About Mums, and a little about Dads too.

Mother hen with chicks02
Mother hen with chicks02 (Photo credit: Wikipedia)

I wrote about cuteness a couple of posts ago, and this started me thinking about mothers, both human and animals and how the bonds that they form with their offspring.

Many animals do not look after their offspring, just casting their fertilised eggs into the seas like many fish or placing their eggs on a food plant as butterflies and moths mostly do. However, many animals do look after their eggs and young offspring, often for extended periods of time.

Danaus Plexippus, Monarch Butterly, picture ta...
Danaus Plexippus, Monarch Butterly, picture taken in NewYork, October 2008 (Photo credit: Wikipedia)

It is common for the mother of an animal to look after it rather than the father, but it is not uncommon for the father to look after the offspring, and more commonly both parents will look after their progeny.

For example, the egg laid by most species of Kiwi is incubated by the male member of a pair of birds. Also, in the Seahorse, the female deposits her eggs inside the male’s “brood pouch” and the young of the Seahorse develop there.


Once young animals are born, often the female parent will take care of them for some time after they are born, but this is not a definite rule. Sometimes the male parent is around and provides some support and protection, and even if the male parent is around, he may remain fairly distant, with the female doing most of the caring for the young animals.

A common sight is a mother hen closely followed by her chicks, with the aloof cockerel strutting around the farmyard. In a pride of lions, the nucleus of the pride consists of the females and offspring while the associated males remain close.

English: Four Lionesses take down a bull cape ...
English: Four Lionesses take down a bull cape buffalo in the central Serengeti (Photo credit: Wikipedia)

In humans, the so called “nuclear family” is common, at least in Western cultures. A nuclear family usually consists of a couple and their children living in a single house, and is a relatively recent phenomenon, with extended families being common in many cultures, including Western cultures, until fairly recently.

In such a nuclear family, the father goes out to earn money for the family every day, leaving the children in the care of the mother for the day. Such role separation and assignment could be seen as “natural” and “obvious”. This can be problematic when the couple are not male and female, when role assignment is trickier.

Guarani nuclear family of Mato Grosso do Sul, ...
Guarani nuclear family of Mato Grosso do Sul, Brasil (Photo credit: Wikipedia)

It is likely that there is an instinctive drive for a mother to care for her children and for the father to be assigned the role of provider for the family. Certainly this tendency for children to be cared for by the mother and for the father to fill another role can be seen in most societies, even those without the concept of the nuclear family.

In a family consisting of a couple of same sex parents, this role division is not well defined and indeed such couples may decide to share both provider and carer roles within the family group, which could speculatively lead to kids who are unclear about the distinction between the carer and provider roles.

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Kids are resilient though, and being more willing to share the roles when they grow up and form their own, probably heterosexual, relationships and families may even be an advantage. That’s not to say that the father in a heterosexual couple whose parents are also a heterosexual couple are not capable of caring! The roles in Western societies are not so strictly defined that a father cannot be a carer for some of the time, and that a mother cannot be a provider.

Regardless of such quibbles, mothers tend to be more caring and nurturing than fathers in Western societies. Both boys and girls tend to go first to Mum when a knee is scraped or an elbow bashed, and they go to Dad for the resolution of disputes, such as when a sibling has stolen a favourite toy and won’t return it.


This is probably because the mother has more invested in the children than the father. She has carried the child for nine months, culminating in a painful delivery, while the father has watched on and the only pain that he has suffered was when his spouse squeezed his hand too hard during a contraction and left nail marks in it. Of course, I am drastically under valuing the support that the mother has received from her spouse.

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Mother have a close bond with their children, and we can see it in modern society, where a couple is not always “till death do us part”. When a couple splits the children more often seem to go with the mother, although there are blended families where some of the kids are the father’s and some are the mother’s.

Mothers can be particularly close to their daughters, but they are even close to their sons. No other person has changed your nappy (diaper), clothed you, nursed you through minor ailments, and fed you from the moment of birth until you leave home. Step daughters and sons can sometimes have difficulty getting as close to step mothers, and this can cause issues.

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Poor old Dad. He gets the affection, the love, but usually not to the same depth as the children love their mother. Actually, I think that the bonds that form between a man and his kids are just as deep as mother love, but they are manifested in different ways. Dad is the one that the kids look to for protection much of the time, Dads tend to be the ones who encourage the kids to stand on their own feet.

The difference is that Dads are in general more able to form relationships at a distance. He may mainly see his kids in the evenings and at weekend. Modern life has pretty much forced a hands off approach to parenting for Dads. When a split up comes it is frequently easier for a father to move away from his children, painful though it may be than for a mother to move away from them.

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Breaking up a family is always difficult, but with nuclear families it is more difficult. In an extended family there are always granddads. grandmas, cousins and aunties and uncles to take up the slack. The modern child doesn’t have quite so much support. It’s a wonder that, in general, they still turn out OK.

English: This is the photograph of an extended...
English: This is the photograph of an extended family belonging to the Pais-Prabhu, a Mangalorean Catholic clan hailing from Falnir in Mangalore. (Photo credit: Wikipedia)

The Banach Tarski Theorem

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There’s a mathematical theorem (the Banach Tarski theorem) which states that

Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.

This is, to say the least, counter intuitive! It suggests that you can dissect a beach ball, put the parts back together and get two beach balls for the price of one.

This brings up the question of what mathematics really is, and how it is related to what we loosely call reality? Scientists use mathematics to describe the world, and indeed some aspects of reality, such as relativity or quantum mechanics, can only be accurately described in mathematics.

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So we know that there is a relationship of some sort between mathematics and reality as our maths is the best tool that we have found to talk about scientific things in an accurate way. Just how close this relationship is has been discussed by philosophers and scientists for millennia. The Greek philosophers, Aristotle, Plato, Socrates and others, reputedly thought that “all phenomena in the universe can be reduced to whole numbers and their ratios“.

The Banach Tarski theorem seems to go against all sense. It seems to be an example of getting something for nothing, and appears to contravene the restrictions of the first law of thermodynamics. The volume (and hence the amount of matter) appears to have doubled, and hence the amount of energy contain as matter in the balls appears to have doubled. It does not appear that the matter in the resulting balls is more attenuated than that in the original ball.

The Banach–Tarski paradox: A ball can be decom...
The Banach–Tarski paradox: A ball can be decomposed and reassembled into two balls the same size as the original. (Photo credit: Wikipedia)

Since the result appears to be counter intuitive, the question is raised as to whether or not it is merely a mathematical curiosity or whether it has any basis in reality, It asks something fundamental about the relationship between maths and reality.

It’s not the first time that such questions have been asked. When the existence of the irrational numbers was demonstrated, Greek mathematicians were horrified, and the discoverer of the proof (Hippasus) was either killed or exiled, depending on the source quoted. This was because the early mathematicians believed that everything could be reduced to integers and rational numbers, and their world did not have room for irrational numbers in it. In their minds numbers directly related to reality and reality was rational mathematically and in actuality.

English: Dedekind cut defining √2. Created usi...
English: Dedekind cut defining √2. Created using Inkscape. (Photo credit: Wikipedia)

These days we are used to irrational numbers and we see where they fit into the scheme of things. We know that there are many more irrational numbers than rational numbers and that the ‘real’ numbers (the rational and irrational numbers together) can be described by points on a line.

Interestingly we don’t, when do an experiment, use real numbers, because to specify a real number we would have write down an infinite sequence of digits. Instead we approximate the values we read from our meters and gauges with an appropriate rational number. We measure 1.2A for example, where the value 1.2 which equals 12/10 stands in for the real number that corresponds to the actual current flowing.

English: A vintage ampere meter. Français : Un...
English: A vintage ampere meter. Français : Un Ampèremètre à l’ancienne. (Photo credit: Wikipedia)

We then plug this value into our equations, and out pops an answer. Or we plot the values on a graph read off the approximate answer. The equations may have constants which we can only express as rational numbers (that is, we approximate them) so our experimental physics can only ever be approximate.

It’s a wonder that we can get useful results at all, what with the approximation of experimental results, the approximated constants in our equations and the approximated results we get. If we plot our results the graph line will have a certain thickness, of a pencil line or a set of pixels. The best we can do is estimate error bounds on our experimental results, and the constants in our equations, and hence the error bounds in our results. We will probably statistically estimate the confidence that the results show what we believe they show through this miasma of approximations.

Image of simulated dead pixels. Made with Macr...
Image of simulated dead pixels. Made with Macromedia Fireworks. (Photo credit: Wikipedia)

It’s surprising in some ways what we know about the world. We may measure the diameter of a circle somewhat inaccurately, we multiply it by an approximation to the irrational number pi, and we know that the answer we get will be close to the measured circumference of the circle.

It seems that our world resembles the theoretical world only approximately. The theoretical world has perfect circles, with well-defined diameters and circumference, exactly related by an irrational number. The real world has shapes that are more or less circular, with more or less accurately measured diameters and circumferences, related more or less accurately by an rational number approximating the irrational number, pi.

Pi Animation Example
Pi Animation Example (Photo credit: Wikipedia)

We seem to be very much like the residents of Plato’s Cave and we can only see a shadow of reality, and indeed we can only measure the shadows on the walls of the cave. In spite of this, we apparently can reason pretty well what the real world is like.

Our mathematical ruminations seem to be reflected in reality, even if at the time they seem bizarre. The number pi has been known for so long that it no longer seems strange to us. Real numbers have also been known for millennia and don’t appear to us to be strange, though people don’t seem to realise that when they measure a real number they can only state it as a rational number, like 1.234.

English: The School of Athens (detail). Fresco...
English: The School of Athens (detail). Fresco, Stanza della Segnatura, Palazzi Pontifici, Vatican. (Photo credit: Wikipedia)

For the Greeks, the irrational numbers which actually comprise almost all of the real numbers, were bizarre. For us, they don’t seem strange. It may be that in some way, as yet unknown, the Banach Tarski theorem will not seem strange, and may seem obvious.

It may be that we will use it, but approximately, much as we use the real numbers in our calculations and theories, but only approximately. I doubt that we will be duplicating beach balls, or dissecting a pea and reconstituting it the same size as the sun, but I’m pretty sure that we will be using it for something.

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I see maths as descriptive. It describes the ideal world, it describes the shape of it. I don’t think that the world IS mathematics in the Pythagorean sense, but numbers are an aspect of the real world, and as such can’t help but describe the real world exactly, while we can only measure it approximately. But that’s a very circular description.

English: Illustrates the relationship of a cir...
English: Illustrates the relationship of a circle’s diameter to its circumference. (Photo credit: Wikipedia)