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[Reuben Hersh]

	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 

aren't numbers.

	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 

religion.

	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 

neither.

	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 

pure intuition.

	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

quasi-empiricism.

	Objection!

	"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 

humanity?"

	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.

	Why?

	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 

students learn.