FOM: Objectivity and Truth in Maths
rhersh at math.unm.edu
Sat Jan 17 11:57:42 EST 1998
On Sat, 17 Jan 1998, Edwin Mares wrote:
> Ruben Hersh says:
> "Is the Pythagorean theorem about the material world or about
> some ideal immaterial things? Euclid already knew it was about ideal
> things. But the meaning of it, the interest of it, the use of it, all
> have to do with material things. For instance, right triangles
> carefully drawn on the ground. No matter how big. But if a triangle
> is big enough, the curvature of the Earth will have a measurable
> effect, and the Pythagorean theorem becomes false. Moreover, there
> is no reasonable way of deciding when it becomes false. To a high
> enough degree of precision, it is false even for the most carefully
> drawn right triangle on the ground."
> It is true that we can't draw perfect Euclidean triangles on a curved (and
> bumpy) surface like that of the earth.
No physical surface can be known to be perfectly flat and smooth.
It's the high degree of precision that is the issue, not whether
it is the earth or a table top or a blackboard.
But what are we to conclude from
> this? Consider the laws of nature. If they have mathematical form, as modern
> physics would suggest, it would seem that at least some of maths is about an
> extramental world. Now, you might want to say that the "laws" are themselves
> human contrivances that only approximate physical phenomena. This is
> suggested by the passage quoted above. The meaning of "approximate" here,
> however, is hard to understand. If phenomena have real measurements, then
> there is at least a mapping from mathematical structures like the real
> numbers onto physical phenomena.
THe notion of mapping I think you are using refers to
mathematical objects. A mathematical mapping onto a
physical object doesn't make sense. Maybe you mean a
mapping onto a mathematical model of a physical object.
But that overlooks the unavoidable discrepancies between
any physical object and any mathematical model of it that
involves measuring ( not just counting up to a relatively small number.)
Thus, it would seem that at some parts of
> maths are (perhaps indirectly) about phenomena that we didn't create. If you
> want to deny that real things have magnitudes that are independent of our
> measuring them, then you are buying into a very strong form of anti-realism
> about maths and science. If you accept that they have magnitudes,
They have magnitudes in the sense of lesser and greater. They
do not have *numerical* magnitudes until a scale of measurement is chosen.
"Some parts of maths are (perhaps indirectly) about phenomena
we didn't create." DEpemds on what you mean by "about". Certainly
counting comes from coins, pebbles, fingets, etc, which we didn't create.
Then pure arithmetic comes about by leaving behind those coins etc to
create a purely mental realm. But the interpretations connecting
counting numbers with pure numbers persist, and come into play
anytime we use arithmetic on real objects.
There is an empirical Pythagorean theoem, a very old one,
that is approximately valid for carefully constructed real right
triangles. There is a pure Pythagorean theorem, which applies to
ideal right triangles, whihc are mental objects. (But when I
say mental object I don't mean subjective, I mean intersubjective.)
The pure theorem is about shared concepts. The empirical one is
abouat physical objects. The pure one is proved, relative to
some axiom set. The empirical one is proved inductively, like
other empirical facts (the sky is blue.) Of course the pure one
springs from the empirical one historically. However, the formulation
of the pure theorem is not completely determined by the empirical one.
For instance, we now understand that a theorem on spherical triangles
(the Earth being nearly a sphere) might in some senses be a better
abstract or pure version of the empirical Pythagoras theorem than the
In modern mathematics, Pythagoras blows up into a huge mathematical
structure called Hilbert space. Hilbert space has its ancestry
in the two dimensional Pythagoras theorem. But it has its major
significance as a model for quantum mechanics. Few Would say that
a state of a quantum system really "is" a vector in Hilbert space.
We have theorems about Hilbert space. These theorems are about
the Hilbert space in mathematicians heads. Just as in arithmetic,
there is a mutual interpretation between the physics and the mathematics,
which enable one to say something about physics based on theorems in
Hilbert space. But of course these interpretations are valid only
to the extent that the Hilbert space model of quantum mechnaics is
accurate. Historical experience suggests that, close as it is, it
may not be perfectly accurate.
Perhaps I misinterpreted you in criticizing your use of
the mathematical term "mapping." Perhaps you just meant interpretation,
in which case perhaps we agree.
> seem that your position is much less radical than you make out.
I do not call my position "radical."
That does seem to be the unfortunte reaction of some of my critics.
> resembles, at least at first glance, Hartry Field's fictionalism.
> I see what you mean, they are both anti-Platonist.
I'm not thoroughtly conversant with Field's publications.
For me, the point is not just to reject Platonism, but
to provide another, reality-based explanation of the
nature of mathematical entities, and their truth and
> Ed Mares
> Ed Mares
> Department of Philosophy
> Victoria University of Wellington
> P.O. Box 600
> Wellington, New Zealand
> Ph: 64-4-471-5368
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