3 apples
collect 3 more apples.
i have now 6 apples.
this is the concept of addition.
+
this i like.
2 exp 0 = 1
can anyone explain me this with apples?
ilias wrote:
Do you understand what (expt 2 2) means? 2 X 2 = 4.
Next, (expt 2 1) means (/ (expt 2 2) 2) => 2.
Following the pattern we reach (expt 2 0), which is (/ (expt 2 1) 2) => 1
This has nothing to do with Lisp but a lot to do with elementary school
mathematics.
> Do you understand what (expt 2 2) means? 2 X 2 = 4.
> Next, (expt 2 1) means (/ (expt 2 2) 2) => 2.
> Following the pattern we reach (expt 2 0), which is (/ (expt 2 1) 2) => 1
>
> This has nothing to do with Lisp but a lot to do with elementary school
> mathematics.
you have confused me.
i've requested an explanation based on apples.
to cleanup my mind.
> i've requested an explanation based on apples.
(expt apple 0) => seed (of power).
thi
3  1 = 2
3 apples.
i ate 1 apple.
the i have 2 apples.
2 exp 0 = 1
with apples.
explain.
your explanation.
was not for me.
it was for you.
> (expt apple 0) => seed (of power).
This is brilliant. Now I understand exponentiation. Thank you.[1]
Could you please also explain the HawkingPenrose collapse theorem in
terms of oranges? You see I only understand oranges, although I read
somewhere that oranges are not the only fruit and so I am moreorless
comfortable with apples too. I would prefer you to stick to oranges
though.
tim
Footnotes:
[1] This should read:
Brilliant, this.
Exponentiation.
Understood.
readin your text...
bananas comes to my mind.
You cannot multiply apples with apples. Here is
a more mathematical explanation. ^ is the expt
operator.
a^b / a^c = a^(b  c) Check this one for yourself
if you don't believe me
a^0 = a^(b  b) b = c since they must subtract
to zero
= a^b / a^b A number divided by itself is always 1
= 1
This is not a proof untill you prove the first equation.
That has been done time and time again though and is very
intuitive so that should be no problem.
HTH.

Thomas Stegen
> 2 exp 0 = 1
>
> can anyone explain me this with apples?
but of course. Suppose you have just planted a small apple tree which
gives you only apple this year, but it is of a fabolous breed: For
every year from now on, it will give you twice as many apples as the
preceding year. So, after n years it will give you (expt 2 n)
apples. And after 0 years it will of course give you 1 apple  (expt 2
n) = 1.
Now please explain why you think it's appropriate to ask questions of
elementary math on comp.lang.lisp.

(espen)
3 piles with 3 apples each.
3 times 3 apples.
9 apples.
3 apples * 3 apples =
i see.
you are right.
> a more mathematical explanation. ^ is the expt
> operator.
>
> a^b / a^c = a^(b  c) Check this one for yourself
> if you don't believe me
2^5 / 2^3 = 2^(53) = 2^2
2*2*2*2*2 / 2*2*2 = 2*2
8*2*2/8 = 2*2
ok this seems to fit.
>
> a^0 = a^(b  b) b = c since they must subtract
> to zero
>
> = a^b / a^b A number divided by itself is always 1
>
> = 1
understand.
>
> This is not a proof untill you prove the first equation.
> That has been done time and time again though and is very
> intuitive so that should be no problem.
a^b / a^c = a^(b  c)
how do i know that this equation is correct?
and even if:
isn't there a reallifeexample for 2^0 = 1 ?
> Suppose you have just planted a small apple tree which
> gives you only apple this year, but it is of a fabolous breed:
> For every year from now on, it will give you twice as many apples as the
> preceding year.
> So, after n years it will give you (expt 2 n) apples.
> And after 0 years it will of course give you 1 apple  (expt 2 n) = 1.
the story does not fit.
>
> Now please explain why you think it's appropriate to ask questions of
> elementary math on comp.lang.lisp.
i was working on assimilating backquotesyntax.
while doing this, this question raised.
so this question is directly associated with Lisp.
but i don't know how.
sorry!
> i was working on assimilating backquotesyntax.
>
> while doing this, this question raised.
>
> so this question is directly associated with Lisp.
>
> but i don't know how.
>
> sorry!
I'm trying to assume that you're not here just to entertain or
annoy us, so here's a seriously meant piece of advice: If you're
going to ever learn programming (not to mention understand the
underpinnings of lisp!), you need a minimum of mathematics.
When you have that minimum knowledge, you'll know _several_
reasons why 1 is the right value when you extend the exptfunction
from positive naturals (or reals) to naturals (or reals).

(espen)
i have enouth.
+  * / exp root
>
> When you have that minimum knowledge, you'll know _several_
> reasons why 1 is the right value when you extend the exptfunction
> from positive naturals (or reals) to naturals (or reals).
no i don't know.
and you don't know to explain me this.
seems htat your knowledge does not help you.
i have to sleep know.
nestedmacro is finished.
Then please go away (and take pick up your syntactic waste before
you leave, please).

(espen)
You can prove the following using mathematical induction,
I am not sure exactly how, and I am not about to try very
hard so if you want a rigid proof of thisa explanation a
book or websearch is in place.
(n > 0)
a^n = a * a * a * a ... * a (n times)
a^m = a * a * a * a ... * a (m times)
for example:
(a * a * a) / (a * a) = a
You see that the a's in the denomiator and numerator
cancel each other out. Suppose that. a^m is the
denomiator and a^n is the numerator; a^n / a^m. You see
that m a's will cancel out hence a^n / a^m = a^(n  m).
If you want proof, use mathematical induction or read a
book.
> and even if:
>
> isn't there a reallifeexample for 2^0 = 1 ?
Espen Vestre provided one with apples and apple trees.

Thomas Stegen
Who cares about reals? What we want is an explanation in terms of
counting oranges (or apples will do, bananas if pushed) of why
$e^{i\pi} = 1$. We demand complex fruit[1]!
tim
Footnotes:
[1] since we seem to be dealing with a simple nut.
You obviously don't have any clue about complex numbers, or about
complexity in general. It should be !e^{i\pi} = 1!, and not $e^{i\pi} =
1$. ! looks much nicer than $. I will shortly give a proof...
Pascal
P.S.: ;)

Pascal Costanza University of Bonn
mailto:cost...@web.de Institute of Computer Science III
http://www.pascalcostanza.de Römerstr. 164, D53117 Bonn (Germany)
> 2 exp 0 = 1
>
> can anyone explain me this with apples?
2 ^ 3 means multiply with 2, do it 3 times.
Thus
4 * 2 ^ 3 = 4 * 2 * 2 * 2
4 * 2 ^ 2 = 4 * 2 * 2
4 * 2 ^ 1 = 4 * 2
4 * 2 ^ 0 = 4
or
4 * 2^0 = 4 * 1
That is 2^0 = 1.

Jens Axel Søgaard
Various forms of consumer fruits are now marketed as "seedless". This is,
of course, misleading. They really have imaginary seed. If you rotate a
seedless fruit 90°, you get a fruitless seed. This can provide hours of fun.

Erik Naggum, Oslo, Norway
Act from reason, and failure makes you rethink and study harder.
Act from faith, and failure makes you blame someone and push harder.
2^2 => two dimensions => square => four apples touching arranged in a
small, lumpy square
2^3 => three dimensions => cube => eight apples, one layer of four
precariously balanced atop another.
2^0 => zero dimensions => point => one frickin' apple
kenny
clinisys
Suppose I put you an orange on the table. Now I ask you to demonstrate me
that there is only a single orange on the table.
Cheers

Marco Antoniotti ========================================================
NYU Courant Bioinformatics Group tel. +1  212  998 3488
715 Broadway 10th Floor fax +1  212  995 4122
New York, NY 10003, USA http://bioinformatics.cat.nyu.edu
"Hello New York! We'll do what we can!"
Bill Murray in `Ghostbusters'.
> Come on, Lispers! Don't you see? that this has been sent unto us for
> our sins: arrogance, verbiage, and lack of restraint. Repent and pray.
You mean `ilias' is Catholic?
No. Only explain in terms of avocados.

(concatenate 'string "cbbrowne" "@ntlug.org")
http://cbbrowne.com/info/linuxdistributions.html
"I think there's something to be said for a completely random
userinterface policy  it keeps the users guessing, making life more
interesting for people who deal with the casual X users. As for the
screen display, at least you can't claim that people are mesmorized by
sexy glitz which distracts them from the work at hand."
 Keith Packard
Oh, no... "A dweem within a dweem, ow a nightmawe!!!"

(concatenate 'string "cbbrowne" "@cbbrowne.com")
http://cbbrowne.com/info/finances.html
Rules of the Evil Overlord #139. "If I'm sitting in my camp, hear a
twig snap, start to investigate, then encounter a small woodland
creature, I will send out some scouts anyway just to be on the safe
side. (If they disappear into the foliage, I will not send out another
patrol; I will break out napalm and Agent Orange.)"
<http://www.eviloverlord.com/>
* Christopher Browne
 No. Only explain in terms of avocados.
Ouch. That is such a harsh request. I think you will need approximately
6.02252E23 avocados to complete these proofs, a.k.a the avocado number.
tsk, Erik, I expected better from you. That's the *Avogadro* number,
often known as the bunchofgrapes number. The *Avocado* number is
9.585138E22.
tim
You should know by now how harsh I am. And you got the fourth decimal
place wrong, too, in the avocado number. It is _not_ the same as
Avogadro's Number. Come on, get it right, Eric!

(reverse (concatenate 'string "moc.enworbbc@" "enworbbc"))
http://cbbrowne.com/info/oses.html
"Unless you used NetInfo. _Then_ changing network settings could
often require torching of the existing system, salting of the ground
it had rested on, and termination of anyone who used it."
 JFW <jwi...@biff.com> on comp.sys.next.advocacy
I rather like this old joke: ``The number you have dialed is
imaginary; please rotate your telephone ninety degrees and try
again''. ;)
> * Tim Brashaw
>  Could you please also explain the HawkingPenrose collapse theorem in terms
>  of oranges? You see I only understand oranges, although I read somewhere
>  that oranges are not the only fruit and so I am moreorless comfortable
>  with apples too. I would prefer you to stick to oranges though.
>
> * Christopher Browne
>  No. Only explain in terms of avocados.
>
> Ouch. That is such a harsh request. I think you will need approximately
> 6.02252E23 avocados to complete these proofs, a.k.a the avocado number.
Aha! Ilias is a mole!
Peace,
Petr
> * Tim Brashaw
>  Could you please also explain the HawkingPenrose collapse theorem in terms
>  of oranges? You see I only understand oranges, although I read somewhere
>  that oranges are not the only fruit and so I am moreorless comfortable
>  with apples too. I would prefer you to stick to oranges though.
>
> * Christopher Browne
>  No. Only explain in terms of avocados.
>
> Ouch. That is such a harsh request. I think you will need approximately
> 6.02252E23 avocados to complete these proofs, a.k.a the avocado number.
Yes, but I don't like moles, they scare me, so could you please avoid
them in the proof?

/_ ..
,' .\ /  No to Imperialist war 
,' _,'  Wage class war! 
/ / `'
( . 
 ) 
(`. '.)
`. )'
> Espen Vestre wrote:
>
> > I'm trying to assume that you're not here just to entertain or
> > annoy us, so here's a seriously meant piece of advice: If you're
> > going to ever learn programming (not to mention understand the
> > underpinnings of lisp!), you need a minimum of mathematics.
>
> i have enouth.
>
> +  * / exp root
Okay, I know I shouldn't take Ilias seriously, but ...
Programming takes a *lot* of math, if you want to be good at it. Your
average production programmer *should* have a broader mathematical
education than your average production chemist. This isn't true at
the moment, and that's the cause of a fair amount of software problems
and inefficiencies.
However, all production programmers have at *least* a knowledge of
middleschool math (by American standards). You're not even close.
That's very unfriendly of you. Why aren't you more gentle?
Moles are nice creatures.
And I can even prove it, based on the ANSI Lisp Standard. I'll be
back soon with a set of macros to prove it, so long as you're
gentle...

(reverse (concatenate 'string "gro.mca@" "enworbbc"))
http://cbbrowne.com/info/lsf.html
Evil Overlords tend to get overthrown due to overly baroque plans with
obvious fatal errors. Follow the "Rules of the Evil Overlord," and
you need not fear heroic opposition, whether that hero be James Bond,
Flash Gordon, or a little hobbit named Frodo.
> In the last exciting episode, t...@conquest.OCF.Berkeley.EDU (Thomas F. Burdick) wrote::
>
> > Yes, but I don't like moles, they scare me, so could you please avoid
> > them in the proof?
>
> That's very unfriendly of you. Why aren't you more gentle?
>
> Moles are nice creatures.
Coming to the end of The Exponentiated Vegetable i just recognize some Violence.
The Violence of The Mole.
What does it violate?
The Spirit of The Soil
comming soon.
Oh no, its spreading. some one call the CDC quick?!?!??!
Christopher and Thomas,
please step away from your computers, sit on your hands
and wait for the nice men in the bubble suits to show up.
marc
Math is a formal system, and you should be good with reasoning over formal
systems. I don't think there is a requirement for math per se; formal logic
is probably better suited for most programming tasks. Philosophy is also
very important since analysis is essentially the task of choosing a
representational system which entails ontological commitments, making
decisions as to how your software will garner data entailing epistemelogical
tradeoffs etc.
Personally, I don't deal much with sin and cos much, but "necessary that"
and "possibly"; "forall" and "there exists"; "wants" and "believes" come up
every day.
> >>> can anyone explain me this with apples?
to which "Kenny Tilton" <kti...@nyc.rr.com> replied:
> 2^1 => one dimension => line => two apples sidebyside forming a short,
> lumpy line.
>
> 2^2 => two dimensions => square => four apples touching arranged in a
> small, lumpy square
>
> 2^3 => three dimensions => cube => eight apples, one layer of four
> precariously balanced atop another.
>
> 2^0 => zero dimensions => point => one frickin' apple
>
Bravo! But will that explanation port to MacIntosh? ;)

Coby Beck
(remove #\Space "coby 101 @ bigpond . com")
No. Math is being able to see patterns and think in terms of abstractions
that focus on the patterns and discard everything else. The result is a
massive formal system that has grown in size and complexity with tremendous
speed over the past 400 years or so. But mathematics starts with looking at
a box of a dozen apples and see the number 12, at a crate of a dozen boxes
and see the number 12, at a truck that holds a dozen crates and see the
number 12, and then realize that there are 144 boxes and 1728 apples without
ever counting to more than 12 because you worked this out by putting three
matches each in three matchboxes, and then you repeated this thrice and put
the three sets of three matchboxes aside and noticed that you had used up 27
matches and 9 matchboxes. Mathematics is watching something move at 1 foot
per second and noticing that after 5 seconds, it had traveled 5 feet, then
watching something accelerate at 1 foot per second per second and noticing
that after 5 seconds, its speed was 5 feet per second and that it had
traveled 12.5 feet and that in both cases the distance traveled was the area
under the graph of its speed. Mathematics is noticing that two marbles can
be laid out in two patterns, three marbles laid out in three times the two
patterns of the two marbles and reason that the number of patterns is the
product of all the whole numbers from 1 to the number of marbles.
Mathematics is watching a yardstick rotate around one end to describe an
area that is half as large as its circumference and that the relationship to
the length of the yardstick is a constant that is present in circumferences,
areas, and volumes of all things circular. If you think mathematics is only
the formal system that describes these discoveries, you have missed out on
all the exciting discoveries.
 and you should be good with reasoning over formal systems.
You should be good at finding the relevant and ignoring the irrelevant
aspects of things that are vastly different, yet still similar in some ways.
What I _really_ ought to do is to put together a collection of all the
relevant postings to generate a "corpus," and run it through
"parsepoetry.lisp".
It can cope with both poetry (where line breaks are highly
significant) and prose (where they aren't).
It then basically builds Markov chains, a big hash table of how often
various "substrings" of some specified length occur within the corpus.
For instance, the above sentence breaks down to something like:
(("thecorpus." . #<PSEQUENCE #x20381485>)
("withinthe" . #<PSEQUENCE #x2037EBB9>)
("occurwithin" . #<PSEQUENCE #x2037EB61>)
("lengthoccur" . #<PSEQUENCE #x2037EB09>)
("specifiedlength" . #<PSEQUENCE #x2037E875>)
("somespecified" . #<PSEQUENCE #x2037E811>)
("ofsome" . #<PSEQUENCE #x2037E7A9>)
("\"substrings\"of" . #<PSEQUENCE #x2037E751>)
("various\"substrings\"" . #<PSEQUENCE #x2037E6F5>)
("oftenvarious" . #<PSEQUENCE #x2037E681>)
("howoften" . #<PSEQUENCE #x2037E611>) ("ofhow" . #<PSEQUENCE #x2037E5B5>)
("tableof" . #<PSEQUENCE #x2037E569>)
("hashtable" . #<PSEQUENCE #x2037E521>)
("bighash" . #<PSEQUENCE #x2037E4D1>) ("abig" . #<PSEQUENCE #x2037E481>)
("chains,a" . #<PSEQUENCE #x2037E439>)
("Markovchains," . #<PSEQUENCE #x2037E3ED>)
("buildsMarkov" . #<PSEQUENCE #x2037E391>)
("basicallybuilds" . #<PSEQUENCE #x2037E32D>)
("thenbasically" . #<PSEQUENCE #x2037E2C5>)
("Itthen" . #<PSEQUENCE #x2037E25D>)
("###START###It" . #<PSEQUENCE #x2037E205>))
where the PSEQUENCE objects contain a list of permissible successor
words, and statistics of how often they occurred in the corpus.
You can then go off and generate your own prose/poetry (as the case
may be) by random selection, and produce a text with the same
statistical characteristics as the corpus.
Based on a corpus of poetry by a certain poet (who shall remain
nameless, as he apparently monitors references to his name and plagues
newsgroups where he gets referenced), this program generates the
following:
IN MUDDY FIELDS FORGOTTEN
Above Muddy Fields Forgotten,
O'er Fools Thrown
There Astrew
Raptors Soar So Patiently,
For Soldier Meat Anew
Their Caw A
Certain Tempo,
Eyes Glancing Down The Wait
For Bullets Through Vacant Minds
And Carrion For Their Plates.
God! I Pray!
Damn Such Fools! Such Brainless,
Useless Twits!
Send Them To A Special Hell!
Where Together They Can Frit!
About Their Flags And Colours,
And Reasons Long Forgot!
And Why They Died So
Uselessly, Their Shrouded Faces Frought,
With The Price Of Their Stupidity!
And Their Crowns All Agilt!
Axe Them All!
The problem with the comp.lang.lisp "literary genius" is that it's a
pain to collect corpus material.

(concatenate 'string "chris" "@cbbrowne.com")
http://www.ntlug.org/~cbbrowne/sap.html
Dijkstra probably hates me
(Linus Torvalds, in kernel/sched.c)
No; that requires highschool level mathematics.
> What I _really_ ought to do is to put together a collection of all the
> relevant postings to generate a "corpus," and run it through
> "parsepoetry.lisp".
>
for "he who must not be mentioned" would not "parsedrivel.lisp"
be a better fit?
No flippin way. I read that in jr. highschool(grades 79). I doubt yutz
boy was around then the 20+ years ago.
Thanks for posting it though I enjoyed reading it again. The sad part
is that we are getting some first had experience with it again.
marc
ps It would be nice if we could figure out a way to make peace
work better.
OK. You're an idiot. How do you like them apples?
> to cleanup my mind.
I'd learn Perl, instead.
faa
No wonder you are trying to cleanup your mind...
faa
P.S. I've heard that, if you use certain readtables, mole can rhyme with
trole, but only if 'l' and 'e' are equivalent.
i extract:
"
the minimum knowledge for beeing in cll is knowing _several_ reasons why
1 is the right value when you extend the exptfunction from positive
naturals (or reals) to naturals (or reals).
"
"please go away..."
rejected.
of course i'm not.
and i don't want to be.
zero apples
any dimensions  still zero apples.
one apple.
3 dimensions  still one apple.
2 dimensions  still one apple.
1 dimension  still one apple.
0 dimensions  still one apple
2 apples.
3 dimensions  8 apples
2 dimensions  4 apples
1 dimension  2 apples
0 dimensions  1 apple ( linguistic remark: notice: dimension_s_ )
3 2 1 0

OO
OOO OO OO O
OO OO

assimilation fault:
zero dimensions = zero space
zero dimensions = zero space ?
Apples need space.
Dimension 3, 2, 1 give space.
Dimension 1 gives space as a line.
The line has length, but now wideness and so no space.
But the apples fit in the line.
The space the apples need e.g. in dimension 1 is not given by the
dimensionline itself.
 => OO
 OO
 => OO
Thus 0dimensions does not have to provide space for the apple, too.
. => O
The "0" in "0 dimensions" apply not to space.
...
"2^0 = 1"
assimilation passed.

You gave me the fundamental part to assimilate the construct "2^0 = 1" !
Thank you *very* much.
hope you've fun.
i had this time.
could someone be so kindly to confirm the error in the specs i've found?
this one about backquotesyntax i mean.
in topic:
am72i2$eov$1...@usenet.otenet.gr
The Challenge of Nested Macros  [#V0.9]
as the thread gets big.
just for this is not missed:
> Kenny Tilton wrote:
> 2^1 => one dimension => line => two apples sidebyside forming a short, lumpy line.
>
> 2^2 => two dimensions => square => four apples touching arranged in a small, lumpy square
>
> 2^3 => three dimensions => cube => eight apples, one layer of four precariously balanced atop another.
>
> 2^0 => zero dimensions => point => one frickin' apple
zero apples
Apples need space.
 => OO
 OO
 => OO
.. => O
The "0" in "0 dimensions" apply not to space.
....
[ ... Suggestions that observations of patterns are an essential part
of mathematics ... ]
> Mathematics is watching a yardstick rotate around one end to
> describe an area that is half as large as its circumference
... and noticing that if you choose to call it a threefootstick the
area is now oneandahalf times as large as its circumference ? :)
(Not trying to detract from your general point at all here)
> No. Math is being able to see patterns and think in terms of abstractions
> that focus on the patterns and discard everything else.
> [...]
> If you think mathematics is only the formal system that describes
> these discoveries, you have missed out on all the exciting
> discoveries.
Every CS graduate should have this tattooed onto their forehead.
tim
Erik> * Brad Miller  Math is a formal system
Erik> No. Math is being able to see patterns and think in terms
Erik> of abstractions that focus on the patterns and discard
Erik> everything else.
Reading, or almost any task you accomplish that uses sensory input
has this property.
Erik> The result is a massive formal system that
Erik> has grown in size and complexity with tremendous speed over
Erik> the past 400 years or so. But mathematics starts with
Erik> looking at a box of a dozen apples and see the number 12, at
Erik> a crate of a dozen boxes and see the number 12, at a truck
Erik> that holds a dozen crates and see the number 12, and then
Erik> realize that there are 144 boxes and 1728 apples without
Erik> ever counting to more than 12 because you worked this out by
Erik> putting three matches each in three matchboxes, and then you
Erik> repeated this thrice and put the three sets of three
Erik> matchboxes aside and noticed that you had used up 27 matches
Erik> and 9 matchboxes. Mathematics is watching something move at
Erik> 1 foot per second and noticing that after 5 seconds, it had
How do you watch something move at 1 foot per second? Most people
would watch something move a certain distance during a certain
interval and then calculate the average speed by division. But if you
can watch speed, distance and timeinterval simultaneously the relation
between them might have come as a suprise.
Erik> traveled 5 feet, then watching something accelerate at 1
Erik> foot per second per second and noticing that after 5
Erik> seconds, its speed was 5 feet per second and that it had
Erik> traveled 12.5 feet and that in both cases the distance
Erik> traveled was the area under the graph of its speed.
And you can watch accelerations too? Do you see forces?
Erik> Mathematics is noticing that two marbles can be laid out in
Erik> two patterns, three marbles laid out in three times the two
Erik> patterns of the two marbles and reason that the number of
Erik> patterns is the product of all the whole numbers from 1 to
Erik> the number of marbles. Mathematics is watching a yardstick
Erik> rotate around one end to describe an area that is half as
Erik> large as its circumference and that the relationship to the
Erik> length of the yardstick is a constant that is present in
Erik> circumferences, areas, and volumes of all things circular.
This is engineering. Mathematics is: given a (un)reasonable set of
axioms can I prove that the this is a constant, or what happens if I
take it as an axiom and use that, or what happens if I assume it is
not a constant?
Erik> If you think mathematics is only the formal system that
Erik> describes these discoveries, you have missed out on all the
Erik> exciting discoveries.
I think you might have missed out on the most exiting discoveries if
you think mathematics is a branch of experimental physics. Tell me:
How many parallels to given line can you draw through a point? Have
you observed the ratio or 2 to root(2) to be immesurable? How do you
observe that a^x+b^x = c^z has no nonzero interger solutions for z > 2?
The distinguishing feature of mathematics is the use of a formal
system for reasoning even though the axioms in this system might not
be in concordance with our perception.
Immanuel
> >>>>> "Erik" == Erik Naggum <er...@naggum.no> writes:
>
> Erik> Mathematics is watching a yardstick rotate around one end
> Erik> to describe an area that is half as large as its
> Erik> circumference and that the relationship to the length of
> Erik> the yardstick is a constant that is present in
> Erik> circumferences, areas, and volumes of all things circular.
>
> This is engineering.
Well, it's an _observation_. Just as ``watching somebody move at 1
feet per second'' is an _observation_. Then comes abstraction and:
> Mathematics is: given a (un)reasonable set of axioms can I prove
> that the this is a constant, or what happens if I take it as an
> axiom and use that, or what happens if I assume it is not a
> constant?
No. Even mathematicians have only a finite amount of time on their
hands. They certainly do /not/ take any arbitrary set of axioms and
see what they can prove with them. That's what it possibly appears to
be at first sight, but it's not the whole story. There is always
something that /motivates/ people to investigate /certain/ abstract
systems but not others. Axiomatic formalism is the /technique/ used
by mathematics, but not the whole thing. Even Bourbaki did /not/ just
take any arbitrary axiomatic system and tried to prove as much as
possible; they chose axiomatic systems that would finally be useful
for finding answers to the /real interesting/ questions.
> Erik> If you think mathematics is only the formal system that
> Erik> describes these discoveries, you have missed out on all the
> Erik> exciting discoveries.
>
> I think you might have missed out on the most exiting discoveries if
> you think mathematics is a branch of experimental physics.
Nonsense. All this has nothing to do with physics. Physics, among
other fields, only gives rise to interesting abstractions which then
motivate mathematicians to investigate them.
> Tell me: How many parallels to given line can you draw through a
> point? Have you observed the ratio or 2 to root(2) to be
> immesurable? How do you observe that a^x+b^x = c^z has no nonzero
> interger solutions for z > 2?
By playing around with some numbers first? Planar geometry certainly
was motivated by human observations of the real world. Then
mathematicians tried to abstract out the key elements, which is not an
entirely trivial matter, as it turned out, leading to projective and
hyperbolic geometry, too.
> The distinguishing feature of mathematics is the use of a formal
> system for reasoning even though the axioms in this system might not
> be in concordance with our perception.
But the point is that they are not arbitrarily chosen, either.
Regards,

Nils Goesche
"Don't ask for whom the <CTRLG> tolls."
PGP key ID 0x0655CFA0
An elementary school level treatment of apple arithmetics
can be found in (Lindgren, 1945). It does not assume any
prior knowledge or the possession of an apple computer.
References
Lindgren (1945)
Lindgren, Astrid. Pippi Långstrump. Raben & Sjogren, 1945.
Vassil.
ilias <at_...@pontos.net> writes:
> of course i'm not.
>
> and i don't want to be.
Ignorant! And proud of it!
This is plainly false.
 How do you watch something move at 1 foot per second?
Sigh. Get the point already.
 And you can watch accelerations too? Do you see forces?
I can see the force of intelligence coming to get you.
 Mathematics is: given a (un)reasonable set of axioms can I prove that the
 this is a constant, or what happens if I take it as an axiom and use that,
 or what happens if I assume it is not a constant?
You have not understood mathematics. I recently discovered the works of
Keith Devlin, and would very strongly recommend his book «Mathematics: The
Science of Patterns: The Search for Order in Life, Mind and the Universe
(Scientific American Paperback Library)», ISBN 0716760223. It may well
change the way you see mathematics. That would not be for the worse.
 I think you might have missed out on the most exiting discoveries if you
 think mathematics is a branch of experimental physics.
Good lord. How could you possibly have failed to see the point?
 The distinguishing feature of mathematics is the use of a formal system for
 reasoning even though the axioms in this system might not be in concordance
 with our perception.
This idiocy is why children do not learn mathematical thinking in "modern"
schools and why we have politicians, journalists, and social scientists who
suffer from what John Allen Paulos calls innumeracy. (Get some of his books,
too; he is a hilarious writer for thinkers. Start with «I Think, Therefore I
Laugh», then «Mathematics and Humor» and «A Mathematician Reads the
Newspaper». You need to get rid of the misconception that mathematics /is/
formal systems. It is like saying that poetry is an exercise in grammar.)
Speaking of books on mathematics, this is the book that led me to discover
Richard Courant: «What is Mathematics? An Elementary Approach to Ideas and
Methods ». It is a most spectacular work. The brilliance of Courant's mind
could not possibly fail to capture your fascination, if you have any.
> > The distinguishing feature of mathematics is the use of a formal
> > system for reasoning even though the axioms in this system might not
> > be in concordance with our perception.
>
> But the point is that they are not arbitrarily chosen, either.
yes, but the initial intended application of new theories is often
purely "internal" to mathematics and very far from realworld
applications (but the nice thing is that the new theories may later
turn out to have "engineering" applications as well, I AFAIR
nonstandard analysis is an example of this).

(espen)
Right. I think what Erik was describing was how new abstractions stem
from observation. Of course, what you observe is very often something
mathematical already, which gets then abstracted to greater heights;
but I don't think this changes the overall point.
> Speaking of books on mathematics, this is the book that led me to discover
> Richard Courant: «What is Mathematics? An Elementary Approach to Ideas and
> Methods ». It is a most spectacular work. The brilliance of Courant's mind
> could not possibly fail to capture your fascination, if you have any.
I recall reading an interview in "More Mathematical People" with
Herbert Robbins, coauthor of "What is Mathematics?". In it,
he claims that he wrot the book and Courant, his superior, merely
stuck his name on it at the end. The book is at home; I will
try to look up the specifics of his allegation and post them
here. (A quick Google search turned up nothing relevant.)