The reader will find in this book a number of reformulations of well-known theorems and theories intentionally made in a “toy” language—a typical example is my “palindrome” formulation of the Coxeter–Tits Theorem in Section 3.4, see footnote 5 in Chapter 3. These are instances of cryptomorphism, the remarkable capacity of mathematical concepts and facts for a faithful translation from one mathematical language to another. Such translations consti- tute an important but underrated part of mathematical practice; they remain virtually unknown outside professional circles. It can be felt that many teachers of mathematics view “multiple represen- tations” of mathematical objects as a hindrance, see a more detailed discussion in Section 7.5.
One of the reasons why the “language” aspect of mathemat- ics is ignored in mainstream mathematical education is that most translations, as much of mathematical work generally, never make it from scratch paper to publication. The situation is different in “Olympiad” or “competition” mathematics which pays more atten- tion to what is happening on scratch paper, and which also needs a steady flow of new original (or attractively disguised old) problems for higher level competitions.
To give you the flavor, here is a “double” problem from a classical Olympiad problem book [261, Problem 145]. [?]
Solve the both versions of the problem.
a. Two people play on a chessboard, moving, in turns, the same piece, the King. The following moves are allowed: one square left, one square down, or the diagonal move
4.2 Cryptomorphism 67
left-down. The player who places the King in the left- most square in the first (bottom) row, wins. At what ini- tial positions does the first player have a winning strat- egy?4
b. Two players take, in turns, stones from two heaps. They are allowed either to take one stone from one heap, or to take one stone from both heaps. The player who picks the last stone wins. At what initial numbers of stones in the two heaps does the first player have a winning strategy?
Is all that just a game? Does the bewildering variety of mathe- matical languages which can be express the same fact matter? Let us listen to two expert opinions.
4.2.1 Israel Gelfand on languages and translation
My position on the issues of cryptomorphism and “multiple pre- sentation” is much influenced by my conversations with Israel Gelfand. He once said to me:
Many people think that I am slow, almost stupid. Yes, it takes time for me to understand what people are saying to me. To understand a mathematical fact, you have to trans- late it into a mathematical language which you know. Most mathematicians use three, four languages. But I am an old man and know too many languages. When you tell me some- thing from combinatorics, I have to translate what you say in the languages of representation theory, integral geome- try, hypergeometric functions, cohomology, and so on, in too many languages. This takes time.
It is amusing to watch how fellow mathematicians, not accus- tomed with the peculiarities of Gelfand’s style, speak to him the first time. Very soon they become bewildered why he insists on their giving him really basic, everyone-always-knew-it kinds of def- initions; then they are taken aback when he becomes furious at the merest suggestion that the definition is easier to write down than to say orally (“I know, you want to cheat me, do not try to cheat me!”). Next morning, their second conversation is usually even more en- tertaining, because Gelfand starts it with the demand to repeat all the definitions; then he proceeds by questioning everything which was agreed yesterday, and eventually settles for a definition given in a completely different language.
Mathematical languages unstoppably develop towards ever increasing degree of compression of information.
I have observed such scenes many times and came to the conclusion that, for him, a definition of some simple basic concept, or a clear for- mulation of a very simple example, is a kind of synchronization marker which aligns together many different
languages and makes possible the
translation of much more complex mathematics.
4.2.2 Isadore Singer on the compression of language
Mathematicians are so sensitive to math- ematical language issues because they can see dramatic changes in the lan- guages used over their working life.
Another aspect of the Babel of math- ematical languages, their unstop- pable development towards ever in- creasing degree of compression of in- formation, is succinctly expressed by Isadore Singer in a recent interview [74]:
I find it disconcerting speak- ing to my young colleagues, because they have absorbed, reorganized, and simplified a great deal of known material into a new language, much of which I don’t understand. Often I’ll finally say, “Oh; is that all you meant?” Their new conceptual framework allows them to encompass succinctly considerably more than I can express with mine. Though impressed with the progress, I must confess impatience because it takes me so long to un- derstand what is really being said. [74, p. 231]
One of the reasons why research mathematicians are so sensi- tive to mathematical language issues is that they can see dramatic changes in the languages used over their working life.
4.2.3 Cognitive nature of cryptomorphism
Returning to the running example of this book, we see that palin- dromes and mirrors are, essentially, cryptomorphic objects. They are sufficiently basic and “atomic” to belong simultaneously to two different realms of cognition, the verbal/symbolic and the visual, although the status of palindromes is clearly borderline: to appre- ciate a palindrome, you have toseethe symmetry of its presenta- tion in type. Remember the peculiar typesetting of “ABBA” on their posters? And a question to experimental psychologists: do blind people (especially if they are blind from birth) aesthetically appre- ciate palindromes when they read them in Braille? In more general terms, is there any significant difference in perception of symmetry by blind people from that of sighted?
Wilfrid Hodges aged 10
A similar observation appears to be valid also in the case ofmu- sical palindromes, where the visual symmetry of the score is of im- portance, see the analysis of symmetry in music in Wilfrid Hodges’ paperThe geometry of music[49]. (In particular, Hodges discusses the paradoxical results of playing a recording of palindromic music (Haydn’sMenuet al Rovescio for piano)backwards: the individual notes, as produced by musical instruments, are not reversible in time. For example, piano notes start with a bang and then fade away.