles have to do with the sum of reciprocals of squares? Is this not truly miraculous?

Sometimes when I look at this equation with fresh eyes, I’m amazed and in awe all over again. Equations like this represent why I fell in love with mathematics. There are so many unexpected connections, so much order when you would expect none, a mostly hidden tapestry into which we get a few limited glimpses through the efforts of our brightest minds.

Who made these connections? Why do they exist?

The proof unfortunately is not as beautiful as the result. It justifies rather than explains

--Gaurav Suri

Ayn Rand Misused Axioms

2005-04-29T21:52:00.000-07:00

When I was in the 11th grade I read Ayn Rand’s, The Fountainhead. At that time I was convinced that Rand was on to something great and I spent two years in her cult, being an objectivist. But the more I lived, the more I realized that life has too much grey, too much complexity and too much randomness, to fit in to Rand’s framework. She was though—in her way—a good writer. Even though I no longer agree with much of what she said, I continue to believe that she could write with unequaled charisma.

What she most certainly didn’t have though was a good understanding of how mathematicians use axioms. An axiom is an irreducible primary. It is a starting point from which theorems are deduced deductively. From time to time Rand implies that her philosophy (Objectivism) is deductively derivable from axioms. Here are two she explicitly mentions:

Axiom of Existence: Existence exists is an axiom which states that there is something, as opposed to nothing.

Axiom of Identity: Everything that exists has a specific nature. Each entity exists as something in particular and it has characteristics that are a part of what it is.

Huh? Even if one gave her the benefit of doubt and allowed that these axioms are not as content free as they look, we would still have to acknowledge that it is not possible to deduce anything from these axioms; to say nothing of an entire philosophical edifice.

--Gaurav Suri

Spinoza Following Euclid

2005-04-28T13:21:00.000-07:00

A fascinating entry I found trolling around:

Upon opening Spinoza’s masterpiece, the Ethics, one is immediately struck by its form. It is written in the style of a geometrical treatise, much like Euclid’s Elements, with each book comprising a set of definitions, axioms, propositions, scholia, and other features that make up the formal apparatus of geometry. One wonders why Spinoza would have employed this mode of presentation. The effort it required must have been enormous, and the result is a work that only the most dedicated of readers can make their way through.
Some of this is explained by the fact that the seventeenth century was a time in which geometry was enjoying a resurgence of interest and was held in extraordinarily high esteem, especially within the intellectual circles in which Spinoza moved. We may add to this the fact that Spinoza, though not a Cartesian, was an avid student of Descartes’s works. As is well known, Descartes was the leading advocate of the use of geometric method within philosophy, and his Meditations was written more geometrico, in the geometrical style. In this respect the Ethics can be said to be Cartesian in inspiration.
While this characterization is true, it needs qualification. The Meditations and the Ethics are very different works, not just in substance, but also in style. In order to understand this difference one must take into account the distinction between two types of geometrical method, the analytic and the synthetic. Descartes explains this distinction as follows:
Analysis shows the true way by means of which the thing in question was discovered methodically and as it were a priori, so that if the reader is willing to follow it and give sufficient attention to all points, he will make the thing his own and understand it just as perfectly as if he had discovered it for himself. . . . . Synthesis, by contrast, employs a directly opposite method where the search is, as it were, a posteriori . . . . It demonstrates the conclusion clearly and employs a long series of definitions, postulates, axioms, theorems and problems, so that if anyone denies one of the conclusions it can be shown at once that it is contained in what has gone before, and hence the reader, however argumentative or stubborn he may be, is compelled to give his assent. (CSM II,110-111)The analytic method is the way of discovery. Its aim is to lead the mind to the apprehension of primary truths that can serve as the foundation of a discipline. The synthetic method is the way of invention. Its aim is to build up from a set of primary truths a system of results, each of which is fully established on the basis of what has come before. As the Meditations is a work whose explicit aim is to establish the foundations of scientific knowledge, it is appropriate that it employs the analytic method. The Ethics, however, has another aim, one for which the synthetic method is appropriate.
As its title indicates, the Ethics is a work of ethical philosophy. Its ultimate aim is to aid us in the attainment of happiness, which is to be found in the intellectual love of God. This love, according to Spinoza, arises out of the knowledge that we gain of the divine essence insofar as we see how the essences of singular things follow of necessity from it. In view of this, it is easy to see why Spinoza favored the synthetic method. Beginning with propositions concerning God, he was able to employ it to show how all other things can be derived from God. In grasping the order of propositions as they are demonstrated in the Ethics, we thus attain a kind of knowledge that approximates the knowledge that underwrites human happiness. We are, as it were, put on the road towards happiness. Of the two methods it is only the synthetic method that is suitable for this purpose.

I do understand Spinoza's quest and I admire him for it. Unfortunately life repells all attempts at purely deductive understanding. Nevertheless, deduction does happen, mixed in with all the half-guesses and induction driven inferences that we get to every day.

--Gaurav Suri

Mindscapes vs. Landscapes

2005-04-27T13:25:00.000-07:00

The universe is tremendously complex; whatever aspect we humans have attempted to study has shown glimpses of never ending complexity and eternally subtle (and delicate) mystery. Take gravity: Newton understood that the force that pulls a small object to our planet’s surface (I refuse to say apple; the story is a myth), is the same force that is responsible for the moon’s orbit around earth. He further realized that the gravitational force is directly proportional the masses of the 2 objects, and inversely proportional to the square of the distance between them. It was a remarkable theory, one that is fully of providing all the precision we need for travel within the solar system. But then Einstein came up with the General theory of relativity which postulated that space is ‘curved’ near heavy objects (the more the mass of the object, the greater the curvature) and that falling objects merely follow these curves in space.

The picture is suggestive only. It shows a 2-dimesional plane curving into the 3rd dimension. It is not a true representation of how space curves.

The General Theory was more precise than Newton’s (we’re talking ‘teeny’ differences here), and so now it has become the gospel truth. But the fact is that the General Theory is just that—a theory. It is a model of how the universe behaves. There may be another theory that explains gravity better than the General Theory (although a part of me hopes not—for it is a thing of great beauty).

Which brings me to the point: a lot of mathematics (applied mathematics in particular) models the world; it (or anything else) can’t say much about what the ‘real’ word is. At best we can model it, and some models are more useful—and beautiful—than others.

Admittedly there are branches of mathematics (often labeled ‘pure’ mathematics) where we are not modeling anything; Take Number Theory or Abstract Algebra or Set Theory. These branches work with the very stuff they build their knowledge base on. When we prove that there an infinite number of primes, we can be absolutely sure that there are, in fact, an infinite number of primes. Similarly if you accept Cantor’s rules for comparing infinite sets, then you must accept that the infinity of the Reals is greater than the infinity of the Integers. It is not a model of how those infinities ‘actually’ are, rather it is the terminus of a deductive argument that begins with the axioms of Set Theory.

While Set Theory (or Number Theory) are not models of the Real world, they also have nothing to do with it. Rather they are manifestations of how human brains work. They are creatures in our mindscape. We have certain (biological) intuitions about how sets (or numbers) should work and then we go ahead and derive less obvious properties about these objects.

Euclid’s geometry is an interesting case. Precision in the mindscape ran into the stringent demands of reality. Euclid developed believing that his mindscape picture of lines on a plane, applied to actual lines in space; and for over two thousand years there was no reason to doubt him. Then general relativity happened and Euclid’s Geometry lost it’s perch of a precise theory developed for objects in our mindscape that also applied to the real world.

Mathematics can either be imprecise about reality, or precise about our mindscape; but not both.

--Gaurav Suri

Madness

2005-04-26T16:47:00.000-07:00

I'm struck by how many of the truly great mathematicians had nervous breakdowns:

Georg Cantor fell in to a deep depression after his repeated attempts at cracking the continuum hypothesis (CH) did not yield any answers (I wish there was a way we could reach back through time and tell him about the futility of his endeavors -- Godel and Cohen proved that CH is neither provable, nor disprovable from the axioms of Set Theory).

Everyone knows about the madness of John Nash

Godel himself was a recluse and a fairly extreme conspiracy theorist in his later years. He is said to have found a logical flaw in the American constitution -- a point he almost brought up during his naturalization ceremony. An intervention by his friend Albert Einstein prevented a scene.

Ramanujan too had a tendency towards depression, though in his case it may just be because found himself stuck in the UK for months on end.

There are other examples (the Unabomber for one, though he was not a great mathematician, just a good one), enough to make one self ask the question - is there a co-relation?

--Gaurav Suri

Euclid Alone Has Looked on Beauty Bare (Not Really)

2005-04-25T21:32:00.000-07:00

To the best of my knowledge very little is known about Euclid himself. We know he came to work at the Alexandria library around 300 BCE. We also know that he founded a school of mathematics and was quite familiar with the mathematical results of his predecessors. Not much more is known about his life. We know a few things he is supposed to have said. One story says that a student who had just learned his first theorem asked Euclid what he could gain by studying such things. Euclid is said to have asked his slave to give the student three obols, “since he must make gain out of what he learns”. In another story Ptolemy is said to have asked Euclid if there was a quick way to learn geometry, to which Euclid is said to have replied, “there is no royal road to geometry”.

From his presentation in ‘Elements’ it is clear that Euclid was a thoughtful, patient man, with a wonderful eye for detail. He was of-course a very good mathematician, but he was probably an even better teacher. He wrote 'Elements' so that people could study mathematics methodically. And such was his passion that he recorded almost all of the basic mathematics known at his time. He must have had a lot of energy for ‘Elements’ is a collection of 13 books that contain no fewer than 465 separate propositions from plane and solid geometry and from number theory. Euclid’s genius was not that he created every single one of these 465 propositions – indeed it is known that he borrowed heavily from the works of his predecessors. Instead, Euclid’s genius was that he understood and illustrated the concept of proof. He founded, or at the very least propagated the notion of mathematical rigor. He was passionate about certainty. He begins each book within ‘Elements’ with a set of definitions and axioms. He then constructs a first proposition based exclusively on the definitions and axioms. Succeeding propositions build on previous results as well as the definitions and the axioms. And what emerge are beautiful, (seemingly) certain facts about the world around us. To this day mathematicians follow the same structure that Euclid laid out over two thousand years ago!

In a real sense, ‘Elements’ has had an impact comparable to the Bible. When it was written, it accelerated the pace of Greek mathematics. It was translated into Arabic and it had a real impact on that culture. Translated into Italian, the ‘Elements’ was one of the shaping influences of the Renaissance period. I know that Issac Newton was an admirer of the book and Abraham Lincoln himself was an ardent student of the ‘Elements’.

Here's a poem about Euclid by Edna St. Vincent Millay

EUCLID ALONE HAS LOOKED ON BEAUTY BARE

Euclid alone has looked on Beauty bare.
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth and cease
To ponder on themselves, the while they stare
At nothing, intricately drawn nowhere
In shapes of shifting lineage; let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.
O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.

The poem doesn't do much for me. I wish Edna had been able to talk about Euclid's real contribution: he invented the axiomatic method. The title though has a nice ring to it; "Euclid alone has looked on beauty bare", that phrase really has a lot of poetry to it; it hints at heroism (he alone), and rich secrets (beauty bare).

The irony is that the title is surely untrue. Very few of the theorems are original to Euclid -- he catalogued and organized theorems that were known by prior geometers, notably Thales and Pythagoras. Surely they too saw the beauty in their original discoveries, and Euclid was not alone in glimpsing the beauty. Nevertheless, I like Edna's phrasing and commend her for her choice of subject.

Here's an analysis on the poem itself

--Gaurav Suri

Where does mathematics come from?

2005-04-25T19:07:00.000-07:00

Here's is one of the better definitions I've seen:

Mathematics is the part of science you could continue to do if you woke up tomorrow and discovered the universe was gone.

Yep--it has the ring of truth, but here's thing: I do believe that the universe (aka the real world) is the only source of new problems. If the universe ceased to exist we could only do mathematics because the universe existed at one point. All mathematics comes from things around us: our awareness of numbers came from the discreteness in the universe; our geometry came from our ability to draw on lines on planes; even highly abstract branches of mathematics are eventually traceable back to something in the real world.

So yes it's true -- if the universe disappears from around us, we would still be able to do mathematics -- but only because the universe once existed. The larger point is that new problems must come from the world around us, they couldn't possibly come from anywhere else. No?

--Gaurav Suri

Bishop Berkeley on Calculus

2005-04-25T17:24:00.000-07:00

Newton and Leibniz's calculus was developed 'illogically'. In other words theorems were developed without a proper axiomatic foundation. If a rule appeared to work then it was -- for the most part -- accepted as a theorem. Bishop George Berkeley who feared that the deterministic nature of mathematics would undermine religion had the following point to make:

And if the first [fluxions] are incomprehensible, what should we say of the second and the third [derivatives of derivatives] etc.? He who can conceive the beginning of a beginning, or the end of an end...may perhaps be sharpsighted enough to conceive of these things. But most men will, I believe, find it impossible to understand them in any sense whatever...He who can digest a second or a third fluxion...need not, methinks, be squeamish about any point in Divinity.

Since Berkeley's assessment, calculus has been satisfactorily axiomatized. But once again the axioms followed the theory -- which tends to support the argument that at its core mathematics is an empirical science.

The Berkeley quote is from Kline's Mathematics: The loss of certainty, one of the most stimulating books I've read.

--Gaurav Suri

The Carvaka's search for proof

2005-04-25T11:37:00.000-07:00

Around the 8th century BCE Hinduism was mourning the lack of justice in the world. Evil deeds going unpunished and saintly acts going unrewarded likely made for a philosophically intolerable position. How could God allow a world without justice?

Perhaps not wanting to doubt the very existence of God, the Hindus came up with the concept of karma and afterlife. Very loosely the idea was that if one did good in this life he (or his soul) would be rewarded in the next, and if one was wicked his soul could well be born as a mouse in the next life. Doing one’s duty--karma--ensured upward progress.

The idea of afterlife neatly resolved the dilemma of an unjust world. Yes, perhaps God didn’t hand down rewards or punishments in this life, but he did keep score, and in the end all actions were counted. The idea stuck and the afterlife became an integral part of Hinduism.

The Carvaka thought that the afterlife concept was nonsensical. Their movement called the Lokayata took hold in the 7th century BCE and it was the first serious rebellion in Hinduism. The Carvaka held a completely materialist doctrine: we are our bodies, we think and feel with them, and eventually our bodies die out. And then it’s over. There is no afterlife whatsoever and whoever thinks otherwise is an ‘ignorant, uncivilized fool.’ They found claims that there is a soul, or spirit, separate from one’s body to be dishonest, and actively sought to counter the claims of the religious establishment.

Alas the establishment won out and systematically destroyed all works of the Carvaka. The only reason we now know of their existence is because later Hindus quoted their arguments in order to rebut them. But for them the Carvaka would have been lost to history. Ironic.

What little has come down is absolutely fascinating. The Lokayata movement did not believe in karma, God, the soul or virtue—unless performed for it’s own sake. There was no heaven to look forward to, or hell to fear. It was all here on earth. All this is somewhat standard fare in the history of doubt. But the Carvaka took it a step further: they doubted the possibility of inference. This notion is mentioned in later texts because it seems to make their position appear nonsensical. In my opinion it was their highest insight, one that brought them to the doorstep of mathematical proof, several centuries before the Greeks got there.

The Carvaka said that it was a fallacy to use dependent ideas. Just because water is wet every time we touch it, does not mean that it will be wet the next time as well. All we can know is that all the instances of water we have encountered have been wet. Similarly, they argued that just because all swans appear to be white, does not constitute knowledge that all swans are in fact white.

So, really the Carvaka rejected the inductive form of knowledge as a path to certainty. It is a brilliant notion far ahead of its time. In fact it is the same sentiment that led to Euclid axiomatizing geometry. In effect the Carvaka argued that only deduction is acceptable, and simple cause and effect does not constitute deductive proof. They were *this* close to developing a deductive foundation for mathematics.

Note: To learn more about the Carvaka and the history of disbelief check out ‘Doubt’, a fine book by Jennifer Michael Hecht.

More on the Carvaka

--Gaurav Suri

The Continuum Hypothesis

2005-04-24T17:22:00.000-07:00

Cantor showed that not all infinities are equal to each other. In particular the infinity of Natural Numbers (1, 2, 3...) is less than the infinity of Real Numbers (Real numbers include all integers, fractions and Irrational numbers such as the square root of 2).

A natural question arises: is there an infinity whose order is greater than that of the Natural numbers, but less than that of the Reals. Cantor guessed that the answer is 'no'; his guess has come to be known as the Continuum Hypothesis (CH), one of the most celebrated problems in mathematics.

Other mathematicians believe that the infinity of the Reals is incredibly rich and that there are infinities of intermediate order. In other words they believe that the continuum hypothesis is false.

Interestingly, it has been shown that the commonly accepted axioms of set theory are insufficient to prove or disprove CH. Some other, richer axioms are required. CH then becomes an interesting test case of one’s philosophical beliefs about mathematics. One could believe that 1) Of-course CH is either true or false; we just need to find a simple enough axiom that can help us prove this; or that 2) It is meaningless to ask whether CH is true or false. One can only say that CH is unprovable in the current set theoretic framework. In other words CH has no independent truth or falsehood of its own, it only derives meaning from the axiom set.

I deeply believe that CH is either true or false. In fact I’m almost sure (despite all the recent writings on the other side) that CH is true. Here’s why: If CH was false, wouldn’t we have found some natural intermediate set? God knows we’ve tried everything—yet whatever set we imagine-it turns out to have the cardinality of the Naturals or of the Reals.

I realize that this is not a proof--far from it; but it is what I believe.

More on CH: http://www.ii.com/math/ch/

--Gaurav Suri

Would Jovians Know Numbers?

2005-04-24T17:03:00.000-07:00

Mathematics was long considered to be an absolute body of knowledge. Its truths were held to be universal, eternal and absolute. Euclid’s geometry for example was believed to be the gold standard in human certainty. But over time mathematicians came to realize that the truth of their theorems is contingent on underlying axioms—and underlying axioms, even the seemingly ‘obvious’ ones were often seen to be false. As such, Einstein and Eddington were able to prove that Euclid’s parallel postulate did not hold for space, and therefore theorems based on the parallel postulate were not true of space.

The question then arose: What is mathematics? One school of thought (The Platonists) continued to believe that mathematics ‘discovers’ absolute truths about the universe. They held that the errors made by Euclid and others did not take-away from the essential quest of mathematics: absolute certainty. An opposing group of mathematicians (The Formalists) pointed to the fallibility of axioms and held that mathematics is a game, somewhat like chess, where symbols were manipulated according to rules legislated by axioms. Neither the theorems nor the axioms were true (or false) in any sense.

These schools—established in the early part of the twentieth century—hardened their positions and nothing much happened to resolve questions about the essential nature of mathematics. Then, a few years ago the mathematician-philosopher Hersh wrote that mathematics is a human activity, and the truths we discover about it are a function of our culture, values and biology. According to Hersh a different intelligence from ours would come up with radically different mathematics.

I understand Hersh’s argument and motivation. Large parts of mathematics seem to be sociologically driven, and he neatly avoids the Platonist-Formalist conundrum. But there is a natural question that arises: What about numbers? Are they a human invention or must any intelligence eventually find their way to 1, 2, 3…

It certainly seems that consciousness forces us to distinguish the ‘I’ from the other. As soon as an intelligence becomes aware of this separateness, they will need and invent numbers. But wait, Hersh might say—perhaps individual consciousness is a human peculiarity; perhaps there are intelligences that are collective in some sense and would find our notion of I-ness to be peculiar. Perhaps their bodies are intermingled with each other’s and they have a group-consciousness rather than a self-consciousness. For example, their intelligence could live in a gaseous cloud on some Jupiter like planet.

Hard to argue with that, but let us push a little. This gaseous, intelligent cloud, wouldn’t it observe things around it—wouldn’t it want to count stars for example? No, Hersh would argue. This gaseous cloud might live in a some homogenous, featureless cloudy soup where there were no discreteness of any kind.
But what about time? Wouldn’t any intelligence evolve to be aware of time? And as soon as they become aware of time, they will need to measure it, and once they need to measure it they will invent numbers. With numbers they would discover that 2 + 2 = 4, and with that they would get the richness of our number theory. Is it possible, Mr. Hersh, to visualize an intelligence that does not need time?
“Well,” he might say. “Why not?”

‘Why not’, indeed. There is no way to be sure either way. But surely one can see that only a barren intelligence—one unaware of time, distance, size and shape—could conceive of a mathematics that does not include numbers. Could they then even qualify as an intelligence?

Check out http://www.edge.org/q2005/q05_7.html#sabbagh for a related post

--Gaurav Suri