Philosophy

Quasi-empirical formalism and rationality

thesamovar — Thu, 21 Aug 2008 15:12:35 +0100

Note: I wrote this in May 2003. It's unfinished but still of some interest I think.

Formalist philosophies of mathematics isolate themselves from foundational problems whilst retaining something like certainty. Unfortunately, the side effect of this is that they cannot account for the meaning or practice of mathematics. They cannot account for the meaning of mathematics because for formalism the symbols and rules of inference are purely syntactic and both the choice of axioms and the choice of which propositions to prove are a matter of convention alone. They cannot account for the practice of mathematics because, with a tiny number of exceptions, no field of mathematics lives up to its standards of rigour. The function of formalist philosophies of mathematics is to make mathematics “anarchist” (as Feyerabend would have put it) or “elitist” (as Lakatos would have put it). That is, to bar non-mathematicians from judging mathematics. This probably accounts for its popularity among mathematicians.

A more charitable view is that formalism is a regulative ideal for mathematics. It is an ideal because if we had the capacity to work with mathematics purely formally, and we had the capacity to manipulate symbols at great speed without error, then the only uncertainty in mathematics would be about whether the axioms led to contradiction or not. Certainly this would still leave an empirical question open, and would not constitute a theory of meaning but it would provide a perfect court of arbitration for proofs. This view is expressed by David Hilbert in his essay On the Infinite: “Mathematics in a certain sense develops into a tribunal of arbitration, a supreme court that will decide questions of principle – and on such a concrete basis that universal agreement must be attainable and all assertions can be verified.” This ideal is unattainable, but it is also regulative in the following sense. In practice we can only work with informal proofs which we believe to be formalisable. If there is a dispute over a theorem, the ideal of formalism suggests a relatively unambiguous way to resolve the dispute, or at least to find where the disagreement lies if it cannot be resolved. Suppose I dispute an informal step in your proof, you can respond by further formalising that step (assuming you didn’t make a mistake), getting closer to the axioms of whatever formal system you’re working in. I can then read over your formalisation of that step and either agree that you were right after all or criticise a step in your more formal version. This process either ends with agreement (either you made a mistake or I did) or it ends in a fundamental disagreement about the axioms. Either way, we know, in principle, where the problem lies. So, if we consider formalism as a regulative ideal for the resolution of mathematical disputes it has methodological significance for mathematics and is unobjectionable in the sense that it doesn’t claim to be a theory of meaning.

The concept of a regulative ideal has a precedent in epistemology which suggests its usefulness. In his paper Truth, Rationality and the Growth of Knowledge, Popper used a modification of Tarski’s theory of truth, and his own notion of closeness to truth, as a regulative ideal in defining science. He defined closeness to truth in the following way: A theory S is closer to the truth than a theory T if the set of true consequences of S is a superset of the set of true consequences of T and the set of false consequences of S is a subset of the set of false consequences of T. This defines a partial order on theories. However, he recognises the ideal nature of his definition: “… we have no criterion of truth, but are nevertheless guided by the idea of truth as a regulative principle (as Kant or Peirce might have said); and that … there are criteria of progress towards the truth…”. He later goes on to say “It is only the idea of truth which allows us to speak sensibly of mistakes and of rational criticism, and which makes rational discussion possible… Thus the very idea of error – and of fallibility – involves the idea of an objective truth as the standard of which we may fall short. (It is in this sense that the idea of truth is a regulative idea.)”. And, although he has said that there are “criteria of progress towards the truth” he later goes on to say that “… approximation to truth … has the same objective character and the same ideal or regulative character as the idea of objective or absolute truth.” In other words, there are no such criteria, as he recognises when he says “It always remains possible, of course, that we shall make mistakes in our relative appraisal of two theories, and the appraisal will often be a controversial matter.” He cannot directly deduce methodological rules from these ideal definitions, but it suggests (regulates) them. This can be more clearly seen in Lakatos’ definition of what it means to falsify a theory. For Lakatos a theory T is falsified if there is another theory, S, which predicts (explains) all the corroborated content of T and correctly predicts at least one fact that T fails to predict. Since closeness to truth is only a partial order on theories, there remains a logical possibility that two theories may not be comparable (if S correctly explains a fact that T fails to and T correctly explains a fact that S fails to). Kuhn and Feyerabend showed that this sort of thing happens all the time. This undermines the idea that Popperian or Lakatosian science are complete philosophical theories of science, but doesn’t undermine their status as regulative ideals.

In fact, there is a sense in which philosophical theories can never be more than regulative. The written word obviously has no mystical power that can, in itself, cause things to happen (except for magic spells of course). An individual is free to interpret a philosophical theory however he likes, especially if it involves such vague notions as truth.

But how can we discuss the relative merits of different regulative ideals if philosophical theories can never be more than regulative? This is a difficult question and in general it is probably impossible to do so abstractly. However, if we think about the function of rational discussion and discuss regulative ideals pragmatically we might be able to get somewhere. As an example, consider the regulative ideal of objective truth. I believe that the reason that it is so fruitful in practice can be understood as a phenomenon something (very roughly) like a generalised dialectical process.

Consider the following model scenario. There are two individuals in a shared environment. Each individual has his own heuristic approach to dealing with the world. That is, he has rules of thumb that help him to interact and deal with the environment he finds himself in. The two individuals can communicate with each other, in particular they can, to the best of their abilities, describe their heuristic approaches to one another. Through the process of arguing together about which heuristic approach is better, and by trying out and testing each others approaches on the environment, there emerges an improved approach which better deals with the environment than either of the two individuals’ initial approaches. From the first individual’s point of view, his initial approach is something like a thesis, the second individual’s is like an antithesis, and the better approach they reach together is like a synthesis.

This scenario is simplistic, but it suggests a more sophisticated and general one. Firstly, the two individuals will probably not be able to explain their own heuristic approaches perfectly. Secondly, they probably will not come to an agreement about a third approach, although both of their own approaches will probably be modified by the exchange and there will be an overlap between their modified approaches. Now, increase the number of individuals in the scenario and, over time, the clash of heuristic approaches will tend to make the individuals’ approaches more and more effective. In this account so far there is in one sense an objective fact, the shared environment. However, it may easily be that no individual can describe this shared environment. No statement about the world which an individual in that environment could think of would be true. [Suggest a variety of reasons that this might be the case.] Despite this, by using such statements about the environment as the basic mechanism of communicating their heuristic approaches, they improve, in some sense, these approaches. This scenario suggests why a belief in the possibility of statements about the world expressing objective truths can be so fruitful in practice, even if it is totally mistaken (or even meaningless). [Make some remarks about the apparent meaninglessness of the idea of a proposition expressing a truth about the world. At the very least, it’s implausibility.]

Before I go on to explain my view of mathematics, I want to quickly describe Lakatos’ quasi-empiricism since my view is related to it. Lakatos likened mathematics to science, in that it is empirical in a certain sense. The first sense in which it is empirical is that we can never know if a formal mathematical system is consistent or not. A logical falsifier of a formal system is a proof of the statement pnot-p (i.e. a demonstration of the inconsistency of the system). More interesting is his notion of a heuristic falsifier. If a formal mathematical system is a formalisation of a previously existing informal mathematical system, then a heuristic falsifier consists of an informal proof of the statement p in the informal system, together with a formal proof of the statement not-p in the formal system. This is a falsifier because it says that the formal mathematical system is not an accurate model of the informal system. For example, if in our formal mathematical system we could prove that 2+2=5 this would be a heuristic falsifier (even if the model of arithmetic which proved it was perfectly consistent). Finally, he encourages formalisations which are, in some sense, testable. For example, he mentions that Gödel says that there are consequences of certain infinity axioms in the field of Diophantine equations. together with a proof of the statement

For Lakatos, that is all that quasi-empiricism amounts to, formal mathematics is just a model of informal mathematics: “… we should speak only of formal mathematical theories if they are formalisations of established informal mathematical theories.” However, I think that more can be said about the way in which we choose propositions to try and prove from among the vast array of propositions which can be proved. The definitions and theorems of mathematics are chosen with exquisite care. Indeed, if they were not so chosen mathematics would just be a rather peculiar past-time, a bit like fitting together oddly shaped coloured bricks without any particular scheme. The difficult question is – why are they chosen the way they are? Lakatos provides a first approximation of an answer to this question - formal mathematics is a model of informal mathematics. But this leaves the question, why is informal mathematics the way it is?

First of all, mathematics is a model of our intuitions. For example, arithmetic is a model of an intuitive process which everyone is perfectly familiar with (but it is a learned concept). Our direct intuition of whole numbers, that is our experience of counting objects and comparing collections of objects, only extends to very small whole numbers, probably less than 20 or 30. Our indirect intuition of whole numbers, that is our experience of calculating with numbers, extends much further. Our direct intuition of whole numbers suggests the model we use (say, Peano’s axioms of arithmetic), and our indirect intuition, our experience with calculating using this model, doesn’t give cause for dissatisfaction.

As well as being a model of our intuition about arithmetic, mathematics provides the language and basic concepts used to define physics – real numbers, functions, differentiability, etc. The definition of real numbers in some sense captures our intuitive notions of space. Think of the square root of 2. The discovery that 2 doesn’t have a rational square root led the Greeks to abandon their arithmetical notions of number in favour of geometrical ones. The following two thought experiments suggest that there “ought to be” a square root of 2. First, it seems as though we can approximate the square root of 2 as closely as we like with rational numbers. That is, we can find rational numbers p/q for which (p/q)2-2 is as small as we like. This sort of thinking, although not directly, underlies the definition of real numbers as the Cauchy equivalence classes of sequences of rational numbers. Second, for any rational number p/q you can decide if it is less than the square root of two or bigger than it (just decide if (p/q)2<2 or (p/q)2>2). If you think of the number line, we have a way of breaking it into two pieces, one to the left of the square root of two and one to the right. It seems as though if we can break the number line at this point, there must be a point on the number line here. This thought experiment underlies the definition of real numbers as Dedekind cuts. However reasonable these thought experiments seem, they make intuitive hypotheses about numbers, and these hypotheses determine a formal model of the concept of a number. In the first thought experiment, the hypothesis is about limits, in the second thought experiment the hypothesis is about the continuity of the number line. As the example of hyper-real numbers indicates (the hyper-reals are an extension of the reals which allow for two hyper-reals to differ by an infinitesimal quantity, amongst other things), these intuitive hypotheses can be modelled in more than one way. The definition of the Cauchy equivalence classes of sequences of rational numbers takes no account of the exact way in which the sequence of rational numbers converges, but the equivalent definition underlying hyper-real numbers does. For real numbers, there is a single number that cuts the rational numbers into two classes, but for hyper-real numbers there are an infinite number of numbers that cut the rational numbers in the same way. Thus our definition of number is based on an intuition about the number line, an intuition that future physical theories may cause us to reassess.

The mathematical model of the concepts of physics doesn’t stop there. Even supposing we have a satisfactory definition of the number line, this doesn’t uniquely determine a definition of space. The first step in the modern definition of space is to consider the Cartesian product of the number line with itself three times (to get three dimensions). That is, a point in space is identified with a triple of numbers on the number line (x, y, z). Again, this is a model of space. (At this point I’ve not said anything about the geometry of this Cartesian product.) The next step is to give this Cartesian product a geometric structure, by defining the notion of distance and straight lines. Initially, Euclidean geometry was the model, but later other (curved) geometries were introduced. The last step in the model is the notion of a manifold, which is turn based on notions of continuity, topology, differentiability, and so forth. At each step, models inspired by intuition have been made. [I’d quite like to go through this in more detail highlighting the way in which these steps are modelling steps.] Although the structures formed in this way may not be false in the sense that one can derive contradictions in this system, they may fail to provide useful basic concepts for physics. It is in this sense that mathematics is a model of the basic concepts of physics.

More mysteriously, mathematics can even be a model of itself. The modern concepts of real numbers, Cartesian products and so forth didn’t exist for the ancient Greek mathematicians, and yet Euclidean geometry can be modelled in this language of modern mathematics. This reflects the fact that mathematical structures acquire a meaning independent of what they’re modelling, and become interesting in themselves. [Does this paragraph make any sense?]

In practice then, the definitions and theorems of mathematics have, until recently, been motivated by the idea that mathematical definitions can get at the essence of reality. For the same reasons that the belief in objective truth is so fruitful, the belief, among mathematicians, that their definitions capture the essence of the world has been fruitful. [Proof generated concepts suggest that this is not the last word on this subject. They are motivated by the need for mathematicians to be able to say something?]

I find the term heuristic falsifier slightly misleading because it doesn’t emphasise that we are dealing with the process of rational discussion rather than an isolated judgement of the correctness of mathematics. [Emphasise this point more throughout the essay, it is what distinguishes my view from Lakatos’.] Nonetheless, I will use it in describing a few concrete examples to illustrate the quasi-empirical view of mathematics.

The simplest sort of heuristic falsifier would be an arithmetical one, e.g. if our model implies 2+2=5 or some such. The Banach-Tarski paradox is a disputable heuristic falsifier, because our intuition does not really say anything about the sort of sets involved in it. The interaction between physics and mathematics suggests a more interesting hypothetical heuristic falsifier. Imagine that, for example, a new theory of physics was based on an alternative characterisation of the concept of number or space. For example, suppose that hyper-real numbers (the hyper-reals are an extension of the reals which allow for two hyper-reals to differ by an infinitesimal quantity, amongst other things) were used as the basis of a theory of physics that empirically proved itself useful. Or suppose, like in Jorge Luis Borges’ story Blue Tigers, that in the real world numbers started to behave oddly. [ reread story and put something more in here ]. In the former situation, although our definitions and theorems about the real numbers would remain intact, a great deal of interest would go towards developing the theory of hyper-real numbers. Our model of “number” would change from “real number” to “hyper-real number”. Or, suppose that some theory in physics actually managed to use the Banach-Tarski paradox to deduce a physical consequence which was falsified. The traditional view of mathematics still couldn’t decide between AC and the rejection of AC, because the error could still be in the physics and not the maths. The quasi-empirical view, however, would suggest that the error was in the way that mathematics is like a formal model. That is, that the real numbers (in a system with the axiom of choice) were not an adequate model of the numbers needed for that particular physical theory. [Rewrite this last bit.]

Quasi-empiricism can also throw some light on how mathematical conjectures are made. One of the Clay Mathematics Institute’s Millennium Prize Problems offers one million dollars for a proof that the Navier-Stokes equations always have a smooth solution. Looking at the Navier-Stokes equations [put them in?] it might seem unclear as to why mathematicians believe there is a solution and why they think it so important to prove that there is one. The reason is that mathematicians believe that the Navier-Stokes equations govern fluid flow. Since we believe that the relevant mathematical concepts and equations do indeed accurately model fluid flow, and since we believe that it can never happen that the universe would cease to exist because there is no solution to an equation, we must believe that these equations always have a solution. A counterexample to this conjecture would be a heuristic falsifier, whereas a proof of the conjecture would justify studying the Navier-Stokes equation.

[Say something about the effect of other intellectual disciplines on mathematics. For example, computer science is interested in graphs and networks, economics stimulates interest in stochastic calculus, etc.]

[Say something about how people desire unity in mathematics. This has a pragmatic function. Talk about how the interdeducibility of different mathematical systems means that my take on quasi-empiricism doesn’t mean that mathematics will fragment.]

[Talk about computer proofs. If an artificial intelligence came up with a human style proof would we believe it? What if 10 differently programmed such programs checked an unreadable long, but human style, proof over and agreed with it? Would we still be inclined to disbelieve it even though many proofs in maths are only checked by a few people? If maths is a process, we need not be so concerned with worrying about this sort of thing, which is not to say we should be totally gung-ho about it. There is a high probability of error in human proof (see Davis in New Directions). What about non-constructive proofs of the existence of a proof by formalising a theory within itself? Probabilistic proofs? Problem: probabilistically it is a 100% certainty that x is irrational but every number we can work with is irrational. Analogy between maths and software engineering – see de Millo, Lipton, Perlis & Tymoczko in New Directions.]

[Talk about how the vanishingly small number of propositions which are interesting compared to uninteresting ones undermines Chaitin’s argument for axiom growth in maths. Does the same thing apply to Godel’s theorem?]

[There are typically, not exceptionally, many different formalisations of the same concept in maths. See Thurston in New Directions. Incommensurability?]

[Objection, what I have described is largely social but we can imagine someone using mathematics privately. Talk about how we can come up with personal regulative ideals, regulate our own future behaviour – e.g. writing notes to oneself that nobody else could understand. Compare with computer software not defining its run time behaviour but regulating it.]

[Why is maths so successful? We can’t answer this question because to answer it is to assume that it is since we can imagine a world that any minute now will suddenly cease to be modellable. Integer sequence game, and Gardner’s update (see below). Wittgenstein quote.

Message from sci.math on integer sequence game:

Douglas Hofstadter, author of "Godel, Escher, Bach: An Eternal Golden Braid" wrote a column for Scientific American. In one of his columns he proposed a game. The object of the game was to get a group of people trying to guess what symbols the "dealer" had secretly inserted into, say, an 8x8 grid. The dealer could use any symbols he wanted, he could even make up 64 symbols at random if he wanted. The real genius was in the scoring.

You see, the dealer's maximum score came when he stumped all of the players *except one*. The minimum score came when he either stumped *all* the other players, or *none* of the other players. In other words, if the game was too easy, the dealer got a low score; if too hard, an equally low score. But if the pattern was difficult to see, but not impossible, then he/she got a high score.

I don't remember how the scoring worked - maybe somebody here can help - but it wouldn't be difficult to develop a similar scoring system.

For example - the players would start knowing nothing about the sequence - it could be natural numbers, integers, rationals, reals, complex, Gamma functions, whatever the dealer wanted. They start with a certain number of points - say, 50. Each time they ask for a hint, they lose a point. If they guess the sequence correctly, they get to keep their remaining points. If they guess incorrectly, they lose all their points. Their score for the round will be the number of points they have left, minus the average of the other players' points. (It seems obvious that some players will end up with a negative score, though I haven't checked.)

Then try basing the dealer's score on the sum of squares of the other player's scores. (Not their point counts, but their scores for the round, as described above.) If all of the other players scored the same number of points, then the dealer gets a low score - regardless of whether the problem was too easy or too hard.

That's one suggestion - it probably needs some serious tweaking to get it to work, and you might even want to write a short computer program to calculate the scores for you (to avoid slowing down the game).

But it sure beats the problem of some demented prick coming up with the "sequence" 1, 2, 3, 4, 5, 6, 7, 8, .... and then saying the next number is "(ln(17/pi)^(1/37))" :-)

This should be Martin Gardner’s card game Eleusis not Hofstadter.]

Philosophy of Maths seminar

thesamovar — Sat, 24 Feb 2007 15:35:27 +0000

I've been interested in the philosophy of maths for quite a long time. In the last year of my PhD I started writing a pair of seminars on the subject to be given at the Maths department at the University of Warwick. Unfortunately, writing up my thesis got in the way and I never got round to writing the second part.

The first part is largely finished though, and briefly covers the classical views on mathematical philosophy from the early half of the twentieth century (foundationalism), as well as a slightly more recent and radical alternative called quasi-empiricism, due to Imre Lakatos (a student of Karl Popper). I was planning to go on to talk about Wittgenstein and Putnam.

The second part was going to focus more on mathematical practice and how things might develop in the future. I was going to compare the broadly syntactical approach characterised by computer proof and projects like Mizar, and the broadly semantical approach advocated by people like William Thurston (whose attitude to fully worked out rigorous proofs is notorious).

Finally, I was going to talk about what advances in computer technology and algorithms might do to mathematics in the future. Roughly, I was going to say that proof assistants and guided computer proof will likely become a serious part of mathematics within the next, say, 30 years, and that this will change the nature of mathematical practice considerably. It will become possible to focus less on the mechanics of proof (which will be left to the computers), and more on the meaning and underlying ideas. This sounds like a trivial point, but the implications for the teaching and practice of mathematics are enormous. Almost all current mathematical training at present focuses on the mechanics of proof, and much of the expended effort in mathematical research is geared towards the production and checking of proofs.

You can download my notes in PDF format here. Any comments welcome.

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Essays

thesamovar — Tue, 24 Oct 2006 19:14:57 +0100

Below is a list of all my philosophy essays.

Critical review of Robert Nozick's "Anarchy, State and Utopia"

thesamovar — Wed, 18 Oct 2006 06:20:28 +0100

Robert Nozick's "Anarchy, State and Utopia" is undoubtedly a highly original and intelligent work. It has three major flaws though. Firstly, his idea of a minimal state requires not only that the state itself respects individual liberty to an unparalleled degree, but also that the population uniformly respects others' liberty to a high degree. Secondly, there is no discussion of power relations and the possibility of actually achieving a minimal state; the introduction of a minimal state today would not, and perhaps could not, have the effect Nozick desires, and there is little indication of what sequence of events could lead to a minimal state which would work. Thirdly, the "entitlement theory of justice" is inherently dualistic, a distribution of holdings is either just or unjust, but to defend it against the charge that a just distribution of holdings is impossible to achieve in practice, one would probably have to introduce a variable scale of justness, which would be antithetical to the spirit of the theory.

His entitlement theory of justice says, roughly, that if the initial distribution of holdings is just and every transaction that subsequently occurs is just, then the distribution of holdings will be just at all later times. A just transaction, again roughly, is one which is mutually and voluntarily (i.e. without physical coercion) agreed upon.

The minimal state is one that serves only to provide a monopoly police and military function. The police function is to protect its citizens against violence, coercion or fraud, and to enforce contracts. The military function is obviously to protect the state against outside aggression.

Is it possible for a society to exist whose basic structures contradict the propensities of the people who make up that society? More specifically, is it possible that a minimal state could coexist with a society composed of individuals who do not respect others' liberty? It seems unlikely that it could; there would be no pressure to maintain the minimality of the state if a majority wished to infringe on the liberty of some individuals, and so a minimal state would be unstable or prone to becoming a more than minimal state. The only other way a minimal state could be achieved in a stable way would be for an external power to impose it on an essentially unwilling population, a benevolent (libertarian) dictatorship. Surely this is not a state of affairs Nozick would consider desirable?

There are two major consequences of this requirement on a society with a minimal state. Firstly, it is not as permissive as Nozick would presumably like it to be, since it imposes respect for others' liberty on the populace. Secondly, it is as utopian, in the sense that it requires an enormous change in popular consciousness, as many of the socialist systems that Nozick criticises on these grounds. If we were able to achieve a society in which people did uniformly respect others' liberty to a high degree, without doubt an incredibly desirable state of affairs, then the institutions and legislation of a less than minimal state would either not infringe upon others' liberty (because the institutions and legislation wouldn't be used in that way), or those aspects that would infringe upon them would naturally fade away anyway (because there would be no support for such institutions or legislation).

This brings us on to the second point, the lack of a discussion on how a minimal state could be achieved consistently with the principles that motivate it (mainly liberty). The major stumbling block is probably the requirement that the population uniformly respect others' liberty, and the massive change in popular consciousness that this would require.

One of the major questions that has to be answered to determine the feasibility of achieving this is whether or not is possible, in general rather than in particular cases, for an individual who considers himself or herself superior to another to respect the other's liberty to the same degree as someone he or she considers an equal. For example, if the individual has more wealth, better social status, or whatever, can they respect the liberty of individuals with less wealth, social status, etc.?

If indeed general equality is the prerequisite for a uniform respect for others' liberty then it may be that Nozick's minimal state would, to function as intended, actually require a general equality (in wealth at the very least). This would, I suspect, be much at variance with Nozick's idea of a minimal state.

Finally, Nozick's entitlement theory of justice. There are three observations that undermine the usefulness of this theory of justice. Firstly, there will always be unjust transactions to deal with. Doing so is the purpose of the police, one of the two institutions in a minimal state. Secondly, not all unjust transactions will be detected, or even if they are it will not always be possible to provide compensation (for example if the criminal is not found). Thirdly, the entitlement theory of justice is "all or nothing". Either the entire distribution of holdings is just or it is not. It follows that a single unjust transaction that is not compensated undermines the justness of the entire distribution of holdings for all time. Moreover, this state of affairs is inevitable from the first two observations. So what are we to do with a theory of justice that inevitably says that the distribution of holdings is unjust?

The obvious answer is that, for example, a small theft does not really undermine the justness of the entire distribution of holdings. However, to fix the entitlement theory of justice so that this statement is meaningful one would probably have to introduce either a variable scale of justness, or perhaps introduce the idea of a local injustice in the distribution of holdings. The second attempt at a solution to the problem would not alone be sufficient though. Once a local injustice in the distribution of holdings is introduced, it could spread through the entire system reasonably quickly (because a just transaction with someone who has gained by an unjust transaction would spread the injustice to the other party in the transaction) until the entire distribution was unjust. So, at some point the concept of a sliding scale of justness would have to be introduced. What implications does this have for the theory?

Firstly, a sliding scale of justness is antithetical to the spirit of the original theory. The purpose of the original theory was to rule out moralistic statements like "it isn't fair that some are so rich whilst others are so poor". If the initial distribution of holdings and every subsequent transaction is just then nobody can complain. Once the possibility of a variable scale of justness is introduced, there is a possibility that injustice in the distribution of holdings can accumulate. The possibility that "it isn't fair that some are so rich whilst others are so poor" is now one that cannot be avoided a priori but has to be argued empirically (perhaps an impossible task). More importantly, it is not compatible with Nozick's distaste for "patterned" conceptions of the justness of distributions of holdings. Essentially, a patterned conception of justice in holdings is one that says "From each according to X, to each according to Y" where X and Y are some criteria (e.g. X could be "their ability" and Y could be "their need"). Nozick argues against them because, according to him, they necessarily involve a constant interference in the private lives of individuals. (This is simply stated rather than argued, but that's another matter.) However, unless he could produce an argument to demonstrate that injustice in distributions would not accumulate, then it seems that some sort of interference would be necessary to stop it.

So, in conclusion, there seem to be some deep contradictions inherent in "Anarchy, State, and Utopia". The entitlement theory of justice probably cannot be fixed without introducing some form of constant interference in individuals' private transactions and the minimal state cannot survive as intended without general equality.

Philosophy

thesamovar — Wed, 18 Oct 2006 05:41:22 +0100

I'm interested in epistemology in general, and in particular the foundations and philosophy of mathematics. My major interests at the moment are Wittgenstein, Pragmatism (e.g. William James) and quasi-empiricism as a philosophy of mathematics (e.g. Imre Lakatos). You can read the notes for a seminar I started writing on this subject here, and an unfinished essay here.

I might write up a short, pithy explanation of my views on all of these things at some point, but for the time being I only have a long and rather difficult (not to mention unfinished) essay. The title is 'Epistemology without Truth' because I believe it is possible to have a sensible epistemological theory free from the notion of truth, which I find to be incoherent, but which doesn't fall into the standard errors of overly relativistic thinking.

You can read the full list of philosophy essays here.

Epistemology without truth

thesamovar — Wed, 18 Oct 2006 05:37:18 +0100

The following essay is a work in progress. The [] symbols break up logically coherent sections which I have been moving around when writing this essay. Comments in brackets [like so] are reminders to me to think about or change something. I haven't yet written the last section, so for the moment it's just a tantalising header.

I'm always interested in discussing this sort of thing so email me if you have any comments or questions.

I wish to outline an epistemological theory free from the confusing notion of truth. Firstly, I’ll explain why I think we need to do away with an apparently useful notion. In summary, the reasons for this are the unattainability of absolute truth, the mutability of words and scientific concepts in particular, and the subjectivity of the meanings of many words. Secondly, I’ll discuss why I think the concept of truth has been useful despite being flawed. In summary this is because it is an ideal which inspires concrete conceptions in particular circumstances, and because the idea of truth carries with it the idea of logic which has a use separate from the notion of truth. Thirdly, I’ll introduce a new approach to epistemology based around a simple schema relating action, judgement and theory. The result will be somewhat similar to what Wittgenstein called language games. This alternative approach is very similar to pragmatism, but without the pragmatic conception of truth. Human judgement, in this new approach, is unavoidable, but I will argue that it always was and the absolute notion of truth attempts to hide this fact. I’ll illustrate this new approach with reference to mathematics and law representing two extremes. Finally, I’ll talk about the social and institutional analysis of rational frameworks, with law as an exemplar, and some consequences for science.

Thesis

The concept of absolute truth is flawed

The first argument against the notion of absolute truth is that it is unattainable. This shouldn’t be a contentious point, so I’ll limit myself to illustrating the point by mentioning Popper’s view of science as a process in which theories are never shown to be true but merely as yet unfalsified. For the notion of absolute truth to be useful to us, it must have a human function not derived from the absolute truth values of particular propositions. For Popper, a believer in absolute truth, the reason for this was that absolute truth is a regulative ideal (of which more in the next section).

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The meaning of words changes as our understanding of the world develops, and our theories change. For example, consider the word “inside”. We learn how to use this word by extrapolating from exemplars in the domain of “medium sized objects” (roughly speaking, objects whose size is somewhere between grains of sand and large buildings). As our education progresses, we extend the use of this word to both smaller and larger objects. The extension of the concept beyond the domain in which we learnt and defined it proceeds by analogy. For example, telescopes and microscopes provide an analogy between the very large (but distant) or the very small, and the medium sized. Each allows us to perceive something normally imperceptible by transforming an image into the medium sized domain. The means of transformation, that is, the way in which the analogy works, then becomes part of the extended meaning of the word. This is not to say that the analogy is arbitrary. In this case, we perceive directly that magnification from medium sized objects to medium sized objects (e.g. 2x magnification) preserves the property “inside” and so we extrapolate and assume that magnification generally preserves the property “inside”. In other words, the assumptions that (1) the concept “inside” can be extended, that is the assumption of universal scope, and (2) that magnification leaves the property “inside” invariant, together define the extension of the concept “inside”.

Unfortunately, at very small and very large scales, problems start to appear. At the atomic scale, clear boundaries cease to exist and we have to either reconcile ourselves to a fuzzy conception of “inside”, make an arbitrary decision about what constitutes a boundary, or simply admit that the concept of “inside” only makes sense at certain scales. Likewise, at very large scales, assuming that general relativity is correct, the possibility arises that an object “inside” a boundary could move “outside” the boundary without passing through the boundary. At the very small scale, we cease to be able to look at a situation and simply say whether something is inside or outside. Almost all similarities to our everyday experience have gone. At the very large scale, properties which make the concept “inside” useful cease to hold. In the case of the concept “inside” we can, or we think we can, understand why and when the concept does or doesn’t work in terms of an alternative conceptual framework. We can understand that the concept of “inside” makes sense at medium scale because objects are approximately continuous at this scale, whereas at the atomic scale this ceases to be the case. Similarly, we can understand that the fact that the universe has a locally Euclidean topology but a potentially globally non-Euclidean one explains why at large scales the concept of “inside” might cease to function correctly.

This example is a very simple one, we find it easy to explain why, when and how the concept breaks down by exploiting a different conceptual framework. This is not always possible. The concept of “truth” is a case in point. For a start, the concept of truth contains within it all the difficulties of all other concepts. To analyse the truth of the sentence “A is inside B” requires at the very least the above analysis of the concept “inside”. Importantly, we not only need the analysis above to decide whether the sentence is true for some particular A and B, but even to make sense of the meaning of the sentence. We can insist that we not use the word “inside” in sentences which we wish to discuss the truth of, but is there a language which isn’t subject to revision as we test the limits of our concepts? Could we know it if we had got it? There is a logical possibility that no sentence in any current or future language will not be subject to revision of the meanings of the terms within it. If no word, and hence no sentence, has an unrevisable meaning, can any sentence have a truth value?

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A few examples of the changes in our understanding of fundamental concepts should illustrate that this is not a problem that can be contained. Every concept is potentially subject to change. [ Discuss non-Euclidean geometry, pre-quantum atomic theory (e.g. “inside” example above), quantum theory, and some things from GR like reverse causation and the phenomenon of seeing different numbers of particles in different frames. ]

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We’re all aware that a phrase in one language cannot necessarily be translated perfectly into another. In a way this simple observation alone undermines the concept of truth. Translation is usually understood as a relationship between two sentences in two different languages. Statement X in language L translates to statement X’ in language L’. However, two people can also understand a single sentence in different ways. A good example might be “God exists” which will be understood differently by people of different religions, and even differently by two people of the same religion (compare the conception of God of an educated, scientifically literate modern Christian to the conception of God of someone who believes in the literal truth of the Bible).

We can say “they understand the sentence in different ways” – but what does this amount to? It might be tempting to say that they translate the sentence in the public language into a universal “language of thought” and that they have translated it differently. Such an explanation would explain how misunderstandings arise and would suggest how we should resolve them (by trying to make our public languages similar in structure to the “language of thought”). Unfortunately, it isn’t clear what a “language of thought” would be like and whether or not such a thing makes any sense.

So, what does it mean to say whether or not a statement is true or not? If two people can understand the same sentence in different ways, the truth or otherwise of the statement can depend on who is reading or hearing it. One reaction to this is to say that since there is no way to determine the truth of a sentence anyway, it hardly matters from the point of view of truth that people mean different things by the same sentence. Another is to say that although some statements (like “God exists”) are understood differently by different people, some are not (like “that digital display says ‘13.2’”), and that we should endeavour to only talk of the truth of statements phrased in such unambiguous language. Both views excessively limit our use of language. The former allows for the full richness of language but rules out any possibility of objectivity and rational debate. The latter restricts our language a great deal but allows for an objective and rational debate within that limited scope. It does us no good to pretend that statements like “There is no rhinoceros in this room” are no different from statements like “She is pretty” or “He is smiling”. Equally, it does us no good to pretend that statements like the latter serve no purpose.

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So, the notion of absolute truth is in principle unattainable and since meaning and understanding is mutable and subjective, it isn’t clear that absolute truth could itself have a meaning. I think these considerations are enough to undermine our faith in the notion. In a sense, this shouldn’t be surprising. There’s no reason to think that a phenomena like language, which has emerged haphazardly over millions of years through chance and natural selection, which is necessarily learnt finitely and inductively, should be able to attain the Absolute.

Antithesis

The concept of truth is often useful

In his paper Truth, Rationality and the Growth of Knowledge, Popper used a modification of Tarski’s theory of truth, and his own notion of closeness to truth, as a regulative ideal in defining science. He defined closeness to truth in the following way: A theory S is closer to the truth than a theory T if the set of true consequences of S is a superset of the set of true consequences of T and the set of false consequences of S is a subset of the set of false consequences of T. This defines a partial order on theories. However, he recognises the ideal nature of his definition: “… we have no criterion of truth, but are nevertheless guided by the idea of truth as a regulative principle (as Kant or Peirce might have said); and that … there are criteria of progress towards the truth…”. He later goes on to say “It is only the idea of truth which allows us to speak sensibly of mistakes and of rational criticism, and which makes rational discussion possible… Thus the very idea of error – and of fallibility – involves the idea of an objective truth as the standard of which we may fall short. (It is in this sense that the idea of truth is a regulative idea.)”. And, although he has said that there are “criteria of progress towards the truth” he later goes on to say that “… approximation to truth … has the same objective character and the same ideal or regulative character as the idea of objective or absolute truth.” In other words, there are no such criteria, as he recognises when he says “It always remains possible, of course, that we shall make mistakes in our relative appraisal of two theories, and the appraisal will often be a controversial matter.” He cannot directly deduce methodological rules from these ideal definitions, but it suggests (regulates) them. This can be more clearly seen in Lakatos’ definition of what it means to falsify a theory. For Lakatos a theory T is falsified if there is another theory, S, which predicts (explains) all the corroborated content of T and correctly predicts at least one fact that T fails to predict. Since closeness to truth is only a partial order on theories, there remains a logical possibility that two theories may not be comparable (if S correctly explains a fact that T fails to and T correctly explains a fact that S fails to). Kuhn and Feyerabend showed that this sort of thing happens all the time. This undermines the idea that Popperian or Lakatosian science are complete philosophical theories of science, but doesn’t undermine their status as regulative ideals.

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Pragmatically speaking, the usefulness of the notion of truth is that it makes certain sorts of reasoning tractable. If we agree that every proposition is either true or false we can break a reasoning task into components which can be considered independently. Without the concept of truth, it would seem impossible to break a complex piece of reasoning into simple independent components. However, our estimations of the truth values of all the intermediate propositions involved in a chain of reasoning are always subject to revision. The usefulness of truth then is not in fact its absolute quality, because we can never know an unrevisable truth, but the system as a whole, the way it connects propositions. In other words, the usefulness of truth is due to the framework it provides, not the absolute applicability to propositions.

Synthesis

The action-judgement-theory schema

An appealing way out of these difficulties is Wittgenstein’s notion of “language game”. Language doesn’t exist in a vacuum, its meaning is defined by the way we use it. Here “the way we use it” means the way we use it within the confines of language, and how what we physically do relates to what we say. It follows from this that questions about the meaning of sentences cannot be separated from questions about the relationship between words and actions. Two people are playing the same language game if this relationship is the same in some sense. The concept of a “language game” is itself fluid and doesn’t need to be pinned down precisely to be useful, especially as it is a largely descriptive concept.

As long as we relate to statements relative to a particular game or context in which the statements are used, we don’t get into the difficulties mentioned above. To be sure, the concept of language games doesn’t solve all our difficulties – for example there is always the problem of comparison of language games – but the recognition of the validity and usefulness of the concept of language games enables us to escape from bottles we were previously stuck in.

[The purpose of many, perhaps all, theories (philosophical, scientific or whatever) is to diminish the role of judgement in our lives. We construct theories because we are aware of our own inadequacy in judgement. There couldn’t be a “perfectly rational” being because the process of rationality is all about compensating for the inadequacy in our judgement.

However, we can never get to the point where we need no judgement at all. At the very least, we need enough judgement to decide whether or not to accept a theory. It would be irrational to believe in a theory without applying our own judgement as to whether or not we ought to believe in it. Thus, no theory can be a final substitute for judgement.]

– [rather than including this here, could elaborate on it in much more detail later.]

The construction of explicit language games and formal or partly formal languages is an essential part of our rational behaviour, but we shouldn’t get carried away. A formal language allows two like minded people to communicate more efficaciously, and allows us to refine our judgements, but it doesn’t determine them. In judging a statement of a formal language in a concrete situation we have to judge the applicability of that formal language to that situation. Similarly for any sort of language game. Taking two balls of clay and rolling them into one ball is not a demonstration that 1+1=1, but it shows how we must be as careful relating a formal language to the world as we are about working within that formal language.

These examples bring us to the action-judgement-theory schema. In this schema, there are two domains to consider, the action domain and the theory domain. The action domain consists of the things we actually do, such as kicking a ball, or writing a sentence. The theory domain consists of the products of rule-based, or partially rule-based, intellectual endeavour; books, scribbles on paper or even mental manipulation of symbols according to rules (say, doing a sum in your head). Judgement is that part of thinking which connects these two domains. Considered on its own, the theory domain is meaningless. A book, without a person to understand it, is just a particular physical object like a stone. A true mathematical statement in a deductive system no human has ever used is just a collection of marks on paper. In the extreme, any physical symbol can be interpreted to mean anything by someone so inclined. Another example is Searle’s “Chinese Room”. An individual in a closed room is given a rule book for processing sequences of Chinese characters which are fed in from outside. To someone outside, it appears as if someone inside understands Chinese and is conversing with them. However, the individual inside doesn’t know Chinese and is just following the rules in the book. The activity of the individual inside the box is essentially meaningless. This belongs to the theory domain. The total behaviour of the box and the outside world however, has a meaning. [Do I like this example?] Clearly, the theory domain on its own is meaningless and has to be connected with human behaviour, the action domain, to acquire its meaning.

We shouldn’t underestimate the significance of judgement in this schema. As mentioned above, much human intellectual activity is directed to reducing the amount of judgement required, but it can never be entirely eliminated. To clarify, judgement is not only the application of a theory, e.g. the learning of a technique involving written theory, but more importantly the thought processes involved in choosing to use the technique, and the ability to work within the theory when there are no explicit rules guiding us.

Clearly there is a strong pragmatic element to this thought. Ultimately, the justification of any theoretical activity must come from the action domain, but this route can be very circuitous. Most of the objects of mathematical study, for example, can be understood in this way, despite appearing to be purely theoretical objects. Typically in mathematics, the genesis and history of a concepts is an explanation of its meaning in these terms.

A common criticism of pragmatism comes from the equation of ‘truth’ and ‘usefulness’. James, in Lecture II of Pragmatism, anticipates this criticism: “I am well aware how odd it must seem to some of you to hear me say that an idea is ‘true’ so long as to believe it is profitable to our lives.” A critic might say, it would have been profitable for a Nazi to believe that Jews were subhuman but that doesn’t make it true. One can think of less political examples. For example, it would have been profitable for Newton to believe his laws to be true, but that didn’t make them so as we can see clearly now. Despite having some force, there is a mistake in this criticism which is fundamental. Understanding why this criticism is mistaken, or at least misguided, will help me to explain my own schema. In this schema, and in pragmatism, a statement cannot be considered on its own, but only as part of a whole web of statements. When we apply a theory, we must have used our judgement, and our judgement applies not to individual, isolated statements, but to the system of statements in which a particular one occurs. We judge the usefulness of the statement 2+2=4 not on its own merits as a statement, but as part of the system of mathematics. We judge it favourably because we judge the whole system favourably, and we judge the whole system favourably because of its success in application. (This is a slightly simplified account of our judgement of mathematics and mathematical propositions, but it should suggest the general point.) Now, the criticism is valuable, insofar as it shows that attempting to define truth pragmatically but in the absolutist sense (a proposition must be true or false) is doomed to failure. Rather than talking about truth at all, we should talk about statements occurring within systems of statements, and judgements of those systems of statements. Truth can be defined relative to a theoretical system, but the choice to apply, use and develop any particular system is a pragmatic one. A statement which is true relative to a theoretical system that has fallen into disfavour doesn’t become false, but the usefulness of the sense in which the statement is true fades. This gives us an alternative way of thinking about systems of thought which doesn’t involve using the word truth in a dubious absolute sense.

In the most general setting, we can consider a particular theory activity to be defined by the restrictions it places on what can be said. These restrictions can be explicit (written or learned) or more nebulous (learning by observation for example). The most explicit example would be something like formal mathematical proof, whereby a statement in a proof has to follow from the previous statements by applying one of a small collection of explicitly stated rules of deduction. The least explicit example would be something like philosophy, where there are very few explicit restrictions on what you can say, but there are obviously arguments which no philosopher would accept. The restrictions, such as they are, are learnt in the process of learning philosophy, by example, analogy, and so forth.

We can identify at least two significant parts of the action domain, the individual and the social. As well as individual actions, there are coordinated or social actions. Sometimes, the development of a theory can be motivated purely by the individual action domain. We can imagine a very long lived individual on a desert island developing some theoretical concepts for his own use. Sometimes it makes more sense to consider a theory to be motivated by the need for coordination of individual action. For example, the development of number for finance (e.g. currency, taxation).

Mathematics

Using this schema, we can study the history of mathematical concepts. To go right back, the concepts of number and basic Euclidean geometry serve a fairly direct pragmatic function (in both individual and social action domains). The introduction of calculus was motivated by physics, and so serves a pragmatic function. What about the introduction of, for example, complex numbers? It’s harder to see how this serves any pragmatic function, as no direct application was found for complex numbers for many years. In a way, the introduction of complex numbers exemplifies the unique aspect of pure as opposed to applied mathematics. One of the earliest instances of complex numbers is in the solution of cubic equations. It turns out that there is a very simple method for solving cubic equations, but it involves using complex numbers (even if the final solution happens to be real). For a time, complex numbers were restricted to being used only in intermediate steps, they were considered a short hand, a purely algorithmic concept. However, eventually they became accepted in their own right, and have subsequently found both direct and indirect applications. This can be seen as an example of the introduction of new theory to limit the need for judgement. You can always find just the real solutions to cubic equations without using complex numbers, but that requires a certain level of judgement. You can imagine rewriting an argument to find solutions which would typically be made in terms of complex numbers but being careful to rewrite every step to use real numbers only. It would be possible to have come up with this modified argument without using complex numbers, but it would require more judgement than just introducing and using complex numbers. [Todo: check my history here. Also, there are an enormous number of other examples that come to mind which illustrate this point. It might be interesting to go into much more depth about this with plenty of examples.] In general, we often find that a development in mathematics will be motivated by an intratheoretical (it helps to solve problems within mathematics, e.g. complex numbers), intertheoretical (it helps to solve problems in another theory domain, e.g. calculus), or directly pragmatic reason. In terms of the schema above, we have the following situation: a relatively formal theory domain whose development is guided by three major sorts of judgement – intratheoretical use or potential use (which could conceivably be identified with mathematicians aesthetic sensibility), intertheoretical use or potential use, and direct pragmatic use or potential use.

As well as applying the schema to the development of mathematics, we can also apply it to mathematical practice. Mathematics is sometimes seen as fully formalised, an actual mathematical proof is sometimes imagined to be like an ideal mathematical proof - each step justified, every case considered and so forth. However, real mathematical proofs leave a lot to mathematical judgement. A typical phrase in a mathematical proof is “The other case follows similarly.” This could mean, I’ve checked the other case and it works in the same way, or it could mean, it is completely obvious that the other case is the same and you can just see what changes would need to be made and that it would work without actually doing it. This is very much a case of a highly trained judgement, students often use phrases like this when it isn’t true. So mathematics cannot be considered to be just the written body of work, but also the “living knowledge”, the trained judgement of mathematicians. I’ll come back to this point in a later section.

Ideal Concepts

Earlier, I mentioned Popper’s notion of truth as a regulative ideal, and now I want to discuss ideal concepts in general using this schema. An ideal concept is one which can be referred to explicitly, but for which there are no explicit rules for usage. The general situation is as follows: we have a concept which we feel we can always apply in concrete situations, but which we are unable to account for in general. An ideal concept is thus a model of our own judgement. For example, Lakatos’ Proofs and Refutations can be seen as a development of the ideal concept ‘polyhedra’. Another example from mathematics is the idea of a manifold. There is no general definition of a manifold, but we can identify a great number of types of manifold: for example, there are smooth manifolds, topological manifolds, piecewise linear manifolds, Riemannian manifolds, Lorentzian manifolds, symplectic manifolds, cone manifolds, (G,X)-manifolds (the most general form of manifold which includes many of the others listed here for particular values of G and X), complex manifolds, orbifolds, manifolds with and without boundaries, etc. Following Wittgenstein, the different concrete instantiations of an ideal concept have ‘family resemblances’. In developing a concrete instantiation of an ideal concept, we try and identify and model some aspect of our own judgement. For example, if I want to come up with a theory of democracy, I look at my judgements about which countries are democratic and which aren’t, and try to find general criteria which they must all satisfy. For example, I’ll probably think of voting at first, but it’s conceivable I could imagine a country in which there was no voting but it still seemed to be governed by the people in some sense (possibly a small community of like minded people). Similarly, it’s easy to imagine a country in which there is voting but it doesn’t seem to be a democracy. For example, the elections might not be free, the media might not be free, and a whole host of other possibilities that are impossible to enumerate but which would, if we were to see them, invalidate a country’s claim to being democratic. Voting will then be one of the ‘family resemblances’ amongst concrete examples of democracies. Ideal concepts are important because they are a very significant way in which we talk about our own judgements.

Theory development and the role of judgement

As mentioned above, a large part of theory development is intended to reduce the need for judgement. This can be understood in both the individual and social action domains. First of all, reducing the need for judgement makes certain sorts of analysis tractable in ways that they previously weren’t. Suppose we can say ‘A is B’, ‘All Bs are C’ then we can say ‘A is C’. Suppose that the judgement involved in directly asserting ‘A is C’ is very complex, but that in asserting ‘A is B’ is very simple. Then the theory ‘All Bs are C’ enables us to reduce the complexity and difficulty of our judgements. Secondly, reducing the need for judgement has a social function. Sometimes certain sorts of judgement require a great deal of training to be made effectively. The social cost of providing that training might be too high to be feasible, and so if we can replace the need for that sort of judgement by much simpler ones, we create new possibilities for coordinated action.

There are benefits and problems associated with this sort of thing. In his ‘Wealth of Nations’ Adam Smith discusses something like this. He talks about the economic benefits of production lines, but also warns that this could lead to the degradation of the humanity of the workers involved. Similarly, whenever we reduce the need for certain sorts of judgement, we are likely to also reduce the number of people capable of that sort of judgement, leading to a general cultural impoverishment. However, this is not a necessary consequence of the introduction of new theories which reduce the need for certain forms of judgement – it can be a liberating experience, allowing us to concentrate on what is important. The introduction of calculators into schools has positive as well as negative aspects. They free children from excessive focus on mundane calculations, but over-reliance can lead to an impoverishment of their general understanding of number. In his Democracy in America, Alexis de Tocqueville contrasts the approach of democratic Americans, abstraction and generality, with the British aristocratic approach, refinement and judgement. [Could go into this in more detail.]

For a long time, the trend in science has been towards highly reductionist theories. Presumably inspired by the success of such reductionist fields as physics, other fledgling sciences have attempted to do the same, often with little success. It may be that in the future we will need to develop theories in which judgement plays a more central role and that we will need to confront the problem of how to appraise and develop theories in which judgement, which is largely nebulous and ineffable, plays a much bigger part. In particular, it may be that social or psychological sciences cannot be usefully studied from a reductionist viewpoint. The schema above doesn’t rule out the possibility of a theoretical activity relying heavily on judgement. Indeed, the discussion of mathematics above highlights the way that even in the most objective and explicit subjects judgement is in practice crucial and central.

The problem is that judgement is so nebulous and ineffable that it’s difficult to go beyond a recognition of this possibility. What are the most effective ways of appraising and developing an extremely judgement heavy theoretical activity? Part of the problem in answering this question is that it seems we have few examples to go on. However, this might partly be due to the fact that fields such as mathematics have traditionally not been seen in terms of the sorts of judgement used. Perhaps by studying the role of judgement in very successful theory activities such as mathematics and physics, rather than by studying the form of mathematical and physical knowledge, we can better develop new theory in fields which require more judgement.

One thing immediately jumps out at me. In a mathematics exam, a large part of what is being tested is mathematical judgement (the other part is puzzle solving ability). By the end of a mathematics degree, most successful students will typically agree on a wide class of mathematical judgements. In other words, by the end of a mathematics degree, judgements themselves have become highly coordinated. The possibility of coordinated judgements is what makes complex mathematical arguments possible. If you can’t rely on other mathematicians being able to agree on the validity of certain forms of argument, you have to rely on very explicit arguments. This would make most contemporary mathematics papers impossible to digest. This suggests that we should be trying to develop ways of achieving coordinated judgement, and part of this might be in developing ways of appraising judgements.

[Talk about how mathematics papers could be improved by recognising this aspect. Talk about how teaching this form of judgement more explicitly, e.g. testing students ability to recognise false proofs.]

A possibility is that there is no way of appraising certain sorts of judgement. In this case, we should probably be paying close attention to the way people develop their ability to make judgements. More than anything else, our ability to make judgements depends on how we have been taught rather than just what we have been taught. [True?] This suggests a much more substantial role for pedagogy in epistemology.

Analysis

Social and institutional