Quantity
Quantities are fascinating beings. If I were not a
philosopher, my second career choice would be to spend my days studying
quantities. I still remember the feeling of curiosity when as a child I asked
my mom what the biggest number was. When I was in first grade, I spent my time
on the school bus trying to count to a million. My curiosity about numbers and
their relationships to each other continues to intrigue me. I’ve worked on magic squares and magic cubes, creating new
operators beyond the exponent operator, and my most recent project is to solve
the collatz conjecture.
What is a
quantity? Here’s a recursive definition: a quantity is the number one (itself expressed by a primitive term) or a complex
that contains a quantity as a part. A list of quantities can be represented,
then, as follows:
1 = Being
2 = {Being}
3 = {Being, {Being}}
4 = {Being, {Being}, {Being, {Being}}}
5= {Being {Being}, {Being, {Being}}, {Being,
{Being}, {Being, {Being}}}}
6 = {1, 2, 3, 4, 5}
7 = {1, 2, 3, 4, 5, 6}
…
Quantities are themselves properties.
For example, each and every thing exemplifies oneness. Every pair of objects exemplify twoness. And so on. Quantities are what mathematicians
call natural numbers, or the positive integers.
Philosophers whom I respect deeply have confessed to
me that numbers do not seem to be
properties. To be sure, there are properties of the sort I that I call ‘quantities’,
but those aren’t numbers, they say.
Maybe they are right. I don’t have their intuition about the matter, but then
maybe I’m lacking an intuition that I should have if I were to think about
these matters in the right way. At any rate, I do not know what to say about
numbers that are not themselves quantities. They seem ontologically redundant
to me. But if they are real, our insight into the nature of quantities will
give us insight into their nature, too.
Arithmetic
What proposition is expressed by ”2+3
= 5”? To answer that, we need to know what ‘+’ and ‘=’ mean. Here is a
proposal. First, there is the part of relation, which
is primitive. The relation of contains can be defined as the converse of the part of relation. This allows us to
define the relational property of containing
n, the property of containing
something containing n, containing
something containing something containing n, and so on, where ‘n’ names a quantity.
As the relations increase in complexity, they become more difficult to express.
However, since we’ve already given an account of the numbers themselves, we can
simplify our expressions of these containment properties by introducing the
notion of a quantity of containment.
For example, containing n can be
represented as ‘n+1’, containing
something containing n can be represented as ‘n+2’, containing something containing something containing n, can be
represented as ‘n+3’, and so on. So there are degrees of containment, and we
can represent these degrees by the symbol of the form n+m,
where n and m are names of some quantity.
Given this, we might translate “2+3=5” as follows:
(1) The
property of containing something
containing something containing {Being} is exemplified by {Being {Being},
{Being, {Being}}, {Being, {Being}, {Being, {Being}}}}.
In other words, a statement of equals is a statement
of property exemplification. It says that a certain relational property, itself
understood in terms of quantities, is exemplified by a certain quantity.
Unfortunately, there is one complication: according
to our definitions, 2+3=6 is true, too, because 6 contains
something containing something containing 2 given that 6 contains 5. The way to
fix this is to build into the definition of ‘2+3’ the clause and lacks containing something containing
something containing something
containing 2. In other words,
‘2+3’ expresses the property of containing 2 to degree 3 but not degree 4.
Once addition has been defined, we can define
subtraction in a similar fashion. This time we use ‘is a part of’ rather than
‘contains’. For example, 3-2 expresses the property of being a part of something that’s a part of 3 and lacks being a part of something that’s a part of something that’s a
part of 3. The sentence, “3-2=1”
says this:
(2) The
property of being a part of something
that’s a part of {Being, {Being}} and lacks being a part of something that’s a
part of something that’s a part of {Being, {Being}} is exemplified by
Being.
We can go on to define multiplication and addition
in terms of degrees of addition and subtraction. For example, ‘3*4’ means the
same as ‘3 + 3 + 3 + 3’. That is, 3 gets added to itself 4 minus 1 times. So just as addition comes in degrees of containment, so
multiplication comes in degrees of addition. The exponent operator, ‘^’ is to
multiplication as multiplication is to addition: exponent is degree of
multiplication: for example, ‘3^4’ means the same as ‘3 * 3 * 3 * 3’.
We could go on to define higher order operators. I’m
actually surprised we have yet to generalize in the following way. Let ‘n [0]’
represent a relational property of containing n. Let ‘n [1] m’ represent a
relational property of containing n to degree m (this is addition). Let ‘n [2]
m’ represent the relational property of being added to n to degree m (this is
multiplication). Let ‘n [3] m’ represent the relational property of being
multiplied by n to degree m (this is exponential). And so on. We can let ‘n [-k] m’ represent the inverse
of ‘n [k] m’.
In high school, I once gave my calculus teach an
algebra problem expressed in terms of exponents and multiplication but that
couldn’t be solved without the use of
an expression of the form, ‘n [-4] m’. (The problem was this: X^X * A*B^X = C:
solve for X.) I watched my calculus teacher struggle to solve it using logs and
division. But she tried in vain. I then showed her how to do it by defining the
new operator—the inverse of the operator that comes after exponential. I tell this story because I think that number
theorists could make progress in defining and solving certain problems by
generalizing their operators in the way I just recommended.
Here is a case where insight into the metaphysics of
a discipline can help us make progress in the discipline itself.
Zero
and Negative Quantities
Using addition and subtraction, we can identify
properties that could only be had by negative numbers or 0. For example,
there’s the property of being part of 1
and not part of something that’s part of 1. I’ll call this the essence of 0, since ‘0’ is the name we’ve given
the quantity that would exemplify it. I’m doubtful, however, that there is any
such thing as 0. I accept that there is the essence of 0—the property I just
pointed our minds to. But I doubt anything could exemplify it. The same goes
for the essences of so-called negative numbers. The essences exist, but I
strongly suspect that these essences cannot be exemplified (though they have
parts that can be). Negative numbers exist only in the sense that their
essences do. The mathematics who study negative numbers are really studying
their essences.
Rational
Numbers, Real Numbers, and Beyond
Using the definition for the arithmetical operator
of division—the inverse of multiplication—, we can define essences of rational
numbers. For example, there is the relational property of being half of one. (I
still need to think more carefully about how to make this precise in terms of
degree of subtraction…)
I think it is an open question whether every essence
of a rational number is itself exemplified by something. According to my
account of quantities, it should be clear that some rational essences are not
exemplified by any quantity. For example, being
half of one is not exemplified by any quantity. Being half of two, by contrast, is exemplified by a quantity,
namely one. One reason I’m tentatively skeptical that a property like, being half of one, can be exemplified is
that I have no idea what sort of thing would exemplify it. I grasp quantities
by grasping oneness directly, and then by grasping complexes of one. But what
thing would a half be? Of course,
just because I don’t directly grasp anything that could reasonably exemplify being half of one, doesn’t mean that
there isn’t any such thing. And there might be theoretical reasons to think
there is such a thing. But at present, I’m not aware of any theoretical
motivations to believe that anything exemplifies being half of one. Mathematicians can study properties of rational
essences without assuming that there are things that exemplify those essences.
Perhaps to simplify things we can call these rational essences the rational
numbers. But we shouldn’t be misled by this terminology to think that anything
other than quantities exemplifies a rational essence.
Using the inverse of exponent (the root of), we can
define essences, some of which cannot be exemplified by anything that would
count as a quantity or a rational number (like being the square root of two.) Mathematicians may call these
essences, or else the things that exemplify them, ‘the real numbers’. Again,
I’m not sure that any of these essences are exemplified by anything other than
a quantity.
I wonder if we can define essences using the inverse
of an operator higher than exponent—[3]—, like [4]. And I wonder if some of
these essences would be exemplified by something distinct from a real number
(if they could be exemplified at all). Has this been studied? I hope someone
will let me know.
An imaginary number would be a quantity that
exemplifies the property of being the
square root of a negative quantity. Just as I doubt that there are any
negative quantities, so too, I doubt that there are any imaginary ones.
However, there are the essences of imaginary numbers, and it is often very
useful to think about them (that is, one can gain insights into real quantities
by studying their relationships to unexemplifiable essences
of imaginary numbers).
Algebra
Algebra includes a formal language for expressing
propositions about numbers. For example, on my proposal, ‘2 + X = 3’ expresses
the proposition that 3 exemplifies the property of containing 2 to some degree X and not containing 2 to degree X+1.
It also invites a question, “What value could be substituted in for X to
express a true proposition?”
Note: it is evident that people can learn how to
“work with” Algebraic propositions without having any real insight into the
exact propositions they are supposed to express.
Axioms
Some axioms of quantities are listed here. Given the
metaphysics of quantities, we can see more clearly why those axioms are true.
Well, except for the axiom that says that zero is a quantity. That axiom should
be replaced with the axiom that one is a quantity.