In recent posts I have made a number of claims about language and the brain. Allow me to clarify and develop a couple of points.
I don't really want to buy into the debate about various versions of modularity or other theories of mental functioning. For one thing, I don't know the science well enough. I don't have a theory, but I don't know that I need one either.
Which is not to say that it is not important to have a basic understanding of how our minds work. My point is that such an understanding needn't take the form of a theory. It may simply develop from a general (or specialist) knowledge of pertinent disciplines (such as psychology or linguistics), and as a considered response to various kinds of evidence. I am particularly interested in the evidence provided by injuries and genetic disorders which affect cognitive and emotional functioning.
Certain genetically-caused disorders and brain injuries seem to provide evidence that language is in some sense a distinct system - or rather a set of systems - even if it interacts (as it obviously does) with non-linguistic processes. How else can you account for people who have a language deficit but can think well in other respects, or, conversely, who may be seriously cognitively impaired and yet maintain excellent language abilities?
Take Williams syndrome, for instance. It is a genetic disorder characterized by a range of medical problems, developmental delays and learning disabilities. Children with this condition seek interaction with others but are very vulnerable as they lack normal caution and social understanding. They are typically unable to cope with numbers and abstract reasoning. They also have impaired gross and fine motor skills.
On the positive side, they often have an affinity for music (and perfect pitch). And they also tend to do well linguistically, at least in certain respects.
Williams syndrome, like so many other conditions which impact on brain function, is selective in its effects. If specific aspects of thinking are adversely affected while other specific aspects are not affected or are enhanced, then this certainly supports the view that the brain consists of many (interacting) systems and sub-systems.
Linguists, of course, see language from various points of view corresponding to various sub-disciplines: phonetics (where the focus is on the actual sounds of language), phonology (more abstract), morphology and syntax, semantics, pragmatics, etc. In other words, language has many aspects, so it is misleading to talk about language ability without specifying exactly what one is talking about.
Likewise, it is not particularly helpful to talk about the brain's capacity for language per se. Better to focus on the particular processes which language use requires, like hearing (or seeing in the case of reading); interpreting the raw data (identifying phonemes and lexemes, parsing, etc.) and so understanding; or speaking (which involves not only mentation but also a very complex sequence of fine motor processes).
Children with Williams syndrome are typically slow to start speaking. This is presumably related at least in part to their fine motor problems. Most reference sources say that older children and adults with WS speak fluently and grammatically and have a good concrete, practical vocabulary (though abstract vocabulary remains deficient).
I picked Williams syndrome to focus on in this post because of an anecdotal report I remembered reading about a profoundly retarded girl with WS who nonetheless had an unusually extensive vocabulary and was able to invent strikingly original stories and fantasies. But the more I read about Williams syndrome the more complicated - and equivocal - the picture looks.
For example, consider this (from a recent research report* abstract): 'Williams syndrome (WS) is a neurodevelopmental genetic disorder, often referred [to] as being characterized by dissociation between verbal and non-verbal abilities, although a number of studies disputing this proposal is emerging.'
And in their own study the researchers found significantly more speech disfluencies (hesitations, repetitions, pauses) in the WS group than in a typically-developing group.
So the lesson of my story is that everything concerning the human brain is likely to be more complicated than it seems, and that only scientific findings - rather than models or theories - can give specific answers to specific questions. Of course, science requires its models and theories, but they are always provisional, a means to an end.
And, in the context of such reflections, it is hardly surprising that I find myself becoming more and more skeptical about certain Chomskian assumptions which have been part of my mental furniture since I took a linguistics course taught by one of the Master's protégés a couple of decades ago.
* Rossi, N.F. et al. 'Analysis of speech fluency in Williams syndrome.' Res. Dev. Disabil. 32(6) (2011): 2957-62.
Monday, November 5, 2012
Friday, October 26, 2012
Quantum lemonade
Seth Lloyd's popular book* on quantum computation, life and the universe impressed me when I first read it a few years ago. I had the sense that Lloyd was saying something very important for our understanding of reality, of what ultimately underlies the whole shebang.
I still think the basic thesis of the book - that the cosmos is a quantum computer - is fascinating and maybe even true. Certainly, the parallels between thermodynamics and information theory suggest that information (bits, or qubits, and their operations) is absolutely fundamental to an understanding of the world and - speaking very loosely - the basic stuff out of which we and the cosmos are made.
But this recent article by Seth Lloyd disappointed me in a couple of ways.
Lloyd's book is beautifully written, a model of popular science writing. The science is clearly and simply presented, and there is some good - if at times only tangentially relevant - autobiographical background material. (The story of the death of Heinz Pagels is unforgettable. 'Heartbreaking', one reviewer called it.)
By contrast the article is in large part a rehash of things Lloyd has said many times before (for example, about the recalcitrance of atoms and sub-atomic particles, their reluctance to do what we want them to do and the need for infinite guile and patience on the part of quantum engineers). And unfortunately the metaphors are strained and distracting, in my opinion, and just a touch condescending. I think Lloyd is trying too hard not to sound like a boffin.
But the most significant thing about this recent piece is that in it Lloyd doesn't attempt (as he might well have done) to talk up the prospects for serious quantum computers. On the contrary, the whole program to develop and build useful quantum computers, about which he was so sanguine in his book, is presented as being somewhat problematic.
He writes: "The quantum sensitivity [Nobel Prize-winner Serge] Haroche identified certainly makes quantum computers hard to build, but it's also that very sensitivity that makes funky quantum phenomena such as Schrödinger's cat states the basis for hypersensitive detectors and measurement devices... What's bad for quantum computation is good for precision measurement - if life deals you quantum lemons, make quantum lemonade."
In other words, if we can't have miraculously powerful computers of an entirely new kind, we can at least have very accurate clocks. Mmm.
Guess I was a bit naïve to believe the hype.
•••••••••••••••••
Come to think of it, years ago I was quite excited about artificial intelligence. And they can't even do a convincing natural language interface yet.
Frankly, though, I don't much care about whether these technologies eventuate or not. What interests me more is the light that research into computing - digital and quantum - has thrown on some perennial questions.
The old answers to fundamental questions are just no good any more. And if some of the old answers do get a new lease on life, it will only be, I suspect, because they happened to prefigure an explanation informed by information theory, quantum mechanics and/or other recent theoretical work in physics or related sciences.
There is hype about technology and hype about basic science. But the fact is, though progress seems slow in both spheres, progress is indeed occurring.
Which is more than can be said of perhaps any other area of human life or endeavour.
* Programming the Universe: A Quantum Computer Scientist Takes on the Cosmos (Knopf, 2006).
I still think the basic thesis of the book - that the cosmos is a quantum computer - is fascinating and maybe even true. Certainly, the parallels between thermodynamics and information theory suggest that information (bits, or qubits, and their operations) is absolutely fundamental to an understanding of the world and - speaking very loosely - the basic stuff out of which we and the cosmos are made.
But this recent article by Seth Lloyd disappointed me in a couple of ways.
Lloyd's book is beautifully written, a model of popular science writing. The science is clearly and simply presented, and there is some good - if at times only tangentially relevant - autobiographical background material. (The story of the death of Heinz Pagels is unforgettable. 'Heartbreaking', one reviewer called it.)
By contrast the article is in large part a rehash of things Lloyd has said many times before (for example, about the recalcitrance of atoms and sub-atomic particles, their reluctance to do what we want them to do and the need for infinite guile and patience on the part of quantum engineers). And unfortunately the metaphors are strained and distracting, in my opinion, and just a touch condescending. I think Lloyd is trying too hard not to sound like a boffin.
But the most significant thing about this recent piece is that in it Lloyd doesn't attempt (as he might well have done) to talk up the prospects for serious quantum computers. On the contrary, the whole program to develop and build useful quantum computers, about which he was so sanguine in his book, is presented as being somewhat problematic.
He writes: "The quantum sensitivity [Nobel Prize-winner Serge] Haroche identified certainly makes quantum computers hard to build, but it's also that very sensitivity that makes funky quantum phenomena such as Schrödinger's cat states the basis for hypersensitive detectors and measurement devices... What's bad for quantum computation is good for precision measurement - if life deals you quantum lemons, make quantum lemonade."
In other words, if we can't have miraculously powerful computers of an entirely new kind, we can at least have very accurate clocks. Mmm.
Guess I was a bit naïve to believe the hype.
•••••••••••••••••
Come to think of it, years ago I was quite excited about artificial intelligence. And they can't even do a convincing natural language interface yet.
Frankly, though, I don't much care about whether these technologies eventuate or not. What interests me more is the light that research into computing - digital and quantum - has thrown on some perennial questions.
The old answers to fundamental questions are just no good any more. And if some of the old answers do get a new lease on life, it will only be, I suspect, because they happened to prefigure an explanation informed by information theory, quantum mechanics and/or other recent theoretical work in physics or related sciences.
There is hype about technology and hype about basic science. But the fact is, though progress seems slow in both spheres, progress is indeed occurring.
Which is more than can be said of perhaps any other area of human life or endeavour.
* Programming the Universe: A Quantum Computer Scientist Takes on the Cosmos (Knopf, 2006).
Friday, October 12, 2012
One, two, many
The question of the relevance of natural language to counting and calculating capacities has been raised in recent years in connection with two similar research projects.
In a well-known study published in 2004, the counting abilities of the Piraha people of the Amazon were examined. Their language lacks number words (other than for one and two). The researchers suggested that language (rather than other societal or environmental factors) was the crucial factor in explaining the poor counting abilities of members of this tribe.
Though not everyone was convinced by the researchers' claims, more recent research on several adults in Nicaragua who were born deaf and never learned Spanish or a formal sign language provided some slightly more convincing evidence of the importance of language for counting ability.
Elizabeth Spaepen (of the University of Chicago) and her colleagues conducted experiments involving, for example, the experimenter knocking her fist against the subject's fist a number of times and asking the subject to respond with the same number of knocks. (Iteration, note, rather than objects.)
'So if I were to knock four times on their fist,' commented Dr Spaepen, 'they might knock my fist five times.' *
The earlier research on the Piraha involved similar tests and similar results, but there was nothing to say that language was the crucial factor. A stronger case for language being the key factor can be made on the basis of the more recent research, as the Nicaraguans, unlike the Piraha, were living in a culture rich in counting systems.
Daniel Casasanto (of the Max Planck Institute for Psycholinguistics) points out that the human brain is good at approximating, e.g. distinguishing between ten and twenty objects, but needs a counting system to distinguish between ten and eleven, say.
'What language does,' he explains, 'is give you a means of linking up our small, exact number abilities with our large, approximate number abilities.'
As I see it, language provides for individuals, societies and cultures a kind of bridge to sophisticated forms of counting and calculation. Number words (in conjunction with other aids like fingers) facilitate simple forms of counting and these form a basis for more advanced techniques incorporating symbols and calculating devices.
Though number words are an intrinsic part of language, counting systems by and large are not. And - significantly - the more sophisticated the counting and calculating systems are, the less dependent they are on natural language.
So I don't see any necessary or intrinsic link between natural language and counting systems.
Historically, it may well be that only societies with number words went on to develop sophisticated counting systems and mathematics generally. And it may well be that, for most human children, learning number words is a prerequisite for learning to count and do basic arithmetic.
But this does not mean that arithmetic is in any fundamental way dependent on natural language.
Even in terms of human psychology, the link between language and calculating ability is pretty tenuous.
Think of autistic savants, for example. Are there not many instances of individuals who lack the ability to use and process language and yet whose brains display advanced calculating abilities?
* Wittgenstein would have had a field day with this!
In a well-known study published in 2004, the counting abilities of the Piraha people of the Amazon were examined. Their language lacks number words (other than for one and two). The researchers suggested that language (rather than other societal or environmental factors) was the crucial factor in explaining the poor counting abilities of members of this tribe.
Though not everyone was convinced by the researchers' claims, more recent research on several adults in Nicaragua who were born deaf and never learned Spanish or a formal sign language provided some slightly more convincing evidence of the importance of language for counting ability.
Elizabeth Spaepen (of the University of Chicago) and her colleagues conducted experiments involving, for example, the experimenter knocking her fist against the subject's fist a number of times and asking the subject to respond with the same number of knocks. (Iteration, note, rather than objects.)
'So if I were to knock four times on their fist,' commented Dr Spaepen, 'they might knock my fist five times.' *
The earlier research on the Piraha involved similar tests and similar results, but there was nothing to say that language was the crucial factor. A stronger case for language being the key factor can be made on the basis of the more recent research, as the Nicaraguans, unlike the Piraha, were living in a culture rich in counting systems.
Daniel Casasanto (of the Max Planck Institute for Psycholinguistics) points out that the human brain is good at approximating, e.g. distinguishing between ten and twenty objects, but needs a counting system to distinguish between ten and eleven, say.
'What language does,' he explains, 'is give you a means of linking up our small, exact number abilities with our large, approximate number abilities.'
As I see it, language provides for individuals, societies and cultures a kind of bridge to sophisticated forms of counting and calculation. Number words (in conjunction with other aids like fingers) facilitate simple forms of counting and these form a basis for more advanced techniques incorporating symbols and calculating devices.
Though number words are an intrinsic part of language, counting systems by and large are not. And - significantly - the more sophisticated the counting and calculating systems are, the less dependent they are on natural language.
So I don't see any necessary or intrinsic link between natural language and counting systems.
Historically, it may well be that only societies with number words went on to develop sophisticated counting systems and mathematics generally. And it may well be that, for most human children, learning number words is a prerequisite for learning to count and do basic arithmetic.
But this does not mean that arithmetic is in any fundamental way dependent on natural language.
Even in terms of human psychology, the link between language and calculating ability is pretty tenuous.
Think of autistic savants, for example. Are there not many instances of individuals who lack the ability to use and process language and yet whose brains display advanced calculating abilities?
* Wittgenstein would have had a field day with this!
Friday, September 21, 2012
Numbers and language
Number words come in various categories. There are the cardinal numbers (in English: one, two, three...), the ordinal numbers (first, second, third...) and the adverbial numbers (once, twice, thrice). There are also other number-word categories, but my special interest here is in the adverbials.
On the face of it, the cardinal numbers seem more linguistically primitive in the sense that they constitute the basic form upon which the adverbials are built. For example, the Middle English ones (= once) is an inflected form (genitive) of the Middle English word on (= one).
In Latin the situation is slightly different. The first few adverbials (semel, bis, ter, quater) are not obviously derived from - though all but the first are related to - the equivalent cardinals. Semel comes from the Proto-Indo-European *sem.
But irrespective of which category of number represents the earliest form linguistically, a case could be made that mathematically (and logically) the adverbial form is the basic one.
This is just a preliminary idea and I don't want to make too much of it. Also, I am aware of the use of cardinals and ordinals in set theory which makes it more difficult to make my point clearly. I am not talking set theory here.
The idea that the adverbials are somehow basic attracts me because it seems to provide a way of looking at numbers which tends to undermine (or at least not encourage) mathematical Platonism.
Focusing on the cardinals encourages mathematical Platonism because, even though it was through counting actual things that cardinal number words no doubt arose, their usefulness lies in their not being tied to any one kind of thing and so being applicable to anything countable. Inevitably, the cardinal numbers came to be seen as objects themselves, existing in an abstract realm.
If, on the other hand, we see numbers as being based on, or deriving from, the iteration process, then our focus moves from static objects and a timeless Platonistic realm to the ordinary world we all inhabit, of processes or actions which may (or may not) be repeated.
Interestingly, not only the Latin word semel (= once) but also the English word 'same' derives ultimately from the Proto-Indo-European *sem.
It's odd to think of certain modern English expressions as having such an ancient lineage.
"Same again," for example. (Licensing the re-execution of a previous order, a particular drink at a bar, say, and thus requiring the barman to go through roughly the same motions twice (or thrice...).)
This way of conceptualizing number is quite as natural as counting apples or oranges, and may, as I suggested, provide a good basis for a non-Platonistic and altogether more satisfactory way of understanding mathematics.
Modern mathematical Platonism is a long way from Plato, but it shares with Plato a static view of (mathematical) reality. It is at odds not only with the dynamic character of ordinary life and experience but also with the new ways of looking at things which the digital revolution of the last century has encouraged.
(I am currently looking at how the ideas of one of the great twentieth-century logicians, Alonzo Church, may relate to this notion of number as iteration and to mathematical Platonism. More later, perhaps. It's heavy stuff and may not be worth the trouble!)
On the face of it, the cardinal numbers seem more linguistically primitive in the sense that they constitute the basic form upon which the adverbials are built. For example, the Middle English ones (= once) is an inflected form (genitive) of the Middle English word on (= one).
In Latin the situation is slightly different. The first few adverbials (semel, bis, ter, quater) are not obviously derived from - though all but the first are related to - the equivalent cardinals. Semel comes from the Proto-Indo-European *sem.
But irrespective of which category of number represents the earliest form linguistically, a case could be made that mathematically (and logically) the adverbial form is the basic one.
This is just a preliminary idea and I don't want to make too much of it. Also, I am aware of the use of cardinals and ordinals in set theory which makes it more difficult to make my point clearly. I am not talking set theory here.
The idea that the adverbials are somehow basic attracts me because it seems to provide a way of looking at numbers which tends to undermine (or at least not encourage) mathematical Platonism.
Focusing on the cardinals encourages mathematical Platonism because, even though it was through counting actual things that cardinal number words no doubt arose, their usefulness lies in their not being tied to any one kind of thing and so being applicable to anything countable. Inevitably, the cardinal numbers came to be seen as objects themselves, existing in an abstract realm.
If, on the other hand, we see numbers as being based on, or deriving from, the iteration process, then our focus moves from static objects and a timeless Platonistic realm to the ordinary world we all inhabit, of processes or actions which may (or may not) be repeated.
Interestingly, not only the Latin word semel (= once) but also the English word 'same' derives ultimately from the Proto-Indo-European *sem.
It's odd to think of certain modern English expressions as having such an ancient lineage.
"Same again," for example. (Licensing the re-execution of a previous order, a particular drink at a bar, say, and thus requiring the barman to go through roughly the same motions twice (or thrice...).)
This way of conceptualizing number is quite as natural as counting apples or oranges, and may, as I suggested, provide a good basis for a non-Platonistic and altogether more satisfactory way of understanding mathematics.
Modern mathematical Platonism is a long way from Plato, but it shares with Plato a static view of (mathematical) reality. It is at odds not only with the dynamic character of ordinary life and experience but also with the new ways of looking at things which the digital revolution of the last century has encouraged.
(I am currently looking at how the ideas of one of the great twentieth-century logicians, Alonzo Church, may relate to this notion of number as iteration and to mathematical Platonism. More later, perhaps. It's heavy stuff and may not be worth the trouble!)
Tuesday, August 28, 2012
Science, philosophy, ideology
Previously I have discussed* the implications of studies which indicate that a person's basic political (and religious) orientation is influenced greatly by genetic and early developmental factors. We generally only engage in political or religious debate because we have strong ideological or religious convictions, and those convictions set the general tone and direction of our contributions. Rationality comes in only later - to help us elaborate and defend that general position which feels so true to us (but strangely not to our antagonists).
These facts (as I take them to be) are rather inconvenient. It takes all the fun out of argument if one feels obliged to be skeptical towards one's own deeply felt convictions!
But on the plus side, it allows one (I believe) better to understand what is really going in much ideological, religious and philosophical debate.
In my previous post on this site, I touched on these issues, suggesting that platonists and anti-platonists in the philosophy of mathematics may be caught up in a debate which is superficially rational but ultimately driven by non-rational factors - deep convictions similar to religious or political convictions.
If progress is to be made in any of these areas, I think there has to be an acceptance that we are less rational than we would like to think; and so we need to depend more on scientific methods (which incorporate mechanisms to counter individual biases etc.), and less on convictions (or the elaborate arguments which we have built upon them).
A boring conclusion, I know. Especially for those of us who have strong convictions and a taste for argument and debate about the big questions.
Within the (rather ill-defined) area of philosophy, history certainly seems to indicate that arguments and debates are most fruitful (albeit somewhat constrained) when the dividing line between philosophy and science is blurred or non-existent, and most pointless and futile whenever philosophy is disengaged from science.
I recognize, however, that we are inveterately ideological creatures**, and there will always be a role for those who can identify, articulate and criticize the ideological frameworks we inevitably create and seek to live by.
Could this be what will replace the bits of philosophy which are not swallowed up by the various sciences: the scientifically-informed critique of ideologies?
* For example, here and here.
** There are problems with the term 'ideology', I know. I am using it in a very broad sense to mean something like a system of beliefs involving values and prompting certain forms of action, often in concert with others who share the ideology and sometimes in opposition to those who don't. It may be that I would do better to speak of us being inveterately tribal. 'Ideology' may be just the intellectual's (way of rationalizing) tribalism.
These facts (as I take them to be) are rather inconvenient. It takes all the fun out of argument if one feels obliged to be skeptical towards one's own deeply felt convictions!
But on the plus side, it allows one (I believe) better to understand what is really going in much ideological, religious and philosophical debate.
In my previous post on this site, I touched on these issues, suggesting that platonists and anti-platonists in the philosophy of mathematics may be caught up in a debate which is superficially rational but ultimately driven by non-rational factors - deep convictions similar to religious or political convictions.
If progress is to be made in any of these areas, I think there has to be an acceptance that we are less rational than we would like to think; and so we need to depend more on scientific methods (which incorporate mechanisms to counter individual biases etc.), and less on convictions (or the elaborate arguments which we have built upon them).
A boring conclusion, I know. Especially for those of us who have strong convictions and a taste for argument and debate about the big questions.
Within the (rather ill-defined) area of philosophy, history certainly seems to indicate that arguments and debates are most fruitful (albeit somewhat constrained) when the dividing line between philosophy and science is blurred or non-existent, and most pointless and futile whenever philosophy is disengaged from science.
I recognize, however, that we are inveterately ideological creatures**, and there will always be a role for those who can identify, articulate and criticize the ideological frameworks we inevitably create and seek to live by.
Could this be what will replace the bits of philosophy which are not swallowed up by the various sciences: the scientifically-informed critique of ideologies?
* For example, here and here.
** There are problems with the term 'ideology', I know. I am using it in a very broad sense to mean something like a system of beliefs involving values and prompting certain forms of action, often in concert with others who share the ideology and sometimes in opposition to those who don't. It may be that I would do better to speak of us being inveterately tribal. 'Ideology' may be just the intellectual's (way of rationalizing) tribalism.
Sunday, August 19, 2012
Timothy Gowers and the philosophy of mathematics
Following on from my previous post, here are a few more thoughts on the philosophy of mathematics and on Timothy Gowers's views.
On rereading his talk, I thought the section on what it could mean for 2+2 to equal 5 in an alien world to be too clever by half and ultimately unconvincing. (Gowers virtually concedes this himself, so why include the material in a short talk?) It reminded me of Wittgenstein's equally unconvincing (to me) arguments about deviant forms of counting.
And then there is Gowers's politics, including his role in the campaign against the scientific publisher, Elsevier. (Wittgenstein too had strong moral and social convictions - including a conviction that ideas should not be 'owned' - but, unlike Gowers, he was, so far as I know, never an activist.) I am not aware of Gowers speaking anywhere of his general ideological or political position. I would guess that his views are left-leaning, but I don't know for sure, and I don't know what effect (if any) his general political and moral views might have had on his views on the philosophy of mathematics.
Which brings me back to the main point of the previous post: there seems to be no objective way of deciding on the truth or otherwise of the many and various options within the philosophy of mathematics. So when someone shows a strong commitment to a particular view, I wonder whether extraneous factors - such as ideology - might be playing a role.
I should also say something about neo-Meinongianism, having raised the topic in my previous post. Needless to say, it's complicated, but the general gist of it is something like this. Neo-Meinongians think that you can make statements about mathematical objects (like numbers) without being committed to believing they exist. They would claim that the statement that there are infinitely many prime numbers is literally true even though numbers may not exist as such. This view is based on the (I think) plausible idea that expressions like 'there is' or 'there are' are used in various ways and do not necessarily entail any ontological commitment.
As I understand it, (non-neo) Meinongianism is supposed to countenance gradations or different kinds of existence or being, whereas neo-Meinongians claim that certain uses of expressions such as 'there is' involve no ontological commitment.
In fact, Gowers makes a somewhat neo-Meinongian point in his talk, endorsing Rudolf Carnap's distinction between internal and external questions, only the latter (possibly) involving ontological commitments. So, on this view 'there are two senses of the phrase "there exists". One is the sense in which it is used in ordinary mathematical discourse - if I say that there are infinitely many primes I merely mean that the normal rules for proving mathematical statements license me to use appropriate quantifiers. The other is the more philosophical sense, the idea that those infinitely many primes "actually exist out there". These are the internal and external uses respectively.'
Or again, Gowers writes: 'One view, which I do not share, is that at least some ontological commitment is implicit in mathematical language.'
I'm not sure where I stand on all this. I have doubts about the worthwhileness of much of what goes on under the designation of 'philosophy of mathematics', but see some issues which seem real and important. Though I am certainly drawn to the position outlined by Gowers, I am acutely aware that this general orientation may be, like general religious or political predispositions, largely a function of inherited or early-environmental factors.
It seems to me that platonistic or anti-platonistic intuitions lie behind most philosophical work in the area. But what are these intuitions worth if they are in large part the result of arbitrary genetic and developmental factors?
The fact that little progress has been made in answering apparently real and interesting questions in the philosophy of mathematics and related areas suggests to me that the standard, traditional philosophical approaches are somehow flawed. Or perhaps the questions are ill-conceived, based on an inadequate understanding of the intellectual disciplines in question.
In fact, the advent of digital computers and new ways of conceptualizing information and information processing is changing the way we see mathematics (and much else besides). As new ways of doing and looking at mathematics and science emerge, questions that once seemed meaningful and important may no longer seem so.
On rereading his talk, I thought the section on what it could mean for 2+2 to equal 5 in an alien world to be too clever by half and ultimately unconvincing. (Gowers virtually concedes this himself, so why include the material in a short talk?) It reminded me of Wittgenstein's equally unconvincing (to me) arguments about deviant forms of counting.
And then there is Gowers's politics, including his role in the campaign against the scientific publisher, Elsevier. (Wittgenstein too had strong moral and social convictions - including a conviction that ideas should not be 'owned' - but, unlike Gowers, he was, so far as I know, never an activist.) I am not aware of Gowers speaking anywhere of his general ideological or political position. I would guess that his views are left-leaning, but I don't know for sure, and I don't know what effect (if any) his general political and moral views might have had on his views on the philosophy of mathematics.
Which brings me back to the main point of the previous post: there seems to be no objective way of deciding on the truth or otherwise of the many and various options within the philosophy of mathematics. So when someone shows a strong commitment to a particular view, I wonder whether extraneous factors - such as ideology - might be playing a role.
I should also say something about neo-Meinongianism, having raised the topic in my previous post. Needless to say, it's complicated, but the general gist of it is something like this. Neo-Meinongians think that you can make statements about mathematical objects (like numbers) without being committed to believing they exist. They would claim that the statement that there are infinitely many prime numbers is literally true even though numbers may not exist as such. This view is based on the (I think) plausible idea that expressions like 'there is' or 'there are' are used in various ways and do not necessarily entail any ontological commitment.
As I understand it, (non-neo) Meinongianism is supposed to countenance gradations or different kinds of existence or being, whereas neo-Meinongians claim that certain uses of expressions such as 'there is' involve no ontological commitment.
In fact, Gowers makes a somewhat neo-Meinongian point in his talk, endorsing Rudolf Carnap's distinction between internal and external questions, only the latter (possibly) involving ontological commitments. So, on this view 'there are two senses of the phrase "there exists". One is the sense in which it is used in ordinary mathematical discourse - if I say that there are infinitely many primes I merely mean that the normal rules for proving mathematical statements license me to use appropriate quantifiers. The other is the more philosophical sense, the idea that those infinitely many primes "actually exist out there". These are the internal and external uses respectively.'
Or again, Gowers writes: 'One view, which I do not share, is that at least some ontological commitment is implicit in mathematical language.'
I'm not sure where I stand on all this. I have doubts about the worthwhileness of much of what goes on under the designation of 'philosophy of mathematics', but see some issues which seem real and important. Though I am certainly drawn to the position outlined by Gowers, I am acutely aware that this general orientation may be, like general religious or political predispositions, largely a function of inherited or early-environmental factors.
It seems to me that platonistic or anti-platonistic intuitions lie behind most philosophical work in the area. But what are these intuitions worth if they are in large part the result of arbitrary genetic and developmental factors?
The fact that little progress has been made in answering apparently real and interesting questions in the philosophy of mathematics and related areas suggests to me that the standard, traditional philosophical approaches are somehow flawed. Or perhaps the questions are ill-conceived, based on an inadequate understanding of the intellectual disciplines in question.
In fact, the advent of digital computers and new ways of conceptualizing information and information processing is changing the way we see mathematics (and much else besides). As new ways of doing and looking at mathematics and science emerge, questions that once seemed meaningful and important may no longer seem so.
Tuesday, August 14, 2012
A strange form of amusement
If you look up an encyclopedia entry on the philosophy of mathematics you will usually find yourself presented with a list of competing approaches dating back to Plato. For someone of my temperament this is unsatisfactory. Well, it's fine having competing views, but which one (if any of them) is true?
With areas like ethics or art, the fact that there is no consensus may be explained by the very real possibility that these areas are largely subjective. But mathematics? Surely there is something that mathematics is, some broad understanding at any rate that we can agree on?
In the philosophy of mathematics there is a basic division between realists (or platonists) who believe that mathematical objects exist (but not in time and space); and anti-realists who don't believe this (seeing mathematics simply as a human activity, for instance).
Most mathematicians are thought to embrace some form of realism (mathematical truths are 'out there' to be discovered); but not all do. The distinguished mathematician Timothy Gowers is an anti-realist (aligning himself with Wittgenstein in this matter).
This basic division is just the beginning, however, as there is (as in just about any area in which philosophers are involved) a proliferation of arguments and counter-arguments resulting in an ever-proliferating list of divisions and subdivisions and so of positions to attack or defend.
Amongst which is the gloriously named neo-Meinongianism. (Could anything so called actually be true?)
A part of me says: 'Steer clear of all this, my dear fellow. Life is too short. And it's not about what it seems to be about. In part it's merely a perverse, self-perpetuating amusement for philosophers, in part an attempt by serious, religiously-inclined thinkers to defend a metaphysico-religious (or should that be religio-metaphysical?) view of the world.'
Unfair, no doubt. What of the serious, non-religious and/or anti-platonistic participants? Like Timothy Gowers.
In fact Gowers has himself asked the question of whether mathematics needs a philosophy, and I find his thoughts on the matter very persuasive.
I will probably not be spending a lot of time researching this area, but I would like to follow up on Gowers's views.*
And also, I intend to have a closer look at neo-Meinongianism. Why not?
* Gowers is a very political (and influential) figure within the mathematical and broader intellectual community and seems to have some fairly radical views about the ownership and distribution of ideas.
With areas like ethics or art, the fact that there is no consensus may be explained by the very real possibility that these areas are largely subjective. But mathematics? Surely there is something that mathematics is, some broad understanding at any rate that we can agree on?
In the philosophy of mathematics there is a basic division between realists (or platonists) who believe that mathematical objects exist (but not in time and space); and anti-realists who don't believe this (seeing mathematics simply as a human activity, for instance).
Most mathematicians are thought to embrace some form of realism (mathematical truths are 'out there' to be discovered); but not all do. The distinguished mathematician Timothy Gowers is an anti-realist (aligning himself with Wittgenstein in this matter).
This basic division is just the beginning, however, as there is (as in just about any area in which philosophers are involved) a proliferation of arguments and counter-arguments resulting in an ever-proliferating list of divisions and subdivisions and so of positions to attack or defend.
Amongst which is the gloriously named neo-Meinongianism. (Could anything so called actually be true?)
A part of me says: 'Steer clear of all this, my dear fellow. Life is too short. And it's not about what it seems to be about. In part it's merely a perverse, self-perpetuating amusement for philosophers, in part an attempt by serious, religiously-inclined thinkers to defend a metaphysico-religious (or should that be religio-metaphysical?) view of the world.'
Unfair, no doubt. What of the serious, non-religious and/or anti-platonistic participants? Like Timothy Gowers.
In fact Gowers has himself asked the question of whether mathematics needs a philosophy, and I find his thoughts on the matter very persuasive.
I will probably not be spending a lot of time researching this area, but I would like to follow up on Gowers's views.*
And also, I intend to have a closer look at neo-Meinongianism. Why not?
* Gowers is a very political (and influential) figure within the mathematical and broader intellectual community and seems to have some fairly radical views about the ownership and distribution of ideas.
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