Epistemology without truth

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.


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).


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?


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. ]


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.


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.


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.


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.


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).


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.


Social and institutional

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