Friam

Science and art: where does mathematics stand?

Classic

List

Threaded

1 message

Jack K. Horner

Science and art: where does mathematics stand?

Here are some observations on the recent FRIAM exchange on the nature
of mathematics:

1. It is possible to view mathematics as a set of
uninterpreted theories, together with a set of interpretations of
those theories ([1], [2]). (Whether this view of mathematics is
desirable, is another matter; see, for example, [3].) A theory, on
this view, is just a collection of sentences. A sentence, on this
view, is an expression with no free variables. An expression, on this
view, is a sequence of uninterpreted marks that satisfies a
specification of what sequences of a specific set of uninterpreted
marks are to be admitted as expressions. (The specific meaning of
"free variable" is not important for the present purposes.)

An *interpretation* of a theory, in this ("Hilbertian") view, is a
function that maps the interpreted marks of the theory into another set.

"Theory", as reserved above, corresponds closely to Russ Abbot's
usage of "mathematics as bare equations"; "interpretation", as
sketched above, corresponds to Russ's notion of "concepts applied to
the equations". As Russell Standish insists, there has to be more to
mathematics than "bare equations"; indeed, there is more even within
the austere bounds of the Hilbertian view. (BTW, I'm not aware of
any mathematician or even philosopher of mathematics who restricts
mathematics to an "uninterpreted theory" in the above sense.)

2. Because the "standard" interpretation of the theory (QT;
[8]) of quantum mechanics (QM) produces what from the viewpoint of
classical physics are paradoxes ([4]), it is often urged that the QT
is an uninterpreted calculus that emits predictions that correspond
with experiments. (This "instrumentalist" view of scientific
theories, BTW, is hardly new with QM; it is traceable to at least
Bishop Berkeley ([5]).) A particularly problematic feature of the
instrumentalist view is that it renders the relationship between the
uninterpreted theory, and our descriptions of observations and
experiments, totally opaque. Why, given that view, should or could
we suppose there is any connection between the two? Is such a view
even coherent? Would you let your daughter/son date someone who held
such a view? ( ;-) )

There is a way out of the paradoxes of QM, but it requires taking a
non-classical view of concepts like position, momentum, energy,
etc. The view can be simply put (no matter how hard it might be to
accept): the classical concepts of position, momentum, etc., are just
approximations that work well enough for our purposes in some
physical settings but work very badly in others. In contrast, the
quantum concepts (especially those corresponding to the
"non-commutative observables" of the QT, such as as
position/momentum) of physical quantities are sufficient to
comprehend the behavior of all physical settings, including those for
which the purely classical concepts were good enough. We just have to
accept the idea, this view holds, that what we thought position,
momentum, etc., were was only an approximation and in some physical
settings, a very bad approximation.

It can be argued that a shift away from the classical view of
physical quantities is not a problem unique to quantum
mechanics. Relativity theory ([7]) also forces to abandon the
concepts of absolute space and time Newton imagined were the subject
of the _Principia_ ([6]).

Cheers,

---

[1] Hilbert D. Foundations of Geometry (the first edition was
published in 1899). 10th edition translated by Unger L. Open Court. 1971.

[2] Chang CC and Kiesler JR. Model Theory. North-Holland. 1990.

[3] Benacerraf P and Putnam H. Philosophy of Mathematics: Selected
Readings. Second Edition. Cambridge. 1983.

[4] Hooker CA. The nature of quantum mechanical reality: Einstein
vs. Bohr. In Colodny RG, ed. Paradigms and Paradoxes: The
Philosophical Challenge of the Quantum Domain. Pittsburgh. 1972.

[5] Berkeley G. A Treatise Concerning the Principles of Human
Knowledge. Aaron Pepyat, Dublin. 1710.

[6] Newton I. _The Principia_. Edition of 1726. Translated by
Motte A (1848). Prometheus Books. See especially Book I,
Definitions:Scholium.

[7] Einstein A. Elektrodynamik bewegter Koerper. Annalen der
Physik 17 (1905), pp. 891-921.

[8] Bohm D. Quantum Theory. Dover. 1979. See especially Sections 1-7.

Jack K. Horner
P. O. Box 3827
Santa Fe, NM 87501
Voice: 505-455-0381
Fax: 505-455-0382
email: [hidden email]

---------------------

At 10:00 AM 1/12/2009, you wrote:

>Send Friam mailing list submissions to
> [hidden email]
>
>To subscribe or unsubscribe via the World Wide Web, visit
> http://redfish.com/mailman/listinfo/friam_redfish.com
>or, via email, send a message with subject or body 'help' to
> [hidden email]
>
>You can reach the person managing the list at
> [hidden email]
>
>When replying, please edit your Subject line so it is more specific
>than "Re: Contents of Friam digest..."
>
>Today's Topics:
>
> 1. Re: Fw: art and science (Russell Standish)
> 2. Re: Fw: art and science (Russ Abbott)
> 3. Re: Financial crisis [was bye(?)] (John Kennison)
>
>From: Russell Standish <[hidden email]>
>Precedence: list
>MIME-Version: 1.0
>To: The Friday Morning Applied Complexity Coffee Group <[hidden email]>
>References: <70F3E1563BEC475FADBDD3ECE87B652A@dell8400>
>In-Reply-To: <70F3E1563BEC475FADBDD3ECE87B652A@dell8400>
>Date: Mon, 29 Dec 2008 22:03:46 +1100
>Reply-To: The Friday Morning Applied Complexity Coffee Group
> <[hidden email]>
>Message-ID: <20081229110346.GB2717@bloody-dell>
>Content-Type: text/plain; charset=us-ascii
>Subject: Re: [FRIAM] Fw: art and science
>Message: 1
>
>Russ Abbott wrote >
>
> > > Mathematics is a language of equations and
> > numbers. Of course equations operate within frameworks, which
> > themselves involve concepts--such as dimensionality, symmetry,
> > etc. These are important concepts. But the equations themselves are
> > conceptless. They are simply relationships among numbers that match
> > observation. I suspect that this is one of the reasons the general
> > public is turned off to much of science. The equations don't speak to
> > them. I would say that the equations don't speak to scientists either
> > except to the extent that they manage to interpret them in terms of
> > concepts: this is the strength of this field; this is the mass of this
> > object; etc. But the concepts are not part of the equations. And
> > (famously) quantum mechanics has no concepts for its equations! The
> > equations work, but no one can conceptualize what they mean. So how
> > should one think about quantum mechanics? As a black box with dials
> > one can read? What should the public think about quantum mechanics if
> > that's the best that scientists can do? > > I can think of two
> > primary goals for science: to understand nature and to give us some
> > leverage over nature. Equations give us the leverage; concepts give us
> > the understanding. > > -- Russ > > > > On Sat, Dec 27, 2008 at 7:33
>
>I disagree completely with this. Mathematics is not just about
>equations, but about concepts and expressing those concepts. The
>equations are like the letters and words that make up the play Romeo &
>Juliet. If that is all you see, you miss a fantastic story!
>
>Truly, this is important. When I studied linear algebra in first year
>university, the lecturer could not recommend a single text
>book. Instead, he taught the concepts of linear algebra, and how one
>might imagine them in one's mind's eye. (Linear Algebra is basically
>about rotations and stretching in n-dimensional spaces - we can easily
>imagine the 3D ones, and handle the other dimensions by analogy. Only
>infinite dimensional spaces get a little tricky!). Using this
>technique, significant theorems become obvious. Translating the
>theorems into algebra often required a page or more of terse equations
>to express. I once proved a theorem on a necessary condition for
>"permanence" (an ecological stability concept) in generalised
>Lotka-Volterra equations one sleepless night using this conceptual way
>of thinking about linear algebra. In the morning, I translated the
>proof into algebra, and found it to be correct. Unfortunately, I then
>discovered that the theorem had been proved and published about 15
>years before :(.
>
>In quantum mechanics, the concepts are just that of linear algebra
>(rotations and stretches), complex arithmetic (which are planar
>rotations and stretches) and Fourier transforms (spectral analysis of
>a wave form - familiar to all users of "graphic equalisers" in Hi Fi
>systems.).
>
>
>Cheers
>
>----------------------------------------------------------------------------
>A/Prof Russell Standish Phone 0425 253119 (mobile)
>Mathematics
>UNSW SYDNEY 2052 [hidden email]
>Australia http://www.hpcoders.com.au
>----------------------------------------------------------------------------
>
>
>
>From: "Russ Abbott" <[hidden email]>
>Precedence: list
>MIME-Version: 1.0
>To: "The Friday Morning Applied Complexity Coffee Group" <[hidden email]>
>References: <70F3E1563BEC475FADBDD3ECE87B652A@dell8400>
> <20081229110346.GB2717@bloody-dell>
>In-Reply-To: <20081229110346.GB2717@bloody-dell>
>Date: Sun, 11 Jan 2009 23:21:23 -0800
>Reply-To: [hidden email],
> The Friday Morning Applied Complexity Coffee Group
> <[hidden email]>
>Message-ID: <[hidden email]>
>Content-Type: multipart/alternative;
> boundary="----=_Part_64494_5841501.1231744883301"
>Subject: Re: [FRIAM] Fw: art and science
>Message: 2
>
>I think you're agreeing with me. It's the concepts that are
>important, not the equations. To the extent that you can read the
>equations as statements about concepts the equations talk to you.
>But a computer can read and calculate with those same equations
>without the concepts. The concepts are in the mind of the person
>reading the equations, not in the equations themselves.
>
>-- Russ Abbott
>_____________________________________________
>Professor, Computer Science
>California State University, Los Angeles
>o Check out my blog at
><http://russabbott.blogspot.com/>http://russabbott.blogspot.com/
>
>
>On Mon, Dec 29, 2008 at 3:03 AM, Russell Standish
><<mailto:[hidden email]>[hidden email]> wrote:
>Russ Abbott wrote >
>
> > > Mathematics is a language of equations and
> > numbers. Of course equations operate within frameworks, which
> > themselves involve concepts--such as dimensionality, symmetry,
> > etc. These are important concepts. But the equations themselves are
> > conceptless. They are simply relationships among numbers that match
> > observation. I suspect that this is one of the reasons the general
> > public is turned off to much of science. The equations don't speak to
> > them. I would say that the equations don't speak to scientists either
> > except to the extent that they manage to interpret them in terms of
> > concepts: this is the strength of this field; this is the mass of this
> > object; etc. But the concepts are not part of the equations. And
> > (famously) quantum mechanics has no concepts for its equations! The
> > equations work, but no one can conceptualize what they mean. So how
> > should one think about quantum mechanics? As a black box with dials
> > one can read? What should the public think about quantum mechanics if
> > that's the best that scientists can do? > > I can think of two
> > primary goals for science: to understand nature and to give us some
> > leverage over nature. Equations give us the leverage; concepts give us
> > the understanding. > > -- Russ > > > > On Sat, Dec 27, 2008 at 7:33
>
>I disagree completely with this. Mathematics is not just about
>equations, but about concepts and expressing those concepts. The
>equations are like the letters and words that make up the play Romeo &
>Juliet. If that is all you see, you miss a fantastic story!
>
>Truly, this is important. When I studied linear algebra in first year
>university, the lecturer could not recommend a single text
>book. Instead, he taught the concepts of linear algebra, and how one
>might imagine them in one's mind's eye. (Linear Algebra is basically
>about rotations and stretching in n-dimensional spaces - we can easily
>imagine the 3D ones, and handle the other dimensions by analogy. Only
>infinite dimensional spaces get a little tricky!). Using this
>technique, significant theorems become obvious. Translating the
>theorems into algebra often required a page or more of terse equations
>to express. I once proved a theorem on a necessary condition for
>"permanence" (an ecological stability concept) in generalised
>Lotka-Volterra equations one sleepless night using this conceptual way
>of thinking about linear algebra. In the morning, I translated the
>proof into algebra, and found it to be correct. Unfortunately, I then
>discovered that the theorem had been proved and published about 15
>years before :(.
>
>In quantum mechanics, the concepts are just that of linear algebra
>(rotations and stretches), complex arithmetic (which are planar
>rotations and stretches) and Fourier transforms (spectral analysis of
>a wave form - familiar to all users of "graphic equalisers" in Hi Fi
>systems.).
>
>
>Cheers
>
>----------------------------------------------------------------------------
>A/Prof Russell Standish Phone 0425 253119 (mobile)
>Mathematics
>UNSW SYDNEY
>2052
><mailto:[hidden email]>[hidden email]
>Australia
><http://www.hpcoders.com.au>http://www.hpcoders.com.au
>----------------------------------------------------------------------------
>
>============================================================
>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>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