The evidence for Mises’s misunderstanding of philosophy of mathematics 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.First, Mises’s belief that aprioristic reasoning can deliver new, informative knowledge of the real world fails because it is all dependent on the untenable idea of Kantian synthetic a priori 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)
“Praxeology is a theoretical and systematic, not a historical, science. Its scope is human action as such, irrespective of all environmental, accidental, and individual circumstances of the concrete acts. Its cognition is purely formal and general without reference to the material content and the particular features of the actual case. It aims at knowledge valid for all instances in which the conditions exactly correspond to those implied in its assumptions and inferences. Its statements and propositions are not derived from experience. They are, like those of logic and mathematics, a priori. They are not subject to verification and falsification on the ground of experience and facts. They are both logically and temporally antecedent to any comprehension of historical facts. They are a necessary requirement of any intellectual grasp of historical events” (Mises 2008: 32).
Kant’s belief in the synthetic a priori is false, and we know this now given the empirical evidence in support of non-Euclidean geometry: this damns Kant’s claim that Euclidean geometry – the geometry of his day – was synthetic a priori (Salmon 2010: 395). In addition, despite Gödel’s incompleteness theorems and the failure of Bertrand Russell’s strict logicist program, the consensus today is that most of classical mathematics can nevertheless be derived from pure logic and set theory (Schwartz 2012: 19), just as Russell thought,* and it is arguably just analytic a priori knowledge (and even arithmetic might be conceptually divided into (1) analytic a priori pure arithmetic and (2) synthetic a posteriori applied arithmetic [see Musgrave 1993: 240]).
Furthermore, 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 as (2) applied geometry (for the distinction, see Salmon 2010: 395). Mises’s statements are ignorant and wrong, because he conflates these two distinct forms of geometry. The inability to separate geometry into these senses – pure geometry versus applied geometry – leads to all sorts of philosophical disasters, amongst them Platonic mystical belief in the eternal realm of the forms and aprioristic Rationalism (the derivation of these things from Euclidean geometry is described in Salmon 2010: 393).
Rudolf Carnap explains the difference pure geometry and applied (physical) geometry:
“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 (Musgrave 1993: 236; 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; Hausman 1994: 386; Musgrave 2006: 329). Non-Euclidean geometry is now understood to be a better theory of reality. When confined to its role as a pure mathematical theory, Euclidean geometry is true but vacuous. That is to say, modern apriorist Rationalists can defend the necessary, a priori truth of Euclidean geometry (as in Katz 1998: 49–50), but only as a pure mathematical theory that is vacuous, non-informative and tautologous. It tells us no necessary truth about reality (Salmon 2010: 395).
But isn’t Euclidean geometry still a useful empirical 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 Austrian 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 as pure mathematics is used beyond its tautologous form, 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 the axioms of praxeology are analytic a priori, then praxeology would indeed be a tautological system that is non-informative, and asserts nothing necessarily and apodictically true about the real world. The only viable route left for modern Misesian praxeologists is to accept the empirical nature of the human action axiom (and other axioms) and admit that derived praxeological theorems are empirical.
That is to say, as soon as praxeology is taken as a system that asserts something about the real world of human economic life (and is not simply asserted as a non-informative, tautologous and vacuous system), it must be judged, like applied geometry, as making synthetic a posteriori statements, which – contrary to Mises’s bizarre assertions cited above (Mises 2008: 32) – can certainly be refuted by experience and empirical evidence.
Like Kant, Mises’s project is damned, as is traditional Rationalist epistemology in general, as has been noted by the Popperian philosopher Alan Musgrave:
“The invention of non-Euclidean geometries deprived rationalism of its paradigm. It also suggested to empiricists a new way to deal with mathematics: distinguish pure mathematics from applied mathematics, locate the latter in the synthetic a posteriori compartment of Kant’s box, and the former in the analytic a priori compartment of Kant’s box. One attempt to do the last, logicism, is generally admitted to have failed. Another attempt, if-thenism, is still hotly debated among philosophers. On the other hand, the logical empiricist view of applied mathematics has met with pretty wide acceptance. The rationalist dream, ‘certain knowledge of the objects of experience by means of pure thinking’, is shattered even though the nature of pure mathematics remains problematic indeed.” (Musgrave 1993: 245–246).Note
* Successors of logicism include (1) the formalism of David Hilbert; (2) conditionalism or “if-thenism” (a term coined by Hilary Putnam), which is a deductivist version of formalism (see Musgrave 1977); and (3) various forms of Intuitionism.
Carnap, Rudolf. 1958. “Introduction,” in Hans Reichenbach, The Philosophy of Space and Time (trans. Maria Reichenbach and John Freund). Dover, York.
Elugardo, R. 2010. “Analytic/Synthetic, Necessary/Contingent, and a priori/a posterori: Distinction,” in Alex Barber and Robert J Stainton (eds.), Concise Encyclopedia of Philosophy of Language and Linguistics. Elsevier, Oxford. 10–19.
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.
Katz, Jerrold J. 1998. Realistic Rationalism. MIT Press, Cambridge, Mass.
Mises, L. von. 2008. Human Action: A Treatise on Economics. The Scholar’s Edition. Mises Institute, Auburn, Ala.
Musgrave, Alan. 1977. “Logicism Revisited,” British Journal for the Philosophy of Science 28: 99–127.
Musgrave, Alan. 1993. Common Sense, Science and Scepticism: Historical Introduction to the Theory of Knowledge. Cambridge University Press, Cambridge.
Musgrave, Alan. 2006. “Responses,” in Colin Cheyne and John Worrall (eds.), Rationality and Reality: Conversations with Alan Musgrave. Springer, Dordrecht. 293–334.
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.
Reichenbach, Hans. 1958. The Philosophy of Space and Time (trans. Maria Reichenbach and John Freund). Dover, York.
Salmon, W. C. 2010. “Geometry,” in Jonathan Dancy, Ernest Sosa, and Matthias Steup (eds.), A Companion to Epistemology (2nd edn.). Wiley-Blackwell, Chichester, UK and Malden, MA. 393–395.
Schwartz, Stephen P. 2012. A Brief History of Analytic Philosophy: From Russell to Rawls. Wiley-Blackwell, Chichester, UK.
Ward, Andrew. 2006. Kant: The Three Critiques. Polity, Cambridge.