Mises’s Non Sequitur on synthetic a priori Knowledge

We can find it here in Mises’s book The Ultimate Foundation of Economic Science: An Essay on Method (1962):

“The essence of logical positivism is to deny the cognitive value of a priori knowledge by pointing out that all a priori propositions are merely analytic. They do not provide new information, but are merely verbal or tautological, asserting what has already been implied in the definitions and premises. Only experience can lead to synthetic propositions. There is an obvious objection against this doctrine, viz., that this proposition that there are no synthetic a priori propositions is in itself a—as the present writer thinks, false—synthetic a priori proposition, for it can manifestly not be established by experience.” (Mises 1962: 5).

The proposition that “there are no synthetic a priori propositions” would appear to be synthetic a posteriori, for one could indeed refute it by successfully demonstrating the existence of synthetic a priori knowledge.

Alternatively, it could be verified by demonstrating empirically that all alleged examples of synthetic a priori knowledge are untenable and can be refuted by empirical evidence. And, in fact, the latter has been the fate of all alleged kinds of synthetic a priori knowledge, such as, for instance, Euclidean geometry.

Even the Kantian idea of necessary, deterministic causation as a universal truth known a priori is untrue given modern quantum physics (Melnick 2006: 229; Anscombe 1993), which now separates the notions of causation and determinism (Weinert 2004: 260).

At most, the strict notion of deterministic, necessary causation is probably true of the macroscopic world – a limited domain – but breaks down at the level of the quantum world. But we only know this a posteriori (empirically), and not as an a priori or a necessary truth.

If this isn’t bad enough Mises then proceeds to shoot himself in the foot, and demonstrates how his own statement above is highly dubious, by admitting that modern science has questioned the synthetic a priori status of Euclidean geometry:

“The whole controversy is, however, meaningless when applied to praxeology. It refers essentially to geometry. Its present state, especially its treatment by logical positivism, has been deeply influenced by the shock that Western philosophy received from the discovery of non-Euclidian geometries. Before Bolyai and Lobachevsky, geometry was, in the eyes of the philosophers, the paragon of perfect science; it was assumed that it provided unshakable certainty forever and for everybody. To proceed also in other branches of knowledge more geometrico was the great ideal of truth-seekers. All traditional epistemological concepts began to totter when the attempts to construct non-Euclidian geometries succeeded.

Yet praxeology is not geometry. It is the worst of all superstitions to assume that the epistemological characteristics of one branch of knowledge must necessarily be applicable to any other branch. In dealing with the epistemology of the sciences of human action, one must not take one’s cue from geometry, mechanics, or any other science.

The assumptions of Euclid were once considered as self-evidently true. Present-day epistemology looks upon them as freely chosen postulates, the starting point of a hypothetical chain of reasoning. Whatever this may mean, it has no reference at all to the problems of praxeology.” (Mises 1962: 5).

Come again? The collapse of Euclidean geometry as synthetic a priori knowledge has “[n]o reference at all to the problems of praxeology”? The stupidity of this passage beggars belief.

The issue of whether Euclidean geometry is synthetic a priori knowledge is absolutely of great relevance to the epistemological status of Mises’s praxeology, because Mises needs to prove that synthetic a priori knowledge exists.

But, having assured us that there is good reason to believe in the truth of synthetic a priori knowledge, Mises then immediately seems to concede that Euclidean geometry – the leading paradigm of the synthetic a priori – was not actually synthetic a priori at all, which destroys much of the alleged evidence for the existence of the latter.

Mises’s statement that “[p]resent-day epistemology looks upon [sc. the assumptions of Euclid] … as freely chosen postulates, the starting point of a hypothetical chain of reasoning” seems to be a tacit admission, or veiled reference to, the finding of modern science that Euclidean geometry as a universal theory of space is false.

Without any convincing evidence for the truth of synthetic a priori knowledge, Mises’s whole system of praxeology is left hanging in the air, without foundation, and must come crashing down.

BIBLIOGRAPHY
Anscombe, G. E. M. 1993. “Causality and Determination,” in Ernest Sosa and Michael Tooley (eds.), Causation. Oxford University Press, Oxford.

Melnick, Arthur. 2006. “Kant’s Proof of Substance and Causation,” in Paul Guyer (ed.), The Cambridge Companion to Kant and Modern Philosophy. Cambridge University Press, Cambridge. 203–237.

Mises, Ludwig von. 1962. The Ultimate Foundation of Economic Science: An Essay on Method. Van Nostrand, Princeton, N.J.

Weinert, Friedel. 2004. The Scientist as Philosopher: Philosophical Consequences of Great Scientific Discoveries. Springer, Berlin and London.

Bob Murphy All At Sea on Geometry and Economic Epistemology

A beautiful illustration of the continuing errors of Austrians who support Misesian praxeology can be seen in Robert Murphy’s comments in this video on geometry, and in his debate with David Friedman.*



Robert Murphy, like Mises, cannot properly distinguish between (1) pure geometry and (2) applied geometry (on which, see Salmon 1967: 38). When Euclidean geometry is considered as a pure mathematical theory, it can be regarded as analytic a priori knowledge, and asserts nothing necessarily true of the external, real world, since it is tautologous and non-informative. (An alternative view derived from the theory called “conditionalism” or “if-thenism” holds that pure geometry is merely a set of conditional statements from axioms to theorems, derivable by logic, and asserting nothing about the real world [Musgrave 1977: 109–110], but this is just as devastating to Misesians.)

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. That means that applied Euclidean geometrical statements can be refuted empirically, and we know that Euclidean geometry – understood as a universally true theory of space – is a false theory (Putnam 1975: 46; Hausman 1994: 386; Musgrave 2006: 329).

Murphy’s confusion is also confirmed in these remarks below.



The fact that the refutation of Euclidean geometry understood as an empirical theory leaves pure geometry untouched does not help Murphy, because pure geometry per se says nothing necessarily true about the real-world universe, and is an elegant but non-informative system.

Albert Einstein was expressing this idea in the following remarks about mathematics in an address called “Geometry and Experience” on 27 January 1921 at the Prussian Academy of Sciences:

“One reason why mathematics enjoys special esteem ... is that its laws are absolutely certain and indisputable, while those of all other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts. In spite of this, the investigator in another department of science would not need to envy the mathematician if the laws of mathematics referred to objects of our mere imagination, and not to objects of reality. For it cannot occasion surprise that different persons should arrive at the same logical conclusions when they have already agreed upon the fundamental laws (axioms), as well as the methods by which other laws are to be deduced therefrom. But there is another reason for the high repute of mathematics, in that it is mathematics which affords the exact natural sciences a certain measure of security, to which without mathematics they could not attain. At this point an enigma presents itself which in all ages has agitated inquiring minds. How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things. In my opinion the answer to this question is, briefly, this:- As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”
http://www-history.mcs.st-and.ac.uk/Extras/Einstein_geometry.html

If we were to pursue this analysis further as applied to economic methodology, it would follow that praxeology – if it is conceived as deduced from analytic a priori axioms – is also an empty, tautologous, and vacuous theory that says nothing necessarily true of the real world. And the instant any Austrian asserts that praxeology is making real assertions about the world, it must be judged as synthetic a posteriori, and so is to be verified or falsified by experience or empirical evidence.

What Murphy fails to mention is that the only way to sustain his whole praxeological program is to defend the truth of Kant’s synthetic a priori knowledge, which, as we have seen from the last post, is a category of knowledge that must be judged as non-existent.

Note
* Murphy also conflates (1) the logical positivists’ verifiability criterion for meaningfulness with (2) Popper’s falsifiability criterion for scientific knowledge, but this is an issue I will not bother to pursue here.

BIBLIOGRAPHY
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.

Musgrave, Alan. 1977. “Logicism Revisited,” British Journal for the Philosophy of Science 28: 99–127.

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.

Salmon, Wesley C. 1967. The Foundations of Scientific Inference. University of Pittsburgh Press, Pittsburgh.

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

Mises Fails Philosophy of Mathematics 101

My post below makes a broad point about the intellectual bankruptcy of aprioristic praxeology on the basis of Mises’s misunderstanding of modern epistemology and the philosophy of mathematics.

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.

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).

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.

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.


BIBLIOGRAPHY
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.