## Posts Tagged ‘Mathematics’

### Why did we start to count?

October 12, 2020

Counting and the invention of numbers and the abstractions enabling mathematics are surely cognitive abilities. Counting itself involves an abstract ability. The simple act of raising two fingers to denote the number of stones or lions or stars implies first, the abstract ability to describe an observed quality and second, the desire to communicate that observation.

What led humans to counting and when?

Before an intelligence can turn to counting it must first have some concept of numbers. When and how did our ancient ancestors  first develop a concept of numbers and then start counting? ……..

It seems clear that many animals do distinguish – in a primitive and elementary way – between “more” and “less, and “few” and “many”,and “bigger” and “smaller”, and even manage to distinguish between simple number counts. They show a sophisticated use of hierarchy and precedence.

Some primates show some primitive abilities when tested by humans

…..  Rhesus monkeys appear to understand that 1 + 1 = 2. They also seem to understand that 2 + 1 = 3, 2 – 1 = 1, and 3 – 1 = 2—but fail, however, to understand that 2 + 2 = 4. ……

But even chimpanzees and monkeys rarely, if ever, use counts or counting in interactions among themselves. The abilities for language and counting are not necessarily connected genetically (though it is probable), but they are both certainly abilities which appear gradually as cognition increases. Mathematics is, of course, just another language for describing the world around us. Number systems, as all invented languages, need that a system and its rules be shared before any communication is feasible. It is very likely that the expressions of the abilities to count and to have language follow much the same timeline. The invention of specific sounds or gestures to signify words surely coincided with the invention of gestures or sounds to signify numbers. The step change in the size of brains along the evolutionary path of humans is very likely closely connected with the expressions of the language and the counting abilities.

The ability to have language surely preceded the invention of languages just as the ability to count preceded the expressions of counting and numbering. It is not implausible that the first member of a homo erectus descendant who used his fingers to indicate one of something, or four of something else, to one of his peers, made a far, far greater discovery – relatively – than Newton or Einstein ever did.

We must have started counting and using counts (using gestures) long before we invented words to represent counts. Of course, it is the desire to communicate which is the driving force which takes us from having abilities to expressions of those abilities. The “cooperation gene” goes back to before the development of bipedalism and before the split with chimpanzees or even gorillas (at least 9 million years ago).

The simple answer to the question “Why did we start to count?” is because we could conceive of a count, observed it and wished to communicate it. But this presupposes the ability to count. Just as with language, the ability and the expression of the ability, are a consequence of the rapid increase in brain size which happened between 3 m and 1 m years ago.

I am persuaded that that rapid change was due to the control of fire and the change to eating cooked food and especially cooked meat. The digestion of many nutrients becomes possible only with cooked food and is the most plausible driver for the rapid increase in brain size.

Raw Food not enough to feed big brains

………. our brains would still be the size of an ape’s if H. erectus hadn’t played with fire: “Gorillas are stuck with this limitation of how much they can eat in a day; orangutans are stuck there; H. erectus would be stuck there if they had not invented cooking,” she says. “The more I think about it, the more I bow to my kitchen. It’s the reason we are here.”

### Numeracy and language

December 2, 2013

I tend towards considering mathematics a language rather than a science. In fact mathematics is more like a family of languages each with a rigorous grammar. I like this quote:

R. L. E. SchwarzenbergerThe Language of Geometry, in A Mathematical Spectrum Miscellany, Applied Probability Trust, 2000, p. 112:

My own attitude, which I share with many of my colleagues, is simply that mathematics is a language. Like English, or Latin, or Chinese, there are certain concepts for which mathematics is particularly well suited: it would be as foolish to attempt to write a love poem in the language of mathematics as to prove the Fundamental Theorem of Algebra using the English language.

Just as conventional languages enable culture and provide a tool for social communication, the various languages of mathematics, I think, enable science and provide a tool for scientific discourse. I take “science” here to be analaogous to a “culture”. To follow that thought then, just as science is embedded within a “larger” culture, so is mathematics embedded within conventional languages. This embedding shows up as the ability of a language to deal with numeracy and numerical concepts.

And that means then the value judgement of what is “primitive” when applied to language can depend upon the extent to which mathematics and therefore numeracy is embedded within that language.

According to a recent article by Mike Vuolo in Slate.com, Pirahã is among “only a few documented cases” of languages that almost completely lack of numbers. Dan Everett, a renowned expert in the Pirahã language, further claims that the lack of numeracy is just one of many linguistic deficiencies of this language, which he relates to gaps in the Pirahã culture. …..

The various types of number systems are considered in the WALS.info article on Numeral Bases, written by Bernard Comrie. Of the 196 languages in the sample, 88% can handle an infinite set of numerals. To do so, languages use some arithmetic base to construct numeral expressions. According to Comrie, “we live in a decimal world”: two thirds of the world’s languages use base 10 and such languages are spoken “in nearly every part of the world”. English, Russian, and Mandarin are three examples of such languages. …..

Around 20% of the world’s languages use either purely vigesimal (or base 20) or a hybrid vigesimal-decimal system. In a purely vigesimal system, the base is consistently 20, yielding the general formula for constructing numerals as x20 + y. For example, in Diola-Fogny, a Niger-Congo language spoken in Senegal, 51 is expressed as bukan ku-gaba di uɲɛn di b-əkɔn ‘two twenties and eleven’. Other languages with a purely vigesimal system include Arawak spoken in Suriname, Chukchi spoken in the Russian Far East, Yimas in Papua New Guinea, and Tamang in Nepal. In a hybrid vigesimal-decimal system, numbers up to 99 use base 20, but the system then shifts to being decimal for the expression of the hundreds, so that one ends up with expressions of the type x100 + y20 + z. A good example of such a system is Basque, where 256 is expressed as berr-eun eta berr-ogei-ta-hama-sei ‘two hundred and two-twenty-and-ten-six’. Other hybrid vigesimal-decimal systems are found in Abkhaz in the Caucasus, Burushaski in northern Pakistan, Fulfulde in West Africa, Jakaltek in Guatemala, and Greenlandic. In a few mostly decimal languages, moreover, a small proportion of the overall numerical system is vigesimal. In French, for example, numerals in the range 80-99 have a vigesimal structure: 97 is thus expressed as quatre-vingt-dix-sept ‘four-twenty-ten-seven’. Only five languages in the WALS sample use a base that is neither 10 nor 20. For instance, Ekari, a Trans-New Guinean language spoken in Indonesian Papua uses base of 60, as did the ancient Near Eastern language Sumerian, which has bequeathed to us our system of counting seconds and minutes. Besides Ekari, non-10-non-20-base languages include Embera Chami in Colombia, Ngiti in Democratic Republic of Congo, Supyire in Mali, and Tommo So in Mali. ……

Going back to the various types of counting, some languages use a restricted system that does not effectively go above around 20, and some languages are even more limited, as is the case in Pirahã. The WALS sample contains 20 such languages, all but one of which are spoken in either Australia, highland New Guinea, or Amazonia. The one such language found outside these areas is !Xóõ, a Khoisan language spoken in Botswana. …….

Counting monkey?

In some societies in the ancient past, numeracy did not contribute significantly to survival as probably with isolated tribes like the Pirahã. But in most human societies, numeracy was of significant benefit especially for cooperation between different bands of humans. I suspect that it was the need for social cooperation which fed the need for communication within a tribe and among tribes, which in turn was the spur to the development of language, perhaps over 100,000 years ago. What instigated the need to count is in the realm of speculation. The need for a calendar would only have developed with the development of agriculture. But the need for counting herds probably came earlier in a semi-nomadic phase. Even earlier than that would have come the need to trade with other hunter gatherer groups and that  probably gave rise to counting 50,000 years ago or even earlier. The tribes who learned to trade and developed the ability and concepts of trading were probably the tribes that had the best prospects of surviving while moving from one territory to another. It could be that the ability to trade was an indicator of how far a group could move.

And so I am inclined to think that numeracy in language became a critical factor which 30,000 to 50,000 years ago determined the groups which survived and prospered. It may well be that it is these tribes which developed numbers, and learned to count, and learned to trade that eventually populated most of the globe. It may be a little far-fetched but not impossible that numeracy in language may have been one of the features distinguishing Anatomically Modern Humans from Neanderthals. Even though the Neanderthals had larger brains and that we are all Neanderthals to some extent!

### The Fibonacci spiral applied

July 22, 2013

Mathematics is wonderful but numbers are transcendental.

From twistedswifter

The Fibonacci Series – 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ….

An approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling; this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34.

Approximating the Golden Spiral: Wikipedia

### Mathematical genius?

June 4, 2013

Retraction Watch reports on the retraction of a paper by a Turkish mathematician for plagiarism. The author did not agree with the retraction.

But what struck me was the track record of this amazing Assistant Professor at Ege University.

Ahmet Yildirim Assistant Professor, Ege University, Turkey

Editorial Board Member of International Journal of Theoretical and Mathematical Physics

• 2009       Ph.D      Applied Mathematics, Ege University (Turkey)
• 2005       M.Sc      Applied Mathematics, Ege University (Turkey)
• 2002       B.Sc        Mathematics, Ege University (Turkey)

Since 2007 he has a list of 279 publications!

That’s an impressive rate of about 50 publications per year. Prolific would be an understatement.

All peer reviewed no doubt.

### Algebraic Art

June 1, 2013

Found this while browsing:

### Math and Reading gender differences are not affected by level of gender equality

March 18, 2013

A new paper suggests that  math and reading differences between the genders persist  regardless of a country’s “gender equality” level. Though I am not sure why there should be so much surprise about such a finding. That the genders are fundamentally different – I would have thought – was self-evident. That some of the biological differences between the sexes must lie in the brain also seems obvious. It has always confused me as to why legislation – which should be for ensuring the equality of opportunity between the genders – often tries to suppress or deny gender differences in futile attempts to try and make the genders “equal”. We will – I think – only achieve a real equality of opportunity when we truly understand and acknowledge all the inherent differences between the sexes. Legislation can surely help to address behaviour but it cannot do away with the inherent differences. As Prof. Geary states “Educational systems could be improved by acknowledging that, in general, boys and girls are different.” And I would add that equality of opportunity between the genders has to start by acknowledging that men and women are different. It could well be that “Swedish boys fall behind in reading more so than in most other highly developed nations” just because Sweden spends so much effort to create gender equality by suppressing gender difference.

Stoet G, Geary DC (2013) Sex Differences in Mathematics and Reading Achievement Are Inversely Related: Within- and Across-Nation Assessment of 10 Years of PISA Data. PLoS ONE 8(3): e57988. doi:10.1371/journal.pone.0057988

The University of Missouri has a press release:

… even in countries with high gender equality, sex differences in math and reading scores persisted in the 75 nations examined by a University of Missouri and University of Leeds study. Girls consistently scored higher in reading, while boys got higher scores in math, but these gaps are linked and vary with overall social and economic conditions of the nation.

“Educational systems could be improved by acknowledging that, in general, boys and girls are different,” said David Geary, MU professor of psychological science. “For example, in trying to close the sex gap in math scores, the reading gap was left behind. Now, our study has found that the difference between girls’ and boys’ reading scores was three times larger than the sex difference in math scores. Girls’ higher scores in reading could lead to advantages in admissions to certain university programs, such as marketing, journalism or literature, and subsequently careers in those fields. Boys lower reading scores could correlate to problems in any career, since reading is essential in most jobs.”

Generally, when conditions are good, the math gap increases and the reading gap decreases and when conditions are bad the math gap decreases and the reading gap increases. This pattern remained consistent within nations as well as among them, according to the study by Geary and Gijsbert Stoet of the University of Leeds that included testing performance data from 1.5 million 15-year-olds in 75 nations. The top five percent of scores within nations generally showed girls to be lower in math and boys to be lower in reading. That pattern continued in lower scoring groups until reaching the lowest scoring students, where the math achievement of boys and girls evened out but the reading gap increased, according to Geary.

“The consistent pattern within nations suggests the sex differences are not simply related to socio-economic factors,” said Geary. Socio-economic and cultural factors are important in that they influence the performance of all students, but boys, as a group, respond more strongly than girls, perhaps due to a biological difference in sensitivity to wider conditions.”  For example, in nations with impoverished or violent conditions, boys’ scores tended to fall faster and further than girls. On the other hand, in wealthier, socially stable nations boys’ scores benefitted more than girls. This resulted in boys reducing the reading gap and widening the math gap.

“This finding has important implications for how we interpret the math gap of other countries,” said co-author Gijsbert Stoet of the University of Leeds. “For example, policy makers often take Sweden as an example of being particularly good for reducing the gender gap in science, technology, engineering and math, but they do not realize that Swedish boys fall behind in reading more so than in most other highly developed nations. This is a good example of the inverse relation between the math and reading gaps. This phenomenon urgently needs more attention.” ……

### Maths paper “which makes no sense mathematically” first published and then retracted

December 6, 2012

Acharya Sennimalai Kalimuthu strikes again! And Elsevier as publishers do look like idiots.

Back in April I posted about a paper by Kalimuthu which was first published in Computers & Mathematics with Applications and then retracted because it “lacked scientific content”.

This time he managed to get a paper published in Applied Mathematics Letters

For the origin of new geometry, by S. Kalimuthu, 2/394, Kanjampatti P.O., Pollachi Via, Tamil Nadu 642003, India. http://dx.doi.org/10.1016/j.aml.2010.08.006,

He has 12 references – all self-citations. The paper has now been retracted because it “makes no sense mathematically”. The title itself should have been a give-away but the paper was published in December 2010 and it has taken 2 years to be retracted.

This paper does not meet the minimum research and mathematical standards of Applied Mathematics Letters; for example, some of this paper’s constructions and arguments make no sense mathematically. Though handled by the previous editorial office, the available records lead us to believe its publication was the result of an administrative oversight and apologies are offered to readers of the journal that this was not detected earlier.

In both cases Elsevier was the unfortunate publisher. This does not say much for the “peer-review” process at Elsevier which allowed such rubbish to be published. First I wondered if Kalimuthu might be an unrecognised genius until I read his two papers. You do not need to be an advanced mathematician to appreciate the absurdities. His two papers are

Sivasubramanian and Kalimuthu

kalimuthu 2

After the 5th reading of his second paper I managed to figure out the central claim:

Our constructions and proofs are consistent. We have not introduced any new hypothesis in this work. . ….. But we have pointed out in the abstract that the fifth Euclidean postulate problem is one of the most famous mathematical impossibilities. So, although our finding is consistent, it poses a very serious question about the foundations of geometry.

… we have obtained a challenging result, namely the smaller side of triangle AHJ is equal to the larger side BC of triangle ABC. This is a problematic problem.

Further studies will certainly unlock this mathematical mystery.

No doubt the further studies will be first published and then retracted by Elsevier.

Retraction Watch covers the story and actually took the time to write to Kalimuthu for his comments on the retraction. His reply will surely go down as a classic:

“Please and please note that I do NOT agree with retraction of this relevant paper.Can you tell me WHAT IS THE FLAW? AND WHERE IS THE FLAW? A result is a result, A result is a result, A result is a result, and A result is a result,.Let us recall what Einstein told about simplicity: IF YOU CAN NOT PUT YOUR IDEA IN SIMPLE, IT SHOWS THAT YOU DO NOT KNOW THE SUBJECT. Who is expert? We are all so called experts. Only God is expert. I am going to re write this particular paper in 20 long pages and get published. Kindly note that papers rejected by referees and editors have won the NOBEL PRIZE.”

But what on earth was Elsevier playing at to publish such drivel.

Once was bad enough but twice???

Either Kalimuthu has some kind of genius in being able to get papers without scientific content and which make no mathematical sense published or the Elsevier peer-review process is a farce.

### Mathematics mayhem – paper proving the impossible to be possible retracted for lack of “scientific content”

April 17, 2012

This is hilarious but it does make the Dr.Mahalingam College of Engineering & Technology, Pollachi, Tamil Nadu look ridiculous.  This paper was accepted for publication by Elsevier and has now been retracted by the publishers for not containing any scientific content. It seems that the authors have applied a computer program to a “problematic problem” and have proved a 4 300 year old “impossible proposition to be possible” !

“Computer application in mathematics” [Comput. Math. Appl. 59 (1) (2009) 296–297], by M. Sivasubramanian and S. Kalimuthu, Department of Mathematics, Dr. Mahalingam College of Engineering and Technology, Pollachi, Tamilnadu-642003, India

The paper is here : Sivasubramanian and Kalimuthu

But while the paper itself is remarkably short and is just a nonsense paper, it does not say very much for Elsevier’s editorial acumen or for its peer-review process. Perhaps this journal should be retracted for lack of editorial content? Timothy Gowers will surely get more support for his Elsevier boycott in the UK.

Retraction Watch has the full story: