Lakatos, Imre;
Proofs and Refutations
Thomas Nelson and Sons, Edinburgh. 1964 (Reprint: British J. Philosophy of Science v.XIV:53-56, 1963-64)
ISBN 0521290384
topics: | philosophy | science | logic | math
When I first encountered this book at a reading circle run by Mohini Mullick in the 90s, I found it to be a powerful deconstruction of the myth of logical progression in science. For me this text is the bible of post-modernism - with its destructive critique of formalism as a process, as a de-humanization of ourselves, and with the ultimate realization that concepts like "truth" itself are subjective. It mainly argues against the formalist dogma that mathematics proceeds via logic from proof to proof The book operates at many levels; framed in a story of a teacher unfolding Euler's theorem in geometry with a group of students, the discourse follows a trajectory through the tortuous history of this proof, outlining the creative aspects of mathematical proof - e.g. the role of observation (p.15) or even taste (p.103-4); the human power-play in mathematics - how a proof is not a matter of pure logic but of a majority decision, and how the acceptability of a proof rises with its elegance (p.9) - AM dec 08
'Proofs and Refutations' essay was a much amended and improved version of chapter 1 of Lakatos's 1961 Cambridge Ph.D. thesis. When he died in 1974, he was planning a revised edition. This was ultimately brought out in 1976, edited by his students John Worrall and Elie Zahar. It includes parts of Chapter 2 from his thesis, relating to Poincare's vector-algebraic proof of the Descartes-Euler conjecture, and more details on Cauchy's proof of the theorem on the limit of convergent series.
Carnap demands that (a) 'philosophy is to be replaced by the logic of science ...', (b) 'the logic of science is nothing other than the logical syntax of the language of science ...', (c) 'metamathematics is the syntax of mathematical language'. - p.2-3
[A]ccording to the formalist concept of mathematics, there is no
history of mathematics proper. Any formalist would basically agree with
Russell's 'romantically' put but seriously meant remark, according to
which Boole's Laws of Thought (1854) was the 'first book ever written
on mathematics'....
None of the creative periods... would be admitted into the formalist heaven,
where mathematical theories dwell like the seraphim, purged of all the
impurities of earthly uncertainty... for some 'mixtures of mathematics and
something else' we can find formal systems 'which include them in a certain
sense' then they too may be admitted. On these terms Newton had to wait four
centuries until Peano, Russell, and Quine helped him onto heaven by
formalizing the Calculus. Dirac is more fortunate: Schwartz saved his soul
during his lifetime. - p.3
Tarski uses the term 'deductive sciences' explicitly as a
shorthand for 'formalised deductive sciences'. ... the subject matter
of metamathematics is confined to formalized deductive disciplines
because non-formalised deductive sciences are not suitable objects for
scientific investigation at all. ...
Nobody will doubt that some problems about a mathematical theory can
only be approached after it has been formalised, just as some problems
about human beings (say concerning their anatomy) can only be
approached after their death. But few will infer from this that human
beings are 'suitable for scientific investigation; only when they are
'presented in "dead" form', and that biological investigations are
confined in consequence to the discussion of dead human beings. - p.4-
5, footnote
But formalist philosophy of mathematics has very deep roots. It is the latest
link in the long chain of dogmatist philosophies of mathematics. For
more than two thousand years there has been an argument between {\it
dogmatists} and sceptics. The dogmatists hold that ... we can attain
truth and know that we have attained it. The sceptics on the other hand
either hold that we cannot attain truth at all ... or that we cannot know if
we attain it or that we have attained it. - p.6
Nothing is more characteristic of a dogmatist epistemology than its
theory of error. For if some truths are manifest, one must explain how
anyone can be mistaken about them, in other words, why the truths are
not manifest to everybody. According to its particular theory of error,
each dogmatist epistemology offers its particular therapeutics to purge
minds from error. Cf Popper [1963], Introduction. [IDEA: Useful in
debates against fanatics.] - p.34 foot
"a sick mind, twisting in pain" - part of the Stoic theory of error p.35
[The modest aim of this case-study] is to elaborate the point that
informal, quasi-empirical, mathematics does not grow through a
monotonous increase of the number of indubitably established theorems
but through the incessant improvement of guesses by speculation and
criticism, by the logic of proofs and refutations. - p.6
---
[ELEGANCE: IDEA/EXTENSION: If the informal process of mathematical
discovery can be studied, why not the question of mathematical
elegance or beauty: "After Cauchy's proof it became absolutely
indubitable that the elegant relation V + F = E + 2 applies to all
sorts of polyhedra," [Jonquieres 1890a] - p.9 footnote. What
constitutes elegance? And its opposite: "I turn aside with a
shudder of horror from this lamentable plague of functions which
have no derivatives." [Hermite, in letter to Stieltjes 1893] -
p.21 footnote. Also p.60: Weierstrassian rigour triumphed over its
reactionary monster-barring and lemma-hiding opponents who used
slogans like 'the dullness of rigour','artificiality versus
beauty', etc.
Also, p.103: Why are steps 6 and 7, i.e. incorporation of included
surfaces and ring-shapes - not an increase in depth?
"RHO: not every increase in content is also an increase in depth:
think of (6) and (7)!" followed by the footnote that "Quite a few
mathematicians cannot distinguish the trivial from the non-
trivial." (p.103). There is also the quote from von Neumann about
the 'danger of degeneration' from such trivialities, who thought
that it would not be so bad 'if the discipline is under the
influence of men with an exceptionally well-developed taste'
[1947]. This quote is passed on by Lakatos passim (p.104), but is
this 'taste' not an extremely mysterious yet IMPORTANT aspect of
all scientific advance?
---
TEACHER: I do not think that ['proof'] establishes the truth of the
conjecture... 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. - p.10
"Just send me the theorems, then I shall find the proofs" - Chrysippus
to Cleanthes.
Riemann: "If only I had the theorems! Then I should find the proofs
easily enough." - p.11
TEACHER: You are interested only in proofs which 'prove' what they set
out to prove. I am interested in proofs even if they do not accomplish
their intended task. Columbus did not reach India but he discovered
something quite interesting. - p.15
"As we must refer the numbers to the pure intellect alone, we can
hardly understand how observations and quasi-experiments can be of use
in investigating the nature of the numbers. Yet, in fact, as I shall
show here with very good reasons, The properties of the numbers known
today have been mostly discovered by observation"... editors intro to
Euler [1753]. - p.11.
[The cube-in-cube counterexample was discovered through observation.]
Both Lhulier and Hessel were led to their discovery by mineralogical
collections in which they noticed some double crystals... Lhulier
acknowledges the stimulus of the crystal collection of his friend
Professor Picret [1812]. Hessel refers to lead sulphide cubes enclosed
in transluscent calcium fluoride crystals [1832]. - p.15
DELTA: So really you showed us two polyhedra - two surfaces, one
completely inside the other. A woman with a child in her womb is not a
counterexample to the thesis that human beings have one head. - p.16
"Researches dealing with ... functions violating laws which one hoped
were universal, were regarded almost as the propagation of anarchy and
chaos where past genberations had sought order and harmony" [Saks
1933]. The similarly fierce battle that raged later between opponents
and protagonists of modern mathematical logic and set theory was a
direct continuation of this. - p.21 foot
ALPHA: It is strange to think that once upon a time [V-E+F=2] was a
wonderful guess, full of challenge and excitement. Now, because of your
weird shifts of meaning, it has turned into a poor convention, a
despicable piece of dogma. (He leaves the classroom. - p.23
GAMMA: I think that if we want to learn about anything really deep, we
have to study it not in its 'normal', regular, usual form, but in its
critical state, in fever, in passion.... If you want to know ordinary
polyhedra, study their lunatic fringe. This is how one can carry
mathematical analysis into the heart of the subject. - p.25 [IDEA:
Qualitative reasoning: the importance of tangencies/alignments.]
Poniard [1908]: 'Logic sometimes makes monsters. Since half a century
we have seen arise a crowd of bizarre functions which seem to try to
resemble as little as possible the honest functions which serve some
purpose. No longer continuity, or perhaps continuity, but no
derivatives, etc. Nay more, from the logical point of view, it is these
strange functions which are the most general, those one meets without
seeking no longer appear except as particular cases. There remains for
them only a very small corner.' - p.24
Newton[1717]: If no exception occur from phenomena, the conclusion may
be pronounced generally. But if at any time afterwards any exception
should occur, it may then begin to be pronounced with such exceptions
as occur. - p.30
We first guessed that for all polyhedra V-E+F=2, because we found it to
be true for cubes, octahedra, pyramids, and prisms. We certainly cannot
accept 'this miserable way of inferring from the special to the
general'. - p.30.
Quote from [Able 1826]: "In Higher Analysis very few propositions are
proved with definitive rigor. One finds everywhere the miserable way
of inferring from the special to the general, and it is a marvel that
such procedure leads only rarely to what are called paradoxes. It is
really very interesting to look for the reason. In my opinion the
reason is to be found in the fact that analysts have been mostly
occupied with functions that can be expressed as power series. As
soon as other functions enter - which certainly is rarely the case -
one does not get on any more and as soon as one starts drawing false
conclusions, an infinite multitude of mistakes will follow, all
supporting each other ... " - p.30
Many working mathematicians are puzzled about what proofs are for if
they do not prove.... Applied mathematicians usually try to solve this
dilemma by a shamefaced but firm belief that the proofs of the pure
mathematicians are 'complete'. and so really prove. Pure
mathematicians, however, know better - they have such respect only for
the complete proofs of logicians. [e.g. Hardy 1928]: 'There is
strictly speaking no such thing as mathematical proof; we can, in the
last analysis, do nothing but point; ... proofs are what Littlewood
and I call gas, rhetorical flourishes designed to affect psychology,
pictures on the board in the lecture, devices to stimulate the
imagination of pupils.' ... G.Polya points out that proofs, even if
incomplete, establish connections between mathematical facts and this
helps us keep them in our memory: proofs yield a mnemotechnic system
[1945]. - p.31- 32
BETA: Not 'guesswork' this time, but insight!
TEACHER: I abhor your pretensions 'insight'. I respect conscious
guessing, because it comes from the best human qualities: courage and
modesty. - p.32
Poinsot was certainly brainwashed some time between 1809 and 1858 ...
now he sees examples where he previously saw counterexamples. The self-
criticism had to be surreptitious, cryptic, because in scientific
tradition there are no patterns available for articulating such volte-
faces.
[EXCLUSION BY RE-DEFINITION = Monsterbarring; DOMAIN EXCLUSION =
EXCEPTION BARRING.
Monsterbarring: Using this method one can eliminate any counterexample
to the original conjecture by a sometimes deft bug always ad hoc
redefinition of the polyhedron, of its defining terms, of the defining
terms of its defining terms. - p.25
Our naive conjecture was 'All polyhedra are Eulerian'.
The monster-barring method defends this by reinterpreting its terms
in such a way that at the end we have a monster-barring theorem:
'All polyhedra are Eulerian'. But the identity of the linguistic
expressions of the naive conjecture and the monster-barring theorem
hides, behind surreptitious changes in the meaning of the terms, an
essential improvement.
The exception-barring method introduced an element which is really
extraneous to the argument: convexity. The exception-barring
theorem was: 'All convex polyhedra are Eulerian'.
The lemma-incorporating method relied on the argument - i.e. on the
proof - and on nothing else. It virtually summed up the proof in
the lemma-incorporating theorem: 'All simple polyhedra with simply-
connected faces are Eulerian'.
This shows that ... one does not prove what one has set out to
prove. Therefore no proof should conclude with the words: 'Quod erat
demonstrandum'. - Alpha, p.44
There is an infinite regress in proofs; therefore proofs do not prove.
You should realize that proving is a game, to be played while you enjoy
it and stopped when you get tired of it. -p.43
We already agreed to omit, that is, 'hide', trivially true lemmas. -
p.49
The standard expression for this is 'we assume familiarity with lemmas
of type x.' Cauchy, e.g. did not even notice that his celebrated
[1821] presupposed 'familiarity' with the theory of real numbers.
... Not so Weierstrass and his school: textbooks of formal mathematics
now contain a new chapter on the theory of real numbers where these
lemmas are collected. ... More rigorous textbooks narrow down
background knowledge even further: Landau, in the introduction to his
famous [1930], assumes familiarity only with 'logical reasoning
and German language'. One wonders when 'the author confesses ignorance
about the field x' will replace the authoritarian euphemism 'the author
assumes familiarity with the field x': surely when it is recognized
that knowledge has no foundations. - p.49 foot
Russell, Principles of Mathematics, 1903: It is one of the chief merits
of proofs that they instill a certain scepticism as to the result
proved.'
H.G. Forder [1927]: 'The virtue of a logical proof is not that it compels
belief, but that it suggests doubts'.- p.52 foot
The analogy between political ideologies and scientific theories is
then more far-reaching than is commonly realised: political ideologies
which first may be debated (and perhaps accepted only under pressure)
may turn into unquestioned background knowledge even in a single
generation: the critics are forgotten (and perhaps executed) until a
revolution vindicates their objections. [Provides case studies with
Euclid and Newton. "The peak of Euclid's authority was reached in the
Age of Enlightenment. Clairaut urges his colleagues not to 'obscure
proofs and disgust readers' by stating evident truths: Euclid did so
only in order to convince 'obstinate readers' [1741]." - p.53 foot
I have the right to put forward any example that satisfies the
conditions of your argument and I strongly suspect that what you call
bizarre, preposterous examples are in fact embarrassing examples,
prejudicial to your theorem. (G. Darboux [1874]). - p.54
I am terrified by the hoard of implicit lemmas. It will take a lot of
work to get rid of them. (G. Darboux [1883] - p.54
ALPHA: There is still the irrefutable master-theorem: 'All
polyhedra on which one can perform the thought-experiment, or briefly,
all Cauchy-polyhedra, are Eulerian. My approximate proof analysis
drew the borderline of the class of Cauchy-polyhedra with a pencil that
- I must admit - was not particularly sharp. Now eccentric
counterexamples teach us to sharpen our pencil. But first: no
pencil is absolutely sharp ( and if we overdo sharpening it may
break); secondly, pencil-sharpening is not creative mathematics.
- p.55
But surely 'at each stage of the evolution our fathers also thought
they had reached it [absolute rigor]? If they deceived themselves, do
we not likewise cheat ourselves? - p.56
Changes in the criterion of 'rigor of the proof' engender major
revolutions in mathematics. Pythagoreans held that rigorous proofs can
only be arithmetical. They however discovered a rigorous proof that
root(2) was 'irrational'. When the scandal eventually leaked out, the
Criterion was changed: arithmetical 'intuition' was discredited and
geometrical intuition took its place. This meant a major and
complicated reorganization of mathematical knowledge (e.g. the theory
of proportions). In the eighteenth century 'misleading' figures brought
geometrical proofs into disrepute, and the nineteenth century saw
arithmetical intuition re-enthroned with the help of the cumbersome
theory of real numbers. Today the main dispute is about what is
rigorous and what not in set-theoretical and meta-mathematical
proofs... - p.56
ALPHA: Ever more eccentric counterexamples will be countered by ever
more trivial lemmas - yielding a vicious infinity of ever longer and
clumsier theorems. ...
GAMMA: At a certain point we may reach truth and then the flow of
refutations will stop. But of course we shall not know when. Only
refutations are conclusive - proofs are a matter of psychology.
LAMBDA: I still trust that the light of absolute certainty will flash
up when refutations peter out!
KAPPA: But will they? What if God created polyhedra so that all true
universal statements about them - formulated in human language - are
infinitely long? Is it not blasphemous anthropomorphism to assume that
(divine) true theorems are of finite length? ... Truth is only for God.
- p.58
Different levels of rigor differ only about where they draw the line
between the rigor of proof-analysis and the rigor of proof, i.e. about
where criticism should stop and justification should start. 'Certainty
is never achieved'; 'foundations' are never found - but the 'cunning of
reason' turns each increase in rigor into an increase in content, in
the scope of mathematics. - p.60
[IDEA: Gaussian Sphere: Legendre's proof (p.64) entails mapping the
polyhedron onto a sphere containing the polyhedron. What is the class
of objects for which this can be done? Is this identical to the
Gaussian Sphere models, recently revived by Ziv/Malik et al?]
OMEGA: My quest is not only for certainty but also for finality. The
theorem has to be certain - there must not be any counterexamples {\it
within} its domain; but it has also to be final; there must not be any
examples outside its domain. I want to draw a dividing line
between examples and counterexamples, and not just between a safe
domain of a few examples on the one hand and a mixed bag of examples
and counterexamples on the other. LAMBDA: Or, you want the conditions
of the theorem to be not only sufficient, but also necessary! - p.67
[A proof must explain the Eulerian-ness of the great stellated
duodecahedron]
'More questions may be easier to answer than just one question. A new
more ambitious problem may be easier to handle than the original
problem.' - p.72, Polya[1945]. "Inventor's Paradox"
ZETA: You have fallen in love with the problem of finding out where God
drew the firmament dividing Eulerian from non-Eulerian polyhedra. But
there is no reason to believe that the term 'Eulerian' occured in God's
blueprint of the universe at all. - p.73
ZETA: Like most mathematicians I cannot count. I just tried to count
the edges and vertices of a heptagon. I found first 7 edges and 8
vertices, and then again 8 edges and 7 vertices... - p.77
BETA: Then what suggested V-E+F=2 to me, if not the facts listed in my
table?
TEACHER: I shall tell you. .. You had three or four conjectures which
in turn were quickly refuted. Your table was built up in the process of
testing and refuting these conjectures. Naive conjectures are not
inductive conjectures: we arrive at them by trial and error, through
conjectures and refutations. - p.78
[Polya: Mathematics and Plausible Reasoning, 2 vols, 1954; v.1 contains
a detailed analysis of the Euler polyhedron problem. p.79 foot
discusses how Lakatos improves on Polya; the central point is that
polya 'never questioned that science is inductive, and because of his
correct vision of deep analogy between scientific and mathematical
heuristic he was led to think that mathematics is also inductive.']
[Section IV, p.75-100+, rushes too quickly I felt, through some
momentous changes. Also, would have done better to introduce the
manifold/Holes topological version of the formula, which simplifies the
Sigma(ek) business. The end itself, with the teacher leaving the room,
leaves one unsatisfied. The drama of the earlier sections is also lost
in section IV]
LAMBDA: Do you really think that (1) is the single axiom from which all
the rest follows? That deduction increases content? [one vertex is one
vertex: axiom 1] - p.86
ALPHA: Of course! Isn't this the miracle of the deductive thought-
experiment? If once you have got hold of a little truth, deduction
expands it infallibly into a tree of knowledge. If a deduction does not
increase the content I would not call it deduction, but 'verification';
'verification differs from true demonstration precisely because it is
purely analytic and because it is sterile'. - p.86-87, quote from
[Poniard 1902].
concepts get reformulated as proofs progress.]
PI: By the time the Descartes-Euler conjecture was put forward, the
concept of polyhedron included all sorts of convex polyhedra and even
some concave polyhedra. But it certainly did not include polyhedra
which were not simple, or polyhedra with ringshaped faces. For the
polyhedra that they [the monsterbarrers] had in mind, the conjecture
was true as it stood and the proof was flawless. ... The refutationists
... stretched the concept of polyhedron, to cover objects that were
alien to the intended representation. Their refutation revealed no
error in the original conjecture, no mistake in the original proof: it
revealed the likelihood of a new conjecture which nobody had stated or
thought of before. ... Imagine a different situation, where the
definition fixed the intended interpretation of 'polyhedron' correctly.
Then it would have been up to the refutationists to devise ever longer
monster-including definitions for say, 'complex polyhedra': 'A
complex polyhedron is an aggregate of (real) polyhedra such that each
two of them are soldered by congruent faces'. 'The faces of complex
polyhedra can be complex polygons that are aggregates of (real)
polygons such that each two of them are soldered by congruent edges'...
SIGMA: I never dreamt that concept-formation might lag behind an
unintendedly wide definition! -p.89-91
Often, as soon as concept-stretching refutes a proposition, the refuted
proposition seems such an elementary mistake that one cannot imagine
that great mathematicians could have made it. This important
characteristic of concept-stretching refutation explains why respectful
historians, because they do not understand that concepts grow, create
for themselves a maze of problems. After saving Cauchy by claiming that
he 'could not possibly miss' polyhedra which are not simple and that
therefore he 'categorically' (!) restricted the theorem to the domain
of convex polyhedra, the respectful historian now has to explain why
Cauchy's borderline was 'unnecessarily' narrow.... So [they] explain
away a mistake Cauchy never made.
Other historians proceed in a different way. They say that before
the point where the correct conceptual framework (i.e. the one they
know) was reached there was only a 'dark age' with 'seldom, if ever,
sound' results. This point in the theory of polyhedra is Jordan's proof
(1866) according to Lebesgue [1923]; it is Poincare's (1895) according
to [Bell 1945, p.460]. - p.93
'Mathematics is the art of giving the same name to different things. ... If one chooses the right language, one is surprised to learn that the proofs made for a known object apply immediately to many new objects, without the slightest change - one can even retain the names' Poniard [1908], but "Darboux, in his [1874] came close to this idea." 94 When the physicists started to talk about "electricity," or the physicians about "contagion," these terms were vague, obscure, muddled. The terms that the scientists use today, such as "electric charge," "electric current," "fungus infection," "virus infection," are incomparably clearer and more definite. Yet what a tremendous amount of observation, how many ingenious experiments lie between the two terminologies, and some great discoveries too. - Polya [1954] v.1p.55 - p.95 The problem of universals should be reconsidered in view of the fact that, as knowledge grows, languages change. - p.98
on the need for "mathematical taste" and "mathematical critics" to stem the tide of pretentious trivialities in mathematical literature: quotes Polya [1954, v.1,p.30]: Shallow, cheap generalization is 'more fashionable nowadays than it was formerly. It dilutes a little idea with a big terminology. It would be very easy to quote examples, but I don't want to antagonize people.' p.104. For any proposition there is always some sufficiently narrow interpretation of its terms, such that it turns out true, and some sufficiently wide interpretation such that it turns out false. The first interpretation may be called the dogmatist, verificationist or justificationist interpretation, and the second the sceptical, critical or refutationists interpretation. - p.105
There is a simple pattern of mathematical discovery - or of the
growth of informal mathematical theories. It consists of the following
stages:
(1) Primitive conjecture.
(2) Proof (a rough thought-experiment or argument, decomposing the
primitive conjecture into subconjectures or lemmas).
(3) 'Global* counterexamples {counterexamples to the primitive conjecture)
emerge.
(4) Proof re-examined: the 'guilty lemma' to which the global counterexample
is a 'local' counterexample is spotted. This guilty lemma may have
previously remained 'hidden' or may have been misidentified. Now it is
made explicit, and built into the primitive conjecture as a
condition. The theorem - the improved conjecture - supersedes the
primitive conjecture with the new proof-generated concept as its
paramount new feature.
[first noted by Seidel in 1847, in connection with the Cauchy conjecture]
Starting from the certainty just achieved, that the theorem is not
universallyvalid, and hence that its proof must rest on some extra
hidden assumption, one then subjects the proof to a more detailed
analysis. It is not very difficult to discover the hidden
hypothesis. One can then infer backwards that this condition
expressed by the hypothesis is not satisfied by series which
represent discontinuous functions, since only thus can the agreement
between the otherwise correct proof sequence, and what has been on
the other hand established, be restored.
[Lakatos remarks that these steps are not infallible - sometimes, step 4 may
precede step 3 - an ingenious proof analysis may suggest the counterexample. ]
Subsequent to these four steps, the theorem is further examined, along with
other related theorems. The counterexamples are eventually turned into new
examples for new theories - new fields of inquiry open up.
PRIMITIVE CONJECTURE: the limit of any convergent series of continuous functions is itself continuous. It was Cauchy [1821] who gave the first proof of this conjecture, whose truth had been taken for granted and assumed therefore not to be in need of any proof throughout the eighteenth century. It was regarded as the special case of the ' axiom' according to which 'what is true up to the limit is true at the limit'. We find the conjecture and its proof in Cauchy's celebrated (p. 131). Cauchy's view of Fourier: that talented but woolly and unrigorous dilettante... 130
Abel 1826:
'It seems to me that there are some exceptions to Cauchy's theorem', and
immediately gives the example of the series
sin(x) - sin(2x)/2 + sin(3x)/3+...
[this sequence of continuous functions is discontinuous owing to the
oscillating terms]
Abel adds that 'as it is known, there are many more examples like this*. His
response to these counterexamples is to start guessing: 'What is the safe
domain of Cauchy's theorem ?'
His answer to this question is this: the domain of validity of the theorems
of analysis in general, and that of the theorems about the continuity of the
limit function in particular, is restricted to power series. All the known
exceptions to this basic continuity principle were trigonometrical series,
and so he proposed to withdraw analysis to within the safe boundaries of
power series, thus leaving behind Fourier's cherished trigonometrical series
as an uncontrollable jungle - where exceptions are the norm and successes
miracles. 133
[It was not until 1847 that Seidel looked at the assumptions implicit in
Cauchy's "proof" and came up with the need to distinguish two kinds of
convergence, now called:
* pointwise convergence: given a sequence of functions f1... fn ..., then
if for all x, fn(x) convergest to some function f(x), then f is the
limit of fn (it is independent of x).
* uniform convergence: that for every epsilon there is an N s.t. the sum of
the first N terms differs from the pointwise limit by less than epsilon
the latter is a stronger notion - every uniform convergent sequence is
pointwise convergent but not vice versa - e.g. for x IN [0,1), x^n is
pointwise convergent but is not independent of x, and hence is not
uniformly convergent.
Similarly, Abel's counter-example
sin(x) - sin(2x)/2 + sin(3x)/3+...
is a sequence of continuous functions that is discontinuous because it is not
uniformly convergent.]
Lakatos's contribution to the philosophy of mathematics was, to put it
simply, definitive: the subject will never be the same again. (...) Lakatos
made us think instead about what most research mathematicians do. He wrote
an amazing philosophical dialogue around the proof of a seemingly elementary
but astonishingly deep geometrical idea pioneered by Euler. It is a work of
art - I rank it right up there with the dialogues composed by Hume or
Berkeley or Plato. He made us see a theorem, a mathematical fact, coming
into being before our eyes.
- Ian Hacking, in a review of For and Against Method by Lakatos/Feyerabend
Mathematics is the art of giving the same name to different things... When
the language is well chosen, we are astonished to learn that all te proofs
made for a certain object apply immediately to many new objects; there is
nothing to change, not even the words, since the names have become the same.
- Jules Henri Poincaré [1908, p.375] p. 91
---blurb
Lakatos is concerned throughout to combat the classical picture of
mathematical development as a steady accumulation of established truths. He
shows that mathematics grows instead through a richer, more dramatic process
of the successive improvement of creative hypotheses by attempts to 'prove'
them and by criticism of these attempts: the logic of proofs and
refutations.
---bio
Imre Lakatos was born in Hungary as Imre Lipsitz in 1922. Active in the
Communist Party in Hungary after World War II he worked in the Ministry of
Education. He earned his Ph.D from Debrecen University in 1947. Expelled
from the Communist Party in 1950, he was interned for three years. He fled
Hungary in 1956, and was awarded a Rockefeller Fellowship to study at
Cambridge, where he completed another Ph.D. He became a lecturer at the
London School of Economics where Karl Popper was a great influence on
him. Lakatos died in 1974.
- http://www.complete-review.com/reviews/lakatosi/pandr.htm
from http://www.rbjones.com/rbjpub/philos/bibliog/lakato76.htm The purpose of this book is: to approach some problems of the methodology of mathematics. where methodology qua logic of discovery is intended. The work, though primarily philosophical in intent, appears also to be in part: historical giving logically reconstructed case studies in the development of mathematical theories, with "real history" in the footnotes. sociological studying the behaviour of mathematicians and partially classifying the recurrent features in their practice of mathematics educational analysis and polemic considering the impact of the presentation of mathematical developments on the comprehension by students of the processes involved, arguing the merits of presentations which retain more of the original structure of the discoveries. mathematical methodology arguing that mathematicians practice heuristic rather than deductive methods mathematical philosophy arguing against formalism and "dogmatism". In relation to this objective the material is illuminating, consisting primarily of case studies showing how a putative mathematical discovery can evolve through a series of conjectures, proofs, refutations and reformulations. In this aspect the work has more the character of sociology than philosophy. The case studies are interpreted through the introduction of special terminology describing recurrent features in the examples cited: * local and global counterexamples * monster-barring * exception-barring * piecemeal exclusions * strategic withdrawal * lemma-incorporation * proof-generated theorem * concept-stretching Alongside and interweaved in this perspective on mathematical discovery we also discover doctrines of a more philosophical nature. In particular: the core of this case-study will challenge mathematical formalism Its modest aim is to elaborate the point that informal, quasi-empirical mathematics does not grow through a monotonous increase of the number of indubitably established theorems, but through the incessant improvement of guesses by speculation and criticism, by the logic of proof and refutation. It is clear also that Lakatos is attacking: * Formalism * dogmatist philosophies of mathematics * meta-mathematics * the deductivist approach --- Ernest Gellner described the philosopher's lectures as 'intelligible, fascinating, dramatic and above all conspicuously amusing'. - http://www.lse.ac.uk/resources/LSEHistory/lakatos.htm ---blurb Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations.