semanticextensional interpretation that happens to satisfy all the axioms.


 Miles Reeves
 7 months ago
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1 No axiom, no deduction 1 Where there is no axiomsystem, there is no deduction. I think this is a fair statement (for most of us) at least if we understand (i) "an axiomsystem" in a certain logicalexpressive/normativepragmatical manner, as an expression of a system of deductiveinferential norms as to how to use the "undefined" nonlogical terms in making and taking deductive inferences under the rule of that axiomsystem, and (ii) "deduction" in the modern rigorous sense of the term, as inference in which we are concerned strictly with what follow from what under the rule of an axiomsystem (in the sense of (i) above), independently from any irrelevant specificities of any semanticextensional interpretation that happens to satisfy all the axioms. (Note) Surely, merely logical deductions i.e., deductive inferences whose validity obtains at the level of pure logic, propositional or (firstorder) predicate logic are possible without any axiomsystem (or with an "empty" axiomsystem). 1 But, I exclude them from the range of "deduction" in this writing, for the sake of simpler exposition. The purpose here is not to offer full accounts of an axiomsystem and deduction that take care of all cases or instances of an axiomsystem and deduction, but to offer an idea about how to understand or think about 1 The distinction I draw here, between logical and nonlogical deductions, roughly corresponds to the distinction between "formal" and "material" (deductive) inferences of Wilfrid Sellars and Robert Brandom. I do respect their insight that our language contains in its "grammar" (to put it deliberately loosely for now) certain norms that legitimate deductive inferences which are more substantial than mere logical ones. But, I don't think it's very appropriate to discuss this distinction by calling the logical deductions "formal" and the nonlogical deductions "material," because I think that linguistic norms that legitimate such nonlogical deductive inferences comprise something essentially like an axiomsystem, which I think is an expression of an abstract structure or form. In short, all deductions are formal (or intensional), i.e., abstinent from what is material (or extensional), which is the object of our reference rather than inference. 1
2 the essential connection between axiomatics and deduction, which I think has been "sensed" by all working mathematicians "by intuition" but has evaded elucidation. So, to focus on its purpose, I ignore logical deductions in this blog post. The slogan ("No axiom, no deduction"), with these assumptions, (i) and (ii), is fair because such a level of rigor would be generally impossible (for most of us) to maintain, in making or taking a long chain of deductive inferences (especially under a complicated system of inferential norms), unless the exact content of the inferential commitments, which are conferred on us by the axiomsystemineffect, is thus made explicit (and precise) in the form of an axiomsystem. Understood in this way, an axiomsystem may also be said to define or demarcate a deductive context, in the sense that while, and only while, we take a set of statements as an axiomsystemineffect, we enter a certain normative community or better yet, a tentative linguistic community in which we entitle and oblige one another to use the "undefined" nonlogical terms of the axioms in the way "implicitly defined" (to borrow this Hilbertian phrase) by the axiomsystem (as a whole, to be sure). Many of us may be more accustomed to thinking of an axiomsystem as a precise statement of a set of (each necessary and together sufficient) conditions for a given "intuited" dataset to satisfy in order to count as an instance of an abstractstructural concept, which is in this sense defined by the axiomsystem in question. (Note) By an "'intuited' dataset," I mean a dataset that "presents itself to us" as a concrete and inherently structured dataset, either through "empirical intuition" (i.e., perception) or through some "nonempirical intuition" (e.g., "mathematical intuition"). 2 For those of us, the notion of "inferential commitments conferred on us by an 2 All these terms related to the notion of "intuition" are put in the scarequotes because I'm critical of this passive, spectatortheoretic conception of "intuition." I'm now writing a paper to present a certain naturalistic philosophy of perception (which is a part of my "rethinking of 'mind'" project) in which this criticism is developed more fully. 2
3 axiomsystem" may be paraphrased as the discursive commitments that we incur when we abstract an "intuited" dataset D as nothing more or less than an instance of an abstractstructural concept C (defined or definable by a certain axiomsystem), whose content consists of (i) our entitlements in that context to say whatever can be said about D only in virtue of its being an instance of C, and (ii) our obligations in that context to refrain from saying anything that cannot be said about D only in virtue of its being an instance of C. My explanation here may not be as precisely on the mark as it can be. But, I hope it is enough to get across my point. My point is that the kind of logicalexpressive/normativepragmatical conception of an axiomsystem and deduction that I put forward above has some inherent connection with the traditional modeltheoretic, structural conception of them, and, so, with a version of semantic inferentialism. Anyway, the slogan above ("No axiom, no deduction") presupposes a certain systematic conception of an axiomsystem and deduction, in which the two concepts (an axiomsystem and deductiveinferential discourse/deliberation) are really mutually inseparable: An axiomsystem is what affords us deductiveinferential discourse/deliberation, and the latter is what we do (with words, or with conscious cognitions of comparable explicitness) under the "rule" or "reign" (so to speak) of an axiomsystem. Admittedly, this conception of an axiomsystem and deduction is an outcome of longlasting inspiration by (some chapters of) Robert Brandom's Making It Explicit (and Articulating Reasons), as is suggested by my use of his terminologies above. 3 3 But, I must report that, based on my recent (belated) study of the chapter 6 of the MIE, I now suspect that my conception of an axiomsystem and deduction, which I think demands a broadly Kripkean, or broadly directreferencetheoretic treatment of the phenomenon of reference, seems to oppose Brandom's total philosophy of language in this regard, while it shares with it a broad philosophical orientation or framework (consisting of logicalexpressivism, normativepragmatics, and semanticinferentialism). I believe that making/taking of referential expressions (or singular terms) can be incorporated in this general framework in a way other than suggested by him At this occasion, let me also confess that, although in this website I have mentioned his name as a major influence on my project of 3
4 Meanwhile, this conception owes an insight to Minao Kukita's "Mathematics as speech act" (a handout, in Japanese, at a workshop of JACAP conference), too. It may be fair to say that this conception is a certain broadlybrandomian extension of Kukita's original speechact theoretic approach to axiomaticdeductive discourses/deliberations in mathematics. So, in this blog post, let me call it the KB conception (of an axiomsystem and deduction), for short. Of course, it goes without saying that all responsibility for any error of this conception is mine. Moreover, this tentative designation surely does not imply that this conception has received any approval from Prof. Brandom or Prof. Kukita, or that I expect that it would. (I have no expectation either way, as of now.) The designation is primarily for want of other purely descriptive and short phrase, and, secondarily, for acknowledging my intellectual debts to them. 2 If we adopt this KB conception of an axiomsystem and deduction, then, the history of rigorous axiomatics can be seen as the history of rigorous deductive sciences. From this point of view, two historical incidents seem to stand out. The first is Newton's axiomatization of mechanics, which I have read being described (in Tomonaga, 1979, if I remember correctly) as the first axiomatic theory in the history of physics, which introduced rigorous deduction to the practice of physics. The second is found in the history of mathematics: namely, the gradual emergence in the mathematical community of the formal (in the sense of semanticsfree) treatment of diverse "intuited" datasets (of classical mathematics), by way of axiomatization of a common pattern that recurs across such diverse datasets, by which we abstract away from their formallyirrelevant specificities. The KB "Rethinking 'mind'," I did so while I had read only a few chapters of the two books. My study of his philosophy has since made some progress, which made me recently suspect the aforementioned difference. But, I'm still very much in the middle of my study. So, I may change my opinion about whether and where my philosophy differs from his. 4
5 conception requires us to study these histories anew. The preceding paragraph says all what I want to say in this section 2. But, it contains a few complicated issues, which I must clarify before moving on to the final section 3. First of all, the adjective "formal" above (as in the phrase "formal treatment") is used in the aforementioned sense of semanticsfree or extensionfree. It should not be confused with another sense of it, which has become common in the philosophies of logic, mathematics, and language, at least ever since Frege expressed, in his Begriffsschrift, a strange ideology about deduction and won almost ubiquitous consensus among the experts since then. The ideology may be put as follows: In order for us to be really rigorous in deductive inference, it is not enough, in the final analysis, to make explicit inferential norms by way of axiomatization. On top of that, such a "preformal" axiomsystem ought to be "formalized," in the sense of being turned into a symbolsystem such that (i) the syntactic notion of a statement (or a wellformed formula) is recursively defined on the set of all (combinatorially possible) symbolstrings and (ii) the prooftheoretic notion of a proof (or provability) is similarly recursively defined on the set of all (combinatorially possible) wffstrings. I refrain from any substantive criticisms of this ideology here, beyond the insinuation (that it's strange). Anyway, although I will use the word "formal" (and related words like "form," "formalism," etc.) in this blogpost, my use of it should not be understood in this sense (unless explicitly noted otherwise). Secondly, I distinguish here between an axiomatization of a common pattern that recurs across diverse classical mathematical datasets, on the one hand, and an axiomatization of a specific classical mathematical dataset, on the other. A wellknown example of the former is the axiomatization of the abstract structure of group. A wellknown example of the latter is the 5
6 axiomatization of the natural number arithmetic into the socalled Peano arithmetic. Let me call the former structural axiomatization and the latter foundational axiomatization, borrowing the nomenclature of Feferman (1999). From the standpoint of the KB conception of an axiomsystem and deduction, an axiomsystem does not count as a pure deduction system insofar as it is conceived of as a foundational axiomsystem, i.e., as a system that expresses the "most basic concepts" and the "most basic doctrines" of a body of "intuited" knowledge (empirical or mathematical). For, to conceive of an axiomsystem as a foundational system is to conceive of it as a set of extensionally interpreted statements, whereas deduction is by definition (according to the KB conception) a (discursive or deliberate) act that abstracts away from extensional interpretations. So, the aforementioned "emergence of formal treatment of diverse datasets" refers to the emergence of structural axiomatics, not foundational axiomatics. Of course, to an extent, an axiomsystem that is originally developed as a foundational system can be easily reconceptualized as a structural system. That is to say, once an "intuited" body of knowledge is axiomatized (even in the spirit of foundationalism), we can, in our conception, easily separate its original, intended interpretation from the system itself, so as to render and treat it as a structural axiomsystem from then on. A perfect example of this reconceptualization is actually found in Hilbert's 1899 axiomatization of Euclidean geometry. 4 This fact may seem to detract from the 4 In his 1899 work, Hilbert proved the consistency of his axiomsystem its consistency relative to that of real number arithmetic, to be precise by offering an interpretation to it that is other than the intended one, thereby in effect treating his system as a structural one. So, in his 1899 work, he treated his system in both ways seamlessly, first presenting it as a foundational system, and then treating it as a structural system. Now, I have an impression that the KB conception of axiomatics/deduction is still novel today, and that, behind this situation, there is this widespread underappreciation of the significance of the foundational/structural distinction for the epistemology of mathematics. If this impression is not too far from the truth, I'm tempted to suspect that a remote cause of this total situation may be found in Hilbert's seamless dual treatment of his 6
7 significance of the foundational/structural distinction for the KB conception. (Structuralists may be especially enticed to think that way.) But, I have a serious reservation for genuine formality, in the aforementioned sense, of an axiomsystem that is denumerably categorical. (Note) Let me call an axiomsystem denumerably categorical if it has a unique model "up to isomorphism," whose cardinality is denumerable. I suppose that any such denumerably categorical system, indeed, any system for which the issue of categoricity has ever been raised at all, is originally developed as a foundational system. I do not think that such denumerably categorical systems, e.g., Peano arithmetic, can afford us genuinely abstract, formaldeductive (i.e., semanticsfree) treatment of its isomorphic denumerable models. So, at least some foundational systems are inherently somewhat material, in the sense of defining a "structure" or "form" that is not purely abstract but to an extent concrete (i.e., requiring some definite reference in its identification or specification). 5 (Note) Unfortunately, I'm not yet ready to say anything about nondenumerably categorical axiomsystems, i.e., systems that have a unique nondenumerable model "up to isomorphism." (I even don't know if there is any such system.) Actually, I'm in the middle of developing my own (naturalistic) philosophy of what numbers are, or what we do in making/taking references to numbers of various sorts. And, I think that this (normativepragmaticsbased, naturalistic) philosophy of numbers will demand a fundamental reappraisal of the standard conception (or treatment) of the denumerable/nondenumerable distinction itself, in a way that is inseparable from the issues of foundational/structural distinction and the "ontology" of "mathematical axiomsystem I explain this opinion in some more details in my On the Aufbau, in which I offer a criticism of Carnap's "structuralism" which can be read as a contrary opinion to mine on this issue. (The relevant part comes in pp But, the discussion there presupposes a lot of materials developed by up to that point.) 7
8 entities." Before this reappraisal, I cannot say anything about any metamathematical results which appeals to this distinction. In any case, in the actual history of mathematics, the emergence of structural and foundational axiomatics are seamlessly interconnected (as pointed out in the footnote 4), although only the emergence of structural axiomatics counts as the emergence of genuinely formaldeductive sciences, according to the KB conception. So, in practice, the historical study of how the genuinely formaldeductive science has been developed in the mathematical community, and how this development has been perceived in that community (including the very mathematicians who contributed to this development, such as Hilbert), involves the study of how its development and its perception have interacted with the development of various foundational axiomsystems, as well as with the development of various "formalizations" (in the sense of recursivesyntactification) of deduction. (Note) An SEP article on the FregeHilbert controversy is brief but informative for this study. Benefit may be mutual. That is, just having this article (coincidentally), I'm now inclined to guess that the KB conception, and a normativepragmaticsbased analysis of the difference between foundational and structural axiomsystems, may allow us to put this controversy in a new, and revealing, historical perspective. Finally, having made the second point (about the strict, even cardinalityfree sense of formality that is assumed in the KB conception of an axiomsystem qua a deduction system), I confess that I'm not yet ready to say anything about axiomatics in physics, either. The reason here is related with the reason for which I cannot say anything about the nondenumerably categorical axiomsystems. In my lay understanding, theories of physics are generally such that, in making and taking them, we commit ourselves to the "existence" of some classicalmathematical "intuited" systems, i.e., concrete and inherently structured datasets about which we are supposed to have (intersubjectively shared) "mathematical intuitions" (e.g., number systems and intuitive Euclidean geometry). Perhaps, theories of physics 8
9 may no longer commit us to anything of intuitive Euclidean geometry. (I'm too ignorant to say anything conclusive about this.) But, I presume (despite my ignorance) that they ontologically commit us to various number systems, by making us make/take definite references to numbers of various sorts. As I stated above, I'm still in the middle of developing an account of the pragmatics of making/taking of numberreferences. So, I cannot say anything about the history of rigorous axiomatics in physics But, despite all this, I still think that even (numberreferring) axiomatic theories in physics can be seen in a broadly KB conception's line. 3 Based on the KB conception, I think that it's fair to say that rigorous deduction became available in the mathematical community only as the structural axiomatics was gradually, and spontaneously, emerged in that community. I'm very ignorant of this history of mathematics, as yet. But, as a KB conception subscriber, I'm very much interested in this historical development. Being an ignorant amateur, I have this impression that mathematics (i.e., "pure" mathematics, excluding "applied" mathematics entirely) has radically changed its outlook before and after the emergence of the structural axiomatics. Before it, mathematics was just classical mathematics, studies of various "intuited" datasets or structures. After it, we seem to have a spectrum of mathematical subjectmatters, from the most concrete ones of classical mathematics to the most abstract ones of firstorder theories without definite reference (or singular terms), such as group or lattice. Between these two extremes, mathematicians seem to have developed/discovered a host of mathematical datasets or structures that have both aspects in various mixtures. What happened to the mathematical community, which brought about this outcome? Of course, to develop any fullfledged answer to this question, I must study 9
10 the actual history of mathematics, and, even before that, must first study prerequisite mathematics for this historicalphilosophical enquiry. But, I have a sort of a "big picture" hypothesis, about one thing that I think is the most important ingredient for this historical development. I think that what happened can be described (again, in retrospect) as an episode of language evolution. More specifically, I hypothesize that the mathematical community (as a special kind of linguistic community, united by making and taking of mathematical statements which are often written in the crosslinguistically shared notations) gradually developed a new metalinguistic (or metapragmatic) competence with which to rigorously talk/think about their own talking/thinking about classical mathematical datasets. Yet more specifically, I think that the emergence of the metapragmatic competence in question is centrally observed with the change in the ways in which mathematicians use the locution of quantification, with its inseparable connection with the use of the equality symbol, =. But, to test this hypothesis, I must first clarify the pragmatics of quantification and the equality symbol, by which the current practice of abstract mathematics is enabled. In other words, I must first substantiate the technical details of the KB conception of axiomatics/deduction. This is where I am, as for my philosophy of mathematics. 10
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