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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique

[objections to the logicism view of maths]

40 ideas
Kant taught that mathematics is independent of logic, and cannot be grounded in it [Kant, by Hilbert]
     Full Idea: Kant taught - and it is an integral part of his doctrine - that mathematics treats a subject matter which is given independently of logic. Mathematics, therefore, can never be grounded solely in logic.
     From: report of Immanuel Kant (Critique of Pure Reason [1781]) by David Hilbert - On the Infinite p.192
     A reaction: Presumably Gödel's Incompleteness Theorems endorse the Kantian view, that arithmetic is sui generis, and beyond logic.
If 7+5=12 is analytic, then an infinity of other ways to reach 12 have to be analytic [Kant, by Dancy,J]
     Full Idea: Kant claimed that 7+5=12 is synthetic a priori. If the concept of 12 analytically involves knowing 7+5, it also involves an infinity of other arithmetical ways to reach 12, which is inadmissible.
     From: report of Immanuel Kant (Critique of Pure Reason [1781], B205/A164) by Jonathan Dancy - Intro to Contemporary Epistemology 14.3
Frege's platonism and logicism are in conflict, if logic must dictates an infinity of objects [Wright,C on Frege]
     Full Idea: Frege's platonism seems to be in some tension with logicism: for the thought is unprepossessing that logic should dictate the existence of infinitely many objects of some kind.
     From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Crispin Wright - Frege's Concept of Numbers as Objects Intro
     A reaction: Obviously Frege didn't think this, but then the crux seems to be that Frege believed that there was a multitude of logical truths awaiting discovery, while modern logic just seems to be about the logical consequences of things.
Why should the existence of pure logic entail the existence of objects? [George/Velleman on Frege]
     Full Idea: If a distinguishing features of logic is its complete generality, focusing on truth in general, why should the existence of logic entail the existence of infinitely many objects? ..How can it be completely general if it has ontological commitments?
     From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2
     A reaction: This strikes me as simple and devastating. It depends how you conceive logic, but I only conceive it as the formalised rules of successful reasoning. I can't comprehend the claim that without certain objects, reasoning would be impossible.
Frege's belief in logicism and in numerical objects seem uncomfortable together [Hodes on Frege]
     Full Idea: Frege's views on arithmetic centred on two central theses, that mathematics is really logic, and that it is about distinctively mathematical sorts of objects, such as cardinal numbers. These theses seem uncomfortable passengers in a single boat.
     From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Harold Hodes - Logicism and Ontological Commits. of Arithmetic
     A reaction: This question pinpoints precisely my unease about Frege. I take logic to be the rules for successful reasoning, so I don't see how they can have ontological implications. It is very extreme platonism to say that right reasoning requires logical objects.
Frege only managed to prove that arithmetic was analytic with a logic that included set-theory [Quine on Frege]
     Full Idea: Frege claimed to have proved that the truths of arithmetic are analytic, but the logic capable of encompassing this reduction was logic inclusive of set theory.
     From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Willard Quine - Philosophy of Logic Ch.5
Late in life Frege abandoned logicism, and saw the source of arithmetic as geometrical [Frege, by Chihara]
     Full Idea: Near the end of his life, Frege completely abandoned his logicism, and came to the conclusion that the source of our arithmetical knowledge is what he called 'the Geometrical Source of Knowledge'.
     From: report of Gottlob Frege (Sources of Knowledge of Mathematics [1922]) by Charles Chihara - A Structural Account of Mathematics Intro n3
     A reaction: We have, rather crucially, lost touch with the geometrical origins of arithmetic (such as 'square' numbers), which is good news for the practice of mathematics, but probably a disaster for the philosophy of the subject.
Logic already contains some arithmetic, so the two must be developed together [Hilbert]
     Full Idea: In the traditional exposition of the laws of logic certain fundamental arithmetic notions are already used, for example in the notion of set, and to some extent also of number. Thus we turn in a circle, and a partly simultaneous development is required.
     From: David Hilbert (On the Foundations of Logic and Arithmetic [1904], p.131)
     A reaction: If the Axiom of Infinity is meant, it may be possible to purge the arithmetic from the logic. Then the challenge to derive arithmetic from it becomes rather tougher.
In 'Principia Mathematica', logic is exceeded in the axioms of infinity and reducibility, and in the domains [Bernays on Russell/Whitehead]
     Full Idea: In the system of 'Principia Mathematica', it is not only the axioms of infinity and reducibility which go beyond pure logic, but also the initial conception of a universal domain of individuals and of a domain of predicates.
     From: comment on B Russell/AN Whitehead (Principia Mathematica [1913], p.267) by Paul Bernays - On Platonism in Mathematics p.267
     A reaction: This sort of criticism seems to be the real collapse of the logicist programme, rather than Russell's paradox, or Gödel's Incompleteness Theorems. It just became impossible to stick strictly to logic in the reduction of arithmetic.
Formalists neglect content, but the logicists have focused on generalizations, and neglected form [Ramsey]
     Full Idea: The formalists neglected the content altogether and made mathematics meaningless, but the logicians neglected the form and made mathematics consist of any true generalisations; only by taking account of both sides can we obtain an adequate theory.
     From: Frank P. Ramsey (The Foundations of Mathematics [1925], §1)
     A reaction: He says mathematics is 'tautological generalizations'. It is a criticism of modern structuralism that it overemphasises form, and fails to pay attention to the meaning of the concepts which stand at the 'nodes' of the structure.
Mathematical abstraction just goes in a different direction from logic [Bernays]
     Full Idea: Mathematical abstraction does not have a lesser degree than logical abstraction, but rather another direction.
     From: Paul Bernays (On Platonism in Mathematics [1934], p.268)
     A reaction: His point is that the logicists seem to think that if you increasingly abstract from mathematics, you end up with pure logic.
Wittgenstein hated logicism, and described it as a cancerous growth [Wittgenstein, by Monk]
     Full Idea: Wittgenstein didn't just have an arguments against logicism; he hated logicism, and described is as a cancerous growth.
     From: report of Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921]) by Ray Monk - Interview with Baggini and Stangroom p.12
     A reaction: This appears to have been part of an inexplicable personal antipathy towards Russell. Wittgenstein appears to have developed a dislike of all reductionist ideas in philosophy.
The logic of the world is shown by tautologies in logic, and by equations in mathematics [Wittgenstein]
     Full Idea: The logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics.
     From: Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921], 6.22)
     A reaction: White observes that this is Wittgenstein distinguishing logic from mathematics, and thus distancing himself from logicism. But see T 6.2.
If set theory is not actually a branch of logic, then Frege's derivation of arithmetic would not be from logic [Quine]
     Full Idea: We might say that set theory is not really logic, but a branch of mathematics. This would deprive 'includes' of the status of a logical word. Frege's derivation of arithmetic would then cease to count as a derivation from logic: for he used set theory.
     From: Willard Quine (Carnap and Logical Truth [1954], II)
     A reaction: Quine has been making the point that higher infinities and the paradoxes undermine the status of set theory as logic, but he decides to continue thinking of set theory as logic. Critics of logicism frequently ask whether the reduction is to logic.
Mathematics reduces to set theory (which is a bit vague and unobvious), but not to logic proper [Quine]
     Full Idea: Mathematics reduces only to set theory, and not to logic proper… but set theory cannot claim the same firmness and obviousness as logic.
     From: Willard Quine (Epistemology Naturalized [1968], p.69-70)
Logicists cheerfully accept reference to bound variables and all sorts of abstract entities [Quine]
     Full Idea: The logicism of Frege, Russell, Whitehead, Church and Carnap condones the use of bound variables or reference to abstract entities known and unknown, specifiable and unspecifiable, indiscriminately.
     From: Willard Quine (On What There Is [1948], p.14)
Logic is dependent on mathematics, not the other way round [Heyting, by Shapiro]
     Full Idea: Heyting (the intuitionist pupil of Brouwer) said that 'logic is dependent on mathematics', not the other way round.
     From: report of Arend Heyting (Intuitionism: an Introduction [1956]) by Stewart Shapiro - Thinking About Mathematics 7.3
     A reaction: To me, this claim makes logicism sound much more plausible, as I don't see how mathematics could get beyond basic counting without a capacity for logical thought. Logic runs much deeper, psychologically and metaphysically.
Saying mathematics is logic is merely replacing one undefined term by another [Curry]
     Full Idea: To say that mathematics is logic is merely to replace one undefined term by another.
     From: Haskell B. Curry (Remarks on the definition and nature of mathematics [1954], 'Mathematics')
Set theory isn't part of logic, and why reduce to something more complex? [Dummett]
     Full Idea: The two frequent modern objects to logicism are that set theory is not part of logic, or that it is of no interest to 'reduce' a mathematical theory to another, more complex, one.
     From: Michael Dummett (Frege philosophy of mathematics [1991], Ch.18)
     A reaction: Dummett says these are irrelevant (see context). The first one seems a good objection. The second one less so, because whether something is 'complex' is a quite different issue from whether it is ontologically more fundamental.
The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members [Benacerraf]
     Full Idea: In no consistent theory is there a class of all classes with seventeen members. The existence of the paradoxes is a good reason to deny to 'seventeen' this univocal role of designating the class of all classes with seventeen members.
     From: Paul Benacerraf (What Numbers Could Not Be [1965], II)
     A reaction: This was Frege's disaster, and seems to block any attempt to achieve logicism by translating numbers into sets. It now seems unclear whether set theory is logic, or mathematics, or sui generis.
Logic is definitional, but real mathematics is axiomatic [Badiou]
     Full Idea: Logic is definitional, whereas real mathematics is axiomatic.
     From: Alain Badiou (Briefings on Existence [1998], 10)
Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock]
     Full Idea: If logic is neutral on the number of objects there are, then logicists can't construe numbers as objects, for arithmetic is certainly not neutral on the number of numbers there are. They must be treated in some other way, perhaps as numerical quantifiers.
     From: David Bostock (Philosophy of Mathematics [2009], 5.5)
Many crucial logicist definitions are in fact impredicative [Bostock]
     Full Idea: Many of the crucial definitions in the logicist programme are in fact impredicative.
     From: David Bostock (Philosophy of Mathematics [2009], 8.2)
If Hume's Principle is the whole story, that implies structuralism [Bostock]
     Full Idea: If Hume's Principle is all we are given, by way of explanation of what the numbers are, the only conclusion to draw would seem to be the structuralists' conclusion, ...studying all systems that satisfy that principle.
     From: David Bostock (Philosophy of Mathematics [2009], 9.A.2)
     A reaction: Any approach that implies a set of matching interpretations will always imply structuralism. To avoid it, you need to pin the target down uniquely.
Analyticity avoids abstract entities, but can there be truth without reference? [Kitcher]
     Full Idea: Philosophers who hope to avoid commitment to abstract entities by claiming that mathematical statements are analytic must show how analyticity is, or provides a species of, truth not requiring reference.
     From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 04.I)
     A reaction: [the last part is a quotation from W.D. Hart] Kitcher notes that Frege has a better account, because he provides objects to which reference can be made. I like this idea, which seems to raise a very large question, connected to truthmakers.
Logicism seemed to fail by Russell's paradox, Gödel's theorems, and non-logical axioms [Wright,C]
     Full Idea: Most would cite Russell's paradox, the non-logical character of the axioms which Russell and Whitehead's reconstruction of Frege's enterprise was constrained to employ, and the incompleteness theorems of Gödel, as decisive for logicism's failure.
     From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro)
The standard objections are Russell's Paradox, non-logical axioms, and Gödel's theorems [Wright,C]
     Full Idea: The general view is that Russell's Paradox put paid to Frege's logicist attempt, and Russell's own attempt is vitiated by the non-logical character of his axioms (esp. Infinity), and by the incompleteness theorems of Gödel. But these are bad reasons.
     From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 4.xvi)
     A reaction: Wright's work is the famous modern attempt to reestablish logicism, in the face of these objections.
It seems impossible to explain the idea that the conclusion is contained in the premises [Field,H]
     Full Idea: No clear explanation of the idea that the conclusion was 'implicitly contained in' the premises was ever given, and I do not believe that any clear explanation is possible.
     From: Hartry Field (Science without Numbers [1980], 1)
Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD]
     Full Idea: The thesis that no existence proposition is analytic is one of the few constants in philosophical consciences, but there are many existence claims in mathematics, such as the infinity of primes, five regular solids, and certain undecidable propositions.
     From: William D. Hart (The Evolution of Logic [2010], 2)
Are neo-Fregeans 'maximalists' - that everything which can exist does exist? [Hale/Wright]
     Full Idea: It is claimed that neo-Fregeans are committed to 'maximalism' - that whatever can exist does.
     From: B Hale / C Wright (The Metaontology of Abstraction [2009], §4)
     A reaction: [The cite Eklund] They observe that maximalism denies contingent non-existence (of the £20 note I haven't got). There seems to be the related problem of 'hyperinflation', that if abstract objects are generated logically, the process is unstoppable.
Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro]
     Full Idea: I extend Quinean holism to logic itself; there is no sharp border between mathematics and logic, especially the logic of mathematics. One cannot expect to do logic without incorporating some mathematics and accepting at least some of its ontology.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
     A reaction: I have strong sales resistance to this proposal. Mathematics may have hijacked logic and warped it for its own evil purposes, but if logic is just the study of inferences then it must be more general than to apply specifically to mathematics.
Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
     Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter.
     From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1)
     A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism.
Did logicism fail, when Russell added three nonlogical axioms, to save mathematics? [Linsky,B]
     Full Idea: It is often thought that Logicism was a failure, because after Frege's contradiction, Russell required obviously nonlogical principles, in order to develop mathematics. The axioms of Reducibility, Infinity and Choice are cited.
     From: Bernard Linsky (Russell's Metaphysical Logic [1999], 6)
     A reaction: Infinity and Choice remain as axioms of the standard ZFC system of set theory, which is why set theory is always assumed to be 'up to its neck' in ontological commitments. Linsky argues that Russell saw ontology in logic.
For those who abandon logicism, standard set theory is a rival option [Linsky,B]
     Full Idea: ZF set theory is seen as a rival to logicism as a foundational scheme. Set theory is for those who have given up the project of reducing mathematics to logic.
     From: Bernard Linsky (Russell's Metaphysical Logic [1999], 6.1)
     A reaction: Presumably there are other rivals. Set theory has lots of ontological commitments. One could start at the other end, and investigate the basic ontological commitments of arithmetic. I have no idea what those might be.
We can understand cardinality without the idea of one-one correspondence [Heck]
     Full Idea: One can have a perfectly serviceable concept of cardinality without so much as having the concept of one-one correspondence.
     From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 3)
     A reaction: This is the culmination of a lengthy discussion. It includes citations about the psychology of children's counting. Cardinality needs one group of things, and 1-1 needs two groups.
Equinumerosity is not the same concept as one-one correspondence [Heck]
     Full Idea: Equinumerosity is not the same concept as being in one-one correspondence with.
     From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 6)
     A reaction: He says this is the case, even if they are coextensive, like renate and cordate. You can see that five loaves are equinumerous with five fishes, without doing a one-one matchup.
First-order logic captures the inferential relations of numbers, but not the semantics [Hofweber]
     Full Idea: Representing arithmetic formally we do not primarily care about semantic features of number words. We are interested in capturing the inferential relations of arithmetical statements to one another, which can be done elegantly in first-order logic.
     From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.3)
     A reaction: This begins to pinpoint the difference between the approach of logicists like Frege, and those who are interested in the psychology of numbers, and the empirical roots of numbers in the process of counting.
Logical truth is true in all models, so mathematical objects can't be purely logical [Linnebo]
     Full Idea: Modern logic requires that logical truths be true in all models, including ones devoid of any mathematical objects. It follows immediately that the existence of mathematical objects can never be a matter of logic alone.
     From: Øystein Linnebo (Philosophy of Mathematics [2017], 2)
     A reaction: Hm. Could there not be a complete set of models for a theory which all included mathematical objects? (I can't answer that).
Logicism struggles because there is no decent theory of analyticity [Hanna]
     Full Idea: All versions of the thesis that arithmetic is reducible to logic remain questionable as long as no good theory of analyticity is available.
     From: Robert Hanna (Rationality and Logic [2006], 2.4)
     A reaction: He rejects the attempts by Frege, Wittgenstein and Carnap to provide a theory of analyticity.
It is not easy to show that Hume's Principle is analytic or definitive in the required sense [Jenkins]
     Full Idea: A problem for the neo-Fregeans is that it has not proved easy to establish that Hume's Principle is analytic or definitive in the required sense.
     From: Carrie Jenkins (Grounding Concepts [2008], 4.3)
     A reaction: It is also asked how we would know the principle, if it is indeed analytic or definitional (Jenkins p.119).