Modal logic, ethics, and obligation

Recently, I posted about necessary truth, the logic of belief, and epistemic logic. I would like to follow up on that one more time by discussing deontic logic, the logic of obligation and moral action. We can capture this concept using the 4 rules of D4 modal logic. The first 3 of these are the same as those I used for belief. I am replacing the previous modal operators with  Ⓞ  which is intended to be read as “it is obligatory that” (hence the O in the circle):

  • if P is any tautology, then  Ⓞ P
  • if  Ⓞ P  and  Ⓞ (PQ)  then  Ⓞ Q
  • if  Ⓞ P  then  Ⓞ Ⓞ P
  • if  Ⓞ P  then  ~ Ⓞ ~P

where  ~ Ⓞ ~P  is read as “~ P is not obligatory,” i.e. “P is permissible.” For those who prefer words rather than symbols:

  • if P is any tautology, then P is obligatory
  • if P and (P implies Q) are both obligatory, then Q is obligatory
  • if P is obligatory, then it is obligatory that P is obligatory
  • if P is obligatory, then P is permissible

For these rules as they stand, the only things that are obligatory are necessary truths like 2 + 2 = 4. This is because you can’t get an “ought” from an “is.” Apart from the first rule, there is no way of introducing a  Ⓞ  symbol out of nowhere. Consequently, if we are to reason about ethics and morality, we must begin with some deontic axioms that already contain the  Ⓞ   symbol. For people of faith, these deontic axioms may be given by God, as in the 10 Comandments, which include:

Ⓞ  you do not murder.
Ⓞ  you do not commit adultery.
Ⓞ  you do not steal.
Ⓞ  you do not bear false witness against your neighbor.

Immanuel Kant famously introduced the categorical imperative, a deontic axiom which Kant thought implied all the other moral rules, and thus provided the smallest possible set of deontic axioms:

Ⓞ  [you] act only according to that maxim whereby you can, at the same time, will that it should become a universal law.

Others have suggested the greatest happiness of the greatest number as a principle. Fyodor Dostoevsky, William James, and Ursula Le Guin are among those who have explained the problem with this:

Tell me yourself, I challenge your answer. Imagine that you are creating a fabric of human destiny with the object of making men happy in the end, giving them peace and rest at last, but that it was essential and inevitable to torture to death only one tiny creature – that baby beating its breast with its fist, for instance – and to found that edifice on its unavenged tears, would you consent to be the architect on those conditions?” (Fyodor Dostoevsky, “The Grand Inquisitor,” in The Brothers Karamazov, 1880; 4.35 on Goodreads)

Or if the hypothesis were offered us of a world in which Messrs. Fourier’s and Bellamy’s and Morris’s Utopias should all be outdone and millions kept permanently happy on the one simple condition that a certain lost soul on the far-off edge of things should lead a life of lonely torture, what except a specifical and independent sort of emotion can it be which would make us immediately feel, even though an impulse arose within us to clutch at the happiness so offered, how hideous a thing would be its enjoyment when deliberately accepted as the fruit of such a bargain?” (William James, “The Moral Philosopher and the Moral Life,” 1891)

Some of them understand why, and some do not, but they all understand that their happiness, the beauty of their city, the tenderness of their friendships, the health of their children, the wisdom of their scholars, the skill of their makers, even the abundance of their harvest and the kindly weathers of their skies, depend wholly on this child’s abominable misery.” (Ursula K. Le Guin, “The Ones Who Walk Away from Omelas,” 1973; reprinted in The Wind’s Twelve Quarters, 1975; 4.05 on Goodreads)

The meaning of deontic statements can be described using Kripke semantics, which exploits the idea of possible worlds (i.e. alternate universes). To say that some statement is obligatory is to say that the statement would be true in better possible worlds (we write w1 → w2 to mean that w2 is a better possible world than w1).

In any given world v, the statement  Ⓞ P  is equivalent to :

  • P  is true in all better worlds wi (i.e. all those with v → wi)

Likewise, in any given world v, the statement  ~ Ⓞ ~P  (P is permissible) is equivalent to:

  • P  is true in at least one better world wi (i.e. one with v → wi)

The rules of deontic logic imply two conditions on these arrows between possible worlds:

  • if  w1 → w2 → w3  then  w1 → w3  (i.e. chains of arrows are treated like arrows too)
  • in every world v there is at least one arrow  v → w  (i.e. chains of arrows don’t stop; this includes the case of  v → v)

A number of philosophers have suggested that deontic logic leads to paradoxes. In all cases that I have seen, these “paradoxes” have involved simple errors in the use of deontic logic – errors that become obvious when the deontic statements are translated into statements about possible worlds.

There are limitations to deontic logic, however. For example, if we say that it is obligatory not to steal, this means that, in all better possible worlds, nobody steals. If we also say that it is obligatory to punish thieves, this means that, in all better possible worlds, thieves are punished. However, if it is obligatory not to steal, better possible worlds have no thieves, so the two statements do not combine well.

Some people would, no doubt, suggest that fiction like that of Dostoevsky is a better tool than logic for exploring such issues. In cases where the writer is a genius, they are probably right.


In this post series: logic of necessary truth, logic of belief, logic of knowledge, logic of obligation


Modal logic, knowledge, and an old joke

Recently, I posted about necessary truth and about the logic of belief. I would like to follow up on that by discussing epistemic logic, the logic of knowledge. Knowledge is traditionally understood as justified true belief (more on that below), and we can capture the concept of knowledge using the 4 rules of S4 modal logic. These are in fact the same 4 rules that I used for for necessary truth, and the first 3 rules are the same as those I used for belief (the fourth rule adds the fact that knowledge is true).

Knowledge is specific to some person, and I am replacing the previous modal operators with  Ⓚ  which is intended to be read as “John knows” (hence the K in the circle):

  • if P is any tautology, then  Ⓚ P
  • if  Ⓚ P  and  Ⓚ (PQ)  then  Ⓚ Q
  • if  Ⓚ P  then  Ⓚ Ⓚ P
  • if  Ⓚ P  then  P

For those who prefer words rather than symbols:

  • if P is any tautology, then John knows P
  • if John knows both P and (P implies Q), then John knows Q
  • if John knows P, then John knows that he knows P
  • if John knows P, then P is true

Epistemic logic is useful for reasoning about, among other things, electronic commerce (see this paper of mine from 2000). How does a bank know that an account-holder is authorising a given transaction? Especially if deceptive fraudsters are around? Epistemic logic can highlight which of the bank’s decisions are truly justified. For this application, the first rule (which implies knowing all of mathematics) actually works, because both the bank’s computer and the account-holder’s device can do quite sophisticated arithmetic, and hence know all the mathematical facts relevant to the transaction they are engaged in.

But let’s get back to the idea of knowledge being justified true belief. In his Theaetetus, Plato has Theaetetus suggest exactly that:

Oh yes, I remember now, Socrates, having heard someone make the distinction, but I had forgotten it. He said that knowledge was true opinion accompanied by reason [ἔφη δὲ τὴν μὲν μετὰ λόγου], but that unreasoning true opinion was outside of the sphere of knowledge; and matters of which there is not a rational explanation are unknowable – yes, that is what he called them – and those of which there is are knowable.” (Theaetetus, 201c)

Although he also uses essentially this same definition in other dialogues, Plato goes on to show that it isn’t entirely clear what kind of “justification” or “reason” is necessary to have true knowledge. In a brief 1963 paper entitled “Is Justified True Belief Knowledge?,” the philosopher Edmund Gettier famously took issue with the whole concept of justified true belief, and provided what seemed to be counterexamples.

My personal opinion, which I have argued elsewhere, is that “justified true belief” works fine as a definition of knowledge, as long as the justification is rigorous enough to exclude beliefs which are “accidentally correct.” For analysing things like electronic commerce, a sufficient level of rigour would involve the use of epistemic logic, as described above.

One of Gettier’s supposed counterexamples involves a proposition of the form  P ∨ Q  (P or Q) such that:

  • Smith believes and knows  P ⇒ (PQ)
  • Smith believes P
  • P is false
  • Q is true, and therefore so is  P ∨ Q

From these propositions we can use doxastic logic to infer that Smith believes the true statement  P ∨ Q,  but we cannot infer (using epistemic logic) that Smith knows  P ∨ Q. A famous old joke is perhaps relevant here:

A physicist, a philosopher, and a mathematician are travelling through Scotland by train. Through the window, they observe a black sheep in a field. ‘Aha,’ says the physicist, ‘I see that Scottish sheep are black!’ The philosopher responds, ‘No! Some Scottish sheep are black!’ The mathematician, looking shocked, replies: ‘What are you guys saying? All we know is that at least one sheep in Scotland is black on at least one side.’


In this post series: logic of necessary truth, logic of belief, logic of knowledge, logic of obligation


Modal logic, belief, and the “flat earth”

Recently, I posted about necessary truth. I would like to follow up on that by discussing doxastic logic, the logic of belief. We can capture this concept using the 3 rules of K4 modal logic, which are in fact identical to the first 3 rules for necessary truth. The main difference is that beliefs need not be true.

Since beliefs are specific to some believing person, I am replacing the modal operator  □  with  Ⓙ  which is intended to be read as “John believes” (hence the J in the circle):

  • if P is any tautology, then  Ⓙ P
  • if  Ⓙ P  and  Ⓙ (PQ)  then  Ⓙ Q
  • if  Ⓙ P  then  Ⓙ Ⓙ P

For those who prefer words rather than symbols:

  • if P is any tautology, then John believes P
  • if John believes both P and (P implies Q), then John believes Q
  • if John believes P, then John believes that he believes P

These rules are very useful for helping computer systems (such as autonomous vehicles) reason about the beliefs of other entities (“If John believed a car was coming, he would not cross the road. But he is crossing the road. Therefore he does not believe that a car is coming. We should warn him.”).

As stated above, however, the rules are extremely optimistic about John’s knowledge of mathematics and logic. For some applications, we may need to assume that John believes less of that stuff. There is also a problem in assuming that John accepts the logical consequences of his beliefs. Real people do not always do that. Some years ago, I posted about the idea of a flat earth (an idea that medieval people were too wise to accept). Most believers in a “flat earth” do not accept the logical consequences of their beliefs. In particular, for the most popular “flat earth” model, sunsets would never be observed, because the sun always remains above the “disc of the earth.” Flat-earthers refuse to admit such consequences. Common air travel routes to and from Australia would also be impossible (see below), but flat-earthers generally realise the incompatibility of those routes, and simply deny that they exist. Logic is perhaps not the best tool for describing such patterns of thought.

The meaning of doxastic statements can be described using Kripke semantics, which exploits the idea of possible worlds (i.e. alternate universes). To say that John believes some statement is to say that the statement is true in the alternate universes that John thinks he might be living in. In those alternate universes, the earth might indeed be flat.


In this post series: logic of necessary truth, logic of belief, logic of knowledge, logic of obligation


Modal logic, necessity, and science fiction

A necessary truth is one that is true in all possible universes. We can capture the concept of necessary truth with the 4 rules of S4 modal logic (where □ is read “necessarily”):

  • if P is any tautology, then  □ P
  • if  □ P  and  □ (PQ)  then  □ Q
  • if  □ P  then  □ □ P
  • if  □ P  then  P

For those who prefer words rather than symbols:

  • if P is any tautology, then P is necessarily true
  • if P and (P implies Q) are both necessarily true, then Q is necessarily true
  • if P is necessarily true, then it is necessarily true that P is necessarily true
  • if P is necessarily true, then P is true (in our universe, among others)

The first rule implies that the truths of mathematics and logic (□ 2 + 2 = 4, etc.) are necessary truths (they must obviously be so, since one cannot consistently imagine an alternate universe where they are false). The second rule implies that the necessary truths include all logical consequences of necessary truths. The last two rules imply that  □ P  is equivalent to  □ □ P,  □ □ □ P,  etc. In other words, there is only one level of “necessary” that needs to be considered.

As it stands, these rules only allow us to infer the truths of mathematics and logic (such as  □ 2 + 2 = 4). One must add other necessary axioms to get more necessary truths than that. A Christian or Muslim might, for example, add “Necessarily, God exists,” and spend time exploring the logical consequences of that.

Countless things that are true in our universe are not necessarily true, such as “Water freezes at 0°C” or “Trees are green” or “Bill Clinton was President of the United States in the year 2000.”

For historical truths like the latter, it’s obvious that they are contingent on events, rather than being necessary. There is a substantial body of “alternate history” fiction which explores alternatives for such contingent truths, such as these four novels (pictured above):

  • Fatherland (Robert Harris, 1992): a detective story set in a universe where Hitler won the war; it is the week leading up to his 75th birthday (3.99 on Goodreads)
  • The Peshawar Lancers (S.M. Stirling, 2002): European civilisation is destroyed by the impact of comet fragments in 1878; a new Kiplingesque Anglo-Indian steampunk civilisation arises (3.86 on Goodreads)
  • SS-GB (Len Deighton, 1978): Hitler defeats Britain in 1941; British police face moral dilemmas cooperating with the SS (3.74 on Goodreads)
  • Romanitas (Sophia McDougall, 2005): the Roman Empire is alive and well in present-day London; slaves are still crucified (3.24 on Goodreads; first of a trilogy)

Three plant pigments: green beech, brown kelp, and red gracilaria algae (cropped from photographs by Simon Burchell, Stef Maruch, and Eric Moody)

The truths of biology are just as contingent as the truths of history. Trees are (mostly) green, but even on our own planet, brown and red are viable alternative colours for plants. From an evolutionary perspective, Stephen Jay Gould expresses the contingency this way:

any replay of the tape [of life] would lead evolution down a pathway radically different from the road actually taken.” (Stephen Jay Gould, Wonderful Life: The Burgess Shale and the Nature of History, 1989)

(some of his colleagues would take issue with the word “radically,” but still accept the word “different”). From a Christian point of view, the contingency of biology follows from the doctrine of the “Free Creation” of God, independently of any beliefs about evolution. To quote Protestant theologian Louis Berkhof:

God determines voluntarily what and whom He will create, and the times, places, and circumstances, of their lives.” (Louis Berkhof, Systematic Theology, Part I, VII, D.1.c)

The Catholic Church shares the same view, as none other than Thomas Aquinas makes clear (using the terminology of necessary truth):

It seems that whatever God wills He wills necessarily. For everything eternal is necessary. But whatever God wills, He wills from eternity, for otherwise His will would be mutable. Therefore whatever He wills, He wills necessarily. … On the contrary, The Apostle says (Ephesians 1:11): ‘Who works all things according to the counsel of His will.’ Now, what we work according to the counsel of the will, we do not will necessarily. Therefore God does not will necessarily whatever He wills.” (Summa Theologiae, Part I, 19.3)

Having taken this line, one might ask why mathematical truths are necessary rather than contingent. The astronomer Johannes Kepler resolves this problem this by telling us that they are not created:

Geometry existed before the Creation, is co-eternal with the mind of God.” (Johannes Kepler, Harmonices Mundi)

In fiction, alternative biologies are normally explored in the context of some other planet, because alternate earths are pretty much logically equivalent to other planets. Here are four examples of fictional biology:

  • Out of the Silent Planet (C.S. Lewis, 1938): written from a Christian perspective, this novel has three intelligent humanoid alien species living on the planet Mars (3.92 on Goodreads; see also my book review)
  • The Mote in God’s Eye (Larry Niven and Jerry Pournelle, 1974): this novel is one of the best alien-contact novels ever written (4.07 on Goodreads)
  • the xenomorph from the film Aliens (1986)
  • the Klingon character Worf from the TV series Star Trek: The Next Generation (1987–1994)

The truths of physics are contingent as well; our universe could have been set up to run on different rules. Science fiction authors often tweak the laws of physics slightly in order to make the plot work (most frequently, to allow interstellar travel). Fantasy authors invent alternate universes which differ from ours far more dramatically:

  • Dune (Frank Herbert, 1965): faster-than-light travel is a feature of the plot; it follows that interstellar navigation requires looking into the future (4.25 on Goodreads; see also my book review)
  • Great North Road (Peter F. Hamilton, 2012): “Stargate” style portals are a key feature of this novel (4.07 on Goodreads)
  • The Many-Coloured Land (Julian May, 1981): a science fiction incorporating psychic powers (4.07 on Goodreads; first of a series)
  • Magician (Raymond E. Feist, 1982): a classic fantasy novel which explores some of the internal logic of magic along the way (4.31 on Goodreads; first of a series)

Because mathematical truths are necessary truths, they are potentially common ground with intelligent aliens. This is one theme in the book (later film) Contact:

‘No, look at it this way,’ she said smiling. ‘This is a beacon. It’s an announcement signal. It’s designed to attract our attention. We get strange patterns of pulses from quasars and pulsars and radio galaxies and God-knows-what. But prime numbers are very specific, very artificial. No even number is prime, for example. It’s hard to imagine some radiating plasma or exploding galaxy sending out a regular set of mathematical signals like this. The prime numbers are to attract our attention.’” (Carl Sagan, Contact, 1985; 4.14 on Goodreads)

Of course, Carl Sagan or his editor should have realised that 2 is prime. Even intelligent beings can make mistakes.


In this post series: logic of necessary truth, logic of belief, logic of knowledge, logic of obligation


Fetal development: what about marsupials and birds?

Recently, I posted something about fetal heartbeats. In humans (and in mammals generally), oxygen and nutrients are transferred by the mother’s circulatory system to the placenta, and from there by the separate fetal circulatory system to where they are needed. As I noted in my earlier post, this process is functional in humans at about 21 days after conception.

In order for this process to work, the fetus obviously needs a beating, functional heart (although the heart continues to develop after it starts beating). It also requires a different kind of hemoglobin, which binds more tightly to oxygen than the mother’s hemoglobin does, thus facilitating oxygen transport across the placenta in one direction. Waste products, including carbon dioxide, are transported across the placenta in the other direction. The water-filled lungs, obviously, play no role in absorbing oxygen or getting rid of carbon dioxide.


Human fetal circulatory system, showing the ductus venosus and ductus arteriosus which partially divert blood away from the liver and the water-filled lungs (from American Heart Association)

There are alternatives to this placental system, however. Marsupials, such as kangaroos, do not have the same kind of placenta. Kangaroos are therefore not able to survive in the womb longer than about a month. Instead, they are born in a partially developed state, and crawl to the pouch, where they complete their development drinking milk and breathing air with their still-developing lungs.


Young joey (baby kangaroo) in its mother’s pouch (photo by Geoff Shaw)

Birds have yet another approach, developing inside an egg. Nutrients are packaged inside the egg along with the embryo. Oxygen and carbon dioxide diffuse in and out through the eggshell, and oxygen is absorbed by the embryo through the allantois. The allantois also acts as a dumping ground for nitrogenous waste. When the nutrients in the egg are exhausted, it is time for the bird to hatch.


Chicken embryo on its 9th day (image by KDS4444)


Looking back: 2001

The 1968 film 2001: A Space Odyssey suggested that we would have extensive space flight in 2001. That turned out not to be the case. What we did get was the September 11 attacks on the USA and the military conflicts which followed. Nevertheless, NASA commemorated the film with the 2001 Mars Odyssey orbiter.

Films of 2000 included the superb The Lord of the Rings: The Fellowship of the Ring, several good animated films (including Monsters, Inc., Shrek, and Hayao Miyazaki’s Spirited Away), the wonderful French film Amélie, some war movies (Enemy at the Gates was good, but Black Hawk Down distorted the book too much for my taste), the first Harry Potter movie, and an award-winning biographical film about the mathematician John Nash.

In books, Connie Willis published Passage, one of my favourite science fiction novels, while Ian Stewart explained some sophisticated mathematics simply in Flatterland.

Saul Kripke (belatedly) received the Rolf Schock Prize in Logic and Philosophy for his work on Kripke semantics, while Ole-Johan Dahl and Kristen Nygaard (also belatedly) received the Turing Award for their work on object-oriented programming languages (both these pioneers of computing died the following year).

The year 2001 also saw the completion of the Cathedral of Saint Gregory the Illuminator in Armenia, which I have sadly never visited.

In this series: 1978, 1980, 1982, 1984, 1987, 1989, 1991, 1994, 2000, 2001, 2004, 2006, 2009.


Cycles

The Tropical Year: 31.6888 nHz

One of the most important cycles we live by is the tropical year, measured from equinox to corresponding equinox (or solstice to corresponding solstice). The tropical year lasts, on average, 365.2422 days (365 days, 5 hours, 48 minutes, 45 seconds), which means that it is an oscillation with a frequency of 31.6888 nanohertz (nHz). This is the cycle of the seasons.

Spring, summer, autumn, and winter are the conventional seasons, but the tropical year may be split up into more than or less than four seasons, and these need not be of equal length. In northern Australia, a frequent division is “the dry” (May to September), “the build up” (September to December), and “the wet” (December to April). Local Aboriginal people, however, may recognise as many as six seasons.

The Sidereal Year: 31.6875 nHz

A sidereal year is the time taken by the Earth to orbit the Sun once with respect to the stars. This is the time that it takes for the sun to move through the “signs of the zodiac.” Because of the precession of the equinoxes, the sidereal year is 365.2564 days, which is about 20.4 minutes longer than the tropical year. As a result, ancient rules assigning dates to the signs of the zodiac are now completely wrong. The sidereal year corresponds to an oscillation of 31.6875 nanohertz.

The Synodic Month: 391.935 nHz

A synodic month is a cycle from new moon to new moon or full moon to full moon. This period actually varies by several hours, but it averages out to 29.530588 days (29 days, 12 hours, 44 minutes, 3 seconds).

In 432 BC, Meton of Athens noted that 235 synodic months (6939.7 days) is almost exactly equal to 19 years (6939.6 days). This period is called the Metonic cycle, and is used for predicting solilunar events such as the date of Easter.

The synodic month is also strangely similar to the average menstrual cycle (28 days), and this is reflected in the word (“menstrual” derives from the Latin mēnsis = month).

The Week: 1.65344 µHz

The week has an origin among the ancient Hebrews. It also has a Babylonian origin (the relationship between the two origins is unclear). The Babylonians related the 7 days of the week to the sun, moon, and 5 visible planets. They also related them to various gods. Our days of the week derive from the Babylonian week, via Greece and Rome: Sunday (Sun), Monday (Moon), Tuesday (Tiw, god of war = Mars), Wednesday (Woden = Mercury), Thursday (Thor = Jupiter), Friday (Frigg = Venus), and Saturday (Saturn).

Early Christians related the two week concepts together, pointing out that the day of the Resurrection (the day after the Jewish Sabbath) corresponded to the day of the Sun in the Roman system. The week corresponds to an oscillation of 1.65344 microherz.

The Sidereal Day: 11.6058 µHz

A sidereal day is the time that it takes the earth to rotate once around its axis. It often surprises people to discover that this time is 23 hours, 56 minutes, 4.1 seconds. It can be measured by the time to go from a star being overhead to the same star being overhead again.

The Solar Day: 11.5741 µHz

A solar day (24 hours, give or take some seconds) is the time from noon to noon. It is longer than a sidereal day because, while the earth is rotating around its axis, it is also moving around the sun. To put it another way, the sun is not a fixed reference point for the earth’s rotation. The difference between the sidereal and solar days mean that the stars seem to rise about 3 minutes and 56 seconds earlier every night.


Fast Fibonacci numbers

There was some discussion on reddit recently of the Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1,597, 2,584, 4,181, 6,765, 10,946, 17,711, 28,657, 46,368, 75,025, 121,393, 196,418, 317,811, 514,229, 832,040, …) and efficient ways of calculating them.

One way of doing so is using numbers of the form a + b σ where  σ is the square root of 5. Multiplication of such numbers satisfies:

(a + b σ) × (c + d σ) = ac + 5bd + (ad + bc) σ.

We can define the golden ratio φ = (1 + σ) / 2 and also ψ = 1 − φ = (1 − σ) / 2, in which case the nth Fibonacci number Fn will be exactly (φn − ψn) / σ. This is known as Binet’s formula.

We can use this formula to calculate Fibonacci numbers using only integer arithmetic, without ever evaluating the σ. We will have:

(2 φ)n − (2 ψ)n = (1 + σ)n − (1 − σ)n = 0 + p σ

for some integer p, and division by a power of two will give Fn = p / 2n.

I am using the R language, with the gmp package, which provides support for large integer matrices, and this allows us to use the relationship:

If we call this matrix A and calculate An−1, the first number in the resultant matrix will be the nth Fibonacci number Fn. The following R code calculates F10 = 55 using a combination of multiplication and squaring:

n <- 10

A <- matrix.bigz(c(
	1, 1,
	1, 0), 2)

p <- function(n) {
	if (n == 1) A
	else if (n %% 2 == 1) A %*% p(n-1)
	else {
		b <- p(n/2)
		b %*% b
	}
}

p(n-1)[1,1]

This same code will calculate, for example:

The time taken to calculate Fn is approximately proportional to n1.156, with the case of n = 1,000,000,000 (giving a number with 208,987,640 digits) taking about a minute.


Rods and cones in the human eye

I already posted these images (click to zoom) on Instagram. They illustrate the sensitivity to colour of the rods (lower right) and the three types of cones in the human eye. Cone sensitivity data is from CVRL.

Notice that red light is pretty much invisible to the rods. This is why red light does not interfere with night vision, and is used in e.g. this aircraft cockpit:


Greenhouse emissions in Australia

I thought I would take the opportunity today to talk about energy production and greenhouse gas emissions in Australia. The chart below shows the populations (blue bars) and population densities of the six Australian states plus the Northern Territory. Note that New South Wales, Victoria, and Queensland have the highest populations (8.2, 6.7, and 5.2 million respectively), while the Northern Territory has the lowest. However, given its smaller area, Victoria has the highest population density (29.4 people per sq km), while Western Australia and the Northern Territory have the lowest population densities (1.1 and 0.2 people per sq km respectively).

The next chart shows the per capita electricity production of the six Australian states and the Northern Territory, by type. These figures are adjusted for net electricity transfer between states. For example, Tasmania imports some mainland coal-fired power.

Notice that the totals are high in the less densely populated regions (Western Australia and the Northern Territory). The total is also high in Tasmania, because of the widespread use of hydro-electrically produced electricity for heating there.

Total per capita electricity production is lowest in Victoria, in part because of the widespread use of natural gas for heating and cooking (total gas use in Australia generally is about 4 times its use in electricity production). Victorian electricity is the dirtiest, however, with heavy use of brown-coal-fired production. Brown coal is by far the dirtiest fuel; it produces about 47% more greenhouse gases per MWh than black coal, and triple the greenhouse gases per MWh of natural gas.

South Australia has achieved 50% renewable energy, but this is not without its problems:

  • Wind and solar power are more expensive, so that South Australians pay about $360 per MWh for their electricity: 44% more than the two large states
  • The sun does not always shine and the wind does not always blow: this means that, in the absence of massive-scale energy storage, South Australia has to “borrow” coal-fired power from the East, although this is eventually repaid with interest
  • Solar and wind power cause substantial grid stability and grid synchronisation issues, which become very apparent at the 50% renewable level – good solutions are needed for this; South Australia currently copes by turning solar power off

To avoid “borrowing” electricity, massive-scale energy storage is required. South Australia would need several days worth of demand, at 40 GWh per day. Their famous Tesla battery has been expanded to a capacity of just 0.2 GWh, which is about a thousandth of what is needed. Batteries appear inadequate for energy storage at the required scale, and hydrogen storage is probably what we want.

Tasmania operates at a 92% renewable electricity level, thanks to multiple hydroelectric dams, which do not suffer from the problems of wind and solar (and availability is only an issue during lengthy droughts). In addition, hydroelectric dams can also provide energy storage for solar and wind power, simply by pumping water uphill. It is unfortunate that environmental groups in Tasmania have campaigned heavily against hydroelectric power.

The last chart shows the per capita CO2-equivalent emissions for state electricity generation, plus other emissions (including agriculture, other energy use, industrial processes, waste, forestry, and land use change). Agricultural emissions are highlighted in green. A note of caution, however: the electricity generation data is for 2019, but the total greenhouse emissions are for 2018 (the latest I could find). These numbers cannot be compared to those of other countries, unless the numbers for other countries are equally recent and also include the full range of emissions, per UNFCCC standards (some comparable national averages are shown on the left).

Note that net greenhouse emissions for Tasmania are negative, largely due to tree-planting. Per capita emissions for the large, less densely populated areas are higher than those for New South Wales and Victoria; in part due to transportation requirements (shifting commuters and freight from road to rail would help here). Agricultural emissions per capita are particularly high in the Northern Territory, because the impact of cattle farming is being divided among a tiny population of just 0.2 million people. The overall Australian average of 21.2 tonnes per capita is quite significantly affected by the inevitably high emissions for the large, less densely populated areas. There is also the question of whether emissions due to mining and agriculture should be attributed to the producing country, or to the country of final consumption.

Economically and geographically, Australia is in many ways more like a Central Asian country than a European one, given its large size and its heavy reliance on mining and agriculture (Australia’s greenhouse emissions are comparable to those of Kazakhstan, which produces 21.7 tonnes per capita). However, progress could be made in Australia with more energy-efficient housing and transportation.

It should also be emphasised that, given its small population, Australia’s greenhouse emissions make a neglible contribution to the global and regional climate. If increasing atmospheric CO2 has an effect in Australia’s region, that is due primarily to emissions by the large countries of the world, particularly China (which produces about a third of the world’s CO2). Australia should, no doubt, reduce its greenhouse emissions, but whether Australia does so or not will make no measurable difference to the global or regional climate.