Friday, April 29, 2005

Ayn Rand Misused Axioms

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

Thursday, April 28, 2005

Spinoza Following Euclid

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

Wednesday, April 27, 2005

Mindscapes vs. Landscapes

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.

Curved 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

Tuesday, April 26, 2005


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

Monday, April 25, 2005

Euclid Alone Has Looked on Beauty Bare (Not Really)

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.
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?

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

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

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

Sunday, April 24, 2005

The Continuum Hypothesis

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:

--Gaurav Suri

Would Jovians Know Numbers?

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?

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--Gaurav Suri