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There is an intriguing paper by Easwaran on types of refutations:

Easwaran, Kenny. Rebutting and undercutting in mathematics. Epistemology, 146-162, Philos. Perspect., 29, Wiley-Blackwell, Malden, MA, 2015.

Briefly, rebutting an argument involves showing that its conclusions contradict those reached in other work published in reliable venues, whereas undercutting involves finding gaps in the argument itself.

(We used this distinction for refuting some of Easwaran's own arguments here.)

I did some searches on this site for types/classifications of refutations, without much success. Meanwhile, I am interested in what seems to be a different type of refutation: one that neither contradicts published work, nor goes into analysis of the argument itself, but rather seeks to argue against its coherence from first philosophical principles.

For example, historians and philosophers of math, who sometimes do not have the technical wherewithal to master the methods of Robinson's analysis with infinitesimals, tend to resort to alleged proofs-from-first-principles that nonstandard analysis could not possibly provide a viable interpretation of Leibniz's infinitesimal mathematics.

Has anyone tried to classify rebuttals/undercuttings/refutations with an eye to this particular category?

Note. I mentioned the issue of Leibniz interpretation only as an example to illustrate what I mean by "proofs from first principles", but since a number of users responded to this particular issue, I would like to respond to their comments.

  1. The issue of "hyperreals versus constructible reals" (mentioned by user DoubleKnot): it would certainly be satisfying, philosophically speaking, if one could make do with constructible reals. However, mathematically speaking this is not very feasible because one immediately runs into problems even with basic results such as the intermediate value theorem (proved by Cauchy!). There is no guarantee that this result will will remain valid over the constructible reals, and the simplest solution is to work with the full complete ordered field R of real numbers. As far a the comparison with the hyperreals is concerned: this is superficially a plausible objection, but in fact working with the hyperreal extension of R is only one of the possible approaches to infinitesimal analysis. Another approach, called the axiomatic approach, works within R itself, and finds infinitesimals there via an enrichment of the language by complementing the membership relation by an additional unary predicate "standard". This is not the place to go into technical details; an introductory exposition can be found here. Thus both parts of the challenge from the constructible reals turn out to be debatable.

  2. The issue of "Anachronism" mentioned by Berry: Let's look at some dates. What historians of mathematics would typically learn when doing their undergraduate degree is some version of Weierstrassian analysis culminating in epsilon-delta and such, in the context of naive set theory. This is of course also a modern framework, dating from 1870s at the earliest. Robinson's framework dates from the 1960s. Therefore we are talking about a comparison of two modern frameworks: one dating from 150 years ago, and another - from 60 years ago. As the decades go by, it becomes less and less plausible to attribute apriori validity to analyses of historical infinitesimalists (Leibniz, Euler, Cauchy) based on one modern framework rather than another.

  3. The issue of "model theory": This concern is mitigated by a distinction that we have developed in a number of publications, between foundations and procedures. Granted Robinson used tools such as model theory that were clearly unaccessible to, say, Leibniz, so as to establish the foundations of his theory. On the other hand, the procedures of Robinson's infinitesimal analysis (such as the relation of infinite proximity, infinitesimal partitions, etc.) furnish better proxies for the Leibnizian procedures than do the procedures of Weiertrassian analysis. Furthermore, the axiomatic approaches to infinitesimal analysis mentioned in item 1 require no model theory, and some of them are conservative over ZF (with no need for the axiom of choice). Thus, there is no need to speculate that Leibniz may have "anticipated" model theory - which is unlikely :-)

Mikhail Katz
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  • Who are these historians and philosophers of math who allegedly don't have the ability to understand Robinson's theory? In my experience, philosophers who engage with mathematics tend to be extremely competent in the field. – David Gudeman Aug 21 '23 at 09:21
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    @DavidGudeman, My experience has been otherwise; see [here](https://u.math.biu.ac.il/~katzmik/infinitesimals.html#loc). – Mikhail Katz Aug 21 '23 at 09:24
  • That's just a list of people who have criticized the work. Which ones do you think are not capable of mathematical work and why do you think that? – David Gudeman Aug 21 '23 at 09:51
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    @DavidGudeman, No, it is not just a list of scholars. It is a list of scholars with links to publications where their work is analyzed. If there is anyone in particular you are interested in, I can comment more fully. – Mikhail Katz Aug 21 '23 at 09:55
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    How do the links make a difference to my point? You expect me to read all of the papers to evaluate the mathematical prowess of the authors? You made a broad, condescending claim. I'm asking you to back it up with specific instances or remove the claim. – David Gudeman Aug 21 '23 at 11:00
  • OK. Consider, for example, the last item on the list: Viertel, who writes: "one can find assertions of Cauchy that contradict a consequent NSA-interpretation. With respect to the function $\frac{-m}{x}$ and~$x^{-m}$ and their continuity he remarks: Both become infinite and, as a consequence, discontinuous when $x=0$. {\ldots} In non-standard analysis, however, one has $x^{-1}\simeq\infty$ for all~$x\simeq 0$ which entails the continuity of the function~$\frac{1}{x}$ for the point zero." ... – Mikhail Katz Aug 21 '23 at 11:17
  • ... Viertel concludes: "Thus Cauchy's concept of continuity is not always consistent (or is not always compatible) with the concept of continuity of nonstandard analysis." Here Viertel uses infinite proximity of a pair of infinitesimals to conclude the infinite proximity of their inverses. Cauchy, on the other hand, explicitly speaks of distinct infinite integers, which therefore cannot be infinitely close. A scholar fluent in the procedures of Robinson's framework would be unlikely to commit Viertel's error. @DavidGudeman – Mikhail Katz Aug 21 '23 at 11:17
  • Whether a theory is historically inconsistent with an interpretation of another theory *should*, we would hope, be discernible for the competent historian, without dwelling on the mathematical viability of either theory (which many a historian will not be all-too-familiar with). But I do see intuitionism-minded philosophers of mathematics who *seem* to summarily dismiss (if they're even aware of) higher set theory, on grounds of epistemological deficiencies (perhaps). OTOH the overlap of history of philosophy and philosophy of history makes this a possibly vague (sorites-like) problem. – Kristian Berry Aug 21 '23 at 14:03
  • Not sure if this is along what you’re looking for, Shapiro in *Thinking about Math* writes that philosophy-first fails against classical mathematics. First principles don’t suffice against the progress of classical mathematics. It’s a misguided and inefficient approach. Sometimes they can buttress or support via hindsight, but there’s little to no history of them guiding math meaningfully on their own. Perhaps that will change if classical math stagnates and bifurcates too much. Most philosophers of math seem to acknowledge this and want to understand what math’s doing, not proscribe/prescribe – J Kusin Aug 21 '23 at 15:35
  • @MikhailKatz the classic example in phil math is perhaps Brouwer and his rejection of LEM. Else, your question is best broken into two subquestions- one with a focus on arguments whose focus is with coherence on first principles (of which there are many, this being philosophy) and the other with the specific arguments you are considering from non-standard analysis – emesupap Aug 21 '23 at 16:15
  • in particular, without the flavor of non-standard analysis, you are just asking about classifying arguments from first principles, and the discipline is rife with such arguments – emesupap Aug 21 '23 at 16:16
  • Thank you. However, that argument belongs in the question, not the comment section. Also, unfortunately, philosophy.stackexchange doesn't support Tex math for some reason. What I do is start a fake question in math.stackexchange, write the equations there, and copy and past the output to this stackexchange. – David Gudeman Aug 21 '23 at 16:50
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    I have not seen it much used on Phil SE, but there is an established taxonomy of defeaters in formal epistemology that includes rebutters and undercutters, see [IEP](https://iep.utm.edu/defeaters-in-epistemology/#H6). What you describe seems to be a kind of rebutter: (allegedly) credible information that contradicts the conclusion, be it from other sources or "first principles". IEP gives a finer taxonomy of secondary defeaters, and yours might be what they call "rebutting reason-defeating defeater". – Conifold Aug 22 '23 at 00:17
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    Rebuttal or undercut purely based on first (metaphysical) principle are not eidetically satisfactory since any first principle to carve nature at its joints alone could be refuted. As for the empirical applicability/verification of hyperreals vs (computable) reals vs dense rationals bundle of hypothesis, scientists especially those subscribed to Popper would be more interested to see how each of them could be possibly falsified instead of simply claiming some possibly nontrivial application. So far the contemporary digital/computational (constructor) philosophy prefers computable numbers ... – Double Knot Aug 22 '23 at 06:21
  • @DavidGudeman, I created a *comment* with formulas at math stackexchange and tried to copy it here, but it did not work: one can find assertions of Cauchy that contradict a consequent NSA-interpretation. With respect to the function −mx and~x−m and their continuity he remarks: Both become infinite and, as a consequence, discontinuous when x=0 . … In non-tandard analysis, however, one has~x−1≃∞ for all~x≃0 which entails the continuity of the function~1x for the point zero. Thus Cauchy's concept of continuity is not always consistent (or is not always compatible) with the concept.. – Mikhail Katz Aug 22 '23 at 08:51
  • @Conifold: What you write sounds like a possible *answer* to the main part of this *question*. Could you possibly elaborate and post this as an *answer*? – Mikhail Katz Aug 22 '23 at 08:52
  • @DoubleKnot: I responded to some of your comments in the body of the *question*. – Mikhail Katz Aug 23 '23 at 14:01

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Compartmentalization is a factor, here: if someone has spent x amount of time studying philosophy but y < x amount of time studying mathematics, and their familiarity with the latter is mediated by examples, in academia, of when philosophers tried their hand at mathematics (or when mathematicians tried their hand at philosophy), there will just for that reason be deficiencies in the application. Notwithstanding that Quine invented a system with a universal set, for instance, you will often hear claims in philosophy about "how Russell's paradox shows that there can't be a universal set" or "Cantor's theorem proves that there can't be a largest set." Sensitivity to the minutiae of comprehension and powerset axioms varying from philosophy essayist to philosophy essayist, you will find what seem to be rather naive claims being advanced:

  1. David Lewis, renowned as a philosopher, apparently thought that ℶ2 was a reasonable estimate for the number of his possible worlds. This in spite of the metatheoretic indeterminacy of the natural powerset beforehand, and then that of that powerset's powerset (extensions of or alternatives to ZFC that decide the natural powerset in a nontrivial way, tend to cluster around ℵ1 and ℵ2, but Fuchino, et. al. [20] provide a novel and detailed basis for (hypothetically) inflating the Continuum well into the stratosphere of V; see also Matthews[21] for an inflation of ℶ1 to the size of a proper class). Lewis had his reasons, of course, but I can't help but think that trying to apply a somewhat generic symbol for a transfinite powerset, to a moment in a theory about the extent of possible worlds, was not the most well-grounded thing he ever did.

  2. I read somewhere recently that Boolos, in philosophically analyzing his theory of plural quantification, suggested cutting off the height of V at ℵωωω.... I don't recall his reasoning clearly, but I have a memory-of-a-memory, so to speak, that it seemed to come down to something like, "We just don't need to push the implications of the basic aleph-notation beyond the fixed point." Understandable, perhaps, and I hear an echo of "let's use Occam's razor" in there; but otherwise, it seems like the ability to assign some relatively specific meaning to arbitrarily other and higher cardinals is testimony against the point of stopping our climb up Cantor's mountain at a merely "convenient" resting place.

But so in the case of Lewis, we seem to have a lesser-than-desirable familiarity with the possible sizes of powersets; Boolos is appealing to what could be styled a "first principle," but maybe not an entirely well-motivated one (if notation expenditures are our concern, why not stop at ℵω1, since we'll run out of perspicuous indexes in the midst of the proof-theoretical ordinals anyway?). Is there a particular "strategy" of appeal-to-first-principles in play, usually, though? The SEP article on non-deductive reasoning in mathematics discusses what Peter Koellner calls the "more-evident-than" relation; but I wonder if weak invocations of "more-evident-than-alternatives" are behind the weakness of many (or most?) philosophical faux pas in commentaries on or involving mathematics.

If our example is historians of mathematics, or historians of Leibniz then, trying to dissociate Robinson's picture of infinitesimals from Leibniz's, the issue perhaps comes down to an anathema-against-anachronism, though. Ironically, some writers try to balance the presumed apriority or even "timelessness" of mathematical knowledge with a claim like, "Since Leibniz didn't envision contemporary model theory, he didn't envision objects constructed from the resources of that theory." However, when knowledge of Leibniz's speculation has been updated with relatively new discoveries in his textual record, perhaps we have found reasons to attribute greater farsightedness to Leibniz (I don't know yet, although of late I did peruse some recent histories of Leibnizian analysis where they might've asked about these matters). At any rate, if there are first principles being appealed to, here, are they more those of mathematics-as-a-discipline or of history-as-a-discipline (if more of either)?

Concerning a taxonomy of arguments inclusive of by-first-principles: Neil Barton publishes quite a bit on the topic of philosophical justification in mathematics. This is from his [19]:

... while we have focussed on prediction, it is presumably only one explanatory good among many. In discussing justification more generally, there has been much good work done by Maddy, Koellner, Martin, and others in providing a taxonomy and analysis of different kinds of justification, and we do not wish to discredit their work. From our perspective there is much to be done in explaining how the criteria they provide, many of which can be given precise characterisations (e.g. restrictiveness of theories67, level of theoretical completeness68, convergence69), can be integrated into our own explanatory account.

Hunter[20]'s abstract suggests a graph-theoretic (meta-)taxonomy of defenses, rebuttals, and neither-defenses-nor-rebuttals:

... the flexibility of the epistemic approach allows us to both model the rationale behind the existing semantics as well as completely deviate from them when required. Epistemic graphs can model both attack and support as well as relations that are neither support nor attack. The way other arguments influence a given argument is expressed by the epistemic constraints that can restrict the belief we have in an argument with a varying degree of specificity. The fact that we can specify the rules under which arguments should be evaluated and we can include constraints between unrelated arguments permits the framework to be more context–sensitive. It also allows for better modelling of imperfect agents, which can be important in multi–agent applications.

Although I have not read through that article in full, I bring it up because I have found the graph-theoretic interpretation of mathematical epistemology to be very perspicuous. I assume, more or less, that Aczel shared this kind of sentiment when developing the accessible-pointed-graph semiotics for his parafounded set theory, and so I should like to add that the parafoundation relations seem to coincide with epistemic coherentism and epistemic infinitism, which suggest a taxonomy in which from-first-principles arguments are appropriate with respect primarily to theories about well-founded objects. It is not absolutely obvious that the Continuum must be taken for a well-foundational structure (certainly the endlessly-descending-decomposition picture from the time of Leibniz and Kant testifies against viewing the Continuum as assembled from absolutely smallest parts) and so one might ask, then, if arguments-from-first-principles, in the substantial analysis of the infinitesimal-grounded Continuum, have much (if any) place. (See also Hamkins[22] on ill-founded and nonlinear patterns in reasoning about the "mathematical universe as a whole".)

Kristian Berry
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  • I responded to some of your comments in the body of the *question*. – Mikhail Katz Aug 23 '23 at 14:00
  • @MikhailKatz I think I agree with you overall, here (although I'm not sure what there would be to disagree with anyway!). I don't know if my answer is all-too-helpful: I tried to provide examples of misguided "arguments from first principles," examples of the first-principles framework more generally (the more-evident-than relation re: Russell/Koellner), and then a possible explanation for the shortcomings of historians in this context. But is the information provided, enough to "verify" a supplement to Easwaran's two argument-types? – Kristian Berry Aug 23 '23 at 15:17
  • I did notice now, when double-checking the link to the "non-deductive methods" SEP article, that in the linked-to section, they mention a work of Easwaran's regarding the axiom of choice. But I don't know if the discussion, there, is more for arguments *about* philosophical first principles than it is for arguments *from* them, so again, not sure if it's useful material for your question. I did think back to assorted papers I've read about axiom-justification in set theory, too, though; do you think Maddy's "Believing the Axioms" might be relevant? – Kristian Berry Aug 23 '23 at 15:25
  • These examples are interesting, but I was looking more for some kind of classification of refutations that would include the types mentioned, more than examples of proofs-from-first-principles. As far as Easwaran and AC: he seems to change his mind about this according to the need of the particular article he is currently working on, as we documented in [this article](http://doi.org/10.15330/ms.51.2.200-224). – Mikhail Katz Aug 23 '23 at 15:34
  • @MikhailKatz do you think arguments from first principles are particularly helpful, though? I would suggest that the structuralism-to-category-theory philosophical pipeline suggests that foundationalist themes in mathematics need to be supplemented by coherentism, where coherentism coincides fairly well with Gödel's notion of extrinsic justification. Offhand, object realism seems more foundationalist, truth-value realism seems more amenable to coherentism (if inclusive of some foundationalism too). [Epistemic graph theory](https://bcmullins.github.io/images/ecej.pdf) might be relevant, then. – Kristian Berry Aug 23 '23 at 15:48
  • In the publication I linked above, we analyze Easwaran/Towsner from the viewpoint of truth-value realism (which was Putnam's forte). I think "arguments from first principles" could in principle be useful, but even if they are not, they are constantly being used, as I tried to suggest in my *question*. One can try to classify even bad arguments :-) – Mikhail Katz Aug 23 '23 at 15:54
  • Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/148041/discussion-between-kristian-berry-and-mikhail-katz). – Kristian Berry Aug 23 '23 at 15:57