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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)?

Conifold
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  • In mathematics, no axioms are "true". There are axiom sets that are generally accepted for most mathematics. – David Thornley Nov 01 '18 at 15:42
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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. https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf – user4894 Nov 01 '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](https://en.wikipedia.org/wiki/Intuitionism) (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 01 '18 at 17:43
  • If your question is about the differences between *Conventionalism* and *Holistic empiricism*, why not just ask about those? Your question would be more focused if you removed the parts about Intuitionism. – E... Nov 01 '18 at 17:57
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    Current debates are more often phrased in terms of platonism (including structuralism) vs fictionalism, [Maddy's Second Philosophy](https://ndpr.nd.edu/news/second-philosophy-a-naturalistic-method/) 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](https://spiritual-minds.com/philosophy/assorted/Mancosu%20-%20Philosophy%20of%20Mathematical%20Practice%20(Oxford,%202008).pdf) – Conifold Nov 01 '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 01 '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 01 '18 at 22:23

2 Answers2

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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).

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Conifold
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  • 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 04 '19 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](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.693.9943&rep=rep1&type=pdf). Admittedly, their causally inert abstracta are different from Plato's active forms, but they do assert a platonic existence for them. – Conifold Jan 05 '19 at 06:33
  • At any rate, platonism is neither necessary nor sufficient for logicism, so the two should not be confused. – Max Jan 06 '19 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 06 '19 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 07 '19 at 19:19
  • @Max Traditional view was that semantic consequence is more fundamental, theorems are not true because they follow from axioms, they follow from axioms because they are true. The axioms are themselves true, and inference rules are designed to preserve the truth. Frege was unhappy with the analytic/synthetic division as not exhaustive, his standard was generality of application (as in logic and arithmetic, but not geometry), deriving arithmetic from logic was to confirm its generality, not to impart derived analyticity. Formalistic logicism you describe is Carnap's later mix of him and Hilbert. – Conifold Jan 07 '19 at 20:57
  • Well, I do not think that's true. If logicism doesn't try to explain the truth of mathematical statements via analyticity, it would be pointless to show that the axioms are analytic truths. If the axioms are true because of platonic facts (or because of forms of intuition, as Kant argued), they would not be analytic. Your characterization also doesn't seem to match the explanations of (neo)logicism on SEP, which explicitly refers to analyticity. – Max Jan 08 '19 at 22:37
  • By the way, there is a paper by Erich Reck which casts serious doubt on the common assumption that Frege was a mathematical platonist: https://www.pdcnet.org/harvardreview/content/harvardreview_2005_0013_0002_0025_0040 Part of the argument is similar to what I have written above. – Max Jan 08 '19 at 22:54
  • @Max SEP says "we opt here for the linguistic version of the analyticity claim", i.e. Carnap's. But that parts of mathematics flow from the meanings of concepts means that they hold in virtue of those meanings, and those meanings are objective for many logicists. This is a conceptual version of platonism. The linguistic path to justifying this claim (through axioms and derivations) reflects the virtue in which it holds, synthetic would add extra tools. Reck argues that Frege was not a *naive* platonist, and he certainly was not, he was a conceptual platonist. I think, so are Hale and Wright. – Conifold Jan 09 '19 at 00:30
  • However, platonism about propositions in general ("thoughts" in Frege's terminology) doesn't entail mathematical platonism. Reck even says that for Frege the realms of the physical, propositional (thoughts), and mental are just "three ways of being determinate—objective-empirical (especially spatio-temporal and causal), objective-logical, and subjective—are, in the end, what underlies Frege's talk of 'three realms'." This would just mean that logical truths are objectively true, and (per logicism) that mathematical truths are logical truths. Which says nothing about platonism. – Max Jan 14 '19 at 20:33
  • @Max Remember that many conceptual platonists self-identify as platonists, to *them* logic/meanings being objective has *something* to do with it. It is not very surprising as Plato's forms were, arguably, conceptual objects. Existence of objectively true without an object, of which it is true, existing is like the smile of the Cheshire Cat without the cat. Its mode of existence is intuitively murky, and, whatever its virtues, such position is psychologically unstable. Which is why, I think, many logicists are (and have to be) platonists. Those embracing Carnap's ghosts are a different type. – Conifold Jan 14 '19 at 20:47
  • If this were true, the objectivity of any analytic truth would entail platonism. E.g. "All bachelors are unmarried". I don't think I have ever heard of anyone else defending such a thesis. I can't think of any reason why the objectivity of analyticity would be inconsistent with "subjective" (i.e. internalist) theories of meaning, where to mean something is a mental phenomenon (like believing, wishing etc. are mental phenomena). If the meaning of what you say depends on what you mean, the truth of what you mean can still be objective. (Similarly, the truth of beliefs can be objective.) – Max Jan 15 '19 at 21:23
  • @Max It would mean that logicism without platonism does not attract many followers, in other words one "implies" the other pragmatically. In human affairs, purely logical considerations mean little by themselves. This is what happened: the founder of logicism professed platonism, Carnap's version quickly collapsed (into Quine's holism, Wittgenstein's normativism, fictionalism, etc.), and the revival of logicism is again associated with platonists. For those who do not wish to pin analyticity to platonism, there are simply more attractive options than "objective analyticity" hanging in the air. – Conifold Jan 15 '19 at 21:31
  • First, your claim seems to be that analyticity presupposes semantic platonism. But this is a very strong claim which is far from being a generally accepted view. I have never heard it from anyone except you. Second, that analytic truths are objectively true instead of being simply subjective contingencies is widely accepted. Analyticity is generally considered to imply truth. Quine at one point actually believed that logic is a posteriori (e.g. that quantum physics can lead us to revise it) but I know of no one else who believes this, and even Quine retracted from this claim later. – Max Jan 17 '19 at 17:08
  • @Max "Does not attract many followers" does not amount to "presupposes", I think. And objectivity (intersubjectivity?) that merely bars subjective whim is too weak for someone like Frege, or many other logicists. They want a *ground* for it to hold. And if the ground for meanings ("shadowy entities") is ultimately convention (Carnap), or something like it (Wittgenstein), why bother with logicism? For Quine, analyticity is a figment altogether. What is widely accepted (let's say) leaves logicism as a steam engine without the steam - it can stand there, but it can not get traction. – Conifold Jan 17 '19 at 21:55
  • If any alternative to platonism about meaning (e.g. internalism about meaning) entailed subjectivity, then _everything_ would be subjective, since even the truth of synthetic sentences depend partly on their meaning (as well as on the world). E.g. whether "There is a bank" is true depends partly on what you mean with "bank" and the other terms in this sentence. But to claim that everything is subjective as long as you aren't a semantic platonist is absurd. – Max Jan 17 '19 at 23:30
  • Regarding convention: It's a convention that in English the word "snow" refers to snow, but the fact that the _relation_ between words and meaning is arbitrary doesn't mean that meaning itself is arbitrary or even subjective. – Max Jan 17 '19 at 23:30
  • @Max I am afraid, I do not follow. Even not objective is broader than subjective, it includes intersubjective, for example. And "objective" is a very vague term to begin with. The question is what kind of objectivity makes logicism more attractive than empiricist or pragmatic conceptions. Perhaps, the issue is that we are using "platonism" and "objective" differently. You keep objecting to "implied" or "entailed", which is not even at issue for me. In any case, this was a long exchange, so I think it is time to move on. – Conifold Jan 19 '19 at 18:21
  • If you don't think logicism entails platonism in any way you shouldn't claim that it has something to do with it. – Max Jan 20 '19 at 19:06
  • @Max The common view of things having to do with each other is much broader than entailment. We wouldn't have any science otherwise. – Conifold Jan 21 '19 at 20:21
  • Yeah, but I was arguing that logicism doesn't even make platonism probable or plausible because the objectivity of analyticity doesn't in any way hinge on platonism. – Max Jan 22 '19 at 21:06
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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).

Thomas Klimpel
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