Mises Fails Philosophy of Mathematics 101

Mises FailsPhilosophyofMathematics 101

The evidence is here in Human Action:

“Aprioristic reasoning is purely conceptual and deductive. It cannot produce anything else but tautologies and analytic judgments. All its implications are logically derived from the premises and were already contained in them. Hence, according to a popular objection, it cannot add anything to our knowledge.

All geometrical theorems are already implied in the axioms. The concept of a rectangular triangle already implies the theorem of Pythagoras. This theorem is a tautology, its deduction results in an analytic judgment. Nonetheless nobody would contend that geometry in general and the theorem of Pythagoras in particular do not enlarge our knowledge. Cognition from purely deductive reasoning is also creative and opens for our mind access to previously barred spheres. The significant task of aprioristic reasoning is on the one hand to bring into relief all that is implied in the categories, concepts, and premises and, on the other hand, to show what they do not imply. It is its vocation to render manifest and obvious what was hidden and unknown before.” (Mises 2008: 38).

First, Mises’s belief that aprioristic reasoning can deliver new, informative knowledge of the real world fails because it is all dependent on synethic a priori knowledge, but Kant’s belief in synthetic a priori knowledge is false, and, as I have already shown, the human action axiom cannot be considered to be a synthetic a priori statement.

But the real issue raised by Mises here is the epistemological status of geometry, or, more precisely, Euclidean geometry.

Mises has failed to distinguish between geometry in its role as (1) a pure mathematical theory, and (2) as applied geometry. Mises is ignorant and wrong, because he conflates these two distinct forms of geometry.

Rudolf Carnap explains this difference:

“It is necessary to distinguish between pure or mathematical geometry and physical geometry. The statements of pure geometry hold logically, but they deal only with abstract structures and say nothing about physical space. Physical geometry describes the structure of physical space; it is a part of physics. The validity of its statements is to be established empirically—as it has to be in any other part of physics—after rules for measuring the magnitudes involved, especially length, have been stated. (In Kantian terminology, mathematical geometry holds indeed a priori, as Kant asserted, but only because it is analytic. Physical geometry is indeed synthetic; but it is based on experience and hence does not hold a priori. In neither of the two branches of science which are called ‘geometry’ do synthetic judgements a priori occur. Thus Kant’s doctrine must be abandoned) (Carnap 1958: vi).

When Euclidean geometry is considered as a pure mathematical theory, it is nothing but analytic a priori knowledge, and asserts nothing of the world, since it is tautologous and non-informative.

But, when Euclidean geometry is applied to the world, it is judged as making synthetic a posteriori statements (Ward 2006: 25), which can only be verified or falsified by experience or empirical evidence (or, in the jargon of philosophy, can be known as true only a posteriori).

That is to say, applied Euclidean geometrical statements can be refuted empirically, and, indeed, Euclidean geometry – when asserted as a universally true theory of space – is now known to be a false theory (Putnam 1975: 46; Rosenburg 1994: 386). Non-Euclidean geometry is now understood to be a better theory of reality.

But isn’t Euclidean geometry still a useful theory in certain ways? Yes, but this does not save Mises. Euclidean geometry is useful only because it is an approximation of reality and only at certain levels of space (Ward 2006: 25). But it is still false when judged as a universal theory of space.

Even on the most generous estimate, all you could argue is that Euclidean geometry is true only in a highly limited domain: the relatively small, macroscopic spaces and distances humans normally deal with in everyday life. But, once we move beyond this world, Euclidean geometry is false.

And even this qualification does not save the Misesian and Austrians apriorists, because we can only know that geometry is true in its limited domain a posteriori, that is, by empirical evidence.

As soon as Euclidean geometry is used beyond its pure tautologous form in pure mathematics, it becomes a system making synthetic a posteriori statements, not Kant’s imaginary synthetic a priori.

Since synthetic a priori propositions do not exist, it follows from this that, if Mises thinks that his axioms are analytic a priori, then praxeology would indeed be a tautological system that is non-informative, and asserts nothing about the real world.

As soon as praxeology is taken as a system that asserts something about the real world of human economic life, it must be judged, like applied geometry, as making synthetic a posteriori statements, which can be refuted by experience and empirical evidence.

Carnap, Rudolf. 1958. “Introduction,” in Hans Reichenbach, The Philosophy of Space and Time (trans. Maria Reichenbach and John Freund). Dover, York.

Hausman, Daniel M. 1994. “If Economics Isn’t Science, What Is It?,” in Daniel M. Hausman (ed.), The Philosophy of Economics: An Anthology (2nd edn.). Cambridge University Press, Cambridge. 376–394.

Mises, L. von. 2008. Human Action: A Treatise on Economics. The Scholar’s Edition. Mises Institute, Auburn, Ala.

Putnam, Hilary. 1975. “The Analytic and the Synthetic,” in Hilary Putnam, Mind, Language and Reality. Philosophical Papers. Volume 2. Cambridge University Press, Cambridge. 33–69.

Ward, Andrew. 2006. Kant: The Three Critiques. Polity, Cambridge.

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