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5. Theory of Logic / K. Features of Logics / 1. Axiomatisation

[giving basic truths from which some system is deduced]

31 ideas
Aristotle's axioms (unlike Euclid's) are assumptions awaiting proof [Aristotle, by Leibniz]
     Full Idea: Aristotle's way with axioms, rather than Euclid's, is as assumptions which we are willing to agree on while awaiting an opportunity to prove them
     From: report of Aristotle (Posterior Analytics [c.327 BCE], 76b23-) by Gottfried Leibniz - New Essays on Human Understanding 4.07
     A reaction: Euclid's are understood as basic self-evident truths which will be accepted by everyone, though the famous parallel line postulate undermined that. The modern view of axioms is a set of minimum theorems that imply the others. I like Aristotle.
It is always good to reduce the number of axioms [Leibniz]
     Full Idea: To reduce the number of axioms is always something gained.
     From: Gottfried Leibniz (New Essays on Human Understanding [1704], 4.06)
     A reaction: This is rather revealing about the nature of axioms. They don't have any huge metaphysical status - in fact one might say that their status is epistemological, or even pedagogic. They enable us to get out minds round things.
To understand axioms you must grasp their logical power and priority [Frege, by Burge]
     Full Idea: Understanding the axioms depends not only on understanding Frege's elucidatory remarks about the interpretation of his symbols, but also on understanding their logical structure - their power to entail other truths, and their reason-giving priority.
     From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], 4) by Tyler Burge - Frege on Knowing the Foundations 4
     A reaction: This is a distinctively Burgean spin put on what Frege has to say about axioms, but I like it, and it seems well enough supported in Frege's writings (e.g. 1914).
Tracing inference backwards closes in on a small set of axioms and postulates [Frege]
     Full Idea: We can trace the chains of inference backwards, …and the circle of theorems closes in more and more. ..We must eventually come to an end by arriving at truths can cannot be inferred, …which are the axioms and postulates.
     From: Gottlob Frege (Logic in Mathematics [1914], p.204)
     A reaction: The rival (more modern) view is that that all theorems are equal in status, and axioms are selected for convenience.
The essence of mathematics is the kernel of primitive truths on which it rests [Frege]
     Full Idea: Science must endeavour to make the circle of unprovable primitive truths as small as possible, for the whole of mathematics is contained in this kernel. The essence of mathematics has to be defined by this kernel of truths.
     From: Gottlob Frege (Logic in Mathematics [1914], p.204-5)
     A reaction: [compressed] I will make use of this thought, by arguing that mathematics may be 'explained' by this kernel.
Axioms are truths which cannot be doubted, and for which no proof is needed [Frege]
     Full Idea: The axioms are theorems, but truths for which no proof can be given in our system, and no proof is needed. It follows from this that there are no false axioms, and we cannot accept a thought as an axiom if we are in doubt about its truth.
     From: Gottlob Frege (Logic in Mathematics [1914], p.205)
     A reaction: He struggles to be as objective as possible, but has to concede that whether we can 'doubt' the axiom is one of the criteria.
A truth can be an axiom in one system and not in another [Frege]
     Full Idea: It is possible for a truth to be an axiom in one system and not in another.
     From: Gottlob Frege (Logic in Mathematics [1914], p.205)
     A reaction: Frege aspired to one huge single system, so this is a begrudging concession, one which modern thinkers would probably take for granted.
The truth of an axiom must be independently recognisable [Frege]
     Full Idea: It is part of the concept of an axiom that it can be recognised as true independently of other truths.
     From: Gottlob Frege (On Euclidean Geometry [1900], 183/168), quoted by Tyler Burge - Frege on Knowing the Foundations 4
     A reaction: Frege thinks the axioms of arithmetic all reside in logic.
The facts of geometry, arithmetic or statics order themselves into theories [Hilbert]
     Full Idea: The facts of geometry order themselves into a geometry, the facts of arithmetic into a theory of numbers, the facts of statics, electrodynamics into a theory of statics, electrodynamics, or the facts of the physics of gases into a theory of gases.
     From: David Hilbert (Axiomatic Thought [1918], [03])
     A reaction: This is the confident (I would say 'essentialist') view of axioms, which received a bit of a setback with Gödel's Theorems. I certainly agree that the world proposes an order to us - we don't just randomly invent one that suits us.
Axioms must reveal their dependence (or not), and must be consistent [Hilbert]
     Full Idea: If a theory is to serve its purpose of orienting and ordering, it must first give us an overview of the independence and dependence of its propositions, and second give a guarantee of the consistency of all of the propositions.
     From: David Hilbert (Axiomatic Thought [1918], [09])
     A reaction: Gödel's Second theorem showed that the theory can never prove its own consistency, which made the second Hilbert requirement more difficult. It is generally assumed that each of the axioms must be independent of the others.
Some axioms may only become accepted when they lead to obvious conclusions [Russell]
     Full Idea: Some of the premisses (of my logicist theory) are much less obvious than some of their consequences, and are believed chiefly because of their consequences. This will be found to be always the case when a science is arranged as a deductive system.
     From: Bertrand Russell (Logical Atomism [1924], p.145)
     A reaction: We shouldn't assume the model of self-evident axioms leading to surprising conclusions, which is something like the standard model for rationalist foundationalists. Russell nicely points out that the situation could be just the opposite
Which premises are ultimate varies with context [Russell]
     Full Idea: Premises which are ultimate in one investigation may cease to be so in another.
     From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.273)
The sources of a proof are the reasons why we believe its conclusion [Russell]
     Full Idea: In mathematics, except in the earliest parts, the propositions from which a given proposition is deduced generally give the reason why we believe the given proposition.
     From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.273)
Finding the axioms may be the only route to some new results [Russell]
     Full Idea: The premises [of a science] ...are pretty certain to lead to a number of new results which could not otherwise have been known.
     From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.282)
     A reaction: I identify this as the 'fruitfulness' that results when the essence of something is discovered.
The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner]
     Full Idea: The inherent limitations of the axiomatic method were first brought to light by the incompleteness theorems.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Koellner - On the Question of Absolute Undecidability 1.1
Logic doesn't split into primitive and derived propositions; they all have the same status [Wittgenstein]
     Full Idea: All the propositions of logic are of equal status: it is not the case that some of them are essentially primitive propositions and others essentially derived propositions.
     From: Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921], 6.127)
     A reaction: So axioms are conventional. This specifically contradicts the claims of Frege and the earlier Russell. Their view is that logic has an explanatory essence, found in some core axioms or rules or concepts. I agree with them.
An axiom has no more authority than a frenzy [Cioran]
     Full Idea: This earth is a place where can confirm anything with an equal likelihood: here axioms and frenzies are interchangeable.
     From: E.M. Cioran (A Short History of Decay [1949], 3)
     A reaction: A perceptive and poetic expression of the modern anti-Euclidean and anti-Fregean view of axioms, as purely formal features of a model or system.
Axioms reveal the underlying assumptions, and reveal relationships between different areas [Kline]
     Full Idea: The axiomatic method ....revealed precisely what assumptions underlie each branch [of mathematics] and made possible the comparison and clarification of the relationships of various branches.
     From: Morris Kline (Mathematical Thought from Ancient to Modern Times [1972], p.1027), quoted by Penelope Maddy - Defending the Axioms 1.3
     A reaction: I take this to be the 'fruitfulness' which marks out the discovery of the essence of something.
We come to believe mathematical propositions via their grounding in the structure [Burge]
     Full Idea: A deeper justification for believing in [mathematical] propositions [apart from pragmatism] lies in finding their place in a logicist proof structure, by understanding the grounds within this structure that support them.
     From: Tyler Burge (Frege on Knowing the Foundations [1998], 3)
     A reaction: This generalises to doubting something until you see what grounds it.
'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry]
     Full Idea: The purpose of a 'classificatory' axiomatic theory is to single out an otherwise disparate species of structures by fixing certain features of morphology. ...The aim is to single out common features.
     From: John Mayberry (What Required for Foundation for Maths? [1994], p.406-2)
Axiomatiation relies on isomorphic structures being essentially the same [Mayberry]
     Full Idea: The central dogma of the axiomatic method is this: isomorphic structures are mathematically indistinguishable in their essential properties.
     From: John Mayberry (What Required for Foundation for Maths? [1994], p.406-2)
     A reaction: Hence it is not that we have to settle for the success of a system 'up to isomorphism', since that was the original aim. The structures must differ in their non-essential properties, or they would be the same system.
'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry]
     Full Idea: The purpose of what I am calling 'eliminatory' axiomatic theories is precisely to eliminate from mathematics those peculiar ideal and abstract objects that, on the traditional view, constitute its subject matter.
     From: John Mayberry (What Required for Foundation for Maths? [1994], p.407-1)
     A reaction: A very interesting idea. I have a natural antipathy to 'abstract objects', because they really mess up what could otherwise be a very tidy ontology. What he describes might be better called 'ignoring' axioms. The objects may 'exist', but who cares?
Mathematics must be based on axioms, which are true because they are axioms, not vice versa [Tait, by Parsons,C]
     Full Idea: The axiomatic conception of mathematics is the only viable one. ...But they are true because they are axioms, in contrast to the view advanced by Frege (to Hilbert) that to be a candidate for axiomhood a statement must be true.
     From: report of William W. Tait (Intro to 'Provenance of Pure Reason' [2005], p.4) by Charles Parsons - Review of Tait 'Provenance of Pure Reason' §2
     A reaction: This looks like the classic twentieth century shift in the attitude to axioms. The Greek idea is that they must be self-evident truths, but the Tait-style view is that they are just the first steps in establishing a logical structure. I prefer the Greeks.
Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy]
     Full Idea: At the end of the nineteenth century there was a renewed emphasis on rigor, the central tool of which was axiomatization, along the lines of Hilbert's axioms for geometry and Dedekind's axioms for real numbers.
     From: Penelope Maddy (Defending the Axioms [2011], 1.3)
If two mathematical themes coincide, that suggest a single deep truth [Maddy]
     Full Idea: The fact that two apparently fruitful mathematical themes turn out to coincide makes it all the more likely that they're tracking a genuine strain of mathematical depth.
     From: Penelope Maddy (Defending the Axioms [2011], 5.3ii)
Axioms are either self-evident, or stipulations, or fallible attempts [Brown,JR]
     Full Idea: The three views one could adopt concerning axioms are that they are self-evident truths, or that they are arbitrary stipulations, or that they are fallible attempts to describe how things are.
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch.10)
     A reaction: Presumably modern platonists like the third version, with others choosing the second, and hardly anyone now having the confidence to embrace the first.
The axioms of group theory are not assertions, but a definition of a structure [Linnebo]
     Full Idea: Considered in isolation, the axioms of group theory are not assertions but comprise an implicit definition of some abstract structure,
     From: Øystein Linnebo (Philosophy of Mathematics [2017], 3.5)
     A reaction: The traditional Euclidean approach is that axioms are plausible assertions with which to start. The present idea sums up the modern approach. In the modern version you can work backwards from a structure to a set of axioms.
To investigate axiomatic theories, mathematics needs its own foundational axioms [Linnebo]
     Full Idea: Mathematics investigates the deductive consequences of axiomatic theories, but it also needs its own foundational axioms in order to provide models for its various axiomatic theories.
     From: Øystein Linnebo (Philosophy of Mathematics [2017], 4.1)
     A reaction: This is a problem which faces the deductivist (if-then) approach. The deductive process needs its own grounds.
A compact axiomatisation makes it possible to understand a field as a whole [Walicki]
     Full Idea: Having such a compact [axiomatic] presentation of a complicated field [such as Euclid's], makes it possible to relate not only to particular theorems but also to the whole field as such.
     From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1)
Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki]
     Full Idea: Axiomatic systems, their primitive terms and proofs, are purely syntactic, that is, do not presuppose any interpretation. ...[142] They never address the world directly, but address a possible semantic model which formally represents the world.
     From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1)
Axioms are 'categorical' if all of their models are isomorphic [Colyvan]
     Full Idea: A set of axioms is said to be 'categorical' if all models of the axioms in question are isomorphic.
     From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2)
     A reaction: The best example is the Peano Axioms, which are 'true up to isomorphism'. Set theory axioms are only 'quasi-isomorphic'.