Where do axioms come from and how are they tested?
Axioms are starting points of deductive systems. They are stipulations.
Axioms must avoid the traps of illogic, circularity, and infinite regress.
Axioms in scientific theories are assumptions derived either from induction as generalization from experience (data) or abduction in C. S. Peirce's sense as discovery through "educated guessing."
The objective of scientific inquiry is not to "prove" axioms, since axioms are the basis of proof in a logical system. Axioms function as criteria for syntactical or logical truth, also called "necessity."
Axioms in hypothetical-deductive systems provide the basis for the scientific method.
Assumptions of representational models stand or fall with the testing of models they are used to construct.
Scientific models are used to generate hypotheses that can be tested empirically by deriving hypotheses as theorems from the assumptions that serve as axioms for the system.
Failure of a hypothesis as a theorem of deductive system reveals semantic inconsistency or incompleteness and calls the system into question as a coherent explanation of the data.
This also calls the explanatory model into question as providing a causal explanation based on a representational model purporting to show causal transmission.
Correlation is not causation. Without a theory explicating transmission, there is no properly scientific explanation.
Lars P. Syll’s Blog
Why testing axioms is necessary in economics
Lars P. Syll | Professor, Malmo University
2 comments:
Don Lars and the axiomatic windmill
Comment on Lars Syll on ‘Why testing axioms is necessary in economics’
Lars Syll writes often about the axiomatic-deductive method but it is painfully obvious that he simply cannot get his head around it.
(i) He writes: “For although the economist himself (implicitly) claims that his axiom is universally accepted as true and in no need of proof, that is in no way a justified reason for the rest of us to simpliciter accept the claim.”
It is simply a silly misunderstanding that axioms have to be universally accepted as true. Axiomatization works as follows: “The attempt is made to collect all the assumptions, which are needed, but no more, to form the apex of the system. They are usually called the ‘axioms’ (or ‘postulates’, or ‘primitive propositions’; no claim of truth is implied in the term ‘axiom’ as here used). The axioms are chosen in such a way that all the other statements belonging to the theoretical system can be derived from the axioms by purely logical or mathematical transformations.” (Popper)#1
Repeat: No claim of truth is implied in the term ‘axiom’ as here used. A set of axioms is the point to start with and it is taken as tentatively true. Nobody starts intentionally with false axioms. To establish the empirical truth of a properly axiomatized theory is understood as the indispensable complementary task of the scientific enterprise: “Research is in fact a continuous discussion of the consistency of theories: formal consistency insofar as the discussion relates to the logical cohesion of what is asserted in joint theories; material consistency insofar as the agreement of observations with theories is concerned.” (Klant)
Axioms must satisfy certain formal properties, e.g. independence and consistency.
(ii) It is misleading to retell again and again that ‘axioms must be universally accepted as true’. This has NEVER been the case and the best example is Euclid’s famous fifth postulate: “For two thousand years, many attempts were made to prove the parallel postulate using Euclid’s first four postulates. The main reason that such a proof was so highly sought after was that, unlike the first four postulates, the parallel postulate is not self-evident.”#2
(iii) It is also misleading to retell again and again that “… of course, the rejection of the parallel postulate (or axiom) did come from empirical tests showing that it does not hold in space-time in general due to gravity curving it.” (Rosser)
It was just the other way round. Non-Euclidean geometry had been developed well before relativity theory by mathematicians (Lobachevsky, Bolyai, Gauss, Hilbert, Rieman, Poincare etc). Without non-Euclidean geometry Einstein could not have formulated relativity theory. It is well known by now that he was not particularly good at math.
(iv) With regard to testing it holds as a matter of course: “Whether an axiom is or is not valid can be ascertained either through direct experimentation or by verification through the result of observations, or, if such a thing is impossible, the correctness of the axiom can be judged through the indirect method of verifying the laws which proceed from the axiom by observation or experimentation. (If the axiom is deemed to be incorrect it must be modified or instead a correct axiom must be found.)” (Morishima)#3
See part 2
Part 2
(v) This, of course, holds also for the Walrasian axioms, which are given as follows: “HC1 economic agents have preferences over outcomes; HC2 agents individually optimize subject to constraints; HC3 agent choice is manifest in interrelated markets; HC4 agents have full relevant knowledge; HC5 observable outcomes are coordinated, and must be discussed with reference to equilibrium states.” (Weintraub)
Now it should be pretty obvious that the Walrasian axiom set contains THREE NONENTITIES: (i) constrained optimization (HC2), (ii) rational expectations (HC4), (iii) equilibrium (HC5). Every model that contains a nonentity is A PRIORI false. In practical terms: as soon as the word equilibrium/disequilibrium appears in an economic paper it can be thrown into the waste basket. The same holds for utility maximization and all other nonentities. This simplifies matters considerably.
(vi) Up to this point the critique is on the right tract but then it exhausts itself in futile repetition. There is no ambition to move forward and to tackle the paradigm shift. Thus, economics got stuck in the pluralism of false theories/models.
(vii) Fact is that Walrasianism is a failed approach, so there is no longer any need to test it. What is urgently needed is a replacement. What has to be done is to move from obsolete microfoundations to macrofoundations.
(viii) This is the (tentatively) true set of macro axioms: (A0) The objectively given and most elementary configuration of the economy consists of the household and the business sector which in turn consists initially of one giant fully integrated firm. (A1) Yw=WL wage income Yw is equal to wage rate W times working hours. L, (A2) O=RL output O is equal to productivity R times working hours L, (A3) C=PX consumption expenditure C is equal to price P times quantity bought/sold X.#4
These axioms are objective, behavior-free, certain, true, and primary, and therefore satisfy all methodological requirements. The set of premises is minimal, that is, it cannot be reduced further, only expanded. The set contains no longer nonentities like maximization or equilibrium and no normative assertions.
(ix) Lars Syll concludes: “If theories and models do not directly or indirectly tell us anything of the world we live in ― then why should we waste any of our precious time on them?” True, but the same holds for the critique of mainstream economics.#5 The four main approaches are known to be axiomatically false and the sooner they are replaced and forgotten the better.
Egmont Kakarot-Handtke
#1 See also ‘Keynes, Euclid, and economic methodology’
http://axecorg.blogspot.de/2015/02/keynes-euclid-and-economic-methodology.html
#2 Wikipedia Parallel postulate
https://en.wikipedia.org/wiki/Parallel_postulate
#3 See also ‘Methodological wrong-way drivers’
https://axecorg.blogspot.de/2016/05/methodological-wrong-way-drivers.html
#4 See ‘How to restart economics’
https://axecorg.blogspot.de/2016/01/how-to-restart-economics.html
#5 See also ‘How the mainstream vanished in the gutter’
http://axecorg.blogspot.de/2016/10/how-mainstream-vanished-in-gutter.html
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