There are three closely related — but in my opinion distinct — concepts that ought to be closely scrutinized before discussing metaphor proper. (The post is long but I hope you will bear with me: making the points I wanted to make took many more words than I had anticipated.)

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Truth is the most difficult topic to do any justice to, especially if we start with the supposition that no truth is self-evident. Truth is a topic that has been flogged to death in academic philosophy, but it is far too important to surrender to philosophers. At the outset, let me just say that I subscribe to the view that truth is an attribute of *statements* about the world, and not of the world itself. Rainfall is a process in the world — it is neither true nor false. It just *is*. But a human statement about rain, such as “It is raining” can be either true or false (or undetermined, if you are in a windowless room). Truth is an aspect of communication. (Can animals lie? I’m sure they can intentionally mislead, but I don’t know if we should call this lying.)

How do we assess the truth of a statement? There is no foolproof formula or algorithm. The truth of statements about directly observable events is determined by corroboration. If you say it is raining, then I can go check. End of story. (At least, for most people. Ludwig Wittgenstein, for instance, refused to acknowledge that there was in fact *no* unicorn in the room, much to Bertrand Russell’s exasperation.) Most people trust their senses, especially when they receive multiple corroborations. This is usually adequate evidence to convince themselves that what their senses throw at them is not a private hallucination. Madness, after all, is a minority of one. (Is the world a collective hallucination then? This is a much stranger question, and one we’d better ignore for now.)

As soon as we get to unobservable events, the trouble begins. If you tell me that it rained in Boston, Massachusetts on August 13th, 1947 at 2:53 pm, I will have to use indirect methods to assess the truth of your statement. I might, say, consult newspaper archives looking for weather reports. But I will then need to place some faith in the newspaper’s accuracy, transforming the question of the truth of your statement about rain into an investigation of the trustworthiness of archived newspapers. We often decide on the truth of a statement based on the trustworthiness of the source. If you know someone who is a pathological liar, you take what s/he says with a grain of salt, until you can confirm what they say. Conversely, if you know a scrupulously honest person, you may believe whatever s/he says without seeking confirmation. Note that the liar could be telling the truth, and Honest Abe could be lying or simply mistaken. Without the ability to confirm statements using our own senses, we are at the mercy of other people.

The truth of a statement is often established using an appeal to authority. This is the typical technique of religious fundamentalists. But scriptural literalists are not the only people who appeal to a higher authority. Everyone does this, and it starts early. Children often ask “Who said so?” We implicitly evaluate *what* is said by finding out *who* originally said it. The value of a statement is displaced and relocated in the personality of its first proponent. Popularizers of science frequently refer to the statements of famous scientists. These are the ‘facts’ that Science — usually a disembodied entity — has ‘shown’. One seldom hears about the conceptual or theoretical framework within which the statement makes sense. The authority of a particular scientist may be tested via the systematized extrapolations of confirmation-via-the-senses that we call experiments, but very few nonscientists engage in this kind of activity. Most people are unable or unwilling to confirm statements by scientists and other public authority figures. Society places trust in some people (for reasons that are far from obvious) and individuals often just* inherit *this trust, even if they have the resources to test it themselves. I imagine most people assume that any statement being widely touted as the Truth would be subject to minute examination by persons more qualified and motivated than themselves. Alas…

Other ‘authorities’ we frequently appeal to: parents, teachers, politicians, philosophers, social ‘scientists’, priests, medical doctors, astrologers, and even journalists. There is also depersonalized tradition –”It’s true because this is what we’ve always believed”. Another authority is aesthetics (“Beauty is truth, truth beauty” — a particularly pernicious obscurer of truth, even, or perhaps especially, for scientists. Perhaps more on this in a future post.) But the most mischievous authority we appeal to is ‘common sense’. What on earth is it? It has something to do with reasonableness, and rationality, and sensitivity to ‘evidence’, but beyond words that are themselves vague I can say very little.

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I can, however, say something about validity, which is related to logic and therefore by association, to the popular perception of rationality. Validity — for the purposes of this blog at least — means logical consistency. We often conflate validity with truth. Many true statements also happen to be valid, in that they are consistent with other truths, and can be related to them using the algorithmic processes of logic. (We can include formal mathematics in the term ‘logic’ here, although there are those who might argue that logic should be seen as a subset of mathematics, not the other way round.)

Consider the sort of ‘truth’ arrived at by syllogisms such as the following:

Major premise: All humans are mortal.

Minor premise: All Greeks are human.

Conclusion: All Greeks are mortal.

The conclusion is valid because logic has been applied correctly, and it is also true. Why is it *also* true? Because the major and minor premises are both true *to begin with*. This is not always the case. Consider:

Major premise: All ducks have feathers.

Minor premise: All basketball players are ducks.

Conclusion: All basketball players have feathers.

It is important to recognize that the conclusion is *valid.* Logic has not been violated. But I hope everyone can agree that the conclusion lacks truthiness. The minor premise is *absurd*. I’ve picked a silly example, but you can easily imagine that things get murky very quickly if the premises sound sophisticated, plausible and/or ‘reasonable’. Logic is an internally consistent system that, if used correctly, will always give you valid results. Like a computer program, it’s Garbage In Garbage Out. (Logically *consistent* garbage, of course.) The truth lies *elsewhere*.

Wittgenstein asserted that statements that can be deduced by logical deduction are tautological — empty of meaning. Meaning and truth lie in what we put into the logic machine. We can put true statements (arrived at using the vague rules of thumb hinted at above) into a logic machine and crank out new true statements. I use the machine metaphor consciously: it is not very hard to get a computer to perform logical deductions, because it all boils down to rule-based symbol manipulation. Douglas Hoftstadter demonstrates this very vividly in *Gödel, Escher, Bach* (a strange, entertaining book whose fundamental argument eludes me). Doing a derivation in mathematics or physics resembles this algorithmic process. It’s all symbol-manipulation — moving around x’s and y’s using the laws of algebra and calculus (with much assistance from intuition and clever tricks, without which we would get nowhere). The meaning of a mathematical symbol lies *elsewhere*. The variables in formulae must (eventually) be mapped to observables in the world — this fundamental act of naming and associating is not done *within* mathematics or logic. Variable x can take on any value you give it. What *value* you give it depends on* the problem at hand*. (Even if we somehow had access to all true axioms and a sufficiently powerful logical system, we would still face a problem. Gödel used formal mathematics/logic to subvert itself, yielding his notorious incompleteness theorems. One of the interpretations of the first theorem is that there will always exist statements expressed in a mathematical system whose validity cannot be tested, *even in principle*, however powerful and consistent the system happens to be. Hofstadter uses an effective metaphor. Imagine that a mathematical system is like a phonograph. Playing a record establishes its validity. There will always be one record that carries a frequency that will destroy the player. You can try building a new, stronger phonograph, but you will always be able to make a new record that can destroy it.)

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And now let us cease this academic chatter about truth and validity, and come to *the problem at hand*. The third leg of our tripod is *usefulness.* Statements (and systems of statements) are not merely true or valid. They can also be useful. Newton’s laws allow us to make precise predictions about the movements of bodies both terrestrial and heavenly. We are offered “better living through chemistry”. And quantum mechanics, for all its alleged conceptual incoherence, is the fertile soil from which spring all the wonders of the electronic age. Clearly there are statements that have *power*. Note that I am not concerned with subjective value here. You may hate iPods, railway trains, or vitamin C tablets, but you should be able to acknowledge the efficacy of chemistry, physics or engineering as *tools *for achieving the goals of its practitioners.

You might argue that the power of science comes from its truth. But we must also admit to ourselves that there are powerful, world-changing statements that are not true, or whose truth has not yet been definitively assessed. Questionable beliefs and superstitions have power in that they influence the way people behave. These effects may not be as well understood as phenomena involving inanimate matter, but people regularly find ways to deploy them reliably. Think of proselytizers, PR companies, or politicians.

Many of the statements born out of religious and spiritual tradition do not hold up to scientific standards of truth and validity, but even scientists are capable of recognizing the power of allegedly false beliefs in helping people cope with pain, anxiety, and poor health — and also in spreading hatred, conflict and ignorance. But even truth and validity do not always co-occur. The Schrödinger equation is true and useful, but it cannot be derived from other more “fundamental” principles. (Heuristic derivations are used as didactic aids.) Physics and chemistry are littered with examples of ad hoc rules that do not simply flow logically from first principles. They may be consistent with other rules, but their validity was not the basis for their acceptance into the cannon of true theories. Validity is often discovered after the fact of a true discovery. This was the case with Newton’s calculus. Mathematicians in the 19th century lamented the fact that calculus was on shaky foundations (fluxions, anyone?), and proceeded to place calculus on more firm, rigorous foundations. I have often wondered whether the “foundation” metaphor is even appropriate here, given that calculus was already being *used* to great effect despite the untrustworthiness of its moorings, since the 17 century. (I intend to return to the metaphor of foundations and fundamentals at a later date.) Richard Feynman described mathematics as coming in two varieties: Greek and Babylonian. The Greeks were concerned with deriving all truths from a small set of self-evident axioms. The practical Babylonians, on the other hand, used intuition and heuristics, working out new truths from other truths that happened to be adjacent to the problem at hand. (I recommend reading the full Feynman quote on p. 46 or watching the lecture.)

I am not trying to knock rigor, however. The pursuit of rigor yields many discoveries that are interesting in their own right. And mathematical ideas that are valid and well-formed can sit around for decades before someone discovers a use for them. This was the case with non-Euclidean geometry, which was born of an attempt to prove (or *validate*) Euclid’s fifth postulate using the other four. The method known as proof by contradiction was employed, but no contradiction was forthcoming, so seemingly unearthly geometries — in which parallel lines intersected well before infinity! — lurked in mathematicians’ closets. Such geometries were dusted off and dragged into the daylight when Einstein announced to the world that spacetime is *curved*.

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Usefulness, truth and validity can be used in concert to get us out of philosophical black holes. You might agonize over the question: “If logic establishes truth, then what established the truth of logic?” Worse, you might arrive at some kind of sophomoric nihilism, based on nothing more than the discovery that contrary to hope or expectation, truths are rarely well-founded or absolute. But if we remind ourselves that logic persists in human society because it serves us well, then questions about the validity of validity-establishing systems or the truth of truth-discovering systems lose their fearful circularity. They are still circular — and going around in circles may not be as pointless as it first appears — but the discomfort is gone. With these three ways of measuring, we can perhaps resolve the apparent dichotomy between theory and application. Truth and validity have their uses. And perhaps usefulness is a truth of its own. Each aspect supports the other, but they can and do exist independently of each other.

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Truth, validity and usefulness can also be deployed in less-than-admirable ways. In a debate, you might complain that a truth held axiomatically is ‘invalid’, or that a valid argument is ‘useless’. And all this can be done even if your opponent has stated clearly that he means to establish one thing — truth, validity or usefulness — and not all three simultaneously. This is the debating equivalent of the three-card monte. Point-scoring debates can evolve into genuine opportunities for learning and progress if we cooperate with our partner (no longer an opponent) by accepting premises provisionally, recapitulating arguments, suggesting truth-testing rubrics, or imagining the uses of ideas or techniques. A shared goal can make communication much more interesting; the sharp tools of science and logic can then be made subordinate to a particular orientation, rather than simply being used to shred *particular* statements or churn out valid-but-useless ones.

Usefulness can breathe life into truth and validity. Truth can shine a light on the workings of power, and confer meaning to validity. And validity gives us a way to be cautious about alleged truths or attributions of usefulness. The truth-validity-usefulness triad gives us a kind of Instrumental Reason with which to explore the world and ourselves. Investigating their complex dynamic — separable but interpenetrating — is the key.

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The post is already too long, so I’ll just end with an image of how these three concepts often get tangled together with related and important concepts that are not always relevant.

Wow, you spent some time on this. Nice post.

Glad you liked it!

I guess it was a product of intentional blog-deprivation and winter rumination. And I got some extra enthu because some of my department mates might join as contributors.

Nice post Yohan. I liked the lucid ways in which you can talk about what is otherwise considered profound or hard. However, I guess you can already guess what I am going to say. I will have to restrict myself to much smaller area, becasue Truth and its validity ‘in general’ is way too broad for me at this stage. So, I stick to mathematical truth and its validity, which ought to be differentiated from general perception of what is truth, which often is some sort of the degree of concensus amongst the parties involved.

With in this, I think there are interesting things that might be relevant. Are there true statements that cannot be validated. By validated, I mean being able to ‘decide’ to be either true or false using some procedure. You rightly point out Godel’s theorems, but those theorems are about consistency and completeness. They are not so much about decidability. Let me ramble a bit more. Godel, in trying to solve Hilbert’s second problem, proved that in mathematics, systems like peano arithmetic cannot be both complete and consistent at the same time, there by answering the question of whether mathematics is complete and consistent, in some way. But it was Turing and Church who answered the third and in some sense more general and important question of decidability. The answer is, of course, negative. Not everything is decidable.

This is also very relevant to something that interests me very much, which is one of the important epsiodes in the history of mathematical philosophy, the foundational crisis in mathematics. In some sense, it was a battle of giants, David Hilbert on one side and LEJ Brouwer on the other. One aspect of this debate was to do with the status of existence of mathematical objects. In some sense, it is valid in this discussion too, because in your list of proofs caricuture, there is no mention of the proof by construction, which according me is very important. Unlike for Hilbert, non-constructive methods in proofs, such as the law of excluded middle and Axiom of Choice were invalid in infinitary cases. This is a serious problem, becasue the moment you resort to nonconstructive techniques, there exists no method to decide them using a effective procedure. This makes them inherently uncomputable. I had discussed with you about the ubiquity of these kind of propositions in economics, that too at the heart of economic theory. They are assumed to be true in one kind of mathematics, but there exists no method to decide, atleast till today. To cut the long story short, decidability is something that you might want to think about if you were to validate truths, atleast in mathematics.

I don’t have much to say regarding the usefulness of anything. I am not so concerned about it now, as much as the first two things, which interest me.

This might seem unfair, but what I’ve said about logic/mathematics applies equally to issues like constructability. A mathematical construction is an entity produced by symbol-manipulation, and therefore falls into the set of validity-establishing methods. It’s ‘truth’ in the world

outsidemathematics proper still depends on the correct association of the entities (however constructed) with processes/observations in the world.If economists rely on associating processes in the economic world with unconstructable entities, then I agree, that’s a problem. But perhaps there’s a way usefulness can assist? If the theoretical entities help economists understand the phenomenon, then maybe the question of constructability can be circumvented?

” I’ve said about logic/mathematics applies equally to issues like constructability” – Not true, not for sure, in the Brouwerian variety of constructivism, which is called Intuitionism. This is an approach to mathematics, which is based on mathematical intuition, unlike symbol manipulation based “formalism”. I think your criticism is valid for formalism and formal logic. But in constructivism of the type Brouwer called for, mathematics is independent of both logic and language. Unfortunately, I am not an expert, and I read it out of interest in my free time. So I don’t think I can explain it to you in a convincing manner. I only want to refer you to this approach and I leave it for you to figure it out.

Second, basing a decision on validity based on its usefulness seems a bit bizarre to me! Usefulness as a criterion can swing our belief in a given assertation depending on the need at hand. I think a procedure to validate truth should be independent of its usefulness. Besides, how can you understand a phenomenon – like the ones we encounter in economic theory, like the possibility of existence of a “general equilibrium”, when such a mathematical object is in principle uncomputable, undecidable? The logical consequence is to refine the theory, in such a way as to having more clarity on the phenomena under investigation. Not to circumvent the problem. Existence questions are very hard to answer in complex systems and one has to find theoretical break throughs. Economics is no exception.

I meant that the object is computable, undecidable in the current form. Also, I should not have said “logical consequence” :)

http://www.ams.org/journals/bull/1924-30-01/S0002-9904-1924-03844-0/S0002-9904-1924-03844-0.pdf

This could be useful.

I didn’t mean that usefulness validates. I meant that usefulness can render the validation irrelevant. If the theory offers ways to reliably manipulate the economy, then the unproven/unprovable assumptions can be retained. This is what happens in science.

I’ll try to read up on intuitionism. But one thing I’d like to point out is that I believe mathematics is a language, and that it cannot be separated from natural language.

When I use terms like logic or mathematics I am not only speaking of epsilon delta formal maths. Even the derivations used in physics and engineering are symbol manipulations, whether they conform to mathematicians notions of rigor or not. I mean symbol manipulation in a very literal way — moving x’s and y’s around equal-to signs. This manipulation has an internal set of rules that does not depend on what the x’s and y’s correspond to in the world.

When a derivation steps out of the rules of symbol manipulation, then there’s a difference. I think renormalization falls into this category, although I only got a brief intro in IIT. You derive an expression algebraically as far as you can. Then you get an expression with infinities all over the place, and you replace the complex expression with something more acceptable or realistic. The rules for this acceptance are

externalto the rules of calculus, trigonometry or algebra, and sometimes violate them.In any case, I wasn’t criticizing formal logic or maths. I was saying that they have their internal rules, and science requires other nonmathematical processes in order to align mathematically valid statements with phenomena in the world.

I worry still that, in your portrayal, Truth and validity are far too closely intertwined. Your initial example of rainfall is a poor example. As you stated, Truth is attributed to communication. This communication is, often, not about empirical, testable events. I mean to say, science cannot, in most cases, have a meaningful input on truthiness (inasmuch as science is the process of verifying hypothesis through repeatable actions and observation). And as you’ve intimated our intuition (that is our mental pre-reason activity) has a huge influence on deducing truth. In this way you’ve illustrated the triangle – but left Truth largely alone. As an example, art largely refuses the interpretive grid of science, while attempting to approach and grasp at Truth. Science cannot operate here. Yet Truth exists there as art communicates to us.

The rainfall example was about directly observable events. And I address the problem of events that are only indirectly witnessed. I didn’t really address the whole class of language-games that doesn’t revolve around events. We use the word “true” to describe a variety of subjective feelings, and I am not sure why we do this. Perhaps there should be two words — “factual truth” for the experiences we classify as events, and “resonant truth” for the vast array of feelings that don’t seem to have clearly identifiable sense-content. I am with you that truthiness is outside the scope of logic per se.

However, since I am a neuroscientist I can’t help but wonder if truthiness has a physiological correlate. Truthiness often leads to violence and repression (not just meditative or artistic insight), so we can use science to counter the unreasonable confidence of truthiness. Zizek’s comments of the relationship between poetry and fascism are relevant here. When we associate emotional valence (truthiness) to abstractions like (particular conceptions of) God, nationalism, the Market or Society, we can often allow ourselves to be very violent to particular individuals, in the service of the Romantic Truth/Ideal.

In normal day to day operations of people, resonant truth has been (and will continue to be) far more important to people than factual truth. We live our lives according to resonant truth. I admit that there are downsides to truthiness – but I don’t see getting rid of truthiness as a solution. Rather, the continual critical attack on what we hold to be truthy, seems to me, to be the best way forward. Thinking people. Imagine that.

Yes. I agree. In fact scientific training ideally makes one get a resonant feeling from the factual truths too. Which is why scientists get an almost spiritual satisfaction from their work.

I think we’re in agreement. Criticism must be the yang to counter intuition’s yin. :)

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