I would say that Einstein's consideration of this enigma (quoted in the previous message) was far more insightful, so perhaps all those Wigner references don't do justice to Einstein (to say nothing of all those other inquiring minds through the ages). On the other hand, Einstein gets quite a few undeserved citations for Kant's epigram about comprehensibility, because he happenned to quote it in his famous 1936 essay "Physics and Reality" :

Of course, Kant was famously discredited among physicists by his pronouncements on "final categories" and necessary modes of thought, among which he unluckily listed the framework of Euclidean space. With the advent of non-Euclidean geometry Kant fell into disrepute among physicists and mathematicians, so it was perhaps inevitable that they would find another source for that epigram, which is simply too good to discard just because you've decided the author was an idiot. . . (As usual, Einstein knew better) . . **]

Applied here (re: the usefulness of math to a scientific model of the world we live in), I must say it really should not be so surprising. Just notice what you require of a scientific model: repeatable & consistent (and preferably predictable by consequence) which are precisely the properties that math brings in (since Plato advocated it as a pre-requisite in his academy).

On the other hand, math plays practically no role in the world model of the old Chinese, where the cause-effect syndrome of the West has little grip (read the forword of Carl Gustav Jung to the Richard Wilhelm translation of the I-Ching. This despite the "fact" that the I-Ching is the oldest binary coded State Machine, of 64 = 2^6 states and all 64^64 state-transformations (just one way of looking at it ;-)

So I guess, math is so applicable, because it fits the properties we are looking for in a desired worldmodel (predictable;-). That is precisely why the upcoming "alternative" approaches are so incompatible with science: they want to make other connections than only with nails... For instance the established, and effective, acupuncture method of healing (restoring balance) shows that other ways of looking at our world & our body are possible. --- The whole controversy is a matter of prejudice and programming and selective memory, if you ask me...

Moreover, did you know that " Newton wrote more on alchemy than on any other subject " ? . . I have this on authority of a club of historians who lamented, last year, the intended public auction of the unique "Bibliotheca Philosophica Hermetica" (Amsterdam). Re : item [e] on my homepage. . . (BTW : our Government chipped in to prevent such dispersion of a unique collection).

In a more general tone: My idea is that the continuous and the discrete
(in my profession, integrated circuit design: analog & digital)
are both necessary for a more complete and useful model of our world.
More specifically: applied alternatively in successive levels of a hierarchy.

- Like clouds, which at one level of modeling block sunrays or don't (discrete model). But the edges are vague, and have a continuous density variation (continous model) -- which at a still more detailed level turn out to consist of countable small droplets (discrete). They each in turn, at finer analysis, are a continuous ocean of water with varying temperature distribution and consistency througout the droplet, etcetera, etcetera.

Notice the detailed combinational structure (=discrete math, like Boolean algebra, or integer arithmetic) on the one hand, and statistics (of large numbers, continuous math, calculus, infinitisimals &c) on the other hand, usually go together:
1. Pascal/Fermat (1640): developed the theories of number and of probability.
2. George Boole: in his book (1854) "The Laws of Thought" = (arithmetization of Aristotels' logic;-)
you find in the first half his binary logic - introduced as idempotent arithmetic x²=x . . (x=0 or 1);
- thus not as set theory, but as property calculus - and in the second half you find statistics.
3. Claude Shannon (1948): combined continuous (entropy) principles with discrete (binary bits) data transmission over a channel (Information theory).
-- NB

 Subject:   Re: How sure are theorems? (In reference to Pertti Lounesto)
sci.math:   22 Apr 2002
    From:   Nico Benschop
     Org:   Digital Research : Finite Associative Networks
---------------------
Pertti Lounesto wrote:
>
> Denis Feldmann wrote:
>
> > We all know the (philosophical) problems arising from mistakes
> > found in published proofs (as pointed in the innumerable
> > "counterexamples" threads by Pertti Lounesto).
>
> According to Robin Chapman, all theorems hold.  Such
> a view is a simplification of the cognitive environment
> mathematicians live in.  Mathematics is a result of the
> interaction between environment and human cognition.
> A theorem, a cognitive chart of the environment, is only
> accurate till more accurate atlases will be drawn.  Thus,
> all theorems, which have an effect on the progress of
> mathematics, do not hold.

The clue here is the meaning of "accurate" as property of a theorem
that describes our (human) environment. Now what relevance does this
have, for instance, regarding the integers and their properties -
or more in general: discrete finite (static and dynamic) systems?

My impression is that (finite) integers accurately describe important
aspects of our environment, and that arithmetic allows reliable
results to be computed, or impossibilities - even for 'all' integers -
to be reliably stated, and proven. What is the possible 'inaccuracy'
here that causes, in the short or long term, say the Pythagoras theorem
coupling rectangles in 'flat space' to integer squares (exponent n=2)-
or Fermats inequality (for n>2), to become invalid (=not to hold;-) ?

The realization that we do not live in flat but in curved space,
- or: not on a flat Earth but on a sphere, does of course change the
relevant laws, but only due do a difference in axioma's ( _that_ is
where the 'fitting' to 'reality' takes place, NOT in the theorems;-)

Just curious,

-- NB - http://piazza.iae.nl/users/benschop/math-use.htm

PS: Math = the excitement of "Chipping away at Eternity" (H.Lenstra)

Newsgrp: sci.math - 19 Apr 2002
Subject: Re: About the definition of algebraically closed
   From: Nico Benschop
-----------------------
"W. Dale Hall" wrote:
>
> James Hunter wrote:
>
> > Robin Chapman wrote:
> >
> >>"Nico Benschop"  wrote:
> >>
> >>> Well, such formal definition makes sense for certain abstract
> >>> aspects of a polynomial, I guess. OTH, without the interpretation
> >>> as a function f(x) in a field (+)(.), even f(x)=0, to define its
> >>> roots, does not make sense.   -- NB
> >>>
> >> A polynomial say  x^2 + x + 1  is not a function.
> >> To say it is is to make a category mistake. Sometimes
> >> it can define a function; the assignment x |-> x^2 + x + 1
> >> may (or may not) define a function between a given domain
> >> and codomain, but that function will depend on the  -- RC
> >
> >   To say that x^2 +x+1 is not implicity a function
> >   is merely to say that mathemawanks are more
> >   wanked out and definitionally-challenged
> >   than anybody previously thought.              -- JH
>
> Yes, and to insist that it is merely a function is to expose
> one's inability to do anything other than wank.   -- Dale

To say a polynomial expression is merely an abstract object, and not
a function, is the other extreme. Interesting, this lack of balance.

1. First came 'polynomial' as function.

2. Later, when abstract algebra developed, this word got overloaded -
   or rather: highjacked - to serve other, more abstract, purposes.

3. Then came Chapman, and those who think that abstract is good,
   and more abstract is better [1], going to the extreme of ignoring
   the value of the origin (here: functions, whether in the form
   of a polynomial computation rule - or as mapping between sets)
   - signified by considering the original meaning of 'polynomial'
   as "a juvenile hangover from secondary school". Thus not only
   insulting students, but losing touch with the basis of his trade,
   which is detrimental in many ways.

BTW:
 Pushing down your environment/origin,
 does the opposite of lifting yourself.

-- NB - http://piazza.iae.nl/users/benschop/quotes.htm  [1]

Prediction: any reply from the M_side (math_side) will probably stress
the importance of exact definitions, ignoring the value of context
(the same word being used in different contexts), and of politeness
in discussions, e.g. in a medium such as this NG.


Newsgrp: sci.math - 19 Apr 2002
Subject: Re: About the definition of algebraically closed
   From: Nico Benschop
-------------
Pertti Lounesto wrote:
>
> Arturo Magidin wrote:
>
> >>Does that mean that if we extend the field F_2 = Z_2 by
> >>the root of the polynomial x^2+x+1 = 0, we get the field F_{2^2}?
> >
> > Yes. F_2[x]/(x^2+x+1) is isomorphic to the (unique) field of four
> > elements. The four elements are the classes of 0, 1, x, and x+1,
> > with the relation x^2=x+1.
> >
>
> And going further, F_2[x]/ is isomorphic to F_{2^3}, the 8
> classes being represented by 0, 1,x,x^2, 1+x,x^2+1,x^2+x, x^2+x+1 ?

... which list, I guess, again is equivalent to, or represented by,
   the discussed compact notation of 'polynomials' (in that order):

    000,  001, 010, 100,   011, 101, 110,   111.    ?

Remarkble how one can disguise something as simple as this!
Do the composition rules for (+) and (.) produce no surprise either,
or does an anomaly save the concept ? (re: "Maths is the art
          of disguising something simple as something difficult")

A sharp observer like Carl Gustav Jung never understood mathematics,
   due to the abundant equivalencies in notation of the same thing...
Why not call a rose a rose,
            a bigot a bigot,
       and bygones be bygones.

-- NB - http://piazza.iae.nl/users/benschop/quotes.htm
     (on reaching, by increasing abstractism, the escape velocity -
      never to return to common sense, and how to be proud of that;-

Subject:  Re: About the definition of algebraically closed  (sci.math - 19apr02)
   From:  Nico Benschop
    Org:  Digital Research : Finite Associative Networks
-------------------------
Pertti Lounesto wrote:
>
> mareg@mimosa.csv.warwick.ac.uk wrote:
>
> >> What is the advantage of making a difference between
> >> 1) polynomials, which are elements P(x) of R[x], and
> >> 2) polynomial functions P : R |--> R?
> >
> >This thread is all about one such advantage - namely that you are not
> >lured into falsely believing that Z_2 is an algebraically closed field.
> >
>
> That is only a disadvantage.  It is important to be lured
> to falsely believing something in a new situation.  That way
> one can correct ones misconceptions and learn new things.
>
> In a new situtation, a new    cognitive    environment,  it is
> impossible for an individual to draw an accurate    cognitive
> chart of the new   cognitive    landscape.  The first chart is
> only a preliminary sketch, later to be complemented to a whole
> atlas.  The completion takes place, if the individual is lured
> to use false charts,
                      \scratches his/her head,
                       \starts thinking for him/herself, and..
>                        and begins to draw the atlas himself.  [*]

Now *there* is wisdom for you, indeed!

"Cognito ergo sum" - or was it: "Cogito, ergo sum" ?
             and/or conversely: "Sum, ergo cogito" ?
               or in fact both:   "Cogito = Sum" ?   -- R.Descartes
                                               (Rene of the Charts;-)

Re[*]: and regarding maps & charts, my favourite quote:

"Mathematics is not a careful march down a well-cleared highway,
but a journey into a strange wilderness where explorers often get
lost. - Rigour is a sure sign to the historian that the maps have
been made, and the real explorers have gone elsewhere." (W.S.Anglin)

-- NB - http://piazza.iae.nl/users/benschop/search.htm


 Subj:   Mathematical Continuum and Discrete Models of Space
 From:   Nico Benschop
23dec99: sci.physics, sci.math
-------
"Matthew T. Brenneman" wrote:
>
> Nico Benschop wrote:
> >
> > There's nothing physically imperative about mathematical 'reals',
> > such as irrationals. From a practical point of view:
> >   there's hardly anything more unreal than the 'reals'.
> >  (although the related concept of 'limit' is a useful abstraction;-)
> >
> > Reals are just asymptotically & algebraically defined, and linking
> > this to practice: dividing matter is assumed to be possible
> > 'without limit', hence: there is no smallest particle (cq space).
> > That absurdity is the price for some useful asymptotic properties
> > at another (higher) level.
> >
> > My guess about your conceptual 'problem':
> >       could it be just a matter of thinking frame & programming ?-)
> > In other words:
> >       the problem of the single focus (or: single level reasoning).
> > --NFB
>
> I guess I'm not convinced that the reals are not supposed to be
> physically "real". Mathematicians would claim that the circumference
> of a circle with radius 1/2 is pi. I take that to mean that this is
> the TRUE value of the circumference, i.e. if I wrapped a string
> around the perimeter, then put that length onto the number line with
> one end at the origin, it's endpoint should be exactly on the point
> on the number line whose length is pi.
> If I'm missing something at this stage, let me know,

Well, I think what you are missing is that any TRUTH has a context.
In this case, pi has two contexts: the real world, and the math world.
The latter is an abstraction of the former. A math 'circle' has only
one parameter: its radius (apart from its position in some given plane)
while a 'real_world' circle does not exist in such capacity, as other
responders pointed out, because quanta & inaccuracies of objects and
measuring equipment spoil that simple and ideal abstract model.
So a real_world 'circle' produces NO TRUE value of pi, simply because
it has NO TRUE diameter (more easily measured than the radius).

In a more general tone: My idea is that the continuous and the discrete
(in my profession, integrated circuit design: analog & digital) are both
necessary for a more complete and useful model of our world. And more
specifically: applied alternatively in successive levels of a hierarchy.

-- Like clouds, which at one level of modeling block sunrays or don't,
(discrete model). But the edges are vague, and have a continuous density
variation (continous model) -- which at a still more detailed level turn
out to consist of countable small droplets (discrete). They each in
turn, at finer analysis, are a continuous ocean of water with varying
temperature distribution and consistency througout the droplet, &c &c.

Notice that detailed combinational structure (=discrete math, like
Boolean algebra, or integer arithmetic) on the one hand, and statistics
(of large numbers, continuous math, calculus, infinitisimals &c) on the
other hand, usually go together: Pascal / Fermat developed the theories
of number *and* of probability. And in Boole's book (1854: "The Laws of
Thought" = arithmetization of Aristotels' logic;-) you will in the first
half find detailed binary logic (introduced as idempotent arithmetic!)
and in the second half you find statistics...

> because it's my understanding that when
> we talk about say the hypotenuse of an isosceles triangle with legs
> of unit length, we're saying its ACTUAL TRUE LENGTH, if we put it
> down on the number line would lie on an irrational number (namely
> sqrt(2)). What we measure of course is an approximation, but the
> ONLY thing that causes me to obtain a rational number for my measured
> quantity is not the object, but my measuring device.

No, BOTH the object *and* the measuring device are material,
    hence finite, that is: discrete & rational.

> It seems to me that a physicist who believed space was discrete would
> say that the TRUE circumference of said circle was not pi.

I guess rather that, if he is honest, he would say there is NO TRUE
circumference of any given material circle, because there *is* no true
circle as a "matter of fact". Many such circles of "fixed & equal"
diameter=1 will, upon measuring them, have rational circumferences that
average-out (statistics: going for the  continuous;-) to a value that
approaches the math pi, which he might call some kind of "true" value
for the circumference of this class of circles with diameter supposedly
to be 1 ;-)     To keep it simple: even diameter 1 cannot be measured
exactly, and is an aimed-for value which might average to that value
for the whole circular class...! You don't need pi for that insight.

> It would be close for a finite number of digits, but the true value
> would be some rational number. So if I had that perfect string,
> wrapped it perfectly around this circle, and laid it out as before
> on the number line, its end would lie exactly on an rational number.
> The origin of what causes the measured quantity to be a rational
> number here is the quantity itself. In the end, it seems to me that
> the mathematicians would say the circumference of pi was the true
> (real) length, and the physicist would be saying that the
> circumference of pi was the not the real length. -- Matt.

The word 'length' is defined differently in these cases: the math
length of pi is the (irrational) quotient of circumference/diameter,
otherwise expressible as the limit of a converging series.

The physical 'length' is measured in arbitrarily chosen units, which
have a finite precision, both the diameter and the circumference, who's
quotient cannot but yield a range of rational numbers over any sequence
of repeated measurements on material circles. A hypothesized test for
the accuracy of these many measurements could be to agree on the
math pi (or pie?-) as central value, with certain distribution.
--
Ciao, Nico Benschop  -- http://www.iae.nl/users/benschop/filosofy.htm
                        http://www.iae.nl/users/benschop/math-use.htm

Subj:  Re:  Where do you Matematicians work?       (sci.math - 23mar00)
From:  Nico Benschop
--------
Tralfaz wrote:
>
> Kent wrote in message <8b5gvn$p5r$1@slb6.atl.mindspring.net>...
> >
> >Matti OVERMARK  wrote:
> >> Hello Group!
> >>
> >> I have already presented one question, and here are some to
> >> follow, but first a short explanation. I do have a degree in
> >> engineering but have recently been thinking of pursuing a
> >> career as mathematician. But I need to know more of the following:
> >>
> >> 1. Typical duties for a master level mathematician.
> >
> >        1. Flip the burgers.
> >        2. Mop the floor.
> >        3. Clean the tables.
> >        4. "Would you like fries with that?"
>
> You forgot the "Sir" after "Would you like fries with that?"
>
> Actually, I got my degree in math, but realizing the uselessness of
> my degree in the real world, I also got a minor in computer science
> at the same time. ...[*]
> So now I teach middle school with my math degree and I never get to
> use my post-high school math knowledge.  But my diploma looks cool.
> -Tralfaz

Re[*]: That's the spirit: I always considered doing purely math as
       standing on one leg -- walking & getting ahead is difficult
on one leg. So for balance, combine it with something else, like
engineering, science, psychology if you like (;-) but preferrably
something practical: a nice source for new math-problems/ideas;-)
This tends to prevent suffering from the single-focus syndrome...
--
Ciao, Nico Benschop --- http://www.iae.nl/users/benschop/math-use.htm

Subj:  Re: Falsified theorems, impetus of mathematics  (sci.math - 21aug00)
From:  Nico Benschop
------------
Stephen Montgomery-Smith wrote:
>
> A friend of mine once remarked that theorems should
> come with an expiry date, just like supermarket food.
>
> This made me think that maybe I could invent a
> new math system - a kind of relativistic quantum
> logic, that contained a time dependence.
>
> When a theorem is proved, the effects of the theorem
> would propagate through logic space at finite speed.   ..[*]
> In this way one could have a kind of uncertainty
> principle - that two theorems that contradicted
> each other could co-exist as long as they were only
> present for a very short amount of time.
> -- Stephen Montgomery-Smith

re[*]: Indeed; similarly, I think the amount of _memory_ involved
  in a proof might have a critical threshold (dependent on our
own means/brains/computers) beyond which no such 'proof' would be
testable: possibly 'true', and 'workable/use_able' for APP, but... ;-(

As it were: the 'physics' of math-proof & 'truth', just as there
is a physics of computation (e.g: # basic operations per Joule,
including the wiring needed for transport of data back_and_forth,
testing orderings & equivalences,  &c)
--
Ciao, Nico Benschop -- http://piazza.iae.nl/users/benschop/math-use.htm

 Subj:  Re: I am disappointed in God         (sci.math - 23aug00)
 From:  Nico Benschop
-----------
Pertti Lounesto wrote:
>
> dannyboy@here.com wrote:
>
> > On Mon, 21 Aug 2000 19:19:07 GMT, "Mike N. Christoff"
> >  wrote:
> >
> > >In my experience, having an ego is not neccessarily a bad thing,
> > >as long as you don't put down others w/o provocation to boost
> > >yourself and you respect and credit the good works of others.
> >
> > Look at his reply.  Save your breath to cool your porridge.
>
> For convenience, I copied my reply below.  I understand that
> you insinuate that I did not "respect and credit the good work
> of others" after Cartan's detection of triality in 1925.  Do you
> respect my "good work" on triality?  Or, did you post before
> understanding my "work" and its relation to other "work"?
> How does my "work" relate to other "work" on triality, after
> Cartan?  To give substance to your posting, could you list the
> most important "good work" on triality, after Cartan 1925?
> Explain also why the "work" is "good", please.
>
> Here is explanation why my "work" is "good":  It paved the
> way to final unravelling of triality, several discoveries, made
> by me.  Like the observation that for any u in Spin(8), the
> element trial(u)/trial(u)' is in Spin(7), representing a simple
> rotation.  Or, the fact that any u in Spin(7) can be uniquely,
> up to a triality triple, factored as a product
>
>    u = h0 g0 = h1 g1 = h2 g2,
>
> where g0, g1, g2 are in G_2 and h0, h1, h2 are isoclinic,
> apart their axes. [...]
> The topic is difficult to explore, because no classical
> methods are available, while it is an exceptional case.
> However, I believe that I have completely unravelled
> these expectional phenomena of dimension 8.  Thus,
> I have come to the final frontiers of mathematics, after
> which there is nothing more to be ravelled.

Well, Pertti, all you mention are symmetries and their closure (groups),
and a special case for R^8. Now don't get me wrong: this may be very
exciting stuff, and I guess related to the space we live in;-)
But do you really think this covers all?
You seem to suggest we reached 'The End' ... far from it, because:

... ever heard of 'dissipation' in this universe? Nothing goes without
friction & loss of original purpose. This is NOT modelled by any theory
of symmetries, which is a 'conservative' approach, viz. conserving
energy. While completely ignoring dissipative (like thermal) effects.
The entropy law in physics is indicative of that for continuous_
variable modelling, and semigroups (for instance of state_
transformations) in discrete dynamic modelling (FSM): hardly explored
yet (stuck at Boolean logic synthesis, & 'cripple' heuristic sequential
logic synthsis that include the 'internal-state' concept, cq 'memory'.

So why not generalize (symmetry-) groups, and go for semigroups ..
of function composition, allowing merging of states, hence going beyond
conservation models, being more 'real'. -- And probably a necessary step
to even think of the opposite: 'anti_entropy' = the creative process of
which we are the long-term result, starting with the creation of heavy
elements in supernovae (towards our Atmosphere & DNA building) -- note:
out of thermal chaos! ;-)   [...]


Subj:  Re: At my signal - Give them h.L!       (sci.math - 14sep01)
From:  Nico Benschop
---------
Chan-Ho Suh wrote:
>
> On Thu, 13 Sep 2001, John R Ramsden wrote:
>
> >  Gladiator star Crowe to play Nobel Prize winner Nash
> >  ----------------------------------------------------
> >  Daily Telegraph c. 1 Sept 2001
> >
> > Gladiator star Russell Crowe, 37, sheds his tough guy image
> > for his latest movie role as a mathematician in A Beautiful
> > Mind. The film follows the true story of John Forbes Nash Jr,
> > who beat schitzophrenia to win the Nobel Prize in the Fifties.
>
> I believe he didn't win the prize until several years ago; there's
> a fascinating story behind the reason for the delay, laid out in
> the book "A Beautiful Mind" by Sylvia Nasar. Just started reading
> it a few days ago, not done yet (reviewing cohomology for a qual
> is more important, supposedly), but very good so far.
>
> Hmmm...we'll see how good the movie is, considering that they
> admitted to fictionalizing parts of his life for the on-screen
> version.
>
> >----------------------------------------------------------------
> > John R Ramsden (jr@redmink.demon.co.uk)
> > ----------------------------------------------------------------
> > The new is in the old concealed, the old is in the new revealed.
> >    St Augustine.
> >-----------------------------------------------------------------
>
> Why would a mathematician quote St. Augustine, the man who said,
> "The good Christian should beware of mathematicians and all those
>  who make empty prophecies. The danger already exists that
>  mathematicians have made a covenant with the devil to darken
>  the spirit and confine man in the bonds of Hell."
>         If only that were so, Auggie.

Oh well, maybe he flunked some qualifying math exam, and this was his
'revenge';-)  Mind you, the Devil was probably the only one in those
days - (and who knows also today;-( - who dared to think for himself:
very dangerous indeed for 'the Powers that Be' !   E.G:

 The Hensel Lift *can* be broken, by quadratic analysis mod p^{3k+1}
 Re: the Cubic roots of Unity (& 'Trinity': sounds ominous enough?-)
     http://piazza.iae.nl/users/benschop/sgrp-flt.htm
     http://piazza.iae.nl/users/benschop/cubic.htm
     http://piazza.iae.nl/users/benschop/scimat98.htm

-- NB -- http://www.iae.nl/users/benschop/math-use.htm
         http://www.newwork.com/Pages/Contributors/Shelton/Hammer.html
 (on Maslow: Those who only have a hammer, think in terms of nails...)


Subj:    Re: A subjective graph.        (sci.math - 25sep01)
From:    Nico Benschop
Orgn:    Digital Research
-------------
> Bill Taylor drew (roughly;-):
>
> >              math
> >
> >   comp.sci  stats    physics
> >
> >      |    /     chemistry  geology   astronomy  engineering
> >      |   /          |                                |
> >      O.R.        biology                             |
> >
> >       |        medicine                         architecture
> >       |            |        \              \         |
> >   economics  psychology  anthropology  geography     |
> >        \         /     \ /                  |        |
> >         \       /       X                   |        |
> >          \     /     __/ \                  |        |
> >          sociology  /  education            |        |
> >            / \   \ /      |                 |        |
> >           /   \   X       |                 |        |
> >          /     \ / \      |                 |        |
> >     ethics  history \     |                 |        |
> >       |        |     `----+---------------town planning
> >       |        |          |                     |
> >~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
> > Math and science require reasoning,
> >   the fuzzy subjects just require scholarship. -  R. A. Heinlein.
> >---------------------------------------------------------------
> >      Bill Taylor

Math and Science require fantasy and intuition (J.H. van 't Hoff)
                            (Nobel price 1901 on Stereo Chemistry)

My subjective graph would put psychology on top, as the study of
human behaviour, and math way down as a special & temporarily
overrated type of behaviour -- with the tendency of a single focus
on logic derivations from axioms, and neglecting the sources of these
axioms (intuition < psychology;-) -- the axioma's being the link to
reality, severely weakened in recent times by an over-emphasis on:
  'abstraction' = escape into the infinite = pure syntax =
  = absence of representation = neglecting the finite.

      -- Reflections: only in the Finite.
             Balance: One is Always Halfway Anyway (AHA;-)

-- NB -- http://piazza.iae.nl/users/benschop/math-use.htm  */quotes.htm