PSA2016: The 25th Biennial Meeting of the Philosophy of Science Association

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Science and mathematical truth

Science and mathematical truth

Parts of scientific theories are sometimes considered as mathematical tautologies, thus casting some doubts on their empirical content. An important exemple is the Price equation in evolutionary biology. Called the second principle of evolutionary biology and grounding most of the works on social evolution, evolution of cooperation or group selection, it is nonetheless often attacked as being a mere « mathematical tautology », thus reducing the importance of the insights it provides in evolutionary biology (see Frank 2011). In this poster, I want to highlight the mechanisms of such an interpretation, explain its justitification, and show that it doesn't follows that the empirical content of the equation should be called into question.
My explanation relies on the idea that one should distinguish two kinds of truth : truth in abstract systems, such as mathematics or logics, derives from consistency of well formed formulas or sentences with the axioms and/or inference rules of the system. This kind of truth depends on the definition of formal conventions, which may or may not relate to real states of affairs. It may be called logical or syntactic truth. On the other hand, truth of scientific models or theories relies on the model or theory standing in a representational relation with the world. This kind of truth may be called representational or semantic truth.
Since the axioms of abstract systems are not necessarily meant to represent real states of affairs, they do not warrant that the formal systems they ground will prove representationaly true and hence have empirical content. This explains the pejorative meaning associated with interpretation of formulas, which aim at providing scientific explanations, as 'mere' tautologies. On the other hand the widespread use of mathematics in science suggests that some axiomatic systems at least have representational value, for instance, by offering abstract structures that apply in many real situations.
Equations like the Price equation seem to derive much of their validity through mathematical rules of inference, yet they receive their scientific importance through their applicability and actual application in real scientific cases, thus trully providing scientific explanations. This is why although being a tautology when regarded from a mathematical point of view, the Price equation is not merely a tautology, because it can be interpreted in real situations and thus aquires or justifies of its empirical content. The important distinction here is of being a mere tautology or being an interpreted tautology. If my proposition correctly describes the epistemic dilema surrounding tautologies in science, then it lends support to the idea that there are two kinds of truth and that one should be careful not to conflate them, in particular in debates surrounding the existence and nature of apriori knowledge in the sciences (see for instance Sober 2011).

Frank, S. A. (2012): Natural selection. IV. The Price equation*. In Journal of Evolutionary Biology 25 (6), pp. 1002–1019.
Sober, Elliott (2011): A Priori Causal Models of Natural Selection. In Australasian Journal of Philosophy 89 (4), pp. 571–589.

Word count: 497

Author Information:

Guillaume Schlaepfer    
Department of Philosophy
University of Geneva


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