Friam

Why are there theorems?

Classic

List

Threaded

39 messages Options

Russ Abbott

Why are there theorems?

I have what probably seems like a strange question: why are there theorems? A theorem is essentially a statement to the effect that some domain is structured in a particular way. If the theorem is interesting, the structure characterized by the theorem is hidden and perhaps surprising. So the question is: why do so many structures have hidden internal structures?

Take the natural numbers: 0, 1, 2, 3, 4, ... It seems so simple: just one thing following another. Yet we have number theory, which is about the structures hidden within the naturals. So the naturals aren't just one thing following another. Why not? Why should there be any hidden structure?

If something as simple as the naturals has inevitable hidden structure, is there anything that doesn't? Is everything more complex than it seems on its surface? If so, why is that? If not, what's a good example of something that isn't.

-- Russ Abbott
______________________________________

Professor, Computer Science
California State University, Los Angeles

cell: 310-621-3805
blog: http://russabbott.blogspot.com/
vita: http://sites.google.com/site/russabbott/
______________________________________

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Nick Thompson

Re: Why are there theorems?

Because of the fallacy of induction?

Nick

Nicholas S. Thompson

Emeritus Professor of Psychology and Ethology,

Clark University ([hidden email])

http://home.earthlink.net/~nickthompson/naturaldesigns/

http://www.cusf.org [City University of Santa Fe]

----- Original Message -----

From: [hidden email]

To: [hidden email]

Sent: 4/24/2010 10:48:21 PM

Subject: [FRIAM] Why are there theorems?

I have what probably seems like a strange question: why are there theorems? A theorem is essentially a statement to the effect that some domain is structured in a particular way. If the theorem is interesting, the structure characterized by the theorem is hidden and perhaps surprising. So the question is: why do so many structures have hidden internal structures?

Take the natural numbers: 0, 1, 2, 3, 4, ... It seems so simple: just one thing following another. Yet we have number theory, which is about the structures hidden within the naturals. So the naturals aren't just one thing following another. Why not? Why should there be any hidden structure?

If something as simple as the naturals has inevitable hidden structure, is there anything that doesn't? Is everything more complex than it seems on its surface? If so, why is that? If not, what's a good example of something that isn't.

-- Russ Abbott
______________________________________

Professor, Computer Science
California State University, Los Angeles

cell: 310-621-3805
blog: http://russabbott.blogspot.com/
vita: http://sites.google.com/site/russabbott/
______________________________________

Douglas Roberts-2

Re: Why are there theorems?

In reply to this post by Russ Abbott

Yes, to me a strange question, begging the follow-on question: why would people want to think in such a way as to ask it? Such a vague, immaterial-to-the-point-of-having-no-practical-application kind of a question.

Abstract. Disconnected.

In other words: what's the point of such a question? Seriously...

Oh, and by the way: who is to decide what is interesting, and what is not?

--Doug

On Sat, Apr 24, 2010 at 10:47 PM, Russ Abbott <[hidden email]> wrote:

I have what probably seems like a strange question: why are there theorems? A theorem is essentially a statement to the effect that some domain is structured in a particular way. If the theorem is interesting, the structure characterized by the theorem is hidden and perhaps surprising. So the question is: why do so many structures have hidden internal structures?

Take the natural numbers: 0, 1, 2, 3, 4, ... It seems so simple: just one thing following another. Yet we have number theory, which is about the structures hidden within the naturals. So the naturals aren't just one thing following another. Why not? Why should there be any hidden structure?

If something as simple as the naturals has inevitable hidden structure, is there anything that doesn't? Is everything more complex than it seems on its surface? If so, why is that? If not, what's a good example of something that isn't.

-- Russ Abbott
______________________________________

Professor, Computer Science
California State University, Los Angeles

cell: 310-621-3805
blog: http://russabbott.blogspot.com/
vita: http://sites.google.com/site/russabbott/
______________________________________

Rich Murray

Re: Why are there theorems?

In reply to this post by Russ Abbott

The "so simple" natural numbers only exist within perhaps infinitely complex individualized awarenesses that co-create within a single infinite unity -- so naturally infinite patterns of nested recursive fractals emerge.

The natural numbers comprise a discrete infinity, which reveal a base for infinite sequences that comprise the continuous infinity of real numbers.

Which in turn allow higher order infinities to emerge without limit.

Single infinite unity is awesome.

It is identity.

Rich Murray

----- Original Message -----

From: [hidden email]

To: [hidden email]

Sent: Saturday, April 24, 2010 10:47 PM

Subject: [FRIAM] Why are there theorems?

I have what probably seems like a strange question: why are there theorems? A theorem is essentially a statement to the effect that some domain is structured in a particular way. If the theorem is interesting, the structure characterized by the theorem is hidden and perhaps surprising. So the question is: why do so many structures have hidden internal structures?

Take the natural numbers: 0, 1, 2, 3, 4, ... It seems so simple: just one thing following another. Yet we have number theory, which is about the structures hidden within the naturals. So the naturals aren't just one thing following another. Why not? Why should there be any hidden structure?

If something as simple as the naturals has inevitable hidden structure, is there anything that doesn't? Is everything more complex than it seems on its surface? If so, why is that? If not, what's a good example of something that isn't.

-- Russ Abbott
______________________________________

Professor, Computer Science
California State University, Los Angeles

cell: 310-621-3805
blog: http://russabbott.blogspot.com/
vita: http://sites.google.com/site/russabbott/
______________________________________

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Jochen Fromm-4

Re: Why are there theorems?

In reply to this post by Russ Abbott

Good questions. You are right, a theorem
is a statement that some domain is
structured in a particular way.
The Princeton companion to mathematics
lists 35 major theorems, from the ABC
conjecture and the Atiyah-Singer Index
Theorem to the Weil Conjectures.

Theorems are based on connections in
the structure of mathematical systems:
to find a new theorem is like revealing
a hidden structure. Some connections are
shortcuts between different points, others
are bridges between different areas.

Why do so many structures have hidden
internal structures? Interesting question.
It is the reason why we do Mathematics,
otherwise it would be boring. I would say
because there are systems where simple
elements and rules can produce complex
structures.

The basic mathematical elements and axioms
allow a whole universe of combinations
and connections which is consistent and
complex at the same time. Algebraic and
geometric systems seem to contain an
infinite number of complex structures.

The integers 0,1,2,3,4,.. may be simple,
but there is an infinite number of them.
If we consider only the numbers of the
finite Group with 4 elements, then Number
Theory becomes less interesting.

In general the patterns and structures
which can emerge in a system depend on the
basic axioms, elements and operations,
and on the size of the system. A kind
of emergence again, perhaps..

-J.

----- Original Message -----
From: Russ Abbott
To: The Friday Morning Applied Complexity Coffee Group
Sent: Sunday, April 25, 2010 6:47 AM
Subject: [FRIAM] Why are there theorems?

I have what probably seems like a strange question: why are there theorems?
A theorem is essentially a statement to the effect that some domain is
structured in a particular way. If the theorem is interesting, the structure
characterized by the theorem is hidden and perhaps surprising. So the
question is: why do so many structures have hidden internal structures?

Take the natural numbers: 0, 1, 2, 3, 4, ... It seems so simple: just one
thing following another. Yet we have number theory, which is about the
structures hidden within the naturals. So the naturals aren't just one thing
following another. Why not? Why should there be any hidden structure?

If something as simple as the naturals has inevitable hidden structure, is
there anything that doesn't? Is everything more complex than it seems on its
surface? If so, why is that? If not, what's a good example of something that
isn't.

-- Russ Abbott
______________________________________

Professor, Computer Science
California State University, Los Angeles

cell: 310-621-3805
blog: http://russabbott.blogspot.com/
vita: http://sites.google.com/site/russabbott/
______________________________________

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Jochen Fromm-4

Re: Why are there theorems?

In reply to this post by Russ Abbott

Another reason for "hidden" structures is our
limited capacity for instant in-depth analysis.
They only appear to be "hidden" for us.

Look at this XKCD Cartoon: http://xkcd.com/731/
There seems to be nothing but flat empty
water as far as the eye can see, but there
is a large number of complex structures
below the surface.

In this sense, there is a large number of
hidden structures in some systems because we
have cognitive limitations. There is a limit
in our cognitive abilities to perceive complex
structures. We can recognize certain patterns
and superficial structures at once, but
we are not able to make an instant
in-depth analysis of a complex system.

-J.

----- Original Message -----
From: Russ Abbott
To: The Friday Morning Applied Complexity Coffee Group
Sent: Sunday, April 25, 2010 6:47 AM
Subject: [FRIAM] Why are there theorems?

[..] So the question is: why do so many structures have hidden internal
structures? [..] Why should there be any hidden structure?

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Grant Holland

Re: Why are there theorems?

In reply to this post by Russ Abbott

There are theorems because systems have relationships as well as elements, from which arise emergent properties.

Grant

Russ Abbott wrote:

I have what probably seems like a strange question: why are there theorems? A theorem is essentially a statement to the effect that some domain is structured in a particular way. If the theorem is interesting, the structure characterized by the theorem is hidden and perhaps surprising. So the question is: why do so many structures have hidden internal structures?

Take the natural numbers: 0, 1, 2, 3, 4, ... It seems so simple: just one thing following another. Yet we have number theory, which is about the structures hidden within the naturals. So the naturals aren't just one thing following another. Why not? Why should there be any hidden structure?

If something as simple as the naturals has inevitable hidden structure, is there anything that doesn't? Is everything more complex than it seems on its surface? If so, why is that? If not, what's a good example of something that isn't.

-- Russ Abbott
______________________________________

Professor, Computer Science
California State University, Los Angeles

cell: 310-621-3805
blog: http://russabbott.blogspot.com/
vita: http://sites.google.com/site/russabbott/
______________________________________
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Grant Holland

Re: Why are there theorems?

Russ, I apologize for being so terse. Let me try again. Here is my take on your question...

As we know, systems are more than just components, or elements. A system must also have relationships among its elements before they it is worthy being called a system.

But, when you take these component relationships into account, the possibilities for what characteristics, or properties, a system may exhibit begins to ramify into a potentially large and surprising number, due to combinatorics. With so many possible component relationships, it often becomes non-intuitive as to which potential properties (true statements) of the system are true.

Thus the need for theorems arises due to a system having relationships among its components. And we haven't even mentioned emergent properties yet!

This is simple, of course, because it is elemental, foundational to systemics.

Take care,
Grant

Grant Holland wrote:

There are theorems because systems have relationships as well as elements, from which arise emergent properties.

Grant

Russ Abbott wrote:
I have what probably seems like a strange question: why are there theorems? A theorem is essentially a statement to the effect that some domain is structured in a particular way. If the theorem is interesting, the structure characterized by the theorem is hidden and perhaps surprising. So the question is: why do so many structures have hidden internal structures?

Take the natural numbers: 0, 1, 2, 3, 4, ... It seems so simple: just one thing following another. Yet we have number theory, which is about the structures hidden within the naturals. So the naturals aren't just one thing following another. Why not? Why should there be any hidden structure?

If something as simple as the naturals has inevitable hidden structure, is there anything that doesn't? Is everything more complex than it seems on its surface? If so, why is that? If not, what's a good example of something that isn't.

-- Russ Abbott
______________________________________

Professor, Computer Science
California State University, Los Angeles

cell: 310-621-3805
blog: http://russabbott.blogspot.com/
vita: http://sites.google.com/site/russabbott/
______________________________________
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Steve Smith

Re: Why are there theorems?

Russ -

Another great question.

While Doug and I have an awful lot in common, this is probably where we most notably diverge.   You ask "why", he asks "why ask why", I ask "why ask why ask why".   ("Who dat who say who dat?" might ring a bell for some of the other old timers here).

I don't waste many of my otherwise productive cycles on such questions as "why theorems?" but I do find myself enjoying such questions quite thoroughly when not occupied with making a living or preparing the garden (actually it is a good question to ponder while turning over a garden bed or raking up last falls detritus from the courtyard.

I *do* like Doug's ancillary point, however, of "who decides what is interesting?" and my corollary would be "isn't 'hidden' a relative concept?"   I do suspect that your question is as much about human nature/intelligence as it is about anything intrinsic in theorems (excepting that theorems are human constructs).

I'll give Rich Murray points for a maximal grandiosity to simplicity ratio, though I don't find that

Single infinite unity is awesome.

It is identity.

has much explanatory power.

Grant seems to tease out an important and key point. The numbers themselves seem quite simple (counting, progression) but as we add relationships (the notion of addition or multiplication), the complexity explodes. Perhaps an interesting corollary to your question is why do simple systems exhibit a geometric(?) explosion of properties and relationships as we seek them out? It seems like there ought to be a meta-answer based entirely in combinatorics of the language involved. Kurt Godel would seem to have something to say about this?

It's a good question, I will contemplate it while I complete the digging of the footer for my emergent greenhouse. And I look forward to the mail flurries from this group of deep and broad thinkers.

- Steve

Russ, I apologize for being so terse. Let me try again. Here is my take on your question...

As we know, systems are more than just components, or elements. A system must also have relationships among its elements before they it is worthy being called a system.

But, when you take these component relationships into account, the possibilities for what characteristics, or properties, a system may exhibit begins to ramify into a potentially large and surprising number, due to combinatorics. With so many possible component relationships, it often becomes non-intuitive as to which potential properties (true statements) of the system are true.

Thus the need for theorems arises due to a system having relationships among its components. And we haven't even mentioned emergent properties yet!

This is simple, of course, because it is elemental, foundational to systemics.

Take care,
Grant

Grant Holland wrote:
There are theorems because systems have relationships as well as elements, from which arise emergent properties.

Grant

Russ Abbott wrote:
I have what probably seems like a strange question: why are there theorems? A theorem is essentially a statement to the effect that some domain is structured in a particular way. If the theorem is interesting, the structure characterized by the theorem is hidden and perhaps surprising. So the question is: why do so many structures have hidden internal structures?

Take the natural numbers: 0, 1, 2, 3, 4, ... It seems so simple: just one thing following another. Yet we have number theory, which is about the structures hidden within the naturals. So the naturals aren't just one thing following another. Why not? Why should there be any hidden structure?

If something as simple as the naturals has inevitable hidden structure, is there anything that doesn't? Is everything more complex than it seems on its surface? If so, why is that? If not, what's a good example of something that isn't.

-- Russ Abbott
______________________________________

Professor, Computer Science
California State University, Los Angeles

cell: 310-621-3805
blog: http://russabbott.blogspot.com/
vita: http://sites.google.com/site/russabbott/
______________________________________
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

glen e. p. ropella-2

Re: Why are there theorems?

In reply to this post by Grant Holland

Grant Holland wrote circa 04/25/2010 05:42 AM:
> Thus the need for theorems arises due to a system having relationships
> among its components. And we haven't even mentioned emergent properties yet!

But I think Nick's answer is relevant to this point, as well. Even in a
seemingly a priori discrete system like that of the natural numbers,
"components" are psychologically induced, not necessarily embedded in
the system. Beyond the original discrete set, all the other constructs
built on top are part component and part relation, depending on your
(psychological) perspective.

--
glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Marcus G. Daniels

Re: Why are there theorems?

In reply to this post by Steve Smith

Steve Smith wrote:
> You ask "why", he asks "why ask why", I ask "why ask why ask why".
A recursive function definition requires a base case for escape. Doug
provides that case.

Marcus

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Douglas Roberts-2

Re: Why are there theorems?

string why()

{

while (!why())

{

why();

}

(string theory search)

On Sun, Apr 25, 2010 at 10:53 AM, Marcus G. Daniels <[hidden email]> wrote:

Steve Smith wrote:

You ask "why", he asks "why ask why", I ask "why ask why ask why".

A recursive function definition requires a base case for escape. Doug provides that case.

Marcus

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Russ Abbott

Re: Why are there theorems?

I agree that the key has to do with relations -- and that this is related to emergence.

Individual carbon atoms are arguably fairly simple. But carbon atoms in relationship either with each other or with other things form extraordinary structures. In some sense those structures were hidden from us (at least not visible to us) when we looked just at individual carbon atoms (and they may appear surprising when we first encounter them -- one of the less important properties of emergence in my view).

Similarly number theory depends on relationships -- such as the addition relation, the multiplication relation, etc. -- that we impose on the individual numbers.

Having taken the step to acknowledge the importance of relationships, the next question is: what sorts of relationships does a domain allow. That is, what enduring structures can be imposed on a domain? For the naturals, a structure is enduring if it can be defined. Once defined there is nothing to break it apart. It doesn't deteriorate with time. For physical elements a structure is enduring if it persists without the need to be held together by external imposed forces.

-- Russ

On Sun, Apr 25, 2010 at 9:53 AM, Marcus G. Daniels <[hidden email]> wrote:

Steve Smith wrote:

You ask "why", he asks "why ask why", I ask "why ask why ask why".

A recursive function definition requires a base case for escape. Doug provides that case.

Marcus

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

-- Russ Abbott
______________________________________

Professor, Computer Science
California State University, Los Angeles

cell: 310-621-3805
blog: http://russabbott.blogspot.com/
vita: http://sites.google.com/site/russabbott/
______________________________________

On Sun, Apr 25, 2010 at 10:09 AM, Douglas Roberts <[hidden email]> wrote:

string why()
{
   while (!why())
   {
   why();
   }
}

(string theory search)

On Sun, Apr 25, 2010 at 10:53 AM, Marcus G. Daniels <[hidden email]> wrote:

Steve Smith wrote:

You ask "why", he asks "why ask why", I ask "why ask why ask why".

A recursive function definition requires a base case for escape. Doug provides that case.

Marcus

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Eric Charles

Re: Why are there theorems?

In reply to this post by Russ Abbott

Russ,
Bypassing all the other replies, I find this question very interesting. When faced with questions like this I usually give an answer, am told it is not satisfactory, give another answer, am told it is not satisfactory, etc. Then at some point I ask the questioner to give me examples of the types of answers they would find acceptable. So.... well.... can we skip to that part? For example, would an acceptable answer be in terms of:

Something about people who do math, explaining why they make theorems?
Something about the math people, explaining why theorems are necessary for those activities?
Something about math itself showing theorems to be an essential part of any math?
Something about the history of doing-math, showing why we now do math with theorems when we otherwise might not have?
Something about the virtues of doing math different ways, showing the theorem enhanced way to be virtuous in some respect?
Something about the limitations of human cognition, demonstrating why we need theorems instead of simply knowing the truth?
Etc.

My hunch is that some of those types of answers would be of more interest to you than others.

Eric

On Sun, Apr 25, 2010 12:47 AM, Russ Abbott <[hidden email]> wrote:

I have what probably seems like a strange question: why are there theorems? A theorem is essentially a statement to the effect that some domain is structured in a particular way. If the theorem is interesting, the structure characterized by the theorem is hidden and perhaps surprising. So the question is: why do so many structures have hidden internal structures?

Take the natural numbers: 0, 1, 2, 3, 4, ... It seems so simple: just one thing following another. Yet we have number theory, which is about the structures hidden within the naturals. So the naturals aren't just one thing following another. Why not? Why should there be any hidden structure?

If something as simple as the naturals has inevitable hidden structure, is there anything that doesn't? Is everything more complex than it seems on its surface? If so, why is that? If not, what's a good example of something that isn't.

-- Russ Abbott
______________________________________

Professor, Computer Science
California State University, Los Angeles

cell: 310-621-3805
blog: <a href="http://russabbott.blogspot.com/" onclick="window.open('http://russabbott.blogspot.com/');return false;">http://russabbott.blogspot.com/
vita: <a href="http://sites.google.com/site/russabbott/" onclick="window.open('http://sites.google.com/site/russabbott/');return false;">http://sites.google.com/site/russabbott/
______________________________________
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Eric Charles

Professional Student and
Assistant Professor of Psychology
Penn State University
Altoona, PA 16601

lrudolph

Re: Why are there theorems?

In reply to this post by glen e. p. ropella-2

glen e. p. ropella wrote:

> But I think Nick's answer is relevant to this point, as well. Even in a
> seemingly a priori discrete system like that of the natural numbers,
> "components" are psychologically induced, not necessarily embedded in
> the system.

There is (actually) only *one* (closed) system, that being the
universe, and "psychologically induced" "components" are *also*
"embedded in the system". ... Even if you don't buy my axiom
(from "There" to the following comma), you might be willing to
buy the less expansive claims that (1) the intent of the question
"why are there theorems?" would be better stated as "why do humans
perceive/recognize theorems, and why are they interested in them?",
(2) the perceived/recognized theorems are indeed "psychologically
induced" AND THEREFORE *ARE* "necessarily embedded in the system"
(a non-closed subsystem of the universe) that consists of the
"mind" (or "minds") where the "psychology" is happening.

As to "a priori", I attach an article by Konrad Lorenz in which
he introduced "Evolutionary Epistemology". Nick and Eric have
already had the opportunity to read it and comment on it, but
as far as I can tell have done neither; perhaps it will be of
interest to others here. (This version is a searchable PDF,
not to mention a copyright violation.) In EE terms, one might
say "theorems are there (to us) because we evolved so as to
understand the world we evolved in, and (some) theorems are
a damned good way to understand it (the rest have come along
for the ride)".

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Douglas Roberts-2

Re: Why are there theorems?

In reply to this post by Russ Abbott

> Individual carbon atoms are arguably fairly simple.

The word arguably being key, I believe.

To wit:

Carbon:

Carbon is the chemical element with symbol C and atomic number 6. As a member of group 14 on the periodic table, it is nonmetallic and tetravalent—making four electrons available to form covalent chemical bonds. There are three naturally occurring isotopes, with ¹²C and ¹³C being stable, while ¹⁴C is radioactive, decaying with a half-life of about 5730 years.^[9] Carbon is one of the few elements known since antiquity.^[10]^[11] The name "carbon" comes from Latin language carbo, coal.

There are several allotropes of carbon of which the best known are graphite, diamond, and amorphous carbon.^[12] The physical properties of carbon vary widely with the allotropic form. For example, diamond is highly transparent, while graphite is opaque and black. Diamond is among the hardest materials known, while graphite is soft enough to form a streak on paper (hence its name, from the Greek word "to write"). Diamond has a very low electrical conductivity, while graphite is a very good conductor. Under normal conditions, diamond has the highest thermal conductivity of all known materials. All the allotropic forms are solids under normal conditions but graphite is the most thermodynamically stable.

Neutrons:

The neutron is a subatomic particle with no net electric charge and a mass slightly larger than that of a proton. They are usually found in atomic nuclei. The nuclei of most atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of protons in a nucleus is the atomic number and defines the type of element the atom forms. The number of neutrons is the neutron number and determines the isotope of an element. For example, the abundant carbon-12 isotope has 6 protons and 6 neutrons, while the very rare radioactive carbon-14 isotope has 6 protons and 8 neutrons.

While bound neutrons in stable nuclei are stable, free neutrons are unstable; they undergo beta decay with a mean lifetime of just under 15 minutes (885.7±0.8 s).^[2] Free neutrons are produced in nuclear fission and fusion. Dedicated neutron sources like research reactors and spallation sources produce free neutrons for use in irradiation and in neutron scattering experiments. Even though it is not a chemical element, the free neutron is sometimes included in tables of nuclides.^{[citation needed]} It is then considered to have an atomic number of zero and a mass number of one, and is sometimes referred to as neutronium.

Quarks:

Neutron

The quark structure of the neutron.
Classification:	Baryon
Composition:	1 up quark, 2 down quarks
Statistical behavior:	Fermion
Group:	Hadron
Interaction:	Gravity, Weak, Strong
Symbol(s):	n, n^⁰, N^⁰
Antiparticle:	Antineutron
Theorized:	Ernest Rutherford^[1] (1920)
Discovered:	James Chadwick^[1] (1932)
Mass:	1.67492729(28)×10⁻²⁷ kg 939.565560(81) MeV/c² 1.0086649156(6) u^[2]
Mean lifetime:	885.7(8) s (free)
Electric charge:	0 e 0 C
Electric dipole moment:	<2.9×10⁻²⁶ e·cm
Electric polarizability:	1.16(15)×10⁻³ fm³
Magnetic moment:	−1.9130427(5) μ_N
Magnetic polarizability:	3.7(20)×10⁻⁴ fm³
Spin:	¹⁄₂
Isospin:	¹⁄₂
Parity:	+1
Condensed:	I(J^P) = ¹⁄₂(¹⁄₂⁺)

And so on. I imagine you are starting to get the point.

That point being: we appear to have yet another quest to provide an overly simple "Theory Of Everything" approach to answering a question which is basically meaningless without a solid context.

Unless, of course, the goal is to launch into another round of deeply philosophical discussion that will provide little actual product.

Silly me. Of course that was the goal...

--Doug

On Sun, Apr 25, 2010 at 11:14 AM, Russ Abbott <[hidden email]> wrote:

I agree that the key has to do with relations -- and that this is related to emergence.

Individual carbon atoms are arguably fairly simple. But carbon atoms in relationship either with each other or with other things form extraordinary structures. In some sense those structures were hidden from us (at least not visible to us) when we looked just at individual carbon atoms (and they may appear surprising when we first encounter them -- one of the less important properties of emergence in my view).

Similarly number theory depends on relationships -- such as the addition relation, the multiplication relation, etc. -- that we impose on the individual numbers.

Having taken the step to acknowledge the importance of relationships, the next question is: what sorts of relationships does a domain allow. That is, what enduring structures can be imposed on a domain? For the naturals, a structure is enduring if it can be defined. Once defined there is nothing to break it apart. It doesn't deteriorate with time. For physical elements a structure is enduring if it persists without the need to be held together by external imposed forces.

-- Russ

On Sun, Apr 25, 2010 at 9:53 AM, Marcus G. Daniels <[hidden email]> wrote:

Steve Smith wrote:

You ask "why", he asks "why ask why", I ask "why ask why ask why".

A recursive function definition requires a base case for escape. Doug provides that case.

Marcus

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

-- Russ Abbott
______________________________________

Professor, Computer Science
California State University, Los Angeles

cell: 310-621-3805
blog: http://russabbott.blogspot.com/
vita: http://sites.google.com/site/russabbott/
______________________________________

On Sun, Apr 25, 2010 at 10:09 AM, Douglas Roberts <[hidden email]> wrote:

string why()
{
   while (!why())
   {
   why();
   }
}

(string theory search)

On Sun, Apr 25, 2010 at 10:53 AM, Marcus G. Daniels <[hidden email]> wrote:

Steve Smith wrote:

You ask "why", he asks "why ask why", I ask "why ask why ask why".

A recursive function definition requires a base case for escape. Doug provides that case.

Marcus

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

--
Doug Roberts
[hidden email]
[hidden email]
505-455-7333 - Office
505-670-8195 - Cell

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Nick Thompson

Re: Why are there theorems?

In reply to this post by Russ Abbott

The philosopher Garfinkel was fond of citing Willy Sutton on questions like this:

REPORTER: Mr Sutton, why do you rob banks?

WILLIE: 'Cuz that's where the money is.

Without a theorem, it's impossible to to know what the question is.

Nick

Nicholas S. Thompson

Emeritus Professor of Psychology and Ethology,

Clark University ([hidden email])

http://home.earthlink.net/~nickthompson/naturaldesigns/

http://www.cusf.org [City University of Santa Fe]

----- Original Message -----

From: [hidden email]
To: [hidden email]
Cc: [hidden email]

Sent: 4/25/2010 11:22:42 AM

Subject: Re: [FRIAM] Why are there theorems?

Russ,
Bypassing all the other replies, I find this question very interesting. When faced with questions like this I usually give an answer, am told it is not satisfactory, give another answer, am told it is not satisfactory, etc. Then at some point I ask the questioner to give me examples of the types of answers they would find acceptable. So.... well.... can we skip to that part? For example, would an acceptable answer be in terms of:

Something about people who do math, explaining why they make theorems?
Something about the math people, explaining why theorems are necessary for those activities?
Something about math itself showing theorems to be an essential part of any math?
Something about the history of doing-math, showing why we now do math with theorems when we otherwise might not have?
Something about the virtues of doing math different ways, showing the theorem enhanced way to be virtuous in some respect?
Something about the limitations of human cognition, demonstrating why we need theorems instead of simply knowing the truth?
Etc.

My hunch is that some of those types of answers would be of more interest to you than others.

Eric

On Sun, Apr 25, 2010 12:47 AM, Russ Abbott <[hidden email]> wrote:

I have what probably seems like a strange question: why are there theorems? A theorem is essentially a statement to the effect that some domain is structured in a particular way. If the theorem is interesting, the structure characterized by the theorem is hidden and perhaps surprising. So the question is: why do so many structures have hidden internal structures?

Take the natural numbers: 0, 1, 2, 3, 4, ... It seems so simple: just one thing following another. Yet we have number theory, which is about the structures hidden within the naturals. So the naturals aren't just one thing following another. Why not? Why should there be any hidden structure?

If something as simple as the naturals has inevitable hidden structure, is there anything that doesn't? Is everything more complex than it seems on its surface? If so, why is that? If not, what's a good example of something that isn't.

-- Russ Abbott
______________________________________

Professor, Computer Science
California State University, Los Angeles

cell: 310-621-3805
blog: <A onclick="window.open('http://russabbott.blogspot.com/');return
false;" href="http://russabbott.blogspot.com/">http://russabbott.blogspot.com/
vita: <A onclick="window.open('http://sites.google.com/site/russabbott/');return
false;" href="http://sites.google.com/site/russabbott/">http://sites.google.com/site/russabbott/
______________________________________

============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
Eric Charles

Professional Student and
Assistant Professor of Psychology
Penn State University
Altoona, PA 16601

Russ Abbott

Re: Why are there theorems?

In reply to this post by lrudolph

In answer to Eric and lrudolph, the answer I'm looking for is not related to epistemology. It is related to the domains to which mathematical thinking is successfully applied, where successfully means something like produces "interesting' theorems. (Please don't quibble with me about what interesting mean -- at least not in this thread. I expect that interesting can be defined so that we will be comfortable with the definition.) What is it about those domains that enables that.

-- Russ Abbott
______________________________________

Professor, Computer Science
California State University, Los Angeles

cell: 310-621-3805
blog: http://russabbott.blogspot.com/
vita: http://sites.google.com/site/russabbott/
______________________________________

On Sun, Apr 25, 2010 at 10:39 AM, <[hidden email]> wrote:

"Evolutionary Epistemology"

Sarbajit Roy (testing)

Re: Why are there theorems?

If I start from the Wikipedia "definition" of "theorem" --> "In mathematics, a theorem is a statement which has been proved on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms." I end up looking at a house of cards which will eventually collapse under the weight of its inherent contradictions.

PS: I am omitting all the formal / rigorous steps in between.

On Sun, Apr 25, 2010 at 11:21 PM, Russ Abbott <[hidden email]> wrote:

In answer to Eric and lrudolph, the answer I'm looking for is not related to epistemology. It is related to the domains to which mathematical thinking is successfully applied, where successfully means something like produces "interesting' theorems. (Please don't quibble with me about what interesting mean -- at least not in this thread. I expect that interesting can be defined so that we will be comfortable with the definition.) What is it about those domains that enables that.

-- Russ Abbott
______________________________________

Professor, Computer Science
California State University, Los Angeles

cell: 310-621-3805
blog: http://russabbott.blogspot.com/
vita: http://sites.google.com/site/russabbott/
______________________________________

On Sun, Apr 25, 2010 at 10:39 AM, <[hidden email]> wrote:
"Evolutionary Epistemology"

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Owen Densmore

Re: Why are there theorems?

Administrator

In reply to this post by Nick Thompson

On Apr 24, 2010, at 11:26 PM, Nicholas Thompson wrote:

Because of the fallacy of induction?

Do you mean this induction:

http://en.wikipedia.org/wiki/Mathematical_induction#Description

I.e. are you interested in proofs over the positive integers?

-- Owen