Let me give you one(but very important) example.
Grothendieck used category theory in essential way
in his revolutionary works(schemes, topos, etale cohomology, etc.)
which led to the proof of Weil conjecture and Wiles' proof of FLT.
By essential, I suppose you mean that there is no way to avoid using
category theory??
Also, do you mean that more than just the language of category theory was
used? i.e. some deep fundamental result from category theory.
Ok, I give up.
Category theory is bunk, right?
Not really, but you don't seem to understand.
Take idea of scheme.
It's very simple, almost trivial idea.
I remember Weil objected Grothendieck's scheme
and cohomology theory.
Ironically his conjecture was proved by using it.
> You seemed to decide to object whatever I post. :)
>
> Ok, I give up.
> Category theory is bunk, right?
>
> Not really, but you don't seem to understand.
> Take idea of scheme.
> It's very simple, almost trivial idea.
Ah yes, a scheme.
A locally ringed space locally isomorphic to spectra of rings, that's it.
A locally ringed space: a topological space with a sheaf of rings.
A sheaf (of rings): an assigment to each open set U on a topological space a
ring S(U)
and to each inclusion of open sets U contained in V a ring homomorphism
r_{V U} from S(V) to S(U).
No wait! that's only a presheaf, drat! For a sheaf need also a covering
condition: for each
collection of open sets {U_i} with union V and elements r_i in S(U_i) such that
r_{U_i U_i intersect U_j}(s_i) = r_{U_J U_i intersect U_j}(s_j) = for all i and
j then
there's a unique s in S(V) with r_{V U_i}(s) = s_i for each i.
Good! and locally ringed? The stalks of the sheaf have to be rings. Aaaagggh!
stalks!
What's a stalk? You got a sheaf S on the space X. If x in X then the opens
U containing x form a directed poset under (reverse) inclusion and the S(U)
form
a direct system of rings, and their direct limit is the stalk S_x. For a
locally ringed
space the rings have to be commutative and the stalks local.
Oh, schemes again. The locally ringed space has to be locally isomorphic to
spectra
of rings, so I suppose I gotta tell you what a spectrum of a (commutative) ring
is.
Well let R be a commutative ring. Then Spec R is the set of prime ideals of R
(not just the maximal ones). Easy. Oh, and it has to have a topology. Well the
closed
sets are V(I) = {P in Spec R: P contains I} for ideals I of R. Oh wait, it has
to be
a locally ringed space, so we've gotta put a sheaf of rings on this space too.
Let's start with the stalks. Try to fix this sheaf O such that O_P = (R/P)^{-1}
R
(a localization of R; this is just commutative algebra, really, really trivial)
and then
on open subsets D(f) = {P in Spec R: f not in P} (the complement of V(fR)) we
try to fix
that O(D(f)) = f^{-1}R (another localization). We can do this! Just take for
O(D(I))
[D(I) complement of V(I)] the maps P |--> a_P in O_P which are locally given
(since the D(f) form a basis of the topology of Spec R) by the maps of this
form
arising from elements of f^{-1}R on O(D(f)). OK I have left out the details
that this all works and the "structure sheaf" does have the stalks I claim it
has.
All really trivial, these schemes.
> I remember Weil objected Grothendieck's scheme
> and cohomology theory.
> Ironically his conjecture was proved by using it.
And it's a trivial consequence of the definition of scheme, isn't it?
OH NO! How could I do it! Look at the title of the thread. Categories!
The schemes for a category, and we can't define just the objects of a
category, we have to define the morphisms too, 'cos it's the morphisms
that are the important part! I don't have time now to describe these, but as
Nobuo says, schemes are a simple, almost trivial idea, so all sci.math
readers can see immediately for themselves how to do it.
Robin Chapman
--
Posted from webcacheh01a.cache.pol.co.uk [195.92.67.65]
via Mailgate.ORG Server - http://www.Mailgate.ORG
> I don't have time now to describe these, but as
> Nobuo says, schemes are a simple, almost trivial idea, so all sci.math
> readers can see immediately for themselves how to do it.
Of course. A morphism of ringed spaces is a very natural thing, once
you've defined a ringed space :-).
Btw, maybe Nabuo was refering to a different approach of (at least
affine) schemes. An affine scheme is just a covariant functor
Rings->Sets that can be written as Hom(R,-) for some ring R.
Likewise, schemes form a full subcategory of Sets^{Rings}, but I don't
think there's a reasonably simple characterization of schemes purely as
objects of this category. On the other hand, if you know how to define,
say, Ã©tale morphisms in Rings^{op}, the category of sheaves for the
corresponding Grothendieck "topology" forms a topos, the logic of which
might be used to characterize schemes (I'm not sure of this, though, and
it's actually way above my head).
Mhh, yeah. Simple. Almost trivial.
--
M. TIBOUCHI <med...@mail.com>
"A theorem is not true anymore because one can draw a picture, it is
true because it is functorial." -- Serge Lang, in his review of the EGA,
Bull. Amer. Math. Soc. 67 (1961).
I didn't object to anything. I wanted a clarification, since you presumably
have a good, solid reason for dismissing the opinions of some experienced
mathematicians, who nonetheless claim category theory is "bunk".
> Ok, I give up.
> Category theory is bunk, right?
>
Well, I don't think I would say that.
> Not really, but you don't seem to understand.
Oh, I don't? Well, why don't you clear up some things for me?
Nobuo Saito originally wrote:
> Grothendieck used category theory in essential way
> in his revolutionary works(schemes, topos, etale cohomology, etc.)
> which led to the proof of Weil conjecture and Wiles' proof of FLT.
And Chan-Ho Suh then asked:
"By essential, I suppose you mean that there is no way to avoid using
category theory??
Also, do you mean that more than just the language of category theory was
used? i.e. some deep fundamental result from category theory."
Answering these questions would greatly increase my chances of "seeing the
light".
Sorry, my explanation was not sufficient.
I took for granted the basic notion of sheaves
and traditional algebraic varieties
over an algebraic closed field.
I was thinking about the situation when Grothendieck invented
his idea of schemes. It was late 50's.
There was Serre's paper FAC(coherent algebraic sheaves)
which defined algebraic varieties with sheaves.
Serre's algebraic variety is a ringed space
locally isomorphic to affine algebraic varieties over
a fixed algebraic closed field.
This definition is natural if one knows the definition of
complex manifold with sheaves.
A complex manifold is a ringed space locally isomorphic
to an open subset of complex affine space
with sheaf of analytic functions.
An affine algebraic variety is regarded
as the set of all maximal ideals of coordinate ring A
with sheaf of regular functions on it.
If one wish to replace A to a general commutative ring,
one only needs to consider the set of all prime ideals
instead of the set of maximal ideals
to let the analogy of Hilbert's Nullstellensatz hold on it.
So Grothendieck's idea was in short:
Replace a coordinate ring of an affine varity
to a general commutatrive ring
and consider all set of prime ideals instead of maximal ideals.
Simple, almost trivial, isn't it?
The chance is zero.
Because your questions are pointless.
You are asking like how much celestial mechanics
is useful in biology(this is not very good analogy
but I think people(maybe except you) will know the point).
No my questions aren't pointless. They are direct and to the point: which
is whether or not category theory played a crucial/essential role in
Grothendieck's work you mentioned. If you lack the knowledge to answer such
a question, you should not post such comments.
Well, if you think my questions are the same as asking about the usefulness
of celestial mechanics in biology...then I guess you are saying that
category theory plays a similar useless role in mathematics.
Also, I like to think R -> Spec(R) and X -> Gamma(X)
is ajoint functors, here X is a local ringed space
and Gamma(x) is a ring of global sections on X
Early in my first year of graduate school, I sat in on a course for
the first day, where the instructor presented such a definition (this
was before I knew what Spec(R) was). An affine scheme was defined to
be a representable functor. Somebody asked for an example of a
non-affine scheme, and the instructor spent the rest of the hour
trying to describe the functor from Rings to Sets that is/describes
P^1 (i.e., P^1 over the integers). After class, I went to my cubicle
and tried to produce examples of schemes (without the usual geometric
intuitions to help), and managed to see that the forgetful functor
(i.e., Spec Z) fit the definition, and then that the cross product
of the forgetful functor with itself (i.e., Spec Z x Spec Z) also fit
the definition. So my impression is that whatever definition it was
couldn't have been so bad, if somebody with so little clue as to what
the point of the definition was, could apply it a little bit.
Years later, I tried to reconstruct the definition for myself, and
felt like I got sort of close, perhaps missing just one clause. So I
asked someone I knew, and he was of the opinion that the definition
wasn't so bad, but he couldn't remember it either. We figured it must
be in one of those old classics of scheme-based algebraic geometry
that has been allowed to go out of print.
In order to be really sure, though, somebody ;-) should go find that
definition.
Keith Ramsay
>By essential, I suppose you mean that there is no way to avoid using
>category theory??
No.
>Also, do you mean that more than just the language of category theory
was
>used? i.e. some deep fundamental result from category theory.
No.
Well then, why do you consider category theory to be the "one of the
greatest discoveries of 20th century mathematics"? And why do you consider
it "more fundamental than set theory"? [quotes are from Nobuo Saito's
original posting]. Certainly (with the answers you just gave to my
questions) it doesn't appear that the example you gave is very convincing in
supporting those claims. You say all that and then give as an example
Grothendieck's work, for which you now imply that category theory doesn't
need to be used and that only the language of categories was used. If this
is considered a most important application of category theory, no wonder
some mathematicians call category theory "bunk".
Maybe I'm a complete fool?
>Certainly (with the answers you just gave to my
>questions) it doesn't appear that the example you gave is very
convincing in
>supporting those claims. You say all that and then give as an
example
>Grothendieck's work, for which you now imply that category theory
doesn't
>need to be used and that only the language of categories was used.
>>By essential, I suppose you mean that there is no way to avoid using
>>category theory??
Who can tell there's no way to prove a theorem other than
a particular method?
The prime number theorem was first proved by using Riemann's zeta
function.
After many years(over 50 years?), it was proved by an elementary
method.
Do you think, then, the Riemann's method is not convincingly powerful?
Who can tell there's no way to avoid using Wiles' method
to prove FLT?
Your question is, as I said before, pointless.
It is beginning to sound a lot like it.
>
> >Certainly (with the answers you just gave to my
> >questions) it doesn't appear that the example you gave is very
> convincing in
> >supporting those claims. You say all that and then give as an
> example
> >Grothendieck's work, for which you now imply that category theory
> doesn't
> >need to be used and that only the language of categories was used.
This is what you did, *in answer* to the following query.
>
> >>By essential, I suppose you mean that there is no way to avoid using
> >>category theory??
>
> Who can tell there's no way to prove a theorem other than
> a particular method?
>
This is a complete non sequitur. in first order logic, it would be: "A is
essential" => (definitionnally) no proof of B exists without using A. Then,
you admit something like "All proofs of B can be rewritten without using A".
This prove, at least that (for B) A is non essential. What could be clearer?
And what does the next example prove ?
> The prime number theorem was first proved by using Riemann's zeta
> function.
> After many years(over 50 years?), it was proved by an elementary
> method.
> Do you think, then, the Riemann's method is not convincingly powerful?
To follow your logic, you would (at least) need a case where category theory
was the only method known to solve a problem, at least during a short time.
Moreover, if every result of analytic number theory could be obtained by
elementary method, this would be a strong case to say the anlytic tool is
not so essential after all.
>
> Who can tell there's no way to avoid using Wiles' method
> to prove FLT?
JSH? Then, again, his answer would be negative :-)
>
> Your question is, as I said before, pointless.
Curiouser and curiouser. Asking if something is essential or bunk doesn't
look a poinless question. And , by the way, you are not so well-informed
that you believe. The Weyl conjectures were not proved *first* by the
Grothendieck program, but by a much lighter and more classical approach, by
A.Connes... (which is not to slight the value and the power of the schemes
, topos etc., but category theory, in all this, is not much more that a very
nice language. To give exemples of the essentialness of it, you will have to
search some more.
>
> Curiouser and curiouser. Asking if something is essential or bunk doesn't
> look a poinless question. And , by the way, you are not so well-informed
> that you believe. The Weyl conjectures were not proved *first* by the
> Grothendieck program, but by a much lighter and more classical approach, by
> A.Connes...
I don't know about the Weyl conjectures :-), but I'm pretty sure at
least one of the Weil conjectures, namely the Riemann hypothesis for
varieties over finite fields, was first proved using all the power of
schemes (Deligne, 1973).
> (which is not to slight the value and the power of the schemes
> , topos etc., but category theory, in all this, is not much more that a very
> nice language.
The right language to express a mathematical problem seems to me an
invaluable asset. Just like algebraic structures(*) had provided a very
fruitful language in the late nineteenth/early twentieth century,
category theory can give you tools and insight when attacking some
question, even if it can be formulated without categories. The moduli
problem is a good example of that, I guess.
More generally, the aim of "theory of X" can often be stated as
"classifying the objects of category X up to isomorphism, using
functorial invariants" :-).
On the other hand, arguing categories are essential, or even one of the
most important discoveries of the twentieth century, is probably
exaggerating!
(*) At a bookshop this afternoon, I overheard a middle-aged guy (who
kept saying he was from the Ã‰cole Centrale) explaining to a math Ph.D.
student how abstruse and useless he thought ideals and rings of integers
were in number theory. He told him the Cornell-Silverman Modular Forms
book was no good at making things clear about FLT, with all those
meaningless formulas with H^q all over the place. :-) Of course, he knew
a proof of FLT, but hadn't patented it yet, so he didn't want to send it
in full to the academics he'd been in touch with.
--
M. TIBOUCHI <med...@mail.com>
wondering if JSH is from Centrale Paris...
> In order to be really sure, though, somebody ;-) should go find that
> definition.
If you ever find that somebody, do let me know! :-)
--
M. TIBOUCHI <med...@mail.com>
int a=10000,b,c=8400,d,e,f[8401],g;main(){for(;b-c;)
f[b++]=a/5;for(;d=0,g=c*2;c-=14,printf("%.4d",e+d/a),
e=d%a)for(b=c;d+=f[b]*a,f[b]=d%--g,d/=g--,--b;d*=b);}
>
> Curiouser and curiouser. Asking if something is essential or bunk doesn't
> look a poinless question. And , by the way, you are not so well-informed
> that you believe. The Weyl conjectures were not proved *first* by the
The what?
> Grothendieck program, but by a much lighter and more classical approach, by
> A.Connes...
We were talking about the Weil conjectures (Andre Weil: 1906-1998)
which were *first* proved by Pierre Deligne circa 1972, using the full power
of Grothendieck's theory. [AFAIK the *first* is redundant - I don't think
anyone else has proved them via an essentially different method.]
> (which is not to slight the value and the power of the schemes
> , topos etc., but category theory, in all this, is not much more that a very
> nice language. To give exemples of the essentialness of it, you will have to
> search some more.
?
Sorry for those ridiculous affirmations of mine. Must have been drunk (but
even that is a poor excuse). Will try not to do that again. Going to hide
myself in a corner now.
Well, if you want to say that, you're free to do so. But I find that
response much less interesting than an actual serious response.
> >Certainly (with the answers you just gave to my
> >questions) it doesn't appear that the example you gave is very
> convincing in
> >supporting those claims. You say all that and then give as an
> example
> >Grothendieck's work, for which you now imply that category theory
> doesn't
> >need to be used and that only the language of categories was used.
>
> >>By essential, I suppose you mean that there is no way to avoid using
> >>category theory??
>
> Who can tell there's no way to prove a theorem other than
> a particular method?
>
Presumably logicians can (sometimes). But 'I don't know' is always an
alternative to just saying 'no' (like you did in response to my question)
and implying stuff you don't really intend. Also alternatively, you could
say whether other methods *are* known or are not; that's an incomplete
answer, but still helpful in convincing me of the essentialness of category
theory in the stated example.
> The prime number theorem was first proved by using Riemann's zeta
> function.
> After many years(over 50 years?), it was proved by an elementary
> method.
> Do you think, then, the Riemann's method is not convincingly powerful?
>
Sure, that shows it's powerful. But if every theorem in analytic number
theory can be proved in just a few pages (like the prime number theorem)
using elementary methods, I don't think this bodes well for the future of
analytic number theory, let alone calling it the "greatest discovery of 20th
century mathematics." There are many (now esoteric) fields of math in the
19th century, that are now considered not very important because they were
superceded by better theories.
As for my second question about whether only the *language* of categories
was used, if one can rephrase the proofs of the Weil Conjectures to avoid
categorical language, and it still makes good sense, then I have to question
how essential category theory is to the proof.
So you see, my questions strike at the heart of the question, "Why is
category theory essential to the proof the Weil Conjectures?" Consequently,
since that was the only example you gave in support of your claims that
category theory is one of the greatest discoveries of the 20th century and
is more fundamental than set theory, my questions thus put your claims in
doubt. That makes my questions far from pointless.
[snip]
> There are many (now esoteric) fields of math in the
> 19th century, that are now considered not very important because they were
> superceded by better theories.
Hey, this is interesting. I was always under the impression that mathematics
was very good in keeping old subjects 'alive' in collective memory. Could
you mention some examples of such 'forgotten' theories? (Either 19th century
or not.)
Regards,
Herman Jurjus
> "Chan-Ho Suh" <cs...@math.ucla.edu> wrote in message
> news:9pbivj$4ap$1...@siamese.noc.ucla.edu...
>
> [snip]
> > There were many (now esoteric) fields of math in the
> > 19th century, that are now considered not very important because they were
> > superseded by better theories.
>
> Hey, this is interesting. I was always under the impression that mathematics
> was very good in keeping old subjects 'alive' in collective memory. Could
> you mention some examples of such 'forgotten' theories (either 19th century
> or not)?....
Well, if we can go back to antiquity, then:
(1) Sexagesimal arithmetic. I've tried using it, and it actually lends
itself to lots of nice short cuts, but that hardly makes it a living
technique.
You may object that that's mere notation. O.K. then:
(2) Classical Chinese algorithms for handling two-variable polynomials on
the counting board. They set out the coefficients in a matrix format,
with the coefficient of (x^m)(y^n) in the (m,n) position. Many
calculations then work rather nicely.
You may object that that's mere algorithmics. O.K. then:
(3) The application of areas. This is sometimes called Greek
"geometrical algebra," and indeed its range of application can be seen as
largely overlapping that of elementary algebra, but not coinciding with
it. Certainly the techniques feel very different.
As for the 19th century, I guess Chan-Ho Suh may have been thinking
of something like:
(4) Algebraic invariants. People do still use these, but they may fit
Chan-Ho's desciption as "now considered not very important because they
were superseded by better theories." Perhaps it would be better to say
that they're no longer such a major research area as they were in the 19th
century.
Whether or not others agree with my list above, it might be
interesting if Chan-Ho or someone else could provide a few more examples.
Ken Pledger.
I've heard that some techniques for multiplying using trig tables
(no logarithms) were largely forgotten.
Keith Ramsay
I went looking a bit, and the best I've found so far is the following.
A functor F:Ring->Set is naturally equivalent to a functor
R->Hom(Spec R, S) if and only if it satisfies the following two
conditions:
(1) There exists a set of pairs (R_i,t_i) where R_i is a ring and
t_i is an element of F(R_i), with the property that for each field
K and element t of F(K), there exists an (R_i,t_i) in the family
and a homomorphism s:R_i->K where F(s)(t_i)=t.
That has a flavorful resemblance to the technical condition in the
Freyd adjoint functor theorem. It's no condition except for the
restriction to just a *set's* worth of rings. (If a proper class of
them were allowed, we could just take all rings.) (You can
substitute "small set" and "big set" for "set" and "proper
class" if you were raised on category theory done that way.)
(2) F is a "sheaf under the Zariski topology": given a ring R and
elements {l} of R such that the Spec R_l cover Spec R, and given
elements t_l of F(R_l) which are compatible in the sense that
t_l1 and t_l2 map to the same element of F(R_{l1*l2}), there exists
a unique t in F(R) which maps to each t_l under F(R)->F(R_l).
I'm quoting from memory here. This second condition seems a bit more
like cheating somehow. I've wondered whether it could be considered
as a special case of something more categorical. The condition that
the Spec R_l's cover Spec R can of course be reformulated in a ring-
theoretic way.
Eisenbud and Harris cite Demazure and Gabriel (Algebraic Groups)
as a place where the functor idea is developed in some detail. I
didn't spend much time looking at Demazure and Gabriel, but their
definition of scheme appeared to be essentially the same thing as
this. They explain that one of the key points of Grothendieck's
big program was the idea that one could more easily show that schemes
having given properties exist by defining this functor and then
showing, by some nice criterion for "having the scheme nature",
that they are associated with schemes. Apparently there was some
doubt among others that this would work, but then it turned out to
work better than expected.
I do wonder now whether during that one lecture, the lecturer didn't
just say something like "functor, satisfying some technical conditions
we won't worry about", because the above still doesn't sound like the
kind of definition that wouldn't have sounded too hairy to me back then.
Keith Ramsay
> As for the 19th century, I guess Chan-Ho Suh may have been thinking
> of something like:
>
> (4) Algebraic invariants. People do still use these, but they may fit
> Chan-Ho's desciption as "now considered not very important because they
> were superseded by better theories." Perhaps it would be better to say
> that they're no longer such a major research area as they were in the 19th
> century.
>
Actually I *was* thinking of certain algebraic invariants! When I was first
learning Galois Theory, I tried reading this book by Edwards(?). It
featured a historical "How Galois saw it" approach. The appendix had his
memoirs on the theory and the rest of the book went into detail on Galois'
paper. But to make a long story shorter, I eventually threw the book down
in bewilderment because it used a lot from the theory of resolvents,
something that was really big-time cutting edge stuff at the time. Some
resolvents, like the Lagrange resolvent, are still pretty important,
cropping up in a crucial role in computational Galois theory, however
because of the development of group theory, the importance of resolvent
theory as a whole has lessened.
There are other things (that I will try and think of) that crop up in the
kind of old math books I sometimes like to flip through. The authors will
typically make a big deal out of it, like it's this major area of research,
but when I ask an expert in that subject, I get a shrug and a remark like,
"Oh yeah, they used to do a lot of that, but nobody knows that stuff
anymore."
Actually, I just thought of another: determinant theory. There are whole
books on this stuff. With the advent of the computer though, *some* of it
is becoming considered important in a computational aspect.
>Actually, I just thought of another: determinant theory. There are whole
>books on this stuff. With the advent of the computer though, *some* of it
>is becoming considered important in a computational aspect.
Since I've only been reading dribs and drabs of this thread, I had
been assuming that you were merely trying to get some poster to
learn. But this line implies the opposite. Are you really
looking for people who are trying to use category theory to do
real stuff?
/BAH
Subtract a hundred and four for e-mail.
>I've heard that some techniques for multiplying using trig tables
>(no logarithms) were largely forgotten.
Given x, y in [0,1] consult a table to write as cosines. Then use
cos(a) cos(b) = (1/2) ( cos(a+b) + cos(a-b) )
Napier and I used to have speed competitions doing this. Then the
rapscallion concocted his logarithms and kept beating me. But enough
of these new-fangled things; the old ways are good enough for me!
dave
I'm not sure what you mean by "this line implies the opposite". Nor by
"real stuff".
>Napier and I used to have speed competitions doing this. Then the
>rapscallion concocted his logarithms and kept beating me. But enough
>of these new-fangled things; the old ways are good enough for me!
None of this "These bones gonna rise again" stuff for you, huh?
Lee Rudolph
I was tired. It was a bad choice of words. I started getting
the feeling that you were looking for instances where category
theory was getting used.
> ... Nor by "real stuff".
My apologies. I must have been really pooped. Real work; real
production; using the math as a tool to see if it's more useful
than other branches of math.
From the preface to _Discriminants, Resultants, and Multidimensional
Determinants_, by Gelfand, Kapranov, and Zelevinsky (which got quite
a glowing review in the Bulletin):
This book has expanded from our attempt to construct a general
theory of hypergeometric functions and can be regarded as a first
step toward its systematic exposition. However, this step turned
out to be so interesting and important, and the whole program so
overwhelming, that we decided to present it as a separate work.
Moreover, in the process of writing we discovered a beautiful
area which had been nearly forgotten so that our work can be
considered as a natural continuation of the classical developments
in algebra during the 19th century.
We found that Cayley and other mathematicians of the period
understood many of the concepts which today are commonly thought of
as modern and quite recent. Thus, in an 1848 note on the resultant,
Cayley in fact laid out the foundations of modern homological
algebra. We were happy to enter into spiritual contact with this
great mathematician.
...
After rediscovering hyperdeterminants in connection with
hypergeometric functions, we found that they too, had been
introduced by Cayley in the 1840s. Unfortunately, later on, the
study of determinants was largely abandoned in favor of another,
more straightforward definition.
Keith Ramsay
You could say that. I'm somewhat skeptical of this whole category
business, but willing to keep an open mind.
>
> > ... Nor by "real stuff".
>
> My apologies. I must have been really pooped. Real work; real
> production; using the math as a tool to see if it's more useful
> than other branches of math.
>
Then I guess I would like to see some good examples of "real stuff".
The reason I got involved in the thread: "category theory is bunk?"
is because I was/am very skeptical of the bold claims made by the
original poster, claiming i.e. category theory is one of the greatest
discoveries of the 20th century, and that it is more fundamental than
set theory. I've never met a category theorist who made these kinds
of claims, although I haven't met too many category theorists,
admittedly.
And I thought the example given might be a little weak, so I asked
questions which I thought would be illuminating. But the original
poster, Nobuo Saito, wasn't very cooperative in his answers (or
evasions).
I did learn some things from some emails pro-category theory folks
sent me, so I did get something out of the discussion.
Sorry about the late response, I've had some problems with my
newsreader, and been busy also.
> We were happy to enter into spiritual contact with this
> great mathematician. [Cayley]
My God! There goes the stereotype of the unfailingly rational and
unspiritual mathematician. A mathematical publication remarking that
some mathematicians held a seance! What next?
--
There is no spoon.
> The reason I got involved in the thread: "category theory is bunk?"
> is because I was/am very skeptical of the bold claims made by the
> original poster, claiming i.e. category theory is one of the greatest
> discoveries of the 20th century, and that it is more fundamental than
> set theory. I've never met a category theorist who made these kinds
> of claims, although I haven't met too many category theorists,
> admittedly.
The teacher who taught me about category theory saw it this way:
"Category theory was very popular back in the sixties. Today, it looks a bit
outdated (just like most of what was popular in the sixties.)"
To which he added: "Although some people today still can't get enough of
it."
To which a student replied: "also just like with the sixties..."
Regards,
Herman
A survey was done of the U.S. National Academy of Science members,
asking them whether they believed in the kind of God that people can
engage in meaningful dialogue with (to distinguish between that and
less traditional theological beliefs). The members have a lower rate
of such belief in general the general public in the U.S., but the
section having the highest rate of belief was mathematics. (Lowest
was biology.)
Keith Ramsay
There was a good article in Scientific American on this, in the last
year or two.
I'm not sure what the relevance of this is to the comment you are
replying to though. Believe in God does not imply non-rationality,
and being spiritual does not imply believe in God.
There was a good article in Scientific American on this, in the last
year or two.
But I'm not sure of the relevance of this to the comment you are
replying to. Belief, in the existence of a God we can communicate
with, neither implies one is irrational nor that one is spiritual.
Well, this just goes to show that one shouldn't say that a theory is
dead even if it appears buried well under 6 feet (or a century or
two). Of course, it remains to be seen if this resurgence of
determinant theory will continue well into the future.
Thanks for the reference.
I wrote about a survey concerning belief in God among NAS members.
In article <cf80a174.01101...@posting.google.com>,
cs...@math.ucla.edu (Chan-Ho Suh) writes:
|I'm not sure what the relevance of this is to the comment you are
|replying to though. Believe in God does not imply non-rationality,
|and being spiritual does not imply believe in God.
I don't think a rational theist fits the stereotype of "the
unfailingly rational and unspiritual mathematician" very well. The
existence of spiritual but nontheist mathematicians is just another
reason to distrust the stereotype.
I think the stereotype is a variation on an old stereotype of the
materialist scientist who believes only in what can be rationally
proven from concrete, numerically measurable data. The naive idea
that mathematicians are more apt to think that way seems to be based
on the impression that we'd be even more liable to trust just numbers
than other scientists would.
Gelfand et al. with their "spiritual communication" with Cayley
reflect the often underappreciated artistic side of mathematics.
People who look to mathematics for the threat to not-so-numerically-
measurable values are looking the wrong direction. They should be
looking at the way that economic interests are allowed cheaply to
dispose of goods that one isn't able to make much money off of, like
certain parts of nature. And actually mathematics is another one of
those goods that is a lot more worthwhile than its meagre ability to
earn people money would suggest.
Keith Ramsay
>"Nobuo Saito" <genki...@hotmail.com> wrote in message
>> Chan-Ho Suh wrote:
>>
>> >By essential, I suppose you mean that there is no way to avoid
using
>> >category theory??
>>
>> No.
>>
>> >Also, do you mean that more than just the language of category
theory
>> was
>> >used? i.e. some deep fundamental result from category theory.
>>
>> No.
>>
>
>Well then, why do you consider category theory to be the "one of the
>greatest discoveries of 20th century mathematics"? And why do you
consider
>it "more fundamental than set theory"? [quotes are from Nobuo
Saito's
>original posting]. Certainly (with the answers you just gave to my
>questions) it doesn't appear that the example you gave is very
convincing in
>supporting those claims. You say all that and then give as an
example
The importance of category theory lies in the fact
that it provides a very powerful, revolutionary view point.
The following are my _rough_ translation of a passage from
Grothendieck's long essay(Recoltes et Semailles).
--------------------------------------------------------
I heard about J.S.H. Whitehead's comment
about the snobbery of young students
who think a proposition is trivial because its proof is trivial.
This snobbery belongs not only to young students.
I witnessed famous mathematicians exercised this dayly.
I'm particularly sensitive about this,
because my best results, the most fruitful concepts and structures
I introduced, and basic properties I extracted from them
by patient works, are put under an adjective _trivial_.
--------------------------------------------------------
For people above the age of about 12, it comes pretty damn close!
|> and being spiritual does not imply believe in God.
That's true, I'd guess; whatever such a vague word as "spirituality" means.
My guess is that people who claim some spirituality are just religicos
without the courage of their convictions, just as creationists are really
flat-earthers without the courage of their convictions.
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
Science - believing what you see.
Religion - seeing what you believe.
------------------------------------------------------------------------------
Those with such strong anti-religious bias display an unprovable faith
of their own.
Well, I'm never too sure what those people who have this sort of
stereotype in their heads are thinking, but from experience, I've
noticed some don't seem particularly surprised at learning that
such-and-such mathematician goes to church on Sunday. So I would have
to disagree with your first sentence.
> I think the stereotype is a variation on an old stereotype of the
> materialist scientist who believes only in what can be rationally
> proven from concrete, numerically measurable data. The naive idea
> that mathematicians are more apt to think that way seems to be based
> on the impression that we'd be even more liable to trust just numbers
> than other scientists would.
>
This is tied to the idea that mathematicians only learn about numbers.
I've given up on explaining how untrue that is to people who ask me
what I do.
> Gelfand et al. with their "spiritual communication" with Cayley
> reflect the often underappreciated artistic side of mathematics.
>
Just out of curiousity, ever read the Art of Mathematics, by Jerry
King?
> People who look to mathematics for the threat to not-so-numerically-
> measurable values are looking the wrong direction. They should be
> looking at the way that economic interests are allowed cheaply to
> dispose of goods that one isn't able to make much money off of, like
> certain parts of nature. And actually mathematics is another one of
> those goods that is a lot more worthwhile than its meagre ability to
> earn people money would suggest.
>
Amen. :)
BTW, my post you quote above was not intended for public view. Damn
google messed up; the second one I posted was what I intended:
"But I'm not sure of the relevance of this to the comment you are
replying to. Belief, in the existence of a God we can communicate
with, neither implies one is irrational nor that one is spiritual."
This is not only grammatically better, but more coherent.
Why do you say that? Is there some evidence of the non-existence of
God that you are holding back from the rest of us? Interesting note:
Martin Gardner was a fideist (belief in God one could communicate and
pray to) cf Whys of a Philosophical Scrivener (1983). I don't find
his arguments convincing, but I thought this was an interesting
example of someone over 12 years old.
> |> and being spiritual does not imply believe in God.
>
> That's true, I'd guess; whatever such a vague word as "spirituality" means.
>
Yes it is very vague. Just like spirituality itself :)
> Science - believing what you see.
> Religion - seeing what you believe.
> ------------------------------------------------------------------------------
Science - believing everything you can't see is the same as what you
can see
Religion - believing what you believe even if you see differently
Certainly it'd be dumb to assume that mathematicians are all alike
this way, but there are plenty of people who hold stereotypes very
strongly but who admit that there are exceptions. That doesn't mean
that the exceptions fit the stereotype in their minds.
It may be different elsewhere, but in the U.S. it tends
to be assumed that if you go to church, you're doing it for spiritual
reasons.
|Just out of curiousity, ever read the Art of Mathematics, by Jerry
|King?
No.
Keith Ramsay
In my last posting on this thread, I was tempted to write that the
stereotypical "rational but not spiritual" mathematician was basically
a more rational version of Bill Taylor. :-) I think this is sort of
the attitude some people associate with scientists.
This is getting a bit off topic, but I suppose rationality has some
connection with mathematics, doesn't it? One of the disturbing
features of the world is the existence of so many issues on which
multitudes of people are firmly convinced of the answer-- multitudes
convinced in *each* direction, without there being any obvious
difference in the kind of evidence available to each. I suppose that
this demonstrates a certain lacuna in their rationality. Rational
people presented with the same evidence should, one thinks, arrive at
much closer estimates of the likelihood of things.
But I'd say that the level of rationality and level of freedom from
bias required to stop this from happening is rather high by ordinary
standards, and I'm afraid I don't know anybody who I trust to be as
rational and unbiased as that. It's sort of disappointing, actually.
Keith Ramsay
Chan-Ho Suh wrote:
Well there's a difference between a mathematician
and a logician. The second one doesn't know
anything about logic.
And they are right.
> They are wrong.
No, they are right.
> The category theory is one of the greatest discoveries
> of 20th century mathematics.
No, its not.
> It's more revolutionary and fundamental than the set theory.
No, its not.
> It is so fundamental that it took thousands years(or more?)
> for mankind to recognize it.
Like... tv?
> Even Bourbaki didn't recognize its importance at first
Good for them.
> (later they recognized their mistake).
When someone is getting old... you know...
> Let me give you one(but very important) example.
> Grothendieck used category theory in essential way
> in his revolutionary works(schemes, topos, etale cohomology, etc.)
> which led to the proof of Weil conjecture and Wiles' proof of FLT.
Very important? Is this your very important example? What about all maths?
Or you are just a simple provocateur?
Let's not cross Bill off the spiritual list yet. I'm not convinced that
he is not harboring some deep monistic faith in the universe, as many
avowed atheists tend to do.
> This is getting a bit off topic, but I suppose rationality has some
> connection with mathematics, doesn't it?
Um...ok, if you say so. But just curious: what connection are you
thinking of (vaguely or otherwise)?
One of the disturbing
> features of the world is the existence of so many issues on which
> multitudes of people are firmly convinced of the answer-- multitudes
> convinced in *each* direction, without there being any obvious
> difference in the kind of evidence available to each. I suppose that
> this demonstrates a certain lacuna in their rationality. Rational
> people presented with the same evidence should, one thinks, arrive at
> much closer estimates of the likelihood of things.
>
Most people are not willing to admit they don't know; thus they pick
some viewpoint in an irrational way, and then back it up with either
science, religion, or some mixture of both. I'm reminded of a quote
from Socrates, "The only truth I've discovered is that I know nothing."
(Or something like that.)
|> > |> Believe in God does not imply non-rationality,
|> > For people above the age of about 12, it comes pretty damn close!
|> Is there some evidence of the non-existence of
|> God that you are holding back from the rest of us?
Oh puh-LEEZ!
Surely you must know that this a silly fallacy. Fair debate does not require
one to prove NON-existence of anything, or even adduce evidence for it.
All the onus is on the reverse - the believers. And all their "evidence"
is so incredibly corrupted by subjectivity and weasel-wordedness that
it takes a pathological desire for self-deception to even countenance
two sentences strung together on the topic.
|> Martin Gardner ... ... but I thought this was an interesting
|> example of someone over 12 years old.
Yes, even the great can fall low at times. We've seen that over and over
again. Newton was an alchemist. Pythagoras thought eating peas was evil.
Hugh Dowding, winner of the Battle of Britain, thought there were fairies
at the bottom of his garden!
If Chan-Ho has some non-evidential beliefs, no doubt he will share them with us.
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
The "faith" meme discourages the exercise of the sort of critical
judgment that might decide that the idea of faith was a dangerous one.
------------------------------------------------------------------------------
|> Those with such strong anti-religious bias display an unprovable faith
|> of their own.
What utter rubbish!
It's accumulated frustration at the idiocies of others. Of course we should
all have learned by now that lots, most, people are going to be fairly
idiotic about a lot of things, and be resigned to it. But still every
once in a while the frustration boils over into strong words.
Religious believers are idiots, about one thing at least, (probably a lot more).
To say so in strong words does not imply a faith of one's own, but rather,
a deliberate setting aside of belief in favour of pragmatic common sense.
I firmly believe I have no beliefs whatsoever, :) but I'm open to debate on it.
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
Religious belief is the root of all evil.
------------------------------------------------------------------------------
|> stereotypical "rational but not spiritual" mathematician was basically
|> a more rational version of Bill Taylor. :-)
Heheh! I was going to say just the same about Keith Ramsay!
|> I think this is sort of
|> the attitude some people associate with scientists.
Yes, but even scientists can be loony in their private life, even their
private intellectual life. (Though they know a lot better than to adduce
even the slightest hint of it in their professional work!) Mind you, I
think the religious scientist is a lot rarer now than formerly, (e.g. in
Victorian times). I suspect that unbelief has become far more widespread
in scientists than it has in the population at large. (I speak here of the
populations of sensible countries like Australasia, North-West Europe, or
Canada; not, alas, the USA, where religious belief has unnaccountably
kept a closer grip on the body politic than in those other countries.)
But yes, scientists can even now be pretty loony, especially outside
their topic of expertise. Mathematicians are even more unreliable,
as many people here have observed. Presumably their more distant contact
with physical reality, and their penchant for contemplating the abstract,
leaves them vulnerable to a mystical turn of personality.
One notable exception is statisticians, who, I seem to recall reading in
a report, were the professional group LEAST likely to accept false evidence
or faulty reasoning based on evidence. Mathematicians rated badly on this,
but statisticians rated even higher than scientists proper, (the biological
types rated better than the physics end as well).
However, not being a statistician myself, I can take no comfort there.
I'm very sad that my profession has such a poor record of crackpottery.
|> multitudes convinced in *each* direction, without there being any obvious
|> difference in the kind of evidence available to each.
Yes, quite astonishing. And it's highly amusing to see how they bluster
and wriggle when this fact is pointed out! MOST amusing.
|> I suppose that this demonstrates a certain lacuna in their rationality.
Hah! Very generous! For most people it is not a matter of a lacuna in
their rationality, but an occasional rock in their ocean of irrationality!
|> I'm afraid I don't know anybody who I trust to be as
|> rational and unbiased as that. It's sort of disappointing, actually.
And how!! You sure said a mouthful there!
Or as Winston Churchill put it, (something like...)
"A ten-minute chat with the average man in the street is enough to dispell
any illusions about democracy being a sensible form of government."
It's just a pity that every other form has proved even worse...
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
USA: The only country in the western world where the right to
superstitious ignorance is constitutionally enshrined.
------------------------------------------------------------------------------
How convenient for you. This rule of "fair debate" you quote is not
always applicable. And not in this situation. For example, certain
physicists believe in a universe that has no 'beginning' or 'end'. I
would certainly ask them for some supporting evidence of this
non-existence of a beginning of the universe.
And all their "evidence"
> is so incredibly corrupted by subjectivity and weasel-wordedness that
> it takes a pathological desire for self-deception to even countenance
> two sentences strung together on the topic.
>
Evidence? I wasn't aware that in order to be a theist one must have
accumulated some sort of evidence for the existence of god. After all,
according to you one doesn't need to gather evidence to support one's
atheistic stance.
> |> Martin Gardner ... ... but I thought this was an interesting
> |> example of someone over 12 years old.
>
> Yes, even the great can fall low at times. We've seen that over and over
> again. Newton was an alchemist. Pythagoras thought eating peas was evil.
> Hugh Dowding, winner of the Battle of Britain, thought there were fairies
> at the bottom of his garden!
>
And Bill Taylor frequently goes on an irrational atheistic rampage. So?
> If Chan-Ho has some non-evidential beliefs, no doubt he will share them with us.
>
Oh, I don't know about that. I like to hold my cards close to the vest.
Hmmm...*most* amusing indeed! Especially given that you seem to have
missed Keith's point completely.
Oh, maybe you _did_ get it! My apologies....we are *all* having a
little fun here I suppose.
> Yes, but even scientists can be loony in their private life, even their
> private intellectual life.
It is to the credit of scientists, surely, that they rarely
engage in ranting and raving about the stupidity of religion. We have
alt.atheism for that.
>Surely you must know that this a silly fallacy. Fair debate does not require
>one to prove NON-existence of anything, or even adduce evidence for it.
>All the onus is on the reverse - the believers.
Why should that be the case? It seems to me that people can adopt
default positions however they like. Then the onus of proof is on
whoever is trying to convince someone to abandon his default
position.
--
Daryl McCullough
CoGenTex, Inc.
Ithaca, NY
|One notable exception is statisticians, who, I seem to recall reading in
|a report, were the professional group LEAST likely to accept false evidence
|or faulty reasoning based on evidence. Mathematicians rated badly on this,
|but statisticians rated even higher than scientists proper, (the biological
|types rated better than the physics end as well).
can you tell us more about this report? who wrote it? what
techniques did they use in reaching their conclusions? how did they
decide what constituted "accepting false evidence or faulty reasoning
based on evidence"?
I could be wrong on this, but I bet the statisticians wrote it, as is
usually the case. Of course, this is not a conflict of interest, since
the statisticians are least likely to accept faulty reasoning ;P
|> > Surely you must know that this a silly fallacy. Fair debate does not require
|> > one to prove NON-existence of anything, or even adduce evidence for it.
|> > All the onus is on the reverse - the believers.
|> How convenient for you.
Yes. :-)
|> This rule of "fair debate" you quote is not
|> always applicable. And not in this situation.
Yes it is.
|> For example, certain
|> physicists believe in a universe that has no 'beginning' or 'end'.
They don't "believe" in it. They just put it forward as a possibility,
suggested by the mathematics of the theory. It's not a "belief" in
the religious sense - something held onto for emotional commitment,
in the face of any evidence to the contrary. It's just an in-mind
possibility to be tested by evidence in due course.
Why should I have to be saying all this? Surely you should know this already!
|> Evidence? I wasn't aware that in order to be a theist one must have
|> accumulated some sort of evidence for the existence of god.
No indeed! My point in a nutshell! But if, as a theist, one hopes to
have any influence on others in a reasonable debate, one must adduce some.
|> After all, according to you one doesn't need to gather evidence
|> to support one's atheistic stance.
"Stance". That's an amusing word. At least you didn't say atheistic beliefs,
so there's some hope you've gained some insight here. But anyway, to answer:
Recall the positives and negatives again! The ones so convenient for me
and which you loathe so much. ;-) Or to put the same thing another way...
"Absence of evidence is evidence of absence."
==========================================
|>>If Chan-Ho has some non-evidential beliefs,no doubt he will share them with us
|> Oh, I don't know about that. I like to hold my cards close to the vest.
Chicken!
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
I'm an atheist. It's a simple faith, but it comforts me.
------------------------------------------------------------------------------
Yeah right! <sarcasm>. This is why a scientist will often spend years
defending his/her "in-mind possibility" in the face of more and more
evidence to the contrary. My suggestion is that these physicists
"believe" in their 'proposed' possibilities and theories for the same
reasons as theists (or other religious types) "believe" in theirs: for
emotional reasons.
Albert Einstein "believed" in his theories and rejected quantum
mechanics from the *start* because of his beliefs, even *before* it
became evident that there was something not quite right with the
theory. Need I remind you of his famous quote, "God does not play dice
with the universe."? Surely you should know this already! He is not an
exception. Many scientists today, like Stephen Hawking, accept and
reject proposed theories for similar, emotional reasons. You are being
incredibly naive and ignorant of the history of science if you buy into
what you wrote above.
> Why should I have to be saying all this? Surely you should know this already!
>
> |> Evidence? I wasn't aware that in order to be a theist one must have
> |> accumulated some sort of evidence for the existence of god.
>
> No indeed! My point in a nutshell! But if, as a theist, one hopes to
> have any influence on others in a reasonable debate, one must adduce some.
>
Well, alright. But in a debate such as this, nobody ever becomes
convinced of anything they weren't convinced of before. Perhaps this is
because there is no evidence for any viewpoints. But more likely it is
for emotional reasons.
> |> After all, according to you one doesn't need to gather evidence
> |> to support one's atheistic stance.
>
> "Stance". That's an amusing word. At least you didn't say atheistic beliefs,
> so there's some hope you've gained some insight here. But anyway, to answer:
>
> Recall the positives and negatives again! The ones so convenient for me
> and which you loathe so much. ;-) Or to put the same thing another way...
>
> "Absence of evidence is evidence of absence."
> ==========================================
>
Where do you come up with these ultimate 'rules' (or shall I say
'beliefs)?
> |>>If Chan-Ho has some non-evidential beliefs,no doubt he will share them with us
> |> Oh, I don't know about that. I like to hold my cards close to the vest.
>
> Chicken!
I'm just being a tease I suppose. But I'll take your one word riposte
as equivalent to, "Please Chan-Ho! I want to know all about your
beliefs about life, the universe, and everything! Please!" So let me
begin:
Well, actually, let me begin after my below remark, since it sums up so
much.
> ------------------------------------------------------------------------------
> I'm an atheist. It's a simple faith, but it comforts me.
> ------------------------------------------------------------------------------'
Well, at least you're honest...about your faith. Some people find being
a theist comforting. In the end, people believe what comforts them.
But here's a summary of what comforted me in various times of my life:
1979 - 1992 My life as a fundamentalist Christian
Believed in the whole water to wine bit. This sums up this part of my
life. Also, read most of the Bible repeatedly, giving me ample
ammunition to use in the next phase.
1992 - 1997 My life as a fundmentalist Atheist
I felt good being in the company of luminaries such as Arthur C. Clarke,
James Randi, Bertrand Russell (and unknown to me at the time, Bill
Taylor). Read the standard Atheist texts. Argued with many
fundamentalist Christians, using my ability to recall obscure parts of
the (Christian) Bible to my advantage.
1997 - 1999 My life as an Agnostic
I had realized that I was not as strong in my Atheistic faith as some.
I even had gotten to the point where arguing with Christians was not
fun. Developed some sympathy for all religious folk. Met Christians
that were more rational than some Atheists I knew. Had some doubts
about my Atheism. Even took some science classes to strengthen my
faith.
1999 - 2000 My brief fling with Monism
After reading standard Atheistic texts (such as The Feynman Lectures of
Physics, I realized many Atheists were really Monists. Monism can be
summed up as a belief in the unity of everything, cf the Force in
_Star_Wars_. Embraced Monism fully. Read up on Grand Unified
Theories.
2000 - current My founding of a new religion: Mathematical Zen
After dissatisfaction with the state of GUTs and theoretical physics in
general, I quit reading physics texts and devoted myself wholely to
mathematics. Read some Zen texts. Realized Zen could be summed up as:
don't ask stupid questions, just meditate (or "sit", as Zen masters like
to say). Realized I already performed a variety of meditation, called
theorem-proving. Founded my own personal religion, which I branded
"Mathematical Zen" in order to answer others' stupid questions about my
own religious beliefs with the answer "I am a Mathematical Zennist."
Even came up with a highly original diatribe to go into when pressed for
details (see archived sci.math post:
http://groups.google.com/groups?q=chan-ho+diatribe&rnum=1&selm=9oitp2%241cr%241%40persian.noc.ucla.edu)
>
> Recall the positives and negatives again! The ones so convenient
> for me and which you loathe so much. ;-) Or to put the same thing
> another way...
>
> "Absence of evidence is evidence of absence."
> ==========================================
>
I'm not sure if I read this comment correctly. Are you suggesting
that the onus of evidence somehow belongs to the proponent of an
existential claim, rather than the proponent of a universal claim?
(I interpret "positive" to mean "existential" and "negative" to mean
"negation of an existential", i.e., a universal.)
I have seen this opinion before but I have never understood the
attractiveness of this position. Why should the existential be
regarded as prima facie more dubious than the universal? Indeed, the
universal seems to be a much stronger statement (prima facie, of
course!), just as we view a conjunction as prima facie stronger than a
disjunction. Why is it so obvious that the universal should be
granted until refuted?
(Possible argument in favor of this position: Suppose that, for any
true existential statement, one will eventually discover evidence
which verifies the existential. In this case, a reliable means of
arriving at the tru