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Kripke, Duchamp & the Standard Metre

Stuart Greenstreet isn’t quite sure how long a metre is. Are you?

In the Academy of Science, located in a suburb of Paris, a slim bar of platinum-iridium alloy is kept in its sturdy case in a temperature-controlled chamber. The precise distance between two scratches on the bar defined the Standard Metre, the basic unit of length in the metric system. How is it possible that this humdrum object could be the focus of two upheavals in the twentieth century, one in the visual arts, and the other in philosophy?

In 1913 the French artist Marcel Duchamp poked fun at the Standard Metre in a work of art which led him very quickly to reject ‘retinal’ painting and to invent conceptual art. Much later, in 1972, the Standard Metre was also useful to the American logician Saul Kripke. He turned to it in his book Naming and Necessity to help him show why no previous philosopher, not even Russell or Frege or Quine, had properly understood the fundamental notions of possibility and necessity. It may be easier to see how this simple rod featured in these two revolutions by starting with Kripke’s work.

Saul Kripke

Up to the 1970s all truths were categorized (by philosophers, at least) in three ways:

Necessary versus Contingent. A statement is necessarily true if its opposite is impossible – if what it expresses could not be otherwise, no matter what. (‘Three is a prime number’.) A statement is contingently true if its opposite is possible – if what it says does actually happen to be the case, but could in fact be otherwise. (‘Blair is Prime Minister.’)

A Priori versus A Posteriori. Necessary truths are a priori – knowable independently of experience, without anyone needing to go and look; whereas contingent truths can only be found out by observation, or a posteriori.

Analytic versus Synthetic. Finally, some statements are analytically true – true simply in virtue of the meaning of words they use. ‘All cygnets are young swans’ is an analytic truth because the predicate ‘young swans’ adds nothing to the subject ‘cygnets’. In a synthetic statement, by contrast, the predicate tells us something new about the subject (‘All swans are white’); and what it says is either true or false in virtue of the way the world actually is.

So the table below shows how truths were thought to be classifiable pre-Kripke:

The idea of a statement being a necessary truth was somehow deeply rooted in its truth being knowable a priori. However, as we are about to see, the two classes (of necessity and a priori) do not always coincide.

Kripke agreed that truths are indeed either necessary or contingent. But why, he asked, should we suppose that we have to pick out this single distinction in three ways? Why, in particular, conflate the metaphysical question of whether the truth of a statement is necessary or contingent with the separate epistemological issue of whether its truth can be determined by a priori reflection or requires empirical observation? The mere fact that we have learned something from experience does not of itself show that what we have learned is not necessarily true.

Kripke could find no reason why a claim should not be both empirical and necessary, given a proper conception of necessity. And that proper conception, he argued, is metaphysical, not logical. A statement could be necessarily true even if it were not warranted as such by logic alone. In Naming and Necessity Kripke cites examples of certain truths that are ‘necessary a posteriori’ – necessary truths that are discovered empirically; and of certain other truths that are ‘contingent a priori’ – propositions which might have been false, but whose truth is knowable a priori.

Kripke found a truth that is necessary a posteriori by revisiting Frege’s puzzle about the identity statement “Phosphorus (the morning star) is Hesperus (the evening star).” The statement’s truth was discovered a posteriori, when both the names were found to refer to the one thing, namely the planet Venus. Hence, Frege felt, the statement is contingently true. Not so, says Kripke, for no statement of identity is contingently true: a name, he insists, is a ‘rigid designator’, which picks out the same object in every possible world in which that object exists. If it was ever true that Phosphorus is identical to Hesperus, then it is always and necessarily true. So ‘Phosphorus is Hesperus’ is a necessary a posteriori truth.

It was when Kripke wanted an instance of a contingent a priori truth that he became interested in the peculiar properties of the Standard Metre rod. Since its length might vary in time – depending on temperature, pressure, etc. – he made the definition more precise by stipulating one metre to be the length at fixed time t₀. He then asked:

Is it a necessary truth that stick S [the metal rod] is one meter long at time t₀? Someone who thinks that everything one knows a priori is necessary might think: “This is the definition of a meter. By definition, stick S is a meter long at t₀. That’s a necessary truth.” (Naming and Necessity, pp.54- 55.)

Again, not so. For the definition has only one use: to fix the reference. Kripke explained that there is a difference between the phrase “one meter” and the phrase “the length of S at t₀”.

The first phrase is meant to designate rigidly a certain length in all possible worlds, which in the actual world happens to be the length of S at t₀. On the other hand ‘the length of S at t₀’ does not designate anything rigidly. In some counterfactual situations the stick might have been longer and in some shorter ... (Naming and Necessity, p.55. My italics.)

The metal rod defines ‘one metre’ by determining the reference of that term. It stipulates that ‘one metre’ is to be a rigid designator of the length, which in our world ‘happens to be’ S . Hence, the Standard rod does not make it a necessary truth that S is a metre long. In fact, (as I’ll show at the end) it was quite possible that S would not have been one metre long. It is true that if someone uses the Standard rod to fix the reference of the term ‘one metre’, then as a result of this kind of definition, he knows automatically, without further investigation, that S is one metre long. This procedure has given the statement its epistemological status. On the other hand, even if the rod is so used as the standard of a metre, the metaphysical status of ‘S is one metre long’ will be that of a contingent statement: for it is possible (given fluctuations in pressure and temperature) that the very same rod could expand or contract. So, Kripke concludes, there are contingent a priori truths.

Marcel Duchamp

In 1913-14, nearly sixty years before Kripke’s Naming and Necessity came out, Marcel Duchamp assembled an artwork directly, and transparently, inspired by the Standard Metre, which he called Three Standard Stoppages (Trois Stoppages-Étalon). It is a croquet box containing a fabricated ‘idea’ – the idea of ‘canned chance’ as Duchamp put it – combined with an application of the idea and a law that arose from it. (Notice in passing that an ‘idea’ as a work of art in itself practically defines ‘conceptual art’.)

Duchamp was 26 in 1913, and employed as a librarian at the Bibliothèque Sainte Geneviève in Paris. While working there until May 1914 he earned a small wage and gave up painting in order to pursue a scholarly interest in mathematics and physics, in which recent discoveries were shaking the foundations of science. In 1900 Max Planck revealed the quantum nature of radiation; and in 1906 Einstein published papers on the theory of relativity, the photoelectric effect, and Brownian motion. Duchamp was influenced most of all by the mathematician Henri Poincaré, whose writings were gathered into four popular books between 1903 and 1913. These explained the conceptual changes that flowed from the discovery (by Marie and Pierre Curie around 1900) of X-rays, the phenomenon of radioactivity, of radium, and especially of the electron and its laws.

Poincaré thought that physics was in crisis, brought on by developments that had raised doubts about the very possibility of objective scientific knowledge. He now regarded a scientific theory as merely a ‘convention’ – a professional consensus, for the time being, about what may explain such and such phenomena. Of such a theory, he said, one demands simplicity and universality, but is not concerned with its truth. The laws that we think govern the behaviour of matter are created by the minds that understand them. Therefore no theories could be considered objectively true: “The things themselves are not what science can reach, but only the relations between them. Outside of these relations there is no knowable reality.” This sentence of Poincaré’s came to be the leitmotif of Duchamp’s life as an artist. If science cannot reach the real essence of matter, then analogously, he thought, it is impossible to apprehend some latent essence in a work of art. A work’s meaning consists in relations between it and its beholder, must be in his or her consciousness of it: “It is the regardeurs who make the picture”, Duchamp insisted.

It is hardly surprising, given his immersion in Poincaré’s writings, that Duchamp’s Three Standard Stoppages, which he made while still at the library, has a scientific subject. His own notes on it are as follows:

3 Standard Stops = canned chance

The Idea of Fabrication
- If a straight horizontal thread one meter long falls from a height of one meter straight onto a horizontal plane twisting as it pleases and creates a new image of the measure of length.
- 3 patterns obtained in more or less similar conditions: considered in their relation to one another they are an approximate reconstruction of the measure of length.
The 3 standard stoppages are the meter diminished.

(These notes come from Duchamp’s Box of 1914. His italics.)

Two key phrases are ‘canned chance’ and ‘twisting as it pleases’, both of which denote accidental properties.

To create Three Standard Stoppages, Duchamp performed the actions he described in ‘The Idea of Fabrication’, above. He held each piece of thread horizontally and let it fall from a height of one metre onto a painted canvas surface, where it landed in a random shape. He used varnish to fix the three resultant impressions permanently onto glass plates. Then from each of these impressions he made a template by cutting along the one edge of a thin wooden slat, copying the undulating curves. The complete assemblage comprises the three curvy wooden templates and their container, the croquet box.

Duchamp later explained that by making a work which is ‘a joke about the meter’ he aimed to ‘discredit’ the authority of the metric standard, the rod. Although the one-metre long strings are still one metre long along their curves, their straight linear measurements are all different because each string fell ‘as it pleases’ and is configured by chance. The workings of chance have ‘stopped a standard’ that was merely one convention among countless many. The Stoppages create three alternative standards for the metre. Yes, their lengths were fixed (or ‘canned’) arbitrarily; but so too, Duchamp implies, was the length of the Standard metre: that length just ‘happens to be’, in Kripke’s words, the distance between those scratches on that alloy rod.

When the metric system was created in the 1790s a metre was defined as one ten-millionth of the distance from the equator to the North Pole. To measure this distance, a quadrant of the earth (one fourth of a circumference) was surveyed along the meridian arc from Dunkirk to Barcelona that passes through Paris. This was how the metre defined by the metal rod was originally derived. This does not make it a necessary truth that the rod is one metre long; for, as Kripke said, the length rigidly referred to as ‘one metre’ might have been longer or shorter. Consider: (i) the Earth’s circumference could have been other than what it actually is, so the result of dividing a quadrant by ten million might have been longer or shorter; (ii) the divisor itself might not have been ten million; (iii) the standard metre length might have been derived from something other than the earth’s circumference, as indeed it actually has been since 1983, when ‘one metre’ was redefined as the length of the path travelled by light in a vacuum at a time interval of 1/299,792,458 of a second. Moreover, piling contingency on top of contingency, the original surveyors miscalculated (very slightly) in the 1790s. According to the latest satellite surveys, the length of the meridian from the pole to the equator is not ten million metres, but 10,002,290 metres; and so the ‘standard’ metre was (‘by chance’) a few tenths of a millimetre short. How prescient of Marcel Duchamp to make an ironic spoof of it.

© STUART GREENSTREET 2003

Stuart Greenstreet, a business manager by day, began philosophy in the evenings at Birkbeck College in the 1980s, and has kept at it ever since because he’s ‘afraid to stop’. He’s currently doing postgraduate philosophy with the Open University.

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