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Characterization of Truth in Mathematics

Date: 11/07/2003 at 00:43:48
From: Kiki
Subject: Characterization of Truth in Mathematics

I am interested in researching the correspondence between mathematics 
and reality.  One of the relevant questions I need to answer is "What 
does it mean for a mathematical statement to be true - from the 
viewpoint of a working mathematician?"  In other words, if a non-
mathematician were to ask a mathematician what it means for a 
mathematical fact or idea to be true, how would the mathematician 
reply?  My feeling is that the mathematician would say that a 
mathematical statement is true just in case it adheres to the rules 
of logic, i.e. just in case it can be logically deduced from other 
established truths of mathematics.  But then, where do the rules of
logic come from and what makes the rules of logic true?  So, my
question is really two-fold.

I am looking for a simple, "textbook" answer to this question as 
opposed to a long, involved philosophical discussion about the nature 
of mathematical truth, logic, etc.  To put it another way, in an 
encyclopedia of mathematics, what would the brief entry under
"mathematical truth" say?

(Are there any such encyclopedias or books that could answer my 
question?)

Thanks for your help.

- Kiki



Date: 11/07/2003 at 10:18:05
From: Doctor Vogler
Subject: Re: Characterization of Truth in Mathematics

Kiki,

Your feeling is right, but "logically deduced from other established
truths of mathematics" might cause you to wonder where you start.

Mathematics begins (logically) from a set of assumptions that we call
axioms.  For example, we assume that the set of positive integers (1,
2, 3, ...) is infinite, is in a progression (the first number is 1 and
then every number has a "next" number), and is the smallest such set.
 This gives us a place to start, and then we use these axioms to prove
other things, such as that every subset of positive integers has a
smallest number.

Mathematics continues with definitions.  Actually, you could consider
the above to be the definition of natural numbers (which it is), as
definitions often come equipped with axioms.  We define <name> to be a
set of <something> which satisfies these <axioms>.  For example, after
defining natural numbers, we define addition and multiplication, and
then we define prime numbers.  Then we prove things about what we've
defined, such as that there are infinitely many prime numbers, and
that every number can be factored uniquely into prime factors.

That's how we mathematicians view "mathematical truth."  How does it
relate to reality?  Well, reality doesn't actually come into play in
the logical axioms and definitions, so it only relates because reality
follows many of the laws of mathematics.  It relates because there
really is a set of "natural numbers" that obeys the three axioms I
mentioned previously.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 11/07/2003 at 11:08:31
From: Doctor Ian
Subject: Re: Characterization of Truth in Mathematics

Hi Kiki,

Nice to hear from you again!  Our first conversation, a few years
back, made it into the Dr. Math archives:

  Understanding Mathematics
    http://mathforum.org/library/drmath/view/52289.html 

It mentions a few different books and articles that would be helpful
in researching your question.  One that isn't mentioned, which might
also be useful, is _The Character of Physical Law_, by Richard
Feynman, which is essentially a book-length exploration of the
question "Why does mathematics seem to do such a great job of
explaining what happens in the world?"

I've checked all the resources I have, and none of them provides the
kind of concise, dictionary-length entry that you're looking for.  I
suspect that's because the authors and editors of those resources
realize that an attempt to provide one would generate more questions
than it would answer. 

Having said that, I'll go ahead and state that the questions 

  [W]here do the rules of logic come from and what makes the 
  rules of logic true?

have a single answer: 'Agreement'. 

Mathematicians have agreed on a particular set of rules for deducing
theorems from axioms, in much the same way that chess players have
agreed on a particular set of rules for playing chess. 

The subtle but important insight explored by Lewis Carroll in his
dialog, "What the Tortoise Said to Achilles", 

    http://www.ditext.com/carroll/tortoise.html 

is that there is nothing that _compels_ a particular mathematician to
accept the rules of logic that are being used by anyone else.  Any
mathematician is free to make up his own rules.  All that means is
that he's playing a different game than everyone else. 

The essential difference between mathematical truth and scientific
truth lies in the direction of the reasoning.  In science, there is
(we assume) some set of rules that defines 'truth'; and we try to
guess what they are by observing outcomes.  In math, we _know_ what
the rules are, because we get to make them up; and we then try to see
what outcomes follow from those rules.  

Another way to express this is that truths in mathematics are
'formal', that is to say, self-contained.  No mathematical truth
exists independently of a set of axioms, and a set of agreed-upon
rules for deducing theorems from axioms.  They all come together as a
system; and strictly speaking, they have no necessary connection to
anything outside the system.  

Applying mathematics to the real world is ultimately just a special
case of analogical reasoning.  We notice a resemblance between
relationships that exist between numbers, and relationships that exist
between things in the world, and we take advantage of that resemblance
to do our reasoning using numbers, which are cheap, plentiful, and
very well-behaved.  Then we translate our conclusions back to the
world, and hope that they still apply.  

But the operative word is 'hope'.  All analogies break down when we
try to make use of attributes that exist in one of the analogs, but
not the other.  If we rely on attributes of numbers (e.g., that
intervals of numbers can be subdivided indefinitely) that are not
shared by the things we're using numbers to model, then our
mathematical truths don't correspond to physical truths.  

It doesn't make the mathematical truths any less true!  It just means
that our analogy doesn't stretch as far as we'd like it to. 

I hope this helps!  

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 11/08/2003 at 22:05:21
From: Kiki
Subject: Thank you (Characterization of Truth in Mathematics)

Thank you Doctors Ian and Vogler, for your detailed 
responses!  Good to hear from you again too, Doctor Ian.

What I want to do with this research project is to "zoom 
in" on the connection point between mathematics and the 
real world.  I suspect that there are two types of 
correspondences:  the type exemplified by the correspondence 
between the facts of Euclidean geometry and physical 
objects and the type exemplified by the correspondence 
between differential equations and the processes they 
describe.  For example, the mathematics of Newtonian physics 
would be of this latter type.

What do you think?  Does my idea make sense?

- Kiki



Date: 11/10/2003 at 08:31:49
From: Doctor Ian
Subject: Re: Thank you (Characterization of Truth in Mathematics)

Hi Kiki,
 
That sounds interesting, but be careful about the way your use of
language (and particularly of metaphor) shapes how you think about the
problem.  For example, do you think there's one 'point of connection'
between math and the real world?  What would be the attributes of such
a point?   

We already know that Newtonian physics is wrong, so you may want
to choose a different correspondence as an example.  Within physics,
you have at least two choices: relativistic physics, and quantum
physics, each of which appears to be at odds with the other, in that
the former requires space to be continuous, while the latter is
inconsistent with that requirement.  

I also wonder if you're leaving some correspondences out of the
picture.  For example, what about the correspondence between the facts
of arithmetic and the apparently discrete nature of objects at the
atomic level?  (For example, atoms are made from discrete numbers of
nucleons, which are made from discrete numbers of quarks; electrons
arrange themselves into discrete orbitals; and so on.) 

Also, probability seems to play a large role in processes at the
nuclear level.  I'm not sure that fits neatly into either geometry or
differential equations. 

As to whether your idea makes sense, I'd have to know more about what
your idea _is_.  For example, by 'correspondence', do you mean
something different from 'analogy'?    

If I arrange some physical objects in a circle, they don't really form
a circle in the Euclidean sense. For one thing, circles are made of
points, and points don't have extent.  But objects in the world
certainly have extent!  So what I have is a rough (inexact) analogy
between a purely geometric 'object' and an arrangement of physical
objects in the world.  By reasoning about the geometric object, I can
draw conclusions about the physical one, although - because the
analogy isn't exact - the conclusions may not be valid.  

Similarly, if I strike the surface of a drum, I can draw an analogy
between the drum and a mathematical surface subject to certain
boundary conditions.  By reasoning about what would happen to the
surface under particular deformations, I can draw conclusions about
what will happen to the drum if I strike it with a stick.  But -
becuse the analogy isn't exact - the conclusions may not be valid.

Apart from integer arithmetic, which seems to describe things like
conservation of charge, it's hard to think of a mathematical concept
that doesn't break down at some point when used to describe natural
phenomena.  

If you haven't yet seen Stephen Wolfram's book, _A New Kind of
Science_, you might want to take a look at it.  The thesis of the book
is that as we continue to look more closely at nature, we'll find that
the analogy between (1) continuous mathematical spaces and equations
and (2) what's 'really' going on, will become increasingly strained. 
If I understand him correctly, he's suggesting that the 'proper'
representation of nature will ultimately be not mathematical, but
algorithmic.  

It sounds like you've found a very interesting topic!  I suspect that
the hardest part of pursuing it will be to frame your question in a
way that will make the answer intelligible to others, as well as to
yourself:

  Is Geometry a Language?
    http://mathforum.org/library/drmath/view/55427.html 

But that's an important step.  Don't skip it.  

I hope this helps!

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 11/14/2003 at 00:16:18
From: Kiki
Subject: Characterization of Truth in Mathematics

Hi.  To clarify my ideas:

I want to research the relationship between mathematics and its 
applications to reality (the spatio-temporal world surrounding us).

More specifically, I want to examine the relationship between: a set 
of mathematical concepts and their use in answering a question about 
a real-world phenomenon (object, event or process).

Particularly, for any application of a set of mathematical concepts, 
I want to examine the correspondence that exists/is made between the 
true mathematical propositions comprising those concepts and the real-
world phenomenon that is to be understood.

I believe that there are two types of correspondences, which I've 
termed "opaque" and "transparent." 

(1) "Transparent" refers to correspondences in which the phenomenon 
is actually an instantiation of the mathematical concept(s) (or, 
equivalently, in which the mathematical concept is an abstraction of 
the phenomenon): e.g. Triangular objects in the physical world can be 
viewed as instantiations of the idealized triangles of Euclidean 
geometry, and hence, their geometrical properties can be studied 
simply by studying the geometrical properties of idealized triangles.
Another excellent example is the application of knot theory to the 
study of DNA strands in biology: DNA strands can be seen as 
instantiations of idealized mathematical knots; their 
knotting/tangling properties can be studied simply through studying 
the corresponding idealized knots.

(2) "Opaque" refers to correspondences in which the mathematics 
merely describes or encodes certain properties of the phenomenon, but 
is not a literal abstraction of it.  For example, the growth of a
population of rabbits can be described with a differential equation,
but a population of rabbits is in no way an instantiation of a
differential equation.

I believe that these two types completely characterize all 
correspondences between mathematical ideas and their applications.

Is this clearer?  Is there anything I haven't accounted for?

Thanks for your help!



Date: 02/06/2004 at 15:29:34
From: Doctor Ian
Subject: Re: Characterization of Truth in Mathematics

Hi Kiki,

Sorry for taking so long to get back to you!

To begin with, I would disagree with the statement that 

  A population of rabbits is in no way an instantiation of 
  a differential equation.

Certainly an individual rabbit isn't an instantiation of a
differential equation.  But when you talk about a 'population' of
rabbits, you're not talking about a physical object.  You're talking
about an abstraction.  In particular, you're talking about the
cardinality of an abstract set, which changes over time - i.e., a
function of a continuous independent variable.  A differential
equation is precisely a description of a class of such functions. 

So I think that as you refine the precision of your definitions, you
may find that the distinction between 'transparent' and 'opaque' may
evaporate. 

Also, offhand, I don't see where logic would fit into your scheme.   

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
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