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.

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.

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.

Wittgenstein's anti-metaphysical stance

One thing I share with Ludwig Wittgenstein is a hostility towards metaphysics.

Henry Le Roy Finch argues* that the origin of metaphysics lies in the idea of identity, which he traces to Plato's conception of original or self-existing things, and to Aristotle's (and the Aristotelian tradition's) more systematically logical approach to the notion of self-identity.

That a thing is identical with itself (traditionally referred to as Aristotle's first law of thought) is often seen as the foundation for all logic.

Wittgenstein, on the other hand, thought that there was no more meaningless statement than a statement of self-identity. "To say of one thing that it is identical with itself is to say nothing at all." (Tractatus 5.5303)

Quite.

But Finch goes further and suggests that this skepticism about self-identity is linked to Wittgenstein's rejection of the popular notion of personal identity, the Cartesian thinking self. This is a central theme of Wittgenstein's (subsequently taken up by Gilbert Ryle). As Wittgenstein put it: "There is no such thing as the subject that thinks or entertains ideas." (Tractatus 5.631)

Certainly, both this claim and the previously-cited one are anti-metaphysical. And, significantly, Wittgenstein saw metaphysics and religion (or at least the sort of religion he embraced) as being in opposition to one another rather than as allies.

Finch rightly points out that "identitylessness" is at the heart of some important religious traditions, notably Buddhism, certain forms of Christianity** and Islam in its Sufi aspect.

Wittgenstein said that his goal was to "show the fly the way out of the fly-bottle", which can be interpreted as referring to the freeing of the human being from his or her false self-perception as a thinking self in its own private world. And this view of freedom is quite consistent with the religious traditions listed above.

On the other hand, as Finch points out, it is not consistent with other religious and philosophical approaches:

"The Stoic (and some would say also Judaic) idea of freedom is essentially that of Kant, which is that of the ethical self or free will, in which the self still retains its identity through its capacity to decide."

I remain uncomfortable with religious language and concepts, but I don't think someone like Wittgenstein can be understood (and I think he is worth trying to understand) if one ignores the implicit religious dimensions of his thought.

Also, having grown up (and so having invested a lot) in a religious tradition which I subsequently rejected, it's satisfying to see elements of that tradition coming into play here in a positive way.



* See his Wittgenstein, published by Element Books as part of the series Masters of Philosophy.

** I would single out the tradition known as fideism, and also the various mystical traditions. Pauline themes are important here; and it is worth noting in this connection that Wittgenstein liked the writings of the twentieth-century theologian Karl Barth.

Thinking about Vienna

In his later years, Ludwig Wittgenstein had many insightful and salutary things to say - about language especially. He had freed himself from a rigorous but narrow view of logic and language, and thought his way back to what looks like a very sane and sensible and quite ordinary point of view which respects the fact that human language is embedded in human life in all its forms and activities, and reflects this variety. There is nothing metaphysical about language and meaning, no mystery (though many philosophers continue to operate as if there were*).

But there is another side of Wittgenstein which I find less appealing: his negative attitude towards science, his tendency to play the sage, and his religion.

He was, I think, very close to Tolstoy in his religious views, and very much a Christian. He gave away his share of the family fortune (and in so doing incurred the lifelong enmity of his brother Paul). He prayed. He read the New Testament.

I say that Wittgenstein played the sage. He did so in his writings, which often have an oracular tone, but also in life (as a teacher, etc.). He was a notorious philosophical head-clutcher.

And, as befits a sage, Wittgenstein had and has disciples. Philosophical Wittgensteinians often play down the religious dimension of his thought, but this is not the case with Henry Le Roy Finch, who, having completed a PhD at Columbia, taught philosophy for more than forty years, mainly at Sarah Lawrence College and CCNY (later CUNY).

I am currently reading a short work of Finch's in which he presents Wittgenstein and Heidegger as harbingers of an epochal change in Western civilization.

"We may not expect the change, which seems to be seeping in from many directions, to be forecast or presaged by any one particular philosopher or prophet. However, the thinker who is attuned to his or her own time as well as to deeper currents may pick up the seismic tremors well before others do and express some critical formative ideas in advance of the more general historical changes. Such a thinker, in the opinion of many, is Ludwig Wittgenstein (1889-1951)... [He] was a thinker of such originality that no one claimed to understand him fully in his lifetime and the attempt to comprehend his 'new way of looking at things' and make it available to the mainstream continues."

For many of Wittgenstein's followers - and especially, I would say, for the non-philosophers among them - his forbiddingly complex and beautifully written oeuvre represents a profound and sophisticated defense of the (or a) religious point of view.

And, because he didn't make explicit religious claims ('Whereof one cannot speak ...'), it is difficult to argue against his position.

I try to keep an open mind on these issues, but I do tend to the view that Wittgenstein's religious propensities are inextricably bound up with some very peculiar psychological imperatives and with his family and cultural background. His culture and most of his preoccupations are alien to us today. He was a member of one of the richest and most highly cultured Viennese families and grew up in the declining years of the once-glorious Austro-Hungarian Empire. The Romantic cult of death was in the air.

I have my doubts also about his followers. Finch sounds at times like a bit of an oddball. In an endnote on the discarding of "age-old machinelike aspects of the human mind", he mentions favorably the religious thinker Eric Gutkind (whom I have not read), and recalls attending in New York in October 1949 a talk by the architect Frank Lloyd Wright.

"He came out on the stage and began his lecture with these words, which are imprinted on my memory: 'The greatest man of our time has died today, and probably none of you has ever heard of him.' It was Gurdjieff ..."

If old Vienna was a weird and alien world, the bohemian milieu of mid-twentieth century New York might have given it a run for its money.




* Saul Kripke's work was very influential in re-mystifying the philosophy of language.

How we ought to think

'One of the great pleasures of the philosopher's life,' wrote Jim Hankinson in The Bluffer's Guide to Philosophy, 'is being able to tell everyone (and not just children and dogs) what they ought to do. This is Ethics.'

On this reckoning, logic should afford even greater pleasure to its practitioners than ethics does insofar as it purports - at least on some accounts - to tell everyone how they ought to think. For example, consider this (from a text book for undergraduates): 'Logic is sometimes said to be the science of reasoning, but that assertion is somewhat misleading. Logic is not the empirical investigation of people's reasoning processes or the products of such processes. If it can be called a science at all, it is a normative science - it tells us what we ought to do, not what we do do.'

Or, as Gottlob Frege put it: 'the laws of logic are ... the most general laws, which prescribe universally the way in which one ought to think if one is to think at all.' (The Basic Laws of Arithmetic)

Frege, in fact, was something of a proto-fascist, and the above statement could be interpreted as having an authoritarian, even totalitarian, tenor. It could also be interpreted simply as an honest statement of the constraints of thought, reflecting Frege's noble goal of defining the bedrock of human reasoning.

It's no surprise that most attempts to articulate logic's normative role run into trouble. For what authority can the logician appeal to?

Formal logical systems are often seen as part of an attempt to systematize thinking, to improve (as it were) on ordinary thinking and the ordinary language on which it depends. And it is certainly true that ordinary language often deceives us and obscures the underlying logic (or structure) of an argument. Translating an argument into a formal language can reduce ambiguity, but those who have sought through the study of formal logical systems to illuminate the laws of thought or their foundations have been disappointed. Doubts surround not only the putative authority of a logical system but the very meaning of its symbols.

Technically, the meaning of what Rudolf Carnap called the fundamental mathematico-logical symbols (now usually called logical constants) derives from the explicit rules we lay down for their use, but in fact the question of their meaning remains obscure. One thing is clear: the whole exercise is paradoxically dependent on a prior understanding of the basic logical operations. Ordinary language use is also predicated on such an understanding: anyone lacking it would not be able to use language in anything like a normal way.

The work of Frege and his successors led, of course, to the development of digital computers in the mid-twentieth century, and in this sense it was spectacularly fruitful and successful. But it has not really led to a new understanding of human reasoning or established clear guidelines - as Frege hoped - for how we ought to think.

In fact, the attempt to create formal systems which can do what natural language can do has led to a renewed appreciation of the complexity, power, elegance and logical depth of the latter. Wittgenstein was right to warn against thinking of our everyday language as only approximating to something better, to some ideal language or calculus.

We need formal systems for dealing with mathematics and science and technology, but, as far as the fundamentals of logic are concerned, it's all there - implicitly at least - in the language of a five-year-old child.

The wider significance of Gödel's Incompleteness Theorem

Torkel Franzén [1950-2006] devoted a lot of time and energy to playing down the wider significance of Gödel's Incompleteness Theorem.* And there is no doubt that a great many extravagant claims about its significance have been made, usually along the lines that Gödel has demonstrated some fatal limitation in what scientific research can achieve, and an equal and opposite conclusion about human spirituality and artistic creativity.

I don't want to suggest that all claims for the general (as distinct from the mathematical and logical) significance of Gödel's work are mistaken: there is genuine disagreement between very knowledgeable people on the matter. But Franzén's position is widely respected as being scrupulously rigorous and based on a thorough understanding of the logical and mathematical concepts involved.

I personally found his views refreshingly straightforward when I came across them a couple of years ago. I guess I had read one too many of those contentious claims and wanted to develop a better understanding before looking again at more general questions.

Let me say, however, that I think there is abiding interest in Gödel's results, for example, in the contrast between formal systems like first-order logic (which he proved to be complete**) and stronger systems like the one outlined in Russell and Whitehead's Principia Mathematica (which he proved to be incomplete). Franzén points out that 'the incompleteness of any sufficiently strong consistent axiomatic theory ... concerns only what may be called the arithmetical component of the theory. A formal system has such a component if it is possible to interpret some of its statements as statements about the natural numbers, in such a way that the system proves some of the basic principles of arithmetic.'

We know that the natural numbers have surprising (not to say mysterious) properties, and I am tempted to say that the gulf between the first-order predicate calculus and stronger systems which Gödel's completeness and incompleteness theorems establish underscores an informal distinction between the pedestrian logic of commonsense and everyday life (which holds no surprises, life's surprises arising from complex concatenations of events rather than from our naive analyses thereof) and mathematical thinking.

Be that as it may, Gödel's work has another, perhaps more solid, claim to significance which flows from the discovery that a class of functions (recursive) which Gödel defined in the course of elaborating his famous proof turns out to be equivalent to some apparently quite different concepts developed in subsequent years (in particular by Alonzo Church, Alan Turing and Emil Post). David Berlinski writes:

'The idea of an algorithm had been resident in the consciousness of the world's mathematicians at least since the seventeenth century; and now, in the third [sic***] decade of the twentieth century, an idea lacking precise explication was endowed with four different definitions, rather as if an attractive but odd woman were to receive four different proposals of marriage where previously she had received none. The four quite different definitions ... were provided by Gödel, Church, Turing and Post. Gödel had written of a certain class of functions; Church of a calculus of conversion; and Turing and Post had both imagined machines capable of manipulating symbols drawn from a finite alphabet. What gives this story its dramatic unity is the fact that by the end of the decade it had become clear to the small coterie of competent logicians that the definitions were, in fact, equivalent in the sense that they defined one concept by means of four verbal flourishes. Gödel's recursive functions were precisely those functions that could be realized by lambda-conversion; and the operations performed by those functions were precisely those that could be executed by a Turing machine or a Post machine. These equivalencies, logicians were able first to imagine and then to demonstrate.

... A concept indifferent to the details of its formulation, Gödel asserted, is absolute. And in commenting on the concept to an audience of logicians, he remarked that the fact that only one concept had emerged from four definitions was something of an epistemological "miracle".' ****

I don't know about a miracle, but the equivalence of the various definitions is certainly suggestive that the underlying concept has a certain robustness and depth.



* This article (pdf), written just before his untimely death, gives a concise statement of his point of view.

** A set of axioms is complete if for any statement in the axioms' language either that statement or its negation is provable from the axioms.

*** This is a strange error for a mathematician to make - we're talking of the 1930s!

**** The Advent of the Algorithm (Harcourt 2001), pp. 205-6.