Quote of the Day
What is “absolute truth”?
'absolutely' true, meaning what no farther experience will ever
alter, is that ideal vanishing-point towards which we imagine that
all our temporary truths will some day converge. It runs on all
fours with the perfectly wise man, and with the absolutely
complete experience; and, if these ideals are ever realized, they
will all be realized together. Meanwhile we have to live to-day by
what truth we can get to-day, and be ready to-morrow to call it
falsehood. Ptolemaic astronomy, euclidean space, aristotelian
logic, scholastic metaphysics, were expedient for centuries, but
human experience has boiled over those limits, and we now call
these things only relatively true, or true within those borders of
experience. 'Absolutely' they are false; for we know that those
limits were casual, and might have been transcended by past
theorists just as they are by present thinkers.”
A New Name for Some Old Ways of Thinking. Lecture
6. Pragmatism's Conception of Truth, 1907
online by Project
was convinced that if one had a sufficiently unifying vision of
mathematics, if one could sufficiently penetrate its conceptual
essence, then particular problems would be nothing but tests that
no longer need to be solved for their own sake."
Grothendieck. A Country Known Only by Name. Inference:
International Review of Science,
1(1), October 15, 2014.
Grothendieck's most popular quote on the Internet:
“If there is one thing in mathematics that fascinates me
more than anything else (and doubtless always has), it is neither
"number" nor "size", but always form. And
among the thousand-and-one faces whereby form chooses to reveal
itself to us, the one that fascinates me more than any other and
continues to fascinate me, is the structure hidden in mathematical
Kolmogorov, 1969. Finite creatures
in a finite world may be forced to invent infinity.
"... let us imagine an intelligent creature
living in a world of finite
complexity, [a world] taking only a finite
number of physically distinct states and evolutioning
in a "discrete time". ... It is possible to explain
plausibly how such a creature, because of its
structure, being unable to cover all the complexity of the world
around it and confronted with systems of growing complexity and
consisting of very large numbers of elements, will create during
the process of its entirely practically and reasonably oriented
activities the concept of an infinite
sequence of natural numbers." (Sorry, my own
The original text:
N. Kolmogorov. Scientific foundations
of the school mathematics course. First lecture. Contemporary
views on the nature of mathematics, Matematika
v Shkole, no. 3 (1969), 12–17 (in
Russian). Reprinted in 1988 as pp. 232-233 in (available
Observation/Discovery vs. Creation/Invention
"... the distinction between discovery and observation is
not especially important: if you notice something, then that
something must have been there for you to notice, just as if you
discover it then it must have been there for you to discover. So
let us think of observation as a mild kind of discovery rather
than as a fundamentally different phenomenon. " (p. 4)
"... we do not normally talk of inventing a single work of
art. However, we do not discover it either: a commonly used word
for what we do would be ‘create’. And most people, if
asked, would say that this kind of creation has more in common
with invention than with discovery, just as observation has more
in common with discovery than with invention." (p. 5)
Is mathematics discovered or invented? In
J. C. Polkinghorne (ed.), Meaning
Oxford University Press. 3-12 (2011)
Thus, the distinction observation/creation is more
fundamental than the (rather psychological) one of
discovery/invention. And, the right question to ask is not:
“Is mathematics discovered or invented?”, but “Is
mathematics observed or created?”
2002. We and our models are both part of the universe.
we are not angels, who view the universe from the outside.
Instead, we and our models are both part of the universe we are
and the end of Physics. Public
2009-2013. The carriers of mathematical knowledge are proofs.
brings us mathematical knowledge? The carriers of mathematical
knowledge are proofs,
more generally arguments and constructions, as embedded in larger
contexts.1 Mathematicians and teachers of higher mathematics know
this, but it should be said. Issues about competence and intuition
can be raised as well as factors of knowledge involving the
general dissemination of analogical or inductive reasoning or the
specific conveyance of methods, approaches or ways of thinking.
But in the end, what
can be directly conveyed as knowledge are proofs.”
Knowledge: Motley and Complexity of Proof. Annals
of the Japan Association for Philosophy of Science,
Vol. 21 (2013), pp. 21-35.
cannot spring from experience alone.
that the human mind has first to construct forms independently,
before we can find them in things. Kepler’s marvelous
achievement is a particularly fine example of the truth that
knowledge cannot spring from experience alone, but only from the
comparison of the inventions of the intellect with observed fact."
translation by Sonja
Bargmann published in:
Ideas and Opinions. Crown
New York, 1954, pp. 262-266.
"Es scheint, dass die menschliche Vernunft die
Formen erst selbständig konstruieren muss, ehe wir sie in den
Dingen nachweisen können. Aus Keplers wunderbarem Lebenswerk
erkennen wir besonders schön, dass aus bloßer Empirie
allein die Erkenntnis nicht erblühen kann, sondern nur aus
dem Vergleich von Erdachtem mit dem Beobachteten."
Albert Einstein über Kepler. Frankfurter
9. November 1930, see also online
copy published by Dr.
See also Einstein's manuscript
of this paper in Einstein
cannot spring from experience alone.
Erfindung der Elektrisirmaschine, des Elektrophors, der Leydener
Flasche, der Volta'schen Säule, die drei Kepler'schen Gesetze
sind durch Combinationen der Einbildungskraft erworben worden;
ebenso verhält es sich mit den Verfahrungsweisen zur
Gewinnung der Metalle, welche, wie die des Eisens aus den
Eisensteinen, des Silbers aus den Bleierzen, des Kupfers aus den
Kupfererzen etc., zu den verwickeltsten Processen gehören.
Die Ueberführung des Eisens in Stahl, des Kupfers in Messing,
die Verwandlung der Haut in Leder, des Fettes in Seife, die des
Kochsalzes in Soda und
tausend ähnliche wichtige Erfindungen sind von Menschen
gemacht worden, welche keine oder eine ganz falsche Vorstellung
von der eigentlichen Natur der Dinge oder den Vorgängen
hatten, an die sich ihre Ideencombination knüpfte.
6, marked bold by me, K.P.]
ist die Idee, von der sie ausgingen, ganz falsch, und es wird die
richtige erst in der Untersuchung erweckt. Daher denn die Meinung
mancher der grössten Forscher, dass die Arbeit alles mache,
und dass jede Theorie zu Entdeckungen führe, vorausgesetzt,
dass sie zur Arbeit antreibt.” [p.
8, marked bold by me, K.P.]
von Liebig. Induktion und Deduktion.
Akademische Rede, München 1865 (available online, google for
von liebig induktion).
2010. Human minds...
that is going on in human minds can be best understood as
of the idea:
Model-Based Model of Cognition.The
Vol. 3, N 6, June 2009, pp. 5-6.
W. Anderson, 1972. Reductionism does not imply
the reductionist hypothesis does not by any means imply a
“constructionist” one: The ability to reduce
everything to simple fundamental laws does not imply the ability
to start from those laws and reconstruct the universe.”
constructionist hypothesis breaks down when confronted with the
twin difficulties of scale and complexity. ..., at each level of
complexity entirely new properties appear, and the understanding
of the new behaviors requires research which I think is as
fundamental in its nature as any other.”
Is Different. Science, New Series, Vol. 177, No. 4047.
(Aug. 4, 1972), pp. 393-396.
W. Anderson, 1994. At the frontier of complexity, the
watchword is not reductionism but emergence.
another large fraction [of physicists - K.P.] are engaged in an
entirely different type of fundamental research: research into
phenomena that are too complex to be analyzed straightforwardly by
simple application of the fundamental laws. These physicists are
working at another frontier between the mysterious and the
understood: the frontier of complexity.
frontier, the watchword is not reductionism but emergence.
Emergent complex phenomena are by no means in violation of the
microscopic laws, but they do not appear as logically consequent
on these laws.”
One may make a digital computer using electrical relays, vacuum
tubes, transistors, or neurons; the latter are capable of
behaviors more complex than simple computation but are certainly
capable of that; we do not know whether the other examples are
capable of "mental" phenomena or not. But the rules
governing computation do not vary depending on the physical
substrate in which they are expressed; hence, they are logically
independent of the physical laws governing that substrate.”
The opening to complexity. Proc. Natl. Acad. Sci. USA,
Vol. 92, pp. 6653-6654, July 1995.
Postulates in metaphysics...
oneness of the world is an almost undiscussed postulate of most
metaphysics. ... Yet I believe that it embodies a failure to
effect thoroughly the "Copernican revolution," and that
the apparent oneness of the world is merely the oneness of what is
seen by a single spectator or apprehended by a single mind.”
On Scientific Method in Philosophy. Herbert Spencer
lecture at Oxford in 1914, pp. 97-124 in:
Russell. Mysticism and
Logic: and Other Essays, New
York: Longmans, Green and Co.,
1918 (available online).
would say: moreover, any
metaphysics is a theory based on its own postulates,
no more than that! Otherwise,
we couldn't discuss several different metaphysics simultaneously.
B. Curry, 1939.
One postulates the existence of an external world...
what grounds do we infer the reality say of the table on which I
am writing? I understand that one can consistently maintain the
view, called solipsism, that physical objects have no reality;
i.e., that the sole reality is my sensations. In fact, one does
not prove the existence of an external world, one postulates it.”
See p. 7 of
Outlines of a Formalist Philosophy of Mathematics. North-Holland,
2007. Let Platonism die.
of most Platonists are based on gut instincts – strong
convictions reinforced by years of immersion in their subject.
[scientific investigations of mental processes – K.P.] are
proceeding systematically and are beginning to provide a genuine
understanding of the basis of our mathematical abilities. They owe
nothing to Platonism, whose main function is to contribute a
feeling of security in those who are believers. Its other function
has been to provide employment for hundreds of philosophers vainly
trying to reconcile it with everything we know about the world. It
is about time that we recognised that mathematics is not different
in type from all our other, equally remarkable, mental skills and
ditched the last remnant of this ancient religion.”
Let Platonism die. NEWSLETTER
OF THE EUROPEAN MATHEMATICAL SOCIETY,
64, June 2007, pp. 24-25.
July 2001, large finite numbers only exist in a metaphysical sense
sufficiently large finite numbers only exist in a metaphysical
sense: they play no role in science and our only access to them
depends upon accepting the rules of Peano arithmetic.
... no material
object can be said to contain a precise number of atoms if that
number is greater than 1030.
One may of course consider an ideal
silver cube containing exactly
atoms along each edge and
atoms altogether, but this cube
is then a mental construction and not something in the material
For even bigger
numbers the situation shifts again. The number of massive
elementary particles in the universe is beleived to be less than
... If one regards all sets of particles as candidates for
material entities then 2N
is an upper bound for the
number of different material entities. It is a matter of fact that
physicists do not make use of numbers vastly bigger than this, and
it is difficult to argue that they have any
We consider the
real number field of Dedekind and Weierstrass to be metaphysical
because its adds features to the physical continuum which have no
empirical justification. ... Mathematicians have produced these
idealized versions of the empirical continuum because of their
need to have sharp formalisms if they are to prove theorems, and
because the idealized systems are easier to grasp intuitively.“
Brian Davies. Empiricism
in Arithmetic and Analysis, Philosophia
11 (2003) 53-66.
A similar point of view is expressed in:
Analysis is a Degenerate Case of Discrete Analysis by Doron
Zeilberger (appeared in
"New Progress in Difference Equations"
(Proc. ICDEA 2001), Taylor and Francis, London, 2004)
The idea can be traced to Paul Bernays in 1934, for a
more detailed history, see Section 1.1 of
is Mathematics: Gödel's Theorem and Around, 1997-2012.
Johannes Thomae, 1898, the true founder of the formalist
philosophy of mathematics?
“The formal conception
of numbers requires of itself more modest limitations than does
the logical conception. It does not ask, what are and what shall
the numbers be, but it asks, what does one require of numbers in
arithmetic. For the formal conception, arithmetic is a game with
signs which one may call empty; by this one wants to say that (in
the game of calculation) they have no other content than that
which has been attributed to them concerning their behaviour with
respect to certain rules of combination (rules of the game).
Similarly a chess player uses his pieces, he attributes to them
certain properties which condition their behaviour in the game,
and the pieces themselves are only external signs for this
behaviour. To be sure, there is an important difference between
the game of chess and arithmetic. The rules of chess are
arbitrary; the system of rules for arithmetic is such that by
means of simple axioms the numbers can be related to intuitive
manifolds, so that they are of essential service in the knowledge
of nature. - The formal standpoint relieves us of all metaphysical
difficulties, that is the benefit it offers to us.”
Johannes Thomae in the MacTutor
History of Mathematics archive
at the University
of St Andrews.
English translation may be due to:
Epple. Chapter 10 of: A
history of analysis. Hans Niels Jahnke (ed.). American
The original text:
Elementare Theorie der analytischen Funktionen einer komplexen
Veränderlichen. Zweite erweiterte und umgearbeitete
Auflage. Halle, 1898.
Dirac on interpretations, before 1984
Dirac was never interested in interpretations [of quantum theory –
K.P.]. It seemed to him to be a pointless preoccupation that led
to no new equations.”
text: p. 277 of
Quantum: Einstein, Bohr and the Great Debate About the Nature of
cases, as in the case of quantum mechanics, a very strictly
empiricist position could have circumvented the problem [of
interpretation – K.P.] altogether, by reducing the content
of the theory to a list of predicted numbers. But perhaps science
can offer us more than such a list; and certainly science
needs more than
such a list to find its ways
[marked bold by me
Relational Quantum Mechanics. Stanford
Encyclopedia of Philosophy,
interpretations may be useful, but only when they “lead to
Feynman, between 1918 and 1988,
Electron is a theory...
electron is a theory that we use; it is so useful in understanding
the way nature works that we can almost call it real. I
wanted to make the idea of a theory clear by analogy.
In the case of the brick, my next question was going to be, "What
about the inside of the brick?" - and I would then point out
that no one has ever seen the inside of a brick. Every time you
break the brick, you only see the surface. That
the brick has an inside is a simple theory which helps us
understand things better. The theory of
electrons is analogous.” (Fragments marked bold by me –
Full text – in Chapter 9 of:
You're Joking, Mr. Feynman!: Adventures of a Curious Character,
with contributions by Ralph Leighton, W. W. Norton & Co,
March 1927, We cannot know the present in all detail.
what is wrong in the sharp formulation of the law of causality,
"When we know the present precisely, we can predict the
future," is not the conclusion but the assumption. Even in
principle we cannot know the present in all detail.”
“Aber an der scharfen Formulierung des Kausalgesetzes: "Wenn
wir die Gegenwart genau kennen, koennen wir die Zukunft
berechnen", ist nicht der Nachsatz, sondern die Voraussetzung
falsch. Wir koennen die Gegenwart in allen Bestimmungstuecken
prinzipiell n i c h t kennenlernen.”
p. 198 of: W.
Über den anschaulichen Inhalt der quantentheoretischen
Kinematik und Mechanik, Zeitschrift
Volume 43, pp. 172-198 (1927), online
Quantum Theory and Measurement; Wheeler, J. A.; Zurek, W. H.,
Princeton, NJ, 1983; pp. 62-84.
1989, Deductive logic is the only logic that we have or need!
reasoners seldom, if ever, state all of the premises they are
assuming. We usually, perhaps always, have to reconstruct the
arguments being employed. Deductivism is the view that deductive
logic is the only logic that we have or need. Deductivists can
always reconstruct what look like non-deductive or inductive
arguments as deductive arguments with missing premises of one kind
or another. ... it conduces to clarity, if we do treat them so.
...if accepted, it would enable us to make good Popper's claim
that induction is a myth."
p. 319 of: A.
Essays on Realism and Rationalism. Rodopi,
Amsterdam-Atlanta, 1999, pp. xiii + 373.
Millennium Run – balancing
ambition and outcome...
path dependency immediately puts to the fore the contingency of a
simulation such as the Millennium Run. Had the cosmologists chosen
different options at some stages in the model-building process,
they would have come up with a different picture of the evolution
of cosmic matter. And the point is that those alternative pictures
would be equally plausible in the sense that they would also be
consistent both with the observations at hand and with our current
to Modeling: Balancing Ambition and Outcome in Astrophysics and
Cosmology. In: Simulation
published on June 2008 as doi: 10.1177/ 1046878108319640, Sage
1989, We are, ... modelers...
underlies our ability to think and imagine, to use signs and
language, to communicate, to generalize from experience, to deal
with the unexpected, and to make sense out of the raw bombardment
of our sensations. It allows us to see patterns, to appreciate,
predict, and manipulate processes and things, and to express
meaning and purpose. In short, it is one of the most essential
activities of the human mind. It is the foundation of what we call
intelligent behavior and is a large part of what makes us human.
We are, in a word, modelers: creatures that build and use models
routinely, habitually−−sometimes even compulsively−−to
face, understand, and interact with reality.
text: J. Rothenberg.
NATURE OF MODELING. AI,
Simulation & Modeling, John
Wiley & Sons, 1989, pp. 75−92.
Cartwright, 1983, How the Laws of
“My basic view is that fundamental equations do
not govern objects in reality; they only govern objects in
See p. 129 of:
Cartwright. How the Laws of Physics
Lie. Oxford University Press,
1983, 232 pp.
Bohr, before 1963.
is no quantum world. There is only an abstract quantum physical
description. It is wrong to think the task of physics is to find
out how nature is.
Physics concerns what we can say
The Philosophy of Niels Bohr. Bulletin
of the Atomic Scientists,
Mazur, June 2008, Experience of
"When I’m working I sometimes have the
sense – possibly the illusion – of gazing on the bare
platonic beauty of structure or of mathematical objects, and at
other times I’m a happy Kantian, marvelling at the
generative power of the intuitions for setting what an
Aristotelian might call the formal
conditions of an object. And sometimes I
seem to straddle these camps (and this represents no contradiction
to me). I feel that the intensity of this experience, the
vertiginous imaginings, the leaps of intuition, the breathlessness
that results from “seeing” but where the sights are of
entities abiding in some realm of ideas, and the passion of it
all, is what makes mathematics so supremely important for me. Of
course, the realm might be illusion. But the experience?"
My comment: see
my explanation below - "What is Mathematics? (My Main
Mathematical Platonism and its
Opposites. NEWSLETTER OF THE EUROPEAN
MATHEMATICAL SOCIETY, Issue 68, June
2008, pp. 17-18 (online
Hersh, June 2008, Mathematics before Big Bang?
"I once took a vote in a talk at New Mexico State University
in Las Cruces. The question was, “Was the spectral theorem
on self-adjoint operators in Hilbert space true before the Big
Bang, before there was a universe?” The vote was yes, by a
margin of 75 to 25. But there were no self-adjoint operators, no
Hilbert space, before the twentieth century!"
R. Hersh. On Platonism. NEWSLETTER
OF THE EUROPEAN MATHEMATICAL SOCIETY, Issue 68, June 2008,
pp. 19-21 (online
Skolnick Weisberg on mathematical platonism, about
"To use a phrase suggested by Deena Weisberg, a view of model
systems as imagined concrete things which many scientists can
simultaneously investigate is the "folk ontology" of
model-based science, the ontology that is implicit in the
practitioners' routine behaviors.
... an implicitly platonist outlook is a feature of successful
mathematical practice – in Weisberg's terms again, ...
platonism is the folk ontology of research mathematics."
Godfrey Smith. Models and Fictions in Science.
Philosophical Studies 143 (2009): pp. 101-116.
"Implicitly platonist outlook is a feature of
successful mathematical practice" - I'm
promoting this idea since 1988 when I first presented it at the
Heyting'88 Summer School & Conference on Mathematical Logic
of Abstract, a detailed
Zeilberger, Computer assisted mathematics, January
"... it is very possible that if Andrew Wiles' programming
skills would have been as good as his proving skills, he would
have already proved the Riemann Hypothesis."
Full text: http://www.math.rutgers.edu/~zeilberg/Opinion94.html.
Podnieks. Infinity, January 2006
invention of the axiom of infinity simply be an act of fantasy?"
Full text: [FOM]
Infinity and the "Noble Lie".
Wiedijk. The Future of Formal Mathematics, November
In a few decades it will no longer take one week to formalize a
page from an undergraduate textbook. Then that time will have
dropped to a few hours. Also then the formalization will be quite
close to what one finds in such a textbook.
When this happens we will see a quantum leap, and suddenly all
mathematicians will start using formalization for their proofs.
When the part of refereeing a mathematical article that consists
of checking its correctness takes more time than formalizing the
contents of the paper would take, referees will insist on getting
a formalized version before they want to look at a paper.
However, having mathematics become utterly reliable might not be
the primary reason that eventually
formal mathematics will be
used by most mathematicians. Formalization of mathematics can be a
very rewarding activity in its own right. It combines the pleasure
of computer programming (craftsmanship, and the computer doing
things for you), with that of mathematics (pure mind, and absolute
certainty.) People who do not like programming or who do not like
mathematics probably will not like formalization. However, for
people who like both, formalization is the best thing there is.
Freek Wiedijk. Formal Proof -
Getting Started. Notices of the AMS, 2008, Vol. 55, N 11, pp.
1408-1414 (available online).
Feynman. The Nobel Prize in Physics, 1965.
"The fact that electrodynamics can be written in so many ways
..., was something I knew, but I have never understood. It always
seems odd to me that the fundamental laws of physics, when
discovered, can appear in so many different forms that are not
apparently identical at first, but, with a little mathematical
fiddling you can show the relationship. ... I don't know why this
is - it remains a mystery, but it was something I learned from
experience. There is always another way to say the same thing that
doesn't look at all like the way you said it before. I don't know
what the reason for this is. I think it is somehow a
representation of the simplicity of nature. ... I don't know what
it means, that nature chooses these curious forms, but maybe that
is a way of defining simplicity. Perhaps a thing is simple if you
can describe it fully in several different ways without
immediately knowing that you are describing the same thing."
R. Feynman. The
Development of the Space-Time View of Quantum Electrodynamics.
Nobel Lecture, December 11, 1965.
C. Dennett, 2008
"A point I have often made is that computer science keeps
cognitive science honest. If it weren’t for the practical
possibility of constructing and demonstrating simplified working
models of cognitive processes, we’d still be at the
hand-waving stage. ... most of the good work in computer science
(and related fields such as robotics) enlarges our appreciation
for just how remarkable our brains are.
for "Philosophy of Computing and Information - Five
Questions", edited by Luciano Floridi, forthcoming.
Putnam: Philosophy of Mathematics: Why
Nothing Works? Question asked 1979, the answer of 1997
follows (as put by Wikipedia):
"Under the influence of Ludwig Wittgenstein, he [Putnam -
K.P.] adopted a pluralist view of philosophy itself and came to
view most philosophical problems as nothing more than conceptual
or linguistic confusions created by philosophers by using ordinary
language out of its original context."
Full text: Hilary
Putnam by Wikipedia.
Putnam, H. “Philosophy of Mathematics: Why Nothing
Works” in Words
and Life, Harvard University Press, 1994, pp. 499-512
(written in 1979).
Putnam, H. "A Half Century of
Philosophy: Viewed from Within". Daedalus,
E. Knuth, 20 May 1995
"Science is what we understand well enough to explain to a
computer. Art is everything else we do."
Foreword of the book: A=B, by Marko
Wilf and Doron
Zeilberger, A K Peters Ltd, 1996, 224 pp. (online
Hestenes, 1992, [models
it is their fitness to data that is discovered!]:
"Kepler employed the Copernican reference system in his own
analysis and showed that Tycho's more accurate data could not be
fitted to the Copernican model. ... Since no one before had ever
considered any kinematical alternative to uniform circular motion,
Kepler had to invent his own to fit the data. His
brilliant result is formulated as a system of functional relations
called Kepler's laws. Many physicists would insist that Kepler's
laws were discovered rather than invented. On the contrary, what
Kepler discovered was that these laws fit the data. He
had considered and discarded many alternatives. It would be better
to speak of Kepler's model (rather than laws) and say
that the model has been validated to the precision in
Tycho's observations. We now know that many alternative models
could be invented to fit the same data, but Kepler's is the
simplest of all models in this class. We also know that Kepler's
could not fit the more accurate data collected with telescopes
rather than the naked eye, because the elliptical planetary orbits
are perturbed by gravitational forces from other planets, ... That
fact could never be discovered by Kepler's method; the more
powerful method of Newton was needed. It is no small irony that
Newton's law of gravitation would undoubtedly have been more
difficult to discover if Kepler's model had been quickly
invalidated by more accurate data. Here we have the possibility
that scientific progress might be impeded by greater experimental
D. Hestenes. Modeling Games in the
Newtonian World. American Journal of Physics, 1992,
Volume 60, Issue 8, pp. 732-748.
E J Brouwer, 19XX
"Mathematics is nothing more, nothing less, than the exact
part of our thinking."
by L E J Brouwer in The
MacTutor History of Mathematics archive.
"... I do want to convey
the basic philosophical point that I continue to find the real
puzzle of quantum mechanics. Not the move away from classical
determinism, but the ways in which the standard versions seem to
lie outside the almost universal methodology of modern probability
theory and mathematical statistics. For me it is in this arena
that the real puzzles of quantum mechanics are to be found. I am
philosophically willing to violate classical physical principles
without too many qualms, but when it comes to moving away from the
broad conceptual and formal framework of modern probability theory
I am at once uneasy. My historical view of the situation is that
if probability theory had been developed to anything like its
current sophisticated state at the time the basic work on quantum
mechanics was done in the twenties, then a very different sort of
theory would have been formulated."
Autobiography, 1978, p.5.
"For example, in quantum chemistry there
is, with present intellectual and computing resources, no hope of
making a direct attack on the behavior of complex molecules by
beginning with the first principles of quantum theory. A problem
as easy to formulate as that of deriving from first principles the
boiling point of water under normal atmospheric pressure is simply
beyond solution at the present time and is recognized as such."
Full text: p.15.
"... On the other hand, I strongly believe that
a reduction of psychology to the biological or physical sciences
will not occur and is not intellectually feasible. I am not happy
with leaving the statement of my views at this level of
generality, and I consider it an intellectual responsibility of
methodological behaviorists like myself to reach for a deeper and
more formal statement of this antireductionist position. What are
needed are theorems based on currently reasonable assumptions
showing that such a reduction cannot be made. I think of such
theorems as being formulated in the spirit in which theorems are
stated in quantum mechanics about the impossibility of
deterministic hidden variable theories."
Full text: p.16.
"As Bernays remarks, syntax is a branch of
number theory and semantics the one of set theory."
See p. 470 of
Wang. EIGHTY YEARS OF
FOUNDATIONAL STUDIES. Dialectica,
Vol. 12, Issue 3-4, pp. 466-497, December 1958.
"To the grand philosophical question: "What
is a man?" Aristotle answered: "Man
is a rational animal." Modeling
Theory offers a new answer: "Man is a
modeling animal!" HOMO
on Modeling Theory, Proceedings of the
2006 GIREP conference: Modelling in Physics and Physics Education.
Sent: Tue Jul 31
15:47:24 EDT 2007
Subject: Re: [FOM]
Sazonov on intuitive and formal mathematics
"When trying to formalize some new imaginary world (say, of
infinite objects - sets) we should realize that it is only
imaginary one and nothing is true or false there in the same sense
as in the ordinary physical world or for not so big finite
objects, and no logical laws hold there just because they are
"objectively" true - "do not hold" in this
sense. Thus, when we DECIDE to impose the ordinary (or any other
preferable) logical laws onto this world, it is not because they
are true there. It is because this is OUR decision, assuming it is
sufficiently coherent with our imagination and sufficiently
robust. (The coherence is typically incomplete; also various
surprises - counterexamples - are possible which would rather
"correct" our intuition. And we so much respect these
formalisms that we usually are quite happy with these
corrections.) We create our own worlds and "play" there
by the laws of some logic we choose, let Aristotelian."
Full text at
McCarthy, February 29, 1996
"It turns out that many philosophical problems take new forms
when thought about in terms of how to design a robot."
J. McCarthy. What
has AI in Common with Philosophy?, 1996
Friedrich Gauss to Franz
Adolph Taurinus, Goettingen, November 8, 1824
"... But it seems to me that in spite of the word-mastery of
the metaphysicians, we know really too little, or even nothing at
all, about the true nature of space to be able to confuse
something that seems unnatural with absolutely impossible.
If non-Euclidean geometry is the real one and the constant is
incomparable to the magnitudes that we encounter on earth or in
the heavens then it can be determined aposteriori. I have
therefore occasionally for fun expressed the wish that Euclidean
geometry not be the real one, for then we would have a priori an
Full text: Gauss
And Non-Euclidean Geometry by Stanley
N. Burris (see also the German original at the
Try replacing "Euclidean
geometry" by "large
cardinal axioms", and "non-Euclidean geometry"
- by Petr
Set Theory. (Pictures,
Danilovic Aleksandrov, about mathematics as a kind of
"Mathematics is creating its apparatus, and speaking about
its truth or falsity is senseless: the apparatus is either
working, or not working, and if working, it is working either
productively, or not. A similar nonsense would be asking: "Is
this screwdriver true or false?"; the screwdriver simply
exists, and one can only ask sensibly, how is it working,
and where could it be applied." (continue
here, in Russian, sorry, my own English translation).
A. D. Aleksandrov. Mathematics and
dialectics. Siberian Mathematical Journal, 1970, Vol.11,
Putnam, about "theory-dependence of meaning and
truth", December 29, 1977.
"It may well be the case that the idea that statements have
their truth values independent of embedding theory is so
deeply built into our ways of talking that there is simply no
"ordinary language" word or short phrase which refers to
the theory-dependence of meaning and truth. Perhaps this is why
Poincare was driven to exclaim "Convention, yes! Arbitrary,
no!" when he was trying to express a similar idea in another
context." (p. 471)
"... The language, on the perspective we talked ourselves
into, has a full program of use; but it still lacks
This is the fatal step. To adopt a theory of meaning according to
which a language whose whole use is specified still lacks
something - viz. its "interpretation" - is to accept
problem which can only have crazy solutions. To speak as
if this were my problem, "I know how to use my
language, but, now, how shall I single out an interpretation?"
is to speak nonsense. Either the use already fixes the
"interpretation" or nothing can."
H. Putnam. Models And Reality.
Journal of Symbolic Logic, September 1980, Vol.45, N3, pp.
Mycielski, at the Russell's Paradox Centennial
Conference, Munich, 2001
"... In this state of affairs the existence of Platonists in
this day and age is puzzling to us as it was puzzling to Tarski
... and presumably to Russell. ... Of course there were very
outstanding Platonists, and among them Frege, Zermelo and Gödel.
Russell's Paradox and Hilbert's (much forgotten) View
of Set Theory. One Hundred Years of
Russell's Paradox: Mathematics, Logic, and Philosophy.
Berlin, New York: Walter de Gruyter, 2004, pp. 533-547 (online
Tarski, at the "Tarski Symposium",
University of Berkeley, 1971:
"People have asked me 'How can you, a nominalist, do work in
set theory and logic, which are theories about things you do not
believe in?' . . . I believe there is value even in fairy tales
and the study of fairy tales."
Chihara letter to Anita Burdman Feferman, June
see p.52 of :
Alfred Tarski: Life and Logic.
Burdman Feferman and Solomon
Feferman, Cambridge University Press, Cambridge, UK, 2004, 432
From: Timothy Y.
Sent: Sun Oct 22 16:15:25 EDT 2006
Subject: Re: [FOM]
First-order arithmetical truth
"Any skepticism about the naive (resp. formal) concept of the
integers carries over directly into skepticism about the naive
(resp. formal) concept of a formal system. Anyone who thinks that
formal systems are crystal clear while integers are vague and
suspect---and who tries to argue that the existence of nonstandard
models lends support to that idea---is simply suffering from a
blindspot that prevents him from seeing that his skeptical
arguments apply equally to formal systems. "
Full text at
Sent: Tue Oct 24
18:30:13 EDT 2006
Subject: Re: [FOM]
First-order arithmetical truth
"I definitely do not suffer from such a blindspot. Moreover,
I make clear distinction between naive and abstract
(meta)mathematical concepts discussed...
You are right. Abstract (meta)mathematical formal systems can have
nonstandard formulas and derivations. Exactly the same as for
abstract mathematical numbers.
In place of that person I would replace here "standard formal
system" by "NAIVE, CONCRETE formal system" because
it is continued with "on a sheet of paper". This is not
about an abstract (meta)mathematical concept. This is from real
human activity of writing symbols, symbolically presented rules
and practical ability to follow these rules. No theory explaining
this activity is needed. People just are able to do this in
practice. That is why this activity is both NAIVE and CONCRETE. "
Full text at
as Henry Poincare noticed in his book "Science et
methode" (Paris, 1908, see Volume II, Chapters III and IV):
(in modern terms) the idea of a "formal theory of natural
numbers" is based on petitio
principii. The abstract notion of formal
syntax includes the same induction principle that is formalized in
this "formal theory of natural numbers". Two possible
exits from this situation are: a) regard natural numbers as a
consistent notion that is independent of any definitions
(platonism, a kind of mysticism), b) conclude that natural numbers
represent an inconsistent notion (formalism, accepting its own
Penrose about Gödel's
Incompleteness Theorem, 2005:
"... he [Gödel] demonstrated that, if we
are prepared to accept that the rules of some such formal system F
are to be trusted as giving us only mathematically correct
conclusions, then we must also accept, as correct, a certain
clear-cut mathematical statement G(F), while concluding that G(F)
is not provable by the methods of F alone. Thus, Gödel shows
us how to transcend any F that we are prepared to trust."
R. Penrose. The Road to Reality: A
Complete Guide to the Laws of the Universe. Knopf, 2005. Thanks to
have a simple misunderstanding here. Let's de-mystify the
situation. To "accept G(F) as correct", one doesn't need
trusting F "as giving us only mathematically correct
conclusions". To prove G(F) "as true" we need a
much weaker assumption - that F is syntactically
consistent, i.e. that it does not allow deriving of
contradictions. This assumption can be formalized as a certain
arithmetical statement Con(F). After this, we can prove the
statement Con(F)->G(F) in first order
arithmetic. Thus, here, the only heroic
act is postulation of Con(T) "as true". Is free of
charge postulation of Con(F) an honest way "how to transcend
any F that we are prepared to trust"?
Goldstein about Kurt Gödel,
"I'm saddened by the sense of his isolation, by
how profound it must have been. It's chilling to consider that he
felt the world to be so hostile that he believed his food was
being poisoned and so stopped eating and so starved to death. I've
spent a long time imagining what that must have felt like for such
a man. And I contrast that dark and cold place in which he lived
many long years and in which he ended his life with the sense of
bright wonderment that I experienced that summer before graduate
school, when I first understood Gödel's masterpiece of
reason. He gave that experience to countless people, and we're
GÖDEL AND THE NATURE OF MATHEMATICAL TRUTH
[6.8.05] A Talk with Rebecca Goldstein.
Aschbacher about probably the most complicated
mathematical proof ever, 2004-2005
"To my knowledge the main theorem of [AS] closes the last gap
in the original proof, so (for the moment) the Classification
Theorem can be regarded as a theorem. On the other hand, I hope I
have convinced you that it is important to complete the program by
carefully writing out a more reliable proof in order to minimize
the chance of other gaps being discovered in the future."
M. Aschbacher. The Status of the
Classification of the Finite Simple Groups. Notices of the
AMS, August 2004, vol. 51, N 7, pp. 736-740 (online
(For the formulation of the Classification Theorem, see
of finite simple groups by Wikipedia,
the free encyclopedia.)
"Conventional wisdom says the ideal proof should be short,
simple, and elegant. However there are now examples of very long,
complicated proofs, and as mathematics continues to mature, more
examples are likely to appear. Such proofs raise various issues.
For example it is impossible to write out a very long and
complicated argument without error, so is such a ‘proof’
really a proof? What conditions make complex proofs necessary,
possible, and of interest? Is the mathematics involved in dealing
with information rich problems qualitatively different from more
M. Aschbacher. Highly complex
proofs and implications of such proofs. Philosophical
Transactions of the Royal Society A: Mathematical, Physical and
Engineering Sciences, October 15, 2005, vol. 363, N 1835, pp.
M. Turing about "the real moral of the Gödel
result" (February 20, 1947)
"As Penrose himself notes, this seems to be what Turing
thought was the real moral of the Gödel result. Turing is
worth quoting at length:"
"It might be argued that there is a fundamental contradiction
in the idea of a machine with intelligence. It is certainly true
that 'acting like a machine', has come to be synonymous with lack
of adaptability... It has for instance been shown that with
certain logical systems there can be no machine which will
distinguish provable formulae of the system from unprovable...
Thus if a machine is made for this purpose it must in some cases
fail to give an answer. On the other hand, if a mathematician is
confronted with such a problem he would search around and find new
methods of proof, so that he ought to be able to reach a decision
about any given formula. Against it I would say that fair play
must be given to the machine. Instead of it sometimes giving no
answer we could arrange it so that it gives occasional wrong
answers. But the human mathematician would likewise make blunders
when trying out new techniques. It is easy for us to regard these
blunders as not counting and give him another chance, but the
machine would probably be allowed no mercy. In other words then,
if a machine is expected to be infallible, it cannot also be
intelligent. There are several mathematical theorems which say
almost exactly that. But these theorems say nothing about how much
intelligence may be displayed if a machine makes no pretense at
Rick Grush, Patricia
S. Churchland. GAPS IN PENROSE'S TOILINGS. Journal of
Consciousness Studies, 1995, vol. 2, N 1, pp. 10-29 (online
A. Turing. Lecture to the London
Mathematical Society on 20 February 1947. In A.M. Turing's ACE
report of 1946 and other papers (eds. B.E. Carpenter and R.W.
Doran). The Charles Babbage Institute Reprint Series for the
History of Computing, Cambridge, MIT Press, 1986, vol. 10.
Sent: Thursday, April 06, 2006 1:42
Subject: Re: [FOM]
Clarity in fom and problem solving
" I think that, like in engineering, mathematicians (or just
our ancestors when they invented numbers) are really creators of
mathematical concepts via formal systems, axioms, definitions,
algorithms etc., and, again like in engineering, these creations
are not absolutely free. However, they are *potentially* free.
There is no restriction to create possibly useless/meaningless
formalisms, incorrect proofs, etc. like useless (but may be
amazing) engineering devices. I see mathematics in general as the
engineering of formal tools (formalisms) strengthening our
abstract thought, and it is this what imposes the restriction
under discussion. This can be also compared with software
engineering which however is devoted to mechanising the routine
part of our intellectual activity."
Full text at
Lloyd about computational capacity of the Universe,
"All physical systems register and process information. The
laws of physics determine the amount of information that a
physical system can register (number of bits) and the number of
elementary logic operations that a system can perform (number of
ops). The Universe is a physical system. The amount of information
that the Universe can register and the number of elementary
operations that it can have performed over its history are
calculated. The Universe can have performed 10120 ops
on 1090 bits ( 10120 bits
including gravitational degrees of freedom)."
S. Lloyd. Computational capacity of
the universe. Physical Review Letters, 2002, vol. 88,
issue 23, 4 pages (available online,
see also 17 page extended
From Hendrik J. Boom
Sent: Thursday, February
02, 2006 10:59 PM
"Practicing physicist seem to act as if every set of real
numbers is measurable, for example. This lets them off the
escalator of ascending set-theoretic axioms rather early, as they
never get around to accepting the axiom of choice.
In fact, some physists seem to act as if every total function from
R->R is continuous! Does this make them unwitting
Full text at
Putnam about experimental mathematics, 1975
this paper, I have stressed the importance of quasi-empirical and
even downright empirical methods in mathematics. ... None of this
is meant to downgrade the notion of proof. Rather, Proof and
Quasi-empirical inference are to be viewed as complementary. Proof
has the great advantage of not increasing the risk of
contradiction, where the introduction of new axioms or new objects
does increase the risk of contradiction, at least until a relative
interpretation of the new theory in some already accepted theory
is found. For this reason, proof will continue to be the primary
method of mathematical verification."
H. Putnam. What is Mathematical
Truth? Historia Mathematica 2 (1975): 529-543 (for
reprints see Hilary
Nelson. Mathematics and Faith. Vatican, May 23-24,
"The notion of truth in mathematics is irrelevant to what
mathematicians do, it is vague unless abstractly formalized, and
it varies according to philosophical opinion. In short, it is
formal abstraction masquerading as reality."
Full text at http://www.math.princeton.edu/~nelson/papers.html.
As a regular term, "platonism
in mathematics" is used since
the lecture delivered June 18, 1934, University of Geneva, by Paul
P. Bernays. Sur
le platonisme dans les mathematiques. L'enseignement
mathematique, Vol. 34 (1935), pp. 52-69.
Quoted from English translation by Charles
D. Parsons at
Bernays considers mathematical platonism as a method
that can be - "taking certain precautions" - applied in
mathematics. Some remarkable quotes (fragments marked bold by me -
... allow me to call it "platonism".
... The value of platonistically
inspired mathematical conceptions is
that they furnish models of abstract imagination. These stand out
by their simplicity and logical strength. They form
representations which extrapolate from certain regions of
experience and intuition.
... This brief summary will suffice to characterize
platonism and its application to
mathematics. This application is so
widespread that it is not an exaggeration to say that platonism
reigns today in mathematics.
... Several mathematicians and philosophers interpret
the methods of platonism
in the sense of conceptual realism, postulating the
existence of a world of ideal objects containing all the objects
and relations of mathematics. It is this absolute platonism which
has been shown untenable by the antinomies, particularly by those
surrounding the Russell-Zermelo paradox.
... It is also this transcendent character which
requires us to take certain
precautions in regard to each platonistic assumption.
For even when such a supposition is not at all
arbitrary and presents itself naturally to the mind, it can still
be that the principle from which it proceeds permits only a
restricted application, outside of which one would fall into
contradiction. We must be all the more careful in the face of this
possibility, since the drive for simplicity leads us to make our
principles as broad as possible. And the need for a restriction is
often not noticed. This was the case, as we have seen, for the
principle of totality, which was pressed too far by absolute
platonism. Here it was only the discovery of the Russell-Zermelo
paradox which showed that a restriction was necessary.
[Inspired by reading: Jacques
Bouveresse. On the Meaning of the
Word 'Platonism' in the Expression 'Mathematical platonism'.
Proceedings of the Aristotelian Society,
September 2004, Volume 105, pp. 55-79 (online French
version). Thanks to William
Selbstbiographie, published in 1948
"Die Wahrheit triumphiert nie, ihre Gegner sterben nur
"Truth never triumphs - its opponents just die
Condensed version from the article Max
Plack in Wikiquote,
full text ibid.
The Mathematician, 1947
"I think that it is a relatively good approximation to truth
- which is much too complicated to allow anything but
approximations - that mathematical ideas originate in empirics,
although the genealogy is sometimes long and obscure. But, once
they are so conceived, the subject begins to live a peculiar life
of its own and is better compared to a creative one, governed by
almost entirely aesthetical motivations, than to anyting else and,
in particular, to an empirical science. There is, however, a
further point which, I believe, need stressing.
mathematical discipline travels far from its empirical source, or
still more, if it is a second and third generation only indirectly
inspired from ideas coming from 'reality', it is beset with very
grave dangers. It becomes more and more purely aestheticizing,
more and more purely l'art pour l'art. This need not be
bad, if the field is surrounded by correlated subjects, which
still have closer empirical connections, or if the discipline is
under the influence of men with an exceptionally well-developed
But there is a grave danger that the subject will
develop along the line of least resistance, that the stream, so
far from its source, will separate into a multitude of
insignificant branches, and that the discipline will become a
disorganized mass of details and complexities.
words, at a great distance from its empirical source, or after
much 'abstract' inbreeding, a mathematical subject is in danger of
degeneration. At the inception the style is usually classical;
when it shows signs of becoming baroque the danger signal is up.
It would be easy to give examples, to trace specific evolutions
into the baroque and the very high baroque, but this, again, would
be too technical.
In any event, whenever this stage is
reached, the only remedy seems to me to be the rejuvenating return
to the source: the reinjection of more or less directly empirical
ideas. I am convinced that this was a necessary condition to
conserve the freshness and the vitality of the subject, and
this will remain equally true in the future."
Neumann. The Mathematician, in: The Works of the Mind,
Robert B. Heywood (ed.), University of Chicago Press, 1947,
Quoted after the October 22, 1995 posting by Louis
A. Talman at Math Forum @
Management and Machiavelli, 1967
"In corporation [corporate] religions as in others, the
heretic must be cast out not because of the probability that he is
wrong but because of the possibility that he is right." - as
quoted twice by Edsger
W. Dijkstra, February,
27, 1975 and August
THEOREMS FOR A
PRICE: Tomorrow's Semi-Rigorous Mathematical Culture
"The computer has already started doing to mathematics what
the telescope and microscope did to astronomy and biology. In the
future, not all mathematicians will care about absolute certainty,
since there will be so many exciting new facts to discover:
mathematical pulsars and quasars that will make the Mandelbrot set
seem like a mere Jovian moon. We will have (both human and
machine) professional theoretical mathematicians, who
will develop conceptual paradigms to make sense out of the
empirical data, and who will reap Fields medals along with (human
and machine) experimental mathematicians. Will there
still be a place for mathematical mathematicians?"
Full text appeared in Notices of the AMS, Vol. 40, N8
(October 1993), pp.978-981 (online
Roger Bishop Jones,
On how many things there might be
"We find ourselves at the beginning of the 21st century
facing the prospect that knowledge may no longer be the exclusive
domain of human intelligence, and that questions of ontology,
mathematics, science and engineering may be entertained and
resolved by fabrications in silicon."
Online text in progress, February 11, 2005, at
Mathematical Platonism Reconsidered
"Admittedly, the mathematician's ideas reside in a modest
amount of jelly-like substance which constitutes the
mathematician's brain, ..."
"... because a mathematician's world is a world of ideas as
envisioned by Plato. But what comes out of our discussion is that
these ideas are very specifically human, depending on the very
special organization of our brain, and in particular on its
"The fundamental limitations put by physical law on
computing, or doing mathematics, do not appear to be very well
understood at this time. ... It seems possible, however, that
another crisis of foundations of mathematics may be awaiting us,
and that collision with physical law could cause further damage to
our Platonist conception of mathematics."
Johann Bernoulli lecture, Groningen, April 20, 1999, full text at
Sent: Monday, January 17, 2005 4:09
I asked the panel members whether they were interested in
this line of investigation: no simple axiom settling the continuum
Woodin responded by saying that, overwhelmingly, he
really wanted to know whether the continuum hypothesis is true or
false. He is far more interested in pursuing that, as he is now,
than any considerations of simplicity, which for him, was a side
Martin responded by saying that the projective
determinacy experience showed that one could have axioms with
simple statements, but with very complicated explanations as to
why they are correct.
I did not have an opportunity to respond
to Martin's statement - I would have said that by the standards of
axioms for set theory, projective determinacy is NOT simple. It is
far more complicated than any accepted axiom for set theory.
did ask Cohen specifically to comment on whether simplicity (of a
new axiom to settle the continuum hypothesis) was important for
him. Cohen responded by saying that such an axiom, for him, must
Let me end here with something concrete. In my
papers in Fund. Math., and in J. Math. Logic, it is proved that
all 3 quantifier sentences in set theory (epsilon,=) are decided
in a weak fragment of ZF, and there is a 5 quantifier sentence
that is not decided in ZFC (it is equivalent to a large
axiom over ZFC). All of the axioms of ZF are an at most four
quantifier sentence and an at most five quantifier axiom scheme.
It has been shown that AxC over ZF is equivalent to a five
quantifier sentence (see Notre Dame Journal, not me). Show that
over ZFC, any equivalence of the
continuum hypothesis requires
a lot more quantifiers. Show that over ZFC, any statement
consistent with ZFC that settles the continuum hypothesis,
requires a lot more quantifiers.
Full text at
David Corfield (2004)
... So much effort has been devoted to a thin notion of truth, so
little to the thicker notion of significance. To say that
scientists and mathematicians aim merely for the truth is a gross
distortion. They aim for significant truths.
text - on The Philosophy of Real
Cantor, August 28, 1899
11 Cantor, by contrast, insists in his letter to
Dedekind of August 28, 1899 that even finite multiplicities cannot
be proved to be consistent. The fact of their consistency is a
simple, unprovable truth - "the axiom of arithmetic";
the fact of the consistency of multiplicities that have an aleph
as their cardinal number is in exactly the same way an axiom, "the
axiom of the extended transfinite arithmetic".
Sieg. Hilbert's programs: 1917-1922. The Bulletin
of Symbolic Logic, March 1999, Vol.5, N 1 (online
From: Jeffrey Ketland ...
August 31, 2004 3:30 AM
Subject: Re: [FOM]
Proof "from the book"
... If I remember right, the gist is this. In studying the
consistency problem, Gödel wanted initially to give an
interpretation of second-order arithmetic within first-order
arithmetic, and tried to find a definition of (second-order!)
arithmetic truth in the first-order language. He discovered
however that even first-order arithmetic truth is not
arithmetically definable: i.e., what we now call Tarski's
Indefinability Theorem. But, as he also discovered, the concept
"provable-in-F", with F some fixed formal system, is
arithmetically definable. This implies that arithmetic truth is
distinct from provable-in-F, for any formal system F. This then
gives us the quick proof of Gödel's first incompleteness
Full text at
Peirce & Son
... in 1870 Benjamin Peirce defined mathematics as "the
science that draws necessary conclusions" (see his son
C.S.Peirce 1898/1955, p.137). C.S.Peirce himself described the
work of a mathematician as composed of two different activities
(p.138): (1) framing of a hypothesis stripped of all features
which do not concern the drawing of consequences from it, without
caring whether this hypothesis agrees with the actual facts; (2)
drawing the necessary consequences from the hypothesis. He noted
(Peirce 1902/1955, p.144) the difficulty to distinguish between
two definitions of mathematics, one by its method ("drawing
necessary conclusions"), another by its aim and subject
matter ("the study of hypothetical state of things").
See p.5 of
Alexander Khait. The Definition of
Mathematics: Philosophical and Pedagogical Aspects. Science &
Education 00: 1-23, 2004, Kluwer Academic Publishers
Davis and Reuben
In the real world of mathematics, a mathematical paper does two
things. It testifies that the author has convinced himself and his
friends that certain "results" are true, and presents a
part of the evidence on which this conviction is based.
Automath approach represents an unrealizable dream. ... the
accepted practice of the mathematical community has hardly
changed, except for the enlargement of the computer
... The myth of totally rigorous, totally
formalized mathematics is indeed a myth.
Davis, P. J. &
Hersh, R.. Rhetoric and mathematics. In J. S. Nelson, A.
Mcgill & D. N. McCloskey (Eds.), The rhetoric of the human
sciences. Madison: University of Wisconsin, 1987, pp. 53-69.
(Thanks toWilliam J.
[Added December 7, 2008. Thanks to Maris Ozols.] Now,
20 years later, the situation is changing... See Notices
of the AMS, Special Issue on Formal
Proof, Vol. 55, N 11, 2008 (available online).
Sent: Saturday, May 15, 2004 1:43
Subject: Re: [FOM]
Freeman Dyson on Inexhaustibility
Maybe physics is inexhaustible, but maybe it isn't. Here's
why it might not be. Consider the Life World based on
Conway's Life cellular automaton. It has been shown that
self-reproducing universal computers are possible as
configurations in the Life World. Therefore, one could have
physicists in the Life World, but their physics would not be
inexhaustible. They could discover or at least conjecture
that their fundamental physics was a particular cellular
automaton. However, their mathematics could be the same as
ours - and therefore inexhaustible.
Full discussion thread - see Foundations
of Mathematics (FOM) e-mail list.
The Continuum Hypothesis (I), 2000, by W.
"The current situation is the
We can build models of set theory with
significant control over what is true in the model.
the 35 years since Cohen's work a great number of set theoretical
propositions have been shown to be independent. Further problems
in other areas of mathematics have also been shown to be
This, as of yet, cannot be accomplished for models
of number theory. The intuition of a true model of number
theory remains unchallenged."
"An important point is that neither Cohen's method of
extension nor Godel's method of restriction affects the arithmetic
statements true in the structures, so the intuition of a true
model of number theory remains unchallenged.
It seems that most
mathematicians do believe that arithmetic statements are either
true or false. No generalization of Cohen's method has yet been
discovered to challenge this view. But this is not to say that
such a generalization will never be found."
See p. 568 of
W. Hugh Woodin. The Continuum
Hypothesis, Part I, Notices of the ACM, Vol. 48, N6
(June/July 2001), pp. 567-576 (try online
reading), Part II, Vol. 48, N7 (July 2001), pp. 681-690 (try
Sent: Tue Jan 20 01:17:00 EST 2004
On Foundational Thinking 1
(To avoid confusion, I draw a
distinction between foundations of mathematics and mathematical
logic. The latter consists of various mathematical spinoffs from
foundations of mathematics, where one deemphasizes foundational
thinking, and emphasizes mathematical adventures, including
connections with various branches of mathematics.)
Full text at
Plough, Sword and Book: The Structure
of Human History, University of Chicago Press, 1988,
When knowledge is the slave of social considerations,
it defines a special class; when it serves its own ends only, it
no longer does so. There is of course a profound logic in this
paradox: genuine knowledge is egalitarian in that it allows no
privileged source, testers, messengers of Truth. It tolerates no
privileged and circumscribed data. The autonomy of knowledge is a
Thanks to William
J. Greenberg. Quoted after Anthropological Wit and Wisdom,
by Steve Froemming.
Date: Fri, 26 Dec 2003 14:44:47 -0300
Gonzalez Cabillon ...
quotation from Weyl
Dear Roman Murawski,
that the passage you are seeking in Weyl's writings is:
now come to the decisive step of mathematical abstraction: we
forget about what the symbols stand for. The mathematician is
concerned with the catalogue alone; he is like the man in the
catalogue room who does not care what books or pieces of an
intuitively given manifold the symbols of his catalogue denote. He
need not be idle; there are many operations which he may carry out
with these symbols, without ever having to look at the things they
which is contained in the classic "The
Mathematical Way of Thinking", an address given by Hermann
Weyl at the Bicentennial Conference at the University of
Pennsylvania, in 1940. It was first published in _Science_, in
1940, volume 92, pp. 437-446, and later reproduced in
Newman's "The World of Mathematics" (volume 3).
best wishes to you all for 2004.
Context - at
Zeilberger, November 26, 2001
"... the conventional wisdom, fooled by our misleading
"physical intuition", is that the real world is
continuous, and that discrete models are necessary evils
for approximating the "real" world, due to the innate
discreteness of the digital computer.
Ironically, the opposite is true. The
REAL REAL WORLDS (Physical and MATHEMATICAL) ARE
Continuous analysis and geometry are just degenerate
approximations to the discrete world, made necessary by the very
limited resources of the human intellect. While discrete analysis
is conceptually simpler (and truer) than continuous analysis,
technically it is (usyally) much more difficult."
Analysis is a Degenerate Case of Discrete Analysis by Doron
Zeilberger (appeared in "New
Progress in Difference Equations" (Proc. ICDEA 2001), Taylor
and Francis, London, 2004)