Number: The Language Of Science

This review is part my series on Progress in Mathematics. The posts aren’t linearly connected but if you’re interested you can read the first two here and here

Geometric representation of the complex numbers from the book

Number is a book about the history of mathematics. It was written in the 1930s by mathematician and part time lumberjack: Tobias Dantzig. It is a birds-eye view of the most important developments in the field from humanity’s first insights to what Dantzig knew as modern mathematics in the early 20th century. Dantzig spiritedly carries the reader along continuous threads of mathematical thought which stretch from the ancient Greeks, through the Renaissance, and into his modern day. Number is a great book, but don’t take my word for it. Albert Einstein has an effusive blurb on the cover: “This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands. If people know how to treasure the truly good, this book will attain a lasting place in the literature of the world.”

There are dozens of fascinating topics and anecdotes in the book, so I’ll just focus on one connection I made while reading that I found particularly interesting: how Zeno’s paradox and the Principle of Position (or lack thereof) stalled mathematics in place for thousands of years, despite supportive societal institutions, great mathematical minds, and sometimes only tiny conceptual leaps to important ideas.

The Principle of Position

The Principle of Position is a simple idea: allow the meaning of a symbol to change based on it’s position in a numeral. This allows the symbol 2 to encode different meanings in 200, 1020, and 12. With this principle we can express all the uncountably infinite numbers with just 9 (or 2) symbols. Despite its simplicity and universal power, the principle of position evaded some of humanity’s greatest minds for thousands of years. Mathematician Pierre Laplace says of the Principle of Position:

But its very simplicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.

In the several millennia between the invention of written numbers and this principle, everyday algebra was confined to what could be done on ten fingers. Expert mathematicians simply expanded the tally system to include special symbols for greater and greater quantities. Archimedes, for example, when enumerating the number of grains of sand needed to fill the entire celestial sphere, added symbols to the Greek set of magnitudes. This set traditionally ended with M = 10,000. He created a symbol for (M*M)^M or about 10^800,000, but he continued to just tally with this new symbol instead of extending the symbolic system to include any possible number without the need for adding new symbols.

The difference between these numeration systems is significant. Although humanity’s computing hardware has not changed much, the principle of position greatly expanded our computational ability. Dantzig reports that:

Computations which a child can now perform required then the services of a specialist, and what is no only a matter of a few minutes meant in the twelfth century days of elaborate work

Calculations which exceeded the computing power of our fingers had to be assisted with a counting board or abacus. This invention is found in a similar form across practically all agricultural cultures. Groups of counters are divided among parallel rows with each row representing a decimal class. So a number like 1020 would be represented on an abacus with 1 counter in the 1000s row and 2 counter in the tens row with no counters anywhere else. It is immediately clear to us that this is exactly the Principle of Position in action. Counters or beads which are otherwise equivalent take on different meanings when they are in different rows. One need only translate the number of counters in each row into a written symbol to make the jump to a universal positional number system. To this Dantzig says:

True! But there is one difficulty. Any attempt to make a record of a counting board that such an entry as I, II may represent any one of several numbers: 12, 102, 120, 1020 among others. In order to avoid ambiguity it is essential to have some method of representing the gaps, i.e., what is needed is a symbol for an empty column. The concrete mind of the ancient Greeks could not conceive the void as a number, let alone endow the void with a symbol.

So Europe stuck with Greek and Roman numerals for centuries until the Hindu positional numbering system that we use today was introduced to Italy. On introduction to this system you might think that all European mathematicians would look at positional numerals, look at their abaci, look back at the numerals, and smack themselves on the forehead for not realizing it sooner. In actual fact, the spread of the positional numbering system was a centuries long fight which should sound familiar to all of us:

The struggle between the Abacists, who defended the old traditions, and the Algorists, who advocated reform lasted from the eleventh to the fifteenth century and went through all the usual stages of obscurantism and reaction. In some places, Arabic numerals were banned from official documents; in others, the art was prohibited altogether. And, as usual, prohibition did not succeed in abolishing but mainly served to spread bootlegging, ample evidence of which is found in the thirteenth century archives of Italy, where, it appears, merchants were using the Arabic numerals as a sort of secret code.

Zeno’s Paradox

You have probably heard of at least some parts of Zeno’s paradox before. As quick context, Zeno’s paradoxes are a group of statements about motion. The punchline is that, commonly held assumptions about the infinite divisibility of space and time lead to apparent contradictions like “a moving arrow is always at rest in every instant” and “the faster Achilles will never overtake the slower turtle.” Read this for a deeper explanation of the paradoxes and interesting connections to the mathematics of relativity and this for a resolution to them with modern mathematics.

There are thousands of translations, interpretations, and resolutions to Zeno’s paradox but I want to focus on the effect it had on contemporary Greek mathematicians. Dantzig says:

The subsequent course of Greek science shows clearly how great was the influence which the crisis precipitated by the Arguments of Zeno exercised on the mathematical thought of the Hellenes . . . By instilling into the mind of the Greek geometers the horror infiniti, the Arguments had the effect of a partial paralysis of their creative imagination. The infinite was taboo, it had to be kept out, at any cost . . . Greek mathematicians stopped short of an algebra in spite of Diophantus, stopped short of an analytic geometry in spite of Apollonius, stopped short of an infinitesimal analysis in spite of Archimedes. I have already pointed out how the absence of a notational symbolism thwarted the growth of Greek mathematics; the horror infiniti was just as great a deterrent.

Later on he adds:

In the method of exhaustion Archimedes possessed all the elements essential to an infinitesimal analysis . . . the idea of the limit as conceived by Archimedes was adequate for the development of the calculus of Newton and Leibnitz

Calculus is the mathematical study of motion and change over time. Zeno was correct to point out that motion and change are impossible to understand with discrete elements, even with an infinite number of them. But his paradox loomed over any Greek mathematician who might have modeled these things with the infinite sequences that define the continuum. The horor infiniti that it instilled in Greek intellectual culture was enough to persuade the Greeks to stick with what they were already comfortable with: static geometric models.

Progress Studies Perspective

Learning that humanity was so close to these mathematical innovations and yet ended up waiting centuries to truly understand them is simultaneously heart-wrenching and inspiring. The alternate historian in me wonders where we might be today if the civilization which built the Antikythera Mechanism had been empowered by calculus and algebra. It seems unlikely that these advances would be sufficient for something like an industrial revolution to begin ~200 BC. But the expanded computational ability, improved physical understanding and mathematical modeling that come from positional numeration and calculus seem to be necessary conditions for such an advance. Ancient Greece had many other important catalysts for industrial revolution as well. Competitive governments willing to invest in science and engineering to outpace their neighbors, substantial political and economic freedoms, a culture that supported scientific inquiry, and well-developed international trade networks. They probably needed more advanced physics and the printing press to have a good chance at industrial revolution, but both of these things seem well within their grasp. It is difficult to view this as anything but an ironic tragedy. In want of a few simple ideas which were truly right under our noses all along, humanity had to endure centuries of extreme poverty.

Lamenting the past won’t help anyone though, and the forward-looking lessons from these observations are inspiring. The study of progress in these ideas prove it is possible to bridge the gap between the present and a wildly optimistic future just with some new ways of looking at the world. We don’t need superhuman strength or utopian global coordination. It may be that all we are missing is a particularly useful mathematical model. The challenges facing the ancient Greeks in mathematics are very different from our own. Mathematicians today are far more willing to play with infinities and dive into deeply abstract symbolism. So we probably won’t be able to take direct advice from the shortcomings of the Greeks. However, it is impossible to know what we might be missing right under our noses. We should learn from the mistakes of the past and stay alert for new ideas at the foundations of mathematics.

Number is short and fun to read. I highly recommend it for anyone interested in the history of math!