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In the introduction of a very nice book by M. Giaquinto, called Visual Thinking in Mathematics, he investigates the conditions that give rise to mathematical knowledge - the following ideas are described (according to my interpretation):

In the 19th and 20th centuries mathematicians were looking for a set of axioms from which any proof could be derived. But the problem was: Where did those axioms come from? There were 4 different approaches:

  1. Conventionalism (Carnap)
  2. Holistic Empiricism (Quine)
  3. Platonism, intuitionism (Gödel, Brouwer)
  4. Pragmatism, social constructivism (suggested in comments, late Wittgenstein)

(You can add dates if you want). I'll expose my understanding of the first two, derived from the book, before asking the question.

Conventionalism: axioms which are useful/fruitful are taken as valid. And those axioms are sort of conventions, in the sense that they are language, and language is a convention.

Holistic empiricism: again, what I understand is that those axioms are accepted if they are empirically proven. The only slight difference with conventionalism seems that, in the former, usefulness is more important than anything else. But the author says they are almost opposite points of view.

Questions

Would you summarize the most important difference between those two views? Is there any new advance on this subject (namely, the justification of the building blocks of mathematics)?

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    See Believing the Axioms parts I and II by Maddy for a discussion of the philosophical principles behind the adoptop of the ZFC axioms. cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf – user4894 Nov 1 '18 at 16:49
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    Gödel was not an intuitionist in the usual sense of the word, your description of "intuitionism" is ambiguous. For Gödel axioms come through intuition, which he treats as an analog of perception, but not from intuition, they instead reflect something like platonic reality the way perception reflects physical reality. For intuitionists (Brouwer, Weyl) intuition is rather mind's quasi-Kantian ability to construct mathematical objects in something like imagination, modern version of this view is more often called constructivism. – Conifold Nov 1 '18 at 17:43
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    Current debates are more often phrased in terms of platonism (including structuralism) vs fictionalism, Maddy's Second Philosophy gives a recent survey. A fourth alternative, not mentioned, can be roughly called pragmatism (goes back to Peirce, late Wittgenstein and Lakatos), often referred to as philosophy of mathematical practice – Conifold Nov 1 '18 at 17:59
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    I edited the question to focus it some more and remove inaccuracies. If it does not meet your intentions you can roll back the edit. – Conifold Nov 1 '18 at 20:35
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    @santimirandarp I appreciate your edits. We're now pretty much agreed, and it wouldn't be productive to discuss the remaining minor differences here. – David Thornley Nov 1 '18 at 22:23
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"Conventionalism" was the original position of positivists, which came to be seen as a failure after Quine's criticisms of truth by convention and the analytic/synthetic distinction. Wittgenstein abandoned it even earlier. The idea was that science uses what Carnap called "linguistic frameworks" based on conceptual schemes, axiomatizing the concepts used, and the empirical "protocol sentences". What derived from the scheme only was empirically independent, and called analytic, what also depended on "protocol sentences" in essential way (not as in "this protocol sentence is a sentence") was called synthetic. The scheme was adopted by convention, logic and mathematics were analytic. It was a convenient position for empiricists, for it explained the necessity of mathematics and its applicability to science without metaphysical baggage of platonic realm, or mystical powers of intuition.

Quine showed in Truth by Convention that logic by convention was circular:"In a word, the difficulty is that if logic is to proceed mediately from conventions, logic is needed for inferring logic from the conventions". Then in Two Dogmas of Empiricism he sharply criticized the analytic/synthetic distinction as impossible to draw. The idea was that there is no clean "observation language" that steers clear of the conceptual scheme and can provide theory-neutral protocol sentences. Conversely, the scheme, including mathematics, was not immune to revision based on empirical pressures either, although such revision has no direct relation to any particular observations. It is rather a reaction to the scheme's inadequacy as a whole. Mathematics and logic then are neither analytic nor necessary, they are just more "entrenched". This is Quine's empiricist holism from Two Dogmas:

"The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man-made fabric which impinges on experience only along the edges. Or, to change the figure, total science is like a field of force whose boundary conditions are experience. A conflict with experience at the periphery occasions readjustments in the interior of the field. Truth values have to be redistributed over some of our statements. Re-evaluation of some statements entails re-evaluation of others, because of their logical interconnections -- the logical laws being in turn simply certain further statements of the system, certain further elements of the field.

Having re-evaluated one statement we must re-evaluate some others, whether they be statements logically connected with the first or whether they be the statements of logical connections themselves. But the total field is so undetermined by its boundary conditions, experience, that there is much latitude of choice as to what statements to re-evaluate in the light of any single contrary experience. No particular experiences are linked with any particular statements in the interior of the field, except indirectly through considerations of equilibrium affecting the field as a whole."

Late Wittgenstein's reaction was different, and can be called normative pragmatism, see Steiner's Empirical Regularities in Wittgenstein's Philosophy of Mathematics. He argued against lumping logic and mathematics with the rest of the "field" due to their normative import. In their genesis they are indirectly entangled with experience, perhaps "derived" from it in a loose sense (think of arithmetic and geometry). But in the mature form they are "hardened", "promoted to the dignity of a rule", not just "entrenched".

"We have invented multiplication up to 100; that is, we’ve written down things like 81 × 63 but have never yet written down things like 123 × 489... Well, suppose that 90 percent do it all one way. I say, “This is now going to be the right result.” The experiment was to show what the most natural way is — which way most of them go. Now everybody is taught to do it — and now there is a right and wrong. Before there was not."

These rules, however, are no conventions, they are rather customs that manifest themselves in rule-governed activities. Thinking otherwise leads to the well-known rule-following regress: we need a convention, or "interpretation", to tell us how to follow the rule's convention. And so on, ad infinitum. Hence, "there is a way of grasping a rule which is not an interpretation, but which is exhibited in what we call „obeying the rule‟ and „going against it‟ in actual cases." Thus, despite the appearances, logic and mathematics do not express any truths in need to be justified, they are rather "grammar" of a "language game", communal practice that makes other, empirical, truths expressible.

For more recent developments see Fictionalism (probably the closest surviving heir to conventionalism), Mathematical Naturalism (an heir to Quine's holism), Neologicism (an heir to Frege style platonism), Mathematical Structuralism (perhaps closer to Plato's platonism), and Mathematical Social Constructivism. Maddy's Second Philosophy and Gold edited Proof and Other Dilemmas give good overviews of the current landscape. See also Mancosu edited Philosophy of Mathematical Practice on the "practical turn" of the last three decades (Giaquinto is a contributor).

  • Interesting answer. One point though: Neither neologicism nor Frege's original logicism had anything to do with mathematical platonism. According to (neo)logicism, the axioms of a mathematical system have to be analytic truths, i.e. should be analyzable as logical tautologies. (This seems to be a quite different sense of "analytic" than the one employed by Carnap.) – Max Jan 4 at 20:58
  • @Max Frege's explicitly motivated his logicism by platonism, and some neologicists, it seems, have similar motivations. See, for example, Hale and Wright defending what they call neo-Fregean platonism in Benacerraf's Dilemma Revisited, pp.18-26. Admittedly, their causally inert abstracta are different from Plato's active forms, but they do assert a platonic existence for them. – Conifold Jan 5 at 6:33
  • At any rate, platonism is neither necessary nor sufficient for logicism, so the two should not be confused. – Max Jan 6 at 11:01
  • @Max Sure. But logical equivalence is too strong a relation for interesting connections in philosophy, motivations, presuppositions, and empirical plausibility are not covered by it. Two positions can be plausible and attractive when taken together, and not so much apart from each other. Or vice versa. Which is why we rarely see subjective materialists, for example. – Conifold Jan 6 at 11:21
  • @Confold I don't see the plausible connection between the two. If mathematical statements are true because they follow from axioms which are analytic truths, platonic facts are not involved. The theorems are then true in virtue of the axioms, and the axioms are true in virtue of themselves / their meaning / their being tautological. If the axioms where instead true in virtue of platonic facts, they would be synthetic, not analytic. Just as descriptive sentences are synthetic because they are true in virtue empirical facts. – Max Jan 7 at 19:19
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  • first order logic
  • Turing machine
  • ordinal analysis
  • bounded Zermelo / MacLane set theory
  • EFA / PRA / PA
  • convention: ZFC / NBG
  • reverse mathematics and ontological commitments
  • de res / de dicto axioms

Those words are written on a 9 cm x 9 cm note in front of me. They encapsulate the different ways I could think of (at the moment I wrote them down) to justify belief in the consistency of (rather weak) formal systems. I wrote them down after reading Timothy Chow's expository article The Consistency of Arithmetic, which tries to address the problem of Convincing Edward Nelson that PA is consistent.

The typical ZFC justifications (like those from Maddy's Believing the Axioms) are absent above, but they had been already mentioned in a previous exchange with Chow:

In addition to the consistency line of defence related to Turing machines, also the von Neumann cumulative hierarchy justification can be evoked.


Chow had challenged FOM researchers with Kripkenstein doubting rule-following. Part of my defence was:

If it would really show that, than it would indeed be a major achievement. But then it should be clarified how it is different from the skeptic denying any possibility to communicate meaning at all.

Bill Taylor's reaction that Kripkenstein is no more convincing than Goodman's "paradox" about grue and bleen clarifies this, if we are willing to admit that Goodman is right. Goodman's "Fact, Fiction, and Forecast" was published shortly after Wittgenstein's "Philosophical Investigations," and successfully clarifies the part of rule-following doubt which resists refutation, at least refutation by pointing out that Turing machines nail down the meaning of rules. (Note that Conifold's answer also mentions rule-following doubt.)


Chow's article offers Gentzen's consistency proof (=ordinal analysis), Friedman's reversal from Bolzano–Weierstrass for ℚ (=reverse mathematics), and a trivial argument that can be formalized in ZFC (=convention: ZFC) as main proofs for consistency of PA, and a finite approximation to the consistency of PA as an alternative justification for the supposed arch formalist. (Nelson withdrew his claim that primitive recursive arithmetic is inconsistent, but the introduction by Sarah Jones Nelson and afterword by Sam Buss and Terence Tao to his two posthumously published works tell us that this was not the end of the story, and that he continued his project to show the inconsistency of arithmetic the very next day.)


Back to my words on that 9 cm x 9 cm note. Chow's trivial argument had confused me with respect to his own position, and I wrote them down while composing an email to him. I didn't use those words in the end, but we had a nice discussion. Using ZFC to formalize his trivial argument felt like a cheap opt-out to me. However, Chow pointed out that as long as he doesn't know anything about my "belief system", the best he can do is to rely on standard assumptions (which in math means ZFC). The dubious part of his trivial argument is:

Now let me ask if a first-order formula in the language of arithmetic (e.g., ∃y:y+y=x) defines a mathematical property of the integers. Again, the answer is so obviously yes that you must wonder if it's a trick question. But it's not a trick question. The answer is yes.

But is the answer obviously yes, because first-order formulas always define mathematical properties (i.e. also in set theory), or because the integers are somehow special? Chow answered that at least historically a mathematical concept is legitimate provided that the language that we use to define the object is sufficiently precise. Before answering, he queried my beliefs with questions like Do you believe that "halting" is a mathematical property of a Turing machine? My answer was yes.

However, I also indicated I am more critical about the same statement for oracle Turing machines. At least one would have to clarify which property going beyond computability is shared by the sets used as oracles. Maybe they are somehow "absolutely definable"? Those types of questions are investigated by recursion theory. The computable sets are Δ01, the limit computable sets are Δ02, the hyperarithmetic sets are Δ11. The hyperarithmetic sets are closely related to the Church-Kleene ordinal ω1CK, but Arithmetical transfinite recursion ATR0 proves Δ11-comprehension, even so its proof-theoretic ordinal is the Feferman–Schütte ordinal Γ0, which is smaller than ω1CK.

Determining the proof-theoretic ordinal of an axiom system is the subject of ordinal analysis. Dmytro Taranovsky talks on FOM about ordinal analysis, and recommends The Art of Ordinal Analysis as a base reference. Personal opinion: Ordinal analysis provides real justifications for mathematics, and it would be nice to understand why it can do that (philosophically speaking). Probably what it does is to boil down the implications of consistency of an axiomatic system to the bare bones on a level where it becomes possible to understand it intuitively. (And this intuition can still be wrong, thereby resolving the paradox how we can know anything nontrivial for certain.)


The words "bounded Zermelo / MacLane set theory", "EFA / PRA / PA", and "de res / de dicto axioms" were not discussed above. Even (full) second order arithmetic is still out of reach for ordinal analysis. Now "bounded Zermelo / MacLane set theory" correspond to (full) higher order logic in a certain sense (as well as many other independent axiom systems for mathematics), and has been defended in the early days: Frank P. Ramsey and Rudolf Carnap accepted the ban on explicit circularity, but argued against the ban on circular quantification. "EFA / PRA / PA" are related to the fact that if we use mathematics to justify mathematics, then it is a good idea to have base-systems allowing us to do mathematics in the first place. "de res / de dicto axioms" is about the difference between asserting the existence of a concrete object fully specified (de res), and the mere claim of existence (de dicto) without specifying any particular object (like the axiom of choice).

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