We are witnessing the birth of a new phenomenon I haven’t seen named before. I propose to call it “algorithmic scholarly communication”.
Alan Kay describes the birth of a new genre of media as proceeding in a series of stages:
A new genre is established. A few years later a significant improvement is made. After a few more years the improvement is perceived as not just a “better old thing” but an “almost new thing” that leads directly to the next stable genre.
This essay identifies an “almost new thing” which is beginning to transform the nature of scholarly communication. First I provide some background, including my personal view of the principles underlying scholarly communication; then, I give some thought-provoking examples of the current state of the system; and then, I project where it seems likely to take us.
A hundred and thirty generations ago, we began to understand formal mental processes when Thales and the Pythagoreans including Socrates formulated proofs in geometry in terms of logical arguments. Over the next ten generations, Mozi, the Pythagoreans, and Aristotle developed the study of logic as an object in itself.
The academy as we know it is, in name, an homage to the estate of Plato, Socrates’ student; in structure it descends from medieval seminaries of the Roman Catholic Church, and indeed some of those seminaries are universities today. It has always aimed at acquiring and diffusing knowledge, both in the sense of practical skill (the arts) and knowledge of what is objectively true (the sciences), though modern epistemology has demoted Christian theology and holy writ in general to the status of myth and metaphor rather than an objectively-true cosmology, tempting us to scoff at the debates of the European medieval Scholastics; it’s easy for us to forget that angels and demons seemed as real to them as states, money, and corporations do to us.
XXX etymology of "university"?
After 2600 years of work, we at last succeeded in fully formalizing logical reasoning and other formal mental processes during the 20th century; the product of this formalization was the digital computer, which executes processes called “algorithms”.
We have already seen executable scholarship happen in many aspects of practical mathematics. I have on my bookshelf a 1965 book by R.S. Burington entitled Handbook of Mathematical Tables and Formulas, first published 1933, LCCN 63-23531. It tells me things like this, neatly organized into tables:
log cos x ≈ -x²/2 - x⁴/12 - x⁶/45 - 17x⁸/2520 - ... when x² < π²/4
roughly 53030 Americans of every 100 000 live to age 63; 1800 die that year
the cotangent of 38.5° (i.e. the reciprocal of the tangent) ≈ 1.2572
∇²S is defined as ∂²S/∂x² + ∂²S/∂y² + ∂²S/∂z²
the logarithm of the cotangent of 17°21' is 0.50526
an endomorphism is a homomorphism of a system onto itself or part of itself
0.15° ≈ 0°9'00"
the definition of the cumulative discrete probability as a sum Σᵢf(xᵢ)
formulas for the sides & angles of a triangle given 2 angles & 1 side
∫dx/(x√(ax + b)) = 1/√b log((√(ax+b)-√b)/(√(ax+b)+√b)) for b > 0
The first half of the book is a brief but complete summary of the most widely useful branches of mathematics; the second half consists of numerical tables. The first “mathematical table” of this sort was assembled by Aryabhata in the fifth century CE; it consisted of 24 first differences of the sine function. Calculating and disseminating such tables — crucial for practical manual celestial navigation, artillery warfare, trigonometric calculation in general, and statistics — was a major effort of mathematicians for centuries. It inspired Babbage’s original plan to automate computation in 1822; computing mathematical tables was the justification for funding one of the three World War II projects that did successfully automate computation, the ENIAC. The ENIAC was designed to compute mathematical tables for artillery firing.
The other two projects were Konrad Zuse hacking in his Berlin apartment (and later with government support) and the secret British cryptanalytic Colossus machine; arguably the Harvard Mark I ASCC, based on Babbage’s work, was a predecessor to the ENIAC. The Mark I was also used to compute mathematical tables.
After fifteen centuries, the publication of books of mathematical tables essentially ceased in about 1975, at which point the numerical tables were replaced in common use by HP scientific calculators, where they had not been replaced earlier by larger and more expensive computers. (Bowditch’s American Practical Navigator is one of the few remnants that still includes mathematical tables, even in the 2002 edition, the latest as of this writing; pp.565–727 are devoted to tables, most of which are mathematically calculated rather than empirically measured.)
The kind of mathematical scholarship formerly devoted to publishing mathematical tables is nowadays devoted to publishing software instead. Pages 325 to 344 of Burington’s handbook are devoted to a table of n², n³, √n, √(10n), and the cube roots of n, 10n, and 100n; given a computer that can multiply integers, the square-root columns of this table can be replaced in practice with a single line of C code, which implements the "Babylonian method" of computing square roots given by Hero of Alexandria (Ἥρων ὁ Ἀλεξανδρεύς) a hundred generations ago:
g,h;q(n){h=n;if(n)do g=h,h=(g+n/g)/2;while(g-h+2&~3);return h-=h*h>n;}
However, this line of C code, although it works and replaces two sevenths of those twenty pages of numbers, falls short of many of the ideals of scholarship, as I will explore below.
Several times a week, I use GNU units(1), the GNU version of the BSD (?) units(1) program, which is a simple desk calculator program which contains a comprehensive database of over two thousand units of measurement, compiled over decades by the tireless scholarship of Adrian Mariano at Cornell. The original uses were things like this:
You have: 60 miles per hour
You want: furlongs per fortnight
* 161280
/ 6.2003968e-06
But units(1) also allows me to immediately calculate things like how many minutes of arc the sun subtends on average:
You have: radians (2*sunradius)/sundist
You want: arcminute
* 31.988013
/ 0.03126171
Or, in case I’ve forgotten, that a mole of ideal gas at STP occupies 22.7ℓ:
You have: gasconstant
You want:
Definition: 8.314472 J / mol K = 8.314472 kg m^2 / K mol s^2
You have: gasconstant * 1 mole * tempC(0) / 100 kPa
You want: liters
* 22.71098
/ 0.044031565
Or that Israel’s new Sorek reverse-osmosis desalination plant, which is reported to use 70 atmospheres of pressure for the reverse- osmosis process step, is doing 7.1kJ of mechanical work per liter, putting a lower bound on the economic cost of the freshwater obtained:
You have: 70 atmospheres
You want: kilojoules per liter
* 7.09275
/ 0.14098904
And that that is US$200 per megaliter or US$240 per acre foot if the energy used to drive it costs US$100 per megawatt hour:
You have: 70 atmospheres * US$ 100/MWh
You want: US$ / megaliter
* 197.02083
/ 0.0050756054
You have: 70 atmospheres * US$ 100/MWh
You want: US$ / acrefoot
* 243.02308
/ 0.0041148356
I can calculate that 10 amperes of electrical current through a round
22-gauge wire amounts to 31 amps per square millimeter; I have to be
careful here, because units’s wiregauge unit gives me the wire
diameter, but its circlearea unit wants the radius:
You have: 10 amps / circlearea(wiregauge(22)/2)
You want: amperes / mm^2
* 30.718763
/ 0.032553394
If I include the resistivity of copper and the equation P = I²R in the calculation, I can get the heat production:
You have: (10 amps)^2 * 16.78 nanoohms * meter / circlearea(wiregauge(22)/2)
You want: watts/meter
* 5.1546084
/ 0.19400116
(That’s enough heat to be dangerous if used as electrical wiring, but not enough to use as a heating element, except maybe in an electrical blanket or something.)
That was actually my second try at that calculation; on the first try, I accidentally calculated the voltage drop per meter instead, which units was able to catch, but not provide a useful error message for:
You have: 16.78 nanoohms * meter * 10 amps / circlearea(wiregauge(22)/2)
You want: watts / meter
conformability error
0.51546084 kg m / A s^3
1 kg m / s^3
You have: 16.78 nanoohms * meter * 10 amps / circlearea(wiregauge(22)/2)
You want:
Definition: 0.51546084 kg m / A s^3
It happens that kg m / A s³ is equivalent to volts per meter, but that’s not at all obvious from looking at the result.
You can see that as the examples get more complex, more and more of the knowledge is in my head, and less and less in the software.
Actually, I didn’t have the resistivity of copper memorized in my head; I looked it up on Wikipedia and typed it in. There’s a Wikidata entry for copper but it doesn’t yet include electrical resistivity as a property. It does include its density, but in the format “8.94±0.01 Q15639371”, which units(1) cannot parse; Q15639371 is the Wikidata name for “grams per cubic centimeter.” There was an entry for copper in Freebase.com too, but I don’t know if it included electrical resistivity.
Stephen Wolfram also has a broad vision of part of what I am calling executable scholarship. He calls it a “computational knowledge engine”, and being Stephen Wolfram, he plans to own it, and his version doesn’t cite any sources.
The Wolfram version is called “Wolfram|Alpha”, but I decline to use marketing-provided punctuation in the middle of proper nouns like “Yahoo!” and “Re/Code”, because I don’t hate my readers enough. Instead I will spell out the term as “Wolfram Vertical Line Alpha” when necessary.
Wolfram Vertical Line Alpha does include the electrical resistivity of copper, and given the input “resistivity of copper * (10 amps)^2 / 0.326 mm^2”, it correctly computes 5.2 watts per meter. It also claims that the size of 22AWG wire is 0.324 mm², which differs by about 0.5% from the value given by units(1), and will return it given the input “22awg”, but I haven’t found a way to get Alpha to combine these pieces of knowledge in a single formula.
What is scholarship?
valid trustworthy well-known
reproducibility self-correcting interlinked honest objective stable readable public reveals preferred embodiment not ignorant creative? documentable? replicable? peer-reviewable?
[Boyer 1990] proposed expanding the definition of “scholarship” to include not only the discovery of knowledge but also teaching, application (“the use of new knowledge in solving society's problems”), and what he calls “integration” (“where new relationships among disciplines are discovered”, a definition broad enough to include any research). Boyer was unable to publish his proposed redefinition in a peer-reviewed venue, publishing it instead through a phony “Carnegie Foundation” whose declared purpose is to promote the interests of teachers, which is to say, university faculty who do not engage in scholarship; not satisfied with this, however, he additionally proposed to expand “scholarship” to include practitioners of a field in general, except those who use only old knowledge.
[Boyer 1990]: E. Boyer, Scholarship reconsidered: Priorities for the professoriate, published by the “Carnegie Foundation for the Advancement of Teaching.”
The line of code I gave above to calculate the square root of an integer falls short of the aims of scholarship for several reasons. It's hard to read (made unnecessarily so by a )
On the computer I’m typing this on, square roots are generally calculated by proprietary Intel hardware designs that I am not able to see, because using specialized hardware makes it much faster, and
tested using a function called sqrt_test.
. But the free software I run the computer on has an option to calculate square roots using addition, subtraction, comparison, bit shifts, and iteration, called _FP_SQRT_MEAT_E.
http://www.uclibc-ng.org/browser/uclibc-ng/test/math/libm-test.inc?rev=fdebbe2044653c5c84172524ed6e036d38716d88#L4446
Consider my