This is a letter I wrote to Michael C. Corballis after reading something he wrote in New Scientist. He wrote a longer article on the same subject in American Scientist (May-June 2007). Michael Corballis is a professor in Psychology at Auckland University. At the end of this posting I include his response. The epiphany I mention (below) is possibly the only original idea in this entire blog. Everything else is either borrowed, stolen or adapted from other people, or independently derived, which still doesn't make it original.
Dear Professor Corballis
I read your article on recursive thinking in New Scientist (1 September 2007) and it reminded me of an epiphany I had when I was studying philosophy about 10 years ago. It occurred to me that what separated us from other species, more than anything else, was our ability to form concepts within concepts ad infinitum, which is similar to what you describe as embedded recursion, though not quite the same.
If one takes writing, as I’m doing now, we have individual words that have their own meanings. But we can create sentences of those words that then have a meaning beyond the individual words, and then we can create a string of sentences that eventually may form an overarching argument or a story. And perhaps it was with storytelling that we first exercised this ability. But we do it with everything: music, architecture, engineering and even mathematics. We take individual parts assemble them mentally to form a larger part that has a different function than the individual parts. I think mathematics is the best example because it is so obviously structured this way while we are learning it. Yet, I believe it is through art that we originally developed this unique skill.
But this was not the epiphany. The epiphany was realising that nature also consists of different levels of entities within entities. If you take an individual organic cell, it is like a miniature world that has a function completely different to the collection of cells, that, combined, create an individual organism like a human, which has another function altogether. And it occurred to me that we are uniquely suited to comprehend nature because we have the ability to conceptualise entities within entities in exactly the same way that nature manifests itself. This is why we have become the self-designated interpreters of the universe, or, at least, the only ones we know of.
On the subject of language, I’ve often wondered how we would think without language, and the obvious answer is we would think in images as we do in our dreams. Again, I wondered if our artwork was our first attempt to project this imagery as a form of expression, communication and social bonding. The language of dreams is imagery and metaphor, so I am not surprised that when we read stories we can so readily create our own images in our heads, and this is one of the reasons that cinema and video, in all its manifestations, hasn’t managed to kill off books. The other reason is that a book can take you inside the character’s head in a way that movies can’t. In the case of a movie you depend on the actor to interpret it for you. I know I’m going off the track a bit here, but I’m speaking from the perspective of someone who writes fiction.
In science and engineering we attempt to visualise things when we explain them or interpret them. Engineers will always draw a picture when they try to explain something. Metaphor is an analogy that allows us to communicate something new by employing something already known. My point being that we are essentially visual creatures, and that is our medium of choice when we strive to comprehend the world. I notice that you believe our earliest language was in sign. I understand that we use our hands when we talk because it helps us to create the concept in our head that we are trying to communicate, rather than for the benefit of the listener. Is this the basis of your thesis: that thinking with our hands preceded thinking with language?
Below is Michael Corballis's response.
I agree with you entirely, and I also like your epiphany, which I’ll think more about.
The idea that language arose from manual gestures is based on a number of considerations: (1) apes can be taught something approximating sign language, but can’t be taught to speak; (2) the brain areas involved in speech in humans are involved in manual action in primates; (3) the sign languages of the deaf are fully expressive languages; and (4) we all gesture as we speak. I agree that gesturing may help us form concepts while we speak, but I suspect that our gesturing also reflects an earlier mode of communication.
Tuesday, 18 September 2007
Saturday, 15 September 2007
Free Will
Below is an argument that I formed and submitted to American Scientist in response to an essay by Gregory Graffin and William Provine, who conducted a survey amongst biology students on their beliefs in religion, God and free will. It was their argument on free will that evoked my response. When they say: 'it adds nothing to the science of human behaviour' (quoted below) they are right. As far as science is concerned, if human behaviour can't be explained by a combination of genetics and environment, then invoking 'free will' won't help. It's a bit like invoking God to explain evolution (see my blog posting on Intelligent Design), so I can understand their argument.
When it comes to studying anything to do with consciousness, we can only examine the consequences caused by a conscious being interacting with its environment. It's not unlike the dilemma we face in quantum mechanics where we don't know what's happening until we take a measurement or make an observation. If we didn't experience consciousness as individuals we would probably claim that it didn't exist, because there is no direct evidence of it except through our own thoughts. And this also applies to free will, which, after all, is a manifestation of consciousness. Effectively, Graffin and Provine are saying that free will is an illusion created by the fact that we are conscious beings, but, if one takes their argument to its logical conclusion, all conscious thoughts are caused by an interaction of our genetic disposition with our environment. So what is the evolutionary purpose of consciousness if our thoughts are just an unnecessary by-product?
Below is my original argument that I submitted to American Scientist.
In the July-August 2007 issue of American Scientist (Evolution, Religion and Free Will) Gregory W. Graffin and William B. Provine contend that free will is non-existent because it ‘adds nothing to the science of human behaviour.’ This would follow logically from the premise that any idea, concept or belief that can’t be scientifically examined, measured or hypothetically tested, must be an illusion or a cultural relic. They point out that evolutionary biologists, who believe in free will, suffer from the misconception that choice and free will are synonymous. One always has a choice – it’s just that when it’s made it’s predetermined. I sense a contradiction. So there is no ‘intentionality’, which lies at the heart of consciousness as we experience it, and is discussed by John Searle in his book, MiND (2004). This leads to a conundrum: if all intentionality is predetermined, then why has evolution given us consciousness? It's hard to escape the conclusion that the 'illusion' of free will must therefore have evolutionary value – maybe that’s its contribution to the science of human behaviour.
Living in the 21st Century
This is in response to a one page essay by William Laurance, a biologist at the Smithsonian Tropical Research Institute in Balboa, Panama, published in New Scientist (1 September 2007), entitled: Cursing Condoms. This is a very good article that discusses the most important issue of the 21st Century: human population growth. Laurance attacks both the Catholic Church and the current American Administration for their backward and morally irresponsible attitudes towards birth control, and towards condoms in particular. He remarks, ‘With a different leadership, the US could become part of the solution, not the problem.’ What I find strange about this whole issue, is that I was aware of this ‘problem’ when I was a teenager, over 40 years ago. I find it extraordinary that, not only do people not recognise it as ‘The Problem’ facing us, but that we still have to deal with anachronistic policies and criteria at the highest level of global politics in order to confront it.
I will not repeat Laurance’s arguments here, but I recommend this article to all and sundry. He contends that, being a Panamanian, he sees the consequences of this negative policy-making first hand. He rightly spells out all the problems arising from human encroachment: fewer resources, greater conflict, greater division between the rich and the poor on a national and global scale, and the diminution of other species world wide. He also points out the most obvious and effective solution. Greater educational opportunities to women, world wide, is the only truly effective means of achieving a zero population growth. But there are other factors. Our current economic paradigm is based on infinite economic growth which is geared to infinite population growth, and is the reason that America is becoming the 3rd most populated country in the world. America believes, that to achieve parity economic growth, it must maintain population growth. Obviously, this is not sustainable and eventually we will need a new economic paradigm that has sustainability at its core. Will this do away with economic growth? I don’t know. If we can have economic growth with sustainability of the earth’s resources and zero population growth then the answer is no. If we can’t then the answer is yes: economic growth will stop.
What is obvious is that we can’t continue with the status quo. Six years ago I read an article by E. O. Wilson in Scientific American where he said: for everyone in the world to have the same standard of living as America, we would need 4 planet earths. I heard this statement reiterated more recently, but I can’t remember where.
There have been a number of mass extinctions in the course of the earth’s history, the dinosaur extinction is the most well known but there have been at least 2 others that were equally catastrophic. But what is most worrying is that at no time in the earth’s history has the rate of species extinction been as great as it is now.
In 2000, I was lucky enough to be part of a small audience at Oxford University to hear the scientific advisor to the British government (I’ve forgotten his name, but I think his first name was Ron) give an address on this issue. He showed graph after graph in a Power Point presentation demonstrating how water, energy and land was being eaten up by human consumption, as if there was no tomorrow, literally. Why wasn’t his voice heard beyond that small lecture hall? I’ve no idea. Afterwards, friends of mine made the observation that he had told them nothing new, and were disappointed that he had no solutions. I will admit a small secret: I don’t have any either.
The 21st Century faces a number of problems, of which global warming is only one, and they are all caused by us, so we must find the solution or the earth will find its own. We have the technology for global education, as well as for achieving greater efficiencies in all areas of human activity: energy, food and water; but do we have the will? Whilst economic growth based on human growth, and infinite resources, remains the global paradigm for progress and success, we can be certain of failure. The 21st Century will see more change than any other century preceding it, including the 20th, but it is up to us whether this change will be an improvement or a catastrophe.
For a more detailed analysis on this topic read the following article by E.O.Wilson: http://www.cosmosmagazine.com/node/1298
Commentary by responsible scientists like Wilson are unpalatable to most politicians, and this is a major concern for our collective future.
I will not repeat Laurance’s arguments here, but I recommend this article to all and sundry. He contends that, being a Panamanian, he sees the consequences of this negative policy-making first hand. He rightly spells out all the problems arising from human encroachment: fewer resources, greater conflict, greater division between the rich and the poor on a national and global scale, and the diminution of other species world wide. He also points out the most obvious and effective solution. Greater educational opportunities to women, world wide, is the only truly effective means of achieving a zero population growth. But there are other factors. Our current economic paradigm is based on infinite economic growth which is geared to infinite population growth, and is the reason that America is becoming the 3rd most populated country in the world. America believes, that to achieve parity economic growth, it must maintain population growth. Obviously, this is not sustainable and eventually we will need a new economic paradigm that has sustainability at its core. Will this do away with economic growth? I don’t know. If we can have economic growth with sustainability of the earth’s resources and zero population growth then the answer is no. If we can’t then the answer is yes: economic growth will stop.
What is obvious is that we can’t continue with the status quo. Six years ago I read an article by E. O. Wilson in Scientific American where he said: for everyone in the world to have the same standard of living as America, we would need 4 planet earths. I heard this statement reiterated more recently, but I can’t remember where.
There have been a number of mass extinctions in the course of the earth’s history, the dinosaur extinction is the most well known but there have been at least 2 others that were equally catastrophic. But what is most worrying is that at no time in the earth’s history has the rate of species extinction been as great as it is now.
In 2000, I was lucky enough to be part of a small audience at Oxford University to hear the scientific advisor to the British government (I’ve forgotten his name, but I think his first name was Ron) give an address on this issue. He showed graph after graph in a Power Point presentation demonstrating how water, energy and land was being eaten up by human consumption, as if there was no tomorrow, literally. Why wasn’t his voice heard beyond that small lecture hall? I’ve no idea. Afterwards, friends of mine made the observation that he had told them nothing new, and were disappointed that he had no solutions. I will admit a small secret: I don’t have any either.
The 21st Century faces a number of problems, of which global warming is only one, and they are all caused by us, so we must find the solution or the earth will find its own. We have the technology for global education, as well as for achieving greater efficiencies in all areas of human activity: energy, food and water; but do we have the will? Whilst economic growth based on human growth, and infinite resources, remains the global paradigm for progress and success, we can be certain of failure. The 21st Century will see more change than any other century preceding it, including the 20th, but it is up to us whether this change will be an improvement or a catastrophe.
For a more detailed analysis on this topic read the following article by E.O.Wilson: http://www.cosmosmagazine.com/node/1298
Commentary by responsible scientists like Wilson are unpalatable to most politicians, and this is a major concern for our collective future.
Saturday, 8 September 2007
Religion
This is another letter I wrote to New Scientist - in response to an essay by Helen Phillips entitled, Is God good? She discusses various studies done by academics examining the effects of religion on people's behaviour and ethics. The general consensus seemed to divide people between 'extrinsic' (those who are overtly religious and belong to religious organisations) and 'intrinsic' (those whose religious beliefs are more personal and less overt). It was found that the 'extrinsic' tended to have an 'in group' mentality, though it must be emphasised this is a broad generalisation, that made them less tolerant of people of 'other' religious persuasions, whereas the 'intrinsic' were more tolerant of 'others'.
This is a brief synopsis - she also discussed the evolutionary (social) value that may have been inherent in forming religious beliefs, as well as how we may have come to believe in a God or Gods as external supernatural beings. There was also a consensus that, while morality seems to be inherent in humans, it is not dependent on religiosity per se.
As a side issue, Karen Armstrong in The History of God, puts forward a thesis that our idea of God changes over history. In other words, God seems to exist, at least in our writings, in a historical context. But I would reference Augustine who seemed to appreciate that God was part of our 'inner journey' as much as something external. Or to quote 19th century German philosopher, Ludwig Feuerbach: God does not exist independently of humanity.
Reference: New Scientist, 1 September 2007, pp32-6
When I was a young child, my father, who had spent 2.5 years as a POW, told me something I’ve never forgotten. He said: there are 2 types of Christians. There are Christians who go to church every week and wear their religion on their sleeve. Then there are Christians who don’t claim to be Christians yet they behave like Christians. In my now 56 years of living I’ve never seen anything to contradict that statement.
More relevant to Phillips’ topic, there are 2 types of religion: institutional religion that is political in every sense of the word; and religion as a personal experience, that is part of life’s journey, either as a unique, possibly one-off experience, or as an evolvement of one’s spirituality. I believe this is the distinction between ‘extrinsic’ and ‘intrinsic’ religion that is discussed by Phillips.
In the case of intrinsic or ‘quest’ religion, it is experienced as something ‘beyond’ the self. We can’t even explain how consciousness emerges from the neuron activity of our brain, so how can we explain a sense of ‘supra-consciousness’, and why should it be dismissed simply because it is no more explicable than ordinary consciousness? After all, they are both an experience as opposed to an objective observable phenomenon.
This is a brief synopsis - she also discussed the evolutionary (social) value that may have been inherent in forming religious beliefs, as well as how we may have come to believe in a God or Gods as external supernatural beings. There was also a consensus that, while morality seems to be inherent in humans, it is not dependent on religiosity per se.
As a side issue, Karen Armstrong in The History of God, puts forward a thesis that our idea of God changes over history. In other words, God seems to exist, at least in our writings, in a historical context. But I would reference Augustine who seemed to appreciate that God was part of our 'inner journey' as much as something external. Or to quote 19th century German philosopher, Ludwig Feuerbach: God does not exist independently of humanity.
Reference: New Scientist, 1 September 2007, pp32-6
When I was a young child, my father, who had spent 2.5 years as a POW, told me something I’ve never forgotten. He said: there are 2 types of Christians. There are Christians who go to church every week and wear their religion on their sleeve. Then there are Christians who don’t claim to be Christians yet they behave like Christians. In my now 56 years of living I’ve never seen anything to contradict that statement.
More relevant to Phillips’ topic, there are 2 types of religion: institutional religion that is political in every sense of the word; and religion as a personal experience, that is part of life’s journey, either as a unique, possibly one-off experience, or as an evolvement of one’s spirituality. I believe this is the distinction between ‘extrinsic’ and ‘intrinsic’ religion that is discussed by Phillips.
In the case of intrinsic or ‘quest’ religion, it is experienced as something ‘beyond’ the self. We can’t even explain how consciousness emerges from the neuron activity of our brain, so how can we explain a sense of ‘supra-consciousness’, and why should it be dismissed simply because it is no more explicable than ordinary consciousness? After all, they are both an experience as opposed to an objective observable phenomenon.
Sunday, 2 September 2007
Where does mathematics come from?
This is a more serious philosophical discourse than other theses, or mini-theses, I’ve posted so far. It’s an argument I’ve had with a number of philosophers, and non-philosophers. It's a question that most philosophers, indeed most people, seem to have an opinion on.
The short answer is that it’s a mixture of both invention and discovery. Mathematics requires creativity to achieve breakthrough discoveries as does any field of science. But I’m short-circuiting the argument. A good starting point is to reference a book I’ve read, Where Mathematics Comes From, by George Lakoff and Rafael E. Nunez. This is an excellent book on mathematics, covering all the basics and a number of esoteric topics like calculus, transfinite numbers and Euler’s famous equation: eiΠ + 1 = 0. This equation brings together such diverse fields as trigonometry, logarithms, calculus, complex algebra and power series into one simple relationship. The physicist, Richard Feynman, who discovered the equation a month before his 15th birthday, called it ‘the most remarkable formula in math’. Lakoff and Nunez provide a very accessible derivation of this equation as the crowning piece of exposition in their book. I must say at the outset that I have neither the expertise nor the ability to write a book like this. It is a very good book on mathematics. All my arguments and contentions deal with its philosophical content.
Lakoff and Nunez eschew any notion that mathematics is ‘discovered’, which is not an uncommon position. They argue, reasonably enough, that mathematics can only come from an ‘embodied mind’, therefore any suggestion that mathematics ‘already exists a priori’ is a conundrum that defies rational explanation. They argue that the only mathematics we know of comes from the human mind, therefore the onus of proof for any alternative view rests with the proponent of that view. In other words, the default point of view has to be that mathematics only exists as a product of the human mind. There is no evidence to support any other point of view.
Just to address that last point: all scientific discoveries are products of the human mind, nevertheless they exist independently of the human mind as well. The specific problem with giving mathematics the same status is that it doesn’t exist materially independently of the human mind. I will come back to this point later.
But my main problem with Lakoff’s and Nunez’s book is the assertion that all mathematics can be explained by ‘conceptual metaphor’. I’ve since learned that this particular philosophical premise is a brainchild of George Lakoff’s, who has written numerous books explaining the significance of metaphor in human endeavour, including philosophy, science, and, of course, mathematics. George Lakoff is Professor of Linguistics at Berkeley University, and I’ve since had correspondence with him. I’ve come to the conclusion that we agree to disagree, though he never responded to my last correspondence.
Many of my criticisms of Professor Lakoff’s philosophy addressed in this blog (though not all) have been made to him directly. In his book, Philosophy in the Flesh, which he co-wrote with Mark Johnson (not Nunez), Lakoff seems to find fault with every philosopher he’s acquainted with, both living and dead. He does this by employing his own 'philosophy of metaphor' (my terminology, not his) to give the reader his interpretation of their ideas. Much of this posting deals with Lakoff's use of the word metaphor. Its relevance to mathematical epistemology is explained in the next paragraph.
Basically, a conceptual metaphor ‘maps’ from a ‘source domain’ to a ‘target domain’ to use Lakoff’s own nomenclature. In the case of mathematics, the source domain is the grouping of objects, and activities that involve removal or combining elements of groups or complete groups in various ways. The target domain are the concepts inside our heads, which we call numbers, and how we manipulate them to represent events in the real world. Target domains can also be graphical representations like number lines and geometrical figures. This is not a verbatim representation of Lakoff’s and Nunez’s ideas, but my interpretation to ensure brevity of exposition without losing the gist of their philosophical premise. I have no problem with this aspect of their argument. I agree that mathematics is one of the most efficacious mediums we have for bridging the external world with our internal world. I have previously explained that the experiential concept of the external and internal world seems to be the starting point for many of my philosophical discourses. Where I disagree with Lakoff and Nunez is their assertion that this ‘bridge’ is strictly metaphorical.
According to The Oxford Companion to the Mind, metaphor is determined by context. This definition of a metaphor assumes that a word, phrase or term that is used in a metaphorical context must also have a literal context. In the case of Lakoff’s conceptual metaphors, that comprise all of mathematics, the metaphorical and literal contexts appear to be the same. I asked Professor Lakoff: ‘In what context is 2+2=4 metaphorical and in what context is it literal? If I say I want 3 of those, am I talking metaphorically and literally at the same time?’ The impression I got from his book is that mathematics has no literal context, only a metaphorical context. In other words, with ‘conceptual metaphors’, he has created a whole new field of metaphors that are permanently metaphorical. I can see no other interpretation and Lakoff has failed to enlighten me when I challenged him specifically on this. Assuming my interpretation is correct, this begs the question: they are metaphors for what? The obvious answer, going back to the original ideas set out in the ‘source domain’ and the ‘target domain’, is that they are metaphors for reality.
In Philosophy in the Flesh, Lakoff continually talks about metaphor as if it’s the progenitor of all ideas and concepts. He analyses a philosophical idea by reducing it to metaphor then presents it as if the metaphor came first. I will discuss an example that’s relevant to the topic: time and space. Lakoff rightly expounds on how we often use terms associated with distance to talk about time – it’s like we visualise time as distance. In relativity theory, this visualisation is real, due to a peculiar property of light. In ancient cultures and some indigenous cultures, however, the reverse is true: they refer to distance in terms of time. When Eratosthenes calculated the earth’s circumference around 230BC, he measured the distance he traveled from the well in Syene (Aswan) to Alexandria by the number of days he traveled by camel. If this was a metaphor and not literal then his whole enterprise would have failed. As it was, his calculation of the earth’s circumference was out by 15% according to modern measurements (ref: Encyclopaedia Britannica).
Everyone knows that there is a mathematical relationship between distance, time and speed, which is literal and not metaphorical. Now all physicists know that this relationship breaks down at sub-atomic speeds and astronomical distances due to relativity, so how can we say it’s true or literal or real? To add a further spoke in the works, when we have quantum tunneling the relationship ceases to exist altogether. But these anomalies are not resolved by saying that they are all metaphors and not real. They are resolved by finding the correct mathematical relationships that nature follows in these circumstances.
Physicists like Roger Penrose and Paul Davies have written extensively on the remarkable concordance we find between mathematics and the physical world. Lakoff claims that this concordance is purely metaphorical, and by his definition of metaphor (source domain: events in real world; target domain: concepts in our heads) I would agree. Using Lakoff’s own logic, mathematics is a metaphorical representation of the real world, but in this use of the term metaphor there is no distinction between metaphorical language and literal language – metaphor is a direct translation. Lakoff often uses the term metaphor where I would use the word definition. When he defines a concept in terms of other known concepts he calls it a conceptual metaphor or a conceptual blend. Conceptual blend is bringing 2 or more concepts together to form a new concept. Conceptual blend makes sense, but conceptual metaphor doesn’t if there are no distinct literal and metaphorical contexts in which to make it a metaphor. I’ve also argued that where there is a causal relationship between 2 concepts, one is not necessarily, by default, a metaphor for the other. An example of this is periodicity being a direct consequence of rotation; day and night resulting from the earth’s rotation is the best known example. In Where Mathematics Comes From, Lakoff implies that this relationship is metaphorical.
Personally, I call Lakoff’s conceptual metaphors literal metaphors, because if they were literal then my entire argument on this issue would evaporate, which, of course, would be preferable for both of us.
Lakoff also maintains that all theories (in physics at least) are metaphorical, which is not an issue I will pursue here. I did point out to him, however, that some of his metaphorical interpretations (of Einstein’s theories in particular) were incorrect or misleading. I referenced Roger Penrose, who is more knowledgeable on this subject than either of us.
Lakoff argues that physics is effectively mathematical modeling that happens to get very close to what we observe, and there are many who would agree with this interpretation. (Renowned physicist, Stephen Hawking, subscribes to this view.) But many physicists would say that the mathematical concordance we find in nature goes beyond modeling because there are too many cases where the mathematics provides an insight into nature that we didn’t expect to find (for example: Maxwell’s equations giving us the constant speed of light in a vacuum or Dirac’s equation giving us anti-matter). Irrespective of this argument, nature follows mathematical relationships at all observable levels of scale. Lakoff, by the way, argues that there are no ‘laws of nature’, which is another argument, though not altogether irrelevant, that I won’t pursue here.
This has been a lengthy detour, but it brings me back to the point I made about the status of mathematics existing independently of the human mind. Most people struggle with this notion – it’s like believing in God. It evokes the idea of an abstract realm independent of human abstract thought. People call this the Platonic realm after Plato’s fabled realm of perfect ‘forms’. The real world, in which we live, being a shadow of this perfect transcendental world. Roger Penrose calls himself a Platonist (refer The Emperor’s New Mind), mathematically speaking, because he believes the mathematics we discover already exists ‘out there’. Paul Davies eschews the idea of Platonism (refer The Goldilocks Enigma) but in The Mind of God, he devotes a whole chapter to what he calls ‘the mathematical secret’: the way the physical world is ‘shadowed’ by mathematics.
I call myself a Pythagorean because Pythagoras was the first (in Western philosophy at least) who seemed to appreciate that mathematics is an inherent aspect of the natural world, like a latent code waiting for someone like us to decipher it. As our knowledge of physics progresses, this realisation only seems to become more necessary to our comprehension of the universe. I can’t help but feel that Pythagoras had no idea how deep his insight really was. Lakoff calls Pythagoras’s philosophical insight a ‘folk theory’, but it’s a folk theory that launched Western science as we know it, so I would call it a paradigm, and one that has had unparalleled success over a 2,500 year history.
Philosophers, since the time of Russell and Wittgenstein, have mostly argued that mathematics is a sub-branch of logic, but Godel, Turing and Church, have all demonstrated, in various ways, that one cannot create all mathematics from a set of known axioms (Godel’s famous incompleteness theorem). This is one point that Lakoff and I appear to agree upon.
What many people fail to understand, or take into account, is that mathematics is not so much about numbers but the relationship between numbers – just look at how much mathematics is written without numbers. Robyn Arionrhod, who teaches mathematics at Monash University, Melbourne, made a similar point in her book, Einstein’s Heroes (effectively, a well written exposition on Maxwell’s equations). If one looks at mathematics from this perspective, one can see that the relationships cannot be invented – they are discovered. Mathematics is essentially a problem-solving endeavour, and this begs the question: if one is looking for a solution to a puzzle, does the solution already exist before someone finds it, or only after it’s found? Think: Fermat's Last Theorem; solved by Andrew Wiles 357 years after it was proposed. Think: Poincare's Conjecture; proposed 1904 and solved in 2002 by Grigory Perelman (read Donal O'Shea's account). Think: The Reimann Hypothesis; proposed 1859, still unsolved. To answer that question, I suggest, is to answer the philosophical conundrum: is mathematics invented or discovered?
Footnote: I sent a copy of this to Professor Lakoff as soon as I posted it, offering him the right to reply.
For an alternative point of view, read Lakoff’s and Nunez’s book, Where Mathematics Comes From. For a physicist’s perspective, read Penrose’s The Emperor’s New Mind or Davies’ The Mind of God. Arionrhod’s Einstein’s Heroes is a good read that indirectly supports the physicist’s perspective.
See also my later posting, Jan. 08: Is mathematics evidence of a transcendental realm? Amongst other things, I discuss Gregory Chaitin's book, Thinking about Godel and Turing.
The short answer is that it’s a mixture of both invention and discovery. Mathematics requires creativity to achieve breakthrough discoveries as does any field of science. But I’m short-circuiting the argument. A good starting point is to reference a book I’ve read, Where Mathematics Comes From, by George Lakoff and Rafael E. Nunez. This is an excellent book on mathematics, covering all the basics and a number of esoteric topics like calculus, transfinite numbers and Euler’s famous equation: eiΠ + 1 = 0. This equation brings together such diverse fields as trigonometry, logarithms, calculus, complex algebra and power series into one simple relationship. The physicist, Richard Feynman, who discovered the equation a month before his 15th birthday, called it ‘the most remarkable formula in math’. Lakoff and Nunez provide a very accessible derivation of this equation as the crowning piece of exposition in their book. I must say at the outset that I have neither the expertise nor the ability to write a book like this. It is a very good book on mathematics. All my arguments and contentions deal with its philosophical content.
Lakoff and Nunez eschew any notion that mathematics is ‘discovered’, which is not an uncommon position. They argue, reasonably enough, that mathematics can only come from an ‘embodied mind’, therefore any suggestion that mathematics ‘already exists a priori’ is a conundrum that defies rational explanation. They argue that the only mathematics we know of comes from the human mind, therefore the onus of proof for any alternative view rests with the proponent of that view. In other words, the default point of view has to be that mathematics only exists as a product of the human mind. There is no evidence to support any other point of view.
Just to address that last point: all scientific discoveries are products of the human mind, nevertheless they exist independently of the human mind as well. The specific problem with giving mathematics the same status is that it doesn’t exist materially independently of the human mind. I will come back to this point later.
But my main problem with Lakoff’s and Nunez’s book is the assertion that all mathematics can be explained by ‘conceptual metaphor’. I’ve since learned that this particular philosophical premise is a brainchild of George Lakoff’s, who has written numerous books explaining the significance of metaphor in human endeavour, including philosophy, science, and, of course, mathematics. George Lakoff is Professor of Linguistics at Berkeley University, and I’ve since had correspondence with him. I’ve come to the conclusion that we agree to disagree, though he never responded to my last correspondence.
Many of my criticisms of Professor Lakoff’s philosophy addressed in this blog (though not all) have been made to him directly. In his book, Philosophy in the Flesh, which he co-wrote with Mark Johnson (not Nunez), Lakoff seems to find fault with every philosopher he’s acquainted with, both living and dead. He does this by employing his own 'philosophy of metaphor' (my terminology, not his) to give the reader his interpretation of their ideas. Much of this posting deals with Lakoff's use of the word metaphor. Its relevance to mathematical epistemology is explained in the next paragraph.
Basically, a conceptual metaphor ‘maps’ from a ‘source domain’ to a ‘target domain’ to use Lakoff’s own nomenclature. In the case of mathematics, the source domain is the grouping of objects, and activities that involve removal or combining elements of groups or complete groups in various ways. The target domain are the concepts inside our heads, which we call numbers, and how we manipulate them to represent events in the real world. Target domains can also be graphical representations like number lines and geometrical figures. This is not a verbatim representation of Lakoff’s and Nunez’s ideas, but my interpretation to ensure brevity of exposition without losing the gist of their philosophical premise. I have no problem with this aspect of their argument. I agree that mathematics is one of the most efficacious mediums we have for bridging the external world with our internal world. I have previously explained that the experiential concept of the external and internal world seems to be the starting point for many of my philosophical discourses. Where I disagree with Lakoff and Nunez is their assertion that this ‘bridge’ is strictly metaphorical.
According to The Oxford Companion to the Mind, metaphor is determined by context. This definition of a metaphor assumes that a word, phrase or term that is used in a metaphorical context must also have a literal context. In the case of Lakoff’s conceptual metaphors, that comprise all of mathematics, the metaphorical and literal contexts appear to be the same. I asked Professor Lakoff: ‘In what context is 2+2=4 metaphorical and in what context is it literal? If I say I want 3 of those, am I talking metaphorically and literally at the same time?’ The impression I got from his book is that mathematics has no literal context, only a metaphorical context. In other words, with ‘conceptual metaphors’, he has created a whole new field of metaphors that are permanently metaphorical. I can see no other interpretation and Lakoff has failed to enlighten me when I challenged him specifically on this. Assuming my interpretation is correct, this begs the question: they are metaphors for what? The obvious answer, going back to the original ideas set out in the ‘source domain’ and the ‘target domain’, is that they are metaphors for reality.
In Philosophy in the Flesh, Lakoff continually talks about metaphor as if it’s the progenitor of all ideas and concepts. He analyses a philosophical idea by reducing it to metaphor then presents it as if the metaphor came first. I will discuss an example that’s relevant to the topic: time and space. Lakoff rightly expounds on how we often use terms associated with distance to talk about time – it’s like we visualise time as distance. In relativity theory, this visualisation is real, due to a peculiar property of light. In ancient cultures and some indigenous cultures, however, the reverse is true: they refer to distance in terms of time. When Eratosthenes calculated the earth’s circumference around 230BC, he measured the distance he traveled from the well in Syene (Aswan) to Alexandria by the number of days he traveled by camel. If this was a metaphor and not literal then his whole enterprise would have failed. As it was, his calculation of the earth’s circumference was out by 15% according to modern measurements (ref: Encyclopaedia Britannica).
Everyone knows that there is a mathematical relationship between distance, time and speed, which is literal and not metaphorical. Now all physicists know that this relationship breaks down at sub-atomic speeds and astronomical distances due to relativity, so how can we say it’s true or literal or real? To add a further spoke in the works, when we have quantum tunneling the relationship ceases to exist altogether. But these anomalies are not resolved by saying that they are all metaphors and not real. They are resolved by finding the correct mathematical relationships that nature follows in these circumstances.
Physicists like Roger Penrose and Paul Davies have written extensively on the remarkable concordance we find between mathematics and the physical world. Lakoff claims that this concordance is purely metaphorical, and by his definition of metaphor (source domain: events in real world; target domain: concepts in our heads) I would agree. Using Lakoff’s own logic, mathematics is a metaphorical representation of the real world, but in this use of the term metaphor there is no distinction between metaphorical language and literal language – metaphor is a direct translation. Lakoff often uses the term metaphor where I would use the word definition. When he defines a concept in terms of other known concepts he calls it a conceptual metaphor or a conceptual blend. Conceptual blend is bringing 2 or more concepts together to form a new concept. Conceptual blend makes sense, but conceptual metaphor doesn’t if there are no distinct literal and metaphorical contexts in which to make it a metaphor. I’ve also argued that where there is a causal relationship between 2 concepts, one is not necessarily, by default, a metaphor for the other. An example of this is periodicity being a direct consequence of rotation; day and night resulting from the earth’s rotation is the best known example. In Where Mathematics Comes From, Lakoff implies that this relationship is metaphorical.
Personally, I call Lakoff’s conceptual metaphors literal metaphors, because if they were literal then my entire argument on this issue would evaporate, which, of course, would be preferable for both of us.
Lakoff also maintains that all theories (in physics at least) are metaphorical, which is not an issue I will pursue here. I did point out to him, however, that some of his metaphorical interpretations (of Einstein’s theories in particular) were incorrect or misleading. I referenced Roger Penrose, who is more knowledgeable on this subject than either of us.
Lakoff argues that physics is effectively mathematical modeling that happens to get very close to what we observe, and there are many who would agree with this interpretation. (Renowned physicist, Stephen Hawking, subscribes to this view.) But many physicists would say that the mathematical concordance we find in nature goes beyond modeling because there are too many cases where the mathematics provides an insight into nature that we didn’t expect to find (for example: Maxwell’s equations giving us the constant speed of light in a vacuum or Dirac’s equation giving us anti-matter). Irrespective of this argument, nature follows mathematical relationships at all observable levels of scale. Lakoff, by the way, argues that there are no ‘laws of nature’, which is another argument, though not altogether irrelevant, that I won’t pursue here.
This has been a lengthy detour, but it brings me back to the point I made about the status of mathematics existing independently of the human mind. Most people struggle with this notion – it’s like believing in God. It evokes the idea of an abstract realm independent of human abstract thought. People call this the Platonic realm after Plato’s fabled realm of perfect ‘forms’. The real world, in which we live, being a shadow of this perfect transcendental world. Roger Penrose calls himself a Platonist (refer The Emperor’s New Mind), mathematically speaking, because he believes the mathematics we discover already exists ‘out there’. Paul Davies eschews the idea of Platonism (refer The Goldilocks Enigma) but in The Mind of God, he devotes a whole chapter to what he calls ‘the mathematical secret’: the way the physical world is ‘shadowed’ by mathematics.
I call myself a Pythagorean because Pythagoras was the first (in Western philosophy at least) who seemed to appreciate that mathematics is an inherent aspect of the natural world, like a latent code waiting for someone like us to decipher it. As our knowledge of physics progresses, this realisation only seems to become more necessary to our comprehension of the universe. I can’t help but feel that Pythagoras had no idea how deep his insight really was. Lakoff calls Pythagoras’s philosophical insight a ‘folk theory’, but it’s a folk theory that launched Western science as we know it, so I would call it a paradigm, and one that has had unparalleled success over a 2,500 year history.
Philosophers, since the time of Russell and Wittgenstein, have mostly argued that mathematics is a sub-branch of logic, but Godel, Turing and Church, have all demonstrated, in various ways, that one cannot create all mathematics from a set of known axioms (Godel’s famous incompleteness theorem). This is one point that Lakoff and I appear to agree upon.
What many people fail to understand, or take into account, is that mathematics is not so much about numbers but the relationship between numbers – just look at how much mathematics is written without numbers. Robyn Arionrhod, who teaches mathematics at Monash University, Melbourne, made a similar point in her book, Einstein’s Heroes (effectively, a well written exposition on Maxwell’s equations). If one looks at mathematics from this perspective, one can see that the relationships cannot be invented – they are discovered. Mathematics is essentially a problem-solving endeavour, and this begs the question: if one is looking for a solution to a puzzle, does the solution already exist before someone finds it, or only after it’s found? Think: Fermat's Last Theorem; solved by Andrew Wiles 357 years after it was proposed. Think: Poincare's Conjecture; proposed 1904 and solved in 2002 by Grigory Perelman (read Donal O'Shea's account). Think: The Reimann Hypothesis; proposed 1859, still unsolved. To answer that question, I suggest, is to answer the philosophical conundrum: is mathematics invented or discovered?
Footnote: I sent a copy of this to Professor Lakoff as soon as I posted it, offering him the right to reply.
For an alternative point of view, read Lakoff’s and Nunez’s book, Where Mathematics Comes From. For a physicist’s perspective, read Penrose’s The Emperor’s New Mind or Davies’ The Mind of God. Arionrhod’s Einstein’s Heroes is a good read that indirectly supports the physicist’s perspective.
See also my later posting, Jan. 08: Is mathematics evidence of a transcendental realm? Amongst other things, I discuss Gregory Chaitin's book, Thinking about Godel and Turing.
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