There is another world

I’ve been a contributor to Plan for decades now, though my contributions are modest. They send me a magazine from time to time, which I usually ignore, but this time they had a cover story titled: Bringing an end to child marriage.

When I look at all the squabbles we have in domestic politics, not just here, in Australia, but in other Western countries, this issue helps to put things in perspective. In the past week, the Australian government, despite having arguably the most resilient economy in the Western world, did it’s best to self-destruct by publicly brawling over a leadership challenge that had obviously been festering for years. In America, politicians argue over the fundamentals of health care as if it distinguishes a free economy from a State-run monopoly, even though much of the rest of the so-called Free World moved on from that debate decades ago.

There is another world that most of us don’t see or hear about or care about, but it comprises the bulk of the Earth’s population. In this world, our political debates seem downright petty, considering that most of us have a fridge with food in it, running water, electricity and heating, as well as a roof over our head.

The education of women is something we take for granted in the West, yet, in many cultures, young girls are still treated as bargaining chips in a household economy. If we weren’t so egocentric and culturally insulated from the rest of the world we might see how important this issue is and that we are in a position to help.

 I strongly believe that women are the key to the world’s future. I would like to see more aid given to women in developing countries directly because I think they are more likely to use it for their children’s benefit, whether it be in schooling or nutrition. The all-pervading patriarchal society is past its use-by date, not just in the West, but globally. Until it is universally recognised that women deserve exactly the same rights as men, then the disparity in wealth, prosperity and health will continue between the West and the rest.

This report depicts the clash between Western feminist values and traditional culture, where being born a woman is perceived as a liability by both sexes. This attitude is pervasive in much of the world – the Western perspective is not only recent but the exception.

Economics of the future

In March 2010 I wrote a post titled, The world badly needs a radical idea.  Well, last Thursday I heard an interview with Guy Standing, Professor of Economic Security at the University of Bath, UK, who does have at least one radical idea as well as a perspective that coincides with mine.

In particular, he challenges the pervasive definition we give to ‘work’. Essentially, that ‘work’ must contribute to the economy. In other words, in the West, we have a distorted view that work only counts if we earn money from it. He gives the example: if a man hires a housekeeper, whom he pays, she is part of the economy, but if he marries her she effectively disappears, economically. I’ve long argued that the most important job you will ever do, you will never be paid for, which is raising children.

To give another very personal example, I make no money from writing fiction, therefore any time I spend writing fiction is a self-indulgence. On the other hand, if I did make money from writing fiction, then any time I didn’t spend writing fiction would be considered a waste of time. By the way, I don’t consider writing as work, because, if I did, I probably wouldn’t do it or I wouldn’t be motivated to do it. Writing fiction is the hardest thing I’ve ever done and treating it as work would only make it harder.

Standing’s radical idea is that there should be a ‘minimum income’ as opposed to a minimum wage. Apparently, this has been introduced in some parts of Brasil and there is a programme to introduce it in India. In Brasil it was championed by a woman mayor who supported the programme if it was given to women. Standing claims that the most significant and measurable outcome is in the nutrition of babies and young children.

Now, many people will say that this is communism, but it’s not about overthrowing capitalism, it’s about redistribution of wealth, which has to be addressed if we are ever going to get through the 21^st Century without more devastating wars than we witnessed in the last century.

The core of the interview is about a new class, which he calls the ‘precariat’, who are the new disenfranchised in the modern world, partly a result of the concentration of wealth, created by those who still believe in the ‘trickledown fantasy’.

Is mathematics invented or discovered?

I've used this title before in Sep. 2007, even though it was really a discussion of George Lakoff's and Rafael E. Nunez's book, Where mathematics comes from. In fact, it was just my 6th post on this blog. This essay predates that post by 5 years (2002) and I found it by accident after someone returned a USB to me that I had lost. Though there is some repetition, this essay is written in the context of an overall epistemology, whilst the previous one is an argument against a specifically defined philosophical position. To avoid confusion, I will rename the Sep.07 post after the title of Lakoff's and Nunez's book.


I would argue that it is a mixture of both, in the same way that our scientific investigations are a combination of inventiveness and discovery. The difference is, that in science, the roles of creativity and discovery are more clearly delineated. We create theories, hypotheses and paradigms, and we perform experiments to observe results, and we also, sometimes, simply perform observations without a hypothesis and make discoveries, though this wouldn’t necessarily be considered scientific.

But there is a link between science and mathematics, because as our knowledge and investigations go deeper into uncovering nature’s secrets, we become more dependent on mathematics. In fact I would contend that the limit of our knowledge in science is determined by the limits of our mathematical abilities. It is only our ability to uncover complex and esoteric mathematical laws that has allowed us to uncover the most esoteric (some would say spooky) aspects of the natural universe. To the physicist there appears to be a link between mathematical laws and natural laws. Roger Penrose made the comment in a BBC programme, Lords of Time, to paraphrase him, that mathematics exists in nature. It is a sentiment that I would concur with. But to many philosophers, this link is an illusion of our own making.

Stanilas Debaene, in his book, The Number Sense, describes the cognitive aspect of our numeracy skills which can be found in pre-language infants as well as many animals. He argues a case that numbers, the basic building blocks of all mathematics, are created in our minds and that there is no such thing as natural numbers. The logical consequence of this argument is that if numbers are a product of the mind then so must be the whole edifice of mathematics. This is in agreement with both Russell and Wittgenstein, who are the most dominant figures in 20th Century philosophy. I have no problem with the notion that numbers exist only as a concept in the human mind, and that they even exist within the minds of some animals up to about 5 (if one reads Debaene’s book) though of course the animals aren’t aware that they have concepts – it’s just that they can count to a rudimentary level.

But mathematics, as we practice it, is not so much about numbers as the relationships that exist between numbers, which follow very precise rules and laws. In fact, the great beauty of algebra is that it strips mathematics of its numbers so that we can merely see the relationships. I have always maintained that mathematical rules are, by and large, not man made, and in fact are universal. From this perspective, Mathematics is a universal language, and it is the ideal tool for uncovering nature’s secrets because nature also obeys mathematical rules and laws. The modern philosopher argues that mathematics is merely logic, created by the human mind, albeit a very complex logic, from which we create models to approximate nature. This is a very persuasive argument, but do we bend mathematics to approximate nature, or is mathematics an inherent aspect of nature that allows an intelligence like ours to comprehend it?

I would argue that relationships like π and Pythagoras’s triangle, and the differential and integral calculus are discovered, not invented. We simply invent the symbols and the means to present them in a comprehensible form for our minds. If you have a problem and you cannot find the solution, does that mean the solution does not exist? Does the solution only exist when someone has unravelled it, like Fermat’s theorem? This is a bit like Schrodinger’s cat; it’s only dead or alive when someone has made an observation. So mathematical theorems and laws only exist when a cognitive mind somewhere reveals them. But do they also exist in nature like Bernoulli’s spiral found in the structure of a shell or a spider web, or Einstein’s equations describing the curvature of space? The modern philosopher would say Einstein’s equations are only an approximation, and he or she may be right, because nature has this habit of changing its laws depending on what scale we observe it at (see Addendum below), which leads paradoxically to the apparent incompatibility of Einstein’s equations with quantum mechanics. This is not unlike the mathematical conundrum of a circle, ellipse, parabola and hyperbola describing different aspects of a curve.

So what we have is this connection between the human mind and the natural world bridged by mathematics. Is mathematics an invention of the mind, a phenomenon of the natural world, or a confluence of both? I would argue that it is the last. Mathematics allows us to render nature’s laws in a coherent and accurate structure – it has the same infinite flexibility while maintaining a rigid consistency. This reads like a contradiction until you take into account two things. One is that nature is comprised of worlds within worlds, each one self-consistent but producing different entities at different levels. The best example is the biological cells that comprise the human body compared to the molecules that makes up the cells, and then in comparison with an individual human, the innumerable social entities that a number of humans can create. Secondly, that this level of complexity appears to be never ending so that our discoveries have infinite potential. This is despite the fact that in every age of technological discovery and invention, we have always believed that we almost know everything that there is to know. The current age is no different in this respect.

The philosophical viewpoint that I prescribe to does not require a belief in the Platonic realm. From my point of view, I consider it to be more Pythagorean than Platonic, because my understanding is that Pythagoras saw mathematics in nature in much the same way that Penrose expresses it. I assume this view, even though we have little direct knowledge of Pythagoras’s teachings. Plato, on the other hand, prescribed an idealised world of forms. He believed that because we’ve had previous incarnations (an idea he picked up from Pythagoras, who was a religious teacher first, mathematician second), we come into this world with preconceived ideas, which are his ‘forms’. These ‘forms’ are an ideal perfect semblance from ‘heaven’, as opposed to the less perfect real objects in nature. This has led to the idea that anyone who prescribes to the notion that mathematical laws and relationships are discovered, must therefore believe in a Platonic realm where they already exist.

This aligns with the idea of God as mathematician. Herbet Westron Turnbull in his short tome, Great Mathematicians, rather poetically states it thus: ‘Mathematics transfigures the fortuitous concourse of atoms into the tracery of the finger of God.’ But mathematics does not have to be a religious connection for its laws to pre-exist. To me, they simply lie dormant awaiting an intelligence like ours to uncover them. The natural world already obeys them in ways that we are finding out, and no doubt, in ways that we are yet to comprehend.

Part of the whole philosophical mystery of our being and the whole extraordinary journey to our arrival on this planet at this time, is contained in this one idea. The universe, whether by accident or anthropic predestination, contains the ability to comprehend itself, and without mathematics that comprehension would be severely limited. Indeed, to return to my earliest point, which converges on Kant and Eco’s treatise in particular, Kant and the Platypus, our ability to comprehend the universe with any degree of certainty, is entirely dependent on our ability to uncover the secrets and details of mathematics. And consequently the limits of our knowledge of the natural world is largely dependent on the limits of our mathematical knowledge.


Addendum 1: This post has become popular, so I'm tempted to augment it, plus I've written a number of posts on the topic since. When studying physics, one is struck by the significance of scale in the emergence of nature's laws. In other words, scale determines what forces dominate and to what extent. This demonstrable fact, all by itself, signifies how mathematics is intrinsically bound into reality. Without a knowledge of mathematics (often at its most complex) we wouldn't know this, and without mathematics being bound into the Universe at a fundamental level, the significance of scale would not be a factor.

Addendum 2: Given the context of Addendum 1, this is a much later post that might be relevant: The Universe's natural units.