Russell and mathematical certainty

 

 “I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure then the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.”

“Mathematics is, I believe,” says Russell, “the chief source of the belief in eternal and exact truth, as well as in a super-sensible intelligible world. Geometry deals with exact circles, but no sensible object is exactly circular; however carefully we may use our compasses, there will be some imperfections and irregularities. This suggests the view that all exact reasoning applies to ideal as opposed to sensible objects; it is natural to go further, and to argue that thought is nobler than sense, and the objects of thought more real than those of sense-perception. Mystical doctrines as to the relation of time to eternity are also reinforced by pure mathematics, for mathematical objects, such as number, if real at all, are eternal and not in time. Such eternal objects can be conceived as God’s thoughts. Hence Plato’s doctrine that God is a geometer, and Sir James Jeans’ belief that He is addicted to arithmetic. Rationalistic as opposed to apocalyptic religion has been, ever since Pythagoras, and notably ever since Plato, very completely dominated by mathematics and mathematical method.

Mainstream Philosophies of Mathematics

 

Frege

                                                          

The name “foundationism” was invented by a prolific name-giver, Imre Lakatos. It refers to Gottlob Frege in his prime, Bertrand Russell in his full logicist phase, Luitjens Brouwer, guru of intuitionism, and David Hilbert, prime advocate of formalism. Lakatos saw that despite their disagreements, they all were hooked on the same delusion: Mathematics must have a firm foundation. They differ on what the foundation should be. 

Foundationism has ancient roots. Behind Frege, Hilbert, and Brouwer stands Immanuel Kant. Behind Kant, Gottfried Leibniz. Behind Leibniz, Baruch Spinoza, and René Descartes. Behind all of them, Thomas Aquinas, Augustine of Hippo, Plato, and the great grandfather of foundationism—Pythagoras. 

We will find that the roots of foundationism are tangled with religion and theology. In Pythagoras and Plato, this intimacy is public. In Kant, it’s half covered. In Frege, it’s out of sight. Then in Georg Cantor, Bertrand Russell, David Hilbert, and Luitjens Brouwer, it pops up like a jack-in-the-box.

In the twentieth century, we look at Russell, Brouwer, Hilbert, Edmund Husserl, Ludwig Wittgenstein, Kurt Gödel, Rudolph Carnap, Willard V. O. Quine, and a small sample of today’s authors. Philip Kitcher said the philosophy of mathematics is generally supposed to begin with Frege—before Frege there was only “prehistory.” Frege transformed the issues constituting philosophy of mathematics. In that sense earlier philosophy can be called prehistoric. But to understand Frege you must see him as a Kantian. To understand Kant you must see his response to Newton, Leibniz, and Hume. Those three go back to Descartes, and through him to Plato. Plato was a Pythagorean. The thread from Pythagoras to Hilbert and Gödel is unbroken. I aim to tell a connected story from Pythagoras to the present—where foundationism came from, where it left us. 

Instead of going straight through from Pythagoras, I’ve split the story into two parallel streams—the first section is about the “Mainstream.” The second is about the “humanists and mavericks.” 

For the Mainstream, mathematics is superhuman—abstract, ideal, infallible, eternal. So many great names: Pythagoras, Plato, Descartes, Spinoza, Leibniz, Kant, Frege, Russell, Carnap. (For Kant, membership in this group is partial.) 

Humanists see mathematics as a human activity, a human creation. Aristotle was a humanist in that sense, as were Locke, Hume, and Mill. Modern philosophers outside the Russell tradition—mavericks—include Peirce, Dewey, Roy Sellars, Wittgenstein, Popper, Lakatos, Wang, Tymoczko, and Kitcher (a self-styled maverick). There are some interesting authors who aren’t labeled philosophers: psychologist Jean Piaget; anthropologist Leslie White; sociologist David Bloor; chemist Michael Polányi; physicist Mario Bunge; educationists Paul Ernest, Gila Hanna, Anna Sfard; mathematicians Henri Poincaré, Alfréd Rényi, George Pólya, Raymond Wilder, Phil Davis, and Brian Rotman.

Platonism vs Formalism

 

Frege, Russell and Wittgenstein

Two principal views of the nature of mathematics are prevalent among mathematicians—Platonism and formalism. Platonism is dominant, but it’s hard to talk about it in public. Formalism feels more respectable philosophically, but it’s almost impossible for a working mathematician to really believe it.

**

The formalist philosophy of mathematics is often condensed to a short slogan: “Mathematics is a meaningless game.” (“Meaningless” and “game” remain undefined. Wittgenstein showed that games have no strict definition, only a family resemblance.)

What do formalists mean by “game” when they call mathematics a game? Perhaps they use “game” to mean something “played by the rules.” (Now “play” and “rule” are undefined!)

For a game in that sense, two things are needed:
(2) people to play by the rules.
(1) rules.

Rule-making can be deliberate, as in Monopoly or Scrabble—or spontaneous, as in natural languages or elementary arithmetic.

In either case, the making of rules doesn’t follow rules!

Wittgenstein and some others seem to think that since the making of rules doesn’t follow rules, then the rules are arbitrary. They could just as well be any way at all. This is a gross error.

The rules of language and of mathematics are historically determined by the workings of society that evolve under pressure of the inner workings and interactions of social groups, and the physical and biological environment of earth.

**
In real life there are no totally rule-governed activities. Only more or less rule-governed ones, with more or less definite procedures for disputes. The rules and procedures evolve, sometimes formally like amending the U.S. Constitution, sometimes informally, as street games evolve with time and mixing of cultures. Is there totally unruly or ruleless behavior? Perhaps not. Mathematics is in part a rule-governed game. But one can’t overlook how the rules are made, how they evolve, and how disputes are resolved. That isn’t rule governed, and can’t be.

Computer proof is changing the way the game of mathematics is played. Wolfgang Haken thinks computer proof is permitted under the rules. Paul Halmos thinks it ought to be against the rules. Tom Tymoczko thinks it amounts to changing the rules. In the long run, what mathematicians publish, cite, and especially teach, will decide the rules. We have no French Academy to set rules, no cabal of team owners to say how to play our game. Our rules are set by our consensus, influenced and led by our most powerful or prestigious members (of course).

These considerations on games and rules in general show that one can’t understand mathematics (or any other nontrivial human activity) by simply finding rules that it follows or ought to follow. Even if that could be done, it would lead to more interesting questions: Why and whence those rules?

The notion of strictly following rules without any need for judgment is a fiction. It has its use and interest. It’s misleading to apply it literally to real life.

**

Platonism says mathematical objects are real and independent of our knowledge. Space-filling curves, uncountably infinite sets, infinite-dimensional manifolds—all the members of the mathematical zoo—are definite objects, with definite properties, known or unknown. These objects exist outside physical space and time. They were never created. They never change. By logic’s law of the excluded middle, a meaningful question about any of them has an answer, whether we know it or not. According to Platonism a mathematician is an empirical scientist, like a botanist. He can’t invent, because everything is already there. He can only discover.

**

The objections to Platonism are never answered: the strange parallel existence of two realities—physical and mathematical; and the impossibility of contact between the flesh-and-blood mathematician and the immaterial mathematical object. Platonism shares the fatal flaw of Cartesian dualism. To explain the existence and properties of mind and matter, Descartes postulated a different “substance” for each. But he couldn’t plausibly explain how the two substances interact, as mind and body do interact. In similar fashion, Platonists explain mathematics by a separate universe of abstract objects, independent of the material universe. But how do the abstract and material universes interact? How do flesh-and-blood mathematicians acquire the knowledge of number?

To answer, you have to forget Platonism, and look in the socio-cultural past and present, in the history of mathematics, including the tragic life of Georg Cantor.

The set-theoretic universe constructed by Cantor and generally adopted by Platonists is believed to include all mathematics, past, present, and future. In it, the uncountable set of real numbers is just the beginning of uncountable chains of uncountables. The cardinality of this set universe is unspeakably greater than that of the material world. It dwarfs the material universe to a tiny speck. And it was all there before there was an earth, a moon, or a sun, even before the Big Bang. Yet this tremendous reality is unnoticed! Humanity dreams on, totally unaware of it—except for us mathematicians. We alone notice it. But only since Cantor revealed it in 1890. Is this plausible? Is this credible? Roger Penrose declares himself a Platonist, but draws the line at swallowing the whole set-theoretic hierarchy.

Platonists don’t acknowledge the arguments against Platonism. They just reavow Platonism.

Frege’s point of view persists today among set-theoretic Platonists. It goes something like this:

1. Surely the empty set exists—we all have encountered it!
2. Starting from the empty set, perform a few natural operations, like forming the set of all subsets. Before long you have a magnificent structure in which you can embed the real numbers, complex numbers, quaternions, Hilbert spaces, infinite-dimensional differentiable manifolds, and anything else you like.
3. Therefore it’s vain to talk of inventing or creating mathematics. In this all-encompassing, set-theoretic structure, everything we could ever want or dream of is already present.

Yet most advances in mainstream mathematics are made without reference to any set-theoretic embedding. Saying Hilbert space was already there in the set universe is like telling Rodin, “The Thinker is a nice piece of work, but all you did was get rid of the extra marble. The statue was there inside the marble quarry before you were born.” Rodin made The Thinker by removing marble. Hilbert, von Neumann, and the rest made the theory of Hilbert space by analysing, generalizing, and rearranging mathematical ideas that were present in the mathematical atmosphere of their time.

(Reuben Hersh, What is mathematics really?)
**

“Mathematical objects are real. Their existence is an objective fact, independent of our knowledge of them. Infinite sets, uncountably infinite sets, infinite-dimensional manifolds, space-filling curves—all the denizens of the mathematical zoo—are definite objects, with definite properties. Some of their properties are known, some are unknown. These objects aren’t physical or material. They’re outside space and time. They’re immutable. They’re uncreated. A meaningful statement about one of these objects is true or false, whether we know it or not. Mathematicians are empirical scientists, like botanists. We can’t invent anything; it’s there already. We try to discover.”

**

Work and Happiness

 

On Envy

 

Napoleon envied Caesar, who envied Alexander, who probably envied Hercules, who existed only in myth! Success is never the cure for envy. Eliminate envy by enjoying what life brings you, by working hard, and by avoiding comparison with those you think more blessed than you.

Extreme modesty is itself a form of envy. Measured modesty is a virtue, but extreme modesty shouldn’t be similarly regarded. An overly modest person needs reassurance to do things which they’re very capable of doing.

[…]

Envy is closely related to competition. We don’t envy people we think are out of reach. During periods when the social hierarchy was fixed, the lower classes didn’t envy the upper classes because movement between them was impossible. Beggars don’t envy billionaires, they envy other beggars who have a warm spot to sleep for the night.

However, the current instability of social status, combined with democracy’s message that everyone is equal, has made envy accessible to all. Eventually we’ll arrive at a more just social system, but for now, the poor envy the rich, poor nations envy rich nations, women envy men, the chaste envy the promiscuous.

While envy can be productive and lead to justice between classes, nations, and even genders, the justice that results is the worst kind, the kind that hurts the fortunate, rather than helping the unfortunate. If you desire profound, positive societal change, you should root for reasons other than envy to be the driving force.

[…]

Most envy which seems on the surface professional is actually sexual in nature. A man who loves his wife and kids will be much less envious of other men’s success and wealth. What makes people happy is simple, often deceptively so, such that so-called sophisticated people fool themselves into making it complicated.

Source

On Boredom

 

Boredom as a factor in human behaviour has received, in my opinion, far less attention than it deserves. It has been, I believe, one of the great motive powers throughout the historical epoch, and is so at the present day more than ever. 

**

The opposite of boredom, in a word, is not pleasure, but excitement.

**
Perhaps some element of boredom is a necessary ingredient in life. A wish to escape from boredom is natural; indeed, all races of mankind have displayed it as opportunity occurred. When savages have first tasted liquor at the hands of the white men, they have found at last an escape from age-old tedium, and, except when the Government has interfered, they have drunk themselves into a riotous death. Wars, pogroms, and persecutions have all been part of the flight from boredom; even quarrels with neighbours have been found better than nothing. Boredom is therefore a vital problem for the moralist, since at least half the sins of mankind are caused by the fear of it.

**

There is an element of boredom which is inseparable from the avoidance of too much excitement, and too much excitement not only undermines the health, but dulls the palate for every kind of pleasure, substituting titillations for profound organic satisfactions, cleverness for wisdom, and jagged surprises for beauty. I do not want to push to extremes the objection to excitement. A certain amount of it is wholesome, but, like almost everything else, the matter is quantitative. Too little may produce morbid cravings, too much will produce exhaustion. A certain power of enduring boredom is therefore essential to a happy life, and is one of the things that ought to be taught to the young.

All great books contain boring portions, and all great lives have contained uninteresting stretches. 

**
... Confucian classics, the Koran, Marx's Capital, and all the other sacred books which have proved to be bestsellers. Nor does this apply only to sacred books. All the best novels contain boring passages. A novel which sparkles from the first page to the last is pretty sure not to be a great book. Nor have the lives of great men been exciting except at a few great moments. Socrates could enjoy a banquet now and again, and must have derived considerable satisfaction from his conversations while the hemlock was taking effect, but most of his life he lived quietly with Xanthippe, taking a constitutional in the afternoon, and perhaps meeting a few friends by the way. Kant is said never to have been more than ten miles from Konigsberg in all his life. Darwin, after going round the world, spent the whole of the rest of his life in his own house. Marx, after stirring up a few revolutions, decided to spend the remainder of his days in the British Museum. Altogether it will be found that a quiet life is characteristic of great men, and that their pleasures have not been of the sort that would look exciting to the outward eye. No great achievement is possible without persistent work, so absorbing and so difficult that little energy is left over for the more strenuous kinds of amusement, except such as serve to recuperate physical energy during holidays, of which Alpine climbing may serve as the best example.

**
I do not like mystical language, and yet I hardly knows how to express what I mean without employing phrases that sound poetic rather than scientific. Whatever we may wish to think, we are creatures of Earth; our life is part of the life of the Earth, and we draw our nourishment from it just as the plants and animals do. The rhythm of Earth life is slow; autumn and winter are as essential to it as spring and summer, the rest is as essential as motion. To the child, even more than to the man, it is necessary to preserve some contact with the ebb and flow of terrestrial life. The human body has been adapted through the ages to this rhythm, and religion has embodied something of it in the festival of Easter.

I have seen a boy of two years old, who had been kept in London, taken out for the first time to walk in green country. The season was winter, and everything was wet and muddy. To the adult eye there was nothing to cause delight, but in the boy there sprang up a strange ecstasy; he kneeled in the wet ground and put his face in the grass, and gave utterance to half-articulate cries of delight. The joy that he was experiencing was primitive, simple and massive. The organic need that was being satisfied is so profound that those in whom it is starved are seldom completely sane.

Many pleasures, of which we may take gambling as a good example, have in them no element of this contact with Earth. Such pleasures, in the instant when they cease, leave a man feeling dusty and dissatisfied, hungry for he knows not what. Such pleasures bring nothing that can be called joy. Those, on the other hand, that bring us into contact with the life of the Earth have something in them profoundly satisfying; when they cease, the happiness that they brought remains, although their intensity while they existed may have been less than that of more exciting dissipations.

The distinction that I have in mind runs through the whole gamut from the simplest to the most civilised occupations. The two-year-old boy whom I spoke of a moment ago displayed the most primitive possible form of union with the life of Earth. 

**

The special kind of boredom from which modern urban populations suffer is intimately bound up with their separation from the life of Earth. It makes life hot and dusty and thirsty, like a pilgrimage in the desert. Among those who are rich enough to choose their way of life, the particular brand of unendurable boredom from which they suffer is due, paradoxical as this may seem, to their fear of boredom. In flying from the fructifying kind of boredom, they fall a prey to the other far worse kind. A happy life must be to a great extent a quiet life, for it is only in an atmosphere of quiet that true joy can live.

Bertrand Russell, Chapter 4, from his The Conquest of Happiness, 1930

"My current happiness is due to a diminishing preoccupation with myself"

 

“Perhaps the best introduction to the philosophy which I wish to advocate will be a few words of autobiography. I was not born happy. As a child, my favourite hymn was: “Weary of earth and laden with my sin.” At the age of five, I reflected that, if I should live to be seventy, I had only endured, so far, a fourteenth part of my whole life, and I felt the long-spread-out boredom ahead of me to be almost unendurable. In adolescence, I hated life and was continually on the verge of suicide, from which, however, I was restrained by the desire to know only more mathematics.

Now, on the contrary, I enjoy life; I might almost say that with every year that passes I enjoy it more. This is due partly to having discovered what were the things that I most desired, and having gradually acquired many of these things. Partly it is due to having successfully dismissed certain objects of desire — such as the acquisition of indubitable knowledge about something or other — as essentially unattainable. But very largely my current happiness is due to a diminishing preoccupation with myself.

Like others who had a Puritan education, I had the habit of meditating on my sins, follies, and shortcomings. I seemed to myself— no doubt justly — a miserable specimen. Gradually I learned to be indifferent to myself and my deficiencies; I came to centre my attention increasingly upon external objects: the state of the world, various branches of knowledge, individuals for whom I felt affection. External interests, it is true, bring each its own possibility of pain: the world may be plunged in war, knowledge in some direction may be hard to achieve, friends may die. But pains of these kinds do not destroy the essential quality of life, as do those that spring from disgust with self. ”

― Bertrand Russell, The Conquest of Happiness, Part I. Causes of Unhappiness, Chapter I. What Makes People Unhappy?, pg. 9.

Barber Paradox

 

 

 

Source: http://www.logicomix.com/en/index.html