Pacifism is properly pacificism, the making of peace: not a belief or an attitude, but a practice. Mathematics then is pacifist, because learning it means learning that you cannot fight your way to the truth. Might does not make right. If others are going to agree with you, they will have to do it freely. Moreover, you cannot rest until they do agree with you, if you’ve got a piece of mathematics that you think is right.
Such is the ideal. It doesn’t mean there won’t be mathematicians who try to bluff their way to dominance, or suppress the work of their competitors. Such persons can be a problem whenever a practice becomes a remunerative profession, no longer pursued for its own sake alone.
Still, since everybody learns some mathematics at school; since etymologically speaking, mathematics is that which is learned; it has been my dream that everybody could realize that in one field of endeavor at least, fighting is not allowed, and universal agreement is believed to be possible and is actually achieved.
In short, mathematics is pacifist, and that is something good; and yet in the political context, pacifism has been justly condemned.
“Pacifism is war-mongery complicated by defeatism,” wrote Collingwood in 1942, in support of the war of Britain against the Nazis. The sentence is italicized in the chapter of The New Leviathan called “External Politics.”
The present post began as a supplement to my original post about that chapter. I spent five years writing a post about each chapter of The New Leviathan, whose alternative title is Man, Society, Civilization, and Barbarism. The experience helps me deal with life today. I return now to the chapter that takes up pacifism, because to some fellow academics, I am annoying or suspect for trying to understand the thoughts of persons with whom they disagree politically.
You don’t try to understand Nazis, they might say; you fight them.
While being critical of the current adminstration of the United States, some writers issue a disclaimer: America under Donald Trump is not at all like Germany and Austria under Adolf Hitler.
That the disclaimer would be needed is worrying enough. If war is a state of mind, as Collingwood says, then the United States is at war with itself. With its foolish way of trying to make peace, pacifism effectively promotes war,
Not realizing that modern war is a neurotic thing, an effect of terror where there is nothing to fear and of hunger where the stomach is already full …
That was 1942. What is there to be afraid of now?
Not enough people are afraid of climate change—afraid with the kind of fear that provokes responsible action.
In America at least, too many people are afraid of black people. That’s my impression of a country I haven’t lived in since 1998. That country may twice have elected a black man as president; this doesn’t mean it has got over being founded by people who owned black people as slaves.
People may say that what they fear is not black people, but an ideology that they call wokism. I don’t know who specifically is held to embrace this ideology, except perhaps Ibram X. Kendi.
It’s October 29, 2020, and the August issue of Harper’s reached me just the other day, by snail mail in Istanbul; however, from a tweet, I have learned about an essay in the November issue called “Is America Ungovernable?” According to David Bromwich,
“The only remedy to racist discrimination,” Kendi’s central axiom asserts, “is antiracist discrimination. The only remedy to past discrimination is present discrimination.” Present discrimination, that is, against white people is to remedy past discrimination against Black people. The present status of Black people is supposed to be marked at every moment by systemic racism. White people are oppositely marked, so the doctrine maintains, by the reality of white supremacy. In every civic or cultural encounter, race identity supervenes on personal identity.
Bromwich’s quotations are correct; I find them in Kendi’s book, How to Be an Antiracist, as excerpted on the website of his UK publisher. From there I learn that, according to Kendi,
The most threatening racist movement is not the alt right’s unlikely drive for a White ethnostate but the regular American’s drive for a “race-neutral” one … there is no such thing as a not-racist idea, only racist ideas and antiracist ideas.
Philosophically speaking, Kendi’s Manichaeism is at odds with Collingwood’s dialectic, even in the chapter of The New Leviathan that I am reviewing:
29. 52. Dialectic is not between contraries but between contradictories (24. 68). The process leading to agreement begins not from disagreement but from non-agreement.
29. 53. Non-agreement may be hardened into disagreement; in that case the stage is set for an eristic in which each party tries to vanquish the other; or, remaining mere non-agreement, it may set the stage for a dialectic in which each party tries to discover that the difference of view between them conceals a fundamental agreement.
I found reason to quote 29. 52 when taking up Chapter XLI, “What Barbarism Is.” Meanwhile, according to Kendi in a tweet (dated September 21, 2020) that I happened to see,
So what does “not racist” mean? The term has no meaning other than denying when one is being racist. We should not have words in the dictionary that don’t have definitions.
I think we can always suspect people who say, “I am not racist,” or “I am not sexist,” or “I am not transphobic,” or what have you. It’s like saying, “I am not rude”: not something for you to determine on your own.
But “We should not have words in the dictionary that don’t have definitions”? It makes no sense, even though Kendi is now a professor of history at Boston University, where he is to “establish a new University-wide research center that will be titled the ‘BU Center for Antiracist Research,’” according to the announcement (pdf
) from the university provost that is linked to in Wikipedia.
Bromwich too writes foolishly in Harper’s, when he says,
calling someone a racist in 2020 inflicts as sure a wound, with as light a burden of proof, as calling someone a Communist did in 1952; and Democrats in our time have found antiracism as valuable a weapon as anticommunism was for Republicans in the McCarthy period.
Andrew Sullivan tweeted this excerpt, perhaps because he agrees with it. The absurdity was noted by a surgeon whom I follow called Mark Hoofnagle:
Imagine the false persecution complex needed to believe this while: an open racist literally occupies the White House, denying black people the vote has become overt policy of an entire party, and there is nothing resembling a HUAC for racism.
Do they hear themselves talk?
At first I felt embarrassed to be a Harper’s reader and subscriber; but then I actually read Bromwich’s piece. As I said in my own tweet, “[the] essay as a whole seemed good to me as a warning about dealing with Trump.”
I suggested above that a racist society can still elect a black president. It may also be the case that in such a society, an accusation of racism can effect social death. It would not be the same thing as being accused of Communism in the middle of the last century, when you could be jailed for being a Communist or even defending Communists.
In the 1990s I listened to Dorothy Healey’s program on WPFW in Washington; she would emphasize the importance of the Supreme Court, since it had overthrown the five-year sentence of her and her comrades for conspiring to overthrow the government.
No analogy can be expected to be perfect, except in mathematics.
From decades ago, in an article about medical practice, I recall that when a patient’s vital signs are constantly recorded on a roll of paper (or presumably now in a computer file), a malpractice attorney can always pore over the record to find something abnormal that the doctor failed to respond to.
We may also suspiciously pore over an essay, to find a sentence that reveals the writer as being somehow tainted or impaired or dishonest.
Modern war is characterized by neurotic fear. I wrote first about this observation of Collingwood’s in 2018. A year later, I went back to add a quote from Rod Dreher. Writing in The American Conservative, Dreher admitted his fear, but not the pathology of it.
Identifying as a Christian, Dreher says his fear is of the “progressives” who “really do hate us, and wish to see harm done to us.” This fear then is the progressives’ fault.
By Collingwood’s account, Dreher is correct, at least as far as progressives are concerned. If you are somehow terrifying others, then it is on you to do something about it. However, as Collingwood also points out, you need not automatically give the fearful what they want.
Should you even give them a hearing?
I called Dreher “thoughtful” in the earlier post, because he thinks about his life and what he wants, and he shares what he learns—and shares honestly, as far as I can tell. By contrast, it’s dishonest for one Nick Hankoff, born in 1985, to write, in the same magazine as Dreher,
Trump and Biden certainly differ on policy, but the more important and clearer contrast is in their leadership capabilities. The incumbent is emerging as a fatherly figure, while the challenger is fading into the form of an absent father …
When millions of Americans witnessed President Donald Trump remove his mask upon returning to the White House from a brief hospital stay for Covid-19, that was the defining moment for the 2020 election. Along with his subsequent, encouraging remarks, the moment also distinguished Trump as the fatherly leader of the nation.
This is dishonest, because (as I suppose) it does not express the writer’s real reasons for supporting Trump. The rhetoric tries to give others an excuse for supporting him.
Dreher is clear why he effectively supports Trump, whether voting for him or not. He is afraid. I have been willing to listen to why.
In a 2017 post called “Community,” I took issue with the ideas of Dreher that were to be gathered into his book, The Benedict Option; at least I argued that a liberal democracy like the United States was the convenient place for him to try to put his ideas into practice.
Dreher has written a sequel now: Live Not by Lies: A Manual for Christian Dissidents. This is reviewed in The Bias Magazine: The Voice of the Christian Left, October 27, 2020, by Daniel Walden, who says,
Dreher’s considerable personal charm will always afford him a warm reception in a media landscape that anoints conservative intellectuals primarily on the basis of their ability to avoid overt racial slurs during fifteen minute television appearances … Rod Dreher’s Live Not By Lies: A Manual for Christian Dissidents is not worth your time or anyone else’s …
Dreher’s main concern in writing this book is “soft totalitarianism,” which he is quick to point out does not actually exist yet, but which he is absolutely certain poses a civilization-scale threat to the United States. It is important to note that at no point does he actually define “soft totalitarianism” or tell us how we would know it had arrived … soft totalitarianism is supposed to instill the atmospheric dread of an old episode of The Twilight Zone, assuring us that the book’s litany of interviews with Eastern Bloc dissidents has something to teach contemporary Christians about the coming persecution …
Dreher is willing to throw away others’ freedom to assuage his own fears. What shall we do about that? Fight him, in some sense; but what sense?
Yet another writer for The American Conservative, Micah Mattix, is quoted by his university:
In my classes, we read texts carefully and take writing with clarity and nuance seriously. In studying great works of literature, we learn about ourselves, our world, and the God who created both. My hope is that students will not only become clearer thinkers as a result but also come to the value of benefit of contemplating the true and the beautiful in their own right.
I studied great works of literature as an undergraduate at St John’s College. It was one of the best things I have done. I’m pretty satisfied with the words that I wrote about the College in 2012. I am therefore curious what is behind the words of Micah Mattix.
Great-books programs are suspect. Jason Stanley tweeted, just a week ago,
Your regular reminder that “Great Books” programs are not essential to “humanistic education”, and are often part of the problem not the solution.
It’s a bizarre remark, but Stanley must have his reasons for it. However, there is no one great-books program. A lot depends, not just on whom you are reading, but whom you are reading with: your teachers, your fellow students. Do the teachers let the students speak? Do they expect the students to speak? Do the students speak, understanding it as their right, while accepting the responsibility of explaining and revising what they say?
Micah Mattix may be reading great books with his students, but he is at the university founded by Pat Robertson and called originally Christian Broadcasting Network University. He invokes God in his blurb.
I wouldn’t do that, were I teaching literature. In the last quoted sentence of Mattix, perhaps “come to the value of benefit” was supposed to be “come to value the benefit.” I would avoid the suggestion that the books to be read were instances of the true and the beautiful. It is the reading and discussing of them that is a good in itself, as I have said of participating in the activities of the Nesin Mathematics Village.
For the students with whom I am reading Euclid now, albeit remotely, I prepared a page of this blog that says, in my simple Turkish:
Freedom is a right and a responsibility. In mathematics we are free because:
To demand of everybody a reason for their claim is our right.
To give to those who ask a reason for our own claim is our responsibility.
In mathematics truth is
individual, because nobody else can order us to accept a claim;
universal, because we all must agree on the same claim; if not, we cannot fight, but must talk.
That’s mathematics, as I see it. I try to extend the ideas beyond that field. But when in a tweet I said, “I wonder what [Mattix’s] classes are like,” an historian said, “I don’t.”
Then she blocked me from seeing this and all of her other tweets, after I said, “Interesting. I wonder what you are curious about. Or perhaps curiosity is suspect, as it can be where I live, curiosity and anxiety being called by the same word here.” So now I’ve got to wonder what the historian’s classes are like.
]]>Being hosted by WordPress.com, thus using the WordPress.org content management system, this blog has posts, pages, and media. This directory is for the pages and the verbal media (namely pdf
files).
The posts, such as this one, have initial publication dates and can be seen at polytropy.com, in reverse chronological order. I have them listed in forward order on my About page. As I explain there, I try to keep track of posts with tags and categories. Moreover, if one post revisits a theme of another post, I try to link to that post, which will then show at the bottom which posts are linked to it.
A dream, never to be realized, would be to have all of my ideas as well-organized as in Wittgenstein’s Tractatus Logico-Philosophicus.
I don’t know how the random visitor can find pages, although search engines find some of them. Much less do I know how one would find media, although the media allowed by WordPress now include pdf
files. I have uploaded a number of these, and created a number of pages. and in this post (which I hope to remember to keep up to date), I try to classify them, if only to remind myself what they are.
It would be possible to have all top-level pages included automatically in the menu which now forms a horizontal list at the top of each post and page.
The About page mentioned above
“The Armeno-Turkish Alphabet” (I think this needs more work)
“Karadeniz” (more on the 2018 trip summarized in “Eastern Black Sea Yayla Tour”)
“A Season on a Farm” (the “season” was the summer and fall of 1988)
“Van 2003” (an example of travel writing that I used to put on a page hosted by my department, before I had this blog)
Poetry, in the broad etymological sense of something made; call it conceptual art, or whatever you like, but it’s all referred to in the post “Discrete Logarithms”:
a Turkish translation of Claude Closky, “first thousand numbers classified in alphabetical order”
“Cartesianism” (the pdf
file described in “An Exercise in Analytic Geometry”)
A proof of Dirichlet’s theorem on primes in arithmetic progressions, in pdf and html format (as discussed in “LaTeX to HTML”)
Pages describing (as well as listing) categories of posts
For my courses I normally prepare pages on my department’s server; but since I cannot access this from home, I may also use the blog.
Aksiyomatik Kümeler Kuramı Özeti (“summary of axiomatic set theory”: an attempt in fall 2019 to supply just that in html
; I had not yet discovered the usefulness of pandoc
as described in “LaTeX to HTML”)
Analitik Geometri Özeti (“summary of analytic geometry,” for a course in spring 2020; as the Covid-19 lockdown took hold, the page just became the course page)
Ordinal Analiz (“ordinal analysis,” that is, set theory with emphasis on the ordinals as a structure analogous to the linearly ordered set of real numbers studied in so-called real analysis; the post “Ordinals” also takes up the analogy; I made the page for a course in Şirince, in case I wanted to change the page while I was there, though in the event I didn’t; notes from the second week, in English, are on a departmental page, along with the syllabus for a summer course in 2020 that was cancelled)
Öklid (Resources for the course Öklid geometrisine giriş, “introduction to Euclidean geometry,” fall 2020)
Ayşe Berkman’ın yedek sayfasıdır (for her spring 2020 course during the lockdown)
Sometimes annotated by me:
R. G. Collingwood
“Causation” from An Essay on Metaphysics (as described in the post “On Causation”)
chapters, including “Monks and Morals,” on a visit to Santorini in The First Mate’s Log
John Donne, “The Undertaking” (to accompany the post that I wrote about it)
Adam Garfinkle, “The Erosion of Deep Literacy” (referred to in “Reading shallow and deep”)
John Goldthwaite, pages on C. S. Lewis and Narnia from The Natural History of Make-Believe (referred to in “Return to Narnia”)
Euphemia Lofton Haynes, “Mathematics—Symbolic Logic” (supporting “What Mathematics Is”)
Homer, Iliad (some of the books, in Chapman’s translation, to go with my commentaries on this)
Somerset Maugham, “Romance,” from On a Chinese Screen (accompanying my own “Romance” and showing how Maugham plagiarized Herbert Giles)
Poetry sent me in a lockdown “poem exchange”
Raymond Smullyan, “Is God a Taoist?”
Arnold Toynbee, “A Turning Point in History” (referred to in “What It Takes”)
This article gathers, and in some cases quotes and examines, popular articles about R. G. Collingwood (1889–1943).
By articles, I mean not blog posts like mine and others’, but essays by professionals in publications that have editors.
By popular, I mean written not for other professionals, but for the laity.
The following list may grow, if more articles come to light:
Ray Monk, “How the untimely death of RG Collingwood changed the course of philosophy forever,” Prospect, September 5, 2019. I talked about this in the post “Anthropology of Mathematics.”
Jonathan Rée, “R.G. Collingwood on the corruption of democracy,” New Humanist, August 30, 2019:
Collingwood … did not believe that philosophy could put people in contact with eternal principles of politics or anything else, and it was for precisely that reason that he deplored fascism and advocated liberalism and democracy.
Jonathan Rée, “A Few Home Truths,” London Review of Books, Vol. 36, No. 12, June 19, 2014. On R. G. Collingwood, “An Autobiography” and Other Writings, with Essays on Collingwood’s Life and Work (edited by David Boucher and Teresa Smith, Oxford, 2013):
Both [of Collingwood’s parents] were professional painters, and like their friend and near neighbour John Ruskin they regarded art not as a quest for aesthetic perfection but a joyful inquiry into the inexhaustible variety of the world, closely allied with history, natural science and the arguments of everyday life …
[Collingwood’s autobiography] leaves you thinking that the literary form best suited to philosophy is not the treatise, the commentary or even the all-conquering academic paper, but the memoir of a seriously thoughtful life.
Simon Blackburn, “Being and Time,” The New Republic, April 3, 2010. I have written about this in “Re-enactment,” referring back to “What It Takes.” Blackburn is positive, but misunderstanding (in my opinion, obviously).
Mary Beard, “No More Scissors and Paste,” London Review of Books, Vol. 32, No. 6, March 25, 2010. (Added to the list, October 18, 2020.) A review of Fred Inglis, History Man: The Life of R.G. Collingwood, like Blackburn’s article; but Beard is less threatened by Collingwood’s ego. Her theme: like Collingwood himself and other scholars of the man, Inglis ranks Collingwood’s being an historian below his being a philosopher; however,
it is surely crucial that he was a product of the old Oxford ‘Greats’ (that is, classics) course, which focused the last two and a half years of a student’s work on the parallel study of ancient history on the one hand, and ancient and modern philosophy on the other. Most students were much better at one side than the other … Collingwood was not a maverick with two incompatible interests. Given the educational aims of the course, he was a rare success, even if something of a quirky overachiever; his combination of interests was exactly what Greats was designed to promote.
I made that quote without ellipses in “Effectiveness.”
Jonathan Rée, “Life after Life,” London Review of Books, Vol. 22, No. 2, January 20, 2000. The best and most thorough of the list, though (inevitably) passing over some things. I shall come back to it.
I prepare this post, goaded by the friend who suggested that I was “a bit fixated on Collingwood.” Responding to Adam Kirsch, “Philosophy in the Shadow of Nazism” (New Yorker, October 19, 2020), I had written that anybody interested in the Vienna Circle or in analytic philosophy in the UK ought to read Collingwood’s 1940 Essay on Metaphysics.
Kirsch writes,
Since the Greeks, Western thinkers had tried to understand the world using terms such as “being” and “becoming,” “substance” and “essence,” “real” and “ideal.” But these abstractions gave rise to complicated arguments that went around and around, never reaching any definite conclusion …
This sounds like the complaint of people who cannot be bothered to try to understand somebody else. Kirsch makes it in the voice of one inspired by Wittgenstein’s Tractatus Logico-Philosophicus (1921).
There’s an argument that reaches no definite conclusion in the dialogue of Plato called the Thaeatetus. Socrates and the title character try out some definitions of knowledge. All are found wanting. My sense (mentioned elsewhere) is that analytic philosophers actually fail to recognize the unsatisfactory conclusion, or else they fail to consider that Plato had a good reason for writing it.
Effectively on behalf of such philosophers, Kirsch asks,
In an age of triumphant physics, did philosophy still need to bother with metaphysics?
We might as well ask whether we still need bother to be human beings. We need bother with metaphysics, and Collingwood shows this.
Nonetheless, for the Vienna Circle, according to Kirsch,
Philosophy’s role in the search for truth is to examine the form of our statements, to insure that they are syntactically and logically correct.
It seems like such a sad project now; also an authoritarian one, and therefore something to be decried, even though
Nazis and Austria’s Christian fascists were right to see the Vienna Circle as an enemy. In Edmonds’s words, the Circle was “contemptuous of superstitious thinking,” including myths about race and religion.
David Edmonds is the author of The Murder of Professor Schlick, which Kirsch is reviewing. Not having read the book itself, I can respond only to Kirsch’s review. This observes that, when they took over Austria, the Nazis let Moritz Schlick’s murderer out of jail, because they judged him to be inspired by nationalism and anti-Semitism.
In this deranged atmosphere [writes Kirsch], no one was deterred by the fact that Schlick was not Jewish but, rather, a German Protestant … in their eyes Jewishness wasn’t defined only by religion or ethnicity. It was also a mind-set, characterized by the modernism and liberalism they saw as sources of spiritual corruption.
This is why the Vienna Circle were the enemy of “Nazis and Austria’s Christian fascists.” The Circle
included Christians and Jews, but its members’ real creed was what they called “the scientific conception of the world.”
Today many of us decry the failure of political leaders to have a scientific conception of the world, as for example regarding Covid-19 in particular, and climate change in general. I’ll suggest that the “real” problem is authoritarianism, which can in principle be a problem in science, as anywhere else.
Authoritarianism can in principle be a problem in science, because the scientist has to allow the possibility of being wrong and of being shown a better way by others. Perhaps the bigger actual problem is that authoritarians take advantage of the uncertainty of science. They attack changing one’s mind as a weakness.
In my last post I wrote about how any kind of practice, once recognized as a practice, can be abused. For me the locus classicus of this idea is in The Principles of Art (Oxford, 1938), the first of Collingwood’s books that I read. That was in 1987, and the book remains my favorite. It is one of the few that Jonathan Rée does not mention in “Life after Life,” which is ostensibly a review of new editions of the three books that Collingwood wrote last:
An Essay on Metaphysics (edited by Rex Martin, Oxford, 1998);
The New Leviathan (edited by David Boucher, 1999);
The Principles of History (edited by W.H. Dray and W.J. van der Dussen, Oxford, 1999).
Rée concludes, “Let’s hope these fine new editions will give Collingwood the good readers he abundantly deserves.” Meanwhile, he has given an excellent overview of much of Collingwood’s work. I select some choice passages:
… Collingwood had never learned the meaning of academic fear … When he went to study philosophy at Oxford … he found himself drawn to the supposedly obsolete social liberalism of T.H. Green, which he associated with Ruskinian political radicalism in its idealisation of active ‘citizenship’ within a comprehensively caring State …
It was the same unaffected self-confidence, combined with chronic insomnia and an incapacity for lazing around, that enabled Collingwood to sustain a part-time career as an archaeologist. He found it a relief from the idiocies of philosophy …
He regarded [his philosophical colleagues’] attempts to make philosophy an academic plaything … as a betrayal, not only of science and culture, but of society, too … They were making the world safe for political irrationalism, for Fascists and Nazis in particular …
In Oxford in 1938, the idea that there was something deeply wrong with Fascism and Nazism was not quite the polite commonplace it has since become, and some of the Delegates of the Oxford University Press wanted Collingwood to cool his rhetoric down …
The kind of historicised metaphysics that Collingwood advocated … would provide us all with reminders that presuppositions which once served our purposes can quickly turn into obstacles to further progress …
… he is one of the classiest philosophical writers in the English language.
… philosophical authors must always write primarily for themselves … And philosophy, like poetry, required not only a rigorously honest author, but a ‘good reader’ as well: a reader committed to ‘living through the same experience’ as the writer went through, and skilled in sustaining a ‘peculiar intimacy’ in the act of reading.
Rée says of The New Leviathan, the last book Collingwood saw to press (in 1942), that it “bears ugly marks of carelessness and haste,” and a “broad theme is not enough to prevent The New Leviathan from meandering and marking time.” None of that bothers me, who spent five years blogging chapter by chapter about the book. Rée has a good selection of the book’s “unusual variety of intelligent political observations” (these are printed continuously in the text; the links are to my own posts on relevant chapters):
“that the overall goal of politics is the promotion of ‘civility’, or respect for self and others;”
“that there will always be ‘conflicts between one way of life and another’;”
“that pacifism may promote war rather than prevent it, because it is more interested in giving the pacifist a clear conscience than in navigating the rough seas of actually existing hostilities;”
“that deceit may sometimes be a political duty;”
“that education should be provided on the same basis as medicine—always available when needed, but never forced down anyone’s throat; and”
“that the professionalisation of teaching is the enemy of the efficient education of children.”
According to Rée,
The only thing tying these observations together … was Collingwood’s general idea that ‘classical politics’ had failed because of its refusal to ‘think dialectically’ …
I don’t need the observations to be tied together by anything but Collingwood himself, as I don’t need A Week on the Concord and Merrimack Rivers to be tied together by anything but Thoreau.
Rée pays almost no attention to the first part of The New Leviathan, the part about the development of the individual. He need not; he cannot cover everything. But to me the most important passage in the book is in that first part, in Chapter XIII:
The problem of free will is not whether men are free (for every one is free who has reached the level of development that enables him to choose) but, how does a man become free? For he must be free before he can make a choice; consequently no man can become free by choosing.
A related feature of Rée’s review may have to do with its having been written in 2000, before the election of George W. Bush as President of the US, and thus before that his invasion of Iraq. I had a similar concern with the Editor’s Introduction to the 1999 edition of The New Leviathan. Back then, it was easier to think that the problem facing the world when Collingwood wrote was over.
]]>I continue with the mathematics posts, taking up, as I did in the last, material originally drafted for the first.
Designated for its own post, material can grow, as has the material of this post in the drafting. Large parts of it are taken up with
the notion (due to Collingwood) of criteriological sciences, logic being one of them;
Gödel’s logical theorems of completeness and incompleteness.
I have defined mathematics as the science whose findings are proved by deduction. This definition does not say what mathematics is about. We can say however what logic is about: it is about mathematics quâ deduction. This makes logic a criteriological science, since it seeks, examines, clarifies and limits the criteria whereby we can make deductions. As examples of this activity, Gödel’s theorems are that
everything true in all possible mathematical worlds can be deduced;
some things true in the world of numbers can never be deduced;
the latter theorem is one of those things.
The present post has the following sections.
My definition of mathematics is like that of Bertrand Russell in The Principles of Mathematics, whereby the subject consists of deductions in the sense of propositions “p implies q.” However, Russell’s definition leaves us out, as would a definition of physics as consisting of the laws of nature.
We can classify sciences as empirical, normative, or criteriological, according as they examine how things are, how we want them to be, or how we want ourselves to be. If this classification is exhaustive, then mathematics is empirical; but it is also in a class by itself.
As scientists, we judge whether we achieve the goals that we set for ourselves in our work. Thus we have freedom of will, and criteriological sciences study this freedom.
Physicist Sabine Hossenfelder denies our freedom, as in a video, “You don’t have free will, but don’t worry,” released during the composition of the present post. The video comes with this summary:
In this video I explain why free will is incompatible with the currently known laws of nature and why the idea makes no sense anyway. However, you don’t need free will to act responsibly and to live a happy life, and I will tell you why.
Raymond Smullyan has a good account of why there is free will, in his dialogue “Is God a Taoist?” reprinted
in Douglas R. Hofstadter and Daniel C. Dennett (editors), The Mind’s I: Fantasies and Reflections on Self & Soul (Basic Books, 1981);
by me on this blog.
I shall be looking below at Hofstadter’s ideas as expressed in Gödel, Escher, Bach (Basic Books, 1979).
Perhaps Hossenfelder would say Smullyan’s is an argument for responsibility, not freedom. To me it makes no sense to speak of responsibility without freedom, as I said in “Antitheses,” when critiquing Hossenfelder’s 2016 blog post, “Free will is dead, let’s bury it.” By my account now, Hossenfelder is effectively denying that there can be such sciences as history and logic. She might tell me that those sciences do not do what I say they do, or that my understanding of freedom is not hers. My understanding of physics is that it studies things insofar as they do not have free will; thus physics cannot discover an absence of freedom.
Mathematics is empirical, in its own peculiar way; logic is criteriological, for helping us get straight what we are trying to do with it.
Gödel’s one completeness theorem and two incompleteness theorems tell us what we can and cannot hope to prove in mathematics.
Logic has allowed mathematics to come into its own as the deductive science.
Mathematics proves that certain conclusions are necessary conditions of certain assumptions. The other way around, the assumptions are sufficient conditions for the conclusions. In other words, certain postulates entail certain theorems; symbolically, for some instances of p and q,
p ⇒ q,
“p implies q.”
Such an account of mathematics resembles Bertrand Russell’s, in The Principles of Mathematics:
Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form.
Timothy Gowers makes that quotation of Russell, along with one more sentence, to be taken up later, in the Preface of the big book called The Princeton Companion to Mathematics (Princeton University Press, 2008; xx + 1034 pages). I referred to this book in
the blog post called “Şirince January 2018,” about time spent in the Nesin Mathematics Village;
the blog page called, and about, “Primes in Arithmetic Progressions”; that page was the subject of the post called “LaTeX to HTML.”
According to Gowers himself,
The Princeton Companion to Mathematics could be said to be about everything that Russell’s definition leaves out.
Russell’s book was published in 1903, and many mathematicians at that time were preoccupied with the logical foundations of the subject. Now, just over a century later, it is no longer a new idea that mathematics can be regarded as a formal system of the kind that Russell describes, and today’s mathematician is more likely to have other concerns. In particular, in an era where so much mathematics is being published that no individual can understand more than a tiny fraction of it, it is useful to know not just which arrangements of symbols form grammatically correct mathematical statements, but also which of these statements deserve our attention.
In short, the book gives an idea of what mathematics is about.
There is a distinction between descriptive and prescriptive science; alternatively, between empirical science and normative science. I have written about a third kind of science, which Collingwood calls criteriological.
Empirical science is about how things are.
Normative science is about how we try to make things.
Criteriological science about how we try to make ourselves.
The empirical sciences are the natural sciences: the sciences of things that are natural. Natural things cannot be any other way than they already are. At least, they are not trying to be any other way; or if they are, as when they are animals as studied by Tinbergen and colleagues, then they don’t know it.
Engineering and medicine are normative sciences, because they study how to make things be some way, according to the aims of those persons who want them to be that way.
A criteriological science like aesthetics or jurisprudence studies our aims as such. Having itself an aim, which involves being true or correct, a criteriological science comes under its own purview.
One may want to emphasize that a normative science not only studies how to do things, but also does them. One may then call the science an art or, if that usage is old-fashioned, a skill.
Any scientific pursuit may have empirical, normative, and criteriological aspects. It will have the last when establishing a code of conduct for itself.
A reason for distinguishing the kinds of science is the hope of doing better what we do, when we know what it is.
Richard Feynman tells a story about confirmation bias in replications of the Millikan Oil Drop Experiment. When researchers followed Millikan’s example in inferring from their experimental data a unit of charge (namely the charge on a so-called electron), they threw out results that were too far from Millikan’s, until they understood that his were off.
The story is currently quoted on Wikipedia. Feynman continues it, in “Cargo Cult Science,” as reprinted in “Surely You’re Joking, Mr. Feynman!” (Norton, 1985):
But this long history of learning how to not fool ourselves—of having utter scientific integrity—is, I’m sorry to say, something that we haven’t specifically included in any particular course that I know of. We just hope you’ve caught on by osmosis.
If the ethics of physics is indeed not taught in physics courses, presumably this is because it is not strictly part of physics. Physics is a natural science; ethics is not. However, every scientist needs to be ethicist enough to understand Feynman’s advice:
The first principle is that you must not fool yourself—and you are the easiest person to fool. So you have to be very careful about that. After you’ve not fooled yourself, it’s easy not to fool other scientists. You just have to be honest in a conventional way after that.
I would like to add something that’s not essential to the science, but something I kind of believe, which is that you should not fool the layman when you’re talking as a scientist.
I quoted that last sentence and more in “Be Sex Binary, We Are Not,” because I thought Feynman’s advice was being ignored in a Scientific American blog post.
All of this puts me in mind of Plato’s Gorgias dialogue, in which the title character agrees with the account of his profession that Socrates offers (455A; Lamb’s translation in the Loeb Classical Library; emphasis mine):
And so the rhetorician’s business is not to instruct a law court or a public meeting in matters of right and wrong, but only to make them believe; since, I take it, he could not in a short while instruct such a mass of people in matters so important.
Gorgias boasts that if a town is hiring a physician, then a rhetorician can pad his résumé better than any actual physician, and get the job; however, teachers of rhetoric such as himself should not be blamed when their students misuse what they have learned (456A–D). Socrates presses him (459B–C):
Then the case is the same in all the other arts for the orator and his rhetoric: there is no need to know the truth of the actual matters, but one merely needs to have discovered some device of persuasion which will make one appear to those who do not know to know better than those who know.
Gorgias again boasts in reply (459C):
Well, and is it not a great convenience, Socrates, to make oneself a match for the professionals by learning just this single art and omitting all the others?
Socrates asks if the rhetorician is as ignorant of justice and goodness as he is of medicine and other arts. Gorgias is reluctant to admit it, but says that the student of rhetoric will have to learn the difference between right and wrong (460A). But now Gorgias has contradicted himself, since nobody can do wrong who knows what it is, and yet Gorgias has admitted that his students may do wrong.
Everything that we do, we do initially for its own sake, because we cannot know what will come of it. As infants we have to utter sounds at random before we can learn to control them and use them to get what we want. We do research out of pure curiosity, before learning how the research can serve other interests.
Then one can find ways to abuse research, or one’s standing as a scientist. “For example,” says Feynman (loc. cit.),
I was a little surprised when I was talking to a friend who was going to go on the radio. He does work on cosmology and astronomy, and he wondered how he would explain what the applications of this work were. “Well,” I said, “there aren’t any.” He said, “Yes, but then we won’t get support for more research of this kind.” I think that’s kind of dishonest.
“Kind of”?
Logic aims to provide, for science in general, and especially mathematics, a code of conduct that is agreeable to the practitioners: a code not for dealing with the public, but for establishing theories and, in mathematics, proving theorems. That is what I would say, though perhaps the current Wikipedia article called “Conceptions of logic” does not really cover this conception.
In the broad sense, “logic is the analysis and appraisal of arguments,” as the main Wikipedia article on the subject says. I would emphasize that the analysis and appraisal are of arguments as such, arguments quâ arguments. These are intended to have a certain effect, and they may succeed or fail in achieving this end, according to the people making the arguments.
For example, in Propositions 5, 6, 18, and 19 of Book I of the Elements, Euclid proves that, in a triangle,
equal sides subtend equal angles,
equal angles are subtended by equal sides,
the greater side subtends the greater angle,
the greater angle is subtended by the greater side.
The last of these follows from the first and third by pure logic, as Euclid’s proof reflects; but so does the second, and it is not needed for the proof of the third. Logically then, Euclid could have dispensed with his proof of the second. If logic were normative, it might convict Euclid of the style error of proving what didn’t need proof. As a criteriological science, logic can only ask whether Euclid would accept style advice on this point.
Logic is called that and not logics, although there are other sciences called ethics and physics, because Aristotle wrote collections of books called, respectively, Ethics and Physics, but not a collection of “logics”; he wrote Analytics instead. Collingwood points this out in a footnote on the first page of An Essay on Metaphysics (Oxford, 1940). He accounts for logic and ethics as criteriological sciences in Chapter X (pp. 108–9; bold emphasis mine):
… the Greeks … constructed a science of theoretical thought called logic and a science of practical thought called ethics. In each case they paid great attention to the task of defining the criteria by reference to which theoretical and practical thought respectively judge of their own success. In view of this … these sciences have been traditionally called normative sciences. But the word ‘normative’ may prove misleading … as if it were for the logician to decide whether a non-logician’s thoughts are true or false and his arguments valid or invalid, and for the student of ethics to pass judgement on the actions of other people as having succeeded or failed in their purpose. This suggestion is incorrect. The characteristic of thought in virtue of which a science of thought is called normative consists … in the necessity that in every act of thought the thinker himself should judge the success of his own act … I propose to substitute for the traditional epithet ‘normative’ the more accurate term ‘criteriological’.
The chapter is called “Psychology as the Science of Feeling,” because that is what psychology was created to be, when people recognized that, not being self-critical, feeling was not thought and therefore must be studied by a science different from logic and ethics. Psychology is non-criteriological, and you can understand this, even if you think that everything in nature behaves “teleologically.” Things can have purposes or ends, but if the things don’t know this, then you won’t be studying them criteriologically.
Collingwood traces the origin of psychology to the sixteenth century, but gives no details. The Oxford English Dictionary traces it more specifically to sixteenth-century Germany, and in particular to Melanchthon, though the earliest recorded use of the term in English is from a 1693 translation of Steven Blanchard’s Lexicon Medicum. Wikipedia spells the author’s surname as Blankaart, but traces the word to a Croatian humanist, Marko Marulić, author of Psichiologia de ratione animae humanae, 1510–17.
Collingwood’s next chapter is called “Psychology as the Pseudo-science of Thought,” because a program to study thought empirically can only be a mistake or a fraud. Such a program was nonetheless urged in the eighteenth century, and possibly for the good reason that logic and ethics had ceased being strictly criteriological, but had become normative in the sense of imposing standards from outside (pp. 114–5):
It might very well be true that a revolt against the old logic and ethics had been desirable and had proved beneficial; for it might very well be true that people who professed those sciences had misunderstood their normative character, and had claimed a right of censorship over the thoughts and actions of other people; and for the sake of scientific progress such tyranny might very well have to be overthrown. When it is a case of overthrowing tyranny one should not be squeamish about the choice of weapons. But the tyrannicide’s dagger is not the best instrument for governing the people it has liberated.
I have elsewhere criticized Collingwood’s violent language, along with my own grandfather’s writing that “we were too squeamish” to fight the Vietnam War properly.
I find in a recent Guardian Weekly an example where a criteriological science, here economics, is treated as normative. Given courtroom evidence, people were asked to make a case, either for the plaintiff or the defense. Then they were asked to predict what the judge had done. According to Tim Harford in “Feeling is believing” (Guardian Weekly vol. 203 no. 14, 18 September 2020; online as “Facts v feelings: how to stop our emotions misleading us”),
Their predictions should have been unrelated to their role-playing, but their judgment was strongly influenced by what they hoped would be true.
Psychologists call this “motivated reasoning”. Motivated reasoning is thinking through a topic with the aim, conscious or unconscious, of reaching a particular kind of conclusion.
I have said it before: all reasoning is motivated reasoning. We engage in it for a reason! In the present “real court case about a motorbike accident,” it was for the judge to decide what “should have” happened; it was for the experimental subjects of Linda Babcock and George Loewenstein to argue, hypothetically, about what “should have happened”; it was not for the two economists to say what those subjects “should have” decided.
Logic has been wrongly made normative if used to “correct” Mick Jagger for singing,
I can’t get no satisfaction.
The pedant may assert that what Jagger is “really” saying here, “logically,” is,
I always get satisfaction.
That is not what he is saying. He is saying what he means, or what his persona as rock-n-roll singer means; and what that persona means, in “standard” English, is,
I can never get any satisfaction.
And this is only an approximate translation. The full translation, into English, would just be the original lyric, which is already in English.
In the address that I discussed in “What Mathematics Is,” Euphemia Lofton Haynes identifies logic and mathematics; but there’s a difference. If mathematics is one of the sciences fitting the three-part classification above, it is a descriptive or empirical science. However, again, logic is the criteriogical science of how sciences justify their findings.
My specialty within mathematics is model theory, which is said to be a part of mathematical logic. It is just mathematics though, albeit mathematics made possible by the development of symbolic logic.
A senior colleague in model theory once suggested to me that logic is about something, namely reason, as physics is about the physical world; but mathematics as such is about nothing. This sounded reasonable; however, when I asked this person about it in a later year, he did not particularly remember what he had said (and this is why I am not naming him).
Logic is associated with mathematics, because mathematics is the science that logic has best illuminated. I have written about Wigner’s notion of an “unreasonable effectiveness of mathematics in physics.” One might speak also of the unreasonable effectiveness of logic in mathematics.
I ended “What Mathematics Is” by saying that logically possible worlds are worlds that can be deduced from postulates. This was glib. The properties of a mathematical world are deduced from the postulates of that world. The properties cannot be observed in the conventional sense, by sense, such as hearing, seeing, and touching. To say that a mathematical world exists in the first place is to say that its postulates are consistent; and this we can prove, only by assuming the consistency of some other postulates.
Today those other postulates are normally the Zermelo–Frankel axioms of set theory, with the Axiom of Choice, composing the collection called ZFC. However, by Gödel’s Second Incompleteness Theorem, ZFC does not entail its own consistency, if it is consistent.
I have already reviewed both of Gödel’s incompleteness theorems. I do it here now, somewhat differently: usually more tersely, but sometimes in more detail, or in another way.
Gödel’s First Incompleteness Theorem is that there is not, and cannot be, an algorithm for identifying all true statements about numbers. By numbers I mean counting numbers: 1, 2, 3, and so on. By a statement about them, I mean, technically, a formula that has no free variables, in a sense to be made more precise later. Meanwhile, in our context, a formula is an expression built up from polynomial equations by finitely many applications of the logical operations of
conjunction (forming p ∧ q, “p and q,” from p and q);
negation (forming ¬p, “not p,” from p);
disjunction (forming p ∨ q, “p or q”; but this is already ¬(¬p ∧ ¬q));
implication (forming p ⇒ q; but this is ¬p ∨ q);
universal quantification (forming ∀x p, “For all x, p,” from p and a variable x).
existential quantification (forming ∃x p, “For some x, p”; but this is ¬∀x ¬p).
In our polynomial equations, the only parameter is 1. However, from 1 we can build up the other counting numbers as the sums 1 + 1, 1 + 1 + 1, and so on, these themselves being polynomials, namely constant polynomials.
The definition of truth is fairly straightforward. An equation of constant polynomials is true or false in the obvious sense; an equation of arbitrary polynomials is true if it is true for all values of its variables; the definition of truth for more complicated formulas follows accordingly.
There is a stronger notion. A formula is logically true if it is true in the sense above, regardless of how values are assigned to variables, sums, and products. The formula
x = y ⇒ x + z = y + z
is logically true. The statement
∃x ∃y x ≠ y
is true, but not logically true, since it is false if every variable, and in particular x and y, can take only the same value. However,
∃x x = x
is logically true.
There are algorithms for generating some true statements about numbers:
Start with some statements known to be true of numbers: these statements are now postulates.
Derive new formulas by means of some rules of inference that are known to preserve truth. Such rules will normally include
modus ponens: from p and p ⇒ q, derive q;
generalization: from p, derive ∀x p;
logical axioms: from nothing at all, derive
tautologies, such as p ∨ ¬p (these are the logically true formulas that involve no quantifiers);
other formulas that are logically true, such as
laws of equality such as x = x,
∀x p ⇒ p(c), where p(c) is the result of replacing each free occurrence of x in p with c, this being a constant polynomial.
The derived formulas that are statements are theorems, because they have proofs, which show how to derive them from the postulates by means of the rules of inference. The theorems constitute a theory of the counting numbers.
We are now using the word “theorem” in two ways.
Gödel’s theorems are logical theorems.
The statements about numbers that are theorems in the sense just defined are mathematical theorems.
One may distinguish between
the logical axioms, which derive mathematical theorems from nothing;
the other rules of inference, which have to start with something.
The logical axioms and the rules of inference then constitute a proof system. Before the incompleteness theorems came Gödel’s Completeness Theorem, whereby there are proof systems that yield every logically true statement as a theorem.
We shall henceforth assume that our proof system is complete in this sense.
Gödel’s First Incompleteness Theorem is now that from no postulates can all true statements about numbers be derived as theorems.
We are not allowed to take all of those true statements as postulates in the first place. I meant to suggest this by saying that our postulates must be known to be true of the counting numbers. In particular, though the postulates may be infinitely numerous, there has to be a mechanical rule for writing them down. In technical terminology, to be made a bit more precise later, the postulates have to be recursively enumerable.
The proof of Gödel’s First Incompleteness Theorem relies on the possibility of turning every formula into a number itself. This number is the Gödel number of the formula. Strictly, it is the set of Gödel numbers of our postulates that has to be recursively enumerable.
Instead of assigning to every formula a number, we could assign a set; then, by Gödel’s method, we could prove that no theory of sets, such as ZFC, is complete. The method applies to any mathematical structure whose elements we can manipulate in a way that mimics how we form postulates and apply rules of inference to them.
Assigning Gödel numbers to formulas is then like assigning letters to sounds, the way the Phoenicians did in creating the alphabet.
A better metaphor is turning grammatical sentences into nouns, as with use of quotation marks or a determiner such as “that” or “whether”:
Are you coming?
I said, “Are you coming?”
I want to know whether you are coming.
I know that you are coming.
We assign Gödel numbers to formulas quâ strings of symbols, and we can recover the formulas from the numbers. In the same way, we assign Gödel numbers to lists of formulas. Such a list could be a proof. Whether it is a proof is something that can be recognized from the Gödel number. Saying that a certain formula about numbers has a proof now means saying that a number with a certain property exists; that the formula has no proof means the number doesn’t exist.
In describing logical axioms, I said p(c) was the result of replacing each free occurrence of x in p with c. The rules are:
Every occurrence of a variable in a polynomial equation is free.
Being a free occurrence in a formula is preserved when the formula is negated or conjoined with another, but not when the variable is quantified.
Thus in the formula x = x ⇒ ∃x x ⋅ x = x, the first two occurrences of x are free; the remaining four are not free. To say that a variable is free in a formula is to say that it has a free occurrence in the formula; it may also have a non-free occurrence, though we can usually arrange for this not to happen, as for example by writing the formula above as x = x ⇒ ∃y y ⋅ y = y.
A formula with a free variable is effectively a predicate with unspecified subject. Given a formula p with a free variable, we can take any counting number a and, using Gödel numbers, form the statement that p(a) has no proof. We shall be interested in the case where a is the Gödel number of p.
Another level of abstraction is possible. We write the Gödel number of any formula p as ⌜p⌝. There is a formula φ with one free variable such that, for all statements s,
φ(⌜s⌝) is true if and only if s is a theorem.
There is now a formula ψ with a free variable such that, for all formulas p with a free variable,
¬φ(⌜p(⌜p⌝)⌝) is ψ(⌜p⌝).
If we think of forming the Gödel number of a formula as quoting the formula, then ψ is the predicate, “yields an unprovable statement when predicated of the quotation of itself.”
We can predicate ψ of the quotation of itself in this sense, forming the statement ψ(⌜ψ⌝). If we write this statement as σ, then it is also ¬φ(⌜σ⌝). Briefly, σ says, “I have no proof.” It doesn’t really have a first-person pronoun though; what σ says is,
“Yields an unprovable statement when predicated of the quotation of itself” yields an unprovable statement when predicated of the quotation of itself.
Upon reflection, we see that the particular predication that this statement is about—which is a predication that has no proof—is just the statement itself. If σ were false, then φ(⌜σ⌝) would true, and thus σ would be a theorem. Since theorems are true, σ must be true; but then it cannot be a theorem. This proves Gödel’s First Incompleteness Theorem.
My use of the quotation of predicates is based on the dialogue that begins on page 431 of Douglas R. Hofstadter’s book of xxii + 778 pages called Gödel, Escher, Bach: An Eternal Golden Braid (Basic Books, 1979). The dialogue is called “Air on G’s String,” and in it, the Tortoise recounts to Achilles the receiving of an “obscene” phone call. Written out and punctuated correctly, what the caller says is,
“Yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.
It’s a way of saying “I am lying,” without actually saying “I.” It seems I have remembered it from reading Hofstadter’s book when the paperback edition came out, in 1980, when I turned fifteen.
I enjoyed reading the book then. Its main benefit may have been to lead to me to learn more about Zen Buddhism, first through Reps and Senzaki, Zen Flesh, Zen Bones (1957).
I cannot have understood Gödel’s Theorem from Hofstadter. I am still trying to understand it now. In graduate school, a professor told me he hoped Hofstadter himself had not understood Gödel’s Theorem, because teaching it with a book like Hofstadter’s would be unconscionable.
I am not naming the professor, because I may not have thoroughly understood and properly recalled his meaning. For the same reason, I am not naming the tutor who told me, in my freshman year at college, that Gödel’s Theorem need not rely on self-reference, contrary to Hofstadter’s evident belief.
That tutor may have been alluding to how Gödel’s Theorem derives from the Halting Problem, as follows.
In principle we can make a list of all computer programs that can be set to work on single numbers as input. We can denote the nth program on the list by {n}. Working on a number k, this program may halt and produce some output, or it may never stop running. For example, the program may seek the least twin prime that is greater than k; if the Twin Prime Conjecture is false, and k is large enough, then the program will never halt.
The set of all numbers at which a given program halts is, by definition, a recursively enumerable set. For example, the set of twin primes is recursively enumerable, whether it is finite or infinite. We can list the elements of a recursively enumerable set in stages. At stage k, on all numbers that do not exceed k, we have run the program for k steps, and we have written down the numbers for which the program has halted so far.
There is a recursively enumerable set of numbers whose complement is not recursively enumerable. This means, by definition, that the original set is not recursive. I looked at this set at the end of “Hypomnesis,” which is about a logic meeting in Delphi in 2017; but let me define the set again here. There is a program that does at k what {k} does. This program must be {m} for some m. But then the set of numbers where {m} does not halt cannot be recursively enumerable; for if it were, then it would be, for some k, the set of numbers where {k} halted. In this case {k} would halt at k if and only if it did not.
For each number k, there is a statement p_{k} that the program {k} does not halt at k. Then the set of k for which p_{k} is true is not recursively enumerable. Hence also the set of Gödel numbers ⌜p_{k}⌝ of the statements p_{k} that are true is not recursively enumerable. However, the set of Gödel numbers of all of the statements p_{k} is recursively enumerable. Therefore the set of Gödel numbers of all true statements about the counting numbers cannot be recursively enumerable.
Now we have proved Gödel’s First Incompleteness Theorem a second time. The second proof does not involve a proof system. In particular, it does not give us a particular statement that cannot be proved; it just shows that such a statement will always exist, regardless of our proof system. Much less does the proof give us a statement that says, “I cannot be proved.” The proof still uses self-reference, in the sense of applying a program to its own number. It doesn’t really need Gödel numbers in the precise sense; however, the whole notion of a computer program is an analogue of Gödel numbering.
After the dialogue of the Tortoise and Achilles that I have referred to, Hofstadter has a chapter called “On Formally Undecidable Propositions of TNT and Related Systems.” The title comes from that of Gödel’s 1931 paper by substitution of TNT (“typographical number theory”) for Principia Mathematica. Hofstadter says his chapter will be “more intuitive” than Gödel’s article, and
I will stress the two key ideas which are at the core of [Gödel’s] proof. The first key idea is the deep discovery that there are strings of TNT which can be interpreted as speaking about other strings of TNT; in short, that TNT, as a language, is capable of “introspection”, or self-scrutiny. This is what comes from Gödel-numbering. The second key idea is that the property of self-scrutiny can be entirely concentrated into a single string; thus that string’s sole focus of attention is itself. This “focusing trick” is traceable, in essence, to the Cantor diagonal method.
We used the diagonal method to produce a recursively enumerable set that is not recursive. Indeed, we can think of a table in which the entry in row k and column m is
1, if {k} halts at m;
0, otherwise.
Each row then is a string of 0s and 1s. The string of entries on the diagonal is also row m, if again {m} is the program that does at each k what {k} does. The string that we get from that row by interchanging 0 and 1 can therefore occur as no row; thus there is no program that halts precisely where {m} does not.
Hofstadter continues:
In my opinion, if one is interested in understanding Gödel’s proof in a deep way, then one must recognize that the proof, in its essence, consists of a fusion of these two main ideas. Each of them alone is a master stroke; to put them together took an act of genius. If I were to choose, however, which of the two key ideas is deeper, I would unhesitatingly pick the first one—the idea of Gödel-numbering, for that idea is related to the whole notion of what meaning and reference are, in symbol-manipulating systems. This is an idea which goes far beyond the confines of mathematical logic, whereas the Cantor trick, rich though it is in mathematical consequences, has little if any relation to issues in real life.
One could question Hofstadter’s (or anybody’s) presumption to
understand Gödel’s proof “in a deep way,”
be able to recognize a master and a genius,
know all about “real life.”
For one thing, mathematics is a part of real life, unless one harbors illusions about what one knows, or tries to create illusions in others’ minds about what one knows.
In his article “Gödel’s Theorem” in the Princeton Companion, Peter J. Cameron also says that the theorem is based on two ideas. Cameron’s first idea is Hofstadter’s, which is Gödel numbering; but Cameron’s second is just self-reference, as in the forming of the statement ψ(⌜ψ⌝).
I don’t know that either person’s “two ideas” are really two. Gödel numbering is an embedding of the logic of arithmetic in arithmetic itself. Embeddings as such are everywhere in mathematics: there’s the embedding of the plane in space for example, or of the counting numbers in the integers or the positive rational numbers. Gödel numbering is special for letting arithmetic be about not just numbers, but itself. Our second proof of Gödel’s theorem seemingly used diagonalization in place of Gödel numbering; but a diagonal argument relies on the possibility that rows and columns of the same table can “speak” about one another in the sense of bearing the same serial number. Moreover, the numbering of programs is like Gödel numbering in the sense that we can mechanically obtain the program {k} from k; this is why there can be a statement, in our formal sense, that {k} does not halt at k.
It remains to establish Gödel’s Second Incompleteness Theorem, that the consistency of our postulates for the numbers is not a theorem. The consistency is a statement though, namely the statement that Λ has no proof, where Λ is a logically false statement, such as ∃x x ≠ x.
We want then to show that
¬φ(Λ)
is not a theorem, since this is the statement that Λ has no proof. We already know that
¬φ(σ)
is not a theorem, where σ is ψ(⌜ψ⌝), where ψ(⌜p⌝) is ¬φ(⌜p(⌜p⌝)⌝), so that ¬φ(⌜σ⌝) is just σ. Thus it is enough to show that the implication
φ(⌜σ⌝) ⇒ φ(⌜Λ⌝)
is a theorem. There is a proof of this statement from
φ(⌜σ⌝) ⇒ φ(⌜¬σ⌝),
and of this statement from
φ(⌜σ⌝) ⇒ φ(⌜φ(⌜σ⌝)⌝).
This last statement is a theorem, regardless of what σ is. So Gödel’s Second Incompleteness Theorem holds. (The sketch of the proof is based on that of C. Smorynski in “The Incompleteness Theorems” in Jon Barwise, editor, Handbook of Mathematical Logic, Elsevier, 1977.)
As telling us what we can and cannot prove in mathematics, Gödel’s Completeness Theorem and First Incompleteness Theorem are theorems of logic. We can turn the latter into a mathematical theorem, and this is how the Second Incompleteness Theorem is proved.
In the first section, “Entailment,” I left off the last sentence of Gowers’s quotation of Russell:
In addition to these [propositions of the form “p implies q”], mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.
That was in 1903, Russell having been born in 1872; Gödel would be born in 1906.
What Russell says about truth in the book is provoked by a paradox concerning the rule of modus ponens that Lewis Carroll presented in the dialogue, “What the Tortoise Said to Achilles.” Along with his own dialogues involving the same characters, Hofstadter will reprint Carroll’s dialogue in Gödel, Escher, Bach. The paradox is that modus ponens is about implication, but is also itself an implication. It is that p and the implication p ⇒ q together imply q. If we now write the rule as the same kind of implication, namely
p ∧ (p ⇒ q) ⇒ q,
then, to use the rule, we need another rule, that p and p ⇒ q and the rule just given together imply q: in short,
p ∧ (p ⇒ q) ∧ (p ∧ (p ⇒ q) ⇒ q) ⇒ q.
But now this is not enough. If we define propositions p_{n} recursively so that
p_{1} is p ⇒ q,
p_{n+1} is p ∧ p_{1} ∧ … ∧ p_{n} ⇒ q,
then all of these together cannot compel us to accept q.
That is so, I would say, because logic is not normative, but criteriological. It cannot tell us what we must do; it observes what we do do. One thing we do is draw conclusions that accord with the rule called modus ponens.
In the first proposition of the first book of the Elements, given a segment ΑΒ and the possibility of drawing circles, Euclid constructs a triangle ΑΒΓ in which ΑΓ = ΑΒ and ΒΓ = ΒΑ. Since ΑΒ and ΒΑ are the same segment, we know that ΑΓ = ΒΓ. Thus the sides are all equal to one another, and therefore, by definition, the triangle is equilateral.
The sides ΑΓ and ΒΓ are equal to one another, because
each is equal to the same side, written indifferently as ΑΒ or ΒΑ;
equals to the same are equal to one another.
This is only an explanation of something that we already know.
We have to know too that the point Γ exists in the first place as an intersection of the circles, each of which has one of Α and Β as center and passes through the other.
Some modern readers complain that Euclid applies no explicit rule whereby those circles must intersect. Such readers are like the visitors that Ruth Fuller Sasaki describes in “Zen: A Method for Religious Awakening” (1959; reprinted in The World of Zen, edited by Nancy Wilson Ross, 1960; I bought the book in 1981):
How many hours have I not spent in my Kyoto temple listening to people, usually Americans recently come to Japan, tell me just what Zen is. To such visitors I have nothing to say; to those who do not understand, I am always searching for a way to give a clue to what Zen is about.
Mathematicians such as Timothy Gowers are keen on searching for a way to give a clue to what mathematics is about. To me the most important clue lies in a chapter of Gowers’s little book called Mathematics: A Very Short Introduction (Oxford, 2002; xiv + 133 pages). Chapter 3 is called “Proof,” and there Gowers observes,
the steps of a mathematical argument ean be broken down into smaller and therefore more clearly valid substeps. These steps can then be broken down into subsubsteps, and so on. A fact of fundamental importance to mathematics is that this process eventually comes to an end. In principle, if you go on and on splitting steps into smaller ones, you will end up with a very long argument [that] starts with axioms that are universally accepted and proceeds to the desired conclusion by means of only the most elementary logical rules (such as ‘if A is true and A implies B then B is true’).
I would propose a couple of adjustments.
Some axioms, such as the Axiom of Choice, are not universally accepted. If you prove a proposition P using this axiom, then everybody will agree that at least you have proved the proposition that AC implies P.
Possibly not everybody will agree, even then, if for example you have used the logical rule of the excluded middle (“either Q or not-Q”). Intuitionism rejects this rule. But an Intuitionist should still be able to tell whether an argument is correct within a system that does allow use of the rule of the excluded middle. The Intuitionist will then prefer to find an argument, a “constructive” argument, that does not need this rule, and a mathematician of any school can confirm the construction.
Gowers goes on to say:
What I have just said in the last paragraph is far from obvious: in fact it was one of the great discoveries of the early 20th century, largely due to Frege, Russell, and Whitehead (see Further reading). This discovery has had a profound impact on mathematics, because it means that any dispute ahout the validity of a mathematical proof can always be resolved …
… the fact that disputes can in principle be resolved does make mathematics unique. There is no mathematical equivalent of astronomers who still believe in the steady-state theory of the universe, or of biologists who hold, with great conviction, very different views about how much is explained by natural selection, or of philosophers who disagree fundamentally about the relationship between consciousness and the physical world, or of economists who follow opposing schools of thought such as monetarism and neo-Keynesianism.
I have become an exponent of this idea, that there is one field of human endeavor where all disputes can be settled amicably.
My wife used the idea in her courtroom defense against the accusation of being a terror-propagandist.
I would summarize the idea as being that mathematics is the deductive science. Gowers doesn’t say it that way, though he uses the word deduction:
No mathematician would ever bother to write out a proof in complete detail—that is, as a deduction from basic axioms using only the most utterly obvious and easily checked steps.
It seems to me that everybody who learns anything about mathematics should learn its deductive nature. Otherwise they haven’t really learned mathematics. The idea of deduction will however need building up over the years of one’s education. It has already needed building up in the millenia since Euclid.
]]>I continue with the recent posts about mathematics, which so far have been as follows.
“What Mathematics Is”: As distinct from the natural sciences, mathematics is the science whose findings are proved by deduction. I say this myself, and I find it at least implicit in an address by Euphemia Lofton Haynes.
“More of What It Is”: Some mathematicians do not distinguish mathematics from physics.
“Knottedness”: Topologically speaking, there is a sphere whose outside is not that of a sphere. The example is Alexander’s Horned Sphere, but it cannot actually be physically constructed.
“Why It Works”: Why there can be such a thing as the horned sphere.
When I first drafted the first post above, I said a lot more than I eventually posted. I saved it for later, and later is starting to come now.
I still said a lot, even in that first post. I want to review it some more, before moving on.
I spent some time to explain that I had discovered Euphemia Lofton Haynes, because I had ignored the call of a mathematician on Twitter to unfollow a certain account. Other mathematicians supported the call, but were reluctant to explain their support.
I have had other encounters with mathematicians who take positions on a social issue, but aren’t prepared to explain why.
They may see it differently, and indeed I have recognized in a couple of posts that people like Steve Bannon may not debate in good faith. Who knows whether a rando on Twitter is going to be such a person?
Still, anybody may question you in good faith, and mathematics ought to train you to be prepared to answer. Practically speaking, you could have your answer ready on a blog.
I say that mathematics is one science, namely the deductive science, and that deduction is “universally valid reasoning.” From this I draw the conclusion, given in such posts as “Salvation,” that in the case of mathematical disagreement, at least, we have to allow the possibility that we ourselves are wrong.
As I suggested in a post written during the 2019 Logic Colloquium in Prague, we cannot say “My way or the highway”; but I was not even talking about mathematics. There are in fact differences and disagreements among us. The unity of the universe, the possibility of universal truth, is not an observation, but a conviction. A key component of this conviction is that resolution of disputes is possible.
Such resolution does not happen if we dismiss our opponents. This is a lesson that I hope students learn from mathematics, although we mathematicians ourselves have not always learned it.
In “What Mathematics Is,” I took up the theorem that the sum of the first n counting numbers is half the product of n and n + 1. This is the theorem that Gauss is said to have recognized at the age of seven, when tasked by a teacher to add up the first hundred numbers. In the blog post, I gave three proofs.
One proof was by formal mathematical induction.
Another was a diagram of a single case.
A third was an algebraic expression of what the diagram was intended to show.
It could be said that the latter two proofs are not real proofs. I would say that of the first as well, if it is considered as a string of symbols. We call the string or the picture a proof, if we can use it to understand that there really is a proof; but the real proof is seen with the mind’s eye.
The proof is universal, in the sense that calling it a proof expresses our conviction that it is true for everybody, everywhere, for all time.
Not every field has proofs like that. Through my window right now, I seem to see a fine rain falling. I can prove that it is falling by stepping onto the balcony and holding out my hand; but this tells me nothing about how long the rain will continue.
Suppose however I prove the Pythagorean Theorem, and a student seems to accept the proof, but then asks, “Will it still be true tomorrow?” I don’t think I can give an additional proof that it will be. This is just something that you have to understand, in order to able to do mathematics. You have to have something like the object permanence that children somehow pick up, without being explicitly taught.
We do not prove this universal validity of mathematics, any more than experiments prove the universal validity of Newton’s laws of motion. We have to believe that such universal laws are possible in the first place; then experiment may prove that we have found a particular law. Playing with a so-called Newton’s cradle here on Earth, I somehow know it will behave the same on Neptune; but the play alone does not show this.
Mathematics is similar, except that the standard of proof is different: deductive rather than inductive.
Some persons take issue with the putative universality of mathematics. They may cite Alan J. Bishop, “Western mathematics: the secret weapon of cultural imperialism,” Race & Class 32(2), 1990. I learned about this paper from a tweet, though not one of those that you can find by searching on “math is not universal” or “mathematics is not universal.” The people saying this may be correct according to their meaning; those denying it, correct according to their meaning. Working out that meaning needs more than a tweet or even a blog article. I’m trying to do what I can.
There may well be kinds of mathematics different from so-called Western mathematics. Even within Western mathematics, there are differences unseen by those unprepared to look for them. Examples of such differences have been created by
the discovery of non-Euclidean geometry;
the abstraction of numbers from numbers of things, and the resulting treatment of ratios as numbers.
People may overlook the differences; they may also learn to see them.
Bishop acknowledges a kind of universality of mathematics in his fourth paragraph:
There is no doubt that mathematical truths like those [namely “two twos are four, a negative number times a negative number gives a positive number, and all triangles have angles which add up to 180 degrees”] are universal. They are valid everywhere, because of their intentionally abstract and general nature. So, it doesn’t matter where you are, if you draw a flat triangle, measure all the angles with a protractor, and add the degrees together, the total will always be approximately 180 degrees. (The ‘approximate’ nature is only due to the imperfections of drawing and measuring—if you were able to draw the ideal and perfect triangle, then the total would be exactly 180 degrees!) Because mathematical truths like these are abstractions from the real world, they are necessarily context-free and universal.
An additional qualification should be made: the total is 180 degrees for the measures of the angles of “the ideal and perfect triangle” in the Euclidean plane, as distinct from, say, the “Lobachevskian” or hyperbolic plane.
Bishop is aware of non-Euclidean geometry, but only in a different sense:
The conception of space which underlies Euclidean geometry is also only one conception—it relies particularly on the ‘atomistic’ and object-oriented ideas of points, lines, planes and solids. Other conceptions exist, such as that of the Navajos where space is neither subdivided nor objectified, and where everything is in motion.
Perhaps hyperbolic geometry is still to be considered “Euclidean” according to Bishop’s conception, since it comes out of the Western, Euclidean tradition. Still, Bishop ought to know that, within this tradition, people tried for two thousand years to prove that Euclid’s was the only geometry, before Lobachevsky and Bolyai showed that it didn’t seem to be, and then Beltrami confirmed it.
Bishop says,
Recognising symbolisations of alternative arithmetics, geometries and logics implies that we should, therefore, raise the question of whether alternative mathematical systems exist.
They do exist, even within “Western” mathematics. Even on the subject of the counting numbers, Euclid’s way of thinking is not clear to mathematicians today. Today we have a notion of “fraction” that Euclid didn’t, because he made a distinction between numbers and their ratios that we do not. I wrote about my understanding of Euclid in a 2017 post called “The geometry of numbers in Euclid” (reposted at the De Morgan Forum), and more recently, with references to all of the relevant literature that I know of, on pages 15–17 of the notes (in pdf format) to a talk I gave at the University of Maryland in June, 2019. As I wrote in “Salvation” in February of this year (2020), Euclid’s numbers are magnitudes, or at least multitudes of magnitudes, but ratios are dimensionless.
I find a paper called “Navajo spatial representation and Navajo geometry” by Rik Pinxten, a researcher whom Bishop refers to. Greek geometry comes from Egypt, by the account of Herodotus that I referred to in “Thales of Miletus”; the Egyptians needed geometry to measure their fields for taxation after the annual Nile floods. Herodotus’s term γεωμετρίη should be understood in the literal sense of earth-measurement, that is, surveying.
Euclid’s geometry is not that, but a system for deducing theorems and problems (that is, constructions) from simple assumptions. If Navajo geometry has a Euclid, let us learn from him or her. If somebody said that Navajo people could not learn from the Greek Euclid, that would be a remarkable claim.
There’s a remarkable claim that the Pirahã language spoken in Amazonas does not exhibit recursion. However, writing in Harper’s (August 2016), Tom Wolfe quotes Chomsky as saying about this,
The speakers of this language, Pirahã speakers, easily learn Portuguese, which has all the properties of normal languages, and they learn it just as easily as any other child does, which means they have the same language capacity as anyone else does.
I would likewise expect any child to learn mathematics—any particular kind of mathematics—as easily as any other child does.
This is like saying Newton’s cradle behaves on Neptune as on Earth.
It also says nothing about the mathematics that children ought to learn, any more than it says children ought to be taught in Portuguese in Brazil, or Turkish in Turkey, or English in the United States.
Not everybody does learn mathematics with the same ease. In “Why It Works,” I quoted Douglas Hofstadter on not being able to handle graduate mathematics. However, Hofstadter goes on to say:
If, at that crucial moment in my life, someone had suggested that before abandoning mathematics, I take a look at geometry, I might have discovered the works of Donald Coxeter and followed a very different pathway in life.
If students do perform differently in school, that doesn’t mean they have to. With human beings, there is no telling in advance what is not possible.
Pinxten recommends teaching Navajo children according to their experience:
In the case of Navajo children we have the particularity that we should not only start from the world of experience of a child, but from the world of experience of a child in another culture. Therefore, we aim at developing a geometry course by means of terms and expressions which are Navajo, and in a cultural context of representations and choices which are, again, Navajo.
Pinxten proposes having children construct and analyze a model rodeo ground, because “Rodeo has grown to be a major point of interest for Navajo children.” By children he means boys though:
Boys especially are eager to assist rodeo events and not seldom the dream of a boy or a young adult is to become a rodeo rider himself … Moreover, it often happens now that upon the birth of a boy one of the parents buries a string of horsetail under the hooghan “so that he may become a good rodeoman” …
Pinxten’s aim is that students learn geometry:
Starting from there, geometric abstractions will be gradually introduced. Finally, children and teacher alike will search for strong relationships between the terms, leading eventually to a genuine formalized (or even axiomatized) geometry.
Pinxten does refer to a geometry, as if it were one among many. Each of us has a geometry; I think this is our individual understanding of the one thing called geometry.
The image near the top of this post is of an icosahedron inscribed in an octahedron. The vertices of the former divide the edges of the latter in the golden ratio. It must take some training to see the image as being of a three-dimensional object. In “June [2014] in the New World,” I recalled not having been able to see the bull in a painting called “Tijuana” by Elaine de Kooning.
The last post, “Knottedness,” constructed Alexander’s Horned Sphere and proved, or sketched the proof, that
the horned sphere itself is topologically a sphere, and in particular is simply connected, meaning
it’s path-connected: there’s a path from every point to every other point;
loops contract to points—are null-homotopic;
the space outside of the horned sphere is not simply connected.
This is paradoxical. You would think that if any loop sitting on the horned sphere can be drawn to a point, and any loop outside the horned sphere can be made to sit on the sphere and then drawn to a point, then we ought to be able to get the loop really close to the horned sphere, and let it contract it to a point, just the way it could, if it were actually on the horned sphere.
You would think that, but you would be wrong. Mathematics is pathological that way. Some of us may note the pathologies, then pass to areas where intuition works better. Other persons may become obsessed with the pathologies.
If not by just reading this blog, you might find such persons by doing an image search on the Alexander Horned Sphere. I made such a search, but could not find images like the ones I am going to give here. I am trying to clarify why the horned sphere has the properties it does.
The idea of the proof of those properties is that the horned sphere is the intersection of a descending chain,
X_{0} ⊃ X_{1} ⊃ X_{2} ⊃ …,
of sets X_{n} that are not simply connected, but are compact.
I did not try to define compactness in the last post. Perhaps one does not normally learn about this concept until one has spent some time with calculus. Calculus is the practical side of what has the theoretical side called analysis. From high school I have Apostol’s Mathematical Analysis (second edition, Reading, Mass.: Addison-Wesley, 1974); we used selections from it with Mr Brown to learn about uniform convergence, a concept needed also for the proof of Dirichlet’s Theorem that I discussed in “LaTeX to HTML.”
Apostol defines compactness for ℝ^{n} on page 59; for an arbitrary metric space, page 63. Thus I must have learned about compactness when I started to work my way through Apostol before graduate school. I did some of this work at the farm.
Alexander’s Horned Sphere sits in ℝ^{3}, which is an example of ℝ^{n}, which is an example of a metric space, which is an example of a topological space. Such a space was apparently not defined till the 20th century. It is a set of points, and certain subsets are called closed, and these must satisfy certain axioms, namely:
the union of two closed sets is closed;
the intersection of any collection of closed sets is closed (here the intersection of the empty collection is understood to be the whole space);
the empty set is closed.
That’s all.
The friend who gave me the book about Helaman Ferguson that inspired my last post (and now this one) gave me also Siobhan Roberts, King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry (Toronto: Anansi, 2006). There’s a foreward by Douglas R. Hofstadter, whom I discussed in “Writing, Typography, and Nature.” In Roberts’s book, Hofstadter recalls discovering, in adolescence, Kelley’s General Topology. This “did not, in its hundreds of pages, contain a single diagram.” At first, Hofstadter was not dismayed.
Within a few years, however, I discovered that I personally could not survive in such an arid atmosphere. Diagrams … were the oxygen of mathematics to me, and without them I would simply die. And thus, when the air of abstraction for abstraction’s sake became too thin for me to breathe, I wound up with no choice by to bail out of graduate school in mathematics. It was a terrible trauma.
Had he known about Coxeter’s geometry, he might have stayed in mathematics.
There are many kinds of mathematics. Some can be visualized in a two-dimensional diagram; some not. The ellipsis in the quotation of Hofstadter stands for a parenthesis:
Diagrams (or at least mental imagery that could be thought of as personal, inner diagrams) were the oxygen of mathematics to me …
Reading anything gives you mental imagery, unless you are not really reading, but only looking at the pages. However, topology as such is so abstract that drawings on paper may mislead. I was thrilled to learn in graduate school that you could prove a lot of interesting theorems, just from the axioms that I wrote above. I was so thrilled that I wrote about the axioms to a friend in New York who had got a job with a publisher.
The friend was not amused, but he did send me the book of mathematical short stories that his firm published: Rudy Rucker (editor), Mathenauts: Tales of Mathematical Wonder (Arbor House, 1987). A theme of one of the stories (“The New Golden Age,” by the editor) is that, if you figure out a way for people to enjoy mathematics without actually understanding it, then the work they enjoy may be that of cranks.
The friend himself later became a chiropractor.
The notion of a topological space is general enough that the structures of a logic can serve as the points of a topological space. The closed sets are the sets of models of theories. For example, the union of the sets of models of formulas φ and ψ respectively is the set of models of the formula (φ ∨ ψ). The connection between topology and logic was a reason why I decided to specialize in model theory.
In “Boolean Arithmetic,” I worked out the proof that the space of models of propositional logic is compact. The topology of that space is the topology of a Cantor set, discussed in “The Tree of Life.”
As a quotation in the last post noted,
there is a Cantor set of ‘bad’ points on the horned sphere.
We constructed that sphere by, for each finite binary sequence σ, attaching the horns H_{σ0} and H_{σ1} to H_{σ}. Here if σ is the empty sequence, we can understand H_{σ} to be the ball B_{0}. Each infinite binary sequence τ yields the sequence of horns H_{τ|n}, where τ|n is the sequence (τ(k): k < n) of the first n entries of τ. The union (over n) of the horns H_{τ|n} has a unique “tip,” a unique accumulation point that doesn’t belong to it. Such points are the “bad” points mentioned in the quotation.
The same sequence τ determines the model of propositional logic in which the true propositions P_{n} are those for which τ(n) = 1.
In our present situation, working inside Euclidean space, I suggested in the last post that being compact is equivalent to being closed and bounded.
Being bounded means being part of a ball. A ball is the inside of a sphere, this being understood in the usual sense of the set of points sharing a common distance from a single point, which is then the center of the sphere and the ball.
A set F is closed if every point not belonging to F is the center of a ball with no points in common with F. I believe the letter F is commonly used for closed sets because it stands for the French fermé(e).
A subset K of Euclidean space is closed and bounded if and only if, whenever K is covered by a collection of balls, meaning the union of the balls includes K, a finite number of those balls are enough to cover K. This theorem, or the difficult part, the “only if” part, is called the Heine–Borel Theorem. The condition involving coverings is now a standard definition of a compact set (in German, kompakt), and I quoted the definition in “Poetry and Mathematics” as being “so poetic.”
The complement of a ball is a closed set. Then being compact is equivalent to the condition whereby, if the intersection of a family of closed subsets is empty, then some finite number of those closed subsets have empty intersection. This yields that the outside of the horned sphere fails to be simply connected, as in the last paragraph of the previous post.
But why does the proof work? Where does the horned sphere come from?
I used the idea of contorting a lump of clay to capture a loop of string. Freeing the string would mean contracting it to a point. During this contraction, the loop would trace out a surface S. Being closed and bounded, this surface would be compact. Its points would therefore have a minimum distance from the horned sphere. Proof: The horned sphere B being closed, each point of the surface S would be center of a ball that contained no points of B; but finitely many of those balls would cover S, by its compactness. The minimum radius of those finitely numerous balls would be the minimum distance from S to B.
Thus, for any distance, no matter how small, there should be arms of the lump of clay
that are that close together, but
that the loop would have to slip through to be free.
An obvious first attempt is to have two arms that grow indefinitely close, as in the figure below. This attempt fails, because the compactness of figure ensures that the two arms will actually touch.
Still, the figure can have, attached to it somehow, infinitely many pairs of arms, with no lower bound on the gap between them, as below.
But how can such arms trap the string? The string cannot initially be made to pass through all of the hoops that are almost formed by the arms; for some of those hoops will be too narrow. (If they are not, then we are back at the first failed attempt.)
However, two hoops, if linked, can form another hoop, which the string would have to break out of to be free.
Thus we let the gap between two arms be almost closed by two more pairs of arms, and so on. This leads to Alexander’s construction, which I have tried to depict below. The original pdf file has size A1, if you want to see more detail.
]]>If you roll out a lump of clay into a snake, then tie a string loosely around it, can you contort the ends of the snake, without actually pressing them together, so that you cannot get the string off?
You can stretch the clay into a Medusa’s head of snakes, and tangle them as you like, again without letting them touch. If you are allowed to rest the string on the surface of the clay, then you can get it off: you just slide it around and over what was an end of the original snake.
If however no point of the string may touch the snake, then you can trap the string so that it cannot be removed. You let the clay start as an amphisbaena, with heads at both ends; you let each head be a hydra’s head, sprouting two new ones from the stump when cut off; and you cut off all heads as they appear, ad infinitum. The result is a solid binary tree, as discussed in “The Tree of Life.” With the branches of the tree intertwined just so, you get what is called Alexander’s Horned Sphere.
You may object that you cannot actually do something ad infinitum. A person called James R. Meyer has this objection. His website is devoted to showing what’s wrong with mathematics and philosophy. I too sometimes take issue with contemporary professional philosophy in my blog. I like to think that I am saved from being a crank by knowing, as a mathematician, that the truth cannot be given uniquely to me. Meyer himself has a page about cranks, taking issue with how others define them. I found first his page about the horned sphere. The page shows no sign of understanding what the horned sphere is for.
The horned sphere is a topological sphere. It arises as a counterexample to a formerly conjectured three-dimensional version of the Jordan–Schönflies Theorem.
I read about that theorem as a child, or at least about the simpler form, the Jordan Curve Theorem: a simple closed curve divides the plane into an inside and an outside. I did not understand what there was to be excited about. I suppose now the point is the following. A simple closed curve is a continuous function f from the unit interval [0, 1] into ℝ^{2} that repeats a value precisely at 0 and 1, so that, f being the function,
f(t) = f(u) ⇔ t = u ∨ {t,u} = {0, 1}.
A simple example is g, given by
g(t) = (sin (2πt), cos (2πt)),
tracing out the unit circle; but f might not be given by a formula. How are you going to define a function h on ℝ^{2} so that h(x,y) = 1, if (x,y) is “inside” the curve given by f, and otherwise h(x,y) = 0? I just did define h, but only by begging the question of what inside means.
If g is as above, and f is some simple closed curve, then f∘g^{−1} is a homeomorphism between the two curves, just for being continuous in both directions. According to the Jordan–Schönflies Theorem, the homeomorphism extends to a homeomorphism from the whole plane ℝ^{2} to itself.
Passing to three dimensions, one may think that if there is a homeomorphism from a sphere, considered as a surface in ℝ^{3}, to some other surface in space, then that homeomorphism should extend to a homeomorphism from the whole space ℝ^{3} to itself.
The horned sphere shows that one would be wrong. Here I want to work out some details of the proof. It may serve as another example of my recent theme, that mathematics is the science whose findings are proved by deduction. Topology in particular can seem to be a counterexample, since it seems to rely on physical intuition, albeit an intuition that tolerates supposed absurdities like completed infinite processes.
The horned sphere has been the inspiration of some sculptures pictured in Claire Ferguson, Helaman Ferguson: Mathematics in Stone and Bronze (Erie, Penn.: Meridian Creative Group, 1994). A friend recently gave me the book, and the book is a reason for this post.
J. W. Alexander described his construction in “An example of a simply connected surface bounding a region which is not simply connected” (Proc. N. A. S. 10, 1924). A simply connected space is one in which
you can carry a string from any point to any other like Ariadne, and
if you carry the string back to where you started from, then you can hold the two ends and draw the whole string to yourself.
In more technical language, the loop of string must be null-homotopic, meaning there is a continuous function f from the square [0, 1] × [0, 1] into the space in question such that the function t ↦ f(0, t) is the original loop, and t ↦ f(1, t) is constant.
Alexander describes his construction with words and a drawing:
The surface Σ is the limiting surface approached by the sequence Σ_{1}, Σ_{2}, Σ_{3}, .. It will be seen without difficulty that the interior of the limiting surface Σ is simply connected, and that the surface itself is of genus zero and without singularities, though a hasty glance at the surface might lead one to doubt this last statement. The exterior R of Σ is not simply connected, however, for a simple closed curve in R differing but little from the boundary of one of the cells γ_{i} cannot be deformed to a point within R.
The surface Σ is the surface of our clay. To say that it is of genus zero means it has zero holes; a torus has genus one. I myself do see without difficulty that Σ will have no holes. I am not sure what Alexander means by singularities. That the exterior of Σ is not simply connected is not clear without a proof. Alexander himself confesses, at the end of his short article,
This example shows that a proof of the generalized Schönfliess theorem announced by me two years ago, but never published, is erroneous.
If he was wrong then, why is he not wrong now?
The relevant Wikipedia article, “Alexander Horned Sphere,” lists a reference on my shelves, Spivak’s Comprehensive Introduction to Differential Geometry, Volume One (2nd ed., Houston: Publish or Perish, 1979). Defining Alexander’s Σ using a drawing like his, an exercise asks the reader to show what I said was clear, that Σ has no holes. (Spivak calls the surface S and asks, “Show that S is homeomorphic to S^{2},” the latter being the surface, which is two-dimensional, of a sphere.) The exercise then asserts, without explicitly asking for a proof, that the S together with the the region outside it fails to be a manifold-with-boundary, though the student has shown that it would have to be one, if S were differentiable.
I pass to another of the Wikipedia references, Hocking and Young, Topology (Reading, Mass.: Addison-Wesley, 1961), which says on page 175 (where the circumflex on ĥ is a tilde in the original),
Let S be a simple closed surface in [Euclidean space] E^{3}, that is, S is a homeomorph of S^{2}, and let h be a homeomorphism of S onto the unit sphere S^{2} in E^{3}. Is there an extension ĥ of h such that ĥ is a homeomorphism of E^{3} onto itself? … Alexander … gave a famous example, the Alexander horned sphere, showing that the answer must be “no” in the general case. This example is pictured [below]. We can see from the picture alone that it is quite obvious that the complement of the horned sphere is not simply connected. Since the complement of S^{2} in E^{3} is simply connected, it follows that no homeomorphism of E^{3} onto itself will throw the horned sphere onto S^{2}. Note that there is a Cantor set of “bad” points on the horned sphere.
My sense is that in topology nothing is obvious. In Real Analysis II in graduate school, the professor made an “obvious” topological assertion that I questioned and later disproved with the help of a book I found: Steen and Seebach, Counterexamples in Topology. I am going to have to work out a proof of what Hocking and Young think is obvious. I am helped by yet another reference in the Wikipedia article, though a reference to which the link was dead, till I revived it: Allen Hatcher, Algebraic Topology.
We started with a lump of clay and rolled it out into a snake. Alternatively, we pulled two horns from it. Now continuing, from each horn we pull two more horns, ad infinitum.
More precisely, letting the initial lump of clay be B_{0}, we pull two horns out of it to get B_{1}. In general, B_{n} will have 2^{n} horns, and when we make each horn into two, we get B_{n+1}. Each B_{n} is closed and bounded, hence compact.
Perhaps ordinary clay is incompressible, so that our original lump retains its volume throughout the reshaping. We prefer the volume to grow, as if we add horns at each step. Let us say that B_{n} has 2^{n} horns, one for each binary sequence σ of length n. Let the corresponding horn itself be called H_{σ}. To obtain B_{n+1}, to each H_{σ} we attach the horns H_{σ0} and H_{σ1}. Now we have a strictly increasing chain:
B_{0} ⊂ B_{1} ⊂ B_{2} ⊂ …
We may think of B_{n} as a balloon filled with air; when we lower the pressure outside, the balloon expands from B_{n} into B_{n+1}.
Since it depends on physical intuition, this description is perhaps not “universally valid.” We could write down equations in three variables, defining the surfaces of the B_{n}. Then inequalities would define the solids themselves.
There is a homeomorphism h_{n} from B_{n} to B_{n+1}. We “could” define it precisely, but we don’t want to bother. Still there are conditions it must satisfy. For each n, we have that the composite function
h_{n}∘ … ∘h_{0}
is a homeomorphism from B_{0} to B_{n+1}. We want the sequence
(h_{0}, h_{1}∘h_{0}, h_{2}∘h_{1}∘h_{0}, …)
of these functions to converge uniformly to an injective function h. This is not automatic; we have to choose the h_{n} right. The uniform convergence will imply that h itself is continuous. Every closed subset of B_{0} is bounded and therefore compact, so its image under h is compact and therefore closed. Thus the inverse of h is also continuous, so h is a homeomorphism onto its image.
We call that image B. Now this is homeomorphic to the original B_{0}. Why then have we bothered to create B? We want the complement of B in space not to be homeomorphic to the complement of B_{0}. We achieve this by attaching in two steps to H_{σ} the horns H_{σ0} and H_{σ1}, for each binary sequence σ of length n, for each n:
Attach a “handle” to H_{σ}.
Cut out part of the handle, leaving the horns H_{σ0} and H_{σ1}.
The cut-out part will include the handles that will be attached to the horns H_{σ0} and H_{σ1} in the next step. Thus B_{n} gets 2^{n} handles attached, resulting in X_{n}. These form a decreasing chain of compact sets. In particular, we now have
B_{0} ⊂ B_{1} ⊂ B_{2} ⊂ … ⊂ B ⊂ … ⊂ X_{2} ⊂ X_{1} ⊂ X_{0}.
It is not automatic that B is the intersection of the X_{n}, but we can have ensured that this will be so.
The key move really needs three dimensions, and the pictures just above don’t handle this. For each n, for each σ of length n, the handles attached respectively to the horns H_{σ0} and H_{σ1} should be interlocking. This is to ensure that
the complement of X_{n+1} is not simply connected,
every loop in the complement of X_{n} that is null-homotopic in the complement of X_{n+1} was already null-homotopic in the complement of X_{n}.
If one accepts this, then the complement of B also fails to be simply connected; for, by the compactness of X_{0}, any loop in the complement of B that is null-homotopic must already have been null-homotopic in the complement of one of the X_{n}, and we know that that complement has loops that are not null-homotopic. Indeed, suppose we are given a homotopy of loops in the complement of B, namely a continuous function f from the square [0, 1] × [0, 1] into the complement of B such that f(0,0) = f(0,1) and f(1,0) = f(1,1). Let K be the image of the square under f. Then K ∩ B is empty, but is the intersection of the descending chain of closed subsets K ∩ X_{n} of the compact set X_{0}. Therefore, by compactness, some set in the chain must be empty. Thus f is a homotopy in the complement of some X_{n+1}.
]]>I say that mathematics is the deductive science; and yet there would seem to be mathematicians who disagree. I take up two cases here.
In November of 2017 I presumed, as an amateur, to write a post called “What Philosophy Is.” From his Essay on Philosophical Method (Oxford, 1933), I quoted R. G. Collingwood on what is commonly understood as Plato’s “doctrine of recollection.” Bertrand Russell was dismissive of this doctrine, in A History of Western Philosophy (New York: Simon and Schuster, 1945), as I had recalled in a post on the first chapter of Collingwood’s New Leviathan (Oxford, 1942); but in the Essay, Collingwood avers sensibly,
in a philosophical inquiry what we are trying to do is … to know better something which in some sense we knew already.
Thus I didn’t think philosophy should be defined as the taking up of certain specific questions, no matter how “general” or “fundamental” in form, such as “What is consciousness?” What matters is what you are trying to do with your questions. “There is only one philosophical question,” I said: “‘What are we going to do now?’”
What I am going to do now is continue the activity of my most recent post, the activity of asking “What Mathematics Is.”
I say in that post that mathematics is the science whose findings are proved by deduction. Deduction is reasoning that is valid universally. At least the reasoning is intended to be so. I don’t tell you that my mathematical proof is correct for everybody, for all time, everywhere; I aim to make it so. I may fail. If you think I have, you tell me, and I reconsider, or I try to tell you what you have missed.
As evidence about mathematics, I used a talk of unknown date by Euphemia Lofton Haynes. There is always more evidence. Now I want to consider two mathematicians who think our subject is not purely deductive, or else do not distinguish deduction as something special.
I shall take up the words of Doron Zeilberger at the end; meanwhile, according to Vladimir Arnold in a 1997 address,
Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.
The Jacobi identity (which forces the heights of a triangle to cross at one point) is an experimental fact in the same way as that the Earth is round (that is, homeomorphic to a ball). But it can be discovered with less expense.
Delivered in Paris, the address is called “On teaching mathematics”; the full citation is, Arnolʹd, V. I., “On the teaching of mathematics” (Russian), Uspekhi Mat. Nauk 53 (1998), no. 1(319), 229–234; translation in Russian Math. Surveys 53 (1998), no. 1, 229–236.
We shall look presently at Arnold’s mathematical example. Meanwhile, writing in Mathematical Reviews, V. Ya. Kreinovich seems right to point out that the “impact” of Arnold’s article
may be somewhat lessened by arguable statements such as “mathematics is a part of physics” (this is actually the very first sentence of the paper). Surely, the author is right that geometry and physics often clarify and simplify the understanding of mathematical notions, but he may be overemphasizing physics, as other mathematical notions may be better understood in the context of applications to biology, computer science, cryptography, etc.
I would go further, saying that the mathematics illuminated by a science that applies it is not part of that science, be it physics, biology, or any other.
Why should it matter to say this? Arnold’s next paragraph reads:
In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy’s warning that ugly mathematics has no permanent place under the Sun).
This may be correct as history, and beneficially pointed out, as Kreinovich allows in his review:
The author gives a lot of (mainly anecdotal) evidence that there exists a tendency to teach mathematics in an unnecessarily abstract form, without giving students motivation and intuitive examples, and that this tendency often decreases the quality of mathematical education.
Nonetheless, though they may have been indistinguishable for much of their history, physics and mathematics are different. An example of the difference is seen in
the discovery of Lobachevskian or hyperbolic geometry, and
the discovery (attributed to Beltrami) that each of Lobachevskian and Euclidean geometry proves the consistency of the other, regardless of whether any one of them fits the physical world.
Arnold knows these discoveries. In referring to the Jacobi identity and the concurrence of the altitudes of a triangle, he alludes to his own work in hyperbolic geometry. In a paper dedicated to Arnold’s memory, Nikolai V. Ivanov writes that Arnold
showed that the Jacobi identity
[[ A, B], C] + [[B, C], A] + [[C, A], B] = 0
lies at the heart of the theory of altitudes in hyperbolic geometry. In his approach, Arnolʹd used the Jacobi identity for the Poisson bracket of quadratic forms on R^{2} endowed with its canonical symplectic structure.
I quote these words, from “Arnolʹd, the Jacobi identity, and orthocenters,” Amer. Math. Monthly 118 (2011), no. 1, 41–65, mainly to illustrate the sentence that follows them:
Unfortunately, the use of these advanced notions renders Arnolʹd’s approach nonelementary.
That the altitudes of a triangle in the Euclidean plane have a common point, which we now call the orthocenter, was supposed to be known to the readers of the Book of Lemmas, which survives only in Arabic, but is attributed to Archimedes. The proof of Proposition 5, about the arbelos, calls on the existence of the orthocenter. I discussed all of this in “Thales and the Nine-point Conic,” § 2.1.
The same section of that paper reviews a Euclidean proof that the medians of a triangle also concur. Archimedes proved it by physical considerations in On the Equilibrium of Planes.
In an exercise that could be done in school, though I don’t remember doing it myself, if you cut a triangle out of heavy cardboard, and you hang from a vertex both the triangle itself and a plumb-line, then that line will lie along a median of the triangle.
For the same reason, if you hold the line taut horizontally, and you let a median of the triangle lie along the line, then the triangle will balance, albeit unstably. That is true by symmetry, if the median is perpendicular to the side that it meets, so that the triangle itself is isosceles; but in any case, if you cut the triangle into thin strips, parallel to the side met by the median, then the median crosses the center of each strip, which therefore balances at the median; so the whole triangle balances. That is the idea of the proof of Archimedes.
Moreover, the triangle must balance at some point along the median; and then all three medians concur at that point, which is the center of gravity or centroid of the triangle.
Strictly, the strips of the triangle are sure to balance along the median, if you cut triangles from their ends with cuts that are parallel to the median, so that the remainder is a parallelogram. However, you can make the area of the sum of all of those cut-off triangles as small as you like. This would yield a contradiction, if the center of gravity were not on the median.
ΑΔ is a median of triangle ΑΒΓ.
If the centroid of the triangle lies elsewhere, at Θ,
then we can halve ΔΓ, and halve the halves, and so on,
till one division point is closer to ΑΔ than Θ is.
Then the parallelograms ΜΝ, ΚΞ, and ΖΟ together
have their centroid at some Ρ on ΑΔ,
and the remainder of ΑΒΓ must have its centroid out at Χ.
But then
ΡΘ : ΡΧ :: ΣΜ : ΔΓ,
so Χ is beyond Φ, which is absurd.
Archimedes thus uses the technique that we know today as calculus. Euclid before him used the technique to prove that circles are in the ratio of the squares on their diameters: the theorem that today we write blithely as
A = πr².
I wrote in “Salvation” about a misunderstanding of Euclid’s theorem and ratios in general in a recent popular book about calculus. The book or its promoters praised calculus for the technology it made possible. I praised mathematical deduction, for being able to save the world the way Jane Austen can.
Possibly Archimedes discovered by trial the concurrence of the medians of a triangle. Then he proved it, using simple postulates about balancing, in addition to Euclidean geometry. His proof is deductive, like (presumably) Arnold’s proofs about the orthocenter.
By Collingwood’s account in the Prologue of Speculum Mentis (Oxford, 1924), art and philosophy won their independence from religion in the Renaissance. You may judge this independence a catastrophe, as people may do who are nostalgic for the middle ages; but the three pursuits are in fact distinct from one another.
Just so are mathematics and physics distinct. Proof in mathematics is deductive; physics, inductive. There may be a deductive element of a physical proof, but the ultimate test is agreement with experience, and such agreement is always provisional. As Arnold says, one may produce a mathematical model, as for weather forecasting, and draw conclusions from this; and yet,
It is obvious that in any real-life activity it is impossible to wholly rely on such deductions. The reason is at least that the parameters of the studied phenomena are never known absolutely exactly and a small change in parameters (for example, the initial conditions of a process) can totally change the result.
That’s fine. Arnold goes on to say,
In exactly the same way a small change in axioms (of which we cannot be completely sure) is capable, generally speaking, of leading to completely different conclusions than those that are obtained from theorems which have been deduced from the accepted axioms. The longer and fancier is the chain of deductions (“proofs”), the less reliable is the final result.
I don’t know what kind of “small change” there can be in axioms. I think normally the researcher who is curious to see what will happen will change an axiom to its negation. This is what Lobachevsky does.
Proposition 17 of Book I of Euclid’s Elements is that any two angles of a triangle are together less than two right angles. The converse is Euclid’s fifth and last postulate—or axiom, if you like— , the parallel postulate: only in a triangle can two facing angles sharing a side be together less than two right angles.
In “Geometric Researches on the Theory of Parallels” (Geometrische Untersuchungen zur Theorie der Parallellinien, 1840), Lobachevsky assumes the negation of the parallel postulate, leaving the rest of Euclidean geometry intact. In this case, for any particular distance a, there is an acute angle θ such that, if a perpendicular is erected at a point B of a line BC, and a point D on the perpendicular is at distance a from the point B, then lines through the point D making
less than angle θ with the line DB will cut BC;
angle θ itself or greater with the line DB will not cut BC.
Denoting θ by Π(a), Lobachevsky shows that, for some distance u, for every a,
cot (Π(a)/2) = exp (a/u).
In Euclidean geometry, Π(a) is always a right angle, and the cotangent of half of this is 1, which is exp (0/u). Thus Lobachevskian geometry is the more Euclidean, so to speak, the smaller the distances being considered. As Lobachevsky says,
the imaginary geometry passes over into the ordinary, when we suppose that the sides of a rectilineal triangle are very small.
One might also say that “imaginary” or Lobachevskian geometry approaches “ordinary” or Euclidean geometry as u increases without bound; also, a small change in u creates big changes at large-enough distances. This may be the kind of thing that Arnold is referring to, when he says, “a small change in axioms … is capable, generally speaking, of leading to completely different conclusions …” However, Lobachevsky has no such notation as my u; he effectively lets it be the unit of measurement.
When Arnold remarks parenthetically of axioms that “we cannot be completely sure” of them, I have to say that I thought we had learned from Lobachevsky not to expect to be “sure” of axioms anymore. So I don’t know what Arnold is talking about here. Of course we can fail in our mathematical deductions, as a model can fail to predict tomorrow’s thunderstorm, or a pipeline can fail to deliver water. These are not all “exactly” the same kind of failure. (I argued this in “Anthropology of Mathematics.”)
To be able to declare a vaccine safe, you test it on what is assumed to be a representative sample of people, while knowing that your assumption is not 100% correct. To be able to declare that there are no odd perfect numbers at all, there is no representative sample to be tested, unless you can prove it by deductive reasoning.
Mathematical theorems are not sought with such reasoning. Reason is a tool, not for discovery, but for confirmation. How we discover theorems is mysterious. It may involve induction, in the sense of generalization from special cases. Arnold suggests that it always does, but that’s a stretch:
The scheme of construction of a mathematical theory is exactly the same as that in any other natural science. First we consider some objects and make some observations in special cases. Then we try and find the limits of application of our observations, look for counter-examples which would prevent unjustified extension of our observations onto a too wide range of events (example: the number of partitions of consecutive odd numbers 1, 3, 5, 7, 9 into an odd number of natural summands gives the sequence 1, 2, 4, 8, 16, but then comes 29).
Here it will be valuable to look at what Victor V. Pambuccian says in Mathematical Reviews about Ivanov’s article on Arnold’s work.
Pambuccian has reviewed also an article of mine, “Abscissas and Ordinates.” I am glad that Mathematical Reviews was willing to ask somebody to do this, and that Pambuccian was willing to take on the job. He was succinct:
The paper is a walk through the Greek origin of the words “abscissa”, “ordinate”, “upright side” and “latus rectum” of a parabola, “transverse side” of a hyperbola—interspersed with lexicographic and literary (Blake, Orwell) asides—focusing on their meaning in Apollonius’ Conics.
That’s the whole review. I might have wished to hear more about those literary asides, which concerned the treatment of children. Sometimes Mathematical Reviews just prints an author’s own summary of their paper, and my summary was,
In the manner of Apollonius of Perga, but hardly any modern book, we investigate conic sections as such. We thus discover why Apollonius calls a conic section a parabola, an hyperbola, or an ellipse; and we discover the meanings of the terms abscissa and ordinate. In an education that is liberating and not simply indoctrinating, the student of mathematics will learn these things.
Pambuccian writes more than two pages (size A4) on Ivanov’s article, mostly explaining what Ivanov and Arnold leave out. In a way to suggest what is missing from Arnold’s account of the “scheme of construction of a mathematical theory,” here is how Pambuccian sets out:
There are two distinct traditions in mathematics. The first one believes in the unity of all existing mathematics and values very highly unexpected uses of results and techniques of one area of mathematics in another, preferably distant, one. Practitioners of this tradition who look at geometry tend to believe that the continuity of space is one of its essential attributes, and thus believe in an intimate connection between geometry and the real numbers (understood as a unique structure, insensitive to potential disturbances of its uniqueness coming from set theory), take advantage of the manifold benefits to be had from its topological or differential structure, or else assume that the geometry in question is built over special classes of fields, to unleash the powerful results of algebraic geometry.
The second, and oldest one, looks at geometry (and by extension, at any area of mathematics) as consisting of statements on the words of a language, that are not there to be verified in an algebraic or a continuous realm, but to be deduced from other such statements, the main aim of this undertaking being the deduction of statements from the weakest possible set of some special statements that are endowed with a certain hard-to-define quality of simplicity.
Arnold sets himself in the first tradition here by saying,
These discoveries of connections between heterogeneous mathematical objects can be compared with the discovery of the connection between electricity and magnetism in physics or with the discovery of the similarity in the geology of the east coast of America and the west coast of Africa.
The emotional significance of such discoveries for teaching is difficult to overestimate. It is they who teach us to search and find such wonderful phenomena of harmony in the universe.
It is inductive, the initial speculation that “heterogeneous” objects may nonetheless be instances or consequences of one law. One may then be able to proof a theorem about such a law, deductively.
Does this scheme account for one of the first mathematical theorems? According to Proclus in his Commentary on the First Book of Euclid’s Elements (translated by Glenn R. Morrow; Princeton University Press, 1970),
The famous Thales is said to have been the first to demonstrate that the circle is bisected by the diameter.
This theorem does not seem like a generalization from the observation of special cases. As Proclus goes on to say,
If you wish to demonstrate this mathematically, imagine the diameter drawn and one part of the circle fitted upon the other. If it is not equal to the other, it will fall either inside or outside it, and in either case it will follow that a shorter line is equal to a longer. For all the lines from the center to the circumference are equal, and hence the line that extends beyond will be equal to the line that falls short, which is impossible. The one part, then, fits the other, so that they are equal. Consequently the diameter bisects the circle.
Fold a circle along a diameter, and the two halves match up. If you remark on a single occurrence of of this, would you need to try again, in order to be sure that it will always happen?
Perhaps not; but you still have generalized from some number of special cases, at least if you allow one to be a number. Collingwood observes the possibility of such generalization, albeit in a different context:
There is nothing in recurrence that is not already present in the single instance. Indeed some determinists have argued that because a certain man once did a certain action, therefore he was bound to do it. This seems a reductio ad absurdum; and yet if we can argue from frequency to necessity, the question “How often must a thing happen before you know it was bound to happen?” can have only one answer:—“Once is enough.”
That is from the last chapter of Religion and Philosophy (London: Macmillan, 1916). Collingwood is proving that there is no miracle in the the sense of a divine interference with the ordinary course of nature; everything is miraculous that we can see as such, because miracles are those events that “compel us to face reality as it is, free, infinite, self-creative in unpredicted ways.”
I took up Collingwood’s treatment of miracles in “Effectiveness,” which was mainly about Eugene Wigner’s 1960 article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” an article that Arnold also brings up.
When Thales realized that the two halves a circle must match up, perhaps this was like noticing that the coastlines of Africa and the Americas match up. A difference is that the matching of the two halves of a circle is a theorem, θεώρημα, literally a spectacle, something to be seen, as in a theater, θέατρον. The matching of coastlines is a random occurrence, until explained by a theory such as plate tectonics.
Religion and Philosophy was Collingwood’s first book, published when he was 27. The first chapter argues
Theology = philosophy = religion.
That is actually how the Index summarizes several pages. Later Collingwood disavowed such identifications and developed the doctrine of the “overlap of classes,” expounded in An Essay on Philosophical Method. You can find things, such as art and craft, bound up together in every observable instance; it doesn’t make them the same. They need not agree just in “extension,” while differing in “intension”; they may overlap in intension. Collingwood’s example is from ethics:
Duty rejects expediency in the sense of refusing to accept it as even a legitimate kind of goodness, and regarding it rather as the inveterate enemy of morality, but reaffirms it in the sense of accepting it, when modified by subordination to its own principles, as a constituent element in itself. Thus duty and expediency overlap … the overlap is essentially not … an overlap of extension between classes, but an overlap of intension between concepts, each in its degree a specification of their generic essence, but each embodying it more adequately than the one below.
Collingwood alludes here to a progression that he will take up in the “Reason” chapter of The New Leviathan. Historically, we have justified our behavior, first as expedient, then as right, and finally as dutiful.
We might suggest then that the recognition of several divisions within what was formerly called natural philosophy is a progression. The several sciences grew up together, and they are still all studied at single universities; but this is done in different departments.
Members of different departments will publish their research in different ways. Our mathematical findings as such will have no physical component; thus we must justify them deductively, as Archimedes justified his theorem locating the centroid of a triangle.
There is however a mathematician who objects to this requirement. On his webpage he publishes a lot of his opinions, and one of these opinions is that the Euclidean deductive approach ruined mathematics and justified absolutism in politics. See for yourself.
Zeilberger is evidently one of those persons who enjoy being controversial. Here’s an article of his, “What is Mathematics and What Should it Be?” saying,
Today’s Mathematics Is a Religion
Its central dogma is thou should[st] prove everything rigorously.
Call it a dogma, or call it, following Collingwood, an absolute presupposition: the requirement of rigorous deductive proof is not a restriction, but a platform for discovery.
According to Zeilberger, “Scientists, by definition, are trying to discover the truth about the outside world.”
We are trying to find the truth, I would say. That there is an “outside world” at all is a dogma, or absolute presupposition, of many sciences, but not of mathematics.
One may say that what we study in mathematics is outside us, but in a different way from the physical world.
Says Zeilberger,
Mathematicians do not care about discovering the truth about the mathematical world. All they care about is playing their artificial game, called [rigorous] proving, and observing their strict dogmas.
I almost said that was true, until I noticed that Zeilberger said not “outside world,” but “mathematical world.” The second sentence is true, though again Zeilberger introduces a needless qualification. Every game is artificial. Zeilberger is welcome to play another.
]]>Mathematics “has no generally accepted definition,” according to Wikipedia today. Two references are given for the assertion. I suggest that what has no generally accepted definition is the subject of mathematics: the object of study, what mathematics is about. Mathematics itself can be defined by its method. As Wikipedia currently says also,
it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.
I would put it more simply. Mathematics is the science whose findings are proved by deduction.
That is my definition. It has only science in common with a dictionary definition:
măthėmă′tĭcs n. pl. (also treated as sing.) (Pure) ⁓, abstract science of space, number, and quantity; (applied) ⁓, this applied to branches of physics, astronomy, etc.; (as pl.) use of mathematics in calculation etc.
That’s the Concise Oxford Dictionary (6th ed., 1976). The Grolier International Dictionary (1981) does not even refer to science:
math⋅e⋅mat⋅ics (măth′ə-măt′ĭks) n. Abbr. math. Plural in form, used with a singular verb. The study of number, form, arrangement, and associated relationships, using rigorously defined literal, numerical, and operational symbols.
Presumably the dictionaries reflect the common view. People think mathematics is numbers, because that is all they learn about it in school. The teaching of more than this is urged by Euphemia Lofton Haynes, who says,
Mathematics is no more the art of reckoning and computation than architecture is the art of making bricks, no more than painting is the art of mixing colors.
This is in an address to high-school teachers, probably in Washington, D.C. The typescript, which I have digitized, is undated, but Haynes’s dates are 1890–1980.
The ideal example of an explanation of a pursuit is Collingwood’s 1938 book, The Principles of Art (1938). I talked about Collingwood’s procedure in the pdf booklet that is linked to from the post about art and mathematics called “Discrete Logarithms.” In the middle of his book, Collingwood offers the definition,
By creating for ourselves an imaginary experience or activity, we express our emotions; and this is what we call art.
He immediately points out:
What this formula means, we do not yet know. We can annotate it word by word; but only to forestall misunderstandings …
In my formula for mathematics (“the science whose findings are proved by deduction”):
“Science” is used in the general sense of a system of knowledge. The Latin root scient- means knowing, more precisely discerning or separating out. The same root is found in our scission, though not scissors, whose initial ess was added by analogy, because the French original is the plural cisoires of cisoir, from the Latin cisorium, from the verb caedo, meaning cut and giving us words like decide and homicide.
Mathematics is not a deductive science, but the deductive science. Other sciences may use deductive arguments, but not exclusively. For example, a friend pointed out to me an article, Ada Marinescu, “Axiomatical examination of the neoclassical economic model. Logical assessment of the assumptions of neoclassical economic model,” Theoretical and Applied Economics, Volume XXIII (2016), No. 2(607), Summer, pp. 47–64:
We analyze in this paper the main assumptions of the neoclassical theory, considered as axioms, like the rationality of the economic actor, equilibrium of the markets, perfect information or methodological instrumentalism … Correspondence with reality matters less in this strictly abstract approach compared to the possibility to build a coherent and convincing system.
Correspondence with reality may matter less; I think it still matters, unless one is going to say that neoclassical economics is simply mathematics.
The “findings” of any science may be found by any means. The means of finding them may not, and usually will not, be the same as the method of proving them.
“Deduction” is universally valid reasoning. That mathematical proofs are universally valid is a “metaphysical proposition,” in the sense of Collingwood’s Essay on Metaphysics (1940). This means the proposition carries implicitly the “metaphysical rubric,” namely, “Certain persons believe.” As mathematicians, we believe our proofs to be universally valid. In the same way, Anselm proves not that God exists, but that we believe it:
Whatever may have been in Anselm’s mind when he wrote the Proslogion, his exchange of correspondence with Gaunilo shows beyond a doubt that on reflection he regarded the fool who ‘hath said in his heart, There is no God’ as a fool not because he was blind to the actual existence of un nommé Dieu, but because he did not know that the presupposition ‘God exists’ was a presupposition he himself made.
In mathematics, we believe not that our proofs are universally valid, but that they can be and ought to be, and if they are not, then they are not proofs.
“Deduction” can be called “deductive logic.” The address by Euphemia Lofton Haynes is called “Mathematics—Symbolic Logic.” The speaker discusses two “scientific movements,” originating in the 19th century:
in mathematics, to criticize the foundations of calculus and geometry;
in logic, to overcome the inadequacy of Aristotelian logic.
Says Haynes,
Thus originating in apparently distinct domains and following separate parallel paths these two scientific movements led to the same conclusion, viz. that the basis of mathematics is the basis of logic also. Symbolic logic is Mathematics and Mathematics is Symbolic Logic.
On the page with the transcript, I mention that:
Haynes is said to have been the first African American woman to receive a doctorate in mathematics, in 1943 at Catholic University of America;
I learned of her address to teachers through a certain Twitter account.
That account is called Great Women of Mathematics, and it was denounced, earlier this year, apparently for not affirming the slogan “Trans women are women.” Perhaps there were additional failures to toe a line. Twitter users were called on to unfollow the account. Had I heeded the call, I might not have learned of Haynes’s address, or even of herself. Let me note:
I try to follow Twitter accounts that I disagree with, at least if they offer actual arguments (either directly or through links).
I haven’t got great expectations for dialogue on Twitter with people I disagree with; however, I have hoped mathematicians would be different, because, as I say, we aim for universality in our arguments. This means we may be wrong, if somebody disagrees with us.
I had an argument with two other mathematicians over the proposed cancellation of Great Women of Mathematics. I shall not try to review the argument. I do have two related blog posts, called “Sex and Gender” and “Be Sex Binary, We Are Not,” composed respectively before and after the argument. I agree with Brian Earp in his 2016 essay, “In praise of ambivalence—‘young’ feminism, gender identity, and free speech”:
… “no-platforming” can be a bad idea … even when the person you want to exclude is a dyed-in-the-wool ideological opponent …
The political theorist Rebecca Reilly-Cooper, herself a controversial figure in this debate, argues that there is “a creeping trend among social justice activists of an identitarian persuasion” towards what she calls ideological totalism.
This is “the attempt to determine not only what policies and actions are acceptable, but what thoughts and beliefs are, too” …
My worry is that such thought-policing, to the extent that it exists, is unlikely to achieve its aims in the long run.
My aim here is to define mathematics. I say that its findings are proved by deduction. We usually refer to a deductive proof as a proof, simply. However, in her address, Haynes uses the word “proof” only in the context of physical science:
Let us consider for a moment a teacher of physics or chemistry. In order to examine the validity of an hypothesis, he does not rush through the proof, i.e. the experiment, concentrating on the findings, and spending the major portion of his time trying to discover whether or not his pupil has retained the results or findings. No …
I have put “and spending” where the typescript as “but spends.” I think the kind of proof that Haynes refers to is inductive, in one of the senses that I discussed in the post “On Gödel’s Incompleteness Theorem.” Haynes uses the term “incomplete induction” for such proofs, as when she says of the mathematician,
His processes and methods are similar to those of his colleagues in natural science. The importance of observation, experimentation, testing of hypotheses by the laboratory method, and incomplete induction cannot be too strongly emphasized.
I assume the qualifier “incomplete” is meant to distinguish this kind of induction from specifically mathematical induction, which in spite of its name is a method of deductive proof. In mathematics, we may be given a set of counting numbers, such as the set of all n such that twice the sum 1 + … + n is the product of n with its successor n + 1. If that set contains
the successor of its every element, and
1 itself,
then the set contains every counting number. Since indeed
2 ⋅ 1 = 1(1 + 1) and,
if, for some k,
2(1 + … + k) = k(k + 1),
then
2(1 + … + (k + 1)) = 2(1 + … + k) + 2(k + 1)
= k(k + 1) + 2(k + 1) = (k + 1)(k + 2),
we conclude that, for all counting numbers n,
2(1 + … + n) = n(n + 1).
That’s a proof. We call it a proof by induction, but it’s a deductive proof. There’s nothing missing, nothing left to prove; the proof is “complete.” The method might then be called “complete induction,” as I find that it is in one source:
The method of mathematical (or complete) induction is a very strong tool in mathematical proofs. Unfortunately, in secondary school it does not receive the attention which it deserves. Most students have a rather hazy idea concerning this important method.
That’s from page 21 of Dorofeev, Potapov, and Rozov, Elementary Mathematics: Selected Topics and Problem Solving (Moscow: Mir Publishers, 1973). My teacher Donald J. Brown had us use the text for precalculus at St Albans School in Washington, D.C.
I assume George Yankovsky, translator of the Soviet book, knew what he was doing in using the word “complete” to describe mathematical induction. Nonetheless, looking around, I see that some people use the term “complete induction” as an alternative to “strong induction.” This is the method of proof whereby a set of counting numbers contains all of them if it contains
the successor of every number for which the set contains all numbers up to and including that number, and
1 itself.
The two conditions collapse to one, if you follow the grammar: a set of counting numbers contains all of them if it contains
every number than which it contains all numbers less.
The condition of containing 1 then follows, since every set contains every counting number that is less than 1, there being no such numbers.
I find a webpage actually called “Complete Induction,” although it is only about the method that I just described as strong induction. The page is part of an online textbook on the foundations of mathematics, and the previous page in the text is called “Mathematical Induction.” Unfortunately the text teaches the same error that Donald Brown taught us from Spivak’s Calculus: the error that ordinary mathematical induction and “complete” or strong induction are equivalent conditions on the natural numbers. They are equivalent, in the sense that they are both true of the natural numbers; but what is meant is that you can prove either of the conditions after assuming the other. The proofs make tacit assumptions that ought to be explicit. Ordinary induction involves only 1 and the operation of succession, while strong induction involves a linear ordering. The explicit assumptions for the putative proofs of equivalence are that
different numbers have different successors;
1 is the successor of no number.
If we make a third assumption, namely
then we can indeed prove that the counting numbers are linearly ordered; however, the takes a lot of work, which is usually forgotten. One approach is that of Landau, whose Elementary Number Theory I used for my last post, “LaTeX to HTML.” In The Foundations of Analysis (1929/1966), given the three axioms above, Landau proves that there is a unique operation of addition with the property
n + (k + 1) = (n + k) + 1.
Then he can define the relation “<” by the rule
k < n ⇔ ∃x k + x = n.
Alternatively, one can prove the “Recursion Theorem,” or at least the special case that every natural number has a well-defined set of predecessors according to the rules,
1 has no predecessor,
the predecessors of n + 1 are precisely n and its predecessors.
Now one can define k < n to mean precisely that k is a predecessor of n.
Either way, one goes on to prove that “<” is a linear ordering and even a “well-ordering,” meaning every nonempty set of numbers has a least element. This is equivalent to strong induction, at least in the compact form with the single condition that I stated.
Conversely, given a nonempty well-ordered set with no greatest element, we can call
its least element 1,
the least of the elements greater than n the successor (namely n + 1) of n
Every limit ordinal in the sense of my post “Ordinals” is a nonempty well-ordered set with no greatest element, but only ω is isomorphic to the set of counting numbers.
There is more discussion of the logic here in Example 1.2.3 (pages 37–8) of Model Theory and the Philosophy of Mathematical Practice (Cambridge University Press, 2018), by John Baldwin, who calls the general problem my paradox (that is, “Pierce’s paradox”). I mentioned the so-called paradox in “Anthropology of Mathematics,” suggesting that we may learn some things when we are too young to question them, then teach them when we are older without going back to question them.
According to Euphemia Lofton Haynes, as in a quotation I made earlier, we mathematicians also use incomplete induction. Her example is the Pythagorean Theorem:
It was observation of the fact that the squares of certain numbers are each the sum of two other squares; the collection of these sets of numbers by the method of trial; the observation that apparently these and only these triplets are the measures of the sides of a right triangle—that is, observation, experimentation, incomplete induction—processes common to the experimental sciences—that led to the discovery of the Pythagorean Theorem.
That is how I would edit the typescript, which actually reads as follows:
It wasby observation of the fact that the squares of certain numbers are each the sum of two other squares; the collection of these sets of numbers by the method of trial; the observation that apparently these and only these triplets are the measures of the side of a triangle. That is by obser- vation, experimentation, incomplete induction processes, common to the experimental sciences, led to the discovery of the Pythagorean Theorem.
The account is plausible. I’m not sure I didn’t offer a similar account of the Pythagorean Theorem, when an English teacher at St Albans tried to explain the distinction between deductive and inductive logic. I may have been in the eighth grade; at any rate, my classmates and I were too young to have seen official proofs in mathematics. We must have been taught the Pythagorean Theorem somehow, so that our English teacher could give it as an example of a general rule from which specific cases could be deduced. He may even have said that the rule itself was established deductively. In my memory at least, I responded that the rule must have been discovered inductively.
Perhaps before hypothesizing the Pythagorean Theorem itself, in all of its generality, somebody did observe that 3^{2} + 4^{2} = 5^{2}, and 5^{2} + 12^{2} = 13^{2}, and 8^{2} + 15^{2} = 17^{2}, and that the bases in each equation were sides of a right triangle. The observation about the sides of a right triangle may have come not from measuring with a ruler, but from pictures. Pictures make it clear that the triangles are indeed right-angled. But then a single picture can also serve as a proof of the general theorem, and it is conceivable that one may thus discover the theorem, without having considered particular triangles whose sides one knows the measures of.
Likewise may a picture replace our inductive proof that for all counting numbers n,
2(1 + … + n) = n(n + 1).
Here’s the picture:
An array of n rows, each row consisting of n + 1 dots, contains n(n + 1) dots in all. It can also be broken into two triangles as indicated, each triangle consisting of 1 + … + n dots.
You may say that’s not a proof. Dorofeev, Potapov, and Rozov say it’s not. The picture establishes the assertion, only for a special case.
The incompleteness of [such a] proof is obvious. We establish the formula for a few values of n and then draw the conclusion that it is true for any [counting number] n. With that approach, it is possible to “prove” the following assertion: for an arbitrary integer n, n^{2} + n + 41 is prime. Indeed, for n = 1, 2, 3, 4 we have 43, 47, 53, 61—all primes. “Consequently”, the assertion is proved, though it is clear that, for example, when n = 41 the number n^{2} + n + 41 is divisible by 41.
Our picture proof that 2(1 + … + n) = n(n + 1) is based on the case when n = 5; but it should be obvious that there is nothing special about 5 here. This makes the proof different from the false proof of the primality of all numbers n^{2} + n + 41. We could also write our picture proof in algebraic form:
2(1 + … + n)
= (1 + … + n) + (1 + … + n)
= (n + … + 1) + (1 + … + n)
= (n + 1) + … + (n + 1)
= n(n + 1).
However, such a proof may be too obscure, needing too much intuition. In that case, one can always fall back on the proof by induction.
If I understand correctly, in the US there’s an attempt to teach arithmetic in a more intuitive way than by just applying the traditional algorithms. Thus for example to add 79 and 18, instead of first adding 9 and 8 to get 17, then adding the tens digit here to the sum of 7 and 1 to get 9, so that 79 + 18 = 97, you may do better to think
79 = 80 − 1,
18 = 20 − 2,
79 + 18 = 80 + 20 − (1 + 2) = 100 − 3 = 97.
That’s fine, but it seems to me one should have the fall-back algorithm of performing the addition digit by digit, right to left, as I described first. This provides a mechanism for resolving disputes about sums, as well as for not having to think and be creative, which nobody wants to do all the time.
We have suggested three proofs that 2(1 + … + n) = n(n + 1), not all of which may be accepted as proofs by everybody. I have suggested that some standard proofs found in textbooks are bogus. What then of my assertion that mathematical proofs are universally valid?
A proof is not a picture or an arrangement of typographical characters, any more than a work of art is pigment in oil on canvas. The physical things are just the means we use to understand the real thing.
How we come to see the real thing may itself be obscure. The Soviet textbook has some good comments here, though they be about seeing the assertion rather than its proof:
It must be stressed that the induction method is a method of proof of specified assertions and does not serve as a derivation of these assertions. For instance, this method cannot be used to obtain the formula of the general term [of an arithmetic progression or—let us add—of the sum of the first n counting numbers]; however, if we have found the formula in some way, say by trial and error, then the proof of it can be carried out by the induction method … In this process, of course, the method of trial and error, the mode of obtaining a formula or an assertion is not a necessary element of the proof. On the basis of some kind of reasoning or guessing we conjecture an assertion, then we can proceed to proof by induction.
This distinction between finding an assertion and proving it is one that we have seen Euphemia Lofton Haynes discuss for the natural sciences. How then is mathematics to be distinguished from, say, physics? According to Haynes,
Although a mathematical system is syllogistic and postulational in style and form, the assumption that syllogistic reasoning is the very foundation of all mathematical activity is another inherited fallacy which is the result of the reign of methodology.
I think another way to say “syllogistic and postulational” is deductive and axiomatic. Perhaps physics can be this, but the conclusions of the deductions still have to be checked, to see whether they fit the experimental data. By contrast,
the observation of the mathematician transcends that of the natural scientist in that it is not confined to observations of the physical eye …
The mathematician builds … worlds that are possible logically. Whether they are possible in our world of sense is of no concern to him.
That worlds are “possible logically” means they can be deduced from postulates. This is what distinguishes mathematics among the sciences. Mathematics is the deductive science.
Pages 22–3 of Joseph Needham,
Mathematics and the Sciences of Heaven and Earth,
Volume 3 of Science and Civilization in China
(Cambridge University Press, 1959),
concerning the Chou Pei Suan Ching or Zhoubi Suanjing
This is a little about mathematics, and a little about writing for the web, but mostly about the nuts and bolts of putting mathematics on the web. I want to record how, mainly with the pandoc
program, I have converted some mathematics from a LaTeX file into html
. Like “Computer Recovery” then, this post is a laboratory notebook.
The mathematics is a proof of Dirichlet’s 1837 theorem on primes in arithmetic progressions. This is the theorem that, if to some number you keep adding a number that is prime to it, there will be no end to the primes that you encounter in this way.
For some reason, I wanted to learn the proof. Maybe this had to do with having given courses on the Prime Number Theorem of 1896 at the Nesin Mathematics Village, but not having been able to teach there this summer, owing to the Covid pandemic. Dirichlet’s theorem could be part of a course at the Village.
I read the proof in Landau’s Elementary Number Theory, originally published in 1927, ninety years after Dedekind’s theorem. I wrote out the argument, according to my understanding. If you want to read what I wrote, there are:
a pdf file (24 pages, size A5) based on the LaTeX file that I composed;
an html file derived from my LaTeX file by means of the pandoc
program.
Recently I encountered a page of links to over two hundred expository articles by a mathematician. I looked at one article, and I wondered what kind of audience would both
need to be told that an automorphism of a field is a bijective homomorphism from the field to itself,
already know the field of p-adic numbers.
I might have thought somebody who knew the p-adic numbers would also know something of Galois theory; but maybe not.
In my own article on Dirichlet’s theorem, I had already given accounts of
what the reader should know,
what I know as an amateur of number theory.
I become increasingly aware of how webpages can be visited from anywhere, although their composers seem unaware of this. If you are a newspaper, what city are you in? If a university, what state or province? If a business, what country? If you are a blog, what are you trying to do, and how can your visitors decide whether to spend time with it?
I have tried to make my own “About” page useful in this way.
I wrote my article on Dirichlet’s theorem, to satisfy my curiosity. Then I remembered posting on this blog my memoir of life on a farm. I had obtained the html
file from a LaTeX file using pandoc
, and I had been pleased with the results. Notably, pandoc
had kept my footnotes as such: they were at the end of the same html
file.
I decided to try to convert the Dirichlet article to html
.
WordPress allows the embedding of TeX and LaTeX code, but converts the code to images. The tex4ht
program does this as well. I wanted to avoid images.
The pandoc
program does not create images, but tries to express mathematics (along with everything else) as text. However, there are mathematical expressions that LaTeX accommodates, but html
does not, or not so well. The pandoc
program leaves those untouched.
I therefore edited my original LaTeX file, to turn all of the mathematics into something that pandoc
could interpret. I tried to make it as easy as possible to switch between a LaTeX file as such—a file to be compiled by the latex
program—and a file to be converted to html
by pandoc
.
Here is what I did.
Apparently pandoc
can handle the commands that you define, even with arguments; but not default arguments. For example, my
\newcommand{\Zmod}[1][k]{\mathbb Z/#1\mathbb Z}
didn’t work until I removed [k]
.
I like the compactitem
and compactenum
environments of the paralist
package, but apparently pandoc
does not recognize these, so I switched back to itemize
and enumerate
.
Fractions are the big challenge. I used
\renewcommand{\frac}[2]{#1/#2}
.
One will then need to use parentheses if a numerator or denominator is a sum; but I had this problem in only one case. The reader still has to understand a/bc as meaning a/(bc), although I have allowed (a/b)c to appear as a/b⋅c (this happens when c is a summation with ∑).
Summations themselves are a problem; I rewrote each \sum_{j=i}^{n}
as \sum_{i\leq j\leq n}
. (One could also define a new command with three arguments here.)
For one use, I defined:
\DeclareMathOperator{\nCk}{C}
\renewcommand{\binom}[2]{\nCk(#1,#2)}
None of the environments align
, gather
, and multline
for displaying more than one line of mathematics together gets interpreted by pandoc
; therefore I put each line to be displayed into its own equation
environment.
There is a similar problem with the cases
environment; I recast using itemize
.
Apparently pandoc
cannot handle a negated symbol such as \not\equiv
, so I used \nequiv
from txfonts
.
I rewrote \pmod{#1}
as \;(\text{mod }#1)
.
Doing that much gave a file that pandoc
would render as pure html
. However, there were remaining issues.
The program was not dealing properly with the bibliography I had created with BibTeX. Also pandoc
gave my section headings <h1>
tags. I took care of this by running the following command:
pandoc --base-header-level=2 --bibliography ../../../references.bib --filter pandoc-citeproc dirichlet-simple.tex -o dirichlet-simple.html
The online documentation says --base-header-level=
is deprecated, and one should use --shift-heading-level-by=
; but this didn’t work for me. (If pandoc
didn’t come with my Ubuntu Linux installation, I may have installed it when converting the docx
file of somebody else’s philosophy paper to LaTeX.)
At this point, the problem remained that pandoc
did not deal properly with
theorem environments,
labels of equations.
I found discussion of these on the Google group for pandoc
. I have not understood why they should be a problem, or at least why references should be a problem.
For example, pandoc
prints the label of a theorem as plain text. It could print the label of an equation in the same way, but apparently it doesn’t.
One can apparently customize pandoc
, but for now I don’t know how. Therefore I have done the following (probably not something to be done on a regular basis):
formatted and numbered by hand my theorems and lemmas;
formatted the proofs by hand, inserting $\Box$
at the end if this is text, and \qquad\Box
if it’s an equation (pandoc
could not deal with \qed
);
changed \label{#1}
(which I habitually place just after \begin{equation}
) to \mylabel{#1}
, defined as (#1)\qquad
;
redefined \eqref{#1}
as (#1)
;
changed all of my equation labels to the desired serial numbers;
changed every {equation}
to {equation*}
.
After all of this, I cleaned up the html
file created by pandoc
by:
putting blank lines between paragraphs (for ease in reading the html
file);
changing <span class = "math display">
to <div style= "text-align:center;">
;
putting <div style = "text-align:justify; margin-left:10%; margin-right:10%;">
at the head, and </div>
at the foot, as with all of my blog posts and pages.
Now I am using pandoc
to create the present html
file from the plain text file that I originally typed.