On theory and observation (4): Sneed’s structuralism and T-theorecity

As we mentioned in the second entry of this series, the distinction between ‘the observable’ and ‘the theoretical’ focused on the first of these two concepts, so that the notion of ‘theoretical’ was implicitly understood as meaning simply ‘non-observable’. This distinction was submitted to powerful criticisms since the end of the fifties, when it was pointed by authors like N. R. Hanson that every scientific concept is ‘theory laden’ to some extent (meaning something like that there are not ‘neutral empirical terms’), and especially when, just a few years later, Thomas Kuhn argued that each scientific theory sees the world from an idiosyncratic paradigm, or even, that defenders of rival theories ‘lived in different worlds’. This debate made more pressing the question of in what precise sense scientific terms (including supposedly ‘empirical’ terms) are ‘theoretical’. One of the most interesting answers to this question was given one decade later by the philosopher Joseph Sneed, who initiated a small but very active brand in contemporary philosophy of science usually known as ‘structuralism’ (though it had hardly anything to do with the so-called ‘French’ linguistic and philosophical structuralism of Saussure, Lévy-Strauss or Althusser).

Celestial map illustrating Claudius Ptolemy’s model of the Universe. Source: Andreas Cellarius (1661)
Planisphaerium Ptolemaicum, Sive Machina Orbium Mundi Ex Hypothesi Ptolemaica In Planno Disposita / Wikimedia Commons

Sneed’s starting point (and that which gives its name to his approach) is to identify physical systems (his examples are from mathematical physics, though the theory has been applied to many other sciences afterwards) with a kind of mathematical models or formal structures, in the way this notion is employed in mathematical structuralism (e.g., the Bourbaki kind of foundation of mathematics). A ‘structure’ is understood as an ordered set <D1, D2, …, Dn, R1, R2, … Rm>, in which the D’s are domains, i.e., the sets of entities of which the theory is about (for example, in Newtonian mechanics -the case Sneed’s take as his main illustration-, these domains are sets of bodies -or, more exactly, mass-points-, as well as several sets of real numbers, like a three-dimensional space -actually, triples of real numbers- or segments of time -also a subset of real numbers-), and the R’s are different relations (usually mathematical functions) that can be defined over those domains (in particular, for each body it is given its mass, its position at any moment of time, and the forces that are exerted on it at any point of time). Some more elements can be added to a mathematical structure (or ‘model’) depending on the concepts the physical theory in question needs to express (for example, we shall need some element to represent physical constants, like universal gravity constant). Hence, in the case of Newtonian particle mechanics, a physical system -real or merely ideal or imaginary- is represented by a mathematical model, system or structure of the form S = <D, R3, T, R, s, m, f>, where the elements of D represent the physical bodies the system contains (‘particles’, not in the sense of ‘elementary particles’, but just in the sense of bodies represented by a geometrical point in the space, usually the ‘center of masses’ of the body), R3represents the three-dimensional Euclidean space, T represents time, R is just the set of real numbers, s gives the position of each particle at each moment in T, m gives the (invariable) mass of each particle, and f is the force exerted on each particle at each time instant. It is important to insist in that S is not a ‘real’ physical system, but a purely mathematical entity that is employed to represent some physical system, that can be ‘physically real’ (say, the system composed by the Earth and the Moon as we try to understand its mechanical behavior under the influence of gravity), or just imaginary (as when, in a physical textbook, we just imagine an idealized system on which some calculations and reasonings are performed).

The next strategy by Sneed is to discuss what is exactly theoretical in our systems like S, from the point of view of the particular theory we know as ‘Newtonian particle mechanics’ (say, NPM). Sneed’s answer is that, of all the seven concepts represented by the list of elements of S, the first five ones have nothing that depends essentially on NPM: we can just describe a physical system identifying the bodies it contains, and their positions at each moment, without assuming anything similar to Newtonian mechanics. Let’s call Sp (‘partial models’) the systems that result from structures like S when we just drop their last two elements. In the case of NPM, we can call these partial models cinematic systems (abstract mathematical systems that just describe the movements of a set of bodies). The concepts of mass and force (i.e., the functions m and f), instead, depend on a fundamental way of NPM, not in the sense that they refer to some ‘unobservable’ or ‘non-empirical’ magnitudes (for, after all, also the real exact position of a body or of it center of masses at an absolutely precise moment of time is also non-observable if understood literally), but in the sense that we need the conceptual building of Newtonian mechanics to understand what we are talking about when we talk about forces and masses. Force and mass are not merely ‘theoretical’ concepts, but concepts whose meaning depends on a specific theory, NPM, and hence we shouldn’t simply say they are ‘theoretical’, but that they are NPM-theoretical concepts. Sneed invites us, hence, to replace the concept of theoreticity with the concept of T-theoreticity, i.e., theoreticity as dependant of a specific theory T. In this sense, we might guess that all scientific concepts are ‘theoretical’ in the sense that they always depend on some theory (but not all of them on the same theory).

A philosophical paradox according to Sneed

This, however, constitutes a philosophical paradox, because it seems that, taken to its ultimate consequences, any scientific theory like NPM is just a mere tautology, for if the theory is needed in order to know what the statements made with the help of the theory mean, then the theory seems to be just true by decree. This paradox is more transparent if we, as Sneed, formulate the theory in terms similar to those of the Ramsey-sentence we saw in the second entry. According to Sneed, what NPM basically says (as a claim about the real physical world) is that for all real cinematic systems Sp there is a way of enlarging them with functions m and f in a system S such that Newton’s second law is fulfilled (i.e., such that f/m equals the second derivative of function s with respect to time, i.e., the bodies’ accelerations), which is nothing but a structuralist (or ‘semantic’) version of the (‘syntactic’) Ramsey-sentence. But this statement is trivially true! For every cinematic system can be made to obey Newton’s second law if we are free to choose whatever numbers to name the pertinent masses and forces. This is similar to the way in which it can be proved that any set of possible movements of the planets can be reconstructed from a Ptolemaic model, if we allow for any arbitrary set of epicicles.

Hence, the second step in Sneed’s strategy to make of NPM an empirical theory (i.e., one that makes some non-trivial claims about reality) is to show how it constrains the way in which masses and forces are introduced in our models. Basically, empirical masses and forces must be the result of some measurement procedures that, though they presuppose the validity of Newtonian laws (as our paradox claimed), they do it in such a way that the results of one measurement in one system has to be valid when it is ‘exported’ to the same bodies appearing in any other system. For example, if we measure the mass of the Earth on the basis of observations of the system Earth-Moon, the same number of Kgs has to be valid in any other system in which the Earth appears, be it in the system Sun-Earth, or in the determination of the gravity’s acceleration on the Earth’s surface to account for Galilei free fall law. Hence, NPM is not a theory about each mechanical system isolated from all the others, but about the big system consisting in the combination of all real mechanical systems, linked by that type of measurement constraints.

The beauty of the structuralist theory of science consists in this magnificent picture of scientific theories as mathematical systems of systems, linked to each other and to the world by measurement procedures and formal laws (by the way, structuralists tended to reserve the word ‘theory’ for that big mathematical structure, instead as the name of the complex claim about the world that this structure allows to make; this led them to call their own proposal ‘the non-statement view of scientific theories’, meaning that theories are not literally systems of propositions, but systems of formal structures or models; but I think nevertheless that it is more natural to identify a theory like NPM with what Sneed calls ‘the empirical claim of NPM’). This magnificent picture allowed, in the years following the publication of Sneed’s book, to attack in an elegant and fruitful way a lot of classical and new problems in the philosophy of science, though perhaps the practitioners of structuralism became too enamoured of their own formalism in its most baroque and sophisticate versions, to allow that it became a relatively popular school the academia, and in the 21st century it has lost much of the vigor it had reached a few decades ago.


Hanson, Norwood Russell, Patterns of discovery, 1958, Patterns of Discovery: An Inquiry into the Conceptual Foundations of Science. Cambridge University Press.

Kuhn, Thomas S., 1962, The Structure of Scientific Revolutions, The University of Chicago Press.

Sneed, Joseph D., 1971, The Logical Structure of Mathematical Physics, second edition, Dordrecht: D. Reidel Publishing Company.

Stegmüller, Wolfgang, 1976, The structure and dynamics of theories, Springer-Verlag.

‘On theory and observation’ series:

(1): The theoretician’s dilemma
(2): The Ramsey sentence
(3): Scientists selling lemons, a game-theoretic analysis of how scientific facts are constructed

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