Should we be surprised if it were con- firmed indeed that the most comfort- able pace of execution of a recursive algorithm is set by a gene responsible for grooming behavior?
I would suggest that HOXB8 is indeed the Bubble Wrap Gene and is responsible for Sudoku being at- tractive to humans. I would rather hear more on that from geneticists and neurophysiologists.
At last I am in a position to for- mulate the moral of this story. I be- lieve that a real understanding of one
of the key issues of mathematical practice (and especially of math- ematics teaching). Namely,
• why are some objects, concepts and processes of mathematics more intuitive, “natural”, or just more convenient and accept- able than others?
cannot be achieved without taking a hard and close look at the very deep and sometimes archaic levels of the human mind and the human neural system. Indeed, Stanislas Dehaene said in his bookThe Number Sense[167] that
We have to do mathematics using the brain which evolved 30 000 years ago for survival in the African savanna.
In particular, should we be surprised if it were confirmed in- deed that the most comfortable pace of execution of a recursive al- gorithm is set by a gene responsible for grooming behavior?
1.7 What lies ahead?
We have seen how the deceptively simple functiony=|x|launched us on a roller coaster ride through several branches of mathemat- ics. More adventures still lie ahead. They all will follow a similar plot:
• Usually I start by describing a very simple—sometimes ridicu- lously simple—mathematical problem, object or procedure. • Then I discuss possible neurophysiological mechanisms which
might underpin the way we think and work with this object. Sometimes my conjectures are purely speculative, sometimes (for example, in the next chapter) they are based on established neurophysiological research.
• I usually include a brief description of mathematical results which deal withmathematicalanalogues of the conjectural neu- rophysiological mechanisms.
• In the later parts of the book, I will more and more frequently venture into the discussion of possible implications of our find- ings for our understanding of mathematics and its philosophy. • My conjectures are frequently outrageous and sketchy. I have
no qualms about that. The aim of the book is to ask questions, not give answers.
I very much value a “global” outlook at mathematical practice (in recent books best represented by David Corfield’sPhilosophy of Real Mathematics[16]), but, in this book, I prefer to concentrate on the “microscopic” level of study.
Quite often the mathematics discussed or mentioned in this book is very deep and belongs to mainstream mathematical re- search, either recent, or, if we talk about the past, of some historic significance. I believe that this is not a coincidence. Mathematics is produced by our brains, which imprint onto it some of the struc- tural patterns of the intrinsic mechanisms of our mind. Even if these imprints are not immediately obvious to individual mathe- maticians, they are very noticeable when mathematics is viewed at a larger scale—not unlike hidden structures of landscape which emerge in photographs made from a plane or a satellite.
David Corfield, aged 10
Notes
1 SIMPLEST POSSIBLE EXAMPLES. Of course, simplest, in the relative sense, examples can be found at every level of mathematics. Here is one example, due to Gelfand: the simplest non-commutative Lie group is the group of isometries of the real lineR; it is the extension of the additive groupR+by the multiplicative group{ −1,+1}. Its representation theory
is a well-known chapter of elementary mathematics, namely, trigonometry; however, the connection between representation theory and trigonometry is not frequently discussed. But this is not the simplest possible example of a simplest possible example, and its discussion will lead us beyond the scope of this book.
However, it would be useful to record one consequence of the relation between representation theory and trigonometry: the formula for matrix multiplication is more fundamental than almost any trigonometric for- mula. We shall return to that later, see Page 185.
2ANALYTIC FUNCTIONS. A functionf(x)isanalyticatx=a
0 if we can
writef(a0+z)as a power series
f(a0+z) =a0+a1z+a2z2+a3z3+· · ·
converging for all sufficiently smallz. For example, the square root func- tion y = √xis analytic at the point x = 1 since by Newton’s Binomial Formula we have √ 1 +z= 1 +z 2− z2 4·2!+ 3·z3 8·3! − 3·5·z4 16·4! + 3·5·7·z5 32·5! +· · ·
for allzsuch that|z|<1. But a power series expansion fory=√xatx= 0
does not exist: the functiony=√xis not analytic atx= 0. 3D
ISCRETE VS. CONTINUOUS: the two other great divides in mathe- matics are between “finite” and “infinite” and between “geometric” and “formula-based”; we shall discuss them later in the book. It is quite com- mon to associate the “geometric” or visual, and “formula-based” or verbal modes of thinking with the activities of the two hemispheres of of the brain. For lack of space, I cannot go into details; in any case, I am more interested in mechanisms of synthesis of the two modes than reasons for their sepa- ration.
NOTES 19
4O()-NOTATION. A few words aboutO()-notation for orders of magni- tude of functions of natural argument n: we say that f(n) = O(g(n))if there is a constantC such thatf(n) 6Cg(n)for all sufficiently largen. Hencef(n)isO(nd)iff(n)6Cnd.
5K
OBLITZ’STHESIS. Like all general proclamations about mathemat- ics, Koblitz’s thesis has its natural limits of applicability. As usual, we have all possible complications caused by the non-constructive nature of many mathematical proofs: sometimes it is possible to prove that a certain al- gorithm has polynomial complexityO(nd), without having any way to find the actual degree d; one example can be found in [325]. But we do not venture into this exciting, but dangerous, territory.
6C
HOICELESS ALGORITHMS. A few words of warning are due. Any algo- rithm on a clearly described finite set of inputs can be made into a choice- less algorithm by running it on all possible reorderings of the input struc- tures. Therefore the concept of choiceless computing is meaningful only if we assume resource limitations and focus on choiceless polynomial time algorithms.
7An interesting comment from Chris Hobbs: on the formula
x1,2=−b±
√
b2−4ac
2a for the roots of the quadratic equation:
I know that this formula is always written with the plus / minus but, as you’ve argued in the text, it’s not only unnecessary, it’s also wrong. It has worried me since I first met it as a child that, accord- ing to that formula, there are four roots to a quadratic equation: the square root function delivers two and the plus/minus turns them into four (two positive ++ and −−, and two negative −+
and+−). Just a quibble but it’s a shame that we don’t have a no- tation for “the negative square root ofx” and “the positive square root ofx”.
8T
ROPICAL MATHEMATICS. The works [392, 404] usemin, notmaxas a basic operation⊕, but this makes no difference since in the both cases the results are similar. Traditionally,maxis used in works originating in control theory and minin papers motivated by applications to algebraic geometry.
9SUDOKU. Sudoku enthusiasts would not forgive me if I move from the subject without giving a single Sudoku puzzle. Here is the one, kindly pro- vided by Gordon Royle. It contains only 17 clues (filled squares), but is still deterministic, that is, can be filled in in only one way. Apparently the ex- istence of deterministic Sudoku puzzles with less than 17 clues is an open problem.
1 4 2 5 4 7 8 3 1 9 3 4 2 5 1 8 6