## Tutte meets Homfly

Graphs are pretty important objects in mathematics, and in the world — what with every network of any kind being represented by one, from social connections, to road and rail systems, to chemical molecules, to abstract symmetries. They are a fundamental concept in understanding a lot about the world, from particle physics to sociology.

Knots are also pretty important. How can a loop of string get knotted in space? That’s a fairly basic question and you would think we might know the answer to it. But it turns out to be quite hard, and there is a whole field of mathematics devoted to understanding it. Lord Kelvin thought that atoms were knots in the ether. As it turns out, they are not, and there is no ether. Nonetheless the idea was an interesting one and it inspired mathematicians to investigate knots. Knot theory is now a major field of mathematics, part of the subject of topology. It turns out to be deeply connected to many other areas of science, because many things can be knotted: umbilical cords, polymers, quantum observables, DNA… and earphone cords. (Oh yes, earphone cords.) Indeed, knots arise crucially in some of the deepest questions about the nature of space and time, whether as Wilson loops in topological field theories, or as crucial examples in the theory of 3-dimensional spaces, or 3-manifolds, and as basic objects of quantum topology.

Being able to tell apart graphs and knots is, therefore, a pretty basic question. If you’re given two graphs, can you tell if they’re the same? Or if you’re given two knots, can you tell if *they’re* the same? This question is harder than it looks. For instance, the two graphs below look quite different, but they are the “same” (or, to use a technical term, *isomorphic*): each vertex in the first graph corresponds to a vertex in the second graph, in such a way that the connectedness of vertices is preserved.

(Source: Chris-martin on wikipedia)

Telling whether two graphs are the same, or isomorphic, is known as the graph isomorphism problem. It’s a hard problem; when you have very large graphs, it may take a long time to tell whether they are the same or not. As to precisely how hard it is, we don’t yet quite know.

Similarly, two knots, given as diagrams drawn on a page, it is difficult to tell if they are the “same” or not. Two knots are the “same”, or equivalent (or ambient isotopic), if there is a way to move one around in space to arrive at the other. For instance, all three knots shown below are equivalent: they are all, like the knot on the left, unknotted. This is pretty clear for the middle knot; for the knot on the right, have fun trying to see why!

(Source: Stannered, C_S, Prboks13 on wikipedia)

Telling whether two knots are equivalent is also very hard. Indeed, it’s hard enough to tell if a given knot is knotted or knot — which is known as the *unknot recognition* or *unknotting* problem. We also don’t quite know precisely how hard it is.

The fact that we don’t know the answers to some of the most basic questions about graphs and knots is part of the reason why graph theory and knot theory are very active areas of current research!

However, there are some extremely clever methods that can be used to tell graphs and knots apart. Many such methods exist. Some are easier to understand than others; some are easier to implement than others; some tell more knots apart than others. I’m going to tell you about two particular methods, one for graphs, and one for knots. Both methods involve *polynomials*. Both methods are able to tell a lot of graphs/knots apart, but not all of them.

The idea is that, given a graph, you can apply a certain procedure to write down a polynomial. Even if the same graph is presented to you in a different way, you will still obtain the same polynomial. So if you have two graphs, and they give you different polynomials, then they must be different graphs!

Similarly, given a knot, you can apply another procedure to write down a polynomial. Even if the knot is drawn in a very different way (like the very different unknots above), you still obtain the same polynomial. So if you have two knots, and they give you different polynomials, then they must be different knots!

### Bill Tutte and his polynomial

Bill Tutte was an interesting character: a second world war British cryptanalyst and mathematician, who helped crack the Lorenz cipher used by the Nazi high command, he also made major contributions to graph theory, and developed the field of matroid theory.

He also introduced a way to obtain a polynomial from a graph, which now bears his name: the Tutte polynomial.

Each graph has a Tutte polynomial . It’s a polynomial in two variables, which we will call and . So we will write the Tutte polynomial of as .

For instance, the graph below, which forms a triangle, has Tutte polynomial given by .

So how do you calculate the Tutte polynomial? There are a few different ways. Probably the easiest is to use a technique where we simplify the graph step by step in the process. We successively collapse or remove edges in various ways, and as we do so, we make some algebra.

There are two operations we perform to simplify the graph. Each of these two operations “removes” an edge, but does so in a different way. They are called *deletion* and *contraction*. You can choose any edge of a graph, and delete it, or contract it, and you’ll end up with a simpler graph.

First, deletion. To delete an edge, you simply rub it out. Everything else stays as it was. The vertices are unchanged: there are still just as many vertices. There is just one fewer edge. So, for instance, the result of deleting an edge of the triangle graph above is shown below.

The graph obtained by deleting edge from graph is denoted .

Note that in the triangle case above, the triangle graph is connected, and after deleting an edge, the result is still connected. This isn’t always the case: it is possible that you have a connected graph, but after moving an edge , it becomes disconnected. In this case the edge is called a *bridge*. (You can then think of it as a “bridge” between two separate “islands” of the graph; removing the edge, there is no bridge between the islands, so they become disconnected.)

Second, contraction. To contract an edge, you imagine shrinking it down, so that it’s shorter and shorter, until it has no length at all. The edge has vertices at its endpoints, and so these two vertices come together, and combine into a single vertex. So if edge has vertices at its endpoints, then after contracting , the vertices are combined into a single vertex . Thus, if we contract an edge of the triangle graph, we obtain a result something like the graph shown below.

The graph obtained by contracting edge from graph is denoted .

Contracting an edge always produces a graph with 1 fewer edges. Each edge which previous ended at or now ends at . And contracting an edge usually produces a graph with 1 fewer vertices: the vertices are collapsed into .

However, this is not always the case. If the edge had both its endpoints at the *same* vertex, then the number of vertices does not decrease at all! The endpoints and of are then the same point, i.e. , and so they are already collapsed into the same vertex! In this case, the edge is called a loop. Contracting a loop is the same as just deleting a loop.

So, that’s deletion and contraction. We can use deletion and contraction to calculate the Tutte polynomial using the following rules:

- Let’s start with some really simple graphs.
- If a graph has no edges, then it’s just a collection of disconnected vertices. In this case the Tutte polynomial is given by .
- If a graph has precisely
*one*edge, then that it consists of a bunch of vertices, with precisely one edge joining them. If connects two distinct vertices, then it is a bridge, and . - On the other hand, if has precisely one edge which connects a vertex to itself, then it is a loop, and .

- When you take a graph and consider deleting an edge to obtain , or contracting it to obtain , these three graphs have Tutte polynomials which are closely related:
.

So in fact the Tutte polynomial you are looking for is just the

*sum*of the Tutte polynomials of the two simpler graphs and .

However! This rule only works if is*not a bridge or a loop*. If*is*a bridge or a loop, then we have two more rules to cover that situation. - When edge is a bridge, then .
- When edge is a loop, then .

These may just look like a random set of rules. And indeed they are: I haven’t tried to explain them, where they come from, or given any motivation for them. And I’m afraid I’m not going to. Tutte was very clever to come up with these rules!

Nonetheless, using the above rules, we can calculate the Tutte polynomial of a graph.

Let’s do some examples. We’ll work out a few, starting with simpler graph and we’ll work our way up to calculating the Tutte polynomial of the triangle, which we’ll denote .

First, consider the graph consisting of a single edge between two vertices.

This graph contains precisely one edge, which is a bridge, so .

Second, consider the graph consisting of a single loop on a single vertex. This graph also contains precisely one edge, but it’s a loop, so .

Third, consider the graph which is just like the graph , consisting of a single edge between two vertices, but with another disconnected vertex.

This graph also contains precisely one edge, which is also a bridge, so it actually has the same Tutte polynomial as ! So we have .

Fourth, consider the graph which consists of a loop and another edge as shown. A graph like this is sometimes called a lollipop.

Now let be the loop. As it’s a loop, rule 4 applies. If we remove , then we just obtain the single edge graph from before. That is, . Applying rule 4, then, we obtain .

Fifth, consider the graph which consists of two edges joining three vertices. We saw this before when we deleted an edge from the triangle.

Pick one of the edges and call it . (It doesn’t matter which one — can you see why?) If we remove , the graph becomes disconnected, so is a bridge. Consequently rule 3, for bridges, applies. Now contracting the edge we obtain the lollipop graph . That is, . So, applying rule 3, we obtain .

Sixth, let’s consider the graph consisting of two “parallel” edges between two vertices. We saw this graph before when we contracted an edge of the triangle.

Pick one of the edges and call it . (Again, it doesn’t matter which one.) This edge is neither a bridge nor a loop, so rule 2 applies. Removing just gives the graph with one vertex, which has Tutte polynomial . Contracting gives a graph with a single vertex and a loop. Applying rule 4, this graph has Tutte polynomial . So, by rule 2, the Tutte polynomial of this graph is given by .

Finally, consider the triangle graph . Take an edge ; it’s neither a bridge nor a loop, so rule 2 applies. Removing results in the graph from above, which has Tutte polynomial . Contracting results in the graph from above with two parallel edges; and we’ve seen it has Tutte polynomial . So, putting it all together, we obtain the Tutte polynomial of the triangle as

Having seen these examples, hopefully the process starts to make some sense.

However, as we mentioned before, we’ve given no motivation for why this works. And, it’s not even clear that it works at all! If you take a graph, you can delete and contract different edges in different orders and get all sorts of different polynomials along the way. It’s not at all clear that you’ll obtain the same result regardless of how you remove the edges.

Nonetheless, it is true, and was proved by Tutte, that no matter how you simplify the graph at each stage, you’ll obtain the same result. In other word, the Tutte polynomial of a graph is actually well defined.

### H, O, M, F, L, Y, P and T

Tutte invented his polynomial in the 1940s — it was part of his PhD thesis. So the Tutte polynomial has been around for a long time. The knot polynomial that we’re going to consider, however, is considerably younger.

In the 1980s, there was a revolution in knot theory. The excellent mathematician Vaughan Jones in 1984 discovered a polynomial which can be associated to a knot. It has become known as the Jones polynomial. It was not the first polynomial that anyone had defined from a knot, but it sparked a great deal of interest in knots, and led to the resolution of many previously unknown questions in knot theory.

Once certain ideas are in the air, other ideas follow. Several mathematicians started trying to find improved versions of the Jones polynomial, and at least 8 mathematicians came up with similar ways to improve the Jones polynomial. In 1985, Jim Hoste, Adrian Ocneanu, Kenneth Millett, Peter J. Freyd, W. B. R. Lickorish, and David N. Yetter published a paper defining a new polynomial invariant. Making an acronym of their initials, it’s often called the HOMFLY polynomial. Two more mathematicians, Józef H. Przytycki and Pawe? Traczyk, did independent work on the subject, and so it’s often called the HOMFLY-PT polynomial.

Like the Tutte polynomial, the HOMFLY polynomial is a polynomial in *two* variables. (The Jones polynomial, however, is just in one variable.) It can also be written as a *homogeneous* polynomial in *three* variables. We’ll take the 3-variable homogeneous version.

Strictly speaking, to get a HOMFLY polynomial, your knot must be *oriented*: it must have a direction. This is usually represented by an arrow along the knot.

HOMFLY polynomials also exist for links — a link is just a knot with many loops invovled. So even if there are several loops knotted up, they still have a HOMFLY polynomial. (Each loop needs to be oriented though.)

So, if you’re given an oriented knot or link , it has a HOMFLY polynomial. We’ll denote it by . So how do you compute it? By following some rules which successively simplify the knot.

- If the knot is the unknot, then .
- If you take one of the crossings in the diagram and alter it in the various ways shown below — but leave the rest of the knot unchanged — then you obtain three links . Their HOMFLY polynomials are related by
.

A relationship like this, between three knots or links which differ only at a single crossing, is called a skein relation.

- If you can move the link around in 3-dimensional space to another link , then this doesn’t change the HOMFLY polynomial: $latex P_L (x,y,z) = P_{L’} (x,y,z).
- If the oriented link is
*split*, i.e. separates into two disjoint (untangled) sub-links , then you can take the HOMFLY polynomials of the two pieces separately, and multiply them, with an extra factor:.

- If is a
*connect sum*of two links , then you can take the HOMFLY polynomials of the two pieces, and multiply them:.

What is a connect sum? It’s when two knots or links are joined together by a pair of strands, as shown below.

And there you go.

Again, it’s not at all clear where these rules come from, or that they will always give the same result. There might be many ways to change the crossings and simplify the knot, but H and O and M and F and L and Y and P and T showed that in fact you do always obtain the same result for the polynomial.

Let’s see how to do this in a couple of examples.

First of all, for the unknot , by rule 1, its HOMFLY polynomial is .

Second, let’s consider two linked unknots as shown below. This is known as the *Hopf link*. Let’s call it .

Let’s orient both the loops so that they are anticlockwise. Pick one of the crossings and consider the three possibilities obtained by replacing it according to the skein relation described above, . You should find that corresponds to the crossing as it is shown, so . Changing the crossing results in two unlinked rings, that is, two split unknots. By rule 4 above then, ; and as each unknot has HOMFLY polynomial , we obtain . On the other hand, smoothing the crossing into gives an unknot, so .

Putting this together with the skein relation (rule 2), we obtain the equation

,

which gives

and hence the HOMFLY of the Hopf link is found to be

.

### When the Tutte is the HOMFLY

In 1988, Francois Jaeger showed that the Tutte and HOMFLY polynomials are closely related.

Given a graph drawn in the plane, it has a Tutte polynomial , as we’ve seen.

But from such a , Jaeger considered a way to *build an oriented link* . And moreover, he showed that the HOMFLY polynomial of is closely related to the Tutte polynomial of . In other words, and are closely related.

But first, let’s see how to build an link from a graph. It’s called the median construction. Here’s what you do. Starting from your graph , which is drawn in the plane, you do the following.

- Thicken up. You can then think of it as a disc around each vertex, together with a band along each edge.
- Along each edge of , there is now a band. Take each band, and put a full right-handed twist in it. You’ve now got a surface which is twisted up in 3-dimensional space.
- Take the boundary of this surface. It’s a link. And this link is precisely . (As it turns out, there’s also a natural way to put a direction on , i.e. make it an oriented link.)

It’s easier to understand with a picture. Below we have a graph , and the link obtained from it.

A graph (source: wikimedia) and the link (source: Jaeger) obtained via the median construction.

Jaeger was able to show that in general, the Tutte polynomial and the HOMFLY polynomial are related by the equation

where denotes the number of vertices of , and denotes the number of edges of . Essentially, Jaeger showed that the process you can use to simplify the link to calculate the HOMFLY polynomial, corresponds in a precise way to the process you can use to simplify the graph to calculate the Tutte polynomial.

In addition to this excellent correspondence — Tutte meeting HOMFLY — Jaeger was able to deduce some further consequences.

He showed that the four colour theorem is equivalent to a fact about HOMFLY polynomials: for every loopless connected plane graph , .

Moreover, since colouring problems for plane graphs are known to be very hard, in terms of computational complexity — NP-hard — it follows that the computation of the HOMFLY polynomial is also NP hard.

Said another way: if you could find a way to compute the HOMFLY polynomial of a link in polynomial time, you would prove that and claim yourself a Millennium prize!

## Adani: icon of Australian climate infamy

Here we are, in the year 2017. With now 25 years of climate-change international agreements behind us, here we are still trying to build oil pipelines and coal mines.

It is sad. Sad for humanity.

It is no longer a question of reducing the speed at which we are approaching the cliff. It is now a question of counting the metres to the cliff, as it approaches so fast. It is no longer a question of reducing climate emissions to a reasonable level. It is now a question of counting the remaining tons which may be emitted, budgeting them carefully and switching off them with an emergency.

There are various different ways the budget can be calculated. The MCC Carbon Clock calculates that to limit increase in global average temperature to 2 degrees Celsius, the CO2 budget remaining is 734 gigatonnes, on a moderate (neither optimistic nor pessimistic) set of assumptions. At the current rate of emissions, that budget will be exhausted by 2035. That is time at which, continuing as we are now, scientific laws predict failure.

But there is a strong argument that a 2 degrees Celsius increase is too much. It means a vast range of climate impacts – for instance, at 2 degrees tropical coral reefs do not stand a chance. A limit of 1.5 degrees increase is an altogether better goal. Indeed, the Paris agreement aims to hold increase in global average temperature to 2 degrees Celsius, but “to pursue efforts” to limit the increase to 1.5 degrees Celsius. And the MCC Carbon Clock, under the same set of assumptions, calculates the remaining CO2 budget, to limit the increase to 1.5 degrees Celsius, as 41.3 gigatonnes. At the current rate, this budget will be exhausted in September 2018 — in just over a year’s time.

One year. Just one year.

Either way, it is a clear and present danger, urgent, all-absorbing, putting all other tumults to silence. All efforts must be to get off fossil fuels immediately, now, yesterday.

Yet where are we in Australia? We are about to build our biggest coal mine ever.

Adani, corrupt, lawbreaking Adani. They still want to build their Carmichael mine in the Galilee Basin in Queensland.

And Australian governments still fall over themselves to assist them. Approvals and re-approvals flow from the federal Liberal government. The Queensland Labor premier’s interventions have convinced Adani to go ahead. The total emissions of the Carmichael project — producing and burning the coal – will be 4.7 gigatonnes of CO2. That’s over 10% of the remaining budget for 1.5 degree increase. Just this one mine.

The project is itself barely financially viable. Adani says a special loan from the Northern Australian Infrastructure Facility (NAIF) is critical to their financing. Nonetheless contracts were announced in July. They say it will employ 10,000 people: the reality, as given by Adani’s own expert under oath, is closer to 1,500.

Like everything else in Australia, the mine would be built on Aboriginal land. The traditional owners, the Wangan and Jagalingou people, released a statement: Stop Adani destroying our land and culture.

If the Carmichael mine were to proceed it would tear the heart out of the land. The scale of this mine means it would have devastating impacts on our native title, ancestral lands and waters, our totemic plants and animals, and our environmental and cultural heritage. It would pollute and drain billions of litres of groundwater, and obliterate important springs systems. It would potentially wipe out threatened and endangered species. It would literally leave a huge black hole, monumental in proportions, where there were once our homelands. These effects are irreversible. Our land will be “disappeared”.

Native Title claims being too much of an uncertain quantity for Adani – and courts showing an increasing level of respect for indigenous desires to control their land — legislation was dutifully passed by the federal parliament in June to smooth Adani’s way.

Just like the Keystone XL and Dakota Access pipelines in the US, which have seen such inspiring resistance, if the Adani Carmichael mine is built it will be game over for the climate.

It cannot go ahead. It must not.

Adani continues to acquire property along its proposed rail corridor, even as new accusations of fraud emerge against them. But largely they are currently playing a waiting game. The minister responsible for NAIF, Matt Canavan, stepped down over the citizenship farce; and his replacement is Barnaby Joyce. Until the High Court rules, and possibly byelections are held, it may wait.

That gives a crucial opportunity to press the opposition to the mine and decisively stop it. Protests continue. A few days ago, religious leaders promised civil disobedience.

As these leaders argued, it is a simple moral choice.

It is a simple scientific choice too.

Stop Adani’s mine, and switch this sunburnt country to renewable energy now.

## An Off-the-Record Genocide: Global Resource Extraction Economy Provides Incentives to Destroy DR Congo Indigenous Groups

*By Deborah S. Rogers of Initiative for Equality (IfE). Also published at Truthout. I am a member of the Board of Advisors of IfE.
*

On April 27, 2017, a hapless cow wandered off-course during a seasonal cattle drive in the Democratic Republic of the Congo, and ended up over the campfire of some Indigenous hunters. The traditional lands of these groups (Batwa and related groups) are routinely trampled by cattle, cut for old-growth timber, or grabbed for mineral resources including diamonds and coltan — generally illegally. As their wild game diminishes from these impacts, the Batwa have come to view cattle as fair game.

The cattle herders followed their cow’s tracks, and upon learning her fate, agreed to share the meat with the Batwa. But when they returned to their village, a local self-appointed “defense militia” was infuriated, returning to kill and mutilate eight of the Batwa.

The global economy’s demand for hard-to-obtain minerals and tropical timber, coupled with a long history of contempt and exploitation by neighboring tribes, have made these Batwa hunter-gatherers easy targets for land grabs and violence. Specifically targeted during a massive regional conflict to gain control over resources, in the early 2000s, an estimated 70,000 Batwa were tortured, killed and even cannibalized in northeastern Democratic Republic of the Congo (DRC), according to American University’s Inventory of Conflict and Environment Case Studies.

There is only one word for the attempted eradication of an entire group of people through the wholesale slaughter of men, women and children, whatever the reason. That word is genocide.

The conflict is now heating up again, this time in southeastern DRC. Since September 2016, volunteer investigators on the ground have been gathering names and numbers of Indigenous community members killed, injured, raped and displaced. These numbers, no doubt gross underestimates, show that well over 1,200 Batwa have been killed in the past 12 months — primarily in skirmishes with non-Indigenous neighboring communities intent on expanding their access to land and resources.

In one recent case, on July 4, 2017, a national online news source in DRC said that daylong clashes between Batwa and other ethnic groups were triggered after the Batwa killed two adversaries near the provincial capital of Kalemie. No casualty list was provided in the news article, but according to our sources, 189 Batwa people were killed that day, including men, women and children.

In the worst attack we have documented so far, on the night of January 13-14, 2017, there was a nighttime attack against the Batwa near a town called Moba. Six hundred Batwa people were slaughtered outright; at least 1,600 women and girls were brutally raped, and were being cared for using traditional medicines because there are no health centers. No pain-killers; no antibiotics; no urgently-needed surgeries; no forensic evidence; no psychological counseling. More than 40 of those women and girls had already died or were on the verge of death several days after the attack.

A desperately inadequate RFI news report on the event, translated from French, says, “On 13 January, clashes took place … 25 kilometers from the city of Moba. Four villages were partially or totally burned down and the population fled to Moba. In total, 24 people — four Bantus and twenty [Batwa] — lost their lives in one week.”

Is this destined to be an off-the-record genocide?

Knowledgeable sources on the ground say that neighboring tribes are intent on exterminating (yes, a dehumanizing term) the Indigenous people, and that the DRC government is determined to prevent word of this massacre from becoming known internationally. This is to be expected: President Joseph Kabila, who refuses to hold elections as required by DRC’s constitution, prefers to get rid of anything that stands in the way of enriching elites in his kleptocracy. Indigenous people’s traditional land rights are an impediment to uncontrolled resource extraction.

Less expected is the lack of forthright information by the UN’s peacekeeping force in DRC. The UN’s radio station in DRC routinely downplays these incidents, and fails to distinguish between deaths of Indigenous peoples and others. A July 11, 2017, article in IRIN, which reports on crises for the UN, left the dangerous misimpression that conflict and large-scale internal displacements of people in this region are instigated primarily by Indigenous Batwa militias. Without providing any objective breakdown of casualty statistics or detailed descriptions of incidents, the article presents, unchallenged, the anti-Batwa statements of individuals, primarily from the very tribal groups who are engaged in driving the Batwa off their lands.

This opacity is a major contributing factor to the ongoing crisis, providing cover for those looking to profit from the chaos. Local news sources fail to provide acceptable coverage, and international media are (rightly) afraid to send reporters in. The DRC government’s information cannot be trusted. UN investigators have been killed. Local journalists have been killed. Human rights advocates have been killed or barred from entering the country. International NGOs have sounded the alarm about conflicts and conflict minerals in the region, but only one organization has paid close attention to the genocide against the Indigenous Batwa. And on July 19, 2017, the UN announced plans to shut down five of its monitoring and peacekeeping bases in DRC, courtesy of the Trump administration’s refusal to meet US funding commitments.

There is, however, a way to obtain accurate and timely information on the situation: from the locals. My organization works with a network of local groups and individuals who are already on the ground and can tap into sources of information from the various ethnic communities and factions. Their cross-verified Field Reports provide one of the only current sources of insight into the devastation faced by the Batwa in eastern DRC.

With awareness comes the possibility of transformation. On January 16, 2017, just two days after the Moba massacre, delegates from organizations across the region convened along the shores of Lake Kivu to form a multi-ethnic coalition to defend the survival and rights of the Batwa people. With strong Batwa leadership, they developed a plan of action to monitor human rights violations and violent conflict, undertake legal interventions, launch a region-wide public awareness campaign on behalf of Indigenous rights, and implement genuine conflict resolution mechanisms (unlike the feeble government efforts led by Emmanuel Shadary, an internationally sanctioned human rights violator, which have failed to bring necessary issues and actors to the table).

We have a choice: we can either look away in horror, or we can take action to help stop the killing. If Congolese people of all ethnic backgrounds can join together to defend Indigenous rights, despite the horrendous civil and regional conflict of the past two decades, the least we in the international community can do is to back them up where we have influence. We need to educate ourselves and others, then support civil society efforts on the ground, demand that African Union and United Nations peacekeepers do their jobs, and block multinational resource extractive companies from providing financial incentives for genocide.

My colleagues in DRC end many of their communications with the exhortation, “Courage!” Let’s follow their lead.

## At least mathematics is commendable

Today the Australian government announced a proposal to force tech companies to provide government agencies with the contents of encrypted communications.

I don’t think any draft of proposed legislation exists yet — my understanding is that a bill will be introduced later in the year — but the most recent announcement today and the press conferences by Turnbull and Brandis essentially follow on from the G20 statement last week, which has a paragraph including such ideas.

Since there are no specifics, it’s hard to comment beyond generalities. But in general the whole proposal seems to me to be, to the extent it is not technically impossible or entirely misconceived, a threat to the privacy and safety of everyone.

The best thing to come out of the Turnbull’s press conference was that he said

The laws of mathematics are very commendable but the only laws that apply in Australia is the law of Australia.

I am very glad to see that Turnbull thinks mathematics is commendable. In this case, he should, for instance, take seriously the results of applying the laws of mathematics in climate models, which show just how dire the planetary climate situation is. He would be better advised to spend his precious days as Prime Minister bringing the laws of Australia into line with the laws of mathematics as applied to climate, than to try to fight the mathematics behind encryption by legislation.

I am afraid that, however commendable Turnbull thinks they are, the laws of mathematics simply cannot be avoided, whatever he thinks of them, and they cannot be legislated away. That’s the way the universe works.

You cannot legislate that messages sent by properly implemented end-to-end encryption be decrypted any more than you can legislate that pi is 3. Central results in cryptography show that properly implemented encryption schemes make decryption practically impossible. (This is putting aside potential futuristic technologies like quantum computers.)

So, in practice what this means is that the government wants to force tech companies to *not* implement end-to-end encryption properly, but to make some modification, whether by using a weakened implementation or malware or a backdoor of some sort, so that the government can access it. Such proposals by law enforcement and intelligence have a long and ignominious history going back to at least the 1990s and the Clipper Chip. Technical dificulties aside, the important point which has come out of all that history is that there is no way to make encryption subject to government-mandated decryption without making it vulnerable to other attacks as well. If encryption is weak enough that a conversation can be decrypted by someone other than the parties to the conversation, then it is weak enough to be decrypted by many others, hackers, other governments, and so on. If it is implemented through government-mandated malware, then anyone who gains access to that malware has similar power — and we have seen precisely this happen, for instance, with NSA malware and WannaCrypt attacks.

The government’s approach with the present proposal appears to be to transfer responsibility to tech companies. Rather than legislate government backdoors, they seem to want to legislate that the tech companies must do what they can to assist. They want to use the legal language of rendering “proportionate” and “reasonable” assistance. But breaking end-to-end encryption, or implementing backdoors, is not at all proportionate or reasonable. If a company makes such a change, then they no longer implement end-to-end encryption and the promises of privacy provided to their users are null and void. There is no proportionate way to break an algorithm which mathematically provides secure encryption. It is either secure, or it is not.

In recent years there has been a mass takeup of encrypted messaging by people around the world. End-to-end encryption has been implemented by many major technology companies. This is largely sparked by revelations of mass warrantless surveillance by the NSA, not only of individuals, but also of those very tech companies. People are right to be wary of their privacy.

The Australian authorities, I’m afraid, do not inspire a great deal of confidence. They have already been given draconian powers. Quite aside from other draconian laws which, for instance, criminalise government leaks and whistleblowing from within refugee detention centres, metadata laws have come into effect. These metadata laws allow many government agencies, without any court warrant, to access the metadata of almost any Australian’s online activity. These agencies have been invested with great power, and yet even the mild protections for journalists have been violated, as we found out in April, when the Australian Federal Police admitted that a journalist’s data had been accessed. No charges were laid and no action was taken, so far as I’m aware, beyond the Federal Police holding a press conference. Given the approach the AFP takes to journalists — a class of people with special legal protections — one wonders what approach they take to ordinary citizens. How will they then treat whistleblowers, activists, and government critics?

Police and intelligence already have enormous powers of surveillance and monitoring. Terrorism, child pornography and sex trafficking are important issues, but these proposals are not the way to deal with them.

## Holy-principle, Batman!

(With apologies and tribute to the late Adam West)

Here’s a situation known to any beginning skier.

You are at the top of a mountain slope. You want to go down to the bottom of the slope. But you are on skis, and you are not very good at skiing.

Despite your lack of skill, you have been deposited by a ski lift at the top of the slope. Slightly terrified, you know that the only honourable way out of your predicament is *down* — down the slope, on your skis.

Pointing your skis down the slope, with a rush of exhilaration you find yourself accelerating downwards. Unfortunately, you know from bitter experience that the hardest thing about learning to ski is learning to control your speed.

If you are bold or stupid, your incompetent attempt to conquer the mountain likely ends in a spectacular crash. If you are cowardly and have mastered the snowplough, your plodding descent ends with a whimper and worn out thighs from holding the skis in an awkward triangle all the way down.

But you know that the more your skis point *down* the mountain, the faster you go. And the more your skis point *across* the slope, the slower you go.

When your skis point down the slope, they are pointing steeply downwards; they are pointing in quite a *steep* direction. But when your skis point across the slope, they are not pointing very downwards at all; they are pointing in quite a *flat* direction.

As an incompetent skier, the best way to get down the slope without injury and without embarrassment is to go take a path which criss-crosses the slope as much as possible. You want your skis to point in a flat direction as much as possible, and in a steep direction as little as possible.

The problem is that each time you change direction, you temporarily point downwards, and risk runaway acceleration.

Is it possible to get down the mountain while always pointing in a flat direction?

Of course the answer is no. But there is an area of mathematics which says that the answer is *yes, almost, more or less*.

This, very roughly, is the beginning of the idea of Gromov’s *homotopy principle* — often abbreviated to *h-principle*.

* * *

The ideas of the h-principle were developed by Mikhail Gromov in his work in the 1970s, including work with my PhD advisor Yakov Eliahsberg. The term “h-principle” first appeared and was systematically elaborated by Gromov in his (notoriously difficult) book *Partial differential relations*. Gromov, who grew up in Soviet Russia, was not a friend of the authorities and in 1970, despite being invited to speak at the International Congress of Mathematicians in France, was prohibited from leaving the USSR. Later he finally made his “gromomorphism” to the US, and now he works at IHES just near Paris.

The ideas of the h-principle are about a “soft” kind of geometry. If you are prepared to treat your geometry like playdough, and morph various things around, within certain constraints, then the $h$-principle tells you how to morph your playdough-geometry to get the nice kind of geometry you want. Technically, morphing playdough has a fancy name: it’s called *homotopy*.

An example of the h-principle (or, more precisely, of “holonomic approximation”) in the skiing context would be as follows.

Your ideal skiing path down the slope would be to go straight down, but always have your skis pointing flat. That is, your skis should always point horizontally, but you should go straight down the mountain. That is the ideal.

That, of course, is a ridiculous ideal path. But it’s an ideal path nonetheless. You want to go straight down the mountain, and the ideal path does this; and you want to have your skis pointing safely horizontally, and the ideal path does this too. This is the ideal of the incompetent skier: go down the mountain as directly as possible, with your skis always being completely flat.

Now the ideal path cannot be achieved in practice. In practice, if your skis are pointing horizontally, you go horizontally. (We ignore skidding for the purposes of our mathematical idealisation.) In practice, you go in the direction your skis are pointing.

The “ideal path” is a path down the mountain, which also tells you which way your skis should point at each stage (i.e. horizontally), but which doesn’t satisfy the practical principle that you should go in the direction you’re pointing. As such, it’s a *generalisation* of the real sort of path you could actually take down the mountain. (The technical name for this type of path is a “section of a jet bundle”.)

A realistic path down the mountain — one where you go in the direction your skis are pointing — is also known as a *holonomic* path.

One of the first results towards the h-principle, says that if you are prepared to make a few tiny tiny adjustments to your path, then you can take an actual, holonomic path down the mountain, where you are very very close to the ideal path — both in terms of where you go, *and* in the direction your skis point. You stay very very close — in fact, *arbitrarily close* — to the path straight down the mountain. And your skis stay very very close — again, *arbitrarily close* — to horizontal at every instant on the way down.

How is this possible? Well, you have to make some adjustments.

First, you make some adjustments to the path. You might have to make a wiggly path, rather than going in a straight line. Actually, it will have to be *very* wiggly — *arbitrarily* wiggly.

And, second, you’ll have to make some adjustments to the mountain too. You’ll have to adjust the shape of the mountain slope — but only by a very very small, arbitrarily small amount.

Well, perhaps these types of alternations are rather drastic. But without moving the mountain, you won’t be able to go down the mountain and stay very close to horizontal. You must alter the ski slope, and you must alter your path. But these movements are very very small, and you can make them as small as you like.

How do you alter the mountain? Roughly, you can make tiny ripples in the slope — and roughly, you turn it into a *terraced slope*. Just like rice farming in Vietnam, or for growing crops in the Andes.

As you go along a terrace, you remain horizontal! We don’t want our terraces to be *completely* horizontal though — we want them to have a very gentle downwards slope, so that we can stay very close to horizontal, and yet eventually get to the bottom of the mountain.

And also, we’ll need to be able to go smoothly from one terrace down to the next, so each terraces should turn into the next. So perhaps it’s more like Lombard Street, the famously windy street in San Francisco. (Which is not, however, the most *crooked* street in the US — that’s Wall Street, of course. Got you there.)

Perhaps a more accurate depiction of what we want is the figure below, from Eliashberg and Mishachev’s book *Introduction to the h-principle*. We want very fine, very gently sloping terraces, and we want them extremely small, so that the mountain is altered by a tiny tiny amount. And to go down the slope we need to take a very windy path — with many many wiggles in it. To go down the slope is almost like a maze — although it’s a very simple, repetitive maze.

Thus, with a astonomical number of microscopic terraces of the mountain, each nano-scopically sloped downwards, and an astronomical number of microscopic wiggles in your path down the terraces, you can go down the mountain, staying very close to your idealised path. You go very very close to straight down the mountain, and your skis stay very very close to horizontal all the way down.

And then, you’ve done it.

This is the flavour of the h-principle. More precisely this result is called *holonomic approximation*. Holonomic approximation says that even an incompetent skier can get down a mountain with an arbitrarily small amount of trouble — provided that they can do an arbitrarily large amount of work in advance to terrace the mountain and prepare themselves an arbitrarily flat arbitrarily wiggly path.

* * *

The h-principle has applications beyond idealised incompetent skiing down microscopically terraced mountains. Two of the most spectacular applications are sphere eversion, and isometric embedding. In fact they both preceded the h-principle — and Gromov’s attempt to understand and formalise them directly inspired the development of the h-principle.

*Sphere eversion* is a statement about spheres in three-dimensional space. Take a standard unit sphere in, but again regard it as made of playdough, and we will consider morphing (erm, *homotoping*) it in space. We allow the sphere to pass through itself, but never to crease, bend or rip. All the sphere can intersect itself, each point of the sphere must remain smooth enough to have a tangent plane. (The technical name for this is an *immersion* of the sphere into 3-dimensional space.)

Smale’s sphere eversion says that it’s possible to turn the sphere inside out by these rules — that is, by a homotopy through immersions. This amazing theorem is all the more amazing because Smale’s original 1957 proof was an *existence* proof: he proved that there *existed* a way to turn the sphere inside out, but did not say how to do it! Many explicit descriptions have now been given for sphere eversions, and there are many excellent videos about it, including Outside In made in the 1990s. My colleague Burkard Polster, aka the Mathologer, also has an excellent video about it.

Smale has an interesting and admirable biography. He was actively involved in the anti-Vietnam War movement, even to the extent of being subpoenaed by the House Un-American Activities Committee. His understanding of the relationship between creativity, leisure and mathematical research was epitomised in his statement that his best work was done “on the beaches of Rio”. (He discovered his horseshoe map on the beach at Leme.)

Isometric embedding is more abstract; see my article in the conversation for another description, but it is even more amazing. It is a theorem about *abstract* spaces. For instance, you could take a surface — but then, put on it an abstractly-defined metric, unrelated to how it sits in space. *Isometric embedding* attempts to map the surface back into 3-dimensional space in a way that preserves distances, so that the abstract metric on a surface corresponds to the familiar notion of distance we know in three dimensions.

Isometric embedding is largely associated with John Nash, who passed away a couple of years ago and is more well known for his work on game theory, and from the book and movie A Beautiful Mind. The proof is incredible. Gromov describes how he came to terms with this proof in some recollections. He originally found Nash’s proof “as convincing as lifting oneself by the hair”, but after eventually finding understanding, he found Nash’s proof “miraculously, did lift you in the air by the hair”!

The Nash-Kuiper theorem says that if you can map your abstract surface into 3-dimensional space in such a way that it *decreases* distances, then you can morph it — homotope it — to make it preserve distances. (Actually, it need not be a surface but a space of any dimension; and it need not be 3-dimensional space, but space of any dimension.) And, just like on the ski slope, this alteration of the surface in 3-dimensional space can be made very very small — arbitrarily small.

The h-principle is another mathematical superpower, and it comes up in many places where geometry is “soft”, and you can slightly “morph” or “adjust” your geometrical situation to find the situation we want.

## Eighty years ago, Spanish people responded to the far right with social revolution

Eighty years ago to the day, the far right was in its ascendancy, and still rising. Hitler was in complete control of Germany, Mussolini had been in charge of a police state in Italy for a decade. The world failed to stop them, and war was to break out within a few years, consuming the world in the deadliest conflict in human history.

But a little to the southwest, in Spain, war had already broken out.

In July 1936, Franco and his co-conspirators made their coup attempt against the elected Republican government, dividing the country and driving it into war. By the end of the year they controlled several major cities and had laid siege to Madrid.

The Spanish civil war had erupted, and while the supposedly democratic powers refused to come to the assistance of the besieged Republic — indeed Britain made some moves in the opposite direction — Mussolini and Hitler had no such qualms supporting the forces of reaction. Their support expressed itself in, among other military attacks, the bombing of Guernica. This attack, the world’s first targeting of civilians by aerial bombardment, inaugurated a new type of crime, and a new type of terror, from which humanity has been suffering ever since. The terror and destruction – though mild by the bombings that were to come, of London, Dresden, Tokyo, Hiroshima, Nagasaki, and elsewhere – has never really left us. Conscientious, idealistic and passionate volunteers from abroad fought for the Republic, Orwell’s account being perhaps the most well known.

But the various forces of Spanish reaction — monarchists, Carlists, phalangists, fascists and more, along with mercenaries, German Nazis and Italian fascists — were not merely met with resistance on the battlefield. Many groups within Spanish society took that very moment to make a social, political and economic revolution.

As Chomsky wrote in his *Objectivity and Liberal Scholarship*,

During the months following the Franco insurrection in July 1936, a social revolution of unprecedented scope took place throughout much of Spain. It had no “revolutionary vanguard” and appears to have been largely spontaneous, involving masses of urban and rural laborers in a radical transformation of social and economic conditions that persisted, with remarkable success, until it was crushed by force. This predominantly anarchist revolution and the massive social transformation to which it gave rise are treated, in recent historical studies, as a kind of aberration, a nuisance that stood in the way of successful prosecution of the war to save the bourgeois regime from the Franco rebellion. Many historians would probably agree with Eric Hobsbawm that the

failureof social revolution in Spain “was due to the anarchists,” that anarchism was “a disaster,” a kind of “moral gymnastics” with no “concrete results,” at best “a profoundly moving spectacle for the student of popular religion.” … In fact, this astonishing social upheaval seems to have largely passed from memory.

In other words, at the same time, and in the same place as this horrific war, also eighty years ago to the day, an extraordinary experiment was underway — which has arguably seen no parallel before or since. It was not undertaken as a sideshow, or as a footnote to the ongoing war, but *because* of it. It was associated with those most optimistic of political philosophies — anarchism, or libertarian socialism — which hold that human beings can organise their lives without the illegitimate authority of bosses or States, without property in capital, and with equality, freedom, democracy, and free association.

How did this happen? All at once. In Catalonia, the question of how far to push for libertarian revolution under conditions of war against fascism presented itself immediately — indeed, within a day of the initial coup. While the Catalan president Luis Companys had refused to issue arms, the anarchists that had stormed the barracks at Atarazanas, defeating the local military coup plotters, seizing weapons and obtaining de facto power. The result was that the anarchists were forced to respond to the question of taking political power in the most dramatic way: they were literally offered State power. Companys, a Catalan nationalist, but unusually sympathetic to anarchists, presented the anarchist leaders Juan Garcia Oliver, Buenaventura Durruti and Diego Abad de Santillán with a proposition, a mixture of generous support, alliance, realpolitik (for the anarchists were more heavily armed than nearby official Republican forces), self-preservation, and acknowledgment of the justice of their cause.

As Anthony Beevor describes it in his history of the war:

On the evening of July 20, Juan Garcia Oliver, Buenaventura Durruti and Diego Abad de Santillán met with President Companys in the palace of the

Generalidad. They still carried the weapons with which they had stormed the Atarazanas barracks that morning. In the afternoon they had attended a hastily called meeting of more than 2,000 representatives of local CNT [anarchist] federations. A fundamental disagreement arose between those who wanted to establish a libertarian society immediately and those who believed that it had to wait until after the generals were crushed. …At the meeting… Companys greeted anarchist delegates warmly:

… Today you are the masters of the city and of Catalonia because you alone have conquered the fascist military… and I hope you will not forget that you did not lack the help of loyal members of my party… But you have won and all is in your power. If you do not need me as president of Catalonia, tell me now, and I will become just another soldier in the fight against fascism. If, on the other hand… you believe that I, my party, my name, my prestige, can be of use, then you can depend on me and my loyalty as a man who is convinced that a whole past of shame is dead.

To my knowledge this is the only offer of its kind ever made by any State to an anarchist organisation. It was an incredible dilemma for the anarchists:

Garcia Oliver described the alternatives as ‘anarchist dictatorship, or democracy which signifies collaboration’. Imposing their social and economic self-management on the rest of the population appeared to violate libertarian ideals more than collaborating with political parties. Abad de Santillán said that ‘we did not believe in dictatorship when it was being exercised against us and we did not want it when we could exercise it only at the expense of others’. At their Saragrossa conference only seven weeks before, the anarchists had affirmed that each political philosophy should be allowed to develop ‘the form of social co-existence which best suited it’. This meant working alongside other political bodies with mutual respect for each other’s differences. Though genuine, this was a simplistic view, since the very idea of worker-control and self-management was anathema both to businessmen and the communists.

Even if the anarchist leaders sitting in Companys’ ornate office, having just been offered the keys of the kingdom, could have foreseen the future, it is doubtful whether their choice would have been made any easier. They had the strength to turn Catalonia and Aragon into an independent non-state almost overnight. But Madrid had the gold, and unofficial sanctions by foreign companies and governments could have brought them down in a relatively short space of time. However, what influenced their decision the most was concern for their comrades in other parts of Spain. The demands of solidarity overrode other considerations. They could not abandon them in a minority which might be crushed by the Marxists.

Accordingly, a Central Committee of Anti-Fascist Militias was organised, allowing pluralism among the various Republican factions in the government of Catalonia; the anarchists decided to share, rather than take, power.

Being in the majority, the least we can do is to recognize the right of minorities to organize their own lives as they want and to offer them our cordial solidarity.

The result in Barcelona, as described by the journalist John Langdon-Davies, was

the strangest city in the world today, the city of anarcho-syndicalism supporting democracy, of anarchists keeping order, and anti-political philosophers wielding political power.

In December 1936, while Madrid was being attacked, George Orwell arrived in Barcelona and described what he saw.

I had come to Spain with some notion of writing newspaper articles, but I had joined the militia almost immediately, because at that time and in that atmosphere it seemed the only conceivable thing to do. The Anarchists were still in virtual control of Catalonia and the revolution was still in full swing. To anyone who had been there since the beginning it probably seemed even in December or January that the revolutionary period was ending; but when one came straight from England the aspect of Barcelona was something startling and overwhelming. It was the first time that I had ever been in a town where the working class was in the saddle. Practically every building of any size had been seized by the workers and was draped with red flags or with the red and black flag of the Anarchists; every wall was scrawled with the hammer and sickle and with the initials of the revolutionary parties; almost every church had been gutted and its images burnt. Churches here and there were being systematically demolished by gangs of workmen. Every shop and cafe had an inscription saying that it had been collectivised; even the bootblacks had been collectivized and their boxes painted red and black. Waiters and shop-walkers looked you in the face and treated you as an equal. Servile and even ceremonial forms of speech had temporarily disappeared. Nobody said ‘Señor’ or ‘Don’ or even ‘Usted’; everyone called everyone else ‘Comrade’ or ‘Thou’, and said ‘Salud!’ instead of ‘Buenos dias’. Tipping had been forbidden by law since the time of Primo de Rivera; almost my first experience was receiving a lecture from a hotel manager for trying to tip a lift-boy. There were no private motor-cars, they had all been commandeered, and the trams and taxis and much of the other transport were painted red and black. The revolutionary posters were everywhere, flaming from the walls in clean reds and blues that made the few remaining advertisements look like daubs of mud. Down the Ramblas, the wide central artery of the town where crowds of people streamed constantly to and fro, the loud-speakers were bellowing revolutionary songs all day and far into the night. And it was the aspect of the crowds that was the queerest thing of all. In outward appearance it was a town in which the wealthy classes had practically ceased to exist. Except for a small number of women and foreigners there were no ‘well-dressed’ people at all. Practically everyone wore rough working-class clothes, or blue overalls or some variant of militia uniform. All this was queer and moving. There was much in this that I did not understand, in some ways I did not not even like it, but I recognized it immediately as a state of affairs worth fighting for. Also, I believed that things were as they appeared, that this was really a workers’ State and that the entire bourgeoisie had either fled, been killed or voluntarily come over to the workers’ side; I did not realise that great numbers of well-to-do bourgeois were simply lying low and disguising themselves as proletarians for the time being.

Together with all this there was something of the evil atmosphere of war. The town had a gaunt untidy look, roads and buildings were in poor repair, the streets at night were dimly lit for fear of air-raids, the shops were mostly shabby and half-empty. Meat was scarce and milk practically unobtainable, there was a shortage of coal, sugar and petrol, and a really serious shortage of bread. Even at this period the bread-queues were often hundreds of yards long. Yet so far as one could judge the people were contented and hopeful. There was no unemployment, and the price of living was still extremely low; you saw very few conspicuously destitute people, and no beggars except the gypsies. Above all, there was a belief in the revolution and the future, a feeling of having suddenly emerged into an era of equality and freedom. Human beings were trying to behave as human beings and not as cogs in the capitalist machine. In the barbers’ shops were Anarchist notices (the barbers were mostly Anarchists) solemnly explaining that barbers were no longer slaves. In the streets were coloured posters appealing to prostitutes to stop being prostitutes. To anyone from the hard-boiled, sneering civilization of the English-speaking races there was something rather pathetic in the literalness with which these idealistic Spaniards took the hackneyed phrase of revolution. At that time revolutionary ballads of the naivest kind, all about the proletarian brotherhood and the wickedness of Mussolini, were being sold on the streets for a few centimes each. I have often seen an illiterate militiaman buy one of these ballads, laboriously spell out the words, and then, when he had got the hang of it, begin singing it to an appropriate tune.

More generally across the Republican zone, anarchists and socialists found themselves in positions of power. As Chomsky writes:

Workers armed themselves in Madrid and Barcelona, robbing government armories and even ships in the harbor, and put down the insurrection while the government vacillated, torn between the twin dangers of submitting to Franco and arming the working classes. In large areas of Spain, effective authority passed into the hands of the anarchist and socialist workers who had played a substantial, generally dominant role in putting down the insurrection.

Turning more specifically to the economic revolution carried out in the midst of the civil war, Beevor explains the collectives as follows.

The collective in Republican Spain were not the state collectives of Russia. They were based on the joint ownership and management of the land or factory. Alongside them were ‘socialized’ industries, restructured and run by the CNT and UGT as well as private companies under the joint worker-owner control. Co-operatives marketing the produce of individual smallholders and artisans also existed, although these were not new. They had a long tradition in many parts of the country, especially in fishing communities. There were estimated to have been around 100,000 people involved in co-operative enterprises in Catalonia alone before the civil war. The [anarchist] CNT was, of course, the prime mover in this development, but [socialist union] UGT members also contributed to it. The UGT or UGT—CNT organized about 15 per cent of the collectives in New Castile and La Mancha, the majority in Estremadura, very few in Andalucia, about 20 per cent in Aragon, and about 12 per cent in Catalonia.

The regions most affected were Catalonia and Aragon, where about 70 per cent of the workforce was involved. The total for the whole of Republican territory was nearly 800,000 on the land and a little over a million in industry. In Barcelona workers’ committees took over all the services, the oil monopoly, the shipping companies, heavy engineering firms such as Vulcano, the Ford motor company, chemical companies, the textile industry and a host of smaller enterprises.

Any assumption by foreigners that the phenomenon simply represented a romantic return to the village communes of the Middle Ages was inaccurate. Modernization was no longer feared because the workers controlled its effects. Both on the land and in the factories technical improvements and rationalization could be carried out in ways that would previously have led to bitter strikes. The CNT wood-workers union shut down hundreds of inefficient workshops so as to concentrate production in large plants. The whole industry was reorganized on a vertical basis… Similar structural changes were carried out in other industries as diverse as leather goods, light engineering, textiles and baking. … One of the most impressive feats of those early days was the resurrection of the public transport system at a time when the streets were still littered and barricaded. …

At the same time as the management of industry was being transformed, there was a mushroom growth of agricultural collectives in the southern part of Republican territory. They were organized by CNT members, either on their own or in conjunction with the UGT. The UGT became involved because it recognized that collectivization was the most practical method of farming the less fertile

latifundia.

There are lessons here for the present day: the economic consequences of technological innovation — whether “modernization” by mechanisation or digitalisation, whether “automation” by production line or software — need no longer be feared when production is under the control of workers. We should, however, be clear that while collectivization was often voluntary — indeed spontaneous — it was sometimes coerced.

To many people’s surprise the anarchists made attempts to win the trust of the middle classes. If a shopkeeper complained to the CNT that his goods were being taken by workers’ patrols, a sign would be put up stating that the premises belonged to the supply committee. Small firms employing fewer than 50 people were left untouched if the management had a good record. …

In Aragon some collectives were installed forcibly by anarchist militia columns, especially Durruti’s. Their impatience to get the harvest in the feed the cities, as well as the fervour of their beliefs, sometimes led to violence. Aragonese peasants resented being told what to do by over-enthusiastic Catalan industrial workers, and many of them had fears of Russian-style collectives. …

There were few villages which were completely collectivized. The ‘individualists’, consisting chiefly of smallholders who were afraid of losing what little they had, were allowed to keep as much land as a family could farm without hired labour. In regions where there had always been a tradition of smallholding, little tended to change. The desire to work the land collectively was much stronger among the landless peasants, especially in less fertile areas where the small plots were hardly visible.

However, persuasion was often recognised as not just the most principled but also the most effective tactic. The Austrian Marxist writer Franz Borkenau, visiting Spain, wrote:

The anarchist nucleus achieved a considerable improvement for the peasants and yet was wise enough not to try to force the conversion of the reluctant part of the village, but to wait till the example of the others should take effect.

Indeed, as Beevor continues,

the anarchists tried to persuade the middle classes that they were in fact oppressed by an obsession with property and respectability. ‘A grovelling existence,’ they called it. ‘Free yourselves socially and morally from the prejudices that have dominated you until today.’ …

The anarchists continually tried to persuade the peasants that the ownership of land gave a false sense of security. The only real security lay within a community which cared for its own members by providing medical facilities and welfare for the sick and retired.

Collectivization of agriculture was, in economic terms, a qualified success:

whatever the ideology, the self-managed co-operative was almost certainly the best solution to the food-supply problem. Not only was non-collectivized production lower, but the ‘individualists’ were to show the worst possible traits of the introverted and suspicious smallholder. When food was in short supply they hoarded it and created a thriving black market, which, apart from disrupting supplies, did much to undermine morale in the Republican zone. The communist civil governor of Cuenca admitted later that the smallholders who predominated in his province held onto their grain when the cities were starving. …

In terms of production and improved standards for the peasants, the self-managed collectives appear to have been successful. They also seem to have encouraged harmonious community relations. There were, however, breakdowns of communication and disputes between collectives. The anarchists were dismayed that collective selfishness should seem to have taken the place of individual selfishness, and inveighed against this ‘neo-capitalism’.

It should be made clear just how much opposition the anarchists faced — not just from the Nationalists, but also from other factions in the Republican camp, liberals and communists.

The most outspoken champions of property were not the liberal republicans, as might have been expected, but the Communist Party and its Catalan subsidiary, the PSUC. La Pasionaria and other members of their central committee emphatically denied that any form of revolution was happening in Spain, and vigorously defended businessmen and small landowners (at a time when

kulakswere dying in Gulag camps). This anti-revolutionary stance, prescribed by Moscow, brought the middle classes into the communist ranks in great numbers. Even the traditional newspapers of the Catalan business communityVanguardiaandNoticiero, praised ‘the Soviet model of discipline’.…

There were [for collectivized industries] serious problems in obtaining new machinery to convert companies which were irrelevant, like luxury goods, or under-used because of raw-material shortages, like the textile industry. They were caused principally by the Madrid [Republican!] government’s attempt to reassert its control by refusing foreign exchange to collectivized enterprises.

…

[Moreover, for Catalonian industry a] sizeable part of the home market had been lost in the rising. The peseta had fallen sharply in value on the outbreak of the war, so imported raw materials cost nearly 50 per cent more in under five months. This was accompanied by an unofficial trade embargo which the pro-Nationalist governors of the Bank of Spain had requested among the international business community. Meanwhile, the central government tried to exert control through withholding creidts and foreign exchange. [Republican Prime Minister] Largo Caballero, the arch-rival of the anarchists, was even to offer the government contract for uniforms to foreign companies, rather than give it to CNT textile factories. (The loss of markets and shortage of raw materials led to a 40 per cent decline in textile output, but engineering production increased by 60 per cent over the next nine months.)

The communists’ Popular Front strategy of defending commercial interests so as to win over the middle class was perfectly compatible with their fundamental opposition to self-management. As a result their Catalonian affiliate, the PSUC, started to persuade [socialist union] UGT bank employees to use all possible means to interfere with the collectives’ financial transactions.

…

[Prime Minister] Giral’s government in Madrid did not share the anarchists’ enthusiasm for self-managed collectives. Nor did it welcome the fragmentation of central power with the establishment of local committees. Its liberal ministers believed in centralized government and a conventional property-owning democracy…. They were appalled at having no control over the industrial base of Catalonia. But… [the Republican government’s] continued control of supply and credit held out the prospect that concessions might gradually be wrung from the revolutionary organizations”

Facing such obstacles — and of course, eventually, destruction under the victorious military forces of fascism — it is perhaps surprising that the collectives achieved even a fraction of the success they did.

* * *

Apparently there is an old Spanish proverb, “History is a common meadow in which everyone can make hay”. No doubt I am making hay of it in my own way. Let us be clear that all sides in the Spanish Civil War were responsible for atrocities, and many collectivizations were forced. Perhaps the libertarianism of the anarchists is too optimistic for human nature; perhaps, left to run its own course, with increasing complexity of industry, self-management might have floundered. But we do know that the revolution did not fail for those reasons; nor did it fail for inefficiency, or bureaucracy, or authoritarianism — on the contrary. It failed because it was crushed, opposed by every other faction both among both enemies and allies. The situation left them no chance. Between the military attacks of the fascists, the opposition by erstwhile Republican allies, ranging from political interference to outright military attack — supported by all the greatest monsters of European history, Hitler, Mussolini *and* Stalin — and abandonment by the liberal democratic powers, it is impossible to say how they might have developed, leaving fodder for cynic and dreamer alike, and everyone in between.

Let us limit ourselves to a few obvious remarks. People fight harder for something worth defending. When an existing political or economic system is at an ebb, giving rise to the worst forces of reaction, xenophobia, nationalism and authoritarianism, is precisely when changes are most possible. The time for the most realistic utopian thinking is in the time of catastrophe. The history of all great reforms is a span which begins with a demand for the impossible and ends with the acceptance of the inevitable. Even an anarchist can be offered the keys to the kingdom. And even those offered power can decline — and try to build something better instead.

But let us leave the history to speak for itself — a reminder of what can be achieved even under the greatest adversity.

Eighty years ago, Spanish people responded to the rise of the far right with social revolution. What will you do?

## Mathematics, mathematicians, philosophy

*Recently I was asked to talk at a secondary school about mathematics and mathematical philosophy. The following is roughly based on what I talked about…*

I was asked to come and talk to you about mathematics, which is lucky, because I’m a mathematician. But then I wondered what exactly I should tell you.

I thought — well, should I give you a maths class? But I figure that’s probably not the best idea. I might talk about a little bit of actual maths, but don’t worry, there will be no test.

Then I thought — should I try to persuade you to study more maths? A sales pitch for mathematics? Well, I’m sure you’ve all had an excellent education here, in mathematics as in every other subject, so there’s no need for further persuasion. And anyway I’m not much of a salesman. However, I might try to show you a couple of things about maths that I think makes it interesting

Then I thought — should I tell you about all the places maths is useful in everyday life? Well, maths does come up pretty much everywhere, and it’s used by people all the time – sometimes with good effect, sometimes bad, but in any case it’s everywhere. So I might talk a little about that too. And It raises all sorts of big philosophical questions.

And speaking of philosophy — in the philosophy of mathematics there are all sorts of deep, abstract, and interesting questions there. Pretty heavy ones too. What does it mean for a mathematics to be true? What is mathematics anyway? Well, those are pretty deep questions, but since I’m the only professional mathematician in the room, I might say a little about it.

Then I thought — I’m just here to talk to you. And as far as I’m concerned I’m talking to adults.

But then I thought — oh, I should probably introduce myself first! Well it would be rude not to, so OK, I’ll do that first, then I’ll tell you some of the things I think about mathematics, which are hopefully kind of interesting. And then we’ll leave some time for questions, and you can ask me about anything you like.

* * *

So yes, I will start by telling you a little bit about myself. I’m a mathematician, a lecturer at Monash Uni. I’m also, a sometime lawyer, activist, writer, husband, dog owner.

I’ve been into maths since I was very young. At school I enjoyed learning lots of things, including mathematics and science. I’ve always wanted to know how the world works. But actually my favourite subject at school was history. You know, “maths, science, history – unravelling the mystery”. I heard that on TV somewhere. I enjoyed history: learning about how we came to be how we are now – how we got into this mess!

At school I found out I was apparently good at maths, and I got involved in the Maths Olympiad. I actually represented the country at the International Maths Olympiad.

I graduated from school, feeling like I didn’t know anything, and that there was so much more to learn. I went to university. I wanted to learn everything, but they only let me enrol in two degrees. So I studied science and law: the rules that govern the world, and the rules that govern society. And from then on I’ve never really left university, but I still feel like I know nothing. I did a few courses here in Melbourne, then I did a PhD in maths at Stanford University in California. I got involved in some politics as well – I was involved in a website you might have heard of called Wikileaks, although I left it long ago, before it became famous, or rather, infamous. After getting my PhD I then worked in France, at the University of Nantes – Nantes is a city in north-western France which has good mathematicians and good crepes. Then I worked in Boston, at a university called Boston College. So as you can see mathematics is very international. It’s the same with all science really – it’s a very international enterprise.

I now work at Monash Uni, in the maths department there. I am also a fully qualified lawyer, and although I’ve never practised law, I have done volunteer legal work and been involved in some politics here as well. I’m an advisor, for instance, to a wonderful group called the *Initiative for Equality*, which works to build more equal and participatory societies around the world.

But that’s enough about myself. If you want to know anything more you will have to ask.

* * *

So, let’s get down to talking a bit about mathematics. And I thought a first thing to do might be to actually ask – what is mathematics? Because I suspect the average person, and you, and me, might all have different answers.

As a mathematician, I think it’s actually quite a difficult thing to define.

Perhaps the main thing I want to impress upon you about what mathematics is, is just how big it is, how long it’s been going, and how narrow a slice of it most people see. Maths is a lot more than the subject you learn at school called maths!

Studying maths at school, you can kind of get the impression that you study year 7 maths, and then year 8 maths, and so on, and finally you get to year 12 maths, and you are then done. You finished mathematics.

Well, no. At that point you know a little bit of the mathematics that was known up to the 17^{th} century. You’ve seen essentially zero mathematics from the last 300 years – possibly a little bit, if you’ve got good teachers, as I’m sure you do here.

But what would your education be like, if you graduated from school completing all the English units, but never having studied, let alone read or even *seen*, any literature from after 1700? Well with mathematics it’s like that.

Perhaps it’s not quite that bad. Mathematics doesn’t change like English does. Literature from 1700 is now outdated, in a certain sense, but mathematics from 1700 is not. It’s just as true, just as valid now as it was then. And mathematics had made a significant amount of progress by 1700, so it’s not terribly bad that you only get up to 1700.

But it’s useful to place it in context. Mathematics has an enormously long and rich history, and its story goes right back to the beginning of civilization. We have evidence of writing numbers dating from about 3500 BC. The Sumerians were doing their times tables from 2500 BC. The ancient Egyptians knew about prime numbers before 1800 BC.

But mathematics exploded in ancient Greece from the 6^{th} century BC onwards and took the world by storm. And since then, mathematics has been actively advancing.

Told you I liked history.

But whenever you are learning mathematics, you are tapping into one of the deepest continuous strains of human thought, and you are tapping into some of the most ingenious, clever, ideas ever conceived, ideas developed by some of the greatest minds in history.

Think of it this way. We are now so advanced that, in a mere 12 or 13 years of schooling, you’re able to go from a pre-literate level of mathematical understanding, up to the 17^{th} century. That’s quite an achievement. It took humanity millennia, but for you it’s just exercises in your textbook – combined with good teachers.

So maths has been around a long time. It’s not going away – sorry if you don’t like it. It was here before we were born, it will still be here after we are gone. It will even survive Donald Trump. And it will be no less true for that.

* * *

So, what is mathematics? In the end I can’t really define it, but I can say some things that *are* mathematics and some things that *are not*.

Now, there are many things we know that are definitely included in mathematics. Algebra is maths. Geometry is maths. Anything with numbers in it is maths. Anything with exact logic in it is maths – maths has a type of thinking and a type of logic that is characteristic to the subject, and unlike the sort of thinking you get pretty much anywhere else – we might talk more about that later.

Some other things are definitely not mathematics. History is not maths. Biology, chemistry, physics are not maths, although they may *use* it. Physics is an interesting one actually – it’s very *mathematical*, and so much so that sometimes the boundary between physics and mathematics is unclear. I might talk more about that later.

The other sciences are easier to define than maths. You can say at least roughly what these other sciences are about. Physics is about the study of objects in the universe and so on, chemistry is about the properties of atoms and molecules and their reactions and so on.

The other sciences are easier to define because they’re basically limited to things which actually exist in this universe. Maths, on the other hand, has no such limitations. It does have *some* limitations. It shouldn’t be wrong. Two and two is definitely not five. Other than that, it’s limited only by our imagination.

* * *

Let me try to give you an idea of what I think mathematics is by giving you an example of the type of outlook a mathematician has.

Suppose a mathematician walks into a bar. This is not a joke… much…

In the bar there is a pool table. But as you’re all underage, you don’t know this.

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For those of you who haven’t played pool, the idea is to hit these billiard balls into the pockets using a cue stick. The exact rules aren’t important here, the point here is simply that the balls roll around on the table, bounce off the sides, hit each other, and so on.

Anyway, a *lot* of mathematics has been inspired by pool tables. And this is not (just) because mathematicians drink too much.

A mathematician might look at a pool table and see some interesting geometry problems – to get this ball into that pocket, where should you aim? Well perhaps you can do a reflection like this, and aim over there.

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And she might go further, and wonder, what can you say about the path, the trajectory traced out by a billiard ball? Is it possible for a ball to start off in one position, going in a particular direction, and later come back to that position going in the same direction? Can it do so after hitting every wall? Is it possible to hit the wall 17 times and come back to your initial position and direction? Is it possible for a billiard ball to go in such a way that it eventually passes through *every* point on the table?

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What if you change the geometry of the table – make it a triangle, or a square, or a pentagon, or even make it curved? What if it’s a concave heptagon?

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What if it’s 3-dimensional?

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So, by pure imagination, you have a dozen questions all ready to think about. They belong to different fields of mathematics, some of which you may never have seen. But somehow they’re all mathematical questions. Some of them are much harder than others. Some might require more advanced ideas, or ideas that haven’t even been invented yet. Others might be impossible.

But the point I’m making is that mathematician is free to ask whatever questions she pleases. She is limited only by her imagination. She invents her own problems, and tries to solve them. She solves the ones she likes. She solves the ones she can.

In the meantime, everyone else in the bar is having a slightly less nerdy good time.

But in fact, it goes even further. There are several whole fields of mathematics devoted to billiard balls.

For example, mathematicians have shown that you can arrange billiard tables and billiard balls in all sorts of interesting shapes and configurations to get all sorts of interesting results. You could have many billiard tables, joined up by narrow passages, like pipes or tubes, forming patterns. You can have billiard tables designed to keep billiard balls rolling about for a while and then come out through a tube. You can have billiard tables with incoming and outcoming tubes, and a billiard ball comes out this tube if balls comes in both those tubes.

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Here is a picture of a billiard table which functions as an AND gate, which is something you see in electronic circuits and in computers. In fact, you *can* make *computers* out of billiard balls.

Of course, it’s a *hypothetical* computer where we assume that all the billiard balls move without friction and never slow down and don’t spin and so on. But mathematicians are happy to make those assumptions, we are not limited by those kinds of practical considerations. But under those assumptions, you can arrange a billiard table to do the things a computer can. Here’s the Wikipedia article on it.

And *here* is a book on mathematical billiards.

Do you recognise any of the maths there? I’m struggling, and I do this stuff for a living!

It gets into some of the most advanced and cutting-edge areas of mathematics.

And this is just one sub-sub-sub field of contemporary mathematics. Mathematicians take an idea, like a billiard table, think up some problems, solve them, and develop entire theories out of their pure imagination and curiosity.

So maybe some mathematicians have too much time on their hands.

But who knows? Perhaps this mathematics one day will be used to design solar panels – after all that’s all about reflecting light rays cleverly, just like a pool player bounces balls off walls cleverly. Or who knows what else. Or perhaps not. History is full of examples of mathematicians thinking up maths for its own sake, which then later on turns out to be useful in a completely unforeseeable kind of way.

If their maths gets used for something useful, the mathematicians who invented it will be very happy; but, if it doesn’t, they won’t be particularly disappointed.

That’s how mathematics works. It is, by far, the craziest and most unpredictable of the sciences, and I think the most fun and the most profound.

* * *

It’s interesting, for instance, to consider from this point of view a subject that by now most of you – maybe all of you? –have studied: calculus.

Who discovered calculus? These two gentlemen.

Calculus was discovered at roughly the same time by two different people: Isaac Newton (left), and Gottfried Wilhelm Leibniz (right). Newton was English, Leibniz was German, and who came first was a matter of national pride and a lot of controversy.

Online there is a comic called XKCD. It has nerdy maths jokes in it. Here’s one about the discovery of calculus.

(Source)

Get it? Because a derivative is a thing in calculus, but also, when someone takes an idea someone else has, it’s called…

Anyway, so let’s consider the discovery of calculus.

It is an enormous advance in knowledge to be able to use calculus. Using differential calculus, you can figure out, merely from knowing the position of an object, *how fast it is moving*, at any instant of time. You can then calculate the exact trajectory of an object – so you can work out the motions of objects, of any size, from billiard balls, to the motions of the stars, to… Angry Birds.

But the discovery of calculus was very controversial – not only for the priority dispute between Newton and Leibniz. It was also controversial because it made no sense. Do you remember your dy/dx, and wondering what the dx and the dy mean? The derivative dy/dx tells you how much y changes compared to how much x changes. But what is this dx and this dy? They are supposed to represent really small changes in x and y. But how small? *Really* small – *infinitesimally* small – smaller than any positive number – and yet not zero.

Newton and Leibniz both had incredible intuitions about these things. They knew what they were doing, and knew how to use calculus – and yet they couldn’t quite express their theories in a sufficiently rigorous way to satisfy their colleagues. They were mocked for producing these inconsistent things which were infinitesimally small and yet not quite zero and if not quite zero but not any definite number more than zero then what? Their colleagues asked WTF.

Later, the ideas of calculus were put on a logically firm footing – today it’s on firm ground. This was largely done with the idea of *limits* – and today, at school, when you learn calculus, you learn about it via limits. This definition of the derivative, which you’ve hopefully seen, comes much later – it was not how Newton or Leibniz did it.

But if you’ve ever felt that the dy and the dx are really weird and what the hell is going on here and am I really getting the full story – then I applaud your scepticism and you are in excellent company and you can get the full story with further study – although it’s not an easy story and there’s a reason it’s left to university. There’s a whole world of mathematics that comes from considering these ideas further — and that mathematics is crucial in engineering and physics and much more.

Newton invented calculus, actually, not out of pure curiosity, but because he wanted to understand the motion of the planets. It was physics that motivated him. And he made enormous breakthroughs by applying calculus to the problem.

Calculus is an intellectual superpower. And today, everyone who learns calculus receives the benefit of this superpower. You all have the power to calculate how fast something is going merely from knowing its position as a function of time. Tell us where something is, and we can tell you where it is going. Nobody had that power until 300 years ago, and today it has become completely routine. Although the awesomeness of your superpower may become a little lost on you as you differentiate six hundred slightly different functions from your mathematics textbook.

And of course, our mathematical superpowers have grown enormously since the 17^{th} century.

We now know how to use mathematics to see how matter bends space and time. We use mathematical superpowers to move gigabytes of information in mere seconds around the world – and yet invisibly, so that all you see of it is a cat video.

We use mathematical superpowers to encrypt our messages so that even if we broadcast our encrypted message to the world, put it on display, and even broadcast how we encrypted it, then all the computing power in the world is not enough to decode our message.

Mathematics can do all this, and we are still figuring things out.

I like to think of mathematics as a brain extension. Mathematics is a brain extension that allows you to solve some problems you never thought you could. And it’s being developed further all the time.

* * *

And that leads to a philosophical question, which you may have discussed in your theory of knowledge course. What is mathematical knowledge? Whatever it is, it’s a different type of knowledge to almost anything else.

When we say that 1+1=2, we’d say that’s a true statement. It’s not very controversial. That’s mathematical knowledge.

But if we said that 1+1=3, we’d be pretty well justified in saying it’s false. That’s not very controversial either. That’s also mathematical knowledge, a known falsehood.

And this truth and falsehood is in a kind of absolute sense.

And there are of course many other mathematical truths. That’s what mathematics consists of. All the mathematical theorems and facts you’ve learned at school so far are mathematical truths.

So let’s look at one piece of mathematical knowledge, which is really nice: Pythagoras’ theorem.

Remember what Pythagoras’ theorem says?

Pythagoras’ Theorem:In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

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Yes, but can you prove it?

Actually there are many many ways to prove Pythagoras’ theorem. It’s probably the most proved theorem ever. There are books consisting entirely of hundreds of different proofs of Pythagoras’ theorem. I’ll show you one of my favourites, which is similar to a one given by Bhaskara, an Indian mathematician who lived in the 12^{th} century AD.

It’s a proof by animated GIF.

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This animation I think shows really nicely what’s going on. Now to give a proof you should write an actual argument, and explain why when you swing the triangles around they make the arrangement shown. But that’s not too hard. I think you can get a lot out of this picture.

But I think, if you understand this picture – and you might have to stare at it for a while to fully understand it – then you will understand that the certainty with which we believe Pythagoras’ theorem to be true is a similar level of certainty as we believe that 1+1=2.

So this is another example of a mathematical proof – a type of knowledge which is arguably the most certain thing in the world.

* * *

So 1+1=2, Pythagoras’ theorem, and other mathematical theorems, all have this kind of absolute truth about them. The level of knowledge contained in them is very different from other statements.

Are there any statements outside of mathematics which have a similar status of absolute truth?

Well, there are facts about the world, which are pretty certainly true. There is a chair here, today is sunny, Donald Trump is the President-Elect of the United States. The last one may be hard to believe, but that doesn’t make it any less true. They’re all empirical facts.

Still, I wouldn’t say they quite have the same level of certainty as 1+1=2. We might debate what sunny means, about the technicalities of US elections. And we might be wrong about them. We say it’s sunny, but this is Melbourne, so it might spontaneously cloud over and rain in a few minutes. Perhaps it will be discovered tomorrow that there was a giant conspiracy to rig the US election. There are also more radical ways we could be wrong. Perhaps there is not in fact a chair here and my eyes are deceiving me. Perhaps I am hallucinating. Perhaps we are all hallucinating. Perhaps this is all a dream. All these things are very unlikely, but in a certain sense, not entirely inconceivable.

On the other hand, is it ever possible that 1+1=2 could be falsified?

Scientific facts and theories also have a very high status as knowledge. But even scientific facts and theories are falsifiable. In fact, this is the whole point of scientific theories: they should be able to be confirmed or disproved. Even the most well-established theories can turn out to be wrong. Newton’s theory of gravity works to a very high level of precision, but turns out it’s wrong – it was superseded by Einstein’s theory of relativity, which makes different predictions, and when they differ Newton is wrong and Einstein is right. Relativity might be superseded in due course. Every scientific fact is just as weak as the next experiment, and if the next experiment doesn’t agree with the theory, it could all come tumbling down.

But could you imagine that one day an observation will contradict 1+1=2, and that in fact it might turn out to be a tiny bit more than 2? Not really. It’s an interesting question why, and nailing down the certainty of mathematics is a topic which has occupied some of the greatest minds in mathematics and philosophy – people like Descartes, Bertrand Russell, and Kurt Gödel.

So there’s a certain sense in which mathematics is the most certain type of knowledge. One might even define mathematics as the thing that produces certain knowledge.

But the type of knowledge that you get from mathematics is limited. It can tell you many things about numbers, or polynomials, or geometry, but perhaps not so much about, say, morality, or justice, or a good life, or love. Perhaps it might contribute something, but not much.

Actually my favourite nerd-comic, XKCD again, has a good one about mathematics and love.

(Source)

A mathematician has some tools that can be used to analyse many things. But alas, it doesn’t take you very far here. Although there is a recent book by Hannah Fry called “The Mathematics of Love”, which is lots of fun.

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* * *

A related question is about what type of thing mathematics is. When the ancient Greeks wrote down the first proofs of Pythagoras’ theorem, were they inventing it, or discovering it?

With science, we speak of scientific *discovery*, not invention. We discover the law of gravity, or the laws of optics, or whatever. We don’t invent it. The law was already there; humans just figured it out.

When we speak of *invention*, we mean something that is made by humans, created by human ingenuity. We invent the steam engine, the transistor, the iPhone, and so on. The invention wasn’t there before; humans brought it into being.

When mathematicians prove new theorems, is it like discovering the law of gravity, or inventing the iPhone?

A lot of mathematicians, in describing how they solve problems and prove theorems, will talk about *discovery* rather than invention. Pythagoras’ theorem was true before they proved it; they just found a proof.

In that case, mathematics already exists. As we gain more knowledge about it, we discover it, just like the laws of nature.

If you’ve ever solved a hard maths problem, you might have had a feeling that everything finally made sense, that it all became clear, you finally “saw” it. And that kind of feels like you were discovering something, or finally seeing something clearly that was already there. Often in maths we find that there is a nice answer that “really” explains what’s going on – perhaps like that proof of Pythagoras we saw before. And then we talk about “finding” or “discovering” the solution, rather than inventing it.

This sort of approach is often associated with the ancient Greek philosopher Plato – and called *Platonism*. Platonism is then basically the view that *abstract objects exist*. When we say that 1+1=2, what are we saying? It seems that we are saying that there are numbers, 1 and 2, and when we add the 1 to itself we get 2. But then what are these things called numbers? The Platonist says that, you know what they are, they exist and they are ideal objects. They are not objects you can touch, but they exist just as well. They are not just in our minds. If there were no humans in the universe, or anyone to think it, it would still be true that 1+1=2. So, a Platonist would say, the numbers 1 and 2 exist.

And similarly, the Platonist would say, not just 1 and 2, but every single number exists. They are all mathematical things. Similarly, abstract triangles, sets, functions and so on all exist. Every abstract mathematical object exists.

So, that’s one view, Platonism.

A contrary, anti-Platonist view, which sometimes goes by the name of *nominalism*, is that no, there is no heaven somewhere containing all these objects. Where would it begin and where would it stop? Mathematics can think up all manner of abstract objects, and if all of them exist, eternally and regardless of whether we think of them, then this is just an enormous parallel universe littered with mostly useless abstract objects, all adding very little except confusion.

Well, fair enough, but then the nominalist has to answer the question: If 1 and 2 are not things that exist, then what are they?

So, suppose you’re asked to explain why 1+1=2. What do you say?

If you’re a normal person, you might say something like, when I have one thing and I put it together with another thing I get two things. If you do this, it’s pretty much a nominalist point of view. You’re not arguing that the numbers are pre-existing things. On this view, numbers are just a sort of generalisation – an abstraction from everyday observations. We made the generalisation a long time ago, probably in our infancy, from collections of objects, to numbers. On this view, numbers are just these really cool abstract ideas, ideas in our heads, useful to count things.

If you’re not a normal person but a mathematician, and you’re asked to explain why 1+1=2, then you might start by answering like the normal person, but if you’re pressed on the point, you might take a different tack. You might fall back on *definitions*. You might go back to a definition of 1, and a definition of 2, and a definition of addition and equals, and then explain why, when you put all these definitions together and make a few deductions, you get an explanation as to why the statement 1+1=2 is true. So the whole thing becomes a big exercise in definitions, and you have to make sure of what your definitions are and why they all fit together – and in the end, the whole thing is just a tautology. 1+1 = 2 because we defined 1 and 2 and + and = in such a way that 1+1=2 is true. This approach sometimes goes by the name of *formalism*, because it relies on formal definitions.

Now this might seem bizarre. Is it really possible to *define* the number 1? Is it possible to *define* addition? Aren’t these just the basic concepts in mathematics? And also, don’t you have to define things in terms of simpler things? How far back can you go? How simple is simple enough? Where does it all end?

Well many mathematicians have thought about this, and there are mathematicians who have devoted a huge amount of time to building mathematics up from the simplest of foundations. In the early twentieth century, two mathematicians thought about it a lot.

(Sources: Bertrand Russell Society, Wikipedia)

These two guys are Bertrand Russell and Alfred North Whitehead. They were both leading mathematicians of the early 20^{th} century. Russell was also a very famous philosopher and author and socialist and educator and peace activist and many other things – probably one of the most impressive human beings of the 20^{th} century. He arguably literally saved the world during the Cuban Missile crisis in 1962, but that’s a whole other story.

These two mathematicians thought very long and hard about how to define mathematics from the very beginnings, from simple, obvious – even more obvious than 1+1=2 – propositions called *axioms*. And then to *deduce* the whole of mathematics from that.

They wrote an enormous series of books, 3 volumes, 2000 pages. It was called *Principia Mathematica*. There it is.

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And in these 2000 pages they make every single definition and axiom and deduction absolutely clear and explicit. And eventually, they manage, after 700 pages or so, to *prove* that 1+1=2. Here’s the key part of the proof.

Looks like alien hieroglyphics doesn’t it? But yes, after nearly a thousand pages, these mathematicians proved that 1+1=2.

Well you can now relax.

You might wonder if these guys had anything better to do. Well, actually Bertrand Russell as I mentioned had lot of things of other things to do as well, but he thought this was at least as important!

So, anyway, if you asked me to explain why 1+1=2, and pressed the point, I would eventually point you back to this work of Russell and Whitehead – *Principia Mathematica*.

And this is really the modern approach in mathematics. A mathematician these days will often write their proofs by making formal definitions of whatever she’s talking about, and then deducing things about them. These objects might or might not relate to the real world; sometimes they do, and sometimes they do not. When they do that’s great; when they don’t, it’s not a great disappointment.

And that is roughly my view. I’m not a Platonist, I’m a nominalist, and a pretty formalist one when it comes down to it. But there are plenty of mathematicians who are Platonists. When you solve a maths problem, I would say, it might feel like you’re uncovering something from the Platonic heaven of universal ideas, but if you feel that way, I’d say that’s just because mathematics is a wonderful subject.

* * *

OK, well that’s all pretty heavy stuff.

Let’s talk about some actual living mathematicians. Most people don’t know any mathematicians alive today, so let me introduce you to a few leading ones.

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This is Grigori Perelman. He won the Fields Medal in 2006. The Fields Medal is the maths equivalent of a Nobel Prize. He works in a field of mathematics not so far from my own. He is an interesting guy – friend of one of my PhD supervisors.

Anyway, Perelman solved one of the hardest outstanding problems about 3-dimensional space, called the *Poincare conjecture*, about 10 years ago*. *I’m not going to try to explain it to you here, but it’s a fundamental problem, but really hard, really fundamental, and there was actually a million dollar prize for anyone who could solve it. He solved it. He then got famous, was in the media a lot, and he didn’t like being paraded in the media. He decided that the community of mathematicians was corrupt, and turned down the million dollar prize. He currently, so far as I know, lives as a hermit with his mother in St Petersburg.

Mathematicians are often interesting characters.

Here’s another interesting character.

(Source)

This is Edward Witten. He is a physicist, perhaps most famous for his work on string theory. But he also won the Fields medal, in 1990, because in doing physics he also did a lot of mathematics. And it’s mathematics which people are still trying to figure out, myself included. Perhaps you can think of what he’s doing, as like Newton or Leibniz with calculus in the 17^{th} century. I read his papers and I have no idea what he’s talking about. Some mathematicians think Witten is sloppy, and talking a lot of nonsense. But then when it comes down to mathematical statements, and calculations, you tend to find out that he’s right. It’s incredible intuition which somehow almost transcends mathematics. By all accounts he’s a very nice guy, and an absolute machine. Is what he does mathematics? It’s not the traditional way of doing mathematics, and not all mathematicians think it is, but I think it is. Just like calculus was hard to digest back in those days, some of Witten’s work isn’t quite yet digestible as maths.

Ed Witten actually studied history at university. He got a degree in history. Then he dropped out of that and decided he wanted to do… economics. Then he dropped out of that too and worked in…. politics. He worked on a losing campaign. They he turned to… maths, but dropped out of that too. Finally he turned to physics. And today he’s probably the most influential physicist alive.

And here’s another famous mathematician.

(Source)

This is Maryam Mirzkhani. She’s an Iranian mathematician, and another Fields Medallist. She works in a field, again, not so far from what I do. She is a great role model for women wanting to study science and mathematics.

* * *

Let me tell you about some of the things I do in my daily life as a mathematician.

As I said, I’m a lecturer at Monash Uni. So one thing I do is lecturing. I teach classes. I give lectures on mathematics. So if you come to Monash uni and study mathematics, you might be in one of my lectures. And if you go on to major in mathematics, you could take my course in differential geometry. If you go on to do honours in mathematics, I might end up teaching you topology. So that’s one thing I do, which is pretty good fun. At least it is for me! And for students, maybe, sometimes!

I also do mathematical research. I work in pure mathematics. As I’ve mentioned to you, as with the billiard balls, pure mathematical research is research that is done purely for its own intrinsic interest. It may have applications, or it may not; but we think it sufficiently interesting and important to develop new fundamental knowledge.

Mathematical research is like mathematical problem-solving, except you get to think up the problem as well as the answer. And if you solve a problem that nobody has ever solved, or even asked, before, then they have progressed human knowledge. And a lot of mathematics develops in this way.

So those are some of the things I do as a mathematician. But I’m not the only type of mathematician.

Many mathematicians also work on research which is much more applied. They tend to make a lot more progress than I do. They don’t beat their head against a wall with impossible problems like I do. Their research has direct practical applications and can affect people’s everyday lives.

Applied mathematics is often motivated by a concrete practical problem, and devising new practical solutions.

And there are many different types of applications. There are mathematicians who apply their mathematics to all sorts of things: chemistry, biology, data analysis, finance, transport, consulting, programming, economics, energy, engineering, government, health, insurance, meteorology, military, intelligence, logistics, biotechnology, teaching – of course, as your maths teachers do here – and much more. Maths is everywhere.

There are also many people who use maths as a large part of their job, but who may not actually call themselves “mathematicians”. They might be setting up mathematical models to simulate a practical situation. They might be making predictions based on such models. They might be running algorithms. They might be coding. They might be analysing data, or calculating statistics, or optimising a manufacturing process or transport network or energy flow. They’re all using maths to solve real world problems. Mathematical skills, and the creative logical thinking skills of mathematicians, are sought after by employers everywhere

So, maths is powerful. It can be used to do many things, and improve the power that we have to do things. It’s a superpower. Like all superpowers, it can be used to achieve all sorts of social ends, good or bad. It can be used to match up organ transplant recipients and save lives. It can be used in war to produce more efficient death. It can be used to optimise a transport network for maximum efficiency. It can also be used to optimise a coal mine for maximum extraction. It can be used to locate tumours and treat cancers. It can also be used to build bigger and more lethal weapons. It can be used to create a more equitable economic system. It can also be used to crash Wall Street.

It’s a big powerful thing, and like all science it can be used for all sorts of ends. I hope you enjoy it, and that you’ll put it to good use.

## What to do while Rome burns

When the Goths sacked Rome, St. Augustine wrote the “City of God”, putting a spiritual hope in place of the material reality that had been destroyed. Throughout the centuries that followed St. Augustine’s hope lived and gave life, while Rome sank to a village of hovels. For us too it is necessary to create a new hope, to build up by our thought a better world than the one which is hurling itself into ruin. Because the times are bad, more is required of us than would be required in normal times. Only a supreme fire of thought and spirit can save future generations from the death that has befallen the generation which we knew and loved.

— Bertrand Russell,

Principles of Social Reconstruction(1916)

## The Eighteenth Brumaire of Donald Trump

Marx and Hegel remark upon the repeating phases of history. On the Eighteenth Brumaire (9 November) 1799, Napoleon Bonaparte seized power in France. Louis Napoleon did the same in 1851, and Marx wrote about the farcical character of the repetition. First as tragedy, then as farce, he said. Tragedy and farce and much more — with vastly greater consequences — have taken place on the Eighteenth Brumaire 2016.

History’s repetitions are not as cleanly tragedy then farce as Marx claims, but it does repeat, and it repeats each time with more tragedy and more farce. And economic development brings with each repetition more powerful technology, more powerful institutions, and greater means to inflict damage on the world.

It is surely true, as Marx wrote then, that people make their own history — voters vote for who they do with an intentional conscious choice — but they do not make it under self-selected circumstances. The circumstances are given and transmitted from the past — the heritage of left and right, of boom and bust, of global recessions and resentments and racism and nostalgia for supposed national glories. And so short are their memories, so influenced are they by the ideas and ideologies that infect culture and psychology, that history’s repeats carry increased tragicomic impact each time.

After the tragedy of Kennedy’s escalation of the Vietnam war and his near-destruction along with Kruschev of the world in the Cuban missile criss, came the carpet-bombing tragedy — enacted in farce, but not for the victims — of Nixon, merely one of whose crimes was to drop more bombs on Cambodia than the US did on Japan in the second world war. Then came the murderous menace of a terrorist and senile Reagan, the destruction of the threat of post-cold-war peace by Bush the Elder and Clinton, and the criminal invasion of Iraq, and more, by Bush the Younger. After eight years of war as usual by Obama, winner of the Nobel Peace Prize for wars in Iraq, Afghanistan, Pakistan, Yemen, Somalia, Syria, and Libya, we now have a thin-skinned lying vindictive narcissist with his finger on the nuclear button.

* * *

Usually the character of a politician is the least of their problems. Discussion of candidates’ character is usually used to deflect attention from policies. In general most politicians will follow the agenda of their backers — lobbyists or their party. They might follow their party’s policies or the whims of their latest focus group — the usual cynical machinations. But their character is more or less irrelevant — the amount of overt lying may vary, but the outcomes not so much.

But with Trump it is different. The level of vindictiveness, the impossibility of compromise, the outright pussy-grabbing misogyny, the outright racism, the encouragement of violence, the twitter meltdowns, betray precisely the temperament that disqualifies a leader — if only because that sort of temperament of a man with his finger on the nuclear trigger augurs poorly for civilization.

But enough has been said about the disqualifying tendencies of Trump. By rights he should have been eliminated from the electoral field long ago. But enough voters wanted to burn the system down that they voted for him.

One generally hopes that conservative governments are incompetent; and the further right, the more incompetent. There is little one can ask or expect of a far-right leader, but at the very least one might hope that they be sane enough not to destroy the world as they destroy their political enemies. It is not at all clear whether this is true of Trump.

The traditions of all dead generations still weigh like a nightmare on the brains of the living. Trump is too ignorant to know the half of it. But the worst of it still lives on in our culture, as racism, as xenophobia, feeding our resentments and, when exploited by politicians, when whipped up by media and — let us not assume voters have no agency — when chosen by voters, it creates monstrosities.

Who said that history has ended? It has just swung into the most unpredicable, dangerous waters since the 1930s.

* * *

Sometimes one wonders why people vote the way they do. Those who vote for the right, or the far right, may do so out of resentment, out of ideology, out of misogyny, out or racism, out of ideology, out of genuine conviction, the cult of personality, or out of a mere desire to burn the system down and root out the corrupt establishment. Surely all of these factors are present in Trump’s election. Those of us on the left will surely point to the failures of an economic and political system that have left the working and middle classes of an enormous nation with little to show for lifetimes of hard work and effort. The anger of voters at a decadently corrupt and self-serving system that crushes unions, destroys hope, depresses wages, and which at its best delivers a meagre improvement in health insurance coverage, is surely justified. We can hope that a half-decent left could win over such voters with a half-decent programme that at minimum restored some worker rights, delivered some improvements in health and welfare, alleviated extreme inequality, and stimulated jobs and growth. And perhaps it could have — polls suggest that Sanders would have done much better against Trump than Clinton did.

But the results do not indicate economic resentments as the only factor in Trump’s win. Trump’s support skewed towards higher income brackets, towards whites — the classic constituency of fascism. A majority of white female voters voted for him and against the first ever US major party female presidential candidate. All of this is speculation — little more than reading tea leaves — but there is no doubt that the ugliest sides of politics, stirring up the ugliest sides of human nature and of the not-so-distant past, have played a role too.

My own view is that human naure is simultaneously so dark and so good that almost any result is possible in varying circumstances. It can soar to the most beautiful heights of compassion and humanity; but also, there is no limit to how low it can go. Trump, carrying in his rhetoric, if not consciously, the dead weight of history’s far right — the fascists, the authoritarians, the segregationists, the slaveowners and yes also the Nazis, for he did not earn his neo-Nazi endorsements for nothing — has taken it to depths not seen in the West for a long time. The spectacle of the first African-American President handing over the keys to the White House to a KKK-endorsed candidate is nauseating, and the nausea is no less for the fact that Obama will likely do it with grace, while Trump clings to his hateful resentments.

* * *

Of course, such depths of depravity have never really gone away. They have been plumbed, conspicuously, by US governments continually for a long time — at least they are conspicious to those on the receiving end of foreign policy. Casual bombing of people and places far away from the US, but able to be regarded as sufficiently evil, terrorist, Muslim or crazy, is bipartisan and par for the course. Obama bombed seven countries and the US establishment never batted an eyelid, unless to berate him for being too weak. Whole provinces of Pakistan suffer trauma from random drone bombing death; liberals applaud Hillary Clinton’s sensible defence of such policies, indeed expansion of them with her more hawkish stance; conservatives rail as to why they are insufficient.

Talk about Clinton being a progressive or “liberal” candidate should never have been met with anything but derision. She was instrumental in creating the tragedy of Libya, which in turn established conditions for tragedies across the region. She would have provided much of the same as President, probably with slightly more death and destruction. Those depths would have remained well plumbed.

To be fair, some conflicts may well ease or even end under a President Trump. Relations with Russia might warm; life might become easier for Syrians. My understanding is that by and large Syrians would prefer Trump.

But the overall threat to the world of a Trump is vastly greater. Calls for carpet bombing, expanded torture and mass killing of “suspected terrorists” and their families means, if it is to be taken seriously, a vastly expanded war machine. The US military industrial complex is a killing machine constantly primed to bomb some enemy, and Trump will turn the machine up to a higher kill rate. One can hope that it is merely rhetoric, but the consistency of his advocacy of war crimes and international terrorism, combined with his macho aggression, reduce that to hope to a very slim one.

So let us not talk of a turn to savagery. The savagery has been present in US government policy for a long time, and there is no doubt it will continue. But while Clinton was bad and would have provided a predictably worse, outcome, Trump is completely unpredictable and irrational, and the worst outcomes under him are global catastrophes.

* * *

There will be enough finger pointing. Clinton was a terrible candidate, and part of the reason she lost was because she was so: more comfortable with bankers than ordinary workers, coldly cynical, no less corrupt than the rest of the system, and murderously hawkish, even as she genuinely pushed a slightly more progressive economic policy, together with gender equity and liberal feminism. She lost because she was a terrible candidate, but no doubt she also lost because she was a woman. Australians do not have to go far into the past to remember the incredible and irrational level of hatred displayed towards former Prime Minsiter Julia Gillard, in order to understand the misogyny directed at Clinton.

Clinton lost for many reasons, no doubt, including these and many others, that are impossible to untangle because every voter has their own mix of incoherent reasons for voting the way they did.

Sanders may have done better; polls seem to suggest he would have. Sanders pushed a mild form of social democracy; he garnered a following under the banner of the word “socialism” that one might have thought scarcely imaginable in the US. He probably would have done better because he also opposed the establishment. He probably would have done better because misogyny does not apply to him. Stein was a far better candidate again. And with a half-decent voting system, and a a half-decent chance at media airplay, who knows how far she might have gone. But all this speculation is of no use now.

The establishment may burn, and in that case good riddance. If the Democratic establishment, that at every turn undermined and sabotaged Sanders and his supporters, falls apart then that is no great loss.

No great loss, that is, as long as a serious and half-decent alternative can be built in its stead.

* * *

The left should be, of course, terrified at the prospect of a Trump presidency. But it should also be emboldened at the inroads made by a self-proclaimed socialist. In an alternative universe not so far from our own, a socialist might now have been the most powerful person in the world.

What is to be done? I can only offer my own thoughts, meagre as they are.

Here in Australia, we are not unaffected by the result. We are affected by US politics just like the rest of the world. And just like much of the rest of the world, we have our own strain of Trumpism, as it adapts to local conditions.

The Australian Labor Party has long possessed many of the worst tendencies of the Democratic Party: a decaying culture, disconnection from the grassroots and ordinary working people, corruption, moral cowardice, and a long historic retreat from the progressive values it was supposed to stand for. The Liberal Party has a faction no less conservative, no less ignorant, climate-denying, racist or misogynist than the Trumpists, to which the Prime Minister is beholden. And the far right of One Nation is newly empowered and welcoming Trump with open arms.

But Australia has already enacted some of the main planks of Trumpism. The equivalent of Trump’s wall with Mexico already exists in Australia, with cruel turnbacks of boats ensuring that all hope of refugees arriving safely in Australia is destroyed. Trump wants to ban Muslim immigration, a policy which also has significant support in Australia. And not even Trump has advocated holding refugees in offshore jurisdictions where they are sheltered from legal liability in conditions amounting to torture. Australia does it as a matter of course and it is accepted bipartisan pollicy.

If Australians want to fight Trumpism, the fight starts at home, against xenophobia, racism, sexism, and all the ugliness which in many ways is no better here than in the US. It also consists of a fight for better working conditions and against inequality, which is creating a divided country, even if not quite so starkly divided as the US.

If we are scared of an Australian version of Trump, the fight begins, in the short term, by taking on the cultural and economic pillars of support for the far right. It should underscore the urgency to close refugee camps on Nauru and Christmas Island, to redress historic wrongs against Aboriginal people, to promote progressive economic policy, and to stand steadfastly against xenophobes, racists, and misogynists.

But in the long term — not just in the US, not just in Australia, but everywhere — surely the task is the same as always: to build serious progressive left movements. The movements of the disaffected, of the oppressed, the marginalised, minorities, and all those crushed by racism, by capitalism, by sexism, need to build a serious infrastructure and a serious programme for a better world.

One might have thought that, faced with Trumpism, it was enough to not be as crazy as a Trump.

On 9 November 2016, the Eighteenth Brumaire of Donald Trump, we have learned that it is not.

## On the end of the world

Astronomy is the most humbling of the sciences.

It is humbling not only because of the reminders of our insignificance provided by the unfathomable depths of interstellar space, or the eons of time in which galaxies form. It is also humbling because, as we know, all stars die. Our sun, being no different, will die too, and our solar system with it, including our precious planet Earth and everything on it. (Indeed, the Earth will be enveloped by the sun and die its own death a rather long time before the sun dies.)

One can take several possible attitudes to this bleakest of certainties about the future.

Bertrand Russell, in his 1903 essay “A Free Man’s Worship”, took the view of unyielding despair.

Man is the product of causes which had no prevision of the end they were achieving; that his origin, his growth, his hopes and fears, his loves and his beliefs, are but the outcome of accidental collocations of atoms; that no fire, no heroism, no intensity of thought and feeling, can preserve an individual life beyond the grave; that all the labours of the ages, all the devotion, all the inspiration, all the noonday brightness of human genius, are destined to extinction in the vast death of the solar system, and that the whole temple of Man’s achievement must inevitably be buried beneath the debris of a universe in ruins — all these things, if not quite beyond dispute, are yet so nearly certain, that no philosophy which rejects them can hope to stand. Only within the scaffolding of these truths, only on the firm foundation of unyielding despair, can the soul’s habitation henceforth be safely built.

One must face the facts. When pressed to think of something really certain, a person will often say that it is certain that the sun will rise tomorrow. And it will; but what is equally certain is that one day it will not — there will be no day. There is no escape from the ineluctable slide into disorder, entropy, and the dusty, cold heat death of the universe.

Many years later, Russell in 1927 took a somewhat different view. His essay “Why I am not a Christian” broached the topic, and took an entirely different attitude of almost breezy nonchalance to this cosmic angst:

if you accept the ordinary laws of science, you have to suppose that human life and life in general on this planet will die out in due course: it is a stage in the decay of the solar system; at a certain stage of decay you get the sort of conditions of temperature and so forth which are suitable to protoplasm, and there is life for a short time in the life of the whole solar system. You see in the moon the sort of thing to which the earth is tending—something dead, cold, and lifeless.

I am told that that sort of view is depressing, and people will sometimes tell you that if they believed that they would not be able to go on living. Do not believe it; it is all nonsense. Nobody really worries much about what is going to happen millions of years hence. Even if they think they are worrying much about that, they are really deceiving themselves. They are worried about something much more mundane, or it may merely be a bad digestion; but nobody is really seriously rendered unhappy by the thought of something that is going to happen to this world millions of years hence. Therefore, although it is of course a gloomy view to suppose that life will die out — at least I suppose we may say so, although sometimes when I contemplate the things that people do with their lives I think it is almost a consolation — it is not such as to render life miserable. It merely makes you turn your attention to other things.

And this would, it seems to me, be the view of the average practical person, who needs to get on with their life and, even if they are not soothed by the temptingly comforting delusions of religion, have quite enough to worry about in the next few hours or days, and anything on a cosmological timescale is entirely outside their purview.

I cannot accept this view. The argument that we should not think about the bleakest and darkest facts of our existence, simply because they are far away, is in essence no different than the argument that we should ignore other uncomfortable facts about the world, merely because they are remote from everyday considerations. The non-existence of god, the loneliness of individual consciousness, faraway victims of war, or the uninhabitable climate left to future generations — all these are cause for despair, and yet we are superficial, or at least incomplete as human beings, if we do not consider them.

The world is there to be faced. We do better to look it square in the face and understand it for what it is, than to shy away and live an unexamined life. Indeed, another quote of Russell’s seems to be appropriate here:

The secret of happiness is to face the fact that the world is horrible, horrible, horrible.

Perhaps as a matter of practical advice, for everyday cheer and a sunny (pardon the pun) disposition, one can justify a wilfully blind attitude. But from the point of view of one who wants to understand the world fully, live in it fully, one cannot.

And that brings us to the extraordinary poem “On living” by the Turkish communist poet Nazim Hikmet. (This reading by Chris Hedges is stirring, but this version, in the original Turkish, with orchestral accompaniment, is beautiful.)

Concerning himself with the question of how to live, Hikmet, writing in 1948, dedicates the final stanza to the death of the world:

This earth will grow cold,

a star among stars

and one of the smallest,

a gilded mote on blue velvet—

I mean this, our great earth.

This earth will grow cold one day,

not like a block of ice

or a dead cloud even

but like an empty walnut it will roll along

in pitch-black space . . .

You must grieve for this right now

—you have to feel this sorrow now—

for the world must be loved this much

if you’re going to say “I lived”. . .