Human embryology again

Returning to the topic of human embryology, here is a human fetal timeline for the first 16 weeks post fertilisation (obstetricians count from the LMP = last menstrual period, which adds about 2 weeks). It is a little disturbing quite how much scientific misinformation is being circulated in regard to the topic. False information is not conducive to honest debate, and is highly corrosive of the trust people have in professionals such as scientists (it’s also unethical on both religious and Kantian grounds). In particular, contrary to what some have suggested:

Fetal length data in the table is mostly from here. Except where indicated, linked images are subsequent to miscarriage or to surgery to resolve ectopic pregnancy, so may be distressing to some readers.

Week post fertilisation Week post LMP Fetal length Image
1 3 0.01 cm / 0.005 inches 8–cell image
2 4 0.02 cm / 0.008 inches
3 5 0.1 cm / 0.04 inches heart begins beating at 21 days
4 6 0.5 cm / 0.2 inches image on flickr
5 7 1 cm / 0.4 inches image on wikimedia
6 8 1.6 cm / 0.6 inches image on flickr
7 9 2.3 cm / 0.9 inches image on flickr
8 10 3.2 cm / 1.3 inches
9 11 4.1 cm / 1.6 inches
10 12 5.4 cm / 2.1 inches
11 13 6.7 cm / 2.6 inches ultrasound image
12 14 14.7 cm / 5.8 inches
13 15 16.7 cm / 6.6 inches
14 16 18.6 cm / 7.3 inches
15 17 20.4 cm / 8 inches ultrasound image
16 18 22 cm / 8.7 inches

Below (from here) is a chart of heart development:


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


Explaining Doppler ultrasound

Recently, I posted something about fetal heartbeats. This seems to be a hot political topic at the moment in the US. Many people don’t seems to understand that, as I noted in my earlier post, the human fetus has a functional (though not yet fully developed) heart from about 21 days after conception.

There seems to be an even greater confusion about the physics of Doppler ultrasound machines. These do not “detect electrical signals,” as has been suggested, but are essentially “speed cameras” for blood (although they use sound rather than radio waves). They detect movement, because movement causes a change in sound frequency, through the Doppler effect. When the heart contracts and the blood is moving fastest, the velocity signal is greater. The pulsing of the resulting velocity signal matches the pulsing of the heart, and can be made audible, although the timbre of the resulting heartbeat sound will not be identical to what a microphone would detect.

The graphic below summarises the operation of the device, at three moments in time (A, B, C). The sound signal sent and received is shown in blue. At the bottom of the graphic is the velocity signal, with the three moments A, B, C marked.


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)


Fetal heartbeat and political debate

Following the recent Dobbs v. Jackson Women’s Health Organization decision in the US, that nation is struggling with two moral/philosophical questions:

  • Is the unborn human fetus a person, and if so from when?
  • Does the unborn human fetus deserve legal protection, and if so from when?

These are independent questions – a puppy or kitten is not a person, but nevertheless has legal protection from animal cruelty. The now-repealed Roe v. Wade (1973) and Planned Parenthood v. Casey (1992) decisions essentially answered the second question as “yes, from viability.”

These questions are moral and philosophical, not scientific. However, scientific questions do arise in the debate. Is the fetus alive? Yes, obviously. Is it human? Well, it has different hemoglobin from adult humans, but the fetus is obviously Homo sapiens. Does the fetus have a heartbeat? Surprisingly, that seems to be controversial, although every textbook I have seen agrees that the heart is functional very early (how else would the developing fetus get oxygen and nutrients?). To quote some sources:

  • “The heart is the first organ to develop. In the human embryo, the heart begins beating at about 21 days after conception [i.e. 5 weeks after LMP = last menstrual period].” (Anatomy and Physiology of the Circulatory and Ventilatory Systems, page 2)
  • “The fetal–placental circulation begins at about 9 days postfertilization … A functional circulation is established by the end of the third developmental week [i.e. 5 weeks after LMP].” (Fetal MRI, page 405)
  • “In a developing embryo, the heart has developed enough by day 21 post-fertilization to begin beating [i.e. 5 weeks after LMP]. Circulation patterns are clearly established by the fourth week of embryonic life. It is critical to the survival of the developing human that the circulatory system forms early to supply the growing tissue with nutrients and gases, and to remove waste products.” (Anatomy and Physiology 2e)
  • “Circulation of fetal blood in the placental circulation begins approximately 21 days postfertilization in humans [i.e. 5 weeks after LMP].” (Handbook of Developmental Neurotoxicology, page 68)
  • “1. Fetal heart development begins during the first month of gestation. At about 21 days of gestation, the fetal heart begins beating, and blood begins circulating [i.e. 5 weeks after LMP]. Between the second and seventh weeks of gestation [i.e. 4 to 9 weeks after LMP], the primitive fetal heart undergoes a series of changes that create the four-chambered heart and its great arteries. 3. During gestation, the lungs are nonfunctional, and fetal oxygenation occurs via the placenta.” (Pediatric Nursing, page 223)

In my view, accurate discussion of the scientific facts is a necessary preliminary to addressing the moral and philosophical questions.

Doppler ultrasound is routinely used to detect fetal heartbeat and the velocity of fetal blood flow. The scientific principle known as the Doppler effect allows the detection of motion. It is the principle behind speed cameras, and it allows bats to “hear” the fluttering wings of a distant insect. During first-trimester screening (at around 11 to 13 weeks after LMP), professional Doppler ultrasound devices are sensitive enough not only to detect fetal heartbeat, but to detect blood flow abnormalities in various parts of the fetal circulatory system.


Fetal heartbeat at 13 weeks (from here). S is the ventricular systolic wave, D the early diastolic, A the atrial contraction.


Fictional Scientists and Mathematicians

I have been reflecting on fictional mathematicians and scientists. The image above shows four:

None of these are terribly good role models, it seems to me. Literature and cinema have some better examples, but on the whole, mathematicians and scientists are not treated well by fiction authors.

Milton Millhauser, in “Dr. Newton and Mr. Hyde: Scientists in Fiction from Swift to Stevenson” (Nineteenth-Century Fiction, 1973, Vol. 28, No. 3, pp. 287–304) notes that, in literature, “broadly speaking the scientist is either disregarded or held up to contempt and ridicule.” But perhaps that is because some writers (Michael Crichton is a notable example) see dangers and dilemmas that demand exploration.

However, female scientists and mathematicians seem to be portrayed somewhat more positively. For example:

  • Eleanor Arroway from Contact by Carl Sagan.
  • Sarah Harding from The Lost World by Michael Crichton.
  • Catherine Llewellyn from the play (and later film) Proof by David Auburn.
  • Grace Augustine from the film Avatar by James Cameron.


Brouwer and his fixed point theorem

    

The Brouwer fixed-point theorem is one of my favourite mathematical theorems. It is named after the Dutch mathematician Luitzen Egbertus Jan Brouwer (above right). Brouwer is also known for his work in Intuitionism. I have mentioned the Brouwer fixed-point theorem before.

The theorem states that any continuous function f on a compact convex set (and specifically, on a disc in the plane) will have at least one fixed point – that is, there will be at least one point p such that f(p) = p. The picture below is intended to illustrate the theorem; it is explained further down.

In the case of a disc, the theorem can be proved by contradiction. Assume that f(p) ≠ p for every point p. Then the pair of f(p) and p always defines a continuous mapping g from p to the boundary of the disc, as illustrated above (left). However, such a continuous mapping is impossible (for complex reasons, but in simple terms, because it creates a hole, which continuous mappings cannot do).

So what about that picture? It shows a continuous function f from the disc to itself, combining an irregular rotation about the centre (rotating least towards the east of the disc) with a “folding” operation that leaves the centre and boundary untouched. The picture below shows a cross-section of the folding in action. The shades of blue in the picture above show how far each point p is from f(p), with lighter colours representing smaller values. Arrows show the action of the function on 6 randomly chosen points. There are two fixed points, marked with black dots: the centre and one other point where the folding and the irregular rotation cancel each other out.

The three-dimensional version of the theorem tells us that, when I stir my morning cup of coffee, at least one speck of liquid will wind up exactly where it started.