- On "Useless Knowledge" - permalink
Published On: 02/10/17
Written By: Muhammad


It has often been said that certain types of knowledge are "useless". All
sorts of knowledge have been accused of this at some time or another. Despite
the obvious overall utility of mathematics, pure mathematics in
particular has often been the target of these criticisms. It's very easy to
turn up examples of people saying this:


This position can be seen being argued for on Physics Forums of all places.
The blog Math with Bad Drawings saw a need to defend pure mathematics in its own way.
The question of the usefulness of pure mathematics has also been raised on Quora.
The issue has also come up in The Journal of Mathematics and Statistics.
The book A Mathematician's Apology by G.H. Hardy basically centers around the 
notion that much of higher mathematics is "useless" but should be done for its 
own sake regardless.

But here is a case against the uselessness of pure mathematics, leading in
with a passage from the mathematical novel The Man Who Counted, written by
the late Brazilian writer Júlio César de Mello e Souza. In this passage, the
hero of the story, a young Persian savant, named Beremiz Samir, with an
extraordinary ability for calculation, becomes the object of the envy of a
vizier to the fictional caliph of Baghdad. The vizier insults Beremiz, calling
his unique ability frivolous. Beremiz does not become angered, but calmly
responds:

"When the mathematician makes his calculations or looks for new
relations among numbers, he does not look for truth with a practical purpose.
To cultivate science only for its practical purpose is to despoil the soul of
science. The theory that we study today, and that appears to us impractical,
might have implications in the future that are unimaginable to us. Who can
imagine the repercussions of an enigma through the centuries?  Who can solve
the unknowns of the future with the equations of the present? Only Allah knows
the truth.  And it is possible that the theoretical investigations of today may
provide, within one or two thousand years, precious practical uses.

"It is important to bear in mind that mathematics, besides solving problems,
calculating areas, and measuring volumes, also possesses much more elevated
purposes. Because it is so valuable in the development of intelligence and
reason, mathematics is one of the surest ways for a man to feel the power of
thought and the magic of the spirit.

"Mathematics is, in conclusion, one of the eternal truths and, as such,
raises the spirit to the same level on which we contemplate the great
spectacles of nature and on which we feel the presence of God, eternal and
omnipotent. As I have said, O illustrious Vizier Nahum ibn-Nahum, you have
made a slight error. I count the verses of a poem, calculate the height of a
star, measure the size of a country or the force of a torrent, and thus I
apply the formulas of algebra and the principles of geometry, without
concerning myself with the profit I might earn from my calculations and
studies. Without dreams or imagination, science is impoverished. It is
lifeless."

The Man Who Counted seems modern to me, but I was surprised to find
out that it was first published in 1938. And it's managed to be very prophetic
indeed. One year later, World War II ensued and, with it, came the conception
and implementation of the first general computing devices. The German Z3 and
the American ENIAC became the first Turing-complete digital computers. Ever
since then, computers have played an ever larger role in all parts of society,
and the body of number theory, mulled over even into antiquity, but apparently
entirely lacking in any real value for applications, was drastically
transformed. Consider that it was once said famously by the legendary
mathematician Carl Friedrich Gauss: "Mathematics is the queen of sciences and
number theory is the queen of mathematics." Consider also that another
not-quite-as-great—an admittedly very high bar to clear—but still great
mathematician Leonard Dickson later said: "Thank God that number theory is
unsullied by any application." But integers are the essential currency of the
digital computer and it stands to reason that the practical value of the study
of the integers per se could only have exploded as soon as the digital computer
became more widespread. Today there are a great many applications of number
theory to computing, including:


Fast numerical techniques, such as the use of the Chinese remainder
theorem, which make all sorts of other computation more efficient, or
possible, including the others on this list.
Pseudorandom number generation, handy for all sorts of things from games to
evolutionary algorithms.
Hash functions, which make it possible to map data structures other than
single integers to other values in computing (briefly put, a very important
abstraction).
And, of course, modern cryptography, which makes all of e-commerce a
possibility at all.

Even the possibility of contacting alien beings one day is influenced by
number theory: the Arecibo message blasted into deep space in 1974 consisted
of a number of bits equal to a semiprime, the product of the two prime
numbers, in this case 23 columns by 73 rows. This makes for one rectangular
arrangement that forms sensible pictures; the other arrangement of 73 columns
and 23 rows makes no sense, and all others do not result in a perfectly
rectangular shape.

So, anyhow, what's the verdict on "useless knowledge"? There is likely
knowledge that is really genuinely very useless. The exact number of carbon
atoms in my body at any one given time, suggested by the editor, could be one
of these very facts, facts that relate to nothing else and signify so very
little. But the ascription of "useless" to, for example, pure mathematical
knowledge such as number theory is more problematic. Was number theory
useless, but then became useful in the 1940s or so? If so, why should it have
been pursued before then? One might counter that the framework could have been
deemed useless before this time and would have simply been devised on an
as-needed basis. A problem here is that t