Archive for June, 2009

Language and thought

A very interesting essay by Lera Boroditsky on the relationship between language and thought.  Comparing languages and cognitive styles in different cultures, she concludes that the structure of a language may influence what we most attend to, and thus our modes of thinking.  (HT: AS)

Follow me to Pormpuraaw, a small Aboriginal community on the western edge of Cape York, in northern Australia. I came here because of the way the locals, the Kuuk Thaayorre, talk about space. Instead of words like “right,” “left,” “forward,” and “back,” which, as commonly used in English, define space relative to an observer, the Kuuk Thaayorre, like many other Aboriginal groups, use cardinal-direction terms — north, south, east, and west — to define space. This is done at all scales, which means you have to say things like “There’s an ant on your southeast leg” or “Move the cup to the north northwest a little bit.” One obvious consequence of speaking such a language is that you have to stay oriented at all times, or else you cannot speak properly. The normal greeting in Kuuk Thaayorre is “Where are you going?” and the answer should be something like ” Southsoutheast, in the middle distance.” If you don’t know which way you’re facing, you can’t even get past “Hello.”

The result is a profound difference in navigational ability and spatial knowledge between speakers of languages that rely primarily on absolute reference frames (like Kuuk Thaayorre) and languages that rely on relative reference frames (like English). Simply put, speakers of languages like Kuuk Thaayorre are much better than English speakers at staying oriented and keeping track of where they are, even in unfamiliar landscapes or inside unfamiliar buildings. What enables them — in fact, forces them — to do this is their language. Having their attention trained in this way equips them to perform navigational feats once thought beyond human capabilities. Because space is such a fundamental domain of thought, differences in how people think about space don’t end there. People rely on their spatial knowledge to build other, more complex, more abstract representations. Representations of such things as time, number, musical pitch, kinship relations, morality, and emotions have been shown to depend on how we think about space. So if the Kuuk Thaayorre think differently about space, do they also think differently about other things, like time? This is what my collaborator Alice Gaby and I came to Pormpuraaw to find out.

To test this idea, we gave people sets of pictures that showed some kind of temporal progression (e.g., pictures of a man aging, or a crocodile growing, or a banana being eaten). Their job was to arrange the shuffled photos on the ground to show the correct temporal order. We tested each person in two separate sittings, each time facing in a different cardinal direction. If you ask English speakers to do this, they’ll arrange the cards so that time proceeds from left to right. Hebrew speakers will tend to lay out the cards from right to left, showing that writing direction in a language plays a role. So what about folks like the Kuuk Thaayorre, who don’t use words like “left” and “right”? What will they do?

The Kuuk Thaayorre did not arrange the cards more often from left to right than from right to left, nor more toward or away from the body. But their arrangements were not random: there was a pattern, just a different one from that of English speakers. Instead of arranging time from left to right, they arranged it from east to west. That is, when they were seated facing south, the cards went left to right. When they faced north, the cards went from right to left. When they faced east, the cards came toward the body and so on. This was true even though we never told any of our subjects which direction they faced. The Kuuk Thaayorre not only knew that already (usually much better than I did), but they also spontaneously used this spatial orientation to construct their representations of time.

People’s ideas of time differ across languages in other ways. For example, English speakers tend to talk about time using horizontal spatial metaphors (e.g., “The best is ahead of us,” “The worst is behind us”), whereas Mandarin speakers have a vertical metaphor for time (e.g., the next month is the “down month” and the last month is the “up month”). Mandarin speakers talk about time vertically more often than English speakers do, so do Mandarin speakers think about time vertically more often than English speakers do? Imagine this simple experiment. I stand next to you, point to a spot in space directly in front of you, and tell you, “This spot, here, is today. Where would you put yesterday? And where would you put tomorrow?” When English speakers are asked to do this, they nearly always point horizontally. But Mandarin speakers often point vertically, about seven or eight times more often than do English speakers.

POSTSCRIPT (ADDED 2010-08-29):  An article by Guy Desutscher in the NYT covering similar ground is here.

Poem: White Violets

Today’s poem is by Gerry Moll (1900 – 1997), an Australian poet from Murtoa, in the Wimmera, who studied at Harvard University and then lived for many years in Oregon, USA.  Moll was apparently a fine teacher of literature, and students who took his Shakespeare class in the 1920s still remembered him seven decades later.

One cannot read this poem this week without thinking of the brave people of Iran.


White Violets

This Spring the white violets
Came early and everywhere
Greet me with their white expectant faces.
For years I’d kept them sternly in their places:
A rock-bowl here, a thin, neat border there.
They’ve broken lines and every boundary
A gardener sets for such;
Even the lawn, still winter-brown, is a sea
Foamed over by them and the warming air,
Filled with their scent, proclaims their empery.

Why did they come so early? Did they know
That one was the way to still the hand
That bestowed blessings on them with its touch?
And why such masses? Did they fear
She might not see them with her pain-dimmed eyes,
Or, hastening on the way she had to go,
Would miss them at this turning of the year?

Grief to these questions asks for no replies.



Previous poems in this series are here.

Ernest G. Moll [1992]:  The View from a Ninetieth Birthday:  Lyrical Poems of Old Age. La Jolla, CA:  La Jolla Poets Press.  page 48.

The mathematics of jellyfish leaves much to be desired

A reader of Normblog presents a (standard) constructive argument for the counting numbers and then the infinite cardinals:

I happen to be friends with a jellyfish, called Jelly von Neumann. I asked Jelly about what Professor Atiyah said and she replied as follows…

‘Even if one has never seen any fish, crabs or the like, one may proceed as follows. First consider the empty set, { }, the set which has no elements whatsoever. Call that 0. Next, having got 0, consider the set {0}, whose only element is 0. Call that 1. Next consider the set {0, 1}, whose elements are exactly 0 and 1. Call that 2. Next consider the set {0, 1, 2}. Call that 3.

‘And so on. This gives you the infinite sequence 0, 1, 2, 3,… (One can prove that this sequence is infinite, since the operation involved is injective and never maps anything to 0.) You may even consider the whole infinite set, {0, 1, 2, 3,…}. Call this set omega. And you can go further. For consider the set {omega}. Call this omega + 1. Then consider {omega, omega + 1}, and call this omega + 2. Keep going. You get to omega + omega, and then omega + omega + omega. And so on. Eventually omega squared. Then omega cubed. And so on. Then omega to the power omega. And then (omega to the power omega) to the power omega. And then keep going. Eventually, you get to epsilon-zero. It gets a bit complicated after that. The point is that you can do mathematics just by virtue of thinking. Of course, I am a rather special jellyfish in that regard.’

Let us look carefully at the first few lines.  Before we have defined or constructed a single number, we are expected to have available a notion of a set and a notion of an element of a set.

First consider the empty set, { }, the set which has no elements whatsoever.

This is very odd – we are people who apparently know some set theory, but we cannot yet count (since we have not yet constructed the counting numbers).   And not just any set, but a set with no elements.   So maybe we can count!  How else can we tell that there are no elements in the empty set?  Perhaps we can only count zero objects.   And, moreover, this set is called “the empty set”, so presumably we know that there is only one of them.  There’s some pretty advanced set theory right there, in that casual statement of uniqueness, I would say.  (The claim of uniqueness, however, is not required for Jelly’s construction.)

Putting aside the question whether it is possible in principle for anyone, even those us with access to counting numbers, to count zero objects (arguably, counting is by definition an activity which requires the presence of at least one object to occur), let us continue with Jelly’s argument:

Call that 0.

So we can label objects.

Next, having got 0,

Wait a goddam minute, buster!  We just labeled an object “{  }” with the label “0“.    That is something different from getting or having anything.   And surely, in order to label an object “{  }” with a label “0“, we must in some fundamental sense already had had the label  “0“.   If we did not already have it, how else could we use it to label an object?   Jelly is using some pretty sleazy slight-of-hand here to slip from assigning a label that looks like a counting number to having the counting number itself, ready and able to be used for counting.   If the label we had used was (say) the greek letter alpha, then Jelly’s argument would proceed in exactly the same way as before, but we would not end the argument having defined the counting numbers.

Ignoring these problems, let us proceed:

consider the set {0}, whose only element is 0.

So now “0″ is an object, available for use as the element of a set. And we not only know some set theory, we ALSO know how to construct sets!   Just how do we do this?  Do we pick the object (or the label?) called “0″ and put it inside some curly braces?  How do we know when to start and stop picking objects?  For some reason we picked just one object.  Do we know how to count already?  At the next step we construct a set with two objects:

Next consider the set {0, 1}, whose elements are exactly 0 and 1. Call that 2.

From what collection of objects (or labels?) did we select the one called “0″ , or (respectively) the ones called “0″ and “1″? We seem not only able to construct sets and to count objects, but we also know how to select particular objects (not just any old objects, but particular ones) from some undefined collection of objects. Quite some skills we have here, we people who don’t yet know how to count.  And is the object that is here called “0” a different object with the same label as the one called “0” just three sentences before?   If they are different, how many of these different objects with the same label do we have?  And how can we tell them apart?  And, if they are not different, we must be re-using the same object called “0”.  Can we do this?  When last handled by us (two sentences before), the object called “0” was sitting inside the set {0}.  Can we just up and take it out and plonk it down inside the set {0,1}?  There are lots of deep questions here, questions whose possibly-different answers motivate entire branches of pure mathematics (e.g., linear logic, which deals with formal logics where we have available only a fixed and finite number of each mathematical symbol), which our jellyfish-cum-mathematician is glossing over or ignoring.

After a few rounds of this, Jelly hits us with:

And so on. This gives you the infinite sequence 0, 1, 2, 3,…

Well, no, actually. We never get an infinite sequence, since we, in this universe, can only ever complete a finite number of such steps in our lifetimes.  This is true even if all humans who ever lived, who are living, and who ever will live were to add their tuppence-worth of steps to the argument.  It’s hard to have confidence in a jellyfish claiming to construct a collection of infinite cardinals who can’t seem to distinguish between a finite and an infinite sequence.  At best (modulo the flaws identified above) we could get a finite, ever-growing sequence of counting numbers, a sequence that can be proven to exceed any pre-determined numerical threshold (thinking of these labels as real numbers for the moment), provided we allow sufficient time for the steps to be undertaken in the order described.  A finite, ever-growing sequence is not ever an infinite sequence; at best, we might call it potentially-infinite.

I think Mr Jelly ought to forget the peano lessons and adopt a cat.   And Norm, a Zimbabwean by birth, could perhaps remember how difficult it is to count objects in chiShona, with its ostentatious plenitude of noun-classes (21 according to Dale), and associated multitudes of counting words; urban Shona children nowadays usually count in English, even when they know little other English.


D. Dale [1968]:  Shona Companion. Mambo Press, Gweru, Zimbabwe.  Second edition, 1972.

Here we go again! Secret decisions about Iraq

The British Government has this week announced a secret inquiry into the invasion of Iraq in 2003.   [UPDATE: The Government subsequently announced that the enquiry would not be held  in secret.]  How appropriate that a decision made in secret, with only scarce, belated and begrudging justification presented to the citizenry, should now be re-evaluated in secret.   Even though today Gordon Brown says that the decision about secrecy is not his preference, he has delegated the decision about openness to the Chairman of the Inquiry.  For this cowardice, Gordon Brown deserves the widespread contempt in which he is held.

On 14 February 2003, annoyed that the major public policy decision to invade Iraq had apparently already been made, and made in secret without due public consultation, I asked myself if such secrecy could ever be justified.  The text below is what I wrote then. The existential wackawacka hunakuna about weapons of mass destruction since the invasion alters my arguments below not a jot.

In order to avoid re-appearance of comments I received in 2003, let me repeat that I make below no case about the worth of the invasion itself, neither for nor against the invasion.  My case, is as the title says, a case for a justification for a claim, to be presented in public and subject to contestation and debate.  If we’d had such a debate BEFORE the decision to invade had been made (ie, before July 2002) we would have either ended up with no invasion of Iraq at all, or one which many more citizens could have supported.



The Case for the Case for War

14 February 2003

The strange public debate we in the West have been having these last few months about whether and how to undertake military action against Iraq has led me to reflect on the role of argument in public life, especially as it concerns the making of major public policy decisions. While I have strong views on the substantive issues involved here, I am trying not to let them be apparent in my discussion this month of the decision-making processes involved. In particular, in this column, I am not putting the case for military action against Iraq at this time, and nor am I putting the case against such action. This column has no view on the matter. My argument is about the use of argument in decision-making in this domain.

1. The debate has been strange because of the refusal, until recently, of the main proponents of military action against Iraq (which action I’ll call simply “war”) to defend their claim publicly. Only last week, 6 months or so after public debate on this issue began, did the British Prime Minister Tony Blair, meet and debate the issue with ordinary people. Only last week, did the US Government present its intelligence evidence publicly to the UN. Only the week before did the UK Government release a document outlining its case (a document, it turned out, that was mostly plagiarised from public sources). As far as I’m aware, the Australian Prime Minister, John Howard, has still not provided reasons publicly for his Government’s policy of uncritical support for the US position, a refusal which led to him being censured by a majority vote of No Confidence in the Australian Senate, the first such in its history. In Britain, the authorities which operate the House of Commons have recently refused to permit a debate in the House on the question.

2. Why is this? Why have the main protagonists been unable and/or unwilling to defend their position, on an issue of such manifest importance? After all, every bar and every cafe the length of Britain (and elsewhere, if TV news reports here are any guide) is filled with ordinary people discussing the proposed war, so it is not as if people are uninterested in the question.

3. So, I asked myself: What would be good reasons for a Government not to give public justification for its desired action of war against Iraq? I thought of the following possible reasons for not giving reasons (in each case, as perceived by the proponents):

3.1 Revealing the case for war would endanger national security.

3.2 Revealing the case for war would place at peril the lives of, or in other ways compromise, intelligence sources.

3.3 The case for war is weak. For example, this would be the situation if the evidence for Iraq having weapons of mass destruction is only circumstantial.

3.4 The case for war dishonours the proponents. This would be the situation, for example, if the reasons for war were: “To capture Iraq’s oil”, or “To avenge the attempted assassination of George Bush senior.”

3.5 There is no need to put a case for war. In Britain, for example, it seems, as the Defence Secretary reminded us all last week, that the Government can engage in foreign wars simply by convincing the Queen to sign the relevant order; there are no legal or constitutional requirements to convince the House of Commons, or Parliament, or the public at large. I imagine the US War Powers Act, which requires the support of Congress before the President can declare war, may limit the US administration’s freedom somewhat more.

3.6 The case for war is so complex that the public would not understand it.

3.7 The proponents do not respect the other parties in the debate (those opposed to the war, and those still undecided), and so are not bothered to put the case to those others. Many Australians appear to believe that this is the attitude of the Australian Prime Minister on this issue.

To me, speaking personally, reasons 3.1 and 3.2 would be a compelling justification for not revealing the case for war, but I don’t recall any of the proponents giving these as their reasons. None of the other reasons would be compelling to me as reasons for not engaging in public argument on this issue.

4. So, I then asked myself: How would I persuade the proponents of war to give us, the citizenry, their reasons for their proposed actions. Again, I thought of several reasons for giving reasons for war:

4.1 Failure to put any case at all leads people to suspect that the real case is weak or dishonourable. One might call this the Baskerville Argument for giving reasons: If the dogs don’t bark, then why are they silent?

4.2 Engagement in argument enables each side to strengthen their case: to learn of the possible attacks against it, to identify defences and counter-attacks for these, and so to bolster the arguments. The outcome of any comprehensive public debate should be a stronger case for war.

4.3 For complex public policy decisions, such as this one, there are usually many alternative action-options, and many and diverse implications and consequences of those options. In fact, the complexity may be such that no one person, or even no single team of people, could adequately hope to assess and comprehend all these. (This is especially the case for teams of politicians and bureaucrats, out of touch with ordinary reality, as the group think of the CIA in the Bay Of Pigs incident showed.) Only by allowing a full public debate before a decision is made can society be certain that all the relevant issues have been raised and have informed the decision, and thus that the best action-option has been chosen.

4.4 Military action is an example of a public policy decision where ultimate success or failure may depend greatly on the quality of execution, as much as on the particular action-option selected. This in turn may depend on the morale of the military personnel undertaking the action, which in turn may depend on the extent of public support those military personnel have. Without public support for a particular military action, it is much less likely to be successful, at least in a democracy. (I believe this argument is part of the so-called Powell Doctrine, formulated by the US Secretary of State when he was Chairman of the Joint Chiefs of the Defense Forces Staff under US Presidents Bush snr. and Clinton.)

But public support depends crucially on public acceptance of the final decision made, and this in turn depends on the public believing that they have played a part in the decision process. Public debate is necessary, therefore, to establish and sustain public involvement in the decision-making process. People may support a decision outcome even when they disagree with it, if they believe they played an appropriate part in the decision-making process. (I believe this is is real lesson of the experience of the US and Australia in Vietnam: not that the decision to wage war in Vietnam was inherently wrong — it may or may not have been wrong — but rather that the public did not feel they had been sufficiently consulted before it was made, or sufficiently consulted as the military involvement increased. Thus, they did not support it.) Prior and ongoing public debate, rather than being a hindrance to execution quality, may therefore increase execution quality, and may in fact be essential to the ultimate success of the military action itself.

4.5 In a democracy, failure to justify and persuade the citizenry of the wisdom of some major policy is ultimately a mistaken strategy, electorally.

4.6 On important public policy issues in a democracy, consensus is unlikely if not impossible. It is therefore crucial to channel disagreement into public argument and debate, in order to prevent recourse to other forms of expression of opinion, such as mass protests and acts of violence. Public argument thus acts as a “safety valve”.

4.7 In a democracy, politicians have a duty to explain their proposed actions to the citizenry who pay their salaries.

5. Reasons 4.1 – 4.6 are instrumental: they are attempts to show that providing public reasons for war will behoove the proponents of war, and/or improve the quality of decision-making and decision-execution. Reason 4.7 is a moral claim.

6. Some of the arguments listed in Section 4 are not new. For example, argument 4.3 about deliberative processes improving the quality of decision outcomes was made by D. J. Fiorini in 1989, and, in a different form, by Bill Rehg in 2001:

D. J. Fiorino [1989]: “Environmental risk and democratic process: a critical review.” Columbia Journal of Environmental Law, 14: 501-547.

W. Rehg [2001]: “The argumentation theorist in deliberative democracy.” Keynote address to the Conference of the International Debate Education Association (IDEA), Prague, October 2001. Revised version published in Controversia, 1(1): 18-42 (2002).

Similarly, James McBurney and Glen Mills, briefly argued a case similar to my argument 4.6, in:

James H. McBurney and Glen Mills [1964]: Argumentation and Debate: Techniques of a Free Society. New York, USA: Macmillan, Second edition.

Moreover, my argument 4.4 may be a valid inference from the Powell doctrine, as I suggest above.

7. However, these works are all primarily concerned with other issues, and do not aim to present an argument for public argument over matters of importance.  Does anyone know of papers or books which do put such a case?

Postscript 1 (added 17 February 2003): The British Prime Minister, Tony Blair, has just presented a detailed case for taking military action against Iraq, in a speech to the British Labour Party in Glasgow two days ago. I believe this was his first extended public presentation of his arguments for military action; the speech was given on the same day that a million people marched in central London against any war in Iraq. The British House of Commons has still not been permitted to debate the matter.

Postscript 2 (added 17 February 2003): British political commentator, Andrew Rawnsley, wrote in his weekly column in The Observer yesterday:

“There are powerful arguments and there are dreadful arguments in favour of definitively dealing with the Iraqi tyrant, and it has been one of the failures of the British and American governments not to advance the better ones.” (Andrew Rawnsley: “It’s do or die, Prime Minister”, The Observer, 16 February 2003.)

Postscript 3 (added 17 February 2003): From an editorial today in The Guardian, a British daily newspaper:

“In fact, the public is wary of the power of argument because it is attenuated, circumscribed and distorted by political calculations. This may explain why many suspected the government of trying to scare people into war when tanks were placed near airports. The temper of these times is to distrust more than trust.” (“The march of history: A moment of truth for British politics”, The Guardian, 17 February 2003.)

Postscript 4 (added 26 February 2003): Finally, the British House of Commons is permitted to debate this issue. Here is Tony Blair’s statement to the House yesterday.

Postscript 5 (added 12 April 2003): Playwright David Hare is unable still – after three weeks of fighting and the capture of Baghdad – to determine the reasons for the war.

Postscript 6 (added 8 May 2003): At last, an argument I can understand decision-makers in the US and British Governments may have found was compelling: that, although the probability that the Iraqi regime had links with Islamic fundamentalist terrorists may not be large, the consequences of such links may be catastrophic. See the article by Jeffrey Goldberg, “The Unknown: The C.I.A. and the Pentagon take another look at Al Qaeda and Iraq” in The New Yorker magazine, published 10 February 2003. Why did the decision-makers not trust us citizens enough to share such analyses?

Postscript 7 (added 21 June 2003): Author and publisher Jason Epstein, writing in The New York Review of Books, May 1, 2003, in an article entitled “Leviathan” (pp. 13-14), said this about the Second Iraq War:

Meanwhile, Americans are sharply divided over a preemptive assualt whose urgency has not been adequately explained and for which no satisfactory explanation, beyond the zealotry of its sponsors, may exist. (page 13)

Postscript 8 (added 14 September 2003): The Observer’s superb political journalist, Andrew Rawnsley, argues in his column today that Tony Blair “didn’t trust the British people to follow the moral argument for dealing with Saddam. This mistrust in them they now reciprocate back to him. For that, Tony Blair has only himself to blame.”

Postscript 9 (added 28 November 2003): Thomas Powers, in an article entitled “The Vanishing Case for War”, in The New York Review of Books, 50(19): 12-17, 4 December 2003, says this (p. 12):

“The invasion and conquest of Iraq by the United States last spring was the result of what is probably the least ambiguous case of the misreading of secret intelligence information in American history. Whether it is even possible that a misreading so profound could yet be in some sense “a mistake” is a question to which I shall return. Going to war was not something we were forced to do and it certainly was not something we were asked to do. It was something we elected to do for reasons that have still not been fully explained.The official argument for war, pressed in numerous speeches by President Bush and others, failed to convince most of the world that war against Iraq was necessary and just; it failed to soften the opposition to war by longtime allies like France and Germany; and it failed to persuade even a simple majority of the Security Council to vote for war despite immense pressure from Washington. The President’s argument was accepted only by the United States Congress, which voted to give him blanket authority to attack Iraq, and then kept silent during the worldwide debate that followed. The entire process – from the moment it became unmistakably clear that the President had decided to go to war in August 2002, until his announcement on May 1 that “major combat” was over – took about nine months, and it will stand for decades to come as an object lesson in secrecy and its hazards.”

Postscript 10 (added 5 April 2004): Richard A. Clarke in his book, Against All Enemies: Inside America’s War on Terror (New York: Free Press, 2004), lists (page 265) five rationales which have been attributed to senior Bush II Administration officials (GW Bush, D Cheney, D Rumsfeld and P Wolfowitz) for seeking a war against Iraq. I paraphrase these here:

To finish the Gulf War of 1991

To remove a hostile enemy of Israel

To create an Arab democracy as a model for other regional states, especially Egypt and Saudi Arabia

To remove a potentially hostile enemy of Saudi Arabia (and hence enable the withdrawal of US troops stationed there)

To create another friendly source of oil for the US, and so reduce dependency on Saudi oil.

Postscript 11 (added 15 August 2005): George Packer, in an article entitled “The Home Front: A soldier’s father wrestles with the ambiguities of Iraq” (The New Yorker, 4 July 2005, pp. 48-59) says this:

“In the fall of 2002, it still might have been possible for President Bush to construct an Iraq policy that united both parties and America’s democratic allies in defeating tyranny in Iraq. Such a policy, however, would have required the Administration to operate with flexibility and openness. The evidence on unconventional weapons would have had to be laid out without exaggeration or deception. The work of U.N. inspectors in Iraq would have had to be supported rather than undermined. Testimony to Congress would have had to be candid, not slippery. Administration officials who offered dissenting views or pessimistic forecasts would have had to be heard rather than silenced or fired. American citizens would have had to be treated as grownups, and not, as Bush’s chief of staff, Andrew Card, once suggested, as ten-year-olds.” (page 54).

Solidarity with the people of Iran






Events can be followed at Andrew Sullivan’s blog.


Visiting my local dojo this week, I saw an advert for a Workaholics Anonymous meeting that also takes place there.  They meet fortnightly, on Saturdays from 10 am to 12 noon. What  a pity, since Saturday mornings are my most productive work-times of the week!

Achilles and the Tortoise

An amusing account (at least to a mathematician) by Harvey Friedman of an encounter with eccentric Russian mathematician and dissident Alexander Yessenin-Volpin. Friedman supervised the Stanford PhD of John E. Hutchinson, who taught me calculus.  (Hat tip: AB)

Let me give an example. I have seen some ultrafinitists go so far as to challenge the existence of 2^100 as a natural number, in the sense of there being a series of ‘points’ of that length. There is the obvious ‘draw the line’ objection, asking where in

2^1, 2^2, 2^3, . . . , 2^100

do we stop having ‘Platonistic reality’? Here this . . . is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas.

I raised just this objection with the (extreme) ultrafinitist [mathematician Alexander] Yessenin Volpin during a lecture of his.  He asked me to be more specific.  I then proceeded to start with 2^1 and asked him whether this is ‘real’ or something to that effect.  He virtually immediately said yes.  Then I asked about 2^2, and he again said yes, but with perceptible delay. Then 2^3, and yes, but with more delay.  This continued for a couple of more times, till it was obvious how he was handling this objection.  Sure, he was prepared to always answer yes, but he was going to take 2^100 times as long to answer yes to 2^100 then he would to answering 2^1.  There is no way that I could get very far with this. (pp. 4-5).

Note: Of course, Friedman is wrong about the . . . being replaced by 100 items. We would expect it to be replaced with just 96 items, since 4 items in the list of 100 are already listed explicitly.


Harvey M. Friedman [2002]: Lecture Notes on Philosophical Problems in Logic. Princeton University.

The future is not what it was

The Internet, as imagined in 1969, complete with free sexism. (Hat tip:  SW).

Presidential worldiness

He was now, if not yet a man, then at least a youth of more than ordinary experience of the world.  He had traveled exhaustively in Britain, Europe, North Africa, and the Middle East, visiting their great cities time and again and actually living in some for long periods.  He had plumbed the Catacombs and climbed the Great Pyramids, slept in a monastery and toured a harem.  He had hunted jackals on horseback, kissed the Pope’s hand, stared into a volcano, traced an ancient civilization to its source, and followed the wanderings of Jesus.  He had been exposed to much of the world’s greatest art and architecture, become conversant in two foreign languages, and felt as much at home in Arab bazaars as at a German kaffeeklatsch, or on the shaven lawns of an English estate.

Ed Morris’ description of Teddy Roosevelt at 15, appropos 44 and 26.


Edmund Morris [2001]: The Rise of Theodore Roosevelt. New York, USA:  The Modern Library.  Revised and updated edition.  First edition published in 1979.  Except is from page 47.

The Supremes

While on the subject of the US Supreme Court nominations, the New Yorker’s Jeff Toobin has a nice historical analysis of diversity on the US Supreme Court here.

In making nominations to the Supreme Court, Presidents care about diversity, which is a relatively new term for an idea that is nearly as old as the Court itself. In the early days of the republic, when regional disputes were the foremost conflict of the era, nominees were generally defined by their home turfs. So Presidents came to honor an informal tradition of preserving a New England seat, a Virginia seat, a Pennsylvania seat, and a New York seat on the Court. In the nineteenth century, as a torrent of European immigrants transformed American society, religious differences took on a new significance, and Presidents used Supreme Court appointments to recognize the new arrivals’ growing power. In 1836, Andrew Jackson made Roger B. Taney the first occupant of what became known as the Catholic seat on the Court, and that tradition carried forward intermittently for more than a century, with Edward White, Joseph McKenna, Pierce Butler, Frank Murphy, and William J. Brennan, Jr., occupying the chair. In 1916, Woodrow Wilson nominated Louis D. Brandeis, establishing the Jewish seat, which later went, with brief overlapping periods, to Benjamin N. Cardozo, Felix Frankfurter, and Abe Fortas.

. . .

At the Court, as in American life, the rules of diversity have changed. Regional differences faded long ago. The fact that two Arizonans, O’Connor and William H. Rehnquist, served together for almost a quarter century mattered little to anyone. Religious tensions have also cooled. By the time Bill Clinton named Ruth Bader Ginsburg and Stephen G. Breyer to the Court, the fact that both are Jewish (and replaced non-Jewish predecessors) was little more than a curiosity. If Sotomayor is confirmed, there will be six Catholics on the Court, which is also of minor significance. George W. Bush appointed John G. Roberts, Jr., and Samuel A. Alito, Jr., because they are conservative, not because they are Catholic. (The Catholic Brennan was the Court’s greatest liberal.) More than anything, it seems clear that the President saw in Sotomayor a kindred spirit—a high achiever from a humble background who reflects, as best as can be determined, his own brand of progressivism.”