A white horse is not a horse

My attention was brought to this two millennia old paradox:

Horse can be black.

White horse cannot be black.

So white horse not horse.

See Wikipedia,  When a white horse is not a horse for a discussion.

This paradox is of interest to me because I started to think that one of the reasons of collapse of the New Math reforms in the 1960s and 1970s was introduction of sets and not predicates;  in my view, children appear to be more perceptive to the concept of unary predicate, or property. Properties are closely related to sets, but they are not sets. I was unaware that the universe of properties also has its share of paradoxes.


A comment on “Unreasonable ineffectiveness of mathematics in biology”

A colleague wrote to me as a comment on my paper

A. Borovik, A mathematician’s view of the unreasonable ineffectiveness of mathematics in biology. Biosystems 205, July 2021, 104410, https://doi.org/10.1016/j.biosystems.2021.104410 :

On the subject of unreasonable ineffectiveness, here’s a hilarious problem that might have a mathematical solution, but seems well outside the scope of current mathematics. […]

The Centipede Segmentation Problem: Prove that it is impossible to have a centipede that has an even number of pairs of legs, or 17 or 19 pairs of legs.

Apparently, there are fairly good datasets in biology which suggest that this is the case: https://onlinelibrary.wiley.com/doi/abs/10.1046/j.1365-3113.2003.00217.x

This reminds me one of the greatest mysteries of the Forest in Strugatsky Brothers‘ novel “The snail on the slope” from 1960s: the number of pups in a brood is always a prime number.

Their novel is admittedly the greatest book of Russian sci-fi, fantastically multifaceted; in particular, it is a vicious satire of the Soviet system, Soviet society, and Soviet science. The Forest is a symbol of many things, and one of them is unreasonable ineffectiveness  of not only mathematics, but also any kind of science in comprehension of Life.

You may skip most of the fragment from the novel that I included — I highlighted a few key lines at the end of the fragment.

The pups had already gone quite a way, but Stoyan, driving with great skill, keeping the offside wheels on the path and the nearside on the dusty moss, soon overtook them and crawled slowly behind carefully using the clutch to adjust his speed. […]

Pepper stroked his swollen finger and looked at the pups. The children of the forest. Or perhaps its servants. Or maybe its experiments. They were proceeding slowly and tirelessly one after the other in line ahead, as if flowing along the ground; they oozed across rotting tree stumps, crossed ruts, pools of stagnant water in the tall grass, through prickly bushes. The track kept disappearing, diving into evil-smelling mud, hiding itself under layers of tough gray mushrooms that crunched under the wheels, then again appearing, while the pups held their direction and stayed white, clean, smooth; not a blade of grass stuck to them, not a thorn wounded them, they were unstained by the sticky black mud. They oozed along with a kind of stupid unthinking confidence, as if along a road long-known and habitual. There were forty-three of them.

I was dying to get here and now I’ve arrived, at least I’m seeing theforest from inside and I’m seeing nothing. I could have imagined all this sitting in my bare hostel room with its three empty bunks; late night insomnia, everything quiet all about, then right on midnight the piledriver starts thumping on the construction site. I could have thought it all up: mermaids, walking trees and these pups, turning into pathfinder Selivan–the most absurd things, the holiest. And everything there is in the Directorate I can imagine and bring to mind. I could have stayed at home and dreamed this all up, lying on my sofa listening to symphojazz or voices talking unfamiliar languages on the radio. […] But that doesn’t mean a thing. To see and not understand is the same as making it up. I’m alive, I can see and I don’t understand. I’m living in a world someone has thought up without bothering to tell me, or maybe even himself. A yearning for understanding–that’s my sickness, thought Pepper suddenly, a yearning for understanding.

He stuck his hand out of the window and held his aching finger against the cool car-body. The pups were paying the landrover no attention. They probably had no suspicion of its existence. They gave off a sharp unpleasant smell; their membrane now seemed transparent and it was as if wave-like shadows moved beneath.

“Let’s catch one,” suggested Quentin. “It’s simple enough, we’ll wrap it in my jerkin and take it to the lab.”

“Not worth it,” said Stoyan.

“Why not?” Quentin asked. “We’ll have to catch one sooner or later.”

“Doesn’t seem right, somehow,” Stoyan said. “In the first place, God help us, the thing’ll die on us and I’ll have to write a report for Hausbotcher.”

“We’ve had them boiled,” Acey announced suddenly. “I didn’t like the taste, but the boys said it was all right. Bit like rabbit, I can’t touch rabbit, to me a cat and a rabbit’s just the same; can’t bear the stuff… .”

“I’ve noticed one thing,” said Quentin. “the number of pups is always asimple number: thirteen, forty-three, forty-seven… .”

“Nonsense,” objected Stoyan. “I’ve come across groups of six or twelve.”

“That’s in the forest,” said Quentin, “after that groups scatter in different directions. the cesspit always produces a simple number, you can check the log, I’ve put all my conclusions down.”


“Well all right, write an article then,” said Stoyan.

“I already have,” said Quentin.

Translation from Russian by by Alan Meyers, 1980.


Help Leicester Pure Maths fund Legal Support

Sibylle Schroll would like you to support Help Leicester Pure Maths fund Legal Support by making a donation and spreading the word.

About this fundraiserThe faculty of pure mathematics at Leicester needs your help. They would like to hire legal counsel in relation to the circumstances described in the links below:

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Tuna Altinel: detained in Turkey for two years now

Istanbul, April 12, 2021

Friends in solidarity,

Today is April 12, 2021. I have been detained in Turkey for two years.

It was two years ago today that my passport was confiscated in Istanbul, as a way to punish me for exercising my freedom of speech in France. Since that day I had to confront the stubborn silence of an autocracy, imprisonment, unfounded prosecutions, the obstinate recalcitrance in the face of my acquittals, the constant refusal to return my passport. The State that put me through this ordeal did not hesitate to violate its own laws.

During these two years, an international movement of solidarity has grown up around me. I wish to thank all those who have taken part. Before such a massive support of democratic values, the dykes of autocracy have crumbled: unlike most political prisoners in Turkey, I was quickly released from jail; I was acquitted; the Regional Court ruled promptly on the appeals. Finally, on April 2, the administrative court to which I had submitted an appeal of the refusal to return my passport has ruled in my favor.

Friends in solidarity, do not be abused! This is not the end. Before us stands a lawless and arbitrary power. The administrative court actually reached its decision as early as January 25, 2021, that is two months and a half before I was notified. Such a delay has no legal grounds. The State, which forced its own judges to this outrageous deferral, will not neglect to submit its appeal within 30 days (the legal deadline).

I call on all democratic forces in France. A new phase, hopefully the last, of this race of indefinite duration has just begun. We must remain mobilised.

Tuna Altınel

Source:   http://math.univ-lyon1.fr/SoutienTunaAltinel/doc/2021_04_12_MessageTuna_en.pdf


David Attenborough: No matter what we do now, it’s too late to avoid climate change

There was a much advertised (at least in the UK)  high level UN Security Council meeting on Climate Change on 23 February 2021 (online, chaired by Boris Johnson, UK Prime Minister). It appears  that the meeting has been hastily downgraded to  a non-event after Sir David Attenborough, addressing the Security Council in a short speech, said  something unspeakable:

“We have left the stable and secure climatic period that gave birth to our civilisations. There is no going back – no matter what we do now, it’s too late to avoid climate change and the poorest, the most vulnerable, those with the least security, are now certain to suffer.”

This segment of Sir David’s speech was not included in the BBC video clip  “Attenborough gives stark warning on climate change to UN – BBC News”A, https://www.youtube.com/watch?v=Dunm6DF78j4, and reports about the meeting (and Attenborough’s video) almost instantly disappeared from the BBC front page. However, the segment can be found on Sky News, https://news.sky.com/story/attenboroughs-stark-warning-on-climate-change-its-already-too-late-12226694.  The entire 8 minute video of Attenborough’s speech  is on the UN site:   https://news.un.org/en/story/2021/02/1085452.

On the optimistic note,  Sir David said

“I do believe if we act fast enough we can reach a new stable state. It will compel us to question our economic models and where we place value; invent entirely new industries; recognise the moral responsibility that wealthy nations have to the rest of the world and put the value on nature that goes far beyond money.”

It appears that there is a coherent, and well supported by archaeological and geological evidence, theory behind Sir David’s words; his speech cannot be easily dismissed as fantasy.

I summarise some points of this theory which I happened to learn from various sources  over the last 20 or maybe even 30 years. I am not an expert, and I would much appreciate corrections and further details.

  1. The current period of stable climate which allowed the human civilisation to develop, was about 11 thousand years long, and it was abnormally long on the scale of the last 50 or 100 thousand pretty turbulent years.
  2. The last violent episode which preceded our  golden era was a circulation event triggered by a flood of fresh water from the melting glaciers in North America which directed the Gulf Stream to Africa rather than Europe, correspondingly directing the jet streams in atmosphere over North Atlantic to south of Europe. This meant that temperatures at what now is London were like in nowadays Irkutsk. But Sahara got abundant rain and was covered by forests and lakes with hippos and crocodiles, and flamingos — immortalised in cave paintings (made by us, humans, who happened to migrate there at that time).
  3. All that is dated with surprising precisions by pollen from flowering plants preserved in sea sediments. The most interesting bit  is the length of transitional period between the two climate regimes — just about 10 years.
  4. A similar event can be triggered by collapse of glaciers in Greenland as the result of  melting water accumulating  in under  ice lakes and eventually finding its way to the ocean.  I would not claim, however, that this is to happen tomorrow. But something like that has already happened once.
  5. Of course, climate can mutate in many other directions, including Britain becoming, in hte new stable state of climate,  a subtropical country and Scotland replacing thistle, as its national symbol, by artichoke (Wiki:  a variety of a species of thistle cultivated as a food).  I’ve seen plantations of artichoke in Turkey, they look hilarious.
  6. The Gulf Stream example suggests that switching to the new stable period could be short, but interesting.
  7. My last comment is on artichokes and jet streams. The most poignant artichokes I have seen in Turkey were on what was, two millennia ago, the river bed of Meander, the great river of Asia Minor, now almost dry. The history of glorious civilisations of Mediterranean and Middle East was the history of ecological disasters created by people. It is hard to believe now that North Africa and Turkey were covered by  subtropic forests. Well, England not long ago was covered by mighty oaks, and Scotland — by pines, and the famous heather moors is a secondary landscape, an ecological system which replaced forests destroyed by people, and which is much more fragile than forest.
  8. Meander became a verb  (with the meaning `follow a winding course’)  because the river was bending, creating loops, etc. Last time I’ve seen this word — to meander — was  in relation to jet streams over North Atlantic which started to meander. Water flows against the gradient of altitude of the surface and starts to meander, that is, the flow is becoming unstable, if the gradient is too small. It is claimed that the same happens with jet steams: they flow against the gradient of temperature, and are destabilised by warming up of Arctic which is decreasing the gradient.  There is a possibility that the climate of British Isles can change even without dramatic events in Greenland.

The reasons or triggers for climate change are becoming a scholastic issue now.  What matters is

A. Ecological systems around the world are weakened and under stress, and their ability  to cope with changes in climate is compromised.

B. The ability of humans, as species, to cope with the change  is also compromised. On one hand, we reached a fantastic level of technological development  (which,  of course, helps) — but on the other, this technology is not equally shared, and, on the top of that:

C. This is the greatest unspeakable issue — the Earth is overpopulated by humans beyond sustainable levels.

D. In point C, the words “beyond sustainable levels” should of course be understood as “at levels unsustainable under the current socio-economic systems prevalent in the world and interconnected within the single global economy”.  However, any  change of the socio-economic systems on that scale is a task perhaps even more challenging than dealing with the climate change itself. It is likely that the number of people killed in the process will be comparable with loss of life in all natural disasters and pandemics triggered by the climate change.

I do not claim that I am right or correct in every detail. But my message is:

The problems of the Earth and of the human civilisation on the Earth reached the level when they can no longer be entrusted to politicians and journalists — and neither can be left to the social media. The readers of my blog are a few but they tend to be educated people with proven skills of analytic thinking — if you read this, please allocate some of your time to analysis of all that mess.

The 21st century starts now, in 2021 — the same way as the 20th century started in 1914 with World War I.


How do illogical proofs or answers “feel” to mathematicians?


My high school math teacher often made rudimentary mistakes in her equations. What does that say about her?

My answer to a question on Quora: My high school math teacher often made rudimentary mistakes in her equations. What does that say about her?

This could be turned into a good learning experience for you: watch your teacher and try instantly identify her mistakes and correct them – for yourself. I had this experience in my school days, and remember it fondly. It really helps to start mastering mathematics.

It is up to you to decide whether to try to correct your teacher in front of the class, but remember, it could be very cruel to her, you may regret that later. It would be much more useful to quietly provide your classmates with correct solutions.

And maths teachers in schools are frequently overworked. And what do you know about her home life? She could simply suffer from long term sleep deprivation. Especially in the present crazy times…


A look from lockdown at horrors of school mathematics

Kit Yates in The Observer: Home schooling: ‘I’m a maths lecturer – and I had to get my children to teach me’  A few quotes:

A senior lecturer in the department of mathematical sciences at the University of Bath, Yates has a PhD in Maths from Oxford and is the author of The Maths of Life and Death. So when he began home schooling his son Will, five, and daughter Emmie, seven, during lockdown, he was pretty confident he already knew everything they would be expected to learn in maths.

He was wrong. “I’d never heard of a ‘bar model’ or a ‘part-whole model’. I had to get my kids to teach me.” He was shocked by how many of these different, “intimidating” methods and models primary school children are expected to use to solve basic maths problems. “I’ve never needed to use them – you don’t need to know all these different mental models to do maths,” he says. […]

But what he really finds frustrating is the lying. The curriculum is forcing teachers to deliberately teach children lies, he says, which then have to be unpicked later. For example, after years of being taught there are no numbers between zero and one, his seven-year-old is suddenly expected to understand that there are such things as fractions.


Why is the constructivist theory applicable in teaching mathematics?

My answer to a question on Quora: Why is the constructivist theory applicable in teaching mathematics?

I suggest to modify the question a bit:

Why do some people find the constructivist theory applicable in teaching mathematics?

This wold allow me to express my surprise: indeed, why?

I have lived in the world of professional research mathematics for almost 50 years now, and I wonder why constructivist theory in mathematics education so blatantly ignores the experience accumulated in the mathematics research community. I feel that the  constructivist theory talks about some different kind of mathematics, not the one known to me and my many friends and colleagues from all around the world. But I am Vygotskian by my philosophy upbringing, and I can see how Vygotsky’s sociocultural approach explains the invention of this mock image of mathematics. I will look for an opportunity to explain that- I hope Quora sooner or later will give me chance to do that.


I can’t remember anything but sparse memories here and there from the ages of 7 and below. Can anyone help?

A very important question. As it was already explained in this thread, this is a well-known and quite usual phenomenon (called childhood amnesia), caused by re-wiring of the brain at the critically important stage of development. The timing is slightly different in different people, and, I feel, in respect to different kinds of brain activity — for example, you cannot forget how to swim or ride a bicycle, if you learnt these skills at the age covered by amnesia. Also, it appears that children do not unlearn how to read or do arithmetic — but they can eventually forget how did they learn to read.

In general, I think non-one should worry about their childhood amnesia — these were natural changes in one’s brain, and they were to one’s benefit.

I collected hundreds of testimonies from people about their very first memories of learning mathematics — and discovered, that it seems that majority of people simply do not remember anything at all about their earliest encounters with school mathematics — including, it appears, many teachers of mathematics. Unfortunately, I had a day job to do and had no time to run a proper statistical analysis. But I think, this is something that should be taken into account in professional education of future school teachers of mathematics.