My paper entitled “Deification of Science and Its Disastrous Consequences” makes the argument that math, science and humanities each have distinct methodologies suitable for the subject matter. These methodologies cannot be interchanged. In particular, a methodology suitable for study of inanimate objects subject to laws is not suited to the study of human beings and societies. This is because individuals and communities choose goals, and strive for them, and are free to change these goals. My paper “The Methodology of Polanyi’s Great Transformation” shows that an appropriate methodology for “social science” must be sensitive to historical and socio-cultural context, unlike scientific theories, which are universal invariants. This implies that “social science” itself is a misnomer, since the project of applying scientific methodology to the study of human societies is misconceived. I therefore suggest reverting to the older name of “humanities. The paper itself is complex and makes a number of related and linked arguments. Among these, I would like to highlight two Propositions listed below:
- Mathematics uses an axiomatic or hypothetico-deductive methodology, whereas science uses an inductive methodology. These two are dramatically different from each other.
- Economics uses an axiomatic methodology, based on a misunderstanding that this is scientific methodology.
Proposition 2 reflects a DOUBLE mistake. Mistake Number One: Economists believe that scientific methodology is axiomatic and hypothetico-deductive, whereas actual science uses a radically different inductive methodology. Mistake Number Two: Economists believe that scientific methodology is appropriate for economics, whereas economics (and all “social sciences”) require a radically different methodology. Current economic methodology goes back to Lionel Robbins (1935), who argued that “The propositions of economic theory, like all scientific theory, are obviously deductions from a series of postulates. And the chief of these postulates are all assumptions involving in some way simple and indisputable facts of experience…” This quote shows the double mistake clearly – economics uses scientific methodology, and scientific methodology is axiomatic. I will not deal with Proposition 2 and the double mistake in this post. Rather, I am only concerned with Proposition 1, establishing that Science is not based on axiomatic methodology, unlike Mathematics. This is partly to counter assertions of Egmont to the contrary in several comments, but more importantly, to clarify a common confusion among economists. Even though the point is almost obvious, the contrary is widely believed, and so I will give several arguments to establish my point.
DEFINITIONS: The Axiomatic Methodology is familiar to all readers from their study of Euclidean geometry, which starts with some simple postulates and then derives many complex properties of triangles and other figures using logic. Deductions from these axioms, which may be termed Theorems, are logically valid, without reference to the real world. The notion of logical truth was subjected to intense study in the early twentieth century, as a part of an effort to prove the consistency and completeness of axiomatic formulations of mathematical knowledge. This effort ended with the surprising results described in “Gödel’s Theorems & the Limits of Reason“. I will try to provide a simplified account of a complex and tangled story here.
ANALYTIC & SYNTHETIC TRUTHS: Within the axiomatic system of Euclidean geometry, all theorems (including Pythagorean) are analytic truths – true by virtue of definitions and logic alone, without reference to any empirical reality. To study the relation between a formal axiomatic system and the real world, one has to construct a mapping – what is the interpretation of the abstract notions of point, line, and angle in the real world. In some mappings, Euclidean geometry holds in the real world to a good approximation. In other mappings, it does not. For example, if we map the surface of the ocean into a Euclidean plane, we would make serious navigational errors (as actually happened). ALL statements of the type that Euclidean geometry holds in the real world are SCIENTIFIC statements and can only be synthetic – that is subject to empirical verification. Like ALL scientific truths (and unlike mathematical truths), such a statement can never be proven. Measuring triangles from mountaintops is an attempt to establish a synthetic truth that Euclidean geometry holds on the surface of the planet – under certain mappings, this would prove false since the surface is curved. Other mappings, which respect the curvature would validate and verify Euclidean geometry. It is always the validity of the mapping as a whole which is under consideration, and never Pythagorean Theorem as such – The Pythagorean Theorem is an analytic truth, which automatically holds if the world is Euclidean. Whereas initial investigations confirmed the validity of Euclidean geometry in the real world under a suitable mapping, later investigations discovered that the appropriate model for our universe is a Riemannian geometry. This did not make the Pythagorean theorem false, but it did mean that this theorem is irrelevant because the axioms of Euclidean geometry do not hold in the space time continuum.
SUMMARY: Mathematical axiomatic systems lead to analytic truths, which do not require empirical verification, since they are true by virtue of definitions and logic. It is a startling discovery of the twentieth century that sufficiently complex axiomatic systems are undecidable and incomplete. That is, the system of theorem and proof can never lead to ALL the true sentences about the system, and ALWAYS contain statements which are undecidable – their truth values cannot be determined by proof techniques. More relevant to our current purpose is that applying an axiomatic hypothetico-deductive system to the real world can only be done by means of a mapping, which creates a model for the axiomatic system. These mappings then lead to assertions about the real world which require empirical verification. These assertions (which are proposed scientific laws) can NEVER be proven in the sense that mathematical theorems can be proven. Even if we measure thousands of real world triangles and find that they have Euclidean properties, tomorrow we may find one which does not. Induction based on the assumption that an observed pattern will continue to be valid can never lead to logically certain conclusions. Even if the sun rises every day for billions of years, it may fail to rise tomorrow. Indeed, the reason that Kuhn’s Structure of Scientific Revolutions is ranked among the top books of the twentieth century is because he established that historically, scientific laws are frequently replaced by alternatives which make previous formulations obsolete. Post-Kuhnian philosophy of science replaces the earlier naïve understanding that science takes propositions which are certain and builds upon them logically. If this was true, then science would never reach a wrong conclusion, when in fact, scientific journals are full of articles announcing that previous theories are proven wrong, and proposing more viable alternatives. Again this strongly differentiates them from mathematical journals which never state that a new empirical finding has led to the negation of a theorem previously held to be valid.
Many more arguments can be given to explain the difference between analytic and synthetic truths, which corresponds to the difference between mathematical and scientific truths. As I have explained in greater detail in my paper, the scientific method arose as a rejection of the axiomatic method used by the Greeks for scientific methodology. It was this rejection of axiomatics and logical certainty in favour of empirical and observational approach which led to dramatic progress in science. However, this did involve giving up the certainties of mathematical argumentation and learning to live with the uncertainties of induction. Economists need to do the same – abandon current methodology borrowed from (a misunderstanding of) scientific methodology, and develop a new methodology suited for the study of human beings and societies.
PostScript: Some additional arguments as to why the axiomatic method is not suitable as a formalization for scientific knowledge are given in my comment on Egmont’s assertions to the contrary.