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The axiomatic method is today mostly associated with mathematics. However, historically there have been some works, as for example Spinoza's Ethics, that have applied axiomatic method to philosophy, or Woodger's The Axiomatic Method in Biology, which tries to apply axiomatic method do biology.

Are there any modern examples of works in which axiomatic method has been applied to something else than mathematics?

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    Do you have an idea of what would be considered to have been analyzed axiomatically when it's not mathematical? Physics? Law? – Mitch Jun 27 '11 at 20:17
  • Isn't this done by any philosopher that has an epistemological system? Start with (or maybe sometimes work backward to...) axiomatic presuppositions, with logic creating the necessary valid conclusions. – LightCC Dec 29 '15 at 6:42
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Within the subject of rational choice theory, there has been an extensive axiomatic development of various rational decision theories, in which general principles of rational decision-making are put forth in a general context, and then detailed arguments are made to deduce further consequences from them. A major issue had been the extent to which the resulting theories were descriptive or prescriptive of rational decision-making, since experiments have now shown that apparently rational people do not generally conform with the originally proposed axioms, leading researchers to consider alternatives (for example, see this article).

But I would also answer your question with the claim that whenever a theory truly adopts an axiomatic mathematic method, in the style of Euclid, then it in effect becomes a piece of mathematics. Many mathematicians would find the process of reasoning from crisply stated axioms to conclusions as laying at the very heart of mathematics. And so it may be that whenever a subject adopts a truly axiomatic method, it thereby itself becomes to that extent mathematics.

  • And Rational Choice offers an axiomatic basis for a lot of microeconomics too... – Seamus Jun 29 '11 at 10:01
  • Does this mean that the Italian school of Algebraic Geometry were not doing mathematics until Grothendieck placed their work in axiomatic terms? It seems to me that mathematics is more a process that is punctuated with axiomatics that both condense and amplify the previous work as well as closing and opening it up. – Mozibur Ullah Mar 9 '13 at 3:36
  • @MoziburUllah I think you might be reading the above stated implication as an equivalence. What was asserted is "if axiomatic, then mathematical". It was not stated that "if not axiomatic, then not mathematical". – Dennis Mar 29 '13 at 3:57
  • @Dennis: yes, I think you're right. – Mozibur Ullah Mar 29 '13 at 8:34
  • @MoziburUllah: A minor correction --- Zariski and Weil had made algebraic geometry perfectly rigorous before anyone had heard of Grothendieck. – WillO Mar 10 '15 at 3:27
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Another extremely natural answer arises, of course, with Spinoza, many of whose arguments follow a deductive axiomatic style, with formally stated axioms, definitions, theorems and corollaries.

Here is what Lucian Wischik has to say about Spinoza's ethics:

One of the most remarkable features of the Ethics is its axiomatic form. Spinoza sets out at the start a small number of definitions and axioms that are assuredly true, and proceeds to deduce from these the rest of his philosophy. In this respect, the work is an attempt to use a theory of philosophy that is modelled upon Euclid’s Elements.

Here is Charles Jarrett's article for the Canadian Journal of Philosophy on Spinoza's ontological argument.

Here are A. Pruss's lecture notes on Spinoza, in which he asserts:

Spinoza’s approach is geometrical, that is modeled on the reasoning in geometry. Euclid defined various terms, provided axioms, and everything else was to be proved from the axioms and definitions.

  • Thanks; I see now, however, that Spinoza was mentioned already in the question. – JDH Jul 2 '11 at 9:41
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The axiomatic method is fundamental to computer sciences. A good resource and explanation of this is An Axiomatic Basis for Computer Programming.

Today, virtually every field leverages the power of computer software to some degree. Since computer software depends on the axiomatic method, and computer software is used in virtually every field, we can deduce that virtually every field uses the axiomatic method to some degree.

The reason this wasn't obvious to you is because of the role in the user's consciousness that the axiomatic method plays. With computer programs, its use is implicit and a user may be completely unaware of how the processing is taking place. It is not necessary to understand how the axiomatic method is being applied in order to leverage its power with a computer program. By contrast, if you were to read a book about the application of the axiomatic method to biology, then the use of the axiomatic method would be very obvious, because you couldn't understand the book without understanding how it was being applied. The same method/process is occurring in both cases. The only difference is the human participant's awareness of its role.

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    Is it your position that when I make sushi rice with my rice cooker, then I am really using the axiomatic method (perhaps without realizing it)? After all, my rice cooker is running a computer program in its control unit. – JDH Jun 28 '11 at 12:00
  • @JDH, no. You are making sushi rice. The computer in your rice cooker is using an application of the axiomatic method. – smartcaveman Jun 28 '11 at 16:06
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Physics. Hilbert's 6th problem is explicitly the problem of axiomatizing physics, although meant exactly by 'axiomatizing' has been debated, and may not be what you mean by it. However, 21st century work by Hajnal Andréka axiomatizes special and general relativity, and she uses that word in a sense that would satisfy the most demanding of definitions. For an upcoming conference relating to these issues, see:

http://www.renyi.hu/conferences/lrb15/

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There is a limit to the axiomatic approach. Applying it to language has a "deadening" impact. It also does violence to "meaning". See the following in Interpretive Social Science: A Second Look:

    From the interpretive point of view what is most striking about structuralism is not its difference from but its continuity with the older reductionism. That massive continuous theme is the priority and independence of logical structures and rules of inference from the contexts of ordinary understanding. As Lévi-Strauss puts it, one must avoid the "shop-grip's web of subjectivity" or the "swamps of experience" to arrive at structure and science. The ideal or "hope" of the intrinsic intelligibility of structures apart from "all sorts of extraneous elements" is the same animus that propelled the Vienna Circle. Ricoeur, in several of his essays, has drawn the clearest implications of this position. For him, the goals of structuralism can be accomplished, in fact already have been, but at a price the structuralists ignore. The conditions which make the enterprise possible—the establishment of operations and elements, and an algebra of their combinations—assure from the beginning and by definition that one is working on a body of material which is reconstituted, stopped, closed, and in a certain sense, dead.[19] The very success of structuralism leaves behind the "understanding of action, operations and process, all of which are constitutive of meaningful discourse. Structuralism seals its formalized language off from discourse, and therefore from the human world.[20] A high price indeed for the sciences of man, although one the structuralists are explicitly willing to pay in the name of science.[21] (12–13)

[19] See Paul Ricoeur, "Structure, Word, Event" in Conflict of Interpretations: Essays in Hermeneutics (Evanston: Northwestern University Press, 1974), 79.
[20] Ibid.
[21] An enterprise such as that of Jacques Derrida might be termed a "poststructuralism" which conceives an absolute text that refers only to itself and consists in the endless play of signifiers in a closed and again ultimately dead and meaningless system. See Jacques Derrida, "Structure, Sign and Play in the Discourse of the Human Sciences," in The Structuralist Controversy: The Languages of Criticism and the Sciences of Man, ed. Richard Macksey and Eugenio Donato (Baltimore: Johns Hopkins Press, 1970), 247–64.

From the cited Conflict of Interpretations: Essays in Hermeneutics:

    1. I wish to show that the type of intelligibility that is expressed in structuralism prevails in every case in which one can: (a) work on a corpus already constituted, finished, closed, and in that sense, dead; (b) establish inventories of elements and units; (c) place these elements or units in relations of opposition, preferably binary opposition; and (d) establish an algebra or combinatory system of these elements and opposed pairs.
    The aspect of language which lends itself to this inventory I will designate a language [langue]; the inventories and combinations which this language yields I will term taxonomies; and the model which governs the investigation I will call semiotics.
    2. I next wish to show that the very success of this undertaking entails (as a counterpart) an elimination from structural thinking of any understanding of the acts, operations, and processes that constitute discourse. Structuralism leads to thinking in an antinomic way about the relation between language and speech. I will make the sentence or utterance [énoncé] the pivot of this second investigation. I will call semantics the model which governs our understanding of the sentence. (79)

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"physically-motivated axioms for a physical theory".

Is there life beyond quantum mechanics? A. Kapustin

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There's Paninis work in 4th Century BCE, in Gandara; he axiomatised Sanskrit grammar in his Astadyayi.

Notably, today we have such axiomatics for artificial languages - c, or c++ for example.

The axiomatic method is so closely identified with Euclid; that its also worth noting that it was the Pythagorean Arcytas, two centuries before him who write the first Elements - but probably without Euclids perfection of style and substance.

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