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Easy as 1, 2, 3 ? -- Wittgenstein on counting

By Catarina Dutilh Novaes

(Cross-posted at M-Phi)

 

“A B C

It's easy as, 1 2 3

As simple as, do re mi

A B C, 1 2 3

Baby, you and me girl”

45 years ago, Michael Jackson and his troupe of brothers famously claimed that counting is easy peasy. But how easy is it really? (We’ll leave aside the matter of the simplicity of A B C and do re mi for present purposes!)

Counting and basic arithmetic operations are often viewed as paradigmatic cases of ‘easy’ mental operations. It might seem that we are all ‘born’ with the innate ability for basic arithmetic, given that we all seem to engage in the practice of counting effortlessly. However, as anyone who has cared for very young children knows, teaching a child how to count is typically a process requiring relentless training. The child may well know how to recite the order of numbers (‘one, two, three…’), but from that to associating each of them to specific quantities is a big step. Even when they start getting the hang of it, they typically do well with small quantities (say, up to 3), but things get mixed up when it comes to counting more items. For example, they need to resist the urge to point at the same item more than once in the counting process, something that is in no way straightforward! 

The later Wittgenstein was acutely aware of how much training is involved in mastering the practice of counting and basic arithmetic operations. (Recall that he was a schoolteacher for many years in the 1920s!) Indeed, counting and adding objects can be described as a specific and rather peculiar language game which must be learned by training, and which raises all kinds of philosophical questions pertaining to what it is exactly that we are doing when we count things. Perhaps my favorite passage in the whole of the Remarks on the Foundations of Mathematics is #37 in part I: 

Formal Methods in Philosophy: a Brief Introduction (Part II)

By Catarina Dutilh Novaes

(Cross-posted at M-Phi) 

This is the second and final part of my 'brief introduction' to formal methods in philosophy to appear in the forthcoming Bloomsbury Philosophical Methodology Reader, being edited by Joachim Horvath. (Part I is here.) In this part I present in more detail the four papers included in the formal methods section, namely Tarski's 'On the concept of following logically', excerpts from Carnap's Logical Foundations of Probability, Hansson's 2000 'Formalization in philosophy', and a commissioned new piece by Michael Titelbaum focusing in particular (though not exclusively) on Bayesian epistemology. 

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Some of the pioneers in formal/mathematical approaches to philosophical questions had a number of interesting things to say on the issue of what counts as an adequate formalization, in particular Tarski and Carnap – hence the inclusion of pieces by each of them in the present volume. Indeed, both in his paper on truth and in his paper on logical consequence (in the 1930s), Tarski started out with an informal notion and then sought to develop an appropriate formal account of it. In the case of truth, the starting point was the correspondence conception of truth, which he claimed dated back to Aristotle. In the case of logical consequence, he was somewhat less precise and referred to the ‘common’ or ‘everyday’ notion of logical consequence. 

These two conceptual starting points allowed Tarski to formulate what he described as ‘conditions of material adequacy’ for the formal accounts. He also formulated criteria of formal correctness, which pertain to the internal exactness of the formal theory. In the case of truth, the basic condition of material adequacy was the famous T-schema; in the case of logical consequence, the properties of necessary truth-preservation and of validity-preserving schematic substitution. Unsurprisingly, the formal theories he then went on to develop both passed the test of material adequacy he had formulated himself. But there is nothing particularly ad hoc about this, since the conceptual core of the notions he was after was presumably captured in these conditions, which thus could serve as conceptual ‘guides’ for the formulation of the formal theories. 

Reductio arguments from a dialogical perspective: final considerations

By Catarina Dutilh Novaes

(Cross-posted at M-Phi)

 This is the final post in my series on reductio ad absurdum from a dialogical perspective. Here is Part I, here is Part II, here is Part III, here is Part IV, and here is Part V. I now return to the issues raised in the earlier posts equipped with the dialogical account of deduction, and of reductio ad absurdum in particular.

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 A general dialogical schema for reductio ad absurdum, following Proclus’ description but inspired by the Socratic elenchus, might look like this:

1. Interlocutor 1 commits to A (either prompted by a question from interlocutor 2, or spontaneously), which corresponds to assuming the initial hypothesis.
2. Interlocutor 2 leads the initial hypothesis to absurdity, typically by relying on additional discursive commitments of 1 (which may be elicited by 2 through questions).
3. Interlocutor 2 concludes ~A.

The main difference between the monological and the dialogical versions of a reductio is thus that in the latter there is a kind of division of labor that is absent from the former (as noted above). The agent making the initial assumption is not the same agent who will lead it to absurdity, and then conclude its contradictory. And so, the perceived pragmatic awkwardness of making an assumption precisely with the goal of ‘destroying’ it seems to vanish. Moreover, the adversarial component provides a compelling rationale for the general idea of ‘destroying’ the initial hypothesis; indeed, while the adversarial component is present in all deductive arguments (in particular given the requirement of necessary truth preservation, as argued above), it is even more pronounced in the case of reductio arguments, that is the procedure whereby someone’s discursive commitments are shown to be collectively incoherent since they lead to absurdity. There remains the question of why interlocutor 1 would want to engage in the dialogue at all, but presumably she simply wishes to voice a discursive commitment to A. From there on, the wheel begins to spin, mostly through 2’s actions.

Dialectical refutations and reductio ad absurdum

By Catarina Dutilh Novaes

(Cross-posted at M-Phi)

This is the fifth installment of my series of posts on reductio ad absurdum from a dialogical perspective. Here is Part I, here is Part II, here is Part III, and here is Part IV. In this post I discuss a closely related argumentative strategy, namely dialectical refutation, and argue that it can be viewed as a genealogical ancestor of reductio ad absurdum.

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Those familiar with Plato’s Socratic dialogues will undoubtedly recall the numerous instances in which Socrates, by means of questions, elicits a number of discursive commitments from his interlocutors, only to go on to show that, taken collectively, these commitments are incoherent. This is the procedure known as an elenchus, or dialectical refutation. 

The ultimate purpose of such a refutation may range from ridiculing the opponent to nobler didactic goals. The etymology of elenchus is related to shame, and indeed at least in some cases it seems that Socrates is out to shame the interlocutor by exposing the incoherence of their beliefs taken collectively (for example, so as to exhort them to positive action, as argued in (Brickhouse & Smith 1991)). However, as noted by Socrates himself in the Gorgias (470c7-10), refuting is also what friends do to each other, a process whereby someone rids a friend of nonsense. An elenchus can also have pedagogical purposes, in interactions between masters and pupils.

There has been much discussion in the secondary literature on what exactly an elenchus is, as well as on whether there is a sufficiently coherent core of properties for what counts as an elenchus, beyond a motley of vaguely related argumentative strategies deployed by Socrates (Carpenter & Polansky 2002). (A useful recent overview is (Wolfsdorf 2013); see also (Scott 2002).) For our purposes, it will be useful to take as our starting point the description of the ‘Socratic method’ in an influential article by G. Vlastos (1983) (a much shorter version of the same argument is to be found in (Vlastos 1982), and I'll be referring to the shorter version). Vlastos distinguishes two kinds of elenchi, the indirect elenchus and the standard elenchus:

A precis of the dialogical account of deduction

By Catarina Dutilh Novaes

(Cross-posted at M-Phi)

This is the fourth installment of my series of posts on reductio ad absurdum arguments from a dialogical perspective. Here is Part I, here is Part II, and here is Part III. In this post I offer a précis of the dialogical account of deduction which I have been developing over the last years, which will then allow me to return to the issue of reductio arguments equipped with a new perspective in the next installments. I have presented the basics of this conception in previous posts, but some details of the account have changed, and so it seems like a good idea to spell it out again.

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In this post, I present a brief account of the general dialogical conception of deduction that I endorse. Its relevance for the present purposes is to show that a dialogical conception of reductio ad absurdum arguments is not in any way ad-hoc; indeed, the claim is that this conception applies to deductive arguments in general, and thus a fortiori to reductio arguments. (But I will argue later on that the dialogical component is even more pronounced in reductio arguments than in other deductive arguments.)

Let us start with what can be described as functionalist questions pertaining to deductive arguments and deductive proofs. What is the point of deductive proofs? What are they good for? Why do mathematicians bother producing mathematical proofs at all? While these questions are typically ignored by mathematicians, they have been raised and addressed by so-called ‘maverick’ philosophers of mathematics, such as Hersh (1993) and Rav (1999). One promising vantage point to address these questions is the historical development of deductive proof in ancient Greek mathematics,[1] and on this topic the most authoritative study remains (Netz 1999). Netz emphasizes the importance of orality and dialogue for the emergence of classical, ‘Euclidean’ mathematics in ancient Greece:

Greek mathematics reflects the importance of persuasion. It reflects the role of orality, in the use of formulae, in the structure of proofs… But this orality is regimented into a written form, where vocabulary is limited, presentations follow a relatively rigid pattern… It is at once oral and written… (Netz 1999, 297/8)

Problems with reductio proofs: "jumping to conclusions"

By Catarina Dutilh Novaes

(Cross-posted at M-Phi)

This is the third installment of my series of posts on reductio ad absurdum arguments from a dialogical perspective. Here is Part I, and here is Part II. In this post I discuss issues pertaining specifically to the last step in a reductio argument, namely that of going from reaching absurdity to concluding the contradictory of the initial hypothesis.

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One worry we may have concerning reductio arguments is what could be described as ‘the culprit problem’. This is not a worry clearly formulated in the protocols previously described, but one which has been raised a number of times when I presented this material to different audiences. The basic problem is: we start with the initial assumption, which we intend to prove to be false, but along the way we avail ourselves to auxiliary hypotheses/premises. Now, it is the conjunction of all these premises and hypotheses that lead to absurdity, and it is not immediately clear whether we can single out one of them as the culprit to be rejected. For all we know, others may be to blame, and so there seems to be some arbitrariness involved in singling out one specific ingredient as responsible for things turning sour.

To be sure, in most practical cases this will not be a real concern; typically, the auxiliary premises we avail ourselves to are statements on which we have a high degree of epistemic confidence (for example, because they have been established by proofs that we recognize as correct). But it remains of philosophical significance that absurdity typically arises from the interaction between numerous elements, any of which can, in theory at least, be held to be responsible for the absurdity. A reductio argument, however, relies on the somewhat contentious assumption that we can isolate the culprit.

However, culprit considerations do not seem to be what motivates Fabio’s dramatic description of this last step as “an act of faith that I must do, a sacrifice I make”. Why is this step problematic then? Well, in first instance, what is established by leading the initial hypothesis to absurdity is that it is a bad idea to maintain this hypothesis (assuming that it can be reliably singled out as the culprit, e.g. if the auxiliary premises are beyond doubt). How does one go from it being a bad idea to maintain the hypothesis to it being a good idea to maintain its contradictory?

Problems with reductio proofs: assuming the impossible

By Catarina Dutilh Novaes

 (Cross-posted at M-Phi)

This is a series of posts with sections of the paper on reductio ad absurdum from a dialogical perspective that I am working on right now. This is Part II, here is Part I. In this post I discuss issues in connection with the first step in a reductio argument, that of assuming the impossible.

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We can think of a reductio ad absurdum as having three main components, following Proclus’ description:

(i) Assuming the initial hypothesis.

(ii) Leading the hypothesis to absurdity.

(iii) Concluding the contradictory of the initial hypothesis.

I discuss two problems pertaining to (i) in this post, and two problems pertaining to (iii) in the next post. (ii) is not itself unproblematic, and we have seen for example that Maria worries whether the ‘usual’ rules for reasoning still apply once we’ve entered the impossible world established by (i). Moreover, the problematic status of (i) arises to a great extent from its perceived pragmatic conflict with (ii). But the focus will be on issues arising in connection with (i) and (iii).

A reductio proof starts with the assumption of precisely that which we want to prove is impossible (or false). As we’ve seen, this seems to create a feeling of cognitive dissonance in (some) reasoners: “I do not know what is true and what I pretend [to be] true.” (Maria) This may seem surprising at first sight: don’t we all regularly reason on the basis of false propositions, such as in counterfactual reasoning? (“If I had eaten a proper meal earlier today, I wouldn’t be so damn hungry now!”) However, as a matter of fact, there is considerable empirical evidence suggesting that dissociating one’s beliefs from reasoning is a very complex task, cognitively speaking (to ‘pretend that something is true’, in Maria’s terms). The belief bias literature, for example, has amply demonstrated the effect of belief on reasoning, even when participants are told to focus only on the connections between premises and conclusions. Moreover, empirical studies of reasoning behavior among adults with low to no schooling show their reluctance to reason with premises of which they have no knowledge (Harris 2000; Dutilh Novaes 2013). From this perspective, reasoning on the basis of hypotheses or suppositions may well be something that requires some sort of training (e.g. schooling) to be mastered.

Problems with reductio proofs: cognitive aspects

By Catarina Dutilh Novaes

(Cross-posted at M-Phi)

As some readers may recall, I ran a couple of posts on reductio proofs from a dialogical perspective quite some time ago (here and here). I am now *finally* writing the paper where I systematize the account. In the coming days I'll be posting sections of the paper; as always, feedback is most welcome! The first part will focus on what seem to be the cognitive challenges that reasoners face when formulating reductio arguments.

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For philosophers and mathematicians having been suitably ‘indoctrinated’ in the relevant methodologies, the issues pertaining to reductio ad absurdum arguments may not become immediately apparent, given their familiarity with the technique. And so, to get a sense of what is problematic about these arguments, let us start with a somewhat dramatic but in fact quite accurate account of what we could describe as the ‘phenomenology’ of producing a reductio argument, in the words of math education researcher U. Leron:

We begin the proof with a declaration that we are about to enter a false, impossible world, and all our subsequent efforts are directed towards ‘destroying’ this world, proving it is indeed false and impossible. (Leron 1985, 323)

In other words, we are first required to postulate this impossible world (which we know to be impossible, given that our very goal is to refute the initial hypothesis), and then required to show that this impossible world is indeed impossible. The first step already raises a number of issues (to be discussed shortly), but the tension between the two main steps (postulating a world, as it were, and then proceeding towards destroying it) is perhaps even more striking. As it so happens, these are not the only two issues that arise once one starts digging deeper. 

To obtain a better grasp of the puzzling nature of reductio arguments, let us start with a discussion of why these arguments appear to be cognitively demanding – that is, if we are to believe findings in the math education literature as well as anecdotal evidence (e.g. of those with experience teaching the technique to students). This will offer a suitable framework to formulate further issues later on.

Mochizuki's proof of the ABC conjecture: still "in limbo"

By Catarina Dutilh Novaes

(Cross-posted at M-Phi)

Here's a short piece by the New Scientist on the status of Mochizuki's purported proof of the ABC conjecture. More than 2 years after the 500-page proof has been made public, the mathematical community still hasn't been able to decide whether it's correct or not. (Recall my post on this from May 2013; little change seems to have taken place since then.)

Going back to my dialogical conception of mathematical proofs as involving a proponent who formulates the proof and opponents who must check it, this stalemate can be viewed from at least two perspectives: either Mochizuki is not trying hard enough as a proponent, or the mathematical community is not trying hard enough as opponent.

[Mochizuki] has also criticised the rest of the community for not studying his work in detail, and says most other mathematicians are "simply not qualified" to issue a definitive statement on the proof unless they start from the very basics of his theory.

Some mathematicians say Mochizuki must do more to explain his work, like simplifying his notes or lecturing abroad.

(Of course, it may well be that both are the case!). And so for now, the proof remains in limbo, as well put by the New Scientist piece. Mathematics, oh so human!

In memoriam: Ivor Grattan-Guinness

The great historian of logic and mathematics Ivor Grattan-Guinness passed away about a month ago, aged 73. I only heard it yesterday, when Stephen Read posted a link to the Guardian obituary on Facebook. From the obituary: 

He rescued the moribund journal Annals of Science, founded the journal History and Philosophy of Logic, and was on the board of Historia Mathematica from its inception. A member of the council of the Society for Psychical Research, he wrote Psychical Research: A Guide to Its History (1982). In 1971 the British Society for the History of Mathematics was founded: Ivor served as its president (1986-88) and instituted a formal constitution.

Indeed, many of us owe him eternal gratitude for founding the journal History and Philosophy of Logic, which continues to be the main journal for studies combining historical and philosophical perspectives on logic. Ivor's work and scholarship spans over an impressive range of topics and areas, and is bound to continue to influence many generations of scholars to come. It is a great loss.

In memoriam: Patrick Suppes

By Catarina Dutilh Novaes

As readers will have noticed, since yesterday there has been an outpour of expressions of sorrow for the passing of Patrick Suppes everywhere on the internet. He was without a doubt one of the most influential philosophers of science in the 20th century, and so all the love and appreciation is richly deserved. I do not have anything much to add to what others who knew him better have been saying, but I thought of posting a link to the video podcast of a lecture he delivered at the Munich Center for Mathematical Philosophy some two years ago, titled ‘A neuroscience perspective on the foundations of mathematics’. (Back then I even wrote a blog post about the lecture, just before it took place.)

As much as it is sad to see dear, talented people passing away, I think it is comforting to note that Suppes lived his life to the fullest until the very end, for example by flying across the ocean to give a lecture on a new, exciting topic at age 90. The vitality he displays in the lecture is truly impressive, and a lovely reminder of his decisive contributions to philosophy and the sciences.

In memoriam: Alexander Grothendieck

By Catarina Dutilh Novaes

Alexander Grothendieck, who is viewed by many as the greatest mathematician of the 20^th century, has passed away yesterday after years of living in total reclusion. (To be honest, I did not even know he was still alive!) He was a key figure in the development of the modern theory of algebraic geometry, among others, but to philosophers and logicians he is perhaps best known as one of the major forces behind the establishment of category theory as a new foundational framework for mathematics. 

So far, I’ve only seen obituaries in French (Libération and Le Monde), as Grothendieck (of German origin) lived almost all of his life in France (I expect that soon obituaries in English will be available too). His life story is almost as remarkable as his mathematical achievements: his father died in Auschwitz in 1942, while he was sent to a concentration camp in France with his mother. In all of his adult life, he was as passionate about pacifism as he was about mathematics (and perhaps even more), and continuously engaged in a number of activist initiatives like lecturing on category theory in a forrest of bombed Vietnam to protest against the war. He seems to have basically stopped doing any mathematics in the 1970s, but the influence of his work in the field is bound to remain colossal in many years or even centuries to come.

Transferability as the epistemic value of deliberation

By Catarina Dutilh Novaes

Today my research group in Groningen (with the illustrious online participation of Tony Booth, beaming in from the UK) held a seminar session where we discussed Fabienne Peter’s 2013 paper ‘The procedural epistemic value of deliberation’. It is a very interesting paper, which defends the view that deliberation has not only epistemic value (as opposed to ‘merely’ ethical, practical value), but also that it has procedural epistemic value (as the title suggests), as opposed to ‘merely’ instrumental value. I'll argue here that I agree with the thesis, but not for the reasons offered by Peter in her paper.

The paper begins with the following observation:

An important question one can ask about collective deliberation is whether it increases or decreases the accuracy of the beliefs of the participants. But this instrumental approach, which only looks at the outcome of deliberation, does not exhaustively account for the epistemic value that deliberation might have. (Peters 2013, 1253) 

One way to spell out this idea is the following: suppose there were two knowledge-producing procedures with the exact same accuracy, i.e. which would produce the same amount of true beliefs and avoid the same amount of false beliefs. Moreover, procedure D involves deliberation, while procedure O relies entirely on an oracle, for example. If we can show that procedure D is superior to procedure O on purely epistemic grounds, then we can establish that deliberation has procedural epistemic value, rather than merely instrumental epistemic value (i.e. increase accuracy).

What makes a mathematical proof beautiful?

By Catarina Dutilh Novaes

 (Cross-posted at M-Phi)

 

In December, I will be presenting at the Aesthetics in Mathematics conference in Norwich. The title of my talk is Beauty, explanation, and persuasion in mathematical proofs, and to be honest at this point there is not much more to it than the title… However, the idea I will try to develop is that many, perhaps even most, of the features we associate with beauty in mathematical proofs can be subsumed to the ideal of explanatory persuasion, which I take to be the essence of mathematical proofs. 

 

As some readers may recall, in my current research I adopt a dialogical perspective to raise a functionalist question: what is the point of mathematical proofs? Why do we bother formulating mathematical proofs at all? The general hypothesis is that most of the defining criteria for what counts as a mathematical proof – and in particular, a good mathematical proof – can be explained in terms of the (presumed) ultimate function of a mathematical proof, namely that of convincing an interlocutor that the conclusion of the proof is true (given the truth of the premises) by showing why that is the case. (See also this recent edited volume on argumentation in mathematics.) Thus, a proof seeks not only to force the interlocutor to grant the conclusion if she has granted the premises; it seeks also to reveal something about the mathematical concepts involved to the interlocutor so that she also apprehends what makes the conclusion true – its causes, as it were. On this conception of proof, beauty may well play an important role, but this role will be subsumed to the ideal of explanatory persuasion.

Is the human referee becoming expendable in mathematics?

(Cross-posted at M-Phi)

Mathematics has been much in the news recently, especially with the announcement of the latest four Fields medalists (I am particularly pleased to see the first woman, and the first Latin-American, receiving the highest recognition in mathematics). But there was another remarkable recent event in the world of mathematics: Thomas Hales has announced the completion of the formalization of his proof of the Kepler conjecture. The conjecture: “what is the best way to stack a collection of spherical objects, such as a display of oranges for sale? In 1611 Johannes Kepler suggested that a pyramid arrangement was the most efficient, but couldn't prove it.” (New Scientist)

 

The official announcement goes as follows:

We are pleased to announce the completion of the Flyspeck project, which has constructed a formal proof of the Kepler conjecture. The Kepler conjecture asserts that no packing of congruent balls in Euclidean 3-space has density greater than the face-centered cubic packing. It is the oldest problem in discrete geometry. The proof of the Kepler conjecture was first obtained by Ferguson and Hales in 1998. The proof relies on about 300 pages of text and on a large number of computer calculations.

Verrry interesting

Eric Schwitzgebel recently took up the question of whether an infinitely extended life must be boring. The discussion ended (when I looked at it) with Eric’s fruitfully suggesting that we look at various cognitive architectures and their capacities for boredom over the long run.

No doubt there are many kinds of minds. Let’s radically simplify the problem, in hopes of arriving at a precise answer for at least one case. (After all, if a mind without much to think about can escape boredom, then presumably a more amply stocked mind can too.) The mind I want consider thinks only of natural numbers and number theory (algebraic and analytic). Its “perceptions” consist in presentations of random natural numbers. Will it be bored?

"Mathematicians are supposed to prove things *in* arithmetic, not *about* arithmetic!"

Hitler does not like Gödel's theorem one bit. Perhaps surprisingly, he displays a sophisticated understanding of the implications and presuppositions of the theorem. (In other words, there's some very solid philosophy of logic in the background -- I think I could teach a whole course only on the material presupposed here.)

(Courtesy of Diego Tajer, talented young logician from Buenos Aires, giving continuation to the best Monty Python tradition!)

 

Numerical cognition as a social phenomenon

A few weeks ago I had a post on different ways of counting infinities; the main point was that two of the basic principles that hold for counting finite collections cannot be both transferred over to the case of measuring infinite collections. Now, as a matter of fact I am equally (if not more) interested in the question of counting finite collections at the most basic level, both from the point of view of the foundations of mathematics (‘but what are numbers?’) and from the point of view of how numerical cognition emerges in humans. In fact, to me, these two questions are deeply related.

In a lecture I’ve given a couple of times to non-academic, non-philosophical audiences (so-called ‘outreach lectures’) called ‘What are numbers for people who do not count?’, my starting point is the classic Dedekindian question, ‘What are numbers?’ But instead of going metaphysical, I examine people’s actual counting habits (including among cultures that have very few number words). The idea is that Benacerraf’s (1973) challenge of how we can have epistemic access to these elusive entities, numbers, should be addressed in an empirically informed way, including data from developmental psychology and from anthropological studies (among others). There is a sense in which all there is to explain is the socially enforced practice of counting, which then gives rise to basic arithmetic (from there on, to the rest of mathematics). And here again, Wittgenstein was on the right track with the following observation in the Remarks on the Foundations of Mathematics:

This is how our children learn sums; for one makes them put down three beans and then another three beans and then count what is there. If the result at one time were 5, at another 7 (say because, as we should now say, one sometimes got added, and one sometimes vanished of itself), then the first thing we said would be that beans were no good for teaching sums. But if the same thing happened with sticks, fingers, lines and most other things, that would be the end of all sums.

“But shouldn’t we then still have 2 + 2 = 4?” – This sentence would have become unusable. (RFM, § 37)

Counting infinities

(Cross-posted at M-Phi)

In his Two New Sciences (1638), Galileo presents a puzzle about infinite collections of numbers that became known as ‘Galileo’s paradox’. Written in the form of a dialogue, the interlocutors in the text observe that there are many more positive integers than there are perfect squares, but that every positive integer is the root of a given square. And so, there is a one-to-one correspondence between the positive integers and the perfect squares, and thus we may conclude that there are as many positive integers as there are perfect squares. And yet, the initial assumption was that there are more positive integers than perfect squares, as every perfect square is a positive integer but not vice-versa; in other words, the collection of the perfect squares is strictly contained in the collection of the positive integers. How can they be of the same size then?

Galileo’s conclusion is that principles and concepts pertaining to the size of finite collections cannot be simply transposed, mutatis mutandis, to cases of infinity: “the attributes "equal," "greater," and "less," are not applicable to infinite, but only to finite, quantities.” With respect to finite collections, two uncontroversial principles hold:

Part-whole: a collection A that is strictly contained in a collection B has a strictly smaller size than B.

One-to-one: two collections for which there exists a one-to-one correspondence between their elements are of the same size.

What Galileo’s paradox shows is that, when moving to infinite cases, these two principles clash with each other, and thus that at least one of them has to go. In other words, we simply cannot transpose these two basic intuitions pertaining to counting finite collections to the case of infinite collections. As is well known, Cantor chose to keep One-to-one at the expenses of Part-whole, famously concluding that all countable infinite collections are of the same size (in his terms, have the same cardinality); this is still the reigning orthodoxy.

Axiomatizations of arithmetic and the first-order/second-order divide

I'm currently running a series of posts at M-Phi with sections of a paper I'm working on, 'Axiomatizations of arithmetic and the first-order/second-order divide', which may be of interest to at least some of the NewAPPS readership. It focuses on the idea that, when it comes to axiomatizing arithmetic, descriptive power and deductive power cannot be combined: axiomatizations that are categorical (using a highly expressive logical language, typically second-order logic) will typically be intractable, whereas axiomatizations with deductively better-behaved underlying logics (typically, first-order logic) will not be categorical -- i.e. will be true of models other than the intended model of the series of the natural numbers. Based on a distinction proposed by Hintikka between the descriptive use and the deductive use of logic in the foundations of mathematics, I discuss what the impossibility of having our arithmetical cake and eating it (i.e. of combining deductive power with expressive power to characterize arithmetic with logical tools) means for the first-order logic vs. second-order logic debate.

Part I is here, Part II here, and Part III here. I still hope to post Part IV tomorrow, and then the final Part V will have to wait for a while.

Two conceptions of criteria of adequacy for formalization: Tarski and Carnap

(Cross-posted at M-Phi)

Formal/mathematical philosophy is a well-established approach within philosophical inquiry, having its friends as well as its foes. Now, even though I am very much a formal-approaches-enthusiast, I believe that fundamental methodological questions tend not to receive as much attention as they deserve within this tradition. In particular, a key question which is unfortunately not asked often enough is: what counts as a ‘good’ formalization? How do we know that a given proposed formalization is adequate, so that the insights provided by it are indeed insights about the target phenomenon in question? In recent years, the question of what counts as adequate formalization seems to be for the most part a ‘Swiss obsession’, with the thought-provoking work of Georg Brun, and Michael Baumgartner & Timm Lampert. But even these authors seem to me to restrict the question to a limited notion of formalization, as translation of pieces of natural language into some formalism. (I argued in chapter 3 of my book Formal Languages in Logic that this is not the best way to think about formalization.)

However, some of the pioneers in formal/mathematical approaches to philosophical questions did pay at least some attention to the issue of what counts as an adequate formalization. In this post, I want to discuss how Tarski and Carnap approached the issue, hoping to convince more ‘formal philosophers’ to go back to these questions. (I also find the ‘squeezing argument’ framework developed by Kreisel particularly illuminating, but will leave it out for now, for reasons of space.)

Wittgenstein on the barest essentials

(Cross-posted at M-Phi)

( From the graphic novel Logicomix, taken from this blog post by Richard Zach.)

“He doesn’t want to prove this or that, but to find out how things really are.” This is how Russell describes Wittgenstein in a letter to Lady Ottoline Morrell (as reported in M. Potter’s wonderful book Wittgenstein's Notes on Logic, p. 50 – see my critical note on the book). This may well be the most accurate characterization of Wittgenstein’s approach to philosophy in general, in fact a fitting description of the different phases Wittgenstein went through. Indeed, if there is a common denominator to the first, second, intermediate etc. Wittgensteins, it is the fundamental nature of the questions he asked: different answers, but similar questions throughout. So instead of proving ‘this or that’, for example, he asks what a proof is in the first place.

A dialogical analysis of structural rules - Part I

(Cross-posted at M-Phi)

As some of you may have seen, we will be hosting the workshop ‘Proof theory and philosophy’ in Groningen at the beginning of December. The idea is to focus on the philosophical significance and import of proof theory, rather than exclusively on technical aspects. An impressive team of philosophically inclined proof theorists will be joining us, so it promises to be a very exciting event (titles of talks will be made available shortly).

For my own talk, I’m planning to discuss the main structural rules as defined in sequent calculus – weakening, contraction, exchange, cut – from the point of view of the dialogical conception of deduction that I’ve been developing, inspired in particular (but not exclusively) by Aristotle’s logical texts. In this post, I'll do a bit of preparatory brainstorming, and I look forward to any comments readers may have!

Further thoughts on reductio proofs

(Cross-posted at M-Phi)

Some months ago I wrote two posts on the concept of indirect proofs: one presenting a dialogical conception of these proofs, and the other analyzing the concept of ‘proofs through the impossible’ in the Prior Analytics. Since then I gave a few talks on this material, receiving useful feedback from audiences in Groningen and Paris. Moreover, this week we hosted the conference ‘Dialectic and Aristotle’s Logic’ in Groningen, and after various talks and discussions I have come to formulate some new ideas on the topic of reductio proofs and their dialectical/dialogical underpinnings. So for those of you who enjoyed the previous posts, here are some further thoughts and tentative answers to lingering questions. 

Recall that the dialogical conception I presented in previous posts was meant to address the awkwardness of the first speech act in a reductio proof, namely that of supposing precisely that which you intend to refute by showing that it entails an absurdity. From studies in the literature on math education, it is known that this first step can be very confusing to students learning the technique of reductio proofs. On the dialogical conception, however, no such awkwardness arises, as there is a division of roles between the agent who supposes the initial thesis to be refuted, and the agent who in fact derives an absurdity from the thesis.

Re-proving theorems, and the trouble with incorrect proofs of true statements

(Cross-posted at M-Phi)

“That's the problem with false proofs of true theorems: it's not easy to produce a counterexample.”

This is a comment by Jeffrey Shallit in a post on a purported proof of Fermat’s Last Theorem. (Incidentally, the author of the purported proof comments at M-Phi occasionally.) In all its apparent simplicity, this remark raises a number of interesting philosophical questions. (Being the pedantic philosopher that I am, I'll change a bit the terminology and use the phrase 'incorrect proof' instead of 'false proof', which I take to be a category mistake.)

First of all, the remark refers to a pervasive but prima facie slightly puzzling feature of mathematical practice: mathematicians often formulate alternative proofs of theorems that have already been proved. This may appear somewhat surprising on the assumption that mathematicians are (solely) in the business of establishing (mathematical) truths; now, if a given truth, a theorem, has already been established, what is the point of going down the same road again? (Or more precisely, going to the same place by taking a different road.) This of course shows that the assumption in question is false: mathematicians are not only interested in theorems, in fact they are mostly interested in proofs. (This is one of the points of Rav’s thought-provoking paper ‘Why do we prove theorems?’)

There are several reasons why mathematicians look for new proofs of previously established theorems, and John Dawson Jr.’s excellent ‘Why do mathematicians re-prove theorems?’ discusses a number of these reasons. The original proof may be seen as too convoluted or not sufficient explanatory – ideally, a proof shows not only that P is the case, but also why P is the case (more on this below). Alternatively, the proof may rely on notions and concepts alien to the formulation and understanding of the theorem itself, giving rise to concerns of purity. Indeed, recall that Colin McLarty motivates his search for a new proof of Fermat’s Last Theorem in these terms: “Fermat’s Last Theorem is just about numbers, so it seems like we ought to be able to prove it by just talking about numbers”. This is not the case of the currently available proof by Wiles, which relies on much heavier machinery.

LSU A great place to generate low Erdős–Bacon number?

Cogburn's most recent post made me wonder if LSU is a great place to generate low Erdős–Bacon numbers.

"A person's Erdős–Bacon number is the sum of one's Erdős number—which measures the "collaborative distance" in authoring mathematical papers between that person and Hungarian mathematician Paul Erdős—and one's Bacon number—which represents the number of links, through roles in films, by which the individual is separated from American actor Kevin Bacon. The lower the number, the closer a person is to Erdős and Bacon, and this reflects a small world phenomenon in academia and entertainment."--Wikipedia. [HT: Wayne Myrvold] So, for example Bertrand Russell's Bacon number is  the result of an appearance in a Bollywood film. 

Request for literature suggestions on ancient Greek mathematics

Those of you who have been following some of my blog posts will recall my current research project ‘Roots of Deduction’, which aims at unearthing (hopefully without damaging!) the conceptual and historical origins of the very concept of a deductive argument as one where the truth of the premises necessitates the truth of the conclusion. In particular, this past year we’ve been reading the Prior Analytics in a reading group, which has been a fantastic experience (highly recommended!). For next year, the plan is to switch from logic to mathematics, and look more closely into the development of deductive arguments in Greek mathematics.

But here’s the catch: the members of the project are all much more versed in the history of logic than in the history of mathematics, so we can’t count on as much previous expertise for mathematics as we could in the case of (Aristotelian) logic. Moreover, the history of ancient Greek mathematics is a rather intimidating topic, with an enormous amount of secondary literature and a notorious scarcity of primary sources (at least for the earlier pre-Euclid period, which is what we would be interested in). So it seems prudent to focus on a few specific aspects of the topic, and for now I have in mind specifically the connections between mathematics and logic (and philosophy) in ancient Greece. More generally, our main interest is not on the ‘contentual’ part of mathematical theories, but rather on the ‘structural’ part, in particular the general structure of arguments and the emergence of necessarily truth-preserving arguments.

One reason (among many) to like Homotopy Type Theory

(Cross-posted at M-Phi)

Last week I was in Munich for the excellent ‘Carnap on logic’ conference, which brought together pretty much everyone who’s someone in the world of Carnap scholarship. (And that excludes me -- still don’t know exactly why I was invited in the first place…) My talk was a comparison between Carnap’s notion of explication and my own conception of formalization, as developed in my book Formal Languages in Logic. In particular, I proposed a cognitive, empirically informed account of Carnap’s notion of the fruitfulness of an explication.

Anyway, I learned an awful lot about Carnap, and got to meet some great people I hadn’t yet met. But perhaps the talk I enjoyed most was Steve Awodey’s ‘On the invariance of logical truth’ (for those of you who have seen Steve lecturing before, this will come as no surprise…). The main point of Steve’s talk was to defend the claim that the notion of (logical) invariance that is now more readily associated with Tarski, in particular his lecture ‘What are logical notions?’ (1966, published posthumously in 1986), is already to be found in the work of Carnap of the 1930s. This in itself was already fascinating, but then Steve ended his talk by drawing some connections between the invariance debate in philosophy of logic and his current work on homotopy type theory. Now, some of you will remember that I am truly excited about this new research program, and since I’ve also spent quite some time thinking about invariance criteria for logicality (more on which below), it was a real treat to hear Steve relating the two debates. In particular, he gave me (yet another) reason to be excited about the homotopy type theory program, which is the topic of this blog post.

Alternative Foundations in the Faith of Mathematics-land

"The usual implicit assumption is that mathematical English could be formalized in a set-theoretic foundation such as ZFC, and this requires various conventions on what we can and can’t say in mathematical English.  The goal of informal type theory is to develop conventions for a version of mathematical English whose “implicit underlying foundation” is instead type theory — specifically, homotopy type theory."--Mike Shulman.

"Writing a 500 pp. book on an entirely new subject, with 40 authors, in 9 months is already an amazing achievement....But even more astonishing, in my humble opinion, is the mathematical and logical content: this is an entirely new foundation, starting from scratch and reaching to , the Yoneda lemma, the cumulative hierarchy of sets, and a new constructive treatment of the real numbers — with a whole lot of new and fascinating mathematics and logic along the way...But for all that, what is perhaps most remarkable about the book is what is not in it: formalized mathematics.  One of the novel things about HoTT is that it is not just formal logical foundations of mathematics in principle: it really is code that runs in a computer proof assistant... At the risk of sounding a bit grandiose, this represents something of a “new paradigm” for mathematics:  fully formal proofs that can be run on the computer to ensure their absolute rigor, accompanied by informal exposition that can focus more on the intuition and examples.  Our goal in this Book has been to provide a first, extended example of this new style of doing mathematics."--Steve Awodey.

[HT Choice & Inference] And on the future sociology of mathematics:

Weekly Philo Economics: metaphysical indeterminacy and the road not taken by Analytic Philosophy (and Political Economy)

"I believe that [W.E.] Johnson, like McTaggart and Aristotle, deserves commentators." A.N. Prior (1949) MIND.

"Mesmerized by Homo economicus, who acts solely on egoism, economists shy away from altruism almost comically. Caught in a shameful act of heroism, they aver: "Shucks, it was only enlightened self interest." Sometimes it is. At other times it may be only rationalization (spurious for card-carrying atheists): "If I rescue somebody's son, someone will rescue mine.

I will not waste ink on face-saving tautologies. When the governess of infants caught in a burning building reenters it unobserved in a hopeless mission of rescue, casuists may argue; "She did it only to get the good feeling of doing it. Because other-wise she wouldn't have done it." Such argumentation (in Wolfgang Pauli's scathing phrase) is not even wrong. It is just boring, irrelevant, and in the technical sense of old-fashioned logical positivism "meaning-less." You do not understand the logic and history of consumer demand theory — Pareto, W. E. Johnson, Slutsky, Allen-Hicks, Hotelling, Samuelson, Houthakker,... — if you think that is its content."--P. Samuelson (1993), The American Economic Review.

There is a school of thought that locates the origins of analytical philosophy in the Cambridge of the philosopher-economist, Sidgwick and his students. After all, in Sidgwick's writings we find all the analytical virtues, and it is, thus, no surprise that Rawls and Parfit treat him as our vital interlocuter. Those (that is, the circle around Sidwick) recognized in Boole's work -- to quote W.E. Johnson -- "the first great revolution in the study of formal logic...comparable in importance with that of the algebraical symbolists in the sixteenth century." (2.6, p. 136) While it is not the story I tend to tell (say, here and here), I like this approach because it reminds us of the non-trivial overlap between logicians and economists so distinctive of Cambridge between 1870-1940, and thus, puts Keynes (father and son) and Ramsey back into the origin of analytical philosophy.

Now, the logician-economist, W.E. Johnson (1858 – 1931), is a test-case for this school of thought.  (Recall the significance of Johnson to of our very own Mohan [and here].) For, while Johnson does not belong to the British Idealists, he does not figure in the stories we tell about our origins at all (selective evidence: Landini's Russell nor Candlish's The Russel/Bradley Dispute do not even mention Johnson). Even Wikipidia claims that his "Logic was dated at the time of its publication, and Johnson can be seen as a member of the British logic "old guard" pushed aside" by Russell and Whitehead. Wikipedia fits our narrative of progress; yet what to make of Prior's judgment?