Re: Mathematical properties

George Andrews (
Fri, 20 Feb 1998 17:49:39 -0600

Jan de Koning wrote:

> At 12:25 PM 14/02/98 -0600, Tom Pearson wrote:
> >At 12:42 PM 2/12/98 -0500, Jan de Koning wrote:
> >
> >>About math.: I disagree. Everything has mathematical (numerical and
> >>spatial) properties. Denying that these properties exist does not make
> >>them going away. 2+2=4 is not an idealization of reality. That is just a
> >>"real" observation. Even saying "Three in One" is a real mathematical
> >>statement, not an idealization, and I thank God for that.
> >
> >I find these comments fascinating, largely because I am feeble when it comes
> >to mathematics. But they leave me with questions. Can someone explain to
> >me what a "mathematical property" is?
> Actually, to be complete I would have to give a philosophy lecture, which I
> don't have the time for. Since I wrote the sentence, I'll try. 1apple + 1
> apple = two apples is a mthematical statement. The group lying on the
> table then has the mathematical property that there are two members in that
> group. They have a spatial (which is a mathematical concept too) property
> that they need enough room to be in. They have hardness, which is a
> physical property; they did grow on a tree, which is a biological property
> etc. etc.

Tom's question is a good one for sure. Math finds application in science via
representing and/or quantifying the physical world by assigning the abstract
notion of numbers. The examples offered by Jan above serve to demonstrate: an
object (a complete apple) is ASSIGNED the number "one" and "added" to another
similarly assigned quantity to produce the number "two". In this example, the
operation of addition is assumed defined. Hence, abstract (not physical) number
theory provides a way of "counting". Lest you think this is unnecessarily
complicating the simple, it took Bertran Russell some 300+ pages of set theory to
prove that 1 + 1 = 2! A proof which led- or at least inspired him to discover his
famous paradox. But just what is the property in nature that corresponds to the
mathematical notion of "one". This is an age old question that many a great mind
has had to deal with.

The quantum mechanical representation of an apple is different entirely from the
simplistic intuitive concept of number; for it depicts an apple as a
superposition of mathematical concepts known as "wavefunctions"; leaving the
notion of "one" out of the picture. Hence, I would have to say that 1 + 1 = 2 is
an idealization.

> >Since you say that "everything" has
> >them, I assume this means that these are properties of "things," including
> >physical objects. Ordinarily, a property is something that belongs to, or
> >is expressed by, a substance of some sort. So what kind of a thing is a
> >"mathematical property" belonging to a physical substance? Or are these
> >mathematical properties simply assigned to a physical substance for the
> >purpose of giving it a particular kind of description?
> Assigning properties does not need to be a simple task, neither are
> physical things the only things having "properties." You can have a creed
> with 12 articles. Having a certain number of articles is a numerical
> (mathematical) property. Assigning properties does not need to be simple
> assigning.

Are you agreeing here that it is simple quantification no matter how difficult?
Mathematics provides a very powerful "language" in which the human mind can
"commune" with and about nature (to which it belongs!); extracting and
abstracting relational information about experiments and observations; yet the
mathematical notion is not the essence of the object. However, mathematical
notions are real notions that actually exist in the mind of mankind and can be
"tested" via reason and logic. But this is different form the ontological
question Tom raised.

> >
> >And what does it mean to say that "2+2=4" is a "real" observation? Just
> >what is being "observed"?
> See above.
> > Is it the case, for instance, that numbers have
> >an independent existence apart from the objects they measure?
> Yes.

But in the mind as "idea".

George A.

George Andrews Jr.
Assistant Professor Physics
LeTourneau University