rhersh at math.unm.edu
Sat Dec 27 12:58:19 EST 1997
I had been teaching math for twenty years before I
realized I didn't know what I was talking about.
Yes, I knew
2 + 2 = 4.
But what is "2"? What is "4"? What is "plus"? What is "equals"?
Let's concentrate on 2. 2 is just 1 and 1. Or, if you're
a little kid, you just stick two fingers in the air. That's 2.
But as to the first answer, 1 and 1--how will you explain what is
1? And as to the two fingers--fingers are fingers. They
Some people think that 2 is an abstraction that has existed forever,
apart from all material or human reality. And, so, they think, has every other
mathematical object or entity, discovered or undiscovered, known or unknown.
Something like this, I take it, is "Platonism" or "realism."
Others, I understand, think that whereas 2 means 1 and 1, the meaning
of "1" may not be requested. It's an undefined term. And all of
mathematics, they think, is just what you get by doing logic to undefined
terms. 2 is merely a symbol in a logical system, This viewpoint, I
believe, is called "formalism."
Still others, I understand, think that each mathematician individually
creates 2, 3, 4, and the rest of mathematics in her head. It's not clear
to me if she does this once and for all, or if she does it over again
every morning. Something like this, I understand, is called "intuitionism."
My working principle is that the philosophy of mathematics should be
true to the real life of mathematics--in creation
and in action, as well as stocked on library shelves.
By this principle, Platonism (or realism), formalism, and intuitionism
are all unacceptable.
Platonism with its Transcendent Reality is a beautiful story,
but perhaps not much better founded than any other highly intellectual
Formalism says mathematics had no meaning. But having actually
done mathematical work, I know it's meaningful. Even if it's not quite
elear what the meaning is.
And intuitionism, with its isolated mathematician doing it all
in her head, is nothing like mathematical reality.
So, here's how I analyzed it. Traditional philosophy recognizes one
or both of two kinds of earthly (non-transcendental) entities--physical and
mental. Mental is "in your head"--your individual inner consciousness.
Physical is what moves, takes up space, has mass or energy.
What about numbers? They aren't physical. 2 doesn't have
position or volume or mass. Is it mental? No. 2 was here before you or I
were born, and will be here after we perish. It's in your head and my
head, yet it's outside of our heads. Because
2 + 2 = 4
whether you and I know it or not.
Everything can't be either mind or matter, because numbers are
What to do?
Just open your eyes and look around. Or close
them and think about what you did yesterday.
You watched the news of the day. The news is neither
mental nor physical.
You spent and earned money. Money is neither mental nor physical.
You obeyed the law. The law is neither mental nor physical.
The physical/mental dichotomy leaves out most of what concerns
you every day--the social and cultural. We have to enlarge philosophy
to allow three categories of earthly reality: physical, mental, social.
Now ask again, what's 2? It's not physical. It's not mental.
It's social! Number, and all mathematics, is part of culture, part of the
system of thought that humanity creates for itself.
In fact, no one would deny that mathematics is part of society
and culture. No one would deny that it has evolved historically. No one
would deny that it's something we do.
But Platonists, formalists and intuitionists think that has
nothing to do with what mathematics *is*.
What it *is* is something higher and finer than an aspect of
society. Such as a Platonist Idea, or a set of formal derivations, or a
I think that a major historic motivation for all three
classical foundationist philosophies was the felt need to restore certainty
to mathematical knowledge. And a natural objection to the claim
that mathematical reality is social-cultural, is: what
becomes of the absolute certainty of mathematical knowledge?
But whatever certainty we had in olden times--which is also very
dubious--is hard to defend today, in the day of proofs hundreds of pages
long, and computer-aided proofs, and proofs created by
an international network of dozens of finite-group theorists.
Like other fields of knowledge, mathematics can survive
with moral certainty. Certainty strong enough to justify
making important choices--but not absolute certainty, not certainty
guaranteed to last for all time, come what may.
I call my viewpoint "humanism." Others have called
related theories social constructivism, naturalism, fallibilism, and
"There are 9 planets traveling round the sun. There were
those 9 planets before any human being stirred on Earth.
*Ergo*, 9 is not a human creation. It is part of the physical universe."
We have to analyze the word and concept "9". In "There are 9
planets," "9" is an adjective. It's a "counting number." Counting
numbers applied to things like the solar ystem are indeed part of
physical reality. The planets shine, they have certain
masses, momenta, and orbits, and they are nine. Their numerosity is a fact of
astronomy as much as their luminosity.
The problem for the philosophy of math is *the other 9*, the
abstract 9, the "pure" 9. This 9 is part of the number
systems. It has a square, a square root, and countless other
fascinating properties which have nothing to do with planets or peanuts.
Pure 9 and counting 9 are closely related, but they aren't the same.
Difficulties about the existence of numbers come from confusing the two.
For instance, there are infinitely many pure or abstract numbers, but only
finitely many counting numbers. (We can't say just how many, but we can
write down a symbol for a number larger than the highest count
that can ever be done in the life of the human race.)
The pure, abstract numbers, including the pure abstract 9, are a
cultural artifact. So are the other objects and entities of pure
mathematics. When we humans depart from this
universe, they will no longer exist. There will still be 9 planets.
"Well," you may object, "So what? You say numbers are this,
others say they are that. What difference does it make to suffering
One difference is in math teaching. I think that when math teachers
understand that numbers are our social creation, they'll have more
success than by thinking numbers are meaningless, or up in Heaven some place.
Too much blind calculation and drill makes people
hate math. That kind of teaching assumes that math is just
algorithms and calculations. It's a kind of debased formalism.
Some students see math as a huge, mysterious, alien thing,
like it's from outer space. They get this impression
in part from teachers who think math is inhuman or superhuman--a debased
form of Platonism.
Humanist teaching, showing math as a part of human history and
culture, could avoid these two kinds of bad teaching. It could help
More information about the FOM