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Beyond the axioms: The question of objectivity in mathematics
"... I will be discussing the axiomatic conception of mathematics, the modern version of which is clearly due to Hilbert... ..."
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I will be discussing the axiomatic conception of mathematics, the modern version of which is clearly due to Hilbert...
RCF 1 Theories of PR Maps and Partial PR Maps
, 2008
"... Abstract: We give to the categorical theory PR of Primitive Recursion a logically simple, algebraic presentation, via equations between maps, plus one genuine Horner type schema, namely Freyd’s uniqueness of the initialised iterated. Free Variables are introduced – formally – as another names for pr ..."
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Abstract: We give to the categorical theory PR of Primitive Recursion a logically simple, algebraic presentation, via equations between maps, plus one genuine Horner type schema, namely Freyd’s uniqueness of the initialised iterated. Free Variables are introduced – formally – as another names for projections. Predicates χ: A → 2 admit interpretation as (formal) Objects {A  χ} of a surrounding Theory PRA = PR + (abstr) : schema (abstr) formalises this predicate abstraction into additional Objects. Categorical Theory P RA ⊐ PRA ⊐ PR then is the Theory of formally partial PRmaps, having Theory PRA embedded. This Theory P RA bears the structure of a (still) diagonal monoidal category. It is equivalent to “the ” categorical theory of µrecursion (and of while loops), viewed as partial PR maps. So the present approach to partial maps sheds new light on Church’s Thesis, “embedded” into a FreeVariables, formally variablefree (categorical) framework. 0 This is part 1 of a cycle on Recursive Categorical Foundations. There is a still more detailed version (pdf file) equally entitled Theories of PR Maps and
RCF 3 MapCode Interpretation via Closure
, 2008
"... For a (minimal) Arithmetical theory with higher Order Objects, i. e. a (minimal) Cartesian closed arithmetical theory – coming as such with the corresponding closed evaluation – we interprete here map codes, out of ⌈A,B ⌉ say, into these maps “themselves”, coming as elements (“names”) of homObjects ..."
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For a (minimal) Arithmetical theory with higher Order Objects, i. e. a (minimal) Cartesian closed arithmetical theory – coming as such with the corresponding closed evaluation – we interprete here map codes, out of ⌈A,B ⌉ say, into these maps “themselves”, coming as elements (“names”) of homObjects B A. The interpretation (family) uses a Chain of Universal Objects Un, one for each Order stratum with respect to “higher” Order of the Objects. Combined with closed, axiomatic evaluation, these interpretation family gives codeselfevaluation. Via the usual diagonal argument, Antinomie Richard then can be formalised within our minimal higher Order (Cartesian closed) arithmetical theory, and yields this way inconsistency, for all of its extensions, in particular of set theories as ZF, of the Elementary Theory of (higher Order) Topoi with Natural Numbers Object as considered by Freyd as well as already of the Theory of Cartesian Closed Categories with NNO considered by Lambek. 1
RCF 4 Inconsistent Quantification
, 2008
"... We exhibit canonical Choice maps within categorical theories of Primitive Recursion, of partially defined PR maps, as well as for classical, quantifier defined PR theories, and show incompatibility of these choice sections in the latter theories, with (iterative) finitedescent property of ω ω, name ..."
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We exhibit canonical Choice maps within categorical theories of Primitive Recursion, of partially defined PR maps, as well as for classical, quantifier defined PR theories, and show incompatibility of these choice sections in the latter theories, with (iterative) finitedescent property of ω ω, namely within a “minimal ” such quantor defined Arithmetic, Q. This is to give inconsistency of ZF, and even of first order set theory 1ZF strengthened by wellorder property of ω ω. The argument is iterative evaluation of PR map codes, which gets epimorphic definedarguments enumeration by above finitedescent property. This enumeration is turned into a retraction by AC, with PR section in Q + = Q + wo(ω ω), and so makes the evaluation a PR map. But the latter is excluded by Ackermann’s result that such (diagonalised) evaluation grows faster than any PR map within any consistent frame.
RCF 2 Evaluation and Consistency ∗ ε&C ∗ πOR ∗ π •
, 2009
"... Abstract: We construct here an iterative evaluation of all (coded) PR maps: progress of this iteration can be measured by descending complexity, within Ordinal O: = N[ω], of polynomials in one indeterminate, called “ω”. As (well) order on this Ordinal we choose the lexicographical one. Noninfinit ..."
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Abstract: We construct here an iterative evaluation of all (coded) PR maps: progress of this iteration can be measured by descending complexity, within Ordinal O: = N[ω], of polynomials in one indeterminate, called “ω”. As (well) order on this Ordinal we choose the lexicographical one. Noninfinit descent of such iterations is added as a mild additional axiom schema (πO) to Theory PRA = PR + (abstr) of Primitive Recursion with predicate abstraction, out of foregoing part RFC 1. This then gives (correct) ontermination of iterative evaluation of argumented deduction trees as well: for theories PRA and πOR = PRA + (πO). By means of this constructive evaluation the Main Theorem is proved, on Terminationconditioned (Inner) Soundness for Theories πOR, O extending N[ω]. As a consequence we get in fact SelfConsistency for theories πOR, namely πORderivability of πOR’s own freevariable Consistency formula ConπOR = ConπOR(k) =def ¬ProvπOR(k, �false � ) : N → 2, k ∈ N free. Here PR predicate ProvT(k, u) says, for an arithmetical theory T: number k ∈ N is a TProof code proving internally Tformula code u, arithmetised Proof in Gödel’s sense. As to expect from classiccal setting, SelfConsistency of πOR gives (unconditioned) Objective Soundness. Eventually we show TerminationConditioned Soundness “already ” for PRA. But it turns out that present derivation of SelfConsistency, and already that of Consistency formula of PRA from this conditioned Soundness “needs ” schema (˜π) of noninfinit descent in Ordinal N[ω], which is presumably not derived by PRA itself. 0 Legend of LOGO: ε for Constructive evaluation, C for SelfConsistency to be derived for suitable theories πOR, π • OR strengthening in a “mild ” way the (categorical) FreeVariables Theory PRA of Primitive Recursion with predicate abstraction Consideration of implicational version (π) of Descent axiom added
NonStandard Models of Arithmetic: a Philosophical and Historical perspective
, 2010
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©2009 The Aristotelian Society Proceedings of the Aristotelian Society Supplementary Volume lxxxiii doi: 10.1111/j.14678349.2009.00182.x
"... In making assertions one takes on commitments to the consistency of what one asserts and to the logical consequences of what one asserts. Although there is no quick link between belief and assertion, the dialectical requirements on assertion feed back into normative constraints on those beliefs tha ..."
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In making assertions one takes on commitments to the consistency of what one asserts and to the logical consequences of what one asserts. Although there is no quick link between belief and assertion, the dialectical requirements on assertion feed back into normative constraints on those beliefs that constitute one’s evidence. But if we are not certain of many of our beliefs and that uncertainty is modelled in terms of probabilities, then there is at least prima facie incoherence between the normative constraints on belief and the probabilitylike structure of degrees of belief. I suggest that the normgoverned practice relating to degrees of belief is the evaluation of betting odds. I My starting point is a little different from Professor Field’s. I’m going to begin with an area in which the normative force of logic is quite clearly discerned, at least by some authorities, and then work back from there to belief. The starting point I have in mind is assertion. The making of assertions is a rule or convention governed practice. Amongst the conventions governing assertion one stands out dramatically for present purposes: that one stands by the logical consequences of what one asserts. This is a commonplace of rational discourse. Assertion itself conventionally indicates some form of commitment to what one asserts, and this commitment carries over to the logical consequences of what one asserts. Vic Dudman has this to say: I deprecate the idea of explaining discourse in terms of belief, preferring C. L. Hamblin’s notion of individual speakers ’ commitments. To these belief is strictly irrelevant: ‘We do not believe everything we say; but our saying it commits us whether we believe it or not ’ (Fallacies (London, 1970): 264). When a speaker affirms a proposition, for example, I construe that as her incurring public commitments to its truth, not as her confiding private belief in its truth. Commitment has
1 Philosophical Method and Galileo’s Paradox of Infinity
"... You are free, therefore choose—that is to say, invent. Sartre, L'existentialisme est un humanisme Philosophy, and especially metaphysics, has often been attacked on either epistemic or semantic grounds. Anything outside of experience and the laws of logic is said to be unknowable, and accordin ..."
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You are free, therefore choose—that is to say, invent. Sartre, L'existentialisme est un humanisme Philosophy, and especially metaphysics, has often been attacked on either epistemic or semantic grounds. Anything outside of experience and the laws of logic is said to be unknowable, and according to Wittgenstein and the logical positivists, there are no such things to know; metaphysical disputes are either meaningless or merely verbal. This was thought to explain philosophy’s supposed lack of progress: philosophers argue endlessly and fruitlessly precisely because they are not really saying anything about matters of fact (Wittgenstein 1953, Remark 402; Carnap 1950). Since the midtwentieth century, the tide has been against such views, and metaphysics has reestablished itself within the analytic tradition. Ontology, essentialism, and de re necessity have regained credibility in many eyes and are often investigated by excavating intuitions of obscure origin. Relatedly, externalist semantic theories have claimed that meaning or reference has a secret life of its own, largely unfettered by our understanding and intentions (Kripke 1971; 1972; Putnam 1973; 1975a). ‘Water, ’ it is claimed, would denote H2O