Chapter 1, §§1-3
Chapter 1, §4

Imre Lakatos’ Proofs and Refutations: The Logic of Mathematical Discovery (1976) is a fantastically interesting, fictional dialogue that concisely represents a history of modern mathematical discovery, proof methods, and criticisms.  A discussion that takes place in a math classroom displays mathematical history through a case study of the Descartes-Euler conjecture, with particular focus given to Cauchy’s “proof” of the conjecture.  Lakatos pushes back against the formalist method of mathematics dominant at his time, emphasizing the aspects of mathematical methodology which are not described by the formalists’ metamathematical language.  Multiple historical positions of notable mathematicians are represented and labelled, and a heuristic model for mathematical discovery is proposed.

While reading Proofs and Refutations for the first time, I thought it would be useful to have an list of the concepts, methods, and heuristics suggested by its characters. It would make positions easier to compare, and it’s nice to have them all in one place. For my third time going through the text, I have decided to compile such a list.  This is not a replacement for actually reading through the dialogue, as I will not be explaining Cauchy’s proof or its counterexamples.

Sections 1-3 focus on a proposed proof for the Descartes-Euler conjecture (the Cauchy proof) and types of counterexamples (local and/or global counterexamples).  These are the concepts I found notable from these sections.  I don’t claim that my definitions and descriptions are the best—my goal here is to provide something I would have found useful the first time I read through P&R.  I hope writing this out will help my own and others’ understanding.

Name of the Concept / Method / Heuristic
Description or definition of the concept, method, or heuristic.
Context in the text. Also, quotes from the text, if I think something is particularly useful.
If applicable, I will note where this concept, method, heuristic notably reappears in the discussion of other concepts, methods, or heuristics.

§1.   A Problem and a Conjecture

Problem
A gap in knowledge about the mathematical relationship among mathematical concepts.
In the first paragraph, a mathematical problem is described.  There is a gap in knowledge regarding the relationship among polyhedra’s numbers of vertices, numbers of edges, and numbers of faces.  Solving this problem would allow mathematicians to classify polyhedra, similar to how they can classify polygons into triangles, quadrangles, pentagons, et cetera. (Observe that polygons display a constant relationship: # of vertices = # of edges.)
“PROBLEM: is there a relation between the number of vertices V, the number of edges E and the number of faces F of polyhedra – particularly of regular polyhedra – analogous to the trivial relation between the number of vertices and edges of polygons, namely, that there are as many edges as vertices: V = E?” (6)

Proposition
A statement of a constant mathematical relationship among mathematical concepts. A proposition may be either true or false, or it may not be known if the proposition is true or false.  Conjectures are a type of proposition.
Sometimes used interchangeably to refer to lemmas, subconjectures, or conjectures (such as the Descartes-Euler Conjecture). Note: the first mention of ‘proposition’ is in §2, but I mention it here to put it closer to the definition of conjecture.

Conjecture
A proposition which is is not known to be true or false, and is suggested as a potential candidate to fill in a gap in knowledge and solve a problem. To become a theorem, the conjecture must be given a legitimate proof.  If a legitimate counterexample to the conjecture is discovered, that would disprove the conjecture. A legitimate proof that shows the conjecture is false would also disprove the conjecture.
The Descartes-Euler Conjecture is that, for any polyhedra, the relationship between its number of vertices, edges, and faces (V, E, and F) is: V-E+F = 2.  The classroom notices this relationship holds for all regular polyhedra, and then make the conjecture that this relationship holds for any polyhedra.
“After much trial and error they notice that for all regular polyhedra V-E+F = 2. Somebody guesses that this may apply for any polyhedron whatsoever. Others try to falsify this conjecture, try to test it in many different ways – it holds good. The results corroborate the conjecture, and suggest that it could be proved.” (6-7)

Proof (defn i)
The qualities of a ‘proof’ are developed throughout Proofs and Refutations, matching the concept’s historical development.  However, a basic definition might state the requirements of a proof as such: “A mathematical proof justifies that a proposition is true with certainty/rigour.  Through reasoning, a mathematical proof establishes that a mathematical relationship among mathematical concepts is true and incontrovertible.”
The sort of “certainty/rigour” required to be considered a mathematical proof changed throughout history, from hazy argumentation, to thought-examples, to deductive reasoning, to metamathematical foundations. (55-56) We will see this development through the definitions of ‘proof’ and ‘rigour’ made throughout the dialogue.
Unless otherwise specified, when I use ‘proof’ or ‘proven’, I will be referring to this basic definition.

§2.   A Proof

‘Method’ of Satisfactory Demonstration
This is not actually a strict proof of a conjecture, but after seeing a conjecture hold for many cases, one might say “the proposition seems to be satisfactorily demonstrated.” (7)  That is, the conjecture seems to work really well but it’s tough to come up with a proof—instead of spending more time on that, we might want to put emphasis on examining the consequences of treating the conjecture as if it were true.  Of course, it would still be preferable to have a proof.
I have taken some liberty in labeling this a “method”. Lakatos does not explicitly state that this is a method, but includes it as a position of one of his characters. Sigma is directly quoting Euler’s position regarding his conjecture (before Euler eventually proposed a proof). “PUPIL SIGMA: ‘I for one have to admit that I have not yet been able to devise a strict proof of this theorem... As however the truth of it has been established in so many cases, there can be no doubt that it holds good for any solid. Thus the proposition seems to be satisfactorily demonstrated.’ But if you have a proof, please do present it.” (7)
This is reminiscent of an inductive method proposed later on in the dialogue (scientific-style induction, not mathematical induction).

Proof (Thought-Experiment) (defn ii)
The first “proof” we are presented with comes in the form of a thought-experiment which is an informal decomposition of the original conjecture into an argument made up of subconjectures or lemmas. If all of the subconjectures/lemmas are true, the story laid out by the thought-experiment seems to imply that the main conjecture is true. The subconjectures/lemmas are well-integrated in areas of knowledge which we think we understand, and therefore seem to lend certainty to the truth of the main conjecture.
This proof format doesn’t “guarantee certain truth” of the main conjecture—it merely shifts the focus of our uncertainty to its subconjectures. (See the Principle of Retransmission of Falsity) Still, as this has historically been considered a valid proof format, we tentatively label it a proof.
The Teacher describes a proof of the Descartes-Euler Conjecture (this proof was first given by Cauchy). “TEACHER: ... I propose to retain the time–honoured technical term ‘proof’ for a thought–experiment – or ‘quasi–experiment’ – which suggests a decomposition of the original conjecture into subconjectures or lemmas, thus embedding it in a possibly quite distant body of knowledge. Our ‘proof’, for instance, has embedded the original conjecture – about crystals, or, say, solids – in the theory of rubber sheets.” (9)
“TEACHER: ... My interpretation of proof will allow for a false conjecture to be ‘proved’, i.e. to be decomposed into subconjectures.” (23)
That the Cauchy proof qualifies as a legitimate proof is criticized in all sorts of ways throughout the dialogue. This definition of ‘proof’ is also later reexamined and considered to be a test of the conjecture—not actually a proof.

Theorem
A conjecture which has been proven true. It fills in a gap in knowledge and solves a problem.
“PUPIL DELTA: You should now call it a theorem. There is nothing conjectural about it any more.” (8)

Lemma
A theorem, or a proposition which is presumed to be true, used in the context of a proof as part of the proof’s argument. If all the lemmas (and subconjectures) in the argument are true, and the argument is valid, it would imply the truth of the main conjecture that the proof has set out to prove. If a lemma turns out to be false, this does not necessarily show that the main conjecture is false. (See Counterexample which is Local but not Global)
Statements made in the Cauchy proof might initially seem like lemmas, but once doubts are raised about their truthfulness, it might be more precise to consider them subconjectures.

Subconjecture
As part of a proof’s argument, a subconjecture plays essentially the same role as a lemma—however, we realize that it may or may not be true. Because it is a proposition, and we don’t know if it is true or false, it is a conjecture. Because it plays an intermediary role in the proof of a main conjecture (like a lemma), it is more precisely thought of as a subconjecture (subordinate to the main conjecture that the proof is proving). If a subconjecture is shown to be true, it would become a theorem (and in the context of the argument or proof, we would refer to it as a lemma).
‘Lemma’ and ‘subconjecture’ are often used interchangeably because the subconjectures of a proof (thought-experiment; defn ii) are presumed to be true by the proof.
Additionally, it only seems to make sense to talk about subconjectures in the context of proofs as thought-experiments (defn ii). Proofs (defn i)—with their certainty/rigour—aren’t supposed to be conjectural at all, and so shouldn’t allow for the possibility of subconjectures.

§3.   Criticism of the Proof by Counterexamples which are Local but not Global

Counterexample
A counterexample to a conjecture or lemma is a case in which the conjecture’s mathematical concepts fail to be accurately described by the conjecture’s proposed mathematical relationship. The conjecture fails to hold true in the case of the counterexample, and is therefore shown to be false.
This may appear to be a self-evident concept, but much of §4 are debates over what should or should not be considered a legitimate counterexample and how mathematicians should react when presented with a legitimate counterexample. Each of the Methods that take a stance on how we should respond to counterexamples can be read to reject or build on specific sections of the first sentence of the description. (See the Method of Surrender, the Method of Monster-Barring, the Method of Exception-Barring, the Method of Monster-Adjustment, the Method of Lemma-Incorporation, and Concept-Stretching)

Refutation (defn i)
The definition of refutation is also developed throughout Proofs and Refutations, matching the historical development of proof. A refutation of a conjecture, proof, or theorem is when people accept that it is false (when it was initially thought to be true, or at least, potentially true).

Local Counterexample
A counterexample to one of the lemmas or subconjectures in a proof. The does not necessarily mean that the main conjecture is refuted.
“TEACHER: ... I shall call a ‘local counterexample’ an example which refutes a lemma (without necessarily refuting the main conjecture), and I shall call a ‘global counterexample’ an example which refutes the main conjecture itself.” (10-11)

Global Counterexample
A counterexample to the main conjecture of a proof.
The class assumes in §4 that a global counterexample not only falsifies the main conjecture, but at least one of the lemmas as well. (See Counterexample which is both Global and Local) However, in §5.b it is declared by Alpha that it is possible to have a Counterexample which is Global but not Local.
“TEACHER: ... I shall call a ‘local counterexample’ an example which refutes a lemma (without necessarily refuting the main conjecture), and I shall call a ‘global counterexample’ an example which refutes the main conjecture itself.” (10-11)

Counterexample which is Local but not Global
(See local counterexample and global counterexample.) This is a counterexample which is a case that refutes a lemma/subconjecture of an argument, but is not a case which shows the main conjecture to be false. The main conjecture holds true for this case, but the lemma does not. Therefore, a ‘counterexample which is local but not global’ reveals a problem in the proof (defn ii, defn iii), but not necessarily in the main conjecture.
In §3, the Teacher advances a method to handle Counterexamples which are Local but not Global (This eventually becomes part of the Method of Proof and Refutations). In §4, we discuss Counterexamples which are both Global and Local. In §5.b-c, we discuss Counterexamples which are Global but not Local.
“TEACHER: ... You have shown the poverty of the argument – the proof – but not the falsity of our conjecture.” (10)
“OMEGA: ... You have succeeded in getting over a local counterexample which was not global by replacing the refuted lemma. What if you do not succeed next time?” (12)

Proof-Analysis (defn iii)
A proof (defn ii) which is represented as a logical argument. A list of lemmas, logically connected.
“KAPPA: ... You improved the proof–analysis, i.e. the list of lemmas; but the thought–experiment which you called ‘the proof’ has disappeared.” (12)
At the end of §5, Lakatos describes the distinction between ‘proof-thought-experiment’ (defn ii) and ‘proof-analysis’ as characteristic of the difference that formed between the methodology of mathematicians like Cauchy and those of “the nineteenth-century union of logic and mathematics.” (55) The shift towards proof-analysis led to the introduction of deductive methods, and the creation of the Method of Proof and Refutations. “[Non-Euclidean geometry and the Weierstrassian revolution of rigour] brought about the integration of proof (thought–experiment) and refutations and started to develop proof-analysis, gradually introducing deductive patterns in the proof-thought-experiment. What we called the ‘method of proof and refutations’ was their heuristic innovation: it united logic and mathematics for the first time.” (55-56)

Bibliography

Lakatos, Imre. Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press. Kindle Edition.