about Carroll's math

[ermete22]

I am aware that math is not a popular subject and Carroll’s math is even less appealing than the one you find in the average mathematical book; many authors have criticised Carroll’s mathematical works; most of the critics are technically correct but fail in understanding what was mathematics for Carroll and how strongly was mathematics connected with logics in Carroll’s opinion. It is well possible that Carroll’s philosophy can, at least in some sense, be classified as neo-platonism (assuming it is important to always find some classification), but, for sure, his mathematics was also terribly concrete. Digging in his papers (which are very often almost impossible to completely understand for a modern mathematicians because of the use of a strange, personal language ) I believe of having found out that Carroll’s was totally insensible to what could be called “intermediate math”, that is 99% of what is discussed in mathematical papers; rather he was interested in the foundations (philosophically) and in applications (enigmistically). So discussing Carroll’s mathematics is equivalent to discussing about foundations and applications. Most of his applications are extremely bright while his discussions about foundations are feeble and, honestly, sometimes clearly wrong, as, being essentially a logician, he was implicitly looking for some demonstrations of the validity of the basic axioms, which of course do not exist.
Carlo

[johntufail]

Hi carlo.

This is why I have argued that Caroll increasingly moved from mathematics to logic (and thus philosophy?) as a discipline. There has always been confusion about the status of mathematics. Some argue that maths is a ascience in it's own right. This means that is able to reveal truths about the universe and the world in which we materially exist.

Others argue that it is nothing of the kind, that it is a language - only able to describe an infinity of possible truths - but completely unable to valididate which of this infinity is existentially real or valid.

Carroll, it seems to me, started off thinking that mathematics revealed 'truths' but gradually came to understand that the 'truths' they revealed were completely dependent on the premise on which the original question is based. He also came to believe that mathematics is a language, which as such, can only describe a reality partially, dependently and ultimately subjectively.

Hence his search for the 'validity' of basic axioms.

This is why many who read Caroll's mathematical works miss the point.

[bettyboop]

So, was Carroll trying to uncover some deep truth? Or just amusing himself? He seemed to approach mathematics almost like a kind of game. He enjoyed solving problems for the sake of doing so.

[ami]

I always feel very uneasy with general statements when it comes to Carroll's (or any other thinker's) philosophy, mathematics, logic, etc. I think that every text should be discussed within its context.
For instance, we can say with confidence that Lewis Carroll invented some mathematical games but one cannot say that he conceived mathematics as a game because he also wrote many very serious texts.
In the same manner, when we discuss the relationship of mathematics and logic in his work, we always have to explain which mathematics and which logic we are talking about, especially in the nineteenth century.
My own opinion is simply that Carroll was a mathematical teacher, as they were hundreds in the Victorian period. Only few however did like him explore successfully new topics and make some significant though non-influential works in advanced mathematics.
But he was hardly a "professional" mathematician. For instance, he doesn't seem to be a regular reader of the mathematical literature of his time, and he never joined any of the numerous mathematical societies of his time.
Surely also he believed mathematics were useful and from time to time contributed to a public debate involving mathematical issues (Vaccination, etc). Surely, he believed there might be some amusement in mathematical practice (See the Tangled Tale).
But again, mathematics was first his job...

[johntufail]

Hi Ami,

I find your mail puzzling in various respects.

You describe him as 'simply a mathematical teacher', yet all the evidence suggests that this was the least of his aspitations. he gave up the teaching of mathematics as soon as he was able. Hos books on mathematics were not intended as text books, rather were polemical. There is absolutely no evidence that Carroll (or anyone else) saw Carroll as a teacher of mathematics. The teaching of mathematics was just a phase he was obliged to go though and quit as soon as decently possible.

Having established this (false) premise, you then go on to castigate him for not reading the key mathematical magazines - on what evidence? or having membership of mathematical societies. Yet his correspondence shows that throughout his life he had personal and in depth corrspondence with some of the most important mathematicians and logicians of the Victorian era.

I think Duncan Black's book on Carroll's mathematical/statistical work certainly challenges your views.

I am interested in how you form these views?

[ami]

Hi John,
First, my post might have been misunderstood because I made a typographic mistake. Instead of “explore unsuccessfully new topics”, you should read “successfully”. I now corrected it.
Second, I never said that Carroll enjoyed teaching. I just said that he was a teacher. I don’t know whether he enjoyed that but I think it was important for him.
Let me now reply to some of your questions and explain my position:
- First, I would like to defend my “premise”. True, he gave up the teaching in 1881. I don’t know however whether it was “as soon as he was able”. His diaries show that he was not against continuing to teach some Euclid (see 31 Oct. 1881, 19 Nov. 1881, 23 Nov. 1881). In the 23 Nov. 1881 entry, he clearly wrote that he will now have the time for some “useful” work, among which he included a textbook on Euclid. Finally, remember that he taught logic later in some schools in Oxford and elsewhere, though on a non-regular basis. Most of his mathematical books were not polemical but merely textbooks. I suppose that the most “polemical” work is “Euclid and his modern rivals” (1879). But, we have to keep in mind that the book is about mathematical teaching, and concerns a wide debate in Victorian Britain on the use of manuals to teach Geometry. The main manual was Euclid’s Elements, and some teachers suggested the use of rival manuals. Carroll defended Euclid’s manual. So, mathematics teaching was important for Carroll as is testified by this book and all the other textbooks that he wrote or planned to write. That doesn’t mean that he enjoyed teaching in practice or that he was a successful teacher. I only claim that he was a teacher and took that job very seriously. He surely defined himself as a mathematics teacher for he signed his works as such.
- My second claim concerned Carroll’s relationship with the “professional” journals and societies of his time. My claim that he was not reading regularly (the word “regularly” is important here) the mathematical literature of his time is based, among other sources, on the catalogue of his library. Of course one has to be careful when using this source. But, one has to observe that most of the mathematical journals he owned was intended for teachers of mathematics and junior mathematicians (Mathematical Questions of the Educational Times, Messenger of Mathematics, etc), even if some renewed mathematicians contributed there. The same can be said regarding the mathematics books in his library. True, many important mathematicians were represented but most of the collection contains textbooks. The most important British mathematicians (not to mention foreign mathematicians) are however absent: Cayley, Sylvester and Clifford. Of course, he might have consulted these advanced journals and books in the library of Christ Church or the Bodleian. But, if yes, that means that he bought essentially educational works. His published works do not help to answer to this question. My own opinion is that he consulted the advanced works in some specific themes which interested him (Determinants, Euclidean geometry, Symbolic Logic, etc.) but was not a regular reader of the mathematical publications of his time. Concerning membership in academic societies, we know that he didn’t belong to any of the numerous mathematical societies of his time, not even the Oxford one. His bank account recently published show that he was not against membership in societies, so why didn’t he participate in any of the mathematical societies of his time? I have no definitive opinion, but just a hypothesis: he considered himself essentially a mathematics teacher (worst, in Oxford). And there were no mathematical teachers associations. The first one to appear was the Association for the Improvement of Geometrical Teaching (1871) which wanted to replace Euclid by rival textbooks. It is obvious why Carroll didn’t belong to this association. Maybe if an association for the defense of Euclid was created, he might have participated…

Sorry for the length of my post. I will in a forthcoming one reply on Carroll’s personal and “in depth” correspondence with the most important mathematicians and logicians of his time…

[ami]

Hi,
Here I continue my argument in the previous post concerning the place of Carroll as a mathematician. One widespread idea is that Carroll corresponded and knew the main mathematical and logical players of his time. In fact, Carroll knew surely many of them enough to destroy the myth of a shy man cloistered in his rooms in Christ Church. But, he was far from being acquainted with them and their work. Only when he explored a particular field, does he went further in his readings and contact some authorities on the subject. In my knowledge, the only important mathematician with whom he was somewhat acquainted was Henry J. S. Smith, the Oxford professor. He accompanied Smith once to lunch with Cayley, he corresponded on determinants with Spottiswoode who told Sylvester about Carroll’s work. Maybe he contacted De Morgan. But all that is not enough…
The case of logic is a good one. It is true that he corresponded with the main British logicians of his time: John Venn, Henry Sidgwick, Francis H. Bradley, John Cook Wilson, etc. But that’s misleading. The bulk of Carroll’s theory of logic was conceived around 1883-1885. Except for Wilson who was professor of logic at Oxford and whom Carroll knew before his appointment in 1889, Carroll didn’t seem to be acquainted with any of all the other logicians. Astonishingly, he never met (in my knowledge) Bradley who was in Oxford too. Carroll’s first references to Keynes or Venn are dated 1894 and I failed to find any earlier contact. The year 1894 corresponds of course to the famous Barber shop controversy that Carroll had with Wilson, during which he contacted other logicians to ask for their opinions. All that I’ve said doesn’t mean that Carroll worked alone or didn’t know the works of his time. In logic for instance, he surely knew the work of Boole quite early, and the work of Jevons and (some) of the work of Peirce at least by 1890.
In conclusion, my opinion on Carroll’s practice of mathematics is that he was a teacher who was mainly interested in educational issues and minor mathematical problems. But, from time to time, he explored some higher branches and made original (but not necessarily important) finds. For his work, he consulted essentially his Oxford colleagues (Sampson, Baynes, etc) and the Oxford Professors Smith (in mathematics) and Wilson (in logic). When, he couldn’t get satisfactory answers, he then wrote to some scholars from Cambridge and elsewhere, or to some journals. But he was far from well-knowing (and being well-known in) the British mathematical community.
Best,

[johntufail]

Hi Ami.

All that you have said is eminently true! Can't fault it. However the point I am making is that Carroll's interest in maths and logic was NOT, in the intricacies and development of mathematics especially pure math. But in the truth-value of mathematics as a way of explaining the existential and spiritual universe. I suggest that if you look at Carroll's works, both fictional and non-fictional, you will find that it is based almost wholly on the question of what is 'Truth'. Mathematical truths, as Carroll often explains, are a priori. This means that any mathematical equation is wholly dependent on the original premise. But if the original premises is at fault, then the entire argument, though perhaps mathematically true or even elegant may not accurately reflect reality. Worse, if one accepts a mathematical truism, then the real world may well adjust itself to comply to this truism.

Regards

JT

[ermete22]

Hello John
Interesting discussion; all facts cited by Ami are correct, but, after me at least, his conclusions are, in some sense, a re-statement of the facts.
I’ll try to add my point of view. The first one refers to mathematics in England in that period: it was a little boring and, for sure, less advanced than the continental one, where subjects like geometry were discussed at a higher level of abstraction, and the connections between logic and mathematics were slowly emerging, (not to criticise UK; I myself studied at the Imperial College in London). Carroll anyhow never missed the point when something really new and promising, appeared in his country, like Babbage and his second machine and Boole’s logics. His method for logical computation was very similar to Boole’s one and equally powerful.
Moreover, my personal opinion is that Carroll could not be an academic authority just because he did not like academy by all his heart, and he did not read what he believed it was not important.
He has always reminded me my theoretical physics’ professor at the Imperial College, John Green, who, if you were trying to discuss with him about something he believed being irrelevant (he was normally right about that) left his office saying “I prefer making sex!”
Why, if you are not interested in academy, you should read and write a lot of “academic papers ?” I wrote myself a number of them when I was young, but I am quite repented, except for my salary of full professor.
I mean that reading and writing academic papers on math does not make you a good mathematician, and the opposite does not make you a bad one or, as Ami says, just a professor of maths. After all Fermat wrote his last theorem on some piece of paper and did not even write down his demonstration. OK, could answer Ami, this is just rhetoric! But, apart that John Green really existed, he should admit that also his messages contain a different rhetoric as well.
Being more serious, the fixed idea of Carroll on Euclid’s geometry is actually rather surprising when you discover it for the first time, but Carroll did not miss the point, that is the role and the nature of the axioms. Lobacewskij-Bolyai changed one of them and discover new geometries, but did not say anything about their nature. Carroll’s efforts had no hope, but the problem was correct and nobody has yet solved it. About logics: my evidence is that Carroll had no objections to Venn’s theory simply because he judged it obvious in some sense, Wilson’s works are quite irrelevant and the subjects of the researches by Henry Sidgwick and Francis H. Bradley were too much academic and simply missed the point Carroll was interested in. On the barber’s shop problem, I think that Carroll’s version is much deeper than the one by Bertrand Russell, but I stop here as I risk John intervenes telling me that I am too technical.
Carlo

[andrei65]

Dear colleagues!

My name is Andrei Moskotelnikov, 43 years old. I live in Minsk, Belarus (former Byelorussia). Dr. Tufail kindly suggested me to visit the Forum and take part in your discussions. My “inaugural lecture” is as follows: if my English and mind are not completely despairing I’ll be glad to tell and listen.

I decided to speak in this mathematical chat first. And first of all I want to tell about our Russian keen interest in some Carroll’s mathematical writings. In fact, apart from the proclaimed virtuosity in constructing complicated logical problems (in some scientific and non-scientific Russian issues, for example in Nina Demurova’s Academician Alice), I had observed my colleagues, physicist and engineers, in Byelorussian University being absorbed in Symbolic Logic.

Despite of many and different translations of Alice books and of the Snark we have in Russian the only but splendid translation of a set of Carrollian works concerning mathematical (and logical) branch of science. This set consists of the Tangled Tales, Pillow Problems, Symbolic Logic, the Game of Logic plus some minor writings such as What the Torture said to Achilles. Translator was the late Yulii Danilov, a physicist, while another well-known Soviet physicist Yakov Smorodinskii often worked with him as a book editor. Generally the publication of books was very responsible affair in the Soviet Union and nobody but approved person can deal with it.

Other Carroll’s scientific and educational writings had been declared as “deeply traditional” id est out of any interest in Academician Alice as well as in dictionaries and encyclopedias. I had not any reason of distrusting Nina Demurova but… But she is a philologist, not a mahtenatician. Is it impossible for her to be mistaken? So I decided to check it up by myself. First, I have got suitable education and second, I appreciate every Carroll’s word. Is it impossible for me to do further discoveries which I can to present to our Carroll’s fans?

Now I can say that I was rewarded for my challenge. I finished a translation of Euclid and his Modern Rivals and collected commentary notes to this book. I ventured to comment it not because I had vast knowledge, but I have got vast knowledge after finishing the commentary. Fortune helped me—time and again she delivered me sources of information, for example A History of Elementary Mathematics with Hints on Methods of Teaching by Florian Cajori in Russian translation 1912. By what miracle did this old book fall in a proper time into one of second-hand bookshops in Minsk? But I have to conclude.

Presently I have got the second volume of Carroll’s pamphlets to proceed with studying his mathematical works. A lot of them indeed were written with the purpose of being “useful for Responsions examinations” and so on. (Cajory is quoted in this volume too! So I am on the right way in selecting my sources.) Of course they have not scientific value at all. But while reading them I am carried in Victorian Oxford and walk among students. By the way, I deal with Oxford pamphlets as well, having translated some of them from the Complete Works.

Andrei M.

[johntufail]

Hi Anrei,

My old Comparative Literature Tutor, Dr Alan MacLauren, once rather forcefully expressed to me at the height of a rather heated 'discussion' that the only western nations that can rival Japan in genuine story telling are Russia and Ireland. Your introduction reminds me of this remark.

Now 'Euclid and his Modern Rivals; is a much dismissed and often maligned work of Carroll's. I have often felt that it is rather a much misunderstood book. My reason for feling this is that it has always been jusdged within the context of it's contribution to mathematics. I do not feel that it was ever intended as a contribution to mathematics, rather as a contribution to the limits of mathematics as the discipline was then understood and, in general, is still understood.

You can imagine that I am eager for you to expand on your commentaries on this work!

[johntufail]

Betty Boop,

You asked a question some time ago that I found very perceptive. 'Was Carroll Trying to uncover some deep truth of was he just trying to amuse himself.' The whole of your mailing appeared to me (correct me if I'm wrong) appeared indicate that you have difficulty reconciling the picture of a serious deeply committed Charles Dodgson, with the apparently frivolous Lewis Carroll (and we have to face it, even in his 'Dodgson' cloak the seeming frivolity does peak out!).

This question goes to the heart of the Dodgson/Carroll dilemma and has led to mumerous appalling (but lucrative) psychobabble books about the schizophenic mythic person that is both Dodgson and Carroll.

To answer your question. There are only three thing that all biographers and students of Dodgson will agree on. That is that he was incredibly self-disciplined. That he was incredibly private about what he actually believed. Only at the end of his life did he begin to reveal only fragmets of his personal beliefs (in the prefaces of the Sylvie and Bruno Books). Even in his diaries he tended to keep his genuine beliefs to himself. And finally that his commitment to his particular belief system was the driving force thatr inspired everything he did.

He was Not, ever, a frivolous person. However he had the intelligence and acuity, derived from his chilhood experiences, to realist using seemingly frivolous games to make important points is a hugely successfull way of educating 'Children of all ages'.

He also realised that many of the issues he wished to raise were, to say the least both controversial and vertually heretical. Thus he needed to develop a medium in which he could express himself in ways that were precise, educational in a positive and challenging way, entertaining, and thought provoking. Yet he needed to do this in a way that would not arouse the ire of the various groups that both his belief and ideas were inimical to.

Hence he used humour, whimsy and 'Nonsense' as his strategies and he did so brilliantly.

That of course why his works continue to tug at both our intellects and our emotions.

Hope this helps.

JT

[ermete22]

When talking about Carroll’s geometry works, one must never forget his fixed interest was in Axioms. In A new theory of Parallels, which I read many years ago, he tried to find a demonstration of a classical Euclid’s conjecture, starting from what he considered a self-evident truth, but which unfortunately was not such.
To understand Carroll’s story one must realise that he believed in the identity

AXIOMS=SELF-EVIDENT TRUTHS

For him the problem was the identification of such self-evident truths. This is his major error in all the development of his logical (and geometrical) efforts. In modern logics (and geometry) the problem id that building an adequate formal system (Axioms included) and then find out interpretations of the formal system. You change, as an example, an axiom, look at the logical consequences and possibly observe that the new system captures some features of a piece of reality.
Carlo

[andrei65]

Dear Dr. Tufail!

I’m afraid my expanding the commentaries to Modern Rivals will disappoint you this time. You see, dear Dr. Tufail, they are not, so to say, scientific but only literary in spite of my notes to the Snark. Now I fully share your opinion concerning Modern Rivals but while translating it I set myself another object. I wanted to understand and to retell to my future Russian readers scientific and educational situation in Carrollian (Victorian) Oxford as well as previous conditions and future trends. So I often cited Cajori, Felix Klein (concerning educational work and history of Association for the Improvement of Geometrical Teaching), H. S. M. Coxeter and John Littlewood. And of course searching and explaining the quotations which begin from the first passage—fascinating activity! (if my use of the word is correct). After all, in addition, including Euclid’s propositions as they are (included by Carroll as their numbers only) with describing of their interpretation before and after according to our Russian fundamental edition of Euclid in 3 volumes with the vast commentary of Morduhkai-Boltovskii.

So it is like this. I worked with hope that all this additional material would make this Carroll’s book clearer and not so dry for modern interesting reader who will be rewarded after finishing the reading with understanding the Carroll’s epoch and anxiety.

Andrei M.

[andrei65]

Now, dear Dr. Tufail and my dear new friends, I strongly want to proceed with this discussion. I will to draw some arguments known to Dr. Tufail already by our correspondence about the Snark and I pray to let me quote some phrase of all you involved at first.

After initial words of Carlo we have got such an exchange of views.

Bishop Ami said, “I always feel very uneasy with general statements when it comes to Carroll's (or any other thinker's) philosophy, mathematics, logic, etc. I think that every text should be discussed within its context… My own opinion is simply that Carroll was a mathematical teacher, as they were hundreds in the Victorian period.”

Johntufail said, “You describe him as 'simply a mathematical teacher', yet all the evidence suggests that this was the least of his aspitations. he gave up the teaching of mathematics as soon as he was able.”

Now let me say, Gentlemen, you are both right! And Bishop Ami proved his case in the next post and Johntufail’s information of course is fully valid. From Nina Demurova’s book I learned that Rev. Dodgson almost hated his job as a teacher and most probably his students at all (most naturally!) but he did the best of him as a teacher according to a lot of his own math pamphlets—and much more than the best: he went into action for most lucid Manual for beginners (see “Modern Rivals”).

But while straggling “for the students rights” Carroll, as Johntufail said, started off thinking that mathematics revealed 'truths' but gradually came to understand that the 'truths' they revealed were completely dependent on the premise on which the original question is based. He also came to believe that mathematics is a language, which as such, can only describe a reality partially, dependently and ultimately subjectively.

Is it mean, by the words of Bettyboop, that Carroll was trying to uncover some deep truth? Or just amusing himself?

It is mean maybe that Carroll was truing to go down to the limits of knowledge in this separate branch of science as Johntufail suggested. And I want to add that it was a major and non-trivial breakthrough—id est first knowing and then seeing the limits. Was Carroll only one of a lot of such persons of that time? I dare to say no. I remember such an author as Jules Vern, my beloved writer once, whose mind didn’t suspect of any limits of human ability in technology and whose heroes wanted to revert the Earth for the sake of triumph of progress. But would it be a progress? No, it will be only a technological progress. A genuine progress is a movement towards our mental limits! But for doing so one must have a clear mind and acute vision and of course Carroll had both.

I had said “trying to go DOWN to limits” in geometry. Indeed we have two paths to our limits—up and down, for our limits are in front of us as well as behind us. And we humans are going now from the second to the first—from naïve to scientific vision of the World—from prehistory to (who knows it!) the End of the History—from non-human to true human…
And one can see this general movement in many particular cases or branches as geometry for example—from the limits of axiomaticity (let me use this word) up to non-Euclidian (id est fictitious) postulates and so on.

And here I dare to give my view on the subject of this theme that is on the essence of Carroll’s search in math itself, in math as language and in language itself. I want to quote dear Bishop Ami, “I think that every text should be discussed within its context”. Which is the context of some Carroll’s math work (for example, Modern Rivals) as a whole? To my mind, it is all the Carroll’s creation. Stating that I can examine, for example, Modern Rivals and, for example, The Hunting of the Snark TOGETHER. And my examination let me tell the follows. In math, maybe logic and of course linguistics Carroll was interesting in limits left behind rather than which are in front of. Id est he examined the very beginning—and thereby the possibility—of our movement to the heights of knowledge (in geometry—Modern Rivals, and in linguistics or even humanity at all—the Snark).

Here I have to explain myself concerning the Snark. Dr. Tufail had been well informed early and now I repeat in brief that to my mind The Snark is, so to say, a poetical analogue to Modern Rivals. As the second, the first text is in fact an investigation of the origin of concrete (not abstract!) notions—first from nothing to a Word (Snark) then from a Word to a Name (Bujoom) and then… Then to the System of the Euclidian Geometry.

Don’t execute me; I’m tired to death…