I finished rereading this book last week, I can tell you my take on heuristic counterexamples.

Imagine you've got this "mathematical theorem" which has been conjectured but not yet given a formal proof...

(1) X is a sandwich if X is a food item consisting a food item (called "the filling") bounded between two pieces of bread.

A so called logical counterexample to it would be something that satisfies all the premises with their intended interpretation but not the conclusion with its intended interpretation. In this case, it would be a food item that has a filling bounded between two slices of bread which is not a sandwich... Perhaps burgers count for that, since they're not clearly sandwiches, but do have a filling (in a sense) and the filling is between two pieces of bread. So burgers perhaps would be a logical counterexample to the characterisation of sandwiches as "sandwiches are all and only those food items that have a filling bounded between two pieces of bread".

Someone like Alpha in the book would present this monster to challenge it:



Because it is a foodstuff whose position is bounded between two pieces of bread, and is thus a sandwich by the definition. A lot of concept stretching has gone into that example; is a banana with the skin on a food item? Does a banana count as a filling if it has the skin on? Do whole pieces of bread separated in space with a banana placed inbetween them count as a sandwich?

The first example of the burger grants the intended interpretation of the terms in (1); the bits of bread bun in the burger resemble slices etc. the second example intentionally stretches the intended interpretation while staying within the letter of the theorem statement. The first looks logical since it grants the intended interpretation and simply blocks the inference; it ain't a sandwich, it's a burger, but it is a food item bounded between two pieces of bread. The second looks more heuristic since it seems to stretch the concepts of the terms involved while staying within the letter of how they were used in the theorem.

Both the burger "logical" counterexample and the monster "heuristic" one actually do the same thing (for Lakatos, I think), they reveal that the intended interpretation had hidden content/assumptions which were not expressed in the verbatim statement of the theorem. The refutations could lead to a revised conjecture:

(2) X is a sandwich if and only if X is a foodstuff, it has a foodstuff Y which is prepared and edible called the filling, the filling is located snugly between two slices of bread, and the two slices of bread enclose the filling.

Which might inspire more counterexamples - Alpha might put Y="a small piece of bread", and now it's a bread sandwich...

In terms of the languages, I'd guess that the theorem statement is in L1.

X is a sandwich if X is a food item consisting a food item (called "the filling") bounded between two pieces of bread.

It doesn't contain a contradiction by itself, it just perhaps doesn't fit what is the intended interpretation of a sandwich. When someone provides an example of a food item which fits the definition but not its intended interpretation, they pass into a meta language L2 to present the discrepancies between the intended interpretation of the terms in the theorem and the counterexamples. And when discussing them, they're already in a meta-meta language L3 in which the intended intepretations of L2 statements and their relationships to L1 terms (like in the theorem statement) are weighed.

I think whenever you start talking about the meaning of the terms in a language, you pass into a meta-language about it.

Compare this statement of semantic completeness of a formal system S.

If X is semantically derivable in S then X is syntactically derivable in S.

You have the level of formulae like X, then you have the level of proofs on formulae and interpretations of formula which syntactic and semantic derivability is talking about, then you have the embedding in another (somewhat informal) metalanguage that spells out the connection (if-then, in which language is that implication relation defined?) between the semantic derivability and syntactic derivability.

Hello, Thank you so much for taking the time to answer and, indeed, I believe it helps!
Especially when you talk about intended interpretation, so this is why there is no contradiction between: (a) all polyhedral are Eulerian, and (b) the picture-frame is not Eulerian.
So then, by changing language i.e., switching from L1 to L2, we're falling into this meta-language. "If we keep to the tacit semantical rules of our original language our counterexamples are not counterexamples" means that switching to L2 or L3 (?) we use concept-stretching to add counter-examples into the proof and contribute to the growth of knowledge (heuristic power). Thus using L1/L2 and L3 is just a way of describing the extension of knowledge/increase in content ?
I'm so sorry, I feel like everything gets mixed up in my head.

And I guess we could add that concept formation cannot be separated from "definition formation" which relates to a more heuristic study of knowledge.

I'm so sorry, I feel like everything gets mixed up in my head. — Twinkle221

Everything is mixed up in my head too, I'm just hoping writing it out will order things for me.

Especially when you talk about intended interpretation, so this is why there is no contradiction between: (a) all polyhedral are Eulerian, (b) the picture-frame is not Eulerian — Twinkle221

I believe that's so, the kinds of polyhedron intuited in "all polyhedron are Eulerian" are conceptually distinct from the picture frame polyhedron (and the other monsters). What "allows the monsters to work as refutations" is a mismatch between how the concept of polyhedron is articulated verbatim; the content of the definition and theorem statement; and the intended content of the articulation; the restricted, not yet characterised class of polyhedra to which Euler's formula applies.

I think I was infelicitous in using "metalanguage" to talk about the shifts between languages, on reflection. I don't think it's strictly speaking "wrong", but I don't think it portrays the language shifts in the above quote very well. I'm sorry!

So then, by changing language i.e., switching from L1 to L2, we're falling into this meta-language. "If we keep to the tacit semantical rules of our original language our counterexamples are not counterexamples — Twinkle221

I think the key word that marks the presence of heuristic reasoning is "tacit" there. Tacit semantical rules; the latent and hitherto unformalised/unarticulated content of the intended interpretation of the terms.

I guess a clearer example of how I see the language shift thing is:

Imagine I have defined rational numbers as: "a number that results from one number being divided by another". Let's call that an L1 statement. It's L1 because it sets out an initial way of using the term rational number.

Someone like Alpha comes along and writes down "$\sqrt{2}=\frac{\sqrt(2)}{1}$ is a rational number", and it indeed satisfies my definition; since $\sqrt{2}$ is a number, $1$ is a number, and dividing $\sqrt{2}$ by $1$ results in $\sqrt{2}$. So by my definition it is a rational number. I could stick to my guns like Pythagoras allegedly did and claim that sqrt(2) is not a number! A consistent position, but a terrible one nevertheless. I could alternatively accept the counterexample... But that would mean adopting new standards of interpretation and ways of talking about the concept of fraction; since it's no longer right to construe it as "a number that results from one number being divided by another". To make that shift, I've got to change how I write and think about fractions, and set out new rules to characterise them. So I'll accept Alpha's concept stretch refutation and institute a new way, an L2.

I come back to Alpha with: "A rational number is the result of dividing one integer by another integer".

To explain why I made that change, I need to explain how my previous articulation of rational numbers was wrong; it just emphasised that you can write one number on top of the other, and Alpha did just that with a monster on the top. So there was something else other than "one number on top of the other" that I needed to include, to a first approximation I'd guess instead of "dividing one number by another number", I'd restrict it to "divide one integer by another integer", that should stop Alpha from coming back with any counterexamples like that - square roots etc are not integers. Perhaps this explanation is in some other language, L3, in which I compare L2 to L1... Comparisons between fraction concepts do not seem to be part of the terms I came back to Alpha with.



Alpha might hate me for monster barring because $\sqrt{2}$ is a perfectly good number and I've barred that by putting in "integers" explicitly into the definition to block it, but I could've also done "A rational number is a number that results from one number being divided by another except $\sqrt{2}$", which doesn't refine the understanding of rational numbers at all, since it doesn't explain why the heuristic refutation forced me to refine my definition - ie, it doesn't spell out how the heuristic refutation clashed with the intended interpretation of the terms. Putting "integer" in there blocks some of the right monsters, so to speak.

And I guess we could add that concept formation cannot be separated from "definition formation" which relates to a more heuristic study of knowledge. — Twinkle221

That's my reading too. Wrestling with definitions is how you pin down the intended interpretation in mathematical language, and that "pinning down" is an articulation of the content of the mathematical structure under study.

Thanks! It's all clearer now! and I think I can fully appreciate the last sentence:

Heuristic is concerned with language-dynamics, while logic is concerned with language-statics. — Twinkle221

And yes, the word "metalanguage" felt odd so I'm glad to see we agree upon its use.

I believe that's so, the kinds of polyhedron intuited in "all polyhedron are Eulerian" are conceptually distinct from the picture frame polyhedron (and the other monsters). — fdrake

So I assume, following your answer, that this is why by switching languages we can introduce "real" counterexamples that were not counterexamples before.

What "allows the monsters to work as refutations" is a mismatch between how concept of polyhedron is articulated verbatim — fdrake

But that works only when we switch between languages and get rid of the "ordinary language" right?

unformalised/unarticulated content of the intended interpretation of the terms. — fdrake

And so when this content gains structure by being incorporated in a language that's when counterexamples (that are all heuristics) get "real" by concept-stretching

logical counterexamples — Twinkle221

And I assume these are heuristic still

That is, we may have two statements that are consistent in L1, but we switch to L2 in which they are inconsistent. Or, we may have two statements that are inconsistent in L1, but we switch to L2 in which they are consistent. As knowledge grows, languages change. — Twinkle221

And this is just to say that concept-stretching does his job through refutations, at least I believe so!

Don't know whether I was clear on this point so I'm writing it again

Usually, when a counterexample is presented, you have a choice: either you refuse to bother with it, since it is not a counter-example at all in your given language L1, or you agree to change your language by concept-stretching and accept the counterexample in your new language L2… — Twinkle221

I think that what appeared to be logical counterexamples were heuristic counterexamples and only when we switched language under some kind of heuristic pressure did we end up with logical counterexamples, used to revise a given theorem.

And so heuristic counterexamples bring about a change in mathematical meaning!

And yes, the word "metalanguage" felt odd so I'm glad to see we agree upon its use. — Twinkle221

:up:

But that works only when we switch between languages and get rid of the "ordinary language" right? — Twinkle221

I don't know? It seems to me when I read mathematics papers, the majority of the reasoning in them is informal by a logician's standards. Suppressed premises and formally invalid inferences everywhere; but assumed competence and familiarity with the concepts papers over the gaps. I would guess that mathematical reasoning's "mother tongue", so to speak, is natural language, and we enter into formal languages to articulate things with higher precision and replicability. EG, the contrast between the whiteboard doodles two mathematicians working on a paper make when discussing its ideas vs what they actually write in a proof in a paper. Terence Tao has written on a similar theme before; he seems to believe mathematical competence in a domain occurs when one's intuitions in it become formalizable and incorporate previously established theory, not when one can "symbol chase" a proof from definitions to conclusion. To put it in other words, mathematical competence in a domain is marked when one's understanding is not merely formal, but can be formalised.

So I assume, following your answer, that this is why by switching languages we can introduce "real" counterexamples that were not counterexamples before. — Twinkle221

I think one can insist that the L1 definition is correct - barring a monster - and be consistent/not entail contradictions. But if the monster is treated as a refutation, that means rejecting the L1 definition/theorem-statement and massaging the concept/intended interpretation with its related space of hypothetical definitions (which would be L2, L3, L4... if put into words/formalised) until a felicitous matching occurs between them. That matching process has changed "the taxonomic, conceptual..." frameworks of (some of) the involved terms, and those frameworks are expressed in the statement of an L2 definition/theorem-statement in which the monster refutes the L1 statement since the monster now unambiguously counts as an example of the term it targets. EG "Polyhedra are Eulerian" with the intended interpretation of convexity and simplicity vs concave polyhedral monsters.

An intuition pump here might be that "polyhedron" in Proofs and Refutations and "rational number" in my example have ambiguous intensions; we roughly know what is intended by the term, but we don't know all the content required to support our claims using the term. If we imagine we have to treat the intension of those terms (despite their ambiguity) as a constant over our investigation, sticking to one's guns in the face of a heuristic refutation commits no one to a logical error; but it does limit the intension in a way it was not limited before the heuristic refutation - or perhaps the original L1 formulation "said too much", so to speak. As an example of "saying too much", like Euler stating his formula held for polyhedra simpliciter. The monster "becomes" a refutation of the L1 statement by including the monster in the intension of L1's terms, like the picture frame in the intension of polyhedron.

However, if one takes the refutation as an opportunity to include the monster in the intension of the original term, it becomes a refutation, but the intension is also altered tacitly by taking this opportunity.

I never knew heuristics was so complicated. I frequently used what mathematicians consider heuristics when teaching, particularly in a course in complex variables. For instance, the maximum modulus principle is easily explained on the blackboard by drawing a simple picture, and this heuristic motivates an actual proof. And when the question of a physical interpretation of the derivative of a complex function arises, another picture on the blackboard demonstrates its meaning as a contractive factor.

As for published papers, heuristics can be quite motivational for tempting other professionals to read deeper into your work where things get more formal.

In mathematics, at an opposite pole are extremely formal proofs by computer algorithms, when that is possible, with nothing presented except printed symbols. Anathema for most mathematicians at the present time, but possibly what lies ahead.

However, if one takes the refutation as an opportunity to include the monster in the intension of the original term, it becomes a refutation, but the intension is also altered tacitly by taking this opportunity. — fdrake

Yes I agree !

That matching process has changed "the taxonomic, conceptual..." frameworks of (some of) the involved terms, and those frameworks are expressed in the statement of an L2 definition/theorem-statement in which the monster refutes the L1 statement since the monster now unambiguously counts as an example of the term it targets. EG "Polyhedra are Eulerian" with the intended interpretation of convexity and simplicity vs concave polyhedral monsters. — fdrake

:up: :100: I think that's the conclusion I too can draw from all this

I really want to thank you for your help, I can really see the bigger picture here now!

In mathematics, at an opposite pole are extremely formal proofs by computer algorithms — jgill

Yes indeed, that's also why Lakatos' book is really interesting! He argues against formal mathematics byt his method of "proofs and reputations" may very well be applied to formal mathematics as well.

I am really interested in the use of heuristics as a research methodology :blush:

Hello! I am a graduate student in philosophy of science — Twinkle221

Hi Twinkle,

Sorry that I can't help you with the book. So this is just a quick note to let you know that it would interest me to be taken in to the world of graduate studies in philosophy of science. Should your time permit that might make an engaging topic for another thread. Good luck!

I really want to thank you for your help, I can really see the bigger picture here now! — Twinkle221

No worries! I'm glad that my ambiguous "meta language" intension has been cleared up. :smile: This was fun.

I think if we've pinned down the heuristic method, there's still a relevant discussion we could have about how it purports to describe mathematical conduct. Game if you are!

In mathematics, at an opposite pole are extremely formal proofs by computer algorithms — jgill

Yes indeed, that's also why Lakatos' book is really interesting! He argues against formal mathematics byt his method of "proofs and reputations" may very well be applied to formal mathematics as well.

I am really interested in the use of heuristics as a research methodology :blush: — Twinkle221

I was speaking of completely automated proofs by computer programs. Formal mathematics is simply that employed by research math people these days. You know all this of course. I've now read a bit about the topic here and I suppose what I have used might be called naive heuristics, in light of all the various types of heuristics described in Wiki.

For me, heuristics in math are not problem-solving rules, although at elementary levels that definition would apply. From my perspective, in research and presentation the word means simplifying by avoiding rigorous logic at various steps, staying holistic to a conclusion. In other words, hand-waving. I guess I never considered it a research methodology other than speculation involving examples.

Formal mathematics is simply that employed by research math people these days. You know all this of course. I've now read a bit about the topic here and I suppose what I have used might be called naive heuristics, in light of all the various types of heuristics described in Wiki. — jgill

I think "formal mathematics" as the book considers is quite similar to "formal mathematics" as practicing mathematicians consider it. There is an aspect of "formal mathematics" - really a philosophical construal of mathematics- that Lakatos strongly criticises, though.

In my experience, practicing mathematicians simply don't expect everything they say or think to already be formalised, nor do they expect a strong mathematical argument to be presented with the required rigour of logic. As an example, this proof is logically invalid as presented but demonstrates what it seeks to to the point of obviousness.

Let $x$ be a real number that satisfies $x^2-1=0$.

$\begin{align}x^2 - 1 = 0\\\implies (x+1)(x-1)=0\\\implies x=\pm 1\end{align}$

It's definitely a true series of implications, but there's no supplementary statement that the factorisation in step 1 is valid , or a statement of the premise that the real numbers as a field have no zero-divisors. To be sure, it clearly demonstrates an understanding of how to solve the quadratic, and how you'd go about doing it, but it's not strictly a valid argument. Those things I highlighted which restore its logical validity are in the background understanding of the terms, and the reader is expected to grok it.

The heuristic analysis of Lakatos is a historical investigation into that kind of background. The book has a historically detailed example of how different background understandings of the concept of polyhedron played out in mathematics. Being able to take a concept out of that background; like polyhedron; and formalise it with precision is an advance in mathematics.

Yes, an expectation of the reader filling in the gaps is common. It would be dreadful if that practice were abandoned! :cool:

Yes, an expectation of the reader filling in the gaps is common. It would be dreadful if that practice were abandoned! :cool: — jgill

Yes! I don't think mathematical research would look anything like it does if we had to reference all conventions we're using. What that does do, however, is make mathematical demonstrations heuristic in Lakatos' sense; more about displaying the concepts to a sufficient degree of obviousness than mandating that all proofs have every step of reasoning spelled out symbolically.

I think that "sufficient degree" has been amplified since the "crisis in analysis"; the whole thing about people not distinguishing continuity from the intermediate value property and various problems I can't remember with integrals; and amplified again since the Hilbert program and the invention of formal languages as mathematical abstractions.

What that does do, however, is make mathematical demonstrations heuristic in Lakatos' sense; more about displaying the concepts to a sufficient degree of obviousness than mandating that all proofs have every step of reasoning spelled out symbolically. — fdrake

Heuristics in mathematical expositions is relative to an intended audience.

Heyyyy

Wow I'm really glad this discussion is interesting you all, sharing is the best really! :blush:

For me, this is a very interesting thread. It's more relevant to my own experiences as a math prof than other threads about mathematics, especially those that argue about the equal symbol and and the Platonic characters. I am curious about heuristics as a research methodology. The research I have done involves speculations about conjectures and imagery that crops up in my computer programs, plus analyzing examples - all very loosely coordinated - until an idea pops into my head. If there's a methodology there it is loathe to reveal itself.

But, then, I was an average math person and not in the upper echelons where things may go differently. To give an example, here is a recent very informal note I posted on researchgate that explains to some extent an intense iteration process in the complex plane. It's of no consequence. I do these things like other old geezers carve wooden ducks, keeps me occupied since retirement twenty years ago. But it does show imagery that sparked the actual math: It is also full of the kind of heuristics I assume might give Lakatos the Vapors.

Tunnels to Centroids of Attractors

Yes, I find Lakatos' book very interesting! Turns your head upsidedown but I enjoyed reading it :smile: I sometimes have some troubles doing thought experiments especially as I am terrible with geometrical concepts...